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Master Thesis in Mathematics and Physics
The equivalence between second-order,Ostrogradsky-free (Galileon) Lagrangians and
regular rst-order theories.
Author:
K.J.D. Hakvoort
Supervisors
Dr. D. Roest
Dr. H. Waalkens
Abstract
Galileon theories are a class of scalar theories, where the second-order derivative ap-
pears in the Lagrangian. Traditionally, such second-order theories have been deemed
unstable, plagued by the so-called Ostrogradsky-Ghost. Recent work however suggests
that Galileon theories are stable, and in fact might be equivalent to ordinary rst-order
Lagrangians. In this thesis, I examine these Galileon theories, as well as other second
order theories. I then use the Cartan Equivalence Algorithm to test the equivalence of
second order theories to rst order theoeries. From this test, I conclude that pure second
order, mechanical theories are equivalent to rst-order theories.
November 28, 2017
Image source: http://www.rug.nl/about-us/how-to-find-us/huisstijl/logobank/logobestandenfaculteiten/logofse
Keywords: Galileons, Ostrogradsky, Cartan Equivalence Algorithm, Equivalence, Jet Bundles, Contact Transformations
Contents
1 Introduction 3
2 Classical Mechanics 4
2.1 Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Constraint Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 A comparison between approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 The Ostrogradsky instability and how to avoid it 15
3.1 The second-order case according to Woodard . . . . . . . . . . . . . . . . . . . . 153.2 The second-order case, using Hamiltonian constraint theory . . . . . . . . . . . . 173.3 The general rth order case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Ostrogradsky-free Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Field Theories 28
4.1 What are eld theories? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Lorentz invariance and Scalar theories . . . . . . . . . . . . . . . . . . . . . . . 294.3 The Ostrogradsky ghosts in eld theories . . . . . . . . . . . . . . . . . . . . . . 324.4 Ostrogradsky-free eld theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Galileons, and other second order eld theories 38
5.1 One Galileon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 Self duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3 Multi Galileons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.4 Shift Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.5 General Scalar Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6 General Relativity 44
6.1 The principles of GR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.2 Scalar gravity and Nordström's theorem. . . . . . . . . . . . . . . . . . . . . . . 446.3 Einstein's tensorial theory of General Relativity . . . . . . . . . . . . . . . . . . 466.4 Lovelock Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7 Modern gravity: both scalars and tensors are important 49
7.1 Vainshtein Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.2 DGP-gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.3 Galileons as scalar-tensor theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.4 Horndeski's general scalar-tensor theory . . . . . . . . . . . . . . . . . . . . . . 547.5 Beyond Horndeski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8 When to call two Lagrangians equivalent 56
8.1 Why equivalence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568.2 Jet bundles and contact forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578.3 Prolongated and allowed transformations . . . . . . . . . . . . . . . . . . . . . . 608.4 The types of equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
1
9 Equivalence of Galileons 66
9.1 Equivalence conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669.2 One Galileon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709.3 One Galileon, many scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719.4 Many Galileons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739.5 Many Galileons, and many scalars . . . . . . . . . . . . . . . . . . . . . . . . . . 749.6 Covariant Galileons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
10 An explanation of Cartan's equivalence algorithm 75
10.1 Strict equivalence of coframes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7510.2 Equivalence with structure-groups . . . . . . . . . . . . . . . . . . . . . . . . . . 7910.3 Involutive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8210.4 Prolonged systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8310.5 Application of Cartan's algorithm to Lagrangians . . . . . . . . . . . . . . . . . 84
11 The case for 1 independent, and q dependent variables, all second order 88
12 Conclusion 97
A Notation 99
A.1 Summation conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99A.2 Jet Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99A.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
B Variational Problems 101
2
1 Introduction
In 1850, Mikhail Ostrogradsky published a paper in which he generalised Hamilton's construc-tion of Hamiltonians to theories containing higher-order (time) derivatives. As a consequenceof this generalization, Ostrogradsky showed that such higher-order theories suer from a linearinstability, making them unsuitable for physics [1].
In 2016, two groups released papers that showed that certain second order Lagrangian the-ories do not suer from this Ostrogradsky instability, or ghost [2, 3]. From this work, thequestion arose if these Ostrogradsky stable Lagrangians are perhaps rst order theories writtenin a poorly chosen coordinate system, i.e. if such Lagrangians are equivalent. If they are equiv-alent, then these higher-order theories have nothing new to tell us. However, if there are somesecond-order, Ostrogradsky stable Lagrangians that are not equivalent to a rst order theory,then these theories open up a whole new class of theories for physics to explore.
Ostrogradsky stable theories include the class of Galileon theories, a class of second-orderLagrangians that only have second-order equations of motion [4]. Galileon theories are aninteresting class of theories, used in General Relativistic theories to modify gravity on largescales. They also exhibit some other properties that make them interesting to study on theirown, regardless of applications [5].
For this master thesis, I was asked to study the types of Ostrogradsky stable theories, andto investigate the possible equivalence between these stable theories and ordinary rst ordertheories. This yields the following research question: Do the conditions that free a higherorder Lagrangian of Ostrogradsky ghosts force such a Lagrangian to be divergence equivalentto a rst-order Lagrangian?, and the following sub questions:
Physics: What types of Ostrogradsky stable theories are there?
Physics: What are the properties of these theories?
Math: What are the conditions to call two Lagrangians divergence equivalent?
Math: How can you determine such equivalence?
In this thesis, I rst discuss the physics behind the relevant theories. In chapter 2 we discussseveral approaches to classical mechanics, in chapter 3 the Ostrogradsky instability is discussedwith regards to classical mechanics [1]. In particular, we discuss why the instability limits thekind of theories allowed and how to avoid it. In chapter 4 we discuss eld theories, and how theOstrogradsky instability aects them. In chapter 5 we then discuss several Ostrogradsky stableeld theories. In particular the class of second order theories called Galileons is discussed [4,5],a class that is both second order and Ostrogradsky stable. After that, we expand to theoriesof General Relativity in chapter 6 and couple these to scalar eld theories in chapter 7.
In the second part of this thesis, we discuss the mathematics of equivalence. First, in chapter8 we set up the mathematical framework of jet bundles and give a denition of equivalenceusing [6, 7]. In chapter 9 we apply this denition and framework to the galileon theories ofchapter, using [8]. Next, we go over the Cartan Equivalence algorithm in chapter 10 [6]. Thisalgorithm can be used to answer the question when two Lagrangians are equivalent, and insubsection 10.5 I show how to apply Cartan's algorithm to Lagrangians [6]. In chapter 11 weapply this algorithm to the case of second order, Ostrogradsky free, mechanical Lagrangiansand check if these Lagrangians are equivalent to rst order theories.
Chapter 12 is the conclusion, in which we sumarize what was discussed and attempt toanswer the research question.
3
2 Classical Mechanics
2.1 Newtonian Mechanics
Physics, like all Natural Sciences, concerns itself with describing the world around us. InTheoretical Physics the goal is to nd a theory that describes the motions and interactions ofparticles, however large or small, fast or slow. One famous physicist who made an attempt atderiving such a theory is Isaac Newton [9, chapter 3]. His second law
F = ma (1)
gives the equation of motion for some object of mass m subject to a force F . It is a secondorder dierential equation, and therefore two pieces of initial data are needed to nd a uniquesolution. Generally, they are chosen as initial position and velocity, or initial and nal positionat a specic time [9, chapter 3]. This method however quickly becomes complicated as moreobjects are described, since for each object a force equation (1) has to be dened. Newton'smethod also doesn't hold outside of classical mechanics [9, chapter 6].
Example 1 (Hooke's Law). Let us give a well know example, that of an object of mass mattached to some spring with spring constant k, like in gure 1. This object experiences a forceF = −kx. Applying Newton's second law, we nd the acceleration to be [9, chapter 4]
x = − kmx. (2)
This is the dierential equation of an harmonic oscillator and has the solution
x (t) = A sin (ωt+ φ) ,
where ω =√
km, and A and φ are an amplitude and phase determined by the initial conditions.
This equation tells us that our object will oscillate with frequency ω [9, chapter 4].
k
m
Figure 1: An object with mass m attached to a horizontal spring with spring constant k.
2.2 Lagrangian Mechanics
The Lagrangian formulation is another method to derive equations of motion. For many me-chanical systems, we can dene a kinetic energy T and a potential energy U [9, chapter 6], [10,chapter 13]. From these energies, we can dene the Lagrangian L = T − V . Next we dene afunctional called the action as the integral over the Lagrangian [9]:
S =
∫ t2
t1
L dt. (3)
Let us now be a bit more general and explicit, and assume that our mechanical system has qdependent coordinates, which we label uα. We can now extremize the action integral, and nd
4
the Euler-Lagrange equations [10]1:
d
dt
∂L∂uα
=∂L∂uα
. (4)
These Euler-Lagrange equations are the equations of motion for each of the uα.So far, for our alternate method we have dened a Lagrangian and an action out of the blue.
Let us now tie them back with Newton's mechanics. Consider that in mechanics the kineticenergy is given by T =
∑αmα2
(uα)2 and the potential energy is just a function of position only,i.e. U = U (uα). Then the Euler-Lagrange equations for this system is given by
mαuα = − ∂U
∂uα. (5)
Furthermore, the force is often conservative, which is precisely Fα = − ∂U∂uα
. Using this substi-tution, we observe that the Euler-Lagrange equations are an equivalent method to Newton'sequations to describe physics.
The Lagrangian method can be generalized. To that end, now consider p independentcoordinates labelled xi, and also consider higher-order derivatives up to order r. The Lagrangiannow does not have to be related to a kinetic and potential energy any more, but instead is anarbitrary function of the independent coordinates xi, the dependent coordinates uα, and theup to rth order derivatives uαI
2. The action then looks like:
S =
∫L(xi, uα, u(r)
)dvolx. (6)
We can still extremize this action integral to again obtain Euler-Lagrange equations. Settingthese Euler-Lagrange equations equal to zero then determines the equations of motion of thedependent coordinates.
Theorem 1. The Euler-Lagrange equations belonging to the action in equation (6) are givenby [11, section 5]
r∑σ=0
∑|I|=σ
(−1)σdσ
dxI
(∂L∂uαI
)= 0. (7)
Proof. To nd the Euler-Lagrange equations, we must extremize the action (6). To that end,we must nd the dierential of S for the functions uα (xi), similar to how we did in appendixB. Take q such functions, and vary them by adding functions hα (xi). These functions have theproperty that hα (xi0) = hα
(xif)
= 0, such that uα = uα + hα agrees with uα at the end pointsx0 and xf . Then the dierence between the actions found from these functions is given by
∆S = S (u+ h)− S (h) =
∫ xf
x0
[L(xi, uα + hα, u(r) + h(r)
)− L
(xi, uα, u(r)
)]dx
=
∫ xf
x0
∂L∂uα
hα +∂L∂uαi
hαi +∑
26|I|6r
∂L∂uαI
hαI
dx+O(h2)
= F (h) +R.
The extremal than follows from those functions uα for which F (h) = 0 for all h [10, chapter12]. To see that this condition equals the Euler-Lagrange equations, we perform integration by
1See appendix B for how to extremize the action2See appendix A for the denitions of this notation.
5
parts on F (h). First, we nd that∫ xf
x0
∂L∂uαi
hαi dx = −∫ xf
x0
hαd
dxi∂L∂uαi
dx+
[hα
∂L∂uαi
]xifxi0
.
Due to the condition that hα = 0 at the boundary of the integral, the boundary term is zero.We can perform this integration by part on each term in the derivative F (h), continuing onuntil we have a term involving hα itself. Again, we get boundary terms which vanish due tothe boundary conditions on h. The result is
∂L∂uαI
hαI = (−1)|I| hαd|I|
dxI
(∂L∂uαI
),
from which we nd that F is equal to
F (h) =
∫ xf
x0
r∑σ=0
∑|I|=σ
(−1)σdσ
dxI
(∂L∂uαI
)hα dx,
which is zero for all h precisely when the Euler-Lagrange equations (7) are equal to zero.Thus, we conclude that the Euler-Lagrange equations (7) are precisely those conditions for thefunctions uα to extremize the action S (6).
Example 2. Let us end the introduction of the Lagrangian formulation with a look at Hooke'slaw in a Lagrangian setting. The potential energy for an object on a spring is given by U =12kx2, and its kinetic energy is T = 1
2mx2 [9, chapter 6]. This potential is conservative, since
−dUdx
= −kx is the equation for the spring force.For this system then the Lagrangian is given by
L =1
2mx2 − 1
2kx2. (8)
For this system (7) reduces to∂L∂x− d
dt
∂L∂x
= 0.
Applied to our Lagrangian, we nd the Euler-Lagrange equation
0 =∂L∂x− d
dt
∂L∂x
= −kx− d
dt(mx)
= −kx−mx,mx = −kx. (9)
This is equal to (2), as expected. The solution is again a harmonic oscilator. This explicitlyshows the equivalence of Newton's and Lagrange's method with respect to Hooke's law.
Last, we shall give a quick example of a theory that can only be written in the Lagrangianformulation.
Example 3. The Dirac Lagrangian is known to be [12, page 43]
L = ψ (iγµ∂µ −m)ψ.
6
Unlike classical physics, in QFT there are 4 independent coordinates, 1 time-like and 3 space-like. Furthermore, each of the spinors ψ and ψ has 4 components, leading to 8 dependentcoordinates. In general, since there are 8 dependent variables, there would be 8 Euler-Lagrangeequations. However, due to the nature of the dependent variables (they are two spinor-elds),we need to write down only 2 equations3. They are
0 =∂L∂ψ− d
dxµ∂L∂∂µψ
= −mψ − ∂µ(ψiγµ
)= −i∂µψγµ −mψ,
and
0 =∂L∂ψ− d
dxµ∂L∂∂µψ
= (iγµ∂µ −m)ψ − ∂µ0
= (iγµ∂µ −m)ψ
This last equation is the Dirac equation [12, page 42]. One can easily observe that neither ofthese equations can be written in the form of Newton's second law (1), since these equationsare rst-order equations and not second order.
2.3 Hamiltonian Mechanics
The third method to formulate classical mechanics is called Hamiltonian Mechanics. WhereLagrangian Mechanics uses a function called the Lagrangian, Hamiltonian Mechanics uses aHamiltonian. The Hamiltonian is a function that depends on the position and momentumcoordinates of the system (as well as potentially any independent coordinates, such as time), andcan be derived from the Lagrangian as follows. Let L (t, uα, uα) be a Lagrangian that depends onq dependent coordinates uα, their time derivatives uα and perhaps even on time t directly. Foreach of the coordinates uα, a momentum coordinate can be dened by pα = ∂L/∂uα [10, chapter3]. With the momentum thus dened, we arrive at the Hamiltonian by performing a Legendretransform on the Lagrangian [10]:
H (t, uα, pα) = pαuα − L (t, uα, uα) . (10)
From this Hamiltonian, equations of motion can be derived. They are given as [10, chapter 3]
pα = − ∂H∂uα
, (11)
uα =∂H
∂pα. (12)
These equations can be derived from the Euler-Lagrange equations (7). First, from pα =∂L/∂uα it follows that
pα =d
dt
∂L∂uα
=∂L∂uα
= − ∂H∂uα
,
3Since the spinors are each (represented by) 4 component vectors, each of the 2 equations can be read as avector equation.
7
where I use the E-L equations to go from the rst to the second line, and the Legendre transform(10) to go from the second to the third line. To derive (12), only the Legendre transform isneeded, since
∂H
∂pα=∂pβu
β − L(t, uβ, uβ
)∂pα
= uα − ∂L∂pα
= uα,
where I have used that the Lagrangian itself does not (explicitly) depend on the momenta pα.The Hamiltonian also has a physical interpretation, it is the energy function. This is easy to
see in classical mechanics, where the Lagrangian is given by L = T −U , with T =∑
αmα2
(uα)2
and U = U (uα) does not depend on the velocities uα. The momenta are then given by
pα = ∂L/∂uα = mαuα,
and if we perform the Legendre transform, we nd
H = pαuα − L (t, uα, uα)
= mαuαuα −
∑ mα
2(uα)2 + U
=∑ mα
2(uα)2 + U
= T + U,
which is precisely the total energy.
Example 4. Let us again discuss Hooke's law, but now using the Hamiltonian. The potentialenergy remains U = 1
2kx2 [9, chapter 6], but we need to rewrite the kinetic energy term in
terms of the momentum p. The Lagrangian is (8)
L =1
2mx2 − 1
2kx2,
from which we can calculate the momentum as
p =dLdx
= mx.
The kinetic energy is thus given by T = p2
2m, and the Hamiltonian for a spring is
H = T + U =p2
2m+
1
2kx2.
The equations of motion are now given by Hamilton's equations (11)(12):
p = −∂H∂x
= −kx
x =∂H
∂p=
p
m.
The rst of these becomes the E-L equation for a spring (9) if we back substitute the momentum,while the second equation is just the denition of the momentum.
8
As shown above, we can derive the equations of motion from the Hamiltonian. We canhowever do something more powerful, we can use the Hamiltonian to directly calculate thetime-evolution of any function F (uα, pα)! Since in the Hamiltonian setting, all functions canonly depend on the `position' coordinates uα and the momentum coordinates pα, this is quitepowerful. To proof this statement, we rst need to dene the Poisson bracket.
Denition 1. Let us work in a Hamiltonian description, with q coordinates uα, and their qmomentum coordinates pα. The Poisson bracket between two functions A and B depending onthese coordinates is given by [10, chapter 8]:
A,B =∂A
∂uα∂B
∂pα− ∂A
∂pα
∂B
∂uα(13)
Theorem 2. Let us work in a Hamiltonian description, with q coordinates uα, and their qmomentum coordinates pα. Let A, B and C be functions depending on these coordinates, anda, b ∈ R be two real numbers. Then the Poisson bracket between these functions has the followingthree properties [10, chapter 8]:
1. The Poisson bracket is skew symmetric: A,B = B,A.
2. The Poisson bracket is linear: aA+ bB,C = a A,C+ b B,C.
3. The Poisson bracket admits the Jacobi Identity:
A,B , C+ B,C , A+ C,A , B = 0.
The proof of this theorem we omit for brevity, but can be found in Arnol'd [10, chapter 8].With the Poisson bracket thus dened, we can calculate the time derivatives from the next
theorem.
Theorem 3. Let us work in a Hamiltonian description, with q coordinates uα, and their qmomentum coordinates pα. Let H be the Hamiltonian function, and let F be an arbitraryfunction that depends on these coordinates and possibly time. Then the time evolution of thisfunction is given by
dF
dt=∂F
∂t+ F,H . (14)
Proof. To proof this theorem, we rst consider the `trivial' functions F = uα and F = pα:
uα, H =∂uα
∂uβ∂H
∂pβ− ∂uα
∂pβ
∂H
∂uβ
=∂H
∂pα= uα,
and
pα, H =∂pα∂uβ
∂H
∂pβ− ∂pα∂pβ
∂H
∂uβ
= − ∂H∂uα
= pα.
These two relations are exactly the equations of motion we derived earlier. Using this, we canproof the theorem for an arbitrary function F . First, by denition, the total time derivative ofF is given by
dF
dt=∂F
∂t+ uα
∂F
∂uα+ pα
∂F
∂pα,
9
and the Poisson bracket between F and H is given by
F,H =∂F
∂uα∂H
∂pα− ∂F
∂pα
∂H
∂uα
=∂F
∂uαuα − ∂F
∂pα· − pα
=∂F
∂uαuα +
∂F
∂pαpα,
ergo∂F
∂t+ F,H =
∂F
∂t+∂F
∂uαuα +
∂F
∂pαpα =
dF
dt.
2.4 Constraint Analysis
Theorem 3 shows the power of using the Hamiltonian formulation. Another thing one can doin the Hamiltonian formulation is counting of degrees of freedom using contraint analysis. Inprinciple, a physical theory has as many degrees of freedom as it has dependent coordinates.However, there could be additional degrees of freedom hiding inside the theory. These hiddendegrees often pop out as ghosts, and are undesired. The Ostrogradsky ghost is one of theseextra degrees of freedom, and we will discuss them in chapter 3.
The other possibility is that there are hidden constraints in the theory that reduce thenumber of degrees of freedom. This is benecial, as it reduces the complexity of the problem.For example, if you are able to reduce the number of degrees of freedom to zero, then the theoryis fully integrable and thus solvable [10, chapter 9], independent of initial conditions.
In order to discover the total number of degrees of freedom, one must perform a constraintanalysis. A constrain analysis can be done in both the Lagrangian and the Hamiltonian for-mulation. In this thesis, we shall focus on the Hamiltonian analysis using [3]. A Lagrangiananalysis can be found in [2, 8].
Example 5. To explain this procedure, we are going to use a simple, but quite general theoryof a particle in the plane. We are going to use an arbitrary Lagrangian: L = L (x, y, x, y). ThisLagrangian is equivalent to a dierent Lagrangian4:
Leq = L (x, y,X, Y ) + λ (x−X) + κ (y − Y ) , (15)
where λ and κ are Lagrange multipliers. The equations of motion for these multipliers ensurethat X and Y behave as the velocities of x and y:
0 = EL (Leq)λ =∂Leq∂λ
= x−X,
0 = EL (Leq)κ =∂Leq∂κ
= y − Y,
4We also give a more strict denition of equivalence in chapter 8, that does not hold for the equivalence wehave here. The Lagrangians here are called equivalent, because they lead to the same physics
10
while the other equations combine to give the equations of motion you would expect from astandard Lagrangian. Let us show this for the equations for x:
0 = EL (Leq)X =∂Leq∂X
=∂L∂X− λ⇔
λ =∂L∂x
0 = EL (Leq)x =∂Leq∂x− d
dt
∂Leq∂x
=∂L∂x− d
dtλ
=∂L∂x− d
dt
∂L∂x
A similar process will yield the equations for y.Now knowing that these Lagrangians give the same physics, let us rewrite Leq into the
Hamiltonian formulation. First, we need to dene 6 momentum variables. Using the Poissonbrackets, they can be dened as precisely those variables that admit the following relations [3]
px, x = py, y = PX , X = PY , Y = ρ1, λ = ρ2, κ = 1, (16)
with all other Poisson brackets equal to zero.Of course, we have already dened the momentum variables through another method, i.e [10,
chapter 3]
px =∂L∂x
= λ, PX =∂L∂X
= 0, ρ1 =∂L∂λ
= 0,
py =∂L∂y
= κ, PY =∂L∂Y
= 0, ρ2 =∂L∂κ
= 0.
The advantage of the Poisson brackets is that they allow one to dene momentum variableseven in the absence of a Lagrangian. Regardless, these 6 equations produce the 6 primaryconstraints:
Φx = px − λ ≈ 0, Px ≈ 0, ρ1 ≈ 0,
Φy = py − κ ≈ 0, Py ≈ 0, ρ2 ≈ 0.
We use an ≈ instead of an = since these are constraints; they are not identically zero, they arezero because we constrain them to be.
Let us now explain the counting of degrees of freedom. The formula is as follows
DOF =1
2(#variables −#secondary class constraints − 2 ·#primary class constraints) .
(17)This formula requires some explanations. First is the 1
2#variables, since for each dependent
coordinate we have 2 variables, i.e. the coordinate itself and its associated momentum, we needto halve #variables in order to obtain the appropriate number of degrees of freedom.
Second are the constraints, these can be divided into two classes, the primary class and thesecondary class. A constraint is of the primary class, if its Poisson bracket commutes with allother constraints and if it commutes with the Hamiltonian5. All other constraints are part of
5We say that two constraints f and g commute, if f, g = 0. Please note, this Poisson bracket must beidentically zero, it should not be zero because we constrain it to be zero. The same holds for commutation withthe Hamiltonian.
11
the secondary class. The constraints of primary class are stronger constraints and as a resultreduce the DOF by 1 each.
To test if the 6 constraints we found thus far are of primary or secondary class, we mustcalculate the Poisson brackets. We nd only two non-zero Poisson brackets
Φx, ρ1 = Φy, ρ2 = −1.
This shows that 4 constraints are already of secondary class. To check the other two, PX andPY , we rst need to dene the Hamiltonian.
One rst denes the Hamiltonian as one would for the regular Lagrangian,
H0 = pxX + pyY − L (x, y,X, Y ) . (18)
This is not the total Hamiltonian however, as it does not include the other variables that wehave dened for our analysis. We arrive at the total Hamiltonian, by adding the constraintstimes a Lagrange multiplier [3]. We add the multipliers that enforce the primary constraints.The total Hamiltonian then becomes [3]
HT = H0 + µiΦi + νiρi + ξiPi. (19)
We now calculate the Poisson brackets of the Hamiltonian and the constraints:
ρi = ρi, Ht = ρi, H0+ µj ρi, Φj= ρi, H0+ µi = µi,
Φi = Φi, Ht = Φi, H0+ νj Φi, ρj= Φi, H0+ νi.
These constraints x the Lagrange multipliers µi and νi, and thus can be made identically zero.The more interesting brackets are those of the constraints of unknown class:
χi = Pi = Pi, HT= Pi, H0+ 0
= Pi, pxX + PyY − L= −pi − Pi,L= −pi + LXi .
These constraints are not identically zero. We can however constrain them to be zero, whichjust enforces that px = ∂L/∂X = ∂L/∂x and py = ∂L/∂y. These additional constraints χi donot commute with the Pi, as we nd out when calculating the Poisson bracket of χi with HT :
χi = χi, Ht = χi, H0+ νj χi, ρj+ ξj χi, Pj= χi, H0+ νj Pi, Ht , ρj+ ξj −pi + LXi , Pj= χi, H0+ ξjLXiXj (20)
If you identify X = x and Y = y, you discover that LXiXj is the kinetic matrix. If this matrix isinvertible, then (20) xes the Lagrange multipliers ξi. In that case, we have identied a total of8 constraints, ρi, Φi, Pi, χi, all of which are second class6, and thus we can calculate the degreesof freedom from (17) as
DOF =1
2(#variables −#secondary class constraints − 2 ·#primary class constraints)
=1
2(12− 8− 2 · 0)
=4
2= 2,
6ρi and Φi do not commute, and neither do Pi and χi, thus they are all second class.
12
precisely as desired.Of course, in a general theory, there is no guarantee that LXiXj is invertible. Proceeding
the analysis like in [3], if LXiXj has a one-dimensional kernel, i.e. only 1 linearly independentnull-vector, then one obtains an additional constraint. Let the null-vector be (a scalar times)(wx, wy), then (20) produces as tertiary constraint
Σ = wx χx, H0+ wy χy, H0 .
Requiring that its time derivative is also zero, i.e. Σ,Ht ≈ 0, generally gives another con-straint, and brings down the degrees of freedom to 1.
Last is the case in which LXiXj vanishes, in which case we have the two constraints
Σi = χi, H0 ≈ 0.
Depending on the time derivative of Σi, these may give rise to at most 2 additional constraints.Thus, we end up with either 1 or 0 degrees of freedom.
The analysis above can be generalised to cases with q dependent variables, simply by addingmore additional elds Xα and Lagrange multipliers λα for each additional dependent variableuα. If the generalised kinetic matrix LXαXβ = Luαuβ is invertible, then one will nd DOF = q.If the matrix is not invertible, one will nd additional constraints, resulting in DOF < q.
The other generalisation is to increase the order of the derivatives appearing in the La-grangian, i.e. add uα. We will do this analysis in chapter 3, and thus bring about the Ostro-gradsky ghost.
2.5 A comparison between approaches
In this chapter, we discussed three possible methods to describe physics and derive equationsof motion, the methods of Newton, Lagrange and Hamilton. It therefore would be good to endthis chapter with a short discussion about each of these methods, and explain why some aremore favoured in physics than others.
First is Newton's method of F = m · a. It has the advantage of being an easy methodthat immediately arrives at the equations of motion for the theory. It is intuitively easy tounderstand: if you apply a force to an object, say you push a couch, that object moves, andhow fast it moves depends on how much force you apply. However, it also quickly becomescomplicated as the amount of objects and sources of forces increases. Furthermore, Newton'sequations are no longer valid when one considers either the really fast ((Special) Relativity) orthe really small (Quantum Mechanics).
The Lagrangian and Hamiltonian method both have the advantage of being just one formulafrom which the equations follow. Any changes to your theory thus require you to modify onlyone equation, instead of however many force equations. The disadvantages are that they aremathematically more dicult to understand, or prove their validity, and are less intuitive tounderstand. Hamilton's method is favoured in non-relativistic quantum mechanics, for example,it is used in the Schrödinger equation [13],
i~∂
∂tΦ(xi, t
)= HΦ
(xi, t
), (21)
which explicitly depends on the Hamiltonian.The Lagrangian method can be used for both quantum mechanics and relativity, and where
both combine in Quantum Field Theory. To explain the importance of the Lagrangian in thelatter, we use the imagery used by David Morin in [9, page 225] and Paul Dirac in [14]. In
13
quantum eld theory, particles don't take only one path between two points, they take allpossible paths. However, some paths are more favoured by others. The weight of a path iscalled the phase of a path, and is given by a factor eiS/~, with S the action dened in (6).The phases from all possible paths have to be added up to determine the amplitude of goingfrom a point A to a point B. The square of the absolute value of the amplitude gives theprobability of a particle going from point A to point B. For non-stationary values of S, andthus for non-stationary paths, the phases for the dierent values of S vary wildly, due to thesmal-ness of ~. Thus, one sums over essentially random vectors in the plane of complex numbersfor the amplitudes, which sums to approximately zero. Thus, the probability of a path with anon-stationary S is essentially zero. However, for paths with stationary values of S, the phaseis constant and no cancelling occurs. Thus it are the paths with stationary actions that weobserve and thus it are the paths with stationary actions that we care about in physics. Seealso Dirac [14] for an argument why this weight factor is given by eiS/~.
It should be noted that in the rest of this thesis we are working on classical, non-quantumtheories, and only do classical eld theory. Thus specics from QFT do not apply to us.
14
3 The Ostrogradsky instability and how to avoid it
In the previous section, we only discussed rst-order theories. Physics in general also only usesrst-order Lagrangians. Why is this the case? A related question might be why Newton'sforce equation (1) is a second order dierential equation, and not a rst order, or third orderequation7? One might say that it is because these theories are what experiments agree with,and one is indeed right. However, such an answer doesn't really oer any new physical insight.Is there a deeper reason for only using rst-order Lagrangians? It turns out there is, and thisreason was discovered by Ostrogradsky [1]. We already hinted at in at the end of chapter 2;higher-order theories harbour an additional, un-physical degree of freedom. This extra degreeof freedom is called the Ostrogradsky ghost, and it causes these higher order theories to beunstable.
In this section, we shall follow R.P Woodard's review and construction of the Ostrogradskyinstability in [1], followed with a Hamiltonian analysis as in [3]. After that, we look into theworks by Klein, Roest in [2] and Motohashi et al. in [3] that derive constraints to eliminate theOstrogradsky instability in second-order classic mechanical theories.
3.1 The second-order case according to Woodard
Consider a Lagrangian that depends on second derivatives. We are primarily concerned with thep = 1, arbitrary q case, thus L = L
(t, u(2)
). For this system, the q Euler-Lagrange equations
(7) are∂L∂uα− d
dt
∂L∂uα
+d2
dt2∂2L∂uα2 = 0. (22)
We assume non-degeneracy. For a second order system, this means that ∂2L∂uα∂uβ
is invertible.Because this matrix is invertible, we can rewrite (22) into a form similar to Newton's equations(1). Due to the last term in (22), this is now a fourth-order dierential equation:
d4uα
dt4= Fα
(t, u(3)
).
Since this is a (system of) fourth-order dierential equations, we need 4q initial data to solvethe system [1]: uα0 = uα (0), uα0 = uα (0), uα0 = uα (0) and
...uα0 =
...uα (0). Since 4q initial data are
needed, there are 4q canonical variables for a Hamiltonian analysis. Following Ostrogradsky,Woodard choose these coordinates as [1]
Uα1 = uα, Pα
1 =∂L∂uα− d
dt
∂L∂uα
,
Uα2 = uα, Pα
2 =∂L∂uα
.
Since this system is non-degenerate, in principle the system can be written in terms of thecanonical variables, i.e. there exist functionsA′β (Uα
1 , Uα2 , P
α1 , P
α2 ) such that uβ = A′β (Uα
1 , Uα2 , P
α1 , P
α2 )
and ∂L∂uα|uβ=Uβ1 ;uβ=Uβ2 ;uβ=A′β = Pα
2 [1].
There now arise two considerations that lead us to change the formula for A′β. The rst isthat the original system only had 3q variables: uα, uα and uα. Thus, Aβ should depend on atmost 3q canonical variables8. The second can be found in the restrictions A has to meet: Adoes not have to be tuned such that the denition for P1 holds. As a result, A need not depend
7Or any order other than 2 really.8This could also be a hint to the astute reader that something is afoot: why do two equivalent descriptions
have dierent numbers of variables?
15
on P1. Thus we can instead dene the acceleration as uβ = Aβ (Uα1 , U
α2 , P
α2 ) and state that it
has to meet uβ = Aβ (Uα1 , U
α2 , P
α2 ) and ∂L
∂uα|uβ=Uβ1 ;uβ=Uβ2 ;uβ=Aβ = Pα
2 .
The above thus dened, Ostrogradsky's Hamiltonian is obtained through the Legendre trans-formation
H (Uα1 , U
α2 , P
α1 , P
α2 ) = P β
1 uβ + P β
2 uβ − L
= P β1 U
β2 + P β
2 Aβ (Uα
1 , Uα2 , P
α2 )− L
(t, Uα
1 , Uα2 , A
(Uβ
1 , Uβ2 , P
β2
)).
The time evolution follows from computing Hamilton's equations [1], [10, chapter 3]. As in therst-order case, the Hamiltonian is the energy function. The interesting thing to note is thatthis Hamiltonian only has a dependence on P β
1 in the P β1 U
β2 term, and thus only depends linearly
on P β1 . As a result, the Hamiltonian is unbounded from below and can not be stable [15]. This
result is quite general, because we only made one assumption on L, that of non-degeneracy.Example 6. Let us consider an example, taken from Woodard [1]. We look at the regular har-monic oscillator, but with an added second order term dependent on a dimensionless parameterε. The Lagrangian is given by [1]
L = − εm2ω2
x2 +m
2x2 − mω2
2x2. (23)
The Euler-Lagrange equation (22) and its solution are given by
0 = −m[ εω2
....x + x+ ω2x
],
x (t) = C+ cos (k+t) + S+ sin (k+t) + C− cos (k−t) + S− sin (k−t) .
The Euler-Lagrange equation is clearly fourth-order, which is problematic. The two frequenciesare given by
k± = ω
√1∓√
1− 4ε
2ε.
The constants C± and S± depend on the initial data, the formulae can be found in [1] and are
C+ =k2−x0 + x0
k2− − k2
+
, S+ =k2−x0 +
...x 0
k+ (k2− − k2
+),
C− =k+x0 + x0
k2+ − k2
−, S− =
k2+x0 +
...x 0
k− (k2+ − k2
−).
Following Ostrogradsky's choice of canonical variables, we can compute the Hamiltonianfor this system. This Hamiltonian can be expressed in terms of the two frequencies and theconstants C± and S±. In this form, it looks like [1]
H =m
2
√1− 4εk2
+
(C2
+ + S2+
)− m
2
√1− 4εk2
−(C2− + S2
−).
In this form, it is clear that C− and S− carry negative energy. If we are to quantize this system,C− and S− would correspond to negative-energy creation operators, allowing for (innite)negative energy particles. As a result, no state can be stable, since each state can always decayinto a state with same energy, just with one additional positive-energy and one additionalnegative-energy particle.
The regular, rst order version of this system can be obtained by taking ε → 0. Then theHamiltonian becomes
H =m
2ω2(C2
+ + S2+
)2.
This system has no negative energy parameters, and is stable [1]. This shows that the instabilityindeed resides in the higher-order part of the Lagrangian.
16
3.2 The second-order case, using Hamiltonian constraint theory
Of course, the above analysis can also be done using Lagrange multipliers and constraint analy-sis, as we did on a rst-order theory in chapter 2. To do so, consider a second order LagrangianL = L (t, uα, uα, uα). To perform the analysis, dene the equivalent Lagrangian
Leq = L (t, Qα, qα, uα) + λβ(uβ − qβ
)+ κβ
(qβ −Qβ
). (24)
Again, we eliminate all derivatives in the Lagrangian itself, in favour of introducing additionalvariables qβ and Qβ and using the Lagrange multipliers λα and κα to force these additionalvariables to behave as derivatives. Unlike the rst order case, where we only had 1 additionalvariable per dependent variable, the appearance of second order derivatives requires us to use2 additional variables per dependent variable9. Since this Lagrangian depends on 5q variables,we must dene 5q pairs of conjugate variables. We choose them such to admit to the followingPoisson brackets,
Qα, Pβ = qα, pβ = uα, πβ =λα, ρ
β
=κα, σ
β
= δαβ ,
with all other Poisson brackets equal to zero. Comparing this denition of the momenta withthe usual denition gives rise to the following equations:
Pα =∂L∂Qα
= 0, pα =∂L∂qα
= κα, πα =∂L∂uα
= λα,
ρα =∂L∂λα
= 0, σα =∂L∂κα
= 0.
These equations produce the 5q primary constraints:
Pα ≈ 0, ρα ≈ 0, σα ≈ 0,
Φα = pα − κα ≈ 0, Ψα = πα − λα ≈ 0.
To compute time-evolution we dene the two Hamiltonians as we did in chapter 2. First theregular Hamiltonian
H0 = pαQα + παq
α − L (t, Qα, qα, uα) , (25)
and the total Hamiltonian
HT = H0 + µαΦα + ναΨα + ξαPα + ζ1αρ
α + ζ2ασ
α. (26)
There are only 2 groups of non-zero Poisson brackets between the constraints:
ρα, Ψβ = σα, Φβ = δαβ .
From this, it is immediate that the Poisson brackets of ρα, Ψβ, σα, Φβ with HT xes the mul-
tipliers µα, να, ζ1α, ζ
2α; in particular µα = να = 0. On the contrary, the time evolution of Pα
gives
χα = Pα = Pα, HT= Pα, H0=Pα, pβQ
β + πβqβ − L
(t, Qβ, qβ, uβ
)= −pα −
Pα,L
(t, Qβ, qβ, uβ
)= −pα + LQα ≈ 0.
9In section II A of [3], Motohashi et al. perform a similar analysis where they only need 1 additional variableper dependent variable for second order theories. They choose not to remove all derivatives from the Lagrangian,only the second order ones. Removing all derivatives is easier to generalize, so we choose that route.
17
We constrain this to be zero, which gives us an additional q constraints. Like in the rst-ordercase, these constraints do not commute with the Pα. As a result, the time derivative χα is givenby
χα, HT = χα, H0+ ξβ χα, Pβ+ ζ1β
χα, ρ
β
+ ζ2β
χα, σ
β
= χα, H0+ ξβ LQα − pα, Pβ= χα, H0+ ξβLQαQβ . (27)
If the matrix LQαQβ is invertible, this equation allows us to x the Lagrange multipliers ξα.With all Lagrange multipliers xed, and all constraints thus determined, the analysis ends.None of the constraints commute with all other constraints, and thus all are of second class.The total number of degrees of freedom then is
DOF =1
2(#variables −#secondary class constraints − 2 ·#primary class constraints)
=1
2(2 · 5q − 6q − 2 · 0)
=4q
2= 2q.
We have only expected q DOF, since we only have q dependent variables. There thus arean additional q ghost DOF. This is because the matrix LQαQβ is invertible, if it is not, it hasnull-vectors which lead to additional constraints, and thus fewer degrees of freedom. If we back-substitute our variables, we identify LQαQβ = Luαuβ . Ergo, the requirement of invertibility isexactly the requirement of non-degeneracy Woodard and Ostrogradsky used in their analysis [1].
We have now shown in two dierent methods that theories having second-order derivativesin their Lagrangian are unfavoured. The dierence between the analyses is that while usingLagrange multipliers only shows that there are additional degrees of freedom, the analysis byWoodard [1] also explicitly shows why these additional degrees of freedom are bad.
3.3 The general rth order case
The construction by Ostrogradsky and Woodard can be extended to a general r-th order case toshow that these cases are also unstable [1]. We could also extend the Hamiltonian analysis, butfor brevity we shall only discuss the analysis by Woodard. Consider a Lagrangian L
(t, u(r)
).
We assume that this Lagrangian is non-degenerate, i.e. ∂2L∂(druα/dtr)∂(druβ/dtr)
6= 0. There are
now 2qr coordinates in phase space. If we generalize Ostrogradsky's choice, the canonicalcoordinates are given by:
Uαi =
di−1uα
dti−1 and Pαi =
r∑j=i
(−1)j−idj−i
dtj−i∂L
∂(djuα/dtj
) .Due to non-degeneracy, we should be able to solve for the druα/dtr in terms of Pα
r and the
Uαi 's. Ergo, we can nd functions Aα
(Uβ
1 , . . . , Uβr , P
βr
)with the property that
∂L∂ druα
dtr
|Uαi =di−1uα/dti−1;Uαr =Aα = Pαr .
With these functions, we can construct the Hamiltonian. It is [1]
H = Pαi
diuα
dti− L,
= Pα1 U
α2 + Pα
2 Uα3 + · · ·+ Pα
r−1Uαr + Pα
r Aα − L(t, Uβ
1 , . . . , Uβr ,Aβ
), (28)
18
where we sum over α. Again, Hamilton's equations will give the time evolution of the system.From (28), one can see that the Hamiltonian is linear in all coordinates Pα
i , for all α andfor all i < r. Ergo, the energy is unbounded from below for most momentum coordinates.Adding more higher derivatives just increases the fraction of unstable coordinates ( r−1
2r) to 1
2
(limr→∞r−12r
= 12), and thus does not x the problem, it rather seems to make it worse [1].
Woodard states 6 principle reasons why this instability is bad. They are summarized below.For a longer explanation, we refer the reader to Woodard's article [1]
1. The Ostrogradsky instability drives the dynamical variables (uα) to a special kind of timedependence, not towards a special numerical value. This is because the Ostrogradskyinstability is a problem with the kinetic energy and not with the potential energy. Inthe latter, more familiar case, energy is released as the dynamical variable approachessome special value. However, in this case the energy is released as the dynamical variableapproaches a special time dependence.
2. The Ostrogradsky dynamical variable carries both positive and negative energy creationand annihilation operators.
3. If a system with this instability interacts, the vacuum decays into a collection of positiveand negative energy excitations. Because there are creators and annihilators for bothpositive and negative energy excitations, both will appear. Since more particles equalsmore entropy, any particle will thus immediately decay into positive and negative energyparticles.
4. If the theory is furthermore a continuum eld theory, then the vast entropy of havingmany particles makes this decay instantaneous.
5. Furthermore, this decay does not decouple from low energy physics. I.e., even at lowenergy the above decay happens and can not be relegated to a high energy sector, higherthan we have observed so far. Therefore, we can not state that our unstable theory ismerely an Eective Field Theory, valid (and stable) up to a UV-cuto scale.
6. A single, global constraint on the energy function does not x the Ostrogradsky instability.
3.4 Ostrogradsky-free Mechanics
In the previous section, we saw how and why many higher-order Lagrangians are unstable.However, not all higher-order Lagrangians are unstable. In 2016 and 2017 two groups, one byCrisostomi, Klein and Roest [2] based in Groningen, and Motohashi, Noui, Suyama, Yamaguchiand Langlois [3] based in Tokyo and Paris, published papers in which they independently derivedconstraints on second-order Lagrangians to remove the Ostrogradsky ghost.
Let us start this subsection with an example for such an Ostrogradsky stable, second-ordertheory.
Example 7. Let us start with the simplest of example, a second order theory with just 1independent and just 1 dependent variable, i.e. L = L (t, u, u, u). If it is non-degenerate [1],i.e. ∂2L
∂u26= 0 it must have the Ostrogradsky ghost. Thus, to avoid the ghost, we at least need
degeneracy, i.e. ∂2L∂u2
= 0. From this requirement, we know that our Lagrangian is of the form
L = uf (t, u, u) + g (t, u, u) , (29)
19
with f 6= 0. We can calculate the Euler-Lagrangian equation (22) for this theory, and nd
0 =d2
dt2∂L∂u− d
dt
∂L∂u
+∂L∂u
=d2
dt2(f)− d
dt(ufu + gu) + ufu + gu
=d
dt(ft + ufu + ufu)−
...ufu − uftu − uufuu − u2fuu − gtu − uguu − uguu + ufu + gu
=d
dt(ft + ufu) + ufu + u
dfudt− ...ufu − uftu − uufuu − u2fuu − gtu − uguu − uguu + ufu + gu
= F (t, u, u) u+G (t, u, u) .
This Euler-Lagrange equation is second order, and thus does not have the ghost. This canalso be seen using the Hamiltonian constraint analysis. We have already done the analysis forgeneral q, ending at (27)
χα = χα, HT = χα, H0+ ξβLQαQβ ,
which could be made identically zero by choosing the appropriate ξβ. For our q = 1 case, (27)becomes
χ = χ,H0+ ξLuu.Since we have degeneracy, Luu = 0 and thus we can not x ξ. Instead, this constraint reads
Ξ = χ,HT = χ,H0= LQ − p, pQ+ πq − L (t, Q, q, u)= LQ, pQ+ πq − L (t, Q, q, u) − p, pQ+ πq − L (t, Q, q, u)= QLqQ + LuQq + π − Lq ≈ 0.
This is not identically zero, and thus produces a new constraint Ξ. To end the analysis, weneed to see calculate the time evolution of this constraint; and constraint it to be zero if it isnot already so:
Σ = Ξ = Ξ,Ht= Ξ,H0= QLqQ + LuQq + π − Lq, pQ+ πq − L (t, Q, q, u)= Q2LQqq + 2qQLQqu +QLQu + q2LQqu + Lu −QLqq − qLqu ≈ 0.
We thus have yet another constraint. Fortunately, this constraint does not commute with P ,and thus we get
Σ,Ht = Σ,H0+ ξ Σ,P= Σ,H0+ ξ (qLQqu + 2LQu − Lqq) ,
which can be tuned to zero by xing ξ. We have thus found all constraints on the theory, wehave derived 8 constraints, and the degrees of freedom of this theory are
DOF =1
2(#variables −#secondary class constraints − 2 ·#primary class constraints)
=1
2(2 · 5− 8− 2 · 0)
= 1.
Thus, a theory with just one dependent variable with only has a linear dependence on thesecond order derivative does not have the Ostrogradsky ghost.
20
In the example, we only had to assume degeneracy to eliminate the Ostrogradsky ghost.However, for theories with multiple dependent variables only assuming this degeneracy is notenough. To continue the analysis, like Klein et al. [2] and Motohashi [3], we will consider twocases. First is the case where all q dependent coordinates appear in second order fashion, i.e.∂L/∂uα 6= 0 ∀α. The second case is where of the q dependent coordinates, only m appear witha second order derivative, a further n = q − m > 1 appear with only rst order derivatives.These n rst-order coordinates can be used to remove the Ostrogradsky ghost from the msecond order derivatives, as we shall see. We will rst discuss the all second-order case.
Our system of all second-order coordinates can be analysed in three dierent methods.We can look at the Euler-Lagrange equations, or apply either a Hamiltonian or Lagrangianconstraint analysis. First, we look at the Euler-Lagrange equations. The Lagrangian is givenby L = L (uα, uα, uα), and the q Euler-Lagrange equations (7) are given by [2]
Eα =d2
dt2∂L∂uα− d
dt
∂L∂uα
+dLduα
= Luαuβ∂4uβ
∂t4+ (Luαuβ − Luβ uα)
∂3uβ
∂t3+ lower order terms.
From Woodard's analysis [1], we know that for the Ostrogradsky ghost to be absent, the Euler-Lagrange equations must contain at most second order time derivatives. To remove the fourthorder derivatives, we need not just degeneracy, i.e. detLuαuβ = 0, we need full degeneracy, i.e.Luαuβ = 0. If the system is not fully degenerate, then there remain(s a linear combination of)Euler-Lagrange equations that contains the fourth order time derivative. The condition
∂2L∂uα∂uβ
= 0 (30)
is called the primary constraint. However, from the E-L equations, we can also see that imposingthe primary constraint is not enough, there still remain third-order terms that need to beremoved. This can be done by imposing the secondary constraint
∂2L∂uα∂uβ
− ∂2L∂uβ∂uα
= 0. (31)
This secondary constraint is anti-symmetric, which explains why we did not encounter it inour example with just 1 dependent variable: it was automatically satised. Imposing bothconstraints upon a second-order Lagrangian produces second-order equations of motion, andthus removes the Ostrogradsky ghost [2].
Next we use a Hamiltonian constraint analysis, such as done by Motohashi et al. in [3]. Wehave already done the rst part of this in section 3.2. We ended up with the time derivative ofthe constraint χα (27),
χα = χα, HT = χα, H0+ ξβLQαQβ .
We discussed that if LQαQβ is invertible, the analysis ends. If not, each linearly independentnull-vector of LQαQβ gives rise to a new constraint. In order to obtain maximal constraints, weneed maximal degeneracy, i.e. the primary constraint
LQαQβ = 0,
which is the same constraint as (30). We thus have q additional constraints
Ξα = χα, HT = χα, H0=LQα − pα, pβQβ + πβq
β − L (Q, q, u)
= QβLQαqβ + qβLQαuβ + πα − Lqα ≈ 0.
21
The time derivative of these constraints is given by
Ξα = Ξα, HT = Ξα, H0+ ξγ Ξα, Pγ+ ζ1γ (Ξα, ρ
γ) ζ2γ (Ξα, σ
γ)
= Ξα, H0+ ξγ (LQαqγ − LQγqα) .
If the matrix LQαqγ − LQγqα now is invertible, we can x ξα and end the analysis. A quickcalculation then tells us that there are 3
2q > q degrees of freedom, still to many. Thus, we
arrive at the secondary constraint
LQαqγ − LQγqα = 0,
which is again the same as (31). This results in q constraints Σα = Ξα, H0 ≈ 0. Timeevolving this constraint then generally xes the ξα, and the analysis ends. Our two additionalsets of constraints reduce the degrees of freedom to
DOF =1
2(#variables −#secondary class constraints − 2 ·#primary class constraints)
=1
2(2 · 5q − 8q − 2 · 0)
= q,
as desired.The third method involves a Lagrangian constraint analysis, and is performed by Klein and
Roest in [2]. We omit it here, but it is no surprise that they nd the same primary andsecondary constraint as we have.
Next we will consider the case where there are m second order and n rst order variables,labelled respectively φa and qi. This case too can be analysed in three separate methods.However, for the sake of brevity, we shall only perform the Hamiltonian constraint analysis.
We consider the Lagrangian L = L(φa, φa, φa, qi, qi
); it is second order in φa, but only rst
order in qi. Assuming no constraints, one will nd that generally this Lagrangian has 2m + ndegrees of freedom, of which m are Ostrogradsky ghosts [3]. We thus need to nd m rst classconstraint, or 2m second class constraint (or a mix of both) in order to reduce the degrees offreedom to the desired q = m+ n.
To perform the analysis and calculate which constraints we need to impose, consider thefollowing equivalent Lagrangian
Leq(Qa
2, Qa1, φ
a, qi2, qi1, λa, κa, ιi
)= L+ λa
(φa −Qa
1
)+ κa
(Q1
a −Qa2
)+ ιi
(q1i − qi2
). (32)
We have dened a total of 5m+ 3n `position' coordinates, and thus we need to dene 5m+ 3nmomentum coordinates. They are dened according to the following Poisson brackets
Qa2, P2b = Qa
1, P1b = φa, φb = δab ,λa, ρ
b
=κa, σ
b
= δab ,qi2, p2j
=qi1, p1j
=ιi, τj
= δij.
Comparing these to the other denition of momenta produces constraints
P2a =∂L∂Qa
2
= 0, P1a =∂L∂Qa
1
= κa, πa =∂L∂φa
= λa,
ρa =∂L∂λa
= 0, σa =∂L∂κa
= 0,
p2i =∂L∂qi2
= 0, p1i =∂L∂qi1
= ιi, τ i =∂L∂ιi
= 0.
22
These directly produce 3m+ 2n constraints, and a further 2m+ n can be found as
Φa = P1a − κa ≈ 0, Ψa = πa − λa ≈ 0, Υi = p1i − ιi ≈ 0.
These constraints admit Poisson brackets, most of which are zero. The non-zero ones areΦa, σ
b
=Ψa, ρ
b
= δab ,Υi, τ
j
= δji .
Next, we dene the two Hamiltonians:
H0 = P1aQa2 + πaQ
a1 + p1iq
i2 − L
(Qa
2, Qa1, φ
a, qi2, qi1
),
HT = µaΦa + νaΨa + υiΥi + ξa2P2a + ξi1p2i + ζ1aρ
a + ζ2aσ
a + ηiτi +H0.
Due to the non-zero Poisson brackets, the time evolution of Φa, Ψa, Υi, σa, ρa, τi xes the La-
grange multipliers ζ2a , ζ
1a , ηi, µ
a, νa, υi. In particular, it xes µa = νa = υi = 0. This leavesthe multipliers ξa2 and ξi1 unxed, because their constraints P2a and p2i commute with all otherconstraints. However, they do evolve non-trivially with time. This gives two new sets of con-straints, χ1i and χ2a:
χ1i = p2i = p2i, HT= p2i, H0=p2i, P1aQ
a2 + πaQ
a1 + p1jq
j2 − L
(Qa
2, Qa1, φ
a, qi2, qi1
)= −p1i + Lqi2 ≈ 0,
and
χ2a = P2a = P2a, HT= P2a, H0=P2a, P1bQ
b2 + πbQ
b1 + p1jq
j2 − L
(Qa
2, Qa1, φ
a, qi2, qi1
)= −P1a + LQa2 ≈ 0.
These constraints do not commute with the P2a and p2i. As such their time derivatives are
χ1i = χ1i, HT= ξb2 χ1i, P2b+ ξj1 χ1i, p2j+ χ1i, H0= ξb2Lqi2Qb2 + ξj1Lqi2qj2 + χ1i, H0 , (33)
and
χ2a = χ2a, HT= ξb2 χ2a, P2b+ ξj1 χ2a, p2j+ χ2a, H0= ξb2LQa2Qb2 + ξj1LQa2qj2 + χ2a, H0 .
These derivatives can be made identically zero if the kinetic matrix
K =
(LQa2Qb2 LQa2qj2Lqi2Qb2 Lqi2qj2
)is invertible, since then we can solve the system(
LQa2Qb2 LQa2qj2Lqi2Qb2 Lqi2qj2
)(ξb2ξj1
)= −
(χ2a, H0χ1i, H0
).
23
If we can solve the system, we have determined all Lagrange multipliers and the time evolutionof all constraints, and thus we are nished. We then compute that we have
DOF =1
2(2 · (5m+ 3n)− (5m+ 3n)−m− n) = 2m+ n
degrees of freedom. We thus need to pose constraints on the matrix K, it can not be invertible.We need to obtain m additional constraints from χ1i and χ2a, which can be achieved if we ndm null-vectors for the matrix K. These can be found by imposing the following condition onthe sub-matrices of K:
LQa2Qb2 − LQa2qi2Lqi2q
j2Lqj2Qb2 = 0, (34)
where Lij is the inverse of Lqi2qj2 . This condition requires that the sub-matrix Lqi2qj2 is invertible.The m null-vectors are then found to be Va =
(δba, V
ia
), where V i
a is given by V ia = −LQa2qj2L
qj2qi2 .
These conditions (34) are the modied primary conditions from before, and they reduce to theprevious primary conditions in the absence of rst order variables.
Having these m null vectors, we can still x n of the multipliers ξ. Using (33), we can x ξi1in terms of ξa2 by xing χ1i = 0. In doing so, we obtain
ξi1 = −Lij χ1j, H0+ V iaξ
a2 . (35)
From the null vectors, we derive the constraints
Ξa = χ2a, H0+ V ia χ1i, H0
= LQa2Qb1Qb2 + LQa2φbQ
b1 + LQa2qj1q
j2 + πa − LQa1
+ V ia
(Lqi2Qb1Q
b2 + Lqi2φbQ
b1 + Lqi2qj1q
j2 − Lqi1
)≈ 0.
These constraints should not evolve with time as well, so we compute
Ξa = Ξa, Ht= ξb2 Ξa, P2b+ ξj1 Ξa, p2j+ Ξa, H0= ξb2
(Ξa, P2b+ V j
b Ξa, p2j)− Lji χ1i, H0 Ξa, p2j+ Ξa, H0 .
We can use this equation to x the remaining multipliers ξb2, and thus end the analysis. Todo so requires that the matrix Lab = Ξa, P2b + V j
b Ξa, p2j is invertible. However, we arestill m constraints short to remove all Ostrogradsky ghosts. We can nd these additional mconstraints if Lab = 0. This condition will give us our secondary conditions, so let us expandLab:
Lab = Ξa, P2b+ V jb Ξa, p2j
=∂Ξa∂Qb
2
+ V jb
∂Ξa
∂qj2= LQb2Qa2Qc1Q
c2 + LQb2Qa2φcQ
c1 + LQb2Qa2qj1q
j2 + LQa2Qb1 − LQb2Qa1
+ V ia
(LQb2qi2Qc1Q
c2 + LQb2qi2φcQ
c1 + LQb2qi2qj1q
j2 + Lqi2Qb1 − LQb2qi1
)+∂V i
a
∂Qb2
(Lqi2Qb1Q
b2 + Lqi2φbQ
b1 + Lqi2qj1q
j2 − Lqi1
)+(Lqj2Qa2Qc1Q
c2 + Lqj2Qa2φcQ
c1 + Lqj2Qa2qi1q
i2 + LQa2qj1 − Lqj2Qa1
)V jb
+ V ia
(Lqj2qi2Qc1Q
c2 + Lqj2qi2φcQ
c1 + Lqj2qi2qk1 q
k2 + Lqi2qj1 − Lqj2qi1
)V jb
+∂V i
a
∂qj2
(Lqi2Qb1Q
b2 + Lqi2φbQ
b1 + Lqi2qj1q
j2 − Lqi1
)V jb .
24
To reduce this equation, we enforce the primary condition (34):
0 = LQa2Qb2 − LQa2qi2LijLqj2Qb2 = LQa2Qb2 + V i
aLqi2Qb2 ,
The time derivatives of (34) are also identically zero, i.e.
0 = LQa2Qb2Qc1 + V iaLqi2Qb2Qc1 +
∂V ia
∂Qc1
Lqi2Qb2
= LQa2Qb2Qc1 + V iaLqi2Qb2Qc1 + Lqj2Qa2Qc1V
jb + V i
aLqi2qj2Qc1Vjb ,
0 = LQa2Qb2φc + V iaLqi2Qb2φc +
∂V ia
∂φcLqi2Qb2
= LQa2Qb2φc + V iaLqi2Qb2φc + Lqj2Qa2φcV
jb + V i
aLqi2qj2φcVjb ,
0 = LQa2Qb2qj1 + V iaLqi2Qb2qj1 +
∂V ia
∂qj1Lqi2Qb2
= LQa2Qb2qk1 + V iaLqi2Qb2qk1 + Lqj2Qa2qk1V
jb + V i
aLqi2qj2qk1Vjb .
Using these constraints, Lab reduces to
Lab = LQa2Qb1 − LQb2Qa1 + V ia
(Lqi2Qb1 − LQb2qi1
)+(LQa2qj1 − Lqj2Qa1
)V jb + V i
a
(Lqi2qj1 − Lqj2qi1
)V jb .
We thus have our secondary condition:
LQa2Qb1 − LQb2Qa1 = −V ia
(Lqi2Qb1 − LQb2qi1
)−(LQa2qj1 − Lqj2Qa1
)V jb − V
ia
(Lqi2qj1 − Lqj2qi1
)V jb . (36)
Again, in the absence of healthy rst-order variables this equation reduces to the previoussecondary condition (31).
Under this condition, we can again constrain Θa = Ξa = Ξa, H0−Lji Σ1i, H0 Ξa, p2j ≈0. The time evolution of Θa will generally x the Lagrange multipliers ξa2 , and we thus endup with m + n degrees of freedom. Else, there is further degeneracy and we end up with lessdegrees of freedom.
In order to make the comparison with eld theories in chapter 4 easier, we can rewrite theconditions (34) and (36). Let ψA be the collection of our dependent variables φa and qi, wherethe index A rst runs over the φa and then over the qi10. The elements of the null vectors Vacan then be labelled by vAa =
(δba, V
ia
). The primary condition can then be written as [8]
0 = vAa∂2L
∂ψA∂ψBvBb = Pab. (37)
Similarly, the secondary condition becomes [8]
0 = vAa∂2L
∂ψ[A∂ψB]
vBb = S[ab]. (38)
Example 8. Let us end with an example of a case where the rst-order sector helps removethe instability. Our example Lagrangian has one second-order variable φ, and one rst-ordervariable q:
L = εφ2 +m
2φ2 − k
2φ2 +
M
2q2 −
√2Mεqφ. (39)
10I.e. A ∈ 1, . . . ,m, 1, . . . , n.
25
For this theory, (37) is given by
∂2L∂φ
2 =∂2L∂φ∂q
·(∂2L∂q2
)−1
· ∂2L
∂φ∂q
To determine if this condition holds we compute the relevant derivatives of the Lagrangian.They are
∂2L∂φ
2 = 2ε,∂2L∂q2 = M,
∂2L∂φ∂q
= −√
2Mε,∂2L∂φ∂φ
= 0,
∂2L∂φ∂q
= 0,∂2L∂φ∂q
= 0,
∂2L∂q∂q
= 0, V = −(∂2L∂q2
)−1
· ∂2L
∂φ∂q= −
√2ε
M
Now, the left hand side of the primary condition (37) is
∂2L∂φ
2 = 2ε,
and the right hand side of (37) is
∂2L∂φ∂q
·(∂2L∂q2
)−1
· ∂2L
∂φ∂q= −√
2Mε · 1
M· −√
2Mε =2Mε
M,
and these two are equal. Thus the primary condition is met.The secondary condition automatically vanishes, since the condition is anti-symmetric and
we have only one second-order variable.To see that the system is indeed Ostrogradsky free, let us look at the Euler-Lagrange equation
(7). They are
0 = Eφ =d2
dt2∂L∂φ− d
dt
∂L∂φ
+∂L∂φ
= 2ε....φ −
√2Mεq −mφ− kφ, (40)
0 = Eq =d
dt
∂L∂q− ∂L∂q
= Mq −√
2Mε...φ. (41)
As can be seen, these equations have higher-order terms. However, we can take combinationsof them that are still zero in order to remove these higher-order terms. One such combinationis Eφ = Eφ + dV Eq
dt
0 = Eφ = 2ε....φ −
√2Mεq −mφ− kφ+
d(√
2εM
[Mq −
√2Mε
...φ])
dt
= −mφ− kφ.
26
This is the equation of motion for a harmonic oscillator. We still have a third order term in the
equations for q. Choose the combination Eq = Eq +BdEφdt
, for a yet to be determined constantB. Then
0 = Eq = Mq −√
2Mε...φ +B
d(−mφ− kφ
)dt
= Mq −(√
2Mε+Bm) ...φ −Bkφ,
which has no third-order terms if we choose B = −√
2Mεm
. It is also in this expression that wecan see the coupling between φ and q: the acceleration of q depends on the velocity of φ. Thesenal equations of motion are up to second order and thus stable. This concludes the example.
27
4 Field Theories
Up until now, we have only discussed (classical) mechanics. We have discussed three separatemethods to describe and solve problems in classical mechanics, and showed why the Ostrograd-sky Instability, or Ghost, forbids higher order theories. However, in physics we have movedon from classical mechanics, into quantum mechanics and relativistic theories, combined intoquantum eld theories. In this thesis, we will not discuss the quantum eld theories, but weshall discuss classical eld theories. In this section, I shall introduce classical eld theory andin particular introduce scalar eld theories. After that, we shall discuss the Ostrogradsky in-stability in eld theories, and the primary and secondary constraints required to remove theseghosts from second order theories. In the sections after this one we will discuss eld theoriesthat are Ostrogradsky free.
4.1 What are eld theories?
Field theories, in the simplest formulation possible, are theories that describe elds. Whereasin classical mechanics, the dependent variables are often the position coordinates of the objectsyou're interested in (1, 2 or 3 coordinates per object, depending on the dimensions of yourtheory), in eld theory the coordinates are promoted to independent variables. Instead, each`object' gets its own `eld'. This eld describes, for each possible position and time, `howmuch' of the object is at that particular time and place. As a comparison, consider the waveson the ocean. In classical mechanics, each wave would be given two coordinates, a latitudeand longitude for example, and we describe the evolution of these waves with 2q equationsof motion, 2 equations for each of the q waves. In eld theory, in contrast, there is only onewave `eld', which describes how high the water level is at each point at each time. The wavesthemselves are the maxima of this eld at each time.
In eld theory, there are multiple independent coordinates. There is generally one timecoordinate t, and d space coordinates xi, depending on the amount of space dimensions con-sidered. In the example of waves, we only needed 2 space coordinates, however, if we wouldwant to describe the winds in the Netherlands at dierent heights, we would also need a thirdspatial coordinate to describe the vertical. These space and time coordinates are together calledspace-time coordinates, and are together denoted by the notation xµ. The index µ runs over0, 1, . . . , d, with x0 = t. For further shorthand, we use ∂µ = ∂
∂xµto describe derivatives w.r.t.
the independent coordinates. If we split up space and time, we shall continue to use dottedderivatives to refer to time derivatives, i.e. ∂φ/∂t = φ, and use ∂i for derivatives explicitlytowards the space coordinates, i.e. ∂φ/∂xi = ∂iφ. In this thesis, we shall restrict ourselvesto three space coordinates. There are theories that use more space or time coordinates, butthose are not relevant to the work we discuss. In either case, most of the work in this sectiongeneralizes to include more spatial coordinates.
Field theories are described using a Lagrangians that depends on, in principle, the indepen-dent coordinates t and xi, the elds φ and their derivatives ∂µφ. They could in principle alsodepend on higher order derivatives, e.g. ∂µ∂νφ, but this generally brings about the OstrogradskyGhost, as we shall see later.
From these Lagrangians, we derive equations of motions using the Euler-Lagrange equations(7). These were derived for a very general case in chapter 2, but we restate them for eldtheories as
0 =∂L∂φ− ∂µ
∂L∂∂µφ
, (42)
28
for rst-order Lagrangians and
0 =∂L∂φ− ∂µ
∂L∂∂µφ
+ ∂µ∂ν
(∂L
∂ (∂µ∂νφ)
), (43)
for second-order Lagrangians.Unlike classical mechanics, the eld theoretical Euler-Lagrange equations produce partial
dierential equations, which are harder to solve [16]. Nonetheless, eld theories are preferredabove classical mechanics. The primary reason is that eld theories treat time and space onequal footing, as independent coordinates. Because of this, eld theories can be used to describerelativistic theories. Field theories that are relativistic are called Lorentz-invariant theories, forreasons that we will discuss in the next subsection.
4.2 Lorentz invariance and Scalar theories
Scalar theories are a particular class of relativistic eld theories. The scalar elds they describeare invariant under Lorentz transformations, which are a particular type of transformations onspace time. Space time transformations allow us to describe the same physics using a dierentcoordinate system. A general space time transformation is given by [17]
xµ = fµ (xν) . (44)
Lorentz transformations are linear space-time transformations, and are precisely those trans-formations that observe Einstein's principle of relativity. Einstein's principle can be caught intwo postulates [18, appendix A]:
1. The speed of light c is the same in all inertial frames.
2. The laws of nature are the same in all inertial frames.
A common example of the second postulate is that of an observer, you for example, placedon board of a sound-proofed train car with no windows. If you were to drop a ball in this traincar, you should not be able to tell if the train is stationary or moving at constant speed fromhow this ball falls. However, if the train is accelerating, you would notice this since the ballwould not drop straight down, but instead fall down to the back-end of the train.
The second postulate was known for a long time. It is preserved under the Galilean space-time transformations. However, the fact that the speed of light is constant only began to beknown around Einstein's time. As a result of this st postulate, the thereunto used Galileantransformations are no longer valid, and instead Lorentz' space-time transformations have tobe used.
Let us set up two observers moving at constant relative velocity v w.r.t. each other Inparticular, this movement is according on the x-axis for both observers; and their y and z axisalso point in the same direction. In this case, the Galilean transformations are given by
t = t, x = x− v · t, y = y, z = z.
In contrast, the Lorentz transformation between these two observers is given by [18]
t =t− vx
c2
1− v2
c2
, x =x− vt1− v2
c2
, y = y, z = z,
where c refers to the speed of light. In 3D-space there are 6 Lorentz transformations: there arethree rotations, one for each axis; and three `boosts'. The boosts are used to transform between
29
observers travelling at dierent velocities, and there is one boost for each space direction. Asimilar set-up can be used in theories with a dierent number of (space) dimensions, and inthose theories one will nd a dierent number of possible transformations11. In any numberof space dimensions, the Lorentz transformations are linear, and can thus be described by theequation
xµ = Λµνxν . (45)
The matrix Λµν is called a Lorentz matrix.These matrices, and thus the Lorentz transformations, can also be dened dierently, as
precisely those matrices that leave the Minkowski metric invariant. The Minkowski metric isgiven by [18]
ηµν =
−c2 0 0 0
0 1 0 00 0 1 00 0 0 1
,
and the invariance requirement is given by
ηµν = ΛρµΛσνηρσ. (46)
Theorem 4. The invariance condition of the Lorentz transformation (46) preserves distancesin space-time.
Proof. Let γ (τ) be a curve in space time, γ = [xµ|xµ (τ) = xµ, τi 6 τ 6 τf ]12. From appendix
B and [10], we know that the length of this curve is given by
Φ (γ) =
∫τiτf
√ηµν xµxν dτ,
where this time xµ = dxµ/dτ . This equation can be written coordinate free, i.e. not dependendon the parametrization τ :
Φ (γ) =
∫γ
√ηµν dxµ dxν .
Next, apply a Lorentz transformation to the coordinates xµ. This transformation transformsthe integrand as
dxµ dxνηµν = Λµρ dxρΛνσ dxσΛκµΛλνηκλ
= ΛκµΛµρ dxρΛλνΛ
νσ dxσηκλ
= δκρ dxρδλσ dxσηκλ
= dxρ dxσηρσ,
and thus
Φ (γ) =
∫γ
√ηµν dxµ dxν =
∫γ
√dxρ dxσηρσ.
Ergo, Lorentz invariance preserves distances.
11Of course, a transformation between two observers now travelling at a velocity v′ instead of v in the samedirection is technically a dierent transformation as well. However, it is the same kind of transformation. Thesame applies to the rotations: rotations of 30 v 135 around the same (x) axis are also technically dierent,but they are of the same kind, a rotation around the x axis.
12We parametrize to variable τ rather than t to avoid confusion.
30
Since we now have dened Lorentz transformations, we can discuss Lorentz invariance orcovariance. An object object is called Lorentz invariant, if it is of the same form before andafter a Lorentz transformation, i.e.
S (xµ) = S (Λµνxν) . (47)
If an invariant object has no indices, then such an object is called a Lorentz scalar or oftenjust scalar. The scalar elds we discuss are called scalar elds because they are Lorentz scalars.Another example of a Lorentz scalar is the Lagrangian of a Lorentz invariant theory, in factsuch a theory is called invariant precisely because the Lagrangian is a Lorentz scalar.
An object can also be Lorentz covariant, such objects are called Lorentz vectors or Lorentztensors, depending on the amount of indices the object has. A particular example of a Lorentzvector is the position vector xµ, since this transforms as xµ = Λµνx
ν by denition. A generalcovariant Lorentz vector Aµ transforms as [18]13
Aµ (xµ) = ΛµνAν (xµ) . (48)
By contrast, a contravariant Lorentz tensor transforms as
Aµ (xµ) =(Λ−1
)νµAν (xµ) . (49)
We can generalize this into a Lorentz tensor T µ1µ2...µn , which transforms as
T µ1µ2...µn (xµ) =
(n∏i=1
Λµiνi
)T ν1...νn (xµ) . (50)
Examples of Lorentz vectors and tensors are the derivatives of Lorentz scalars, i.e. ∂µφ is aLorentz vector and ∂µ∂νφ is a Lorentz 2-tensor.
With these preliminaries out of the way, we can discuss scalar eld theories. As mentionedabove, the `scalar' in `scalar eld theory' refers to the fact that the elds φ are Lorentz scalars.They are not the only Lorentz scalars, the Lagrangian itself also needs to be Lorentz invariant.As a result, the following Lagrangian could be a scalar theory:
L =1
2∂µφ∂µφ−
m2
2φ2, (51)
while this Lagrangian is not
L = φφ− m
2∂i∂iφ.
Equation (51) is one of the most basic Lagrangians in eld theory, and its Euler-Lagrangeequation is the well-known Klein-Gordon equation [12, chapter 2]
∂µ∂µφ = −m2φ. (52)
This theory is called a free theory, because the eld it describes is a free eld, it does notinteract, not with other elds (because there aren't any), nor with itself. This can be provenusing the Feynman path-integral formalism from Quantum Field Theory [12]. An explanationof this formalism would be too far astray of the goal of this thesis, but a short summary of theresult is that a Lagrangian has interactions if the eld φ and/or its derivatives appear morethan quadratically.
The next step from a single-eld theory is a multi-eld theory, describing q scalar elds φa.We can construct a very general multi-eld Lagrangian, L = A (φa, ∂µφ
a). Of course, such aLagrangian is to general to tell us anything. In order to say anything meaningfull, we need torestrain the theory. We can do this by imposing symmetries on the theory.
13This relation is also the denition of a covariant tensor.
31
Denition 2. A symmetry of a Lagrangian theory is a transformation Φ of the (in)dependentcoordinates that leaves the Euler-Lagrange equations invariant.
A standard example of a symmetry to impose upon a theory with q scalar elds is an SO(q)symmetry. Roughly speaking, this symmetry states that nothing changes upon relabelling theelds. E.g. the transformation φi ↔ φj does not change the equations of motion. An exampleof this is the Klein-Gordon Lagrangian (51) generalised to q elds:
L =1
2∂µφa∂µφa −
m
2φaφa.
One could also impose other symmetries. For example, one could split the scalars up intotwo groups, and impose an SO(m) symmetry on the rst m elds and an SO(q −m) on theremaining n. Other symmetries are shift and Galilean symmetries, which we shall discuss inchapter 5. All symmetries lead to a conserved current and conserved quantity, according toNoether's theorem [11,12]. As a result, symmetries are often used or imposed in order to reducethe theory and make it easier to solve the equations of motion.
4.3 The Ostrogradsky ghosts in eld theories
As hinted at in the above, eld theories are not immune to the Ostrogradsky ghost. As inmechanics, in eld theory the Ostrogradsky ghost appears whenever higher order derivativesare used in the Lagrangian. In particular, it is the presence of higher order time derivativesthat causes the ghost to appear, as we shall also see when discussing conditions to remove theghost in the next subsection.
In this subsection and the next, I shall closely follow the derivation by Crisostomi, Kleinand Roest in [8, appendix B]. We shall immediately discuss the most general case, wherethere are m second order elds φa and n rst order elds qα. The Lagrangian is then given asL = L (φa, ∂µφ
a, ∂µ∂νφa, qα, ∂µq
α). Rather than using the full Hamiltonian analysis, in whichevery derivative is replaced by a helper eld, we will only remove the second order derivativesby use of the elds Aµ. The equivalent Lagrangian is then given by
Leq = L(φa, Aaµ, ∂νA
aµ, q
α, ∂µqα)
+ λµa(∂µφ
a − Aaµ). (53)
Through the Euler-Lagrange equations for the multipliers λµa , we can nd the following relations:
0 =∂Leq∂λµa
⇒
∂µφa = Aaµ ⇔
∂νφa = Aaν .
We can take the derivative of the second equation to ∂ν and the third equation to ∂µ and nd
∂νAaµ = ∂ν∂µφ
a = ∂µ∂νφa = ∂µA
aν ,
where for the central equal sign I have used that we can change the order of integration. Inparticular, this equation implies Aai = ∂iA
a0. We can use this relation to split the space and
time components of the Lagrange multipliers and the helper elds, and obtain
Leq = L(φa, Aa0, A
ai , A
a0, ∂iA
a0, ∂iA
aj , q
α, qα, ∂iqα)
+ λ0a
(φa − Aa0
)+ λia (∂iφ
a − Aai ) . (54)
In the next part of the analysis, we dene the conjugate momenta using Poisson brackets.
φa, πb = δab ,Aaµ, P
νb
= δab δ
νµ,
λµa , Λbν
= δbaδ
µν , qα, pβ = δαβ .
32
Next, we compare them with the Lagrangian denition of momenta to derive our primaryconstraints.
πa =∂Leq∂φa
= λ0a, Λaµ =
∂Leq∂λµa
= 0, P ia =
∂Leq∂Aai
= 0,
P 0a =
∂Leq∂Aa0
=∂L∂Aa0
, pα =∂Leq∂qα
=∂L∂qα
.
The top row of these constraints produce in total m+ (D+ 1) ·m+D ·m primary constraints,where D is the amount of spatial dimensions. If there is no additional degeneracy, then thebottom row allows us to express Aa0 and qα in terms of the other variables :
Aa0 = fa(P 0b , φ
b, Ab0, Abi , ∂iA
b0, ∂iA
bj, q
α, pα),
qα = gα(P 0b , φ
b, Ab0, Abi , ∂iA
b0, ∂iA
bj, q
α, pα).
For the next step, we need to time evolve the constraints. This in turn requires us to dene theHamiltonian. Whereas in classical mechanics, the Hamiltonian is just the Legendre transformof the Lagrangian, eld theory is more involved. Legendre transforming the Lagrangian yieldsa function called the Hamiltonian density [12]. It is the integral of this Hamiltonian densityover all space that produces the Hamiltonian. Similar to chapters 2 and 3, we dene twoHamiltonian densities:
H0 = P 0a f
a + pαgα − L,
HT = H0 + aa(πa − λ0
a
)+ b0
aΛa0 + biaΛ
ai + caiP
ia.
The total Hamiltonian is then dened as
HT =
∫HT dDx.
The functions aa, bµa , cai are the Lagrange multipliers that enforce the primary constraints. We
have already split bµa into time and space components, since the time evolution of the constraintsΛaµ produces dierent results for its time and space components. Indeed, the next step in ouranalysis is to compute this time evolution:
Λai , HT = ∂iφa − Aai ≈ 0,
P ia, HT
=
∂L∂Aai
− P 0b
∂f b
∂Aai− λia ≈ 0,
Λa0, HT = Aa0 − aa = 0,πa − λ0
a, HT
=
∂L∂φa
+ P 0b
∂f b
∂φa+ ∂iλ
ia − b0
a = 0.
Again the top row produces constraints, this time 2D ·m secondary constraints. The bottomrow can be used to x the multipliers aa and b0
a. Further time evolving our secondary constraintsallows us to x the remaining multipliers:
∂iφa − Aai , HT = ∂iφa, HT − cai = 0,∂L∂Aai
− P 0b
∂f b
∂Aai− λia, HT
=
∂L∂Aai
− P 0b
∂f b
∂Aai, HT
− bia = 0.
At this point the analysis ends. All constraints are of secondary class, and thus we can calculatethe degrees of freedom using (17):
#DOF =1
2(2 (m+ 2 (D + 1)m+ n)− 2 · 0− (m+ (D + 1)m+Dm)− 2Dm) = 2m+ n.
As we can see, we have m too many degrees of freedom, all of whom are Ostrogradsky ghosts.
33
4.4 Ostrogradsky-free eld theories
We wish to remove the ghost degrees of freedom from our second order theory. To do so, wemust impose additional constraints upon the Lagrangian. In particular, these constraints mustbe imposed upon the m second order elds, since it are those elds that cause the ghost. Wecan impose the following m constraints upon these elds:
χa = P 0a − Fa
(Ab0, ∂iA
b0, q
α, ∂iqα, pα
)≈ 0. (55)
We can not impose this constraint on any Lagrangian. Imposing this constraint requires thatthe relation P 0
a = ∂Leq∂Aa0
is no longer invertible for all P0a. This is the case when the kinetic
matrix
K =
(LAaAb LAaqβLqαAb Lqαqβ
)has m null-vectors in the Ab sector. We can use the same null-vectors as we did for classicalmechanics, and obtain the null vectors Va =
(δba, V
αa
), where V α
a = −LAaqβL−1qβ qα
. The condition
that these vectors are then actual null vectors of K is similar to the primary condition (37) fromclassical mechanics. Using the same notation as we did in chapter 3, let ψA be the collectionof the dependent variables φa and qi, where the index A rst runs over the φa and then overthe qi. In this notation, the primary condition becomes [8]
0 = P(ab) = vAa LψAψBvBb . (56)
The constraints χa need to be added to the total Hamiltonian. Since we have already dealt withthe other constraints in the previous section, we can ignore them for our continued analysisand dene our total Hamiltonian as
HT = H0 +
∫ξaχa dDx,
where H0 =∫H0 dDx. We then compute the time evolution of χa as
χa (x) , HT = χa (x) , H0+
χa (x) ,
∫ξb (y)χb (y) dDy
,
where we have made the dependence on xµ in the integral explicit to avoid confusion. The lastPoisson bracket is composed of several terms, and these are
P 0a ,
∫ξbFb dDy
=
(∂i
∂Fb∂∂iAa0
− ∂Fb∂Aa0
)ξb +
∂Fb∂∂iAa0
∂iξb, (57)
Fa,
∫ξbP 0
b dDy
=∂Fa∂Ab0
ξb +∂Fa∂∂iAb0
∂iξb, (58)
Fa,
∫ξbFb dDy
=
(Fa, Fb+
∂F(a
∂∂iqα∂i∂Fb)∂pα
)ξb +
(∂F(a
∂∂iqα∂Fb)∂pα
)∂iξ
b. (59)
We can group these brackets together, and nd that
χa (x) , HT = χa (x) , H0+ S[ab]ξb +(Si)
(ab)∂iξ
b.
Since we need m additional constraints, we need to remove the dependency of ξb from thisequation, else this equation can be used to x some of the ξb. In that case χa (x) , HT ≈ 0would be identically zero for certain χa and thus we can not make it a constraint. Thus we needall of the dependency removed and therefore we have the constraints S[ab] = 0 and (Si)(ab) = 0.
34
To nd the exact form of these constraints, we use not only the above Poisson brackets, but alsothe constraint (55). This constraint implies Fa ≈ P 0
a = ∂L∂A0
a, and we can use this to calculate
the derivatives of Fa. Doing this computation, we nd that the two sets of secondary conditionsare given by [8]
0 = (Si)(ab) = 2vAa Lψ(A∂iφB)vBb , (60)
and0 = S[ab] = 2vAa Lψ[AψB]
vBb + 2vA[aLψA∂iψB∂ivBb] − δi
(vAa Lψ[A∂iψB]
vBb
). (61)
Note that these secondary conditions reduce to the conditions from classical mechanics in theabsence of space coordinates. In particular, (60) vanishes, while (61) reduces to (38) in theabsence of spatial derivatives.
The conditions (56), (60) and (61) apply to a general eld theory. However, as statedpreviously, we are primarily interested in Lorentz invariant eld theories. We are interestedto see if requiring the constraint of Lorentz invariance causes any changes in the primary andsecondary conditions. Fortunately, Crisostomi et al. also investigated this [8, section 2.3]. LetL be a Lorentz invariant Lagrangian, then after a Lorentz transformation we nd the (generalrelation)
L = L+ δL,
whereδL = Lφaδφa + L∂µφaδ∂µφa + L∂µ∂νφaδ∂µ∂νφa + Lqα + δqα + L∂µqαδ∂µqα.
For a Lorentz invariant theory, a Lorentz transformation should not change the degeneracies.E.g., if the primary condition is satised pre-transformation, then it should also be satisedafter the transformation. As a result, we have
0 = P(ab) = P(ab) + δP(ab).
If we assume that the primary condition is satised, then δP(ab) must also vanish. Let us nowconsider a particular Lorentz transformation, an arbitrary boost in the i-direction. The changein P(ab) is given by
δiP(ab) =(P(ab)
)˙ΨN∂iΨ
N +(P(ab)
)∂iΨN
ΨN + (Si)(ab) , (62)
where the `eld' ΨN runs over all elds, i.e. Ψ ≡ (φa, ∂µφa, qα)14. If the primary condition
is met for a Lorentz invariant theory, this equation equals zero. Further, since P(ab) = 0, itsderivatives should also be zero. As a result, (62) becomes
(Si)(ab) = 0.
Thus, for a Lorentz invariant theory, satisfying the primary condition (56) also satises thesymmetric secondary condition (60).
Further note that, even for a Lorentz invariant theory, the constraints are not Lorentz in-variant. Lorentz invariance however should not be expected, since the Ostrogradsky theoremitself is not invariant. The theorem concerns itself only with second order time derivatives,and the instability only propagates in the time derivatives, so the derivatives in the spatialdirection are irrelevant. This violates Lorentz invariance, and causes non-Lorentz invariantconstraints. This also manifests itself in the `healthy' formulation of the equations of mo-tion, where the manifestly second order formulation of the equations of motion belonging to ahealthy Lagrangian L (∂∂φm, ∂φm, φm, ∂qα, qα) are not Lorentz invariant, and vice-versa: themanifestly Lorentz-invariant equations of motion are not necessarily second-order. We have
14Compare to the denition of ψA above.
35
shown in example 8 that in an Ostrogradsky-free theory we can take certain combinations ofthe Euler-Lagrange equations in order to obtain pure second order equations. This can also bedone for eld theories. In particular, for a eld theory, take the combination [8]
Eφa+d
dt(V α
a Eqα)+∂i(αiαa Eqα
)+Uα
a Eqα = P(ab)φ(4)b +(Si)(ab) ∂iφ
(3)b +S[ab]φ
(3)b +
(φ, q, . . .
), (63)
whereαiαa = −
(L∂iφaqβ + Lφa∂iqβ + 2V γ
a Lq(γ qβ))L−1qβ qα
,
Uαa =
((Ca)qβ − α
iγa ∂iLqγ qβ
)L−1qβ qα
,
andCa = Eφa + V α
a Eqα .
As can be seen from the form of (63), satisfying the constraints (56), (60) and (61) causes thiscombination to be second order in derivatives at the highest. The Euler-Lagrange equationsof a Lorentz invariant theory are also Lorentz invariant. The combination of Euler-Lagrangeequations (63) can only be Lorentz invariant if W µα
m = (V αa , α
iαa ) is a Lorentz vector and Uα
a isa Lorentz scalar, so that (63) becomes
Eφa + ∂µ (W µαa Eqα) + Uα
a Eqα .
This equation would then be Lorentz invariant. Unfortunately, W µαm and Uα
a generally donot meet this requirement of Lorentz (co/in)variance. As a result, we have something of atrade-o: We can either have manifestly Lorentz invariant equations of motion, but that docontain higher-order derivatives, or have manifestly lower-order equations of motion, but arenot Lorentz invariant [8].
Example 9. Let us discuss a simple example of a Lorentz Invariant stable theory with bothone second order and one rst order eld. Our theory is described by the Lagrangian
L (φ, q) = (2φ)2 + 22φ∂µφ∂µq + ∂µφ∂νφ∂µq∂νq. (64)
For this case, the primary constraint is
0 =∂2L∂φ
2 −L2φq
Lqq.
We have∂2L∂φ
2 = 2, Lφq = 2φ and Lqq = 2φ2
so this condition is satised. On account of Lorentz Invariance, the symmetric condition isautomatically satised. The anti-symmetric condition is also satised on account of there onlybeing one second-order eld.
The Euler-Lagrange equations are given by
0 = ∂µ∂ν
(∂L
∂∂µ∂νφ
)− ∂µ
(∂L∂∂µφ
)+∂L∂φ
= 22 [2φ] + 4∂µ∂νφ∂µ∂νq + 2∂µφ∂
µ2q − 22φ2q − 2∂µ∂νφ∂µq∂νq − 2∂µφ2q∂µq − 2∂µφ∂νq∂
µ∂νq,
36
and
0 = ∂µ
(∂L∂∂µq
)− ∂L∂q
= 2∂µ2φ∂µφ+ 22φ2φ+ 22φ∂µφ∂µq + 2∂µφ∂µ∂νφ∂νq + ∂µφ∂νφ∂
µ∂νq.
These equations of motion are clearly Lorentz invariant, but also fourth-order. Let us try torewrite Eφ into second-order form. We have
V =LφqLqq
=1
φ,
αi = −(L∂iφq + Lφ∂iq + 2V Lqq
)L−1qq = −∂
iq
φ− ∂iφ
φ2− 2
φ2.
W µ = (V, αi) is not a Lorentz vector, since αi 6= 1∂iφ
. As a result, the transformed Euler
Lagrange equation Eφ + ∂µ (W µEq) + UEq will not be Lorentz invariant, though it will besecond order. Calculating the exact form of this transformed equation of motion is left as anexercise to the reader.
37
5 Galileons, and other second order eld theories
In the previous section, we have discussed eld theories and how the Ostrogradsky ghost ismanifest in those. We also discuss constraints to avert the ghost. In this section, we shalldiscuss some of the second order theories that are free of the Ostrogradsky ghost. In particular,we shall focus on the Galilean theories, a class of second-order Lagrangian theories that haveonly second-order derivatives in their equations of motion, before generalising to the mostgeneral ghost-free multi-scalar theory.
5.1 One Galileon
Galileon theories are given that name because they obey a so called Galilean symmetry. ThisGalilean symmetry is a symmetry in the elds, and thus a Galilean theory is symmetric underthe transformation
φ = φ+ bµxµ + c. (65)
This symmetry is a generalization of the 3D Galilean symmetry, with the eld φ acting as afourth spacial dimension [5], hence its name. For a theory to be symmetric under the Galileantransformation, only the equations of motion need to be invariant. Clearly, Galilean symmetryrequires that the equations of motion are of strict second order, i.e. only second order termsappear15. Nicolis et al. [4] described the general Lagrangian that obeys the Galilean symmetry.This general Lagrangian is a linear combination of the following ve Lagrangians:
L1 = φ,
L2 = −1
2∂µφ∂µφ,
L3 = −1
2∂µ∂
µφ∂νφ∂νφ, (66)
L4 = −1
2
[(∂µ∂µφ)2 − ∂µ∂νφ∂ν∂µφ
]∂ρφ∂ρφ
L5 = −1
2
[(∂µ∂µφ)3 − 3∂µ∂µφ∂
ν∂ρφ∂ρ∂νφ+ 2∂µ∂νφ∂
ν∂ρφ∂ρ∂µφ
]∂σφ∂σφ.
The total Lagrangian is then given by L =∑5
m=1 αmLm. By computing the associated Euler-
Lagrange equations (7), we can see that these Lagrangians are indeed symmetric under theGalilean symmetry (65). The Euler-Lagrange equations derived from these Lagrangians aregiven by [4, 5]:
E (L1) = 1,
E (L2) = 2φ,
E (L3) = (2φ)2 − ∂µ∂νφ∂ν∂µφ,E (L4) = (2φ)3 − 32φ∂µ∂νφ∂ν∂µφ+ 2∂µ∂νφ∂
ν∂ρφ∂ρ∂µφ,
E (L5) = (2φ)4 − 6 (2φ)2 ∂µ∂νφ∂ν∂
µφ+ 82φ∂µ∂νφ∂ν∂ρφ∂ρ∂µφ
+ 3 (∂µ∂νφ∂ν∂
µφ)2 − 6∂µ∂νφ∂ν∂ρφ∂
ρ∂σφ∂σ∂µφ.
E (L1) is a constant, and thus clearly invariant under any transformation to the eld φ. All otherEuler-Lagrange equations are purely second order, and from (65) it is clear that ∂µ∂
νφ = ∂µ∂νφ.
15This symmetry allows for higher-order terms, but those are blocked by Ostrogradsky's theorem. See chapter4 or [1].
38
There are only 5 Lagrangians, and the question is why not more? From the structure of theLagrangians (66), a general formula can be derived as
Lm = −1
2∂µφ∂
µφδµ3...µmν3...νm∂µ3∂
ν3φ · · · ∂µm∂νmφ, (67)
for m > 3, and where δµ3...µmν3...νmis the generalized Dirac-delta function, see appendix A. This
generalized Dirac-delta function is zero if it has more than p = d + 1 indices, where d is theamount of spatial dimensions. This eliminates Lm for m > 6.
There are other formulations of the Galilean Lagrangians (66), which can be used for dierentgeneralizations [5]. However, they all produce the same Euler-Lagrange equations [5], and thuswe won't discuss them here.
5.2 Self duality
An interesting thing about the theory of Galileons, is that it is dual to itself. That is to say, theGalileon Lagrangians (66) are transformed to another Galilean Lagrangian under the coordinatetransformation [19,20]
xµ = xµ + ∂µφ (x) . (68)
This transformation admits an inverse transformation. Impose that a dual scalar eld π (x)exists, such that the inverse relation is
xµ = xµ − ∂µφ (x) . (69)
In this equation, as well as in the rest of this section, assume that barred coordinates andfunctions refer to coordinates and functions in the dual system.
It is a property of this map that the derivative of φ transforms like a scalar, instead of theeld φ itself, i.e. ∂µφ = ∂µφ. The transformation rules for φ and φ are instead given by [19]
φ (x) = φ (x) +1
2∂µφ∂
µφ (70)
φ (x) = φ (x)− 1
2∂µφ∂
µφ (71)
By comparison, it are the second derivative tensors Πµν = ∂µ∂νφ that transform in a covariant,
vectorial manner: Πµν = JµρΠ
ρν , with J
µν the Jacobian of the coordinate transformation (68),
Jµν = ∂xµ
∂xν= δµν + ∂ν∂
µφ.Next to consider is how the Lagrangians transform. The total Lagrangian is a linear com-
bination of the Lagrangians (66), given as L =∑5
i=2 ciLi (φ, ∂φ, ∂∂φ)16. After performing theduality transformation (68), the Lagrangian becomes L =
∑5i=2 ciLi
(φ, ∂φ, ∂∂φ
)for dierent
constants ci This shows that under this duality transformation, a galileon indeed transformsinto a galileon. The relations between the coecients ci are given by [19]:
c2 = c2,
c3 = c2 + c3,
c4 = c2 + 2c3 + c4,
c5 = c2 + 3c3 + 3c4 + c5.
These relations follow Newton's binomial. An interesting eect of this duality is that it mapsa free scalar theory into a Galilean theory [20]. A free scalar theory is a galileon theory withLagrangian L = c2L2 = c2
2∂µφ∂
µφ, and under the duality it maps to
L = c2L2 + c2L3 + c4L3 + c2L5.
16For this analysis, c1 = 0, i.e. we ignore L0.
39
The nal property to examine is the Galileon transformation. The theory of Galileons isbased on the symmetry under the Galilean transformation. It turns out under the dualitytransformation (68), the Galileon symmetry transformation becomes a mixed transformation,consisting of both a symmetry transformation and a translation [20]. In formula form, if wehave the Galilean symmetry transformation
φ′ = φ (x) + bµxµ,
then its dual eld transforms as
φ′ (x) = φ (x− b)− bµxµ.
This mixing of symmetries can be understood from the origins of Galileon theory, as a decoupledversion of DGP gravity [20,21]. In this decoupling, a galileon transformation on φ is equivalentto a translation of the metric coordinates x in the barred coordinate system, and vice versa.Thus, when a galileon transformation is performed on φ, its dual φ undergoes not just thegalileon transformation, but also the equivalent translation. We will come back to this, afterwe have discussed general relativity and DGP gravity in chapters 6 and further.
5.3 Multi Galileons
We can generalize galileon theory by adding additional galileon elds. The trivial (and un-interesting) is to just add additional the Lagrangians for dierent galileons together, e.g.L = Lgal (φ1) + Lgal (φ2). Mixing galileon elds and thus making them interact is much moreinteresting. Of course, the end result still must both be Galilean invariant and Ostrograd-sky free. Trodden and Hinterbichler [22] and Padilla and Sivanesan [23] all looked into thesemulti-galileon theories. The general Lagrangian for q galileons is given by [23]
L = αaφa + αab∂µφ
a∂µφb +5∑
m=3
αa1...am∂µφa1∂µφa2δµ3...µmν3...νm
∂µ3∂ν3φa3 · · · ∂µm∂νmφam , (72)
where the tensor αa1...am is fully symmetric in all its indices. The αa1...am behave as Lorentzscalars, since the am are not Lorentz indices, i.e. they do not count over the independentvariables. Thus (72) is Lorentz invariant. It is of a similar form to (67), and indeed reduces toit for q = 1.
The Lagrangian (72) is also ghost free. Due to the symmetry of αa1...am , the higher derivativeterms in the Euler-Lagrange equations (7) cancel against each other. We can show this bycomputing the primary (56) and secondary conditions (61)17. Since our theory concerns onlysecond order variables, the conditions reduce to
P(ab) = Lφaφb (73)
andS[ab] = 2Lφ[aφb] − ∂iLφ[a∂iφb] . (74)
We can compute these conditions. For m = 1, 2, 3, P(ab) = 0, simply because there are eitherno second order derivatives, or they appear only linearly. For m = 4, 5 we nd
P(ab) = Lφaφb
=∂
∂φb
5∑m=4
αa1a2a...am (m− 2) ∂µφa1∂µφa2δi4...imj4...jm
∂i4∂j4φa4 · · · ∂im∂jmφam
= 0.
17Lorentz invariance takes care of condition (60).
40
It is due to the anti-symmetric nature of the generalized Dirac-delta matrix that no second-order time derivatives appear in the Lφa , and thus P(ab) = 0. Similarly, we can compute (74).We shall do this in two parts:
Lφ[aφb] = Lφaφb − Lφbφa
=∂
∂φb
(αa1a2a∂µφ
a1∂µφa2 + 2αa1a2aa4∂µφa1∂µφa2∂i∂
iφa4 + 3αa1a2aa4a5∂µφa1∂µφa2δi4i5j4j5
∂i4∂j4φa4∂i5∂
j5φa5)
− ∂
∂φa
(αa1a2b∂µφ
a1∂µφa2 + 2αa1a2ba4∂µφa1∂µφa2∂i∂
iφa4 + 3αa1a2ba4a5∂µφa1∂µφa2δi4i5j4j5
∂i4∂j4φa4∂i5∂
j5φa5)
=(
2αa1baφa1 + 42αa1baa4φ
a1∂i∂iφa4 + 6αa1baa4a5φ
a1δi4i5j4j5∂i4∂
j4φa4∂i5∂j5φa5
)−(
2αa1abφa1 + 42αa1aba4φ
a1∂i∂iφa4 + 6αa1aba4a5φ
a1δi4i5j4j5∂i4∂
j4φa4∂i5∂j5φa5
)= 0,
where the symmetry of the α tensors causes the cancellation. Last, we have
Lφa∂iφb =∂
∂∂iφb
(αa1a2a∂µφ
a1∂µφa2 + 2αa1a2aa4∂µφa1∂µφa2∂i∂
iφa4 + 3αa1a2aa4a5∂µφa1∂µφa2δi4i5j4j5
∂i4∂j4φa4∂i5∂
j5φa5)
= 0.
Again, the anti-symmetric nature of the generalized Dirac-delta matrix avoids mixed derivativesin Lφa . Combining these results, we have S[ab] = 0 for our multi-galileon Lagrangian (72), whichshows that our multi-galileon theories are Ostrogradsky stable.
So far, we have only discussed Galileon theories for their mathematical properties, i.e. asOstrogradsky-free second order theories. However, Galileon theories also have some interestingphysical properties. They are used in scalar-tensor theories of gravity to modify the behaviourof gravity at large distances in order to explain the cosmological constant [4]. We will discussthese modications after we have discussed General Relativity and its scalar, tensorial andscalar-tensorial realizations in chapters 6 and 7.
5.4 Shift Symmetry
We have previously discussed second order scalar theories that obey Galilean invariance. Ofcourse, these are not the only second order Ostrogradsky free theories one can construct. Ratherthan the Galilean symmetry, we impose the shift symmetry φ = φ + c, with c an arbitraryconstant. This symmetry is less restrictive than the Galilean symmetry, clearly any theorythat admits the Galilean symmetry admits the shift-symmetry. This shift symmetry is used incertain `inaton' ination theories [24, 25]. The symmetry that is spontaneously broken in theGoldstone mechanism is also a shift-symmetry [12,26].
Similar to Galileons, it are the equations of motion that should be invariant under the shift-symmetry. Clearly, any derivative is invariant, while an appearance of φ itself would showvariance. Thus, the Euler-Lagrange equations should only depend on derivatives of φ, noton φ itself. Furthermore, we are dealing with Lorentz invariant theories. As a result, the rstderivatives can only appear in pairs. We can thus introduce the variable X = 1
2∂µφ∂
µφ. Clearly,any Lagrangian that is an arbitrary function ofX will be shift-symmetric and Ostrogradsky free.There are more general Lagrangians though, in particular the Galilean invariant Lagrangians(66) are shift-symmetric and ghost free. We can generalize these Lagrangians to obtain the
41
following 5 shift symmetric Lagrangians:
L1 = φ,
L2 = f2(X),
L3 = ∂µ∂µφf3(X), (75)
L4 =[(∂µ∂µφ)2 − ∂µ∂νφ∂ν∂µφ
]f4(X)
L5 =[(∂µ∂µφ)3 − 3∂µ∂µφ∂
ν∂ρφ∂ρ∂νφ+ 2∂µ∂νφ∂
ν∂ρφ∂ρ∂µφ
]f5(X)
where the functions fi(X) are arbitrary functions of X. The total Lagrangian is given byL = αL1 +
∑5m=2 Lm18. All but L1 are shift-symmetric themselves; and due to their second
derivative structure, they are all ghost free. For L1, we have the Euler-Lagrange equationsE1 = 1, which also does not depend on φ.
The question is whether these ve are all shift-symmetric Lagrangians. We shall investigateif there could be other rst-order Lagrangians. For a rst-order Lagrangian, we can rewrite theEuler-Lagrange equations (7) as
E (L) =∂L∂φ−2φ
∂L∂X− ∂µφ∂µ
(∂L∂X
), (76)
written in terms of X rather than ∂µφ. On these equations, we impose the constraint ∂E(L)∂φ
= 0,which translates into the constraint
∂E (L)
∂φ= (1−X)
∂2L∂φ2 − (2φ∂µφ∂νφ∂µ∂νφ)
∂2L∂φ∂X
≈ 0. (77)
If a Lagrangian is linear in φ, this reduces to just
∂L∂X
= 0, (78)
i.e L is not allowed to depend on X. Only L1 and constant multiples of it satisfy this constraint.For all other Lagrangians, (77) reduces to a dierential equation coupling the dependence onX to the dependence on φ:
(1−X)∂L∂φ
= (2φ+ ∂µφ∂νφ∂µ∂νφ)
∂L∂X
+ f (X) , (79)
where f (X) can be any arbitrary function of X. Of course, this derivation only holds is weare looking for a rst-order Lagrangian. Adding second-order terms to the Lagrangian will ofcourse change the condition (77) and thus the dierential equation (79).
Next we generalize our shift-symmetric theory to a multi-scalar, shift-symmetric theory. Wegeneralize the Lagrangians (75), which themselves were derived from the Lagrangians (66) forGalileons. As such, we can derive our shift-symmetric Lagrangians from the multi-GalileonLagrangians (72) by allowing the constants αa1...am to be functions that depend on Xab =12∂µφ
a∂µφb. As a result, we obtain the Lagrangian
L = αaφa + α
(Xab
)+
5∑m=3
αa1...am(Xab
)δµ3...µmν3...νm
∂µ3∂ν3φa3 · · · ∂µm∂νmφam . (80)
The functions αa1...am(Xab
)are still required to be symmetric in all indices ai, and αa remains
a constant to satisfy (78).
18The constants from the Galilean Lagrangian are now made part of the functions fm(X).
42
5.5 General Scalar Theory
In the previous subsections, we have discussed several second-order scalar theories that wereall subject to some constraints or symmetries. In this subsection, we shall state the mostgeneral, second-order, ghost-free, multi-scalar theory. This theory possesses no constraints orsymmetries, other than those required to remove the Ostrogradsky ghost. For a single scalar,this general Lagrangian is given by [23]
L = f (φ,X) +5∑
m=3
fm (φ,X)Lm, (81)
where the Lm are the Galileon Lagrangians from (66). It is a generalization of both (66) and(75). For the Galileons, we required that ci be constants, due to the imposed Galilean symmetry.For shift-symmetric Lagrangians, we eased this requirement since we were only imposed the lessrestrictive shift symmetry. Thus, the cm were allowed to be functions of X and became fm (X).Now that we also drop the shift symmetry, we can allow the fm to be arbitrary functions ofφ and X. The desired absence of Ostrogradsky ghosts still forces a certain dependence onthe second-order derivatives. This dependence can be ensured by the use of the generalizedDirac-delta.
We can use the same `easing of constraints' to generalize (80) into a general multi-scalartheory. The resulting Lagrangian becomes [23]
L = α(φa, Xab
)+
5∑m=3
αa1...am(φa, Xab
)δµ3...µmν3...νm
∂µ3∂ν3φa3 · · · ∂µm∂νmφam . (82)
43
6 General Relativity
In chapters 4 and 5 we discussed eld theories, and in particular relativistic eld theories. Theseeld theories do not concern themselves with gravity. Since gravity is indeed observed, we wouldlike our theories to also be able to describe gravity. When Einstein formulated his theory ofSpecial Relativity, it was quickly noted that Newton's theory of gravity is not relativistic. Thus,work soon began by a variety of scientists to come up with a relativistic theory of gravity. Today,we know that Einstein eventually found the 'correct' theory, in which the gravitational 'force'is mediated by the metric tensor [5, chapter 1]. However, in the early 1900's, a large numberof dierent theories were proposed, and in those gravity was often mediated not by a tensor,but by a vector or even a scalar [27].
In this chapter, we shall discuss the principles behind general relativity and an attemptmade by Nordström and others to generalize Newton's theory of gravity into a scalar theoryof general relativity. This attempt was ultimately unsuccessful, and we shall thus also discussthe tensorial theory that Einstein derived instead. We will conclude with a generalization ofthe tensorial theory, aimed to nd the most general pure tensorial Ostrogradsky free theory ofgeneral relativity, called Lovelock gravity.
6.1 The principles of GR
The idea of relativity is that irrespective of your particular frame, the laws of physics mustremain the same. This was the idea behind Galilean relativity, and also behind Einstein's ideaof special relativity. The dierence between these two are the set of invariant laws, Einstein alsorequired that the speed of light is invariant. The other question is the denition of invarianceor equivalence. To answer this question, three principles of equivalence have been formulated [5,Chapter 1]. They are
• Weak Equivalence Principle (WEP): All uncharged, feely falling test particles follow thesame orbits, for equal initial positions and velocities.
• Einstein Equivalence Principle (EEP): The above is valid, and locally space-time followsthe laws of special relativity, i.e. is at.
• Strong Equivalence Principle (SEP): All uncharged, feely falling test particles, even mas-sive ones, follow the same orbits, for equal initial positions and velocities and locallyspace-time follows the laws of special relativity, i.e. is at.
Of course, it is to question if our universe itself obeys these equivalence principles. All threeof these principles have been experimentally tested, and found to be accurate to order 10−13
or better. As such, valid theories of gravity have to satisfy these principles, at least to theaccuracy of experiments [5, Chapter 2].
6.2 Scalar gravity and Nordström's theorem.
In this subsection, we shall investigate the particular scalar theory of general relativity proposedby Gunnar Nordström. It was one of the more conservative scalar theories, which in this contextmeans that it was closest to Newton's theory. Nordström based his work on a dispute betweenEinstein and Max Abraham regarding the constancy of the speed of light c. The former proposedthat c should depend on the gravitational potential, to avoid a problem in the equation for thegravitational force. Abraham used this hypothesis to derive a theory of gravitation that wasno longer special relativistic [27]. Let us explain the problem.
44
In both Abraham and Nordström's theory, the gravitational potential is a scalar Φ thatfollows the eld equation
2Φ = −4πGν, (83)
where G is the gravitational constant and ν is the rest mass density [27, page 6]. This issimply a Lorentz covariant formulation of Newton's scalar eld equation ∆Φ = −4πGν. Inthese theories, the source of gravity is the mass density distribution. With the force dened asFµ = −∂µΦ, the natural force equation becomes
− ∂µΦ = Fµ = mdUµdτ
, (84)
with Uµ the four-velocity and τ proper time [27, page 6]. Although a simple extension, it failswhen the constancy of c is taken into account. We have c2 = UµU
µ, and the constancy of cleads to the equation
0 =dc2
dτ= 2Uµ
dUµ
dτ. (85)
Inserting (84) into this equation then leads to the condition
0 = mUµdUµ
dτ= F µUµ = −mdΦ
dτ, (86)
which can only be satised in the narrow case that Φ is constant along the world-line of aparticle. Abraham tried to solve this problem, by stating that c is not a constant. Nordströmhowever, in his rst theory of gravity, instead let the inertial massm depend on the gravitationalpotential [27, page 17]. Thus, his new force law is
Fµ = mdUµdτ
+ Uµdm
dτ. (87)
Note the explicit dependence of m on time. It is this extra term that prevents the derivationof condition (86). Instead, the following condition is found
mdΦ
dτ= c2 dm
dτ, (88)
which has as solution for m
m = m0 exp
(Φ
c2
).
Inserting this back into (87) yields the equation of motion of a point particle [27, page 18]
− ∂µΦ =dUµdτ
+Uµc2
dΦ
dτ. (89)
This equation of motion has one drawback, which was Einstein's primary reason to dismissthis theory: an object that also moves in a horizontal direction falls slower than an object thatonly moves in the vertical direction [27, page 21]. Einstein felt that this was a problem for all(Lorentz covariant) scalar theories of gravity.
Later, based on work by Laue on relativistic stretched bodies, Einstein reasoned that thesource for gravity could not be the rest mass density ν, but instead should be the trace of thestress-energy tensor Tµν [27, page 40]. He argued in his 1913 paper, co-written with MarcelGrossmann, that scalar-theories of gravity violate the conservation of energy. See pages 40-42of [27] for a thought experiment that aims to prove this violation.
Nordström amended his theory, in an attempt to get around Einstein's critique. Einsteinhimself, in a 1913 lecture in Vienna, considered only two possible theories of gravity: his own
45
Entwurf theory, that would become General Relativity, and Nordström's amended theory [27,page 53].
The end of scalar gravitational theories then came soon after the publication of Einstein'stheory of General Relativity in 1915. One of the important result of Einstein's theory is theprediction of the anomalous perihelion of Mercury, which matched experiments of the time.Nordström's theory also predicts a perihelion shift, however his perihelion has the wrong sign.Thus, Nordström's theory was eventually dismissed, even by its most ardent supporters, infavour of Einstein's theory of General Relativity [27, chapter 16].
6.3 Einstein's tensorial theory of General Relativity
In the previous subsection, we already gave hints to Einstein's own theory of general relativity.It is a tensorial theory, where the metric is made dynamical in order to `bend' space timearound massive objects and thus give the eect of gravitational attraction. In this subsection,we shall discuss metric geometry and why Einstein arrived at his formulation. Last, we shallcast Einstein's equation into a Lagrangian theory.
Einstein's GR treats space-time as a 4-dimensional manifold. On this manifold, there is ametric tensor called gµν . This metric is used to measure distances in space time. Let γ be acurve in space time, parametrised by λ, i.e. xµ = xµ (λ). The length of this curve is then givenby [5, Chapter 2] [18, Chapter 1]
s =
∫γ
dλ√gµν xµxν , (90)
where xµ = dxµ/dλ. We saw the same thing when discussing Lorentz invariant eld theoriesin chapter 4, where the metric was the static Minkowski metric gµν = ηµν . Furthermore, on ageneral manifold with non-constant metric, dierentiation w.r.t. the space-time coordinates isno longer covariant. Instead, a connection Γ µ
νρ is associated with dierentiation [5, Chapter 2].In particular, the derivative of the metric is given by
∇µgνρ =∂gνρ∂xµ
− Γ σµνgσρ − Γ σ
µρgνσ. (91)
In General Relativity, the connection is chosen to be the Levi-Civita connection, which is givenby [5, Chapter 2]
µρσ
= Γ µ
ρσ =1
2gµν (∂ρgνσ + ∂σgρν − ∂νgρσ) .
As a result of this choice, the covariant derivative of the metric vanishes, ∇µgνρ = 0. Thischoice of connection tells us that everything on the manifold of space-time is governed by themetric tensor. Such a manifold is called a Riemannian manifold. On a Riemannian manifold,we can calculate the curvature of the manifold, i.e. how much it deviates from the `straightplane'. Roughly speaking: a at piece of paper has no curvature, a sphere does. This curvatureis given by the so called Riemann tensor, given by [18, Chapter 3]
Rµνρσ = ∂ρΓ
µνσ − ∂σΓ µ
νρ + Γ τνσΓ
µτρ − Γ τ
νρΓµτσ. (92)
With these preliminaries about the metric tensor discussed, we can discuss Einstein's theory.As mentioned in the previous section, Einstein identied the stress-energy tensor Tµν as thesource of gravity, motivated by Laue's work on relativistic stretched bodies [27, page 40]. Sincethis is a tensor, it makes sense to conclude that gravity itself is mediated by a tensor. Einsteinthen assumed that it was the metric tensor that mediates gravity, with the stress-energy tensorsomehow `bending' the metric, and thus altering the orbits followed by test particles.
46
The next question is how to couple the stress-energy tensor to the metric tensor. The naivemethod is gµν = κTµν , for some coupling constant κ. However this equation does not reduceto Newton's laws in the non-relativistic case [18, section 3.5]. Since Newton's laws have beentested extensively, and found to be a good description of a non-relativistic massive system, thisis a problem.
Newton's law for the gravitational potential is a second order dierential equation, thus thenext obvious candidate is to somehow couple the second order derivatives of gµν to Tµν . Thiscould be done through the Ricci tensor, Rµν = Rρ
µρν . The coupling would then be Rµν = κTµν .Einstein himself proposed this equation in 1915 [18, section 3.5]. However, it is wrong. Oneof the properties of Tµν is that it is divergence free: ∇µTµν = 0, like the metric tensor. TheRicci tensor is, in general, not divergence free, and thus taking the divergence of both sides ofRµν = κTµν leads to an inequality. Realizing this, later in 1915 Einstein proposed the morefamiliar equation [18]
Rµν −1
2gµνR = κTµν . (93)
The left-hand-side of this equation is divergence free. Furthermore, this equation reducesto Newton's laws in the non-relativistic case. This reduction xes the coupling constant toκ = −8πG
c4[18, section 3.5]. Furthermore, (93) satises both the WEP and EEP, part due to
the lack of divergence and in part since the manifold of space time in General Relativity is aRiemannian manifold [5, Chapter 2]. The left-hand-side of this equation is often dubbed theEinstein tensor and is dened as
Gµν = Rµν −1
2gµνR. (94)
Einstein's equation (93) describes how the metric gµν evolves through time and space, ina similar method that the Euler-Lagrange equations (7) are the evolution equations of elds.Thus, one could try to formulate (93) as the Euler-Lagrange equation derived from some La-grangian. This can be done, and the resulting action is called the Einstein-Hilbert action. It isgiven by [5, Chapter 2]
S =
∫ √gR
16πGd4x+
∫Lm (gµν , ψ) d4x. (95)
This action consists of the integral over two Lagrangians. First is the gravitational LagrangianLg =
√gR
16πG. Performing the principle of stationary action produces the left hand side of (93). It
is this gravitational Lagrangian that describes the dynamics of gravity. The second Lagrangian,Lm, contains the so called matter content of the theory. The eld ψ is a stand in for all eldsthat are part of the theory, not just any scalar elds φ, but it also includes all other types ofelds, such as vector elds Aµ. This matter content leads to the stress-energy tensor throughthe denition [5, Chapter 2]
T µν =2√g
∂L∂gµν
. (96)
Through this denition, the Euler-Lagrange equations for the metric gµν for the Lagrangian(95) are the Einstein equations (93) with κ = −8πG
c4.
In considering this procedure, one will note that (95) is a second-order Lagrangian, since Rµν
contains second-order derivatives of the metric. Ostrogradsky's theorem is quite general, anddoes not apply to just scalars. As a result, there should be Ostrogradsky ghost. Fortunately, thisis not the case. Einstein-Hilbert gravity is a degenerate theory, and the additional constraintsremove the Ostrogradsky ghosts. This is already apparent since Einstein's equation (93) is onlysecond-order, not fourth-order as a non-degenerate second order metric theory should be.
47
6.4 Lovelock Gravity
Einstein's theory is perhaps the most famous tensorial gravitational theory, but is not the mostgeneral tensorial theory. In 1971, David Lovelock [28] aimed to generalize Einstein's theoryin order to come up with the most general tensorial theory of gravity yielding second orderequations of motion, not just in our 3+1 dimensional space time, but in arbitrary D space-timedimensions.
Lovelock stated that the most general theory of general relativity involves nding the mostgeneral tensor Aµν that satises the following four properties [28]
a) Aµν is symmetric: Aµν = Aνµ.
b) Aµν is divergence free: ∇µAµν = 0.
c) Aµν is a function only of gµν and its rst two derivatives.
d) Aµν is at most linear in the second derivatives of gµν .
The eld equation in vacuum is given by Aµν = 0, and in general the eld equation is of theform Aµν = κTµν . This explains the rst two conditions; since Tµν is both symmetric anddivergence free, so must Aµν . Conditions c) and d) follow from the Ostrogradsky mechanismand the desire to keep the system ghost-free, see chapters 3 and 4.
According to Lovelock, the most general tensor Aµν satisfying at least conditions a) - c) ingeneral D dimensions is given by [28]
Aµν =
dD2−1e∑
p=1
ap
(δρα1...α2p
µβ1...β2pgρν + δ
ρα1...α2p
νβ1...β2pgρµ
) p∏t=1
Rα2t−1α2t
β2t−1β2t + agµν , (97)
where a and ap are constants. Choosing the constants then ensures condition d) to be met.In 3+1 dimensions, this equation reduces to [28]
Aµν = agµν − bGµν . (98)
Like in general relativity, the eld equation Aµν = κTµν is the Euler-Lagrange equation of aLagrangian. In our general, D-dimensional case, this Lagrangian is given by [28]
L =√g
dD2−1e∑
p=1
2ap
(δα1...α2p
β1...β2p
) p∏t=1
Rα2t−1α2t
β2t−1β2t + 2a√g. (99)
This Lagrangian is similar to the Lagrangian for galileons (72). Both use several copies ofthe second derivatives (Rµ
νρσ and ∂µ∂νφ respectively) of their elds (gµν and φ), multipliedtogether through the use of the generalized Dirac-delta function. The generalized Dirac-deltafunction ensures that there are no higher order derivatives present in the equations of motion.This similarity was noticed by Klein et al., who used this similarity to discuss Galileons as ananalogue to general relativity [29].
In our space-time, the Lovelock Lagrangian (99) becomes
L =√g4apR + 2a
√g. (100)
This is the Lagrangian of the Einstein-Hilbert action (95) with a cosmological constant a [5,Chapter 2]. From this, we conclude that in our regular space-time, Einstein's General Relativityis the most general, purely tensorial, theory of gravity.
48
7 Modern gravity: both scalars and tensors are important
In Einstein's theory of General Relativity, the gravitational 'force' is mediated by the rank-2metric tensor. When quantized, this metric tensor becomes a massless spin-2 particle [5, Chap-ter 3]. General Relativity, with only the rank-2 tensor eld, is generally in good agreement withexperiments, however theories with more than just one rank-2 tensor eld are not necessarilyexcluded by experiment [5]. A common extension of General Relativity then is to add an extrascalar-eld, which is coupled to the metric tensor in such a manner that General Relativity isnot violated at length scales where General Relativity is well-tested, i.e. Solar-system scales.This can be done in various manners, one could employ the Vainshtein mechanism [30], orhave the coupling be very weak [5]. This type of theories are called scalar-tensor theories. Theequations of motion derived from such theories are relatively simple and allows for exact solu-tions, which is another explanation of their popularity. Often a scalar-tensor theory is found asthe dimensionally reduced, eective theory of a higher-dimensional theory. Examples of suchhigher dimensional theories are Kaluza-Klein, string theory [5] and DGP gravity [21].
In this section, we will rst go over the Vainshtein mechanism and how it protects scalar-tensor theories from modifying gravity in the solar system. Then, we shall discuss DGP gravityand how the galileon can be derived from this theory. After having derived Galileons as a scalar-tensor theory, we will discuss Horndeski's general single-scalar-tensor theory, a generalizationfrom the galileon-tensor theory. We will generalize his work to include an arbitrary amount ofscalar elds before moving on to even more general scalar-tensor theories. These most generalscalar-tensor theories are no longer second order, but still Ostrogradsky-free, and are calledBeyond-Horndeski theories.
7.1 Vainshtein Mechanism
The Vainshtein mechanism is a mechanism introduced by Vainshtein in [30] to remove a dis-continuity in the theory of a massive graviton, or metric tensor [39]. If furthermore can be usedto shield scalar-tensor theories from modifying gravity on solar-system scales, which makes itrelevant for our purposes.
In Fierz-Pauli theory, a massive graviton theory, the propagator for a massless graviton isgiven by [39]
Dm=0µνρσ =
1
k2(ηρµησν + ηρνησµ − ηρσηµν) , (101)
while the propagator for a massive metric tensor is given by [39]
Dm6=0µνρσ =
1
k2 +m2
(ηρµησν + ηρνησµ −
2
3ηρσηµν −
2
3ηρσ
kµkνm2
). (102)
The important dierence lies in the third term of these equations. This dierence of 13ηρσηµν
is at the root of the so-called vDVZ discontinuity, and is caused by the additional degree offreedom exhibited by the massive graviton. The discontinuity causes dierent predictions forthe same physical process, of which only the massless case is in agreement with predictions [39].
Fierz-Pauli is a linear theory, however there exist non-linear variants of Fierz-Pauli theory.For regimes very far away from a gravitational source (such as a massive object) these non-lineartheories can be approximated linearly. In this linear regime, the vDVZ discontinuity appears.Vainshtein pointed out that, at some distance rV to the source, this linear regime is no longervalid. For distances closer to the source, the theory must be non-linearly realized [30, 39]. Inorder to solve this non-linear regime, an expansion in e.g. the graviton mass can be used,rather than an expansion in the distance r [39]. In this non-linear regime, GR-predictions arerecovered, and in those cases these extended theories are not in contradiction with experiments.
49
To give a rough explanation, in the non-linear regime there are strong kinetic self-couplings,that hide the extra degrees of freedom generated by a massive graviton [39]. Due to these self-couplings, the eective propagator of the graviton follows (101) and we recover predictions thatmatch solar-scale experiments. Since these self-couplings are strong, their range is short andthus at long distances can be ignored. Thus, at long distances we enter the linear regime, wherethe graviton has all its degrees of freedom, and propagates as in equation (102). Therefore, forthe Vainshtein mechanism to work we need a theory that is non-linear and has strong couplings.
7.2 DGP-gravity
The origin of Galileon theories lies in a pure tensorial theory. The catch is that this tensorialtheory is not dened in regular 4-dimensional space-time, but in a 5-dimensional one [21]. Thistheory is called DGP-gravity, after its inventors Dvali, Gabadadze and Porrati [5, Chapter 5].They claim that gravity is observed to be weak, because we live on a 3-brane [31]19, embeddedin a higher dimensional world [21]. Unlike other higher-dimensional theories [5], the extradimensions are innite in size, and not compactied. When we then reduce to the brane, thisextra fth dimension appears as a scalar eld.
As an analogue to the Brane mechanism, consider the 2-Brane of the surface of the earth inour three-dimensional world. In principle, the earth is described by three coordinates in space.However, when we reduce to the surface, we can describe coordinates using only latitude andlongitude. The third coordinate, height, then acts like a scalar eld on the surface, with a valuefor each lat- and longitude.
Let us now show how to derive the 3-brane from the 5-dimensional `space time'. Let xµ
denote the coordinates on the brane, and y the extra coordinates. Furthermore, I use capitalletters and sup- and subscripts for the 5-dimensional quantities, e.g. XA. If we do not consideradditional (matter) elds, then the DGP-action is given by [21]
S = M35
∫d5X√GR5 +M2
4
∫d4x√gR4, (103)
with Mi the i-dimensional Planck mass, Ri the i-dimensional Ricci scalar, G the metric on5−D-space and g the metric induced on the brane. If we assume that the brane is located aty = 0, then the metric g is given by [21]
gµν (x) = Gµν (x, y = 0) .
With the action thus dened, Dvali et al. [21] found the following equation for the gravita-tional potential on the brane
V (r) = − 1
8π2M24
1
r
[sin
(r
rc
)Ci
(r
rc
)+
1
2cos
(r
rc
)π − 2Si
(r
rc
)], (104)
where Ci(z) = γ + ln(z) +∫ z
0(cos t−1)
tdt, with γ = 0.577 the Euler-Masceroni constant, and
Si (z) =∫ z
0sin tt
dt. The constant rc is dened as rc =M2
4
2M35and is the critical distance below
which the theory behaves like ordinary 4-dimensional gravity and above which the theorybehaves like a 5-dimensional theory of gravity [5, Chapter 5]. We can see this by taking thelimit r << rc, then (104) becomes [21]
V (r) = − 1
8π2M24
1
r
[1
2π +
−1 + γ + ln
(r
rc
)(r
rc
)+O(r2)
].
19A 3-brane is basically a manifold with 3 spatial dimensions.
50
This equation behaves like ordinary 4-dimensional Newtonian gravity, except for the logarithmicterm, which is repulsive.
On the other hand, for large distances, r >> rc, (104) becomes [21]
V (r) = − 1
8π2M24
1
r
[rcr
+O(1
r2)
].
This equation behaves like 1r2, which is expected for a 5-dimensional, Newtonian theory.
The DGP gravity can be described in a strictly four-dimensional theory as mediated by thetensorial metric eld and an additional scalar [5, 21, 32]. Furthermore, the DGP-model has aself-accelerating solution [5,33]. Such a self-accelerating solution could be used to describe theaccelerated expansion of the universe we observe, instead of through a cosmological constant[33]. Unfortunately, the self-accelerating solution has a ghost-instability and is superluminal atlong distances [5, Chapter 5].
7.3 Galileons as scalar-tensor theory
Above, we derived DGP gravity. Nicolis et al. used the fact that DGP can be described as a 4Dtheory with a scalar φ to derive Galileons [4, 32]. As a result, the galileon theory is inherentlya scalar-tensor theory. In chapter 5.1 we discussed galileons as a pure scalar theory, but nowwe have the tools to discuss how Galileons were derived as a scalar-tensor theory.
Nicolis et al. studied the class of scalar-tensor theories with the property that cosmologicalshort distances, the 4-dimensional DGP theory is recovered [4, 5]. These scalar-tensor theoriesare derived on a at, Minkowski background. As a result, the coupling to the metric is notcovariant, i.e. ∂µ is used instead of the covariant ∇µ. Furthermore, we only focus on the4-dimensional part of DGP. Taking this into account, consider the general Hilbert-Einsteingravitational action [5, Chapter 2] (95)
S =
∫d4x
1
2M2
4
√gR4.
To ensure that the modications are long-distance modication to GR, the coupling of φ to themetric should be of a kinetic type. At quadratic order, the action then becomes [4]
S =
∫d4 1
2M2
4
√g (1− 2φ)R4 + Lφ, (105)
where Lφ describes the dynamics of the galileon. We will see that Lφ will be a combination ofthe Galileon Lagrangians in (66). Rather then using the metric itself as a dynamical variable,we will use the deviation from the at metric as our graviton eld. This deviation is given byhµν = gµν − ηµν [5, Chapter 4]. To de-mix the galileons from the metric, the Weyl Rescaling
is used [4]. This rescaling is given by hµν = hµν + 2φηµν . Doing both these computations, theaction 105 becomes [4]
S =
∫d4x
[1
2M2
4
√gR +
1
2hµνT
µν + Lφ + φT µµ
], (106)
where gµν is dened as gµν = ηµν + hµν , and R is the Ricci scalar derived from gµν .To derive the form of Lφ, some conditions are to be met.
1. On cosmological scales the strength of the galileon exchange must be comparable to thegravitational strength. However, on short scales we do not want deviations from Einstein'sGeneral Relativity. This is impossible in a purely linear theory for φ, as per the Vainshteinmechanism. Thus, we need that the dynamics of φ are non-linear.
51
2. Galilean invariance, i.e. both Lφ and the equations of motion should be invariant undertransformations φ → φ + bµx
µ + c. Since hµν only appears in second-order derviative in
(105), hµν admits the Galilean symmetry, and in general there is no condition that breaksthis symmetry. Therefore, we must also allow a general Galileon shift in φ. [4].
3. The equations of motion are only second-order, to avoid Ostrogradsky ghosts.
It are the second and third conditions that x Lφ to be of the form (66).With the Lagrangian given, Nicolis et al. investigated the existence of a self-accelerating
solution to these equations of motion. A self-accelerating solution could explain the observed(accelerated) expansion of the universe [5, Chapter 5]. In galileon theory, such a solution canindeed exist and be stable, for a suitable combination of the Lagrangians (66), but unfortunatelythe speed of propagation for small uctuations in the self-accelerating solution is faster thanthe speed of light [4]. However, these galileons are not coupled to the dynamical metric, andthus this is not the entire story.
In order to properly describe galileons in a gravity context, we need to couple galileontheory to a dynamical, gravitational metric. Deayet et al. described the process of makingthe Lagrangians for the galileon covariant in [34]. The simple replacement ∂µ → ∇µ leads tohigher-order terms in the equations of motion. This can be solved by adding counter-terms, butregardless, Galileon invariance is lost. After adding these counter-terms, the ve Lagrangiansare given by [34]
L1 =√gφ,
L2 = −√g1
2∇µφ∇µφ,
L3 = −√g1
2∇µ∇µφ∇νφ∇νφ,
L4 = −√g∇λφ∇λφ
[2 (2φ)2 − 2 (∇µ∇νφ∇µ∇νφ)− 1
2(∇µφ∇µφ)R
],
L5 = −√g5
2∇λφ∇λφ
[(2φ)3 − 32φ∇µ∇νφ∇µ∇νφ+ 2∇µ∇νφ∇ν∇ρφ∇ρ∇µφ
]+√g
5
2∇λφ∇λφ
[6∇µφ∇µ∇νφ
(Rνρ −
1
2gνρR
)∇ρφ
].
This theory was generalized to the case of general n galileons by Padilla and Sivanesan [23].They started from the non-coupled multi-galileon theory, such as derived in [22] and alsodescribed in chapter 5. Their result is:
L =√g
5∑m=1
αa1...amφa1δµ2...µmν2...νm
∇µ2∇ν2φa2 · · · ∇µm∇νmφam +√g
5∑m=3
m−12∑
n=1
Cnm, (107)
where the counter-terms are
Cmn =
1
(−4)n(m− 1)!
(m− 2n− 1)! (n!)2αa1...amφa1Xa2a3 · · ·Xa2na2n+1 (108)
· δµ2n+2...µmb1c1...bncnν2n+2...νmb1c1...dnen
∇µ2n+2∇ν2n+2φa2n+2 · · · ∇µm∇νmφamRd1e1b1c1· · ·Rdnen
bncn,
and Xab = 12∇µφ
a∇µφb is the rst-derivative variable, as in chapter 5.As discussed above, a single galileon theory can not be self-accelerating, stable and have sub-
luminal propagation uctuations all at the same time. A multi-galileon theory however mighthave all of these properties. In particular, Padilla, San and Zhou constructed a bi-galileon
52
theory from this general Lagrangian [35,36]. Instead of coupling both galileon elds to matter,only one of the two elds, called π, is directly coupled. The other eld, called ξ, is only coupledthrough its coupling with π [5, Chapter 4]. The action is then given by [35]
S =
∫d4x
[1
2M2
4
√gR +
1
2hµνT
µν + Lπξ + πT µµ
], (109)
withLπξ =
∑06m+n64
(αm,nπ + βm,nξ) Em,n (∂µ∂νπ, ∂ρ∂σξ) ,
where
Em,n (∂µ∂νπ, ∂ρ∂σξ) = (m+ n)!δµ1...µmρ1...ρnν1...νmσ1...σn∂µ1∂
ν1π . . . ∂µm∂νmπ∂ρ1∂
σ1ξ . . . ∂ρn∂σnξ.
The functions Em,n are related to the equations of motion, which are [5, 35]
0 =∑
06m+n64
am,nEm,n + T µµ
0 =∑
06m+n64
bm,nEm,n,
with the constants given by
am,n = (m+ 1) (αm,n + βm+1,n1)
bm,n = (n+ 1) (βm,n + αm−1,n+1) ,
with α−1,n = βm,−1 = 0.In such a bi-galileon theory, there can exist a stable, consistent, self-accelerating solution,
given the right parameters [36]. With the proper parameters, we can have a theory thatsatises [5, chapter 4]
(i) there is no L = φ, or tadpole term,
(ii) there is a self-accelerating vacuum,
(iii) there are no ghosts in the self-accelerating solution,
(iv) the spherical symmetric excitations of the vacuum are screened by the Vainshtein mech-anism in at least the solar system, i.e. solar system gravity is unchanged,
(v) the uctuations are never faster than the speed of light,
(vi) the uctuations do not lead to excessive Cherenkov radiation,
(vii) the strong coupling that allows for the scalar-tensor description does not happen at lowmomentum scales and
(viii) the back-reaction of the galileons to the metric does alter the vacuum solution, i.e. theMinkowski metric remains a solution.
Thus a bi-galileon theory can be a suitable alternative to a cosmological constant dark energytheory [5].
Other possible candidates are the Fab-Four theories from Charmousis et al. [37, 38], whichmight have interesting self-tuning properties [35, 36].
53
7.4 Horndeski's general scalar-tensor theory
In 1973, Gregory Horndeski found the general second-order Lagrangian for a scalar-tensortheory that is only second-order in the equations of motion, and thus ghost-free [40]. Thisresult is a generalization of the covariant galileon scalar-tensor theory discussed in the previoussection, though without any (Galilean) symmetry imposed. Horndeski came with his theorywithout any consideration of a galileon symmetry, since galileons were not a thing back then.
Instead, Horndeski was interested in the most general, covariant scalar-tensor theory that hadup-to second order equations of motion in both the scalar and the tensor eld. His derivationrelied upon the observation that the Euler-Lagrange equations for the scalar eld and the tensoreld must be related. If Eµν (L) describes the Euler-Lagrange equations for the metric gµν , andE (L) those for the scalar φ, then this relation is given by [40]
∇νEµν (L) =
1
2∇µE (L) . (110)
Horndeski found that the general Lagrangian was given by a linear combination of La-grangians [5, 40].
L1 = δαβγµνσ
(K1∇µ∇αφR
νσβγ − 4
3K1,X∇µ∇αφ∇ν∇βφ∇σ∇γφ
)(111)
L3 = δαβγµνσ
(K3∇αφ∇µφR νσ
βγ − 4K3,X∇αφ∇µφ∇ν∇βφ∇σ∇γφ)
(112)
L8 = 2δαβµνK8∇αφ∇µφ∇ν∇βφ− 3XK82φ (113)
L9 = K9 (114)
LF = δαβµν(FR µν
αβ − 4F,X∇µ∇αφ∇ν∇βφ)− 6F,φ2φ, (115)
where the Ki are arbitrary functions of φ and X, as is F . However, F has the constraintF,X = K1,φ − K3 − 2XK3,X . The strange labelling of the Ki is due to the generality ofHorndeski's theory, he derived his theory for general D dimensions and found up to 10 arbitraryfunctions [40]. When reduced down to 4 dimensions, only the functions K1, K3, K8 and K9
remain non-zero. It is easy to see that if one chooses F = 14M2
Pl and all other functions zero,the resulting Lagrangian becomes
L = Σ6i=1Li = LF = MPlR4, (116)
which is the standard Einstein-Hilbert action (95) [5], without any scalar component.Horndeski's work can be generalized into a general multi-scalar-tensor theory. This is done
by making the Lagrangian for a general multi-scalar-tensor theory, (82), covariant and then addthe necessary counter-terms to remove any higher-order derivatives [23]. Padilla and Sivanesanfound that this general Lagrangian is given by
L =√gA(Xab, φd
)+ Ac
(Xab, φd
)2φc − 2
∂B2
(Xab, φd
)∂Xc1c2
(2φc12φc2 −∇µ∇νφc1∇µ∇νφc2)
(117)
− 2
3
∂B3,c3
∂Xc1c2εµ1µ2µ3εν1ν2ν3∇µ1∇ν1φ
c1∇µ2∇ν2φc2∇µ3∇ν3φ
c3 +B2R +B3,c2φcR,
where the functions A, B and their derivatives w.r.t. Xab are all symmetric in all of theirindices [23].
54
7.5 Beyond Horndeski
Horndeski's theory is the most general scalar-tensor theory that leads to second order equationsof motion. However, it is not the most general, Ostrogradsky-free, scalar-tensor theory. Thesemore general theories have higher-order equations of motion, but these higher-order termsappear in such a way that they do not propagate any additional degrees of freedom. Thisclass of most general scalar-theories is often simply called beyond-Horndeski theories [41]. Forstarters, when we made the galileon scalar-tensor covariant above, we added counter termsto ensure second order equations of motion. According to Chagoya and Tasinato [19] thesecounter-terms are not necessary to produce a ghost-free theory. The naive covariantization∂µ → ∇µ is enough to produce a ghost-free theory. However, the Galilean symmetry is stilllost, since Γ µ
νρ is not Galilean invariant.Earlier, in 2014, Gleyzes et al. [41] came up with a more general ghost-free scalar-tensor
theory. Their Lagrangian is given by a linear combination of the following Lagrangians
L2 = G2 (φ,X) ,
L3 = G3 (φ,X) 2φ,
L4 = G4 (φ,X)R− 2G4,X (φ,X) δµναβ∇α∇βφ∇µ∇νφ+ F4 (φ,X) δµνραβγ∇
αφ∇µφ∇β∇νφ∇γ∇ρφ,
L5 = G5 (φ,X)
(Rµν −
1
2gµνR
)∇µ∇νφ+
1
3G5,X (φ,X) δµνραβγ∇
α∇µφ∇β∇νφ∇γ∇ρφ
+ F5 (φ,X) δµνρσαβγε∇µφ∇αφ∇ν∇βφ∇ρ∇γφ∇σ∇εφ.
These reduce to Horndeski theory for F4 = F5 = 0 and for appropriate choices of the functionsGi. In their paper, Gleyzes et al. proof this lack of ghosts in two methods, one using anArnowitt-Deser-Misner formulation and a Hamiltonian analysis of the degrees of freedom, theother uses a covariant formulation but does not include L5 due to the extra complexity thisterm generates.
It is however not at all certain that these equations provide the most general scalar-tensortheory that is free from the Ostrogradsky ghost. The conditions found in [2, 3, 8] have notyet been generalised to a (multi) scalar-tensor system, which makes nding the most generalsystem dicult.
55
8 When to call two Lagrangians equivalent
8.1 Why equivalence?
As stated before, the aim of physics is to nd theories that describe the motions and inter-actions of particles. If there are two theories that give the same equations of motions for thesame particles, are these theories then actually dierent? Theories with the same equations ofmotions, will give the same predictions for experiments. There is thus no way to distinguishbetween two such theories on an experimental level, and thus they are equivalent.
It is easy to create two Lagrangians with the same equations of motion. Take an arbitraryLagrangian with p independent and q dependent coordinates L (xi, uα, uαI ), with the highestderivatives appearing at order r, and add to it the (total) divergence of some p-tuple F : L =L +∇F 20. If one computes the Euler-Lagrange equations (7) for both Lagrangians, one shallsee that they are the same [2,6].
Example 10. For an example, again look at the harmonic oscillator. Its Lagrangian is givenby [9]
L =m
2u2 − k
2u2.
We have computed its Euler Lagrange equation (7) before, it is
mu = −ku.
Let us take for our 1-tuple the function f = u3 − 25u, which has the total derivative
∇f = 3u2u− 25u,
and our modied Lagrangian L = L+∇F looks like
L = 3u2u+m
2u2 − 25u− k
2u2.
Since this Lagrangian is of second order, the Euler-Lagrange equation (7) is computed dier-ently, it now also includes a second-order term. The Euler-Lagrange equation is
0 =d2
dt2∂L∂u− d
dt
∂L∂u
+∂L∂u
=d2
dt2(3u2)− d
dt(6uu+mu− 25)− ku
=d
dt(6uu)−
(6u2 + 6u
...u +mu
)− ku
=(6u2 + 6u
...u)−(6u2 + 6u
...u)−mu− ku
= −mu− ku.
This Euler-Lagrange equation equation is identical to the pervious one, as expected.
However, Lagrangians with dierent equations of motions can still be called equivalent.Roughly speaking, if there exists an invertible transformation Φ such that L (Φ(xi), Φ(uα), Φ(uαI )) =L (xi, uα, uαI ), then we can call L and L equivalent. An example of this would be to rewrite aLagrangian in a dierent coordinate system, for example to move between polar and Cartesiancoordinates.
20See also appendix A for more about the notation used.
56
Example 11. For example, consider the system of two masses orbiting each other in free, 2-Dspace. In cartesian coordinates, the Lagrangian describing this system is given by [9, chapter5]
L =1
2m1
(x2
1 + y21
)+
1
2m2
(x2
2 + y22
)+
Gm1m2√(x1 − x2)2 + (y1 − y2)2
. (118)
We can rewrite this system in polar coordinates around the Centre of Mass. In this system,the Lagrangian becomes [9, chapter 7]
L =1
2µ(r2 + r2φ
)+Gm1m2
r,
where µ = m1m2
m1+m2is the reduced mass. These two Lagrangians are dierent, and lead to dierent
(looking) equations of motion. Yet, they describe the same physical system, and thus we callthem equivalent.
Example 10 is interesting, because the modied Lagrangian L is clearly second order. Itis also Ostrogradsky-free, which can be seen both in the computed Euler-Lagrange equationand also by computing the primary (30) and secondary conditions (31). This is an interest-ing observation. Clearly any rst-order Lagrangian can be turned into a second-order La-grangian by adding a suitable divergence. This second-order Lagrangian thus created is clearlyOstrogradsky-free, i.e. healthy, since its equations of motion are only up-to second order.The interesting question is whether this can also be done in reverse, i.e., given any healthysecond-order Lagrangian L, does there exist a p-tuple F such that L = L+∇F is a rst-orderLagrangian? Or do we also need to perform a coordinate transformation? In general then, weask: are all healthy second-order Lagrangians equivalent to a rst-order Lagrangian?
Fortunately, we are not the rst to ask this question on equivalence. Cartan establishedan algorithm to determine when/if this is the case, which Olver and Kamran [6, 17, 42] laterexpanded and they looked at the results of Cartan's algorithm applied to certain cases. We wantto apply their theory to the case of healthy, second-order Lagrangians. Before we can applytheir work, we rst need a more rigorous denition of equivalence and the allowed coordinatetransformations.
8.2 Jet bundles and contact forms
When discussing equivalence, we use coordinate transformations. These transformations aredieomorphisms that map the space of coordinates to itself: Φ : J → J , where J refers tothe space of coordinates. In order to better understand these allowed transformations, werst need to describe this space J . This space is called a jet-space and is a manifold thatcontains the independent and dependent coordinates, as well as the derivatives of the latter.My construction/explanation of jet spaces is based on that of Ivey and Landsberg in 1.9 of [7],but similar constructions can be found by Olver in chapter 4 of [6] or by Kiselev in the rstchapter of [11].
Denition 3 (Equivalence of k-jets). Let f, g : R → R be two smooth functions. They aresaid to have the same k-jet at a point x0 if and only if
f (x0) = g (x0) ,df
dx(x0) =
dg
dx(x0) ,
d2f
dx2 (x0) =d2g
dx2 (x0) , . . . ,dkf
dxk(x0) =
dkg
dxk(x0) . (119)
Further more, let f, g : M → N be smooth functions to and from manifolds of arbitrarydimensions p and q. These functions have the same k-jet at a point x0 ∈ M if and only iff (x0) = g (x0) and if for any functions u : R→ M , v : N → R the compositions v f u andv g u have the same k-jet at t = u−1 (x0) [7].
57
The condition of equivalence of k-jets on functions between general manifolds can also bestated locally. Let xi be local coordinates on M and uα local coordinates on N . Then twofunctions f, g have the same k-jet at x0 if and only if
∂nfα
∂xI(x0) =
∂ngα
∂xI(x0) ,
for 0 6 n 6 k and I a multi-index of length n [7].This relation of equivalence between k-jet allows us to dene the equivalence class jkx0 (f) as
the class of all functions M → N that have the same k-jet as f at the point x0, or, in symboliclanguage21
jkx0 (f) =
g ∈ C∞ (M,N)
∣∣∣∣ ∂nfα∂xI(x0) =
∂ngα
∂xI(x0)∀n 6 k, |I| = n
.
With the equivalence relations and classes thus dened, we can look at the spaces on whichthe k-jets live. These spaces are essentially the space of all functions betweenM and N modulothe equivalence between k-jets. They are called the jet spaces.
Denition 4 (Jet Spaces). Let x0 ∈M and u0 ∈ N be two points. Then Jkx0,u0 (M,N) denotesthe space of all k-jets that map x0 to u0, and J
k (M,N) is the space of all k-jets of all mapsfrom M to N . It is called the kth-order jet space between M and N .
The local coordinates on Jk (M,N) are given by xi, uα, pαi , . . . , pαI . A point jkx0 (x0) ∈
Jk (M,N) then has coordinates xi0, uα = fα (x0) , pαI = ∂|I|fα
∂xI(x0) for 1 6 |I| 6 k [7].
For any l < k, the jet space J l (M,N) is a sub-manifold of the larger space Jk (M,N)and there exists the natural projection πkl : Jk (M,N) → J l (M,N), formed by dropping thederivatives of order greater than l [6]. There exists a similar method to increase the order ofa jet space. One simply appends the derivative coordinates with the higher-order derivativecoordinates pαI necessary [11]. Of course, the thus dened map (π−1)
kl is not well-dened, since
any point in Jk (M,N) maps to many points in Jk+1 (M,N).The spaces Jk (M,N) each form a bundle over both M and M × N , and any functions
f : M → N induces a section p → jkx0(f) of the bundle Jk (M,N) over M , called the lift off [7]. The lift of a function, either from M → N or from M → J l (M,N) to some higher-orderjet space Jk (M,N) (l < k) is also called the prolongation of a function [6].
Unfortunately, the allowed functions are not simply all dieomorphisms that map a jet-spaceJk (M,N) onto itself. The dierential coordinates pαI do not `know' that they are supposed to bederivatives of coordinates uα. As a result, a general dieomorphism may mess up this relation.To explain, take a general function f : M → N and it's lift jk(f) on the bundle Jk (M,N).
The derivative coordinates of this lift are described as pαI = ∂|I|fα
∂xI(x0). Let Φ : Jk (M,N) →
Jk (M,N) be such a general dieomorphism, and barred coordinates refer to the coordinatesafter this transformation: xi = Φ (xi, uα, pαI ), etc. After this transformation, we may have
pαI 6=∂|I|fα
∂xI(x0), and thus jk (f) may no longer be a lift of f . This not desired, and thus we
must reduce the allowed transformations. Note that it is possible that there exists a dierent
function f ′ for which this corodinate transformation leaves the lift intact, i.e. pαI = ∂|I|f ′α
∂xI(x0).
Thus, the reduction in allowed dieomorphisms depends on the functions that we want to bepreserved.
In our case, when talking about Lagrangian equivalence, the previous discussion denedthat equivalence has to do with the equations of motion that follow from the Lagrangian. The
21It is not strictly needed that g lives in C∞ (M,N), as long as g is smooth enough, such that the equivalencerelations are well dened, i.e. g ∈ Ck (M,N) is sucient.
58
functions whose lifts need to be preserved by a transformation are therefore the solutions tothese equations of motion. More general, the to be preserved functions are solutions to asystem of partial dierential equations. Let us therefore dene dierential equations on thesejet spaces. Take
F l
(xi, uα,
∂|I|uα
∂xI
)= 0, (120)
to be a kth-order system of p partial dierential equations for maps f : M → N . Asmentioned, the coordinates on Jk do not know of dierentials and do not know that theyare 'supposed' to be derivatives of coordinates uα that solve these equations. Ergo, theequations F l (xi, uα, pαI ) = 0 do not dene sections in Jk (M,N) (which would be lifts ofsolutions to the PDE), but instead they dene a sub-manifold Σ ⊆ Jk (M,N) by Σ =
(xi, uα, pαI ) ∈ Jk (M,N) |F l (xi, uα, pαI ) = 0.
To obtain suitable sections on this manifold, dene on the jet space manifold the p-form22
Ω = dx1∧· · ·∧dxp and the dierential contact ideal I23. These one-forms are called the contactforms, which is generated dierentially by the 1-forms
θα = duα − pαi dxi,
θαI = dpαI − pαI,j dxj,
with |I| < k. This p-form Ω and contact ideal I together dene a so-called exterior dierentialsystem with independence condition (EDS) on the sub-manifold Σ dened by (120) [7, denition1.9.1, page 27]. Solutions to this EDS are given according to denition 1.9.2 of [7]:
Denition 5. An integral manifold or solution of the system (I, Ω) is an immersed p-foldf : M → Σ such that f ∗ (ω) = 0 ∀ω ∈ I and f ∗ (Ω) 6= 0 at each point of M .
These integral manifolds f are then exactly the lifts of proper solutions to the PDE (120).To see how this works in practise, consider the following example, taken in part from [7].
Example 12. Let M = R2, with coordinates (x, y), and N = R, with coordinate u. Fur-thermore, we work on the rst-order jet space J1 = J1 (R2,R). On this jet-space, dene thefollowing quite general PDE
ux = A (x, y, u) , (121)
uy = B (x, y, u) .
On J1 dierential coordinates don't know that they are supposed to be dierentials, thus theseequations dene the 3-dimensional manifold j : Σ → J1. This is the manifold dened by theequations.
px = A (x, y, u) ,
py = B (x, y, u) .
The contact ideal on J1 is generated by the one-form
θ = du− px dx− py dy,
and the independence condition is given by Ω = dx ∧ dy 6= 0.
22p = dimM , see also appendix A23For a denition of dierential ideals of one-forms, I refer to B.4 of [7]
59
Let i : S → Σ be a surface such that i∗θ = 0 and i∗Ω 6= 0. On this surface x and y arethe only independent coordinates, thus the coordinates on this surface can be expressed asfunctions of the coordinates x and y by the Implicit Function Theorem:
u = u(x, y)
px = px(x, y)
py = py(x, y).
Also, since i∗θ = 0, we havei∗ du = px dx+ py dy.
On the other hand, since u is a function of x and y on S, we have
du = du (x, y) = ux dx+ uy dy.
Since x and y are independent on S by the independence condition i∗Ω 6= 0, the above equationsthen necessarily imply
px = ux,
py = uy,
i.e. on S, the function u is indeed a solution of (121).
So the contact forms/ideal are the important tools, that tell the coordinates on a jet spaceJk (M,N) associated with derivatives (i.e. the pαI ) to actually act like they are the derivativesof the functions uα. Of course, this only works if the independence condition is satised (i∗Ω 6=0), since this ensures that the coordinates uα actually may be expressed as functions of thecoordinates xi.
For the equivalence problem for Lagrangians, the allowed transformations are precisely thosedieomorphism that preserve the contact ideal for this Lagrangian.
8.3 Prolongated and allowed transformations
Continuing down the rabbit-hole of allowed dieomorphism, to nd the allowed dieomorphismswe need to dene their order r, and for that we need to know what the prolongation of adieomorphism is.
Denition 6 (Prolongations). Let Φ be a dieomorphsism on Jr (Rp,Rq): Φ : Jr (Rp,Rq) →Jr (Rp,Rq). This dieomorphism can locally be described as xi = Φi
(xj, u(r)
), uα = Φα
(xj, u(r)
),
pαI = ΦαI(xj, u(r)
), |I| 6 r. The prolongation Φ′ of this dieomorphism to a higher order jet-
bundle Js (Rp,Rq), s > r, is locally given by uα = Φα(xj, u(r)
), pαI = ΦαI
(xj, u(r)
)for |I| 6 r,
and pαI,J =∂pαI∂xJ
for |I| = r, |J | > 1. I.e., the additional higher derivative coordinates are madeto transform according to the chain rule. Please note that pαI , |I| 6 r is not made to abide anychain rule.
With the denition of a prolongation given, we can dene an order for every dieomorphismof the jet-bundle Jr. This can be used to classify dieomorphisms, based on the lowest orderjet-bundle on which they can live.
Denition 7 (Order of a dieomorphism). We dene that a dieomorphism has order r if it isthe prolongation of a dieomorphism on Jr, and can not be described as the prolongation of adieomorphism on any lower order jet bundle Jk, k < r. Thus, all dieomorphisms on the jetbundle Jr are either prolongations of a lower order dieomorphism, or are themselves of orderr.
60
We want to nd dieomorphisms Φ that preserve the contact ideal, i.e. that satisfy Φ∗ (I) =I. According to Bäcklund's Theorem, only zero-th and rst order dieomorphisms can possesthis property [6, Chapter 4]. Kamran and Olver categorised such functions into three classes[17].
1. Fiber-preserving transformations are those dieomorphism where the new independentvariables xi only depend on the old independent variables xi [17]. In this case, Φ has theform
xi = φi(xj), uα = ψα
(xi, uβ
).
The formulas for the derivatives pαI follow from the chain rule. This is a more restrictedclass of the zero-th order dieomorphisms and does not mix dependent and independentvariables.
2. General point transformations. Now the new independent coordinates are allowed todepend on the old dependent coordinates [17]. This is a generalisation of the ber-preserving transformation. The dieomorphisms have the form
xi = φi(xj, uα
), uα = ψα
(xi, uβ
).
Again, the derivatives pαI follow from applying the chain rule.
3. Contact transformations are those transformations, where φ and ψ are allowed to dependon the rst-derivatives of u [17]. Furthermore, they must preserve the contact ideal24.Thus, the dieomorphisms have the form
xi = φi(xj, u(1)
), uα = ψα
(xi, u(1)
), pαi = ψαi
(xi, u(1)
)with the restriction
Φ∗ (I) = I,
where I is the contact ideal generated by the contact forms on Jr.
It is evident that the last category preserves the contact ideal. Next, I shall show thatthe rst and second categories also preserve the contact ideal. It is given that the derivativecoordinates transform according to the chain rule. Thus, these transformations are given by
pαi =∂uα
∂xi=∂xj
∂xi∂ψα
∂xj+∂uβ
∂xi∂ψα
∂uβ,
pαI,j =∂pαI∂xj
=∂xk
∂xj∂pαI∂xk
+∂uβ
∂xj∂pαI∂uβ
+∑|J |6|I|
∂pβJ∂xj
∂pαI
∂pβJ.
To show that this ensures the preservation of the contact ideal, we prove the following theorem.
Theorem 5. The ber-preserving and point transformations preserve the contact ideal:
Φ∗ (I) = I.24Since I enforce this preservation of the contact ideal, I can consider the pαi = uαi as actual derivatives, and
not just coordinates on Jr.
61
Proof. Since the contact ideal is algebraically generated by the contact forms, we only have toprove that the contact forms are mapped to (linear combinations of) contact forms. To remindourselves, we have
Φ∗ (θα) = θα = duα − pαi dxi, (122)
Φ∗ (θαI ) = θαI = dpαI − pαI,j dxj.
In the ber preserving case, we have
dxi =∂φi
∂xjdxj,
duα =∂ψα
∂xidxi +
∂ψα
∂uβduβ, (123)
pαi =∂xj
∂xi∂ψα
∂xj+∂uβ
∂xi∂ψα
∂uβ,
and
dpαI =∂pαI∂xj
dxj +∂pαI∂uβ
duβ +∂pαI
∂pβJdpβJ , (124)
pαI,j =∂xk
∂xj∂pαI∂xk
+∂uβ
∂xj∂pαI∂uβ
+∂pβJ∂xj
∂pαI
∂pβJ,
where 0 < |J | 6 |I|. Substituting (123) into (122) results in
θα =∂ψα
∂xidxi +
∂ψα
∂uβduβ −
(∂xk
∂xi∂ψα
∂xk+∂uβ
∂xi∂ψα
∂uβ
)∂φi
∂xjdxj
=∂ψα
∂xidxi − δkj
∂ψα
∂xkdxj +
∂ψα
∂uβ
(duβ − ∂uβ
∂xi∂φi
∂xjdxj)
=∂ψα
∂uβθβ.
Substituting (124) into (122) results in
θαI =∂pαI∂xj
dxj +∂pαI∂uβ
duβ +∂pαI
∂pβJdpβJ −
(∂xk
∂xj∂pαI∂xk
+∂uβ
∂xj∂pαI∂uβ
+∂pβJ∂xj
∂pαI
∂pβJ
)∂φj
∂xidxi
=∂pαI∂xj
dxj − δki∂pαI∂xk
dxk +∂pαI∂uβ
(duβ − ∂uβ
∂xj∂φj
∂xidxi)
+∂pαI
∂uβJ
(dpβJ −
∂pβJ∂xj
∂φj
∂xidxi
)=∂pαI∂uβ
(duβ − pβi dxi
)+∂pαI
∂pβJ
(dpβJ − p
βJ,i dx
i)
=∂pαI∂uβ
θβ +∑|J |6|I|
∂pαI
∂pβJθβJ .
The above calculations show that ber-preserving transformations map contact forms to (linearcombinations of) contact forms, and thus they preserve the contact ideal.
For point transformations, only the rst equation of (123) changes. It becomes
dxi =∂φi
∂xjdxj +
∂φi
∂uβduβ.
62
Now, substitute the above equation and the other equations of (123) into (122) to obtain
θα =∂ψα
∂xidxi +
∂ψα
∂uβduβ −
(∂xk
∂xj∂ψα
∂xk+∂uβ
∂xj∂ψα
∂uβ
)(∂φj
∂xidxi +
∂φj
∂uγduγ)
=
(∂ψα
∂xi− ∂ψα
∂xk∂xk
∂xj∂φj
∂xi− ∂ψα
∂uβ∂uβ
∂xj∂φj
∂xi
)dxi +
(∂ψα
∂uβ− ∂ψα
∂xk∂xk
∂xj∂φj
∂uβ− ∂ψα
∂uγ∂uγ
∂xj∂φj
∂uβ
)duβ
To continue, we need to introduce some identiers for the Kronecker-delta function.
δij =∂xi
∂xj=∂xi
∂xk∂φk
∂xj+∂xi
∂uα∂ψα
∂xj
δαβ =∂uα
∂uβ=∂uα
∂xi∂φi
∂uβ+∂uα
∂uγ∂ψγ
∂uβ
We can substitute these functions into the ∂xk
∂xj∂φj
∂xiand ∂uγ
∂xj∂φj
∂uβterms.
θα =
(∂ψα
∂xi− ∂ψα
∂xkδki +
∂ψα
∂xk∂xk
∂uβ∂ψβ
∂xi− ∂ψα
∂uβ∂uβ
∂xj∂φj
∂xi
)dxi
+
(∂ψα
∂uβ− ∂ψα
∂xk∂xk
∂xj∂φj
∂uβ− ∂ψα
∂uγδγβ +
∂ψα
∂uγ∂uγ
∂uδ∂ψδ
∂uβ
)duβ
=
(∂ψα
∂xk∂xk
∂uβ∂ψβ
∂xi− ∂ψα
∂uβ∂uβ
∂xj∂φj
∂xi
)dxi
+
(∂ψα
∂uγ∂uγ
∂uδ∂ψδ
∂uβ− ∂ψα
∂xk∂xk
∂xj∂φj
∂uβ
)duβ
Again, use the Kronecker-delta substitutions, this time on ∂ψα
∂xk∂xk
∂uβand ∂ψα
∂uγ∂uγ
∂uδ. This results in
θα =
(δαβ∂ψβ
∂xi− ∂ψα
∂uβ∂uβ
∂uγ∂ψγ
∂xi− ∂ψα
∂uβ∂uβ
∂xj∂φj
∂xi
)dxi
+
(δαδ∂ψδ
∂uβ− ∂ψα
∂xk∂xk
∂uδ∂ψδ
∂uβ− ∂ψα
∂xk∂xk
∂xj∂φj
∂uβ
)duβ
=∂ψα
∂xidxi − ∂ψα
∂uβ∂uβ
∂xidxi +
∂ψα
∂uβduβ − ∂ψα
∂xk∂xk
∂uβduβ
=∂ψα
∂uβ
(duβ − pβi dxi
)− ∂ψα
∂xk
(∂xk
∂uβduβ − δki dxi
)=∂ψα
∂uβθβ − ∂ψα
∂xk∂xk
∂uβθβ
=dψα
duβθβ.
A similar computation can be done on θαI (the second equation of (122)), the result is
θαI =dψαIduβ
θβ +dψαIdpβJ
θβJ ,
where 0 < |J | 6 |I|. Thus, I have shown that for both the ber-preserving transformations andfor the point transformation, the contact ideal is preserved.
63
Example 13. To give an example of a non-trivial contact transformation, take a look at theLegendre transformation on J1 (R,R) [6, chapter 4].
x = p, u = u− xp p = −x. (125)
To cast this in a more familiar setting, let u = L, u = H, x = q and p = ∂L∂q. Then (125)
becomes
H = L − q ∂L∂q
.
Clearly, the transformation rule for p is not simply the chain rule, a factor of ∂u∂p
is missing.The Legendre transformation is thus not a prolonged point transformation, however it doespreserve the contact form. To prove this, rst calculate du and dx:
du = d (u− xp) = du− x dp− p dx,
dx = dp.
Next, substitute these into the zeroth-order contact form:
θ = du− p dx
= (du− x dp− p dx)− (−x) dp
= du− p dx
= θ.
Thus, we have found a contact-transformation on J1 (R,R)
So far, we have dened jet-bundles, contact forms and ideals, and dieomorphisms on jet-bundles. From this, we have found the types of dieomorphism that preserve the contact ideals,and called those the transformations under which two Lagrangians can be equivalent to eachother. Next, we need to dene `equivalence'.
8.4 The types of equivalence
Let L and L be two general Lagrangians dened on a jet-bundle Jr (Rp,Rq). Kamran andOlver describe two types of equivalence between these two general Lagrangians [17,42].
1. First, we can require that the functionals dened by L and L agree on all possible func-tions uα = fα (xi). This is called the standard equivalence problem. Thus, these twoLagrangians are equivalent if they are related by
L(xi, u(r)
)=L(xi, u(r)
)| det J |
, (126)
where (J)ij = dφi
dxj, the Jacobian of the transformation of the independent variables.
2. Second, we can require that the variational problems dened by L and L agree only onextremals. That is to say, they produce the same Euler-Lagrange equations. This is thecase, if and only if the Lagrangians dier by a divergence. In formula form, this can bestated as
L(xi, u(r)
)=L(xi, u(r)
)+∇F
| det J |, (127)
with again (J)ij = dφi
dxjand F
(xi, u(r)
)an arbitrary p-tuple of functions F : Jr → Rp.
This type is called the divergence equivalence problem. It is this type of equivalence thatthe Lagrangians from example 10 exhibit.
64
It is easily seen that any two Lagrangians that are standard equivalent, are also divergenceequivalent, simply use the function F = 0. The divergence equivalence problem is thus lessrestrictive and still meets our goal: preservation of the equations of motion, as mentioned inthe introduction to this section. Therefore, I shall focus on this type of equivalence in the restof this thesis. However, the Cartan Equivalence Algorithm introduced in section 10 can alsobe applied to the standard equivalence problem.
65
9 Equivalence of Galileons
In this section we use the results of Crisostomi et al. [8] and the previous sections to see ifGalileon theories are equivalent to rst order theories. We are still interested in the widestclass of equivalence, divergence equivalent under contact transformations, but we shall see thatthat there is still equivalence in the absence of coordinate transformations.
We rst recap the Ostrogradsky-free conditions, and then go into the classes of Ostrogradsky-stable Lagrangians found by Crisostomi et al. [8] and how these classes are divergence equivalentto rst-order theories. We then apply these results to ever more general Galileon theories, fromjust one Galileon without additional elds, till arbitrary Galileons with arbitrary additionalhelper elds.
9.1 Equivalence conditions
As a recap the conditions found by Crisostomi et al. for Ostrogradsky stability for LorentzInvariant theories are [8] (56)(61)
0 = P(ab) = vAa LψAψBvBb ,
0 = S[ab] = 2vAa Lψ[AψB]vBb + 2vA[aLψA∂iψB∂ivBb] − δi
(vAa Lψ[A∂iψB]vBb
).
They also found certain conditions under which certain Ostrogradsky free theories are di-vergence equivalent to rst order theories. We shall explain the procedure. Let us divide thestable Lagrangians into three classes, based on their null-vector V α
a = −LφaqβL−1qβ qα
.
Class I :V αa = 0 (128)
Class II :V αa = V α
a
(φb, ∂µφ
b, qβ)
(129)
Class III :V αa = V α
a
(φb, ∂µφ
b, qβ, ∂µ∂νφb, ∂µq
β)
(130)
Class I constraints
Both mechanical and Lorentz invariant Class I Lagrangians are divergence equivalent to rst-order theories. In fact, no coordinate transformation is required. We shall proof this statementin two theories, one for classical mechanics and one for eld theories. For both proofs, we shallfollow Crisostomi et al. [8].
Theorem 6. Let L be a mechanical, second-order, Class I Lagrangian. L is divergence equiv-alent to a rst order Lagrangian L under the relation
L = L+dF
dt, (131)
where F = F(t, φa, φa
).
Proof. For a second-order, Ostrogradsky free Lagrangian, L(φa, φa, φa, qα, qα
), the Class I
condition (128) reduces the Ostrogradsky constraints (37) and (38) to
0 = P(ab) = Lφaφb ,0 = S[ab] = Lφ[aφb] .
The primary condition implies linearity in the φa, and thus the Lagrangian can be written as
L = φafa
(φb, φb, qα
)+ g
(φb, φb, qα, qα
). (132)
66
The secondary condition then implies ∂fa/∂φb = ∂fb/∂φa . To remove the second order termsthen, we need to nd a function F such that Fφa = fa. We can dene this function as anintegral on J2 (R,Rq):
F =
∫fa dφa.
This function matches the criteria, since
∂F
∂φa=
∫∂fb
∂φadφb
=
∫∂fa
∂φbdφb
= fa,
where we have used the secondary condition ∂fa/∂φb = ∂fb/∂φa . As a result, the mechanicalClass I Lagrangians can all be written as
L = L+dF
dt
= L+∂F
∂t+∂F
∂φaφa +
∂F
∂φaφa
= L+∂F
∂t+∂F
∂φaφa + faφ
a.
Comparing with (132) shows us that g = L+ ∂F∂t
+ ∂F∂φa
φa, which is a purely rst order function.
Thus L(φa, φa, qα, qα
)is a rst order Lagrangian that is divergence equivalent to L. We
conclude that all mechanical Class I Lagrangians are divergence equivalent to a rst orderLagrangian.
There is a similar theorem and proof for Lorentz Invariant theories:
Theorem 7. Let L be a second-order, Lorentz Invariant, Class I Lagrangian. L is divergenceequivalent to a rst order Lagrangian L under the relation
L = L+ ∂µFµ, (133)
where F µ is 4-tuple of functions, and ∂µFµ its total derivative.
Proof. For eld theories, the Class I condition (128) reduces the constraints (56), (60) and (61)to
0 = P(ab) = Lφaφb ,0 = (Si)(ab) = Lφ(a∂iφb)0 = S[ab] = Lφ[aφb] .
The primary constraint again implies linearity, i.e.
L = φbfb (∂i∂µφa, ∂µφ
a, φa, ∂iqα, qα) + g (∂i∂µφ
a, ∂µφa, φa, ∂iq
α, qα) , (134)
and the secondary constraints become
0 =∂f(a
∂∂iφb),
0 =∂f[a
∂φb]− 1
2∂i
∂f[a
∂∂iφ[b.
67
These constraints are expanded from the mechanical case, and we are now looking for a functionF 0 such that F 0
φa− ∂iF 0
∂iφa= fa. This function F does not generically exist for a general eld
theory, so we turn to the requirement of Lorentz Invariance. Due to Lorentz invariance, wehave ∂fa
∂∂iφb= 0. This causes the symmetric secondary constraint (60) to vanish, and the anti-
symmetric constraint (61) reduces to
∂fa
∂φb− ∂fb
∂φa= 0,
similar to the mechanics case. We thus again have that there exists a function F 0 such thatF 0φa
= fa, and thus we have that
L = L+ ∂µFµ,
where the F i are arbitrary functions F i (∂µφa, φa, qα), e.g. F i = 0. The form of the F i, ensures
that L is rst order. However, there is no guarantee that the tuple F µ is a Lorentz vector,even for certain choices of the F i. Therefore, we have only proven that all Lorentz invariant,Class I Lagrangians are divergence equivalent to a rst order Lagrangian, but not necessarilya Lorentz invariant rst order Lagrangian.
Non-Lorentz invariant theories might still satisfy ∂fa∂∂iφb
= 0. By the above proof, these
theories are also equivalent to a rst order Lagrangian. In particular, theories with only onedependent variable satisfy ∂f
∂∂iφ= 0. However, Crisostomi et al. did not nd a proof that a
function F always exists for the cases ∂fa∂∂iφb
6= 0 [8]. Thus, there is no general theorem for a
eld theoretical Class I Lagrangian.
Class II constraints
Crisostomi et al. proved that all Class II theories are standard equivalent to a Class I Lagrangianunder a contact transformation on the helper elds qα. Since this Class I Lagrangian need notbe Lorentz Invariant, we then do not have that in general all Class II Lagrangians are rstorder.
Theorem 8. All second-order, Class II Lagrangians LII are standard equivalent to a Class ILagrangian LI under the contact transformation
xµ = xµ φa = φa qα = qα(qβ, φb, ∂µφ
b),
Proof. We shall closely follow the proof from Crisostomi et al. [8]. In particular we shall proofthat the specic dependence of the null-vector V α
a = V αa
(φb, ∂µφ
b, qβ), i.e. the Class II condition
(129), requires the existence of this contact transformation and vice-versa, that the existenceof such a transformation for a Class II Lagrangian requires that the transformed Lagrangian isof Class I.
First, assume that there exists this contact transformation between two Lagrangians:
LII (∂i∂jφa, ∂iφ
a, φa, ∂iqα, qα) = LI (∂i∂jφ
a, ∂iφa, φa, ∂iq
α, qα) ,
where LII belongs to Class II and LI belongs to Class I. The null-vectors of both Lagrangiansare related by [8]
V αa =
∂qα
∂φa+ V β
a
∂qα
∂qβ. (135)
68
Since LI belongs to Class I, V αa = 0. Futhermore, qα = qα
(qβ, φB, ∂iφ
b)is invertible. We can
combine these two statements to rewrite (135) as
V βa =
∂qα
∂φa
(∂qα
∂qβ
)−1
,
and thus V βa = V β
a
(φb, ∂µφ
b, qβ).
In the other direction, we assume that we have a Class II Lagrangian LII , with V αa =
V αa
(φb, ∂µφ
b, qβ). Consider the partial dierential equations
∂u
∂φa+ V β
a
(φb, ∂µφ
b, qβ) ∂u∂qβ
= 0. (136)
By Frobenius' theorem [16], these equations have n independent solutions qα if and only if theintegrability conditions
0 =∂V α
[a
∂φb]+ V β
[a
∂V αb]
∂qβ≡ Fαab (137)
are satised. Using the denition of V αa = −LφaqβL
−1qβ qα
and the fact that LII obeys the primary
condition (56) to calculate these conditions, leads to the reduction
Fαab = L−1qβ qα
∂
∂qβS[ab],
which vanishes by construction, since LII also obeys the secondary conditions (61).Thus, we have our n independent solutions qα. Furthermore, in the derivation of the condi-
tions (56) and (61) we have assumed that the qα are non degenerate, i.e. L−1qβ qα
exists. Therefore,
the qα are also independent. Thus, we can conclude that ∂ qα
∂qβis invertible [43] and we have the
invertible contact transformation
xi = xi φa = φa qα = qα(qβ, φB, ∂iφ
b).
Dene the transformed Lagrangian L (∂µ∂νφa, ∂µφ
a, φa, ∂µqα, qα) = LII (∂µ∂νφ
a, ∂µφa, φa, ∂µq
α, qα).The null-vectors transform as in (135)
V αa =
∂qα
∂φa+ V β
a
∂qα
∂qβ,
which is just (136), of which the qα were dened to be solutions. As a result, we have V αa = 0
and thus L belongs to Class I, conluding this direction of the proof.With both the `if' and `only if' proven, we can thus conclude that all second-order, Class II
Lagrangians are standard equivalent to a Class I Lagrangian.
This theorem leads directly to the following corollary.
Corollary 1. All mechanical Class II Lagrangians are divergent equivalent to a regular, rst-order Lagrangian.
Proof. Theorem 8 states that all Class II Lagrangians are standard equivalent to a Class ILagrangian. Since standard equivalence is a subset of divergence equivalence, we thus have thatall Class II Lagrangians are also divergence equivalent to a Class I Lagrangian. From theorem6 we know that all mechanical Class I Lagrangians are divergence equivalent to a rst-orderLagrangian. Combining these two statements then leads to the conclusion that all mechanicalClass II Lagrangians are divergent equivalent to a regular, rst-order Lagrangian.
There does not exist a similar corollary for Lorentz invariant Lagrangians, since the ClassI Lagrangian equivalent to the Class II Lorentz invariant Lagrangian need not be Lorentzinvariant.
69
Class III constraints
The Class III theories can not always be made divergence equivalent to a rst order Lagrangian.Although for some Class III Lagrangians there exists a contact transformation that transformthem into a Class I Lagrangian, Crisostomi et al. were not able to nd a proof that such atransformation is possible for all Class III theories [8]. As we shall not deal with Class IIILagrangians much in the rest of this chapter, we refer the reader to [8] for the Lagrangians thatare equivalent.
9.2 One Galileon
In this subsection, we shall proof that all single-Galileon Lagrangians are explictly divergenceequivalent to a rst-order Lagrangian. In particular, we shall compute the functions F suchthat L = L+ dF
dtis rst-order. As described in section 5.1, the Lagrangian for a single galileon
is a linear combination of (66)
L1 = φ,
L2 = −1
2∂µφ∂µφ,
L3 = −1
2∂µ∂
µφ∂νφ∂νφ,
L4 = −1
2
[(∂µ∂µφ)2 − ∂µ∂νφ∂ν∂µφ
]∂ρφ∂ρφ
L5 = −1
2
[(∂µ∂µφ)3 − 3∂µ∂µφ∂
ν∂ρφ∂ρ∂νφ+ 2∂µ∂νφ∂
ν∂ρφ∂ρ∂µφ
]∂σφ∂σφ.
L1 and L2 are already in rst order form, and thus we need not discuss them. The other threeLagrangians all have no secondary elds, thus for all V α
m = 0 and they belong in Class I. Letus rewrite them in the form of (134):
L3 = φ
(−1
2∂νφ∂
νφ
)− 1
2∂i∂
iφ∂νφ∂νφ,
L4 = φ(−∂iφ∂i∂ρφ∂ρφ
)− 1
2
(∂i∂
iφ∂j∂jφ− 2∂iφ∂
iφ− ∂i∂jφ∂i∂jφ),
L5 = φ
(−3
2∂i∂
iφ∂j∂jφ+
3
2∂i∂jφ∂
i∂jφ
)∂σφ∂
σφ+ 3∂iφ∂jφ∂i∂jφ∂σφ∂
σφ.
From these equations, we nd the following functions `f ':
f3 = −1
2∂νφ∂
νφ,
f4 = −∂iφ∂iφ∂ρφ∂ρφ,
f5 =3
2
(−∂i∂iφ∂j∂jφ+ ∂i∂jφ∂
i∂jφ)∂σφ∂
σφ.
Each of these satises ∂fk∂∂iφ
= 0, with k = 3, 4, 5, and thus each of these Lagrangians can be
turned into a rst order Lagrangian. The relevant functions are
F3 = −1
6φ3 − 1
2∂iφ∂
iφφ,
F4 = −∂iφ∂iφ(
1
3φ3 + ∂jφ∂
jφφ
),
F5 =3
2
(−∂i∂iφ∂j∂jφ+ ∂i∂jφ∂
i∂jφ)(1
3φ3 + ∂jφ∂
jφφ
).
70
Furthermore, the equivalent rst order Lagrangians can be calculated. They are
L3 = L3 −dF3
dt
= −1
2∂i∂
iφ∂νφ∂νφ+ φ∂iφ∂
iφ,
L4 = L4 −dF4
dt
= −1
2
(∂i∂
iφ∂j∂jφ− 2∂iφ∂
iφ− ∂i∂jφ∂i∂jφ)− ∂iφ
(2
3∂iφφ3 + 4∂iφ∂jφ∂
jφφ
)L5 = L5 −
dF5
dt
= 3∂iφ∂jφ∂i∂jφ∂σφ∂
σφ− 3(∂i∂jφ∂
i∂jφ− ∂i∂iφ∂j∂jφ)∂kφ∂
kφφ
+ 3(∂i∂jφ∂
i∂jφ− ∂i∂iφ∂j∂jφ)(1
3φ3 + ∂kφ∂
kφφ
)Clearly, we lose Lorentz invariance, but the resulting Lagrangians are all rst order in time
derivatives, and thus ghost free.
9.3 One Galileon, many scalars
A step up from considering only one galileon, is to consider this galileon together with someadditional scalars. These additional scalars are all healthy, that is, the scalars will only berst-order in derivatives. Unlike the previous subsection, our Lagrangian can now be in anyof the three classes. Thus, we can not show for all of these cases that the Lagrangians aredivergence equivalent to rst-order theories.
The simplest theory with one galileon and many rst-order scalars, is a theory were thesetwo types of elds are not coupled:
L (φ, ∂φ, ∂∂φ, qα, ∂qα) = Lgal (φ, ∂φ, ∂∂φ) + Lscal (qα, ∂qα) (138)
Such a theory is not very interesting, and, due to the decoupled nature of the theory, can infact be treated as in the previous subsection.
Instead, we want the galileon to interact with the galileons. The total theory still has to beghost-free, and thus satises (56):
0 = P11 = Lφφ − LφqαLφqβL−1qβ qα
.
The symmetric secondary condition (60) is satised, since we are working with Lorentz invarianttheories. The anti-symmetric condition (61) is automatically satised, since there is only onesecond-order eld.
Next, we need to consider which class our Lagrangian is in. Class I has the requirement thatV α
1 = LφqβL−1qβ qα
= 0, under which the ghost-free condition becomes
P11 = Lφφ.
The Class I condition can be met by ensuring that there is no coupling between the secondderivative of the unhealthy eld with any rst derivative of the healthy eld. Furthermore,the primary constraint states that the Class I Lagrangians are all linear in the unhealthy eld.Therefore, these Lagrangians are of the form
L = φf(φ, ∂iφ, φ, ∂iφ, ∂i∂jφ, q
α, ∂iqα)
+ g(φ, ∂iφ, φ, ∂iφ, ∂i∂jφ, q
α, ∂iqα, qα
), (139)
71
The functions f and g are almost arbitrary, the only requirement we place upon them is thatthey make the Lagrangian Lorentz invariant.
Since we are calling our unhealthy eld a Galileon, let us give an example of such a Lagrangianthat obeys the Galilean symmetry:
L = LGALf (qα) + g (qα, ∂qα) , (140)
with f and g arbitrary functions and LGAL the Lagrangian for a single galileon (66). ThisLagrangian is equivalent to a rst order Lagrangian by theorem 7. The relevant function F 0 isgiven by F 0 = Fi · f , with the Fi the total derivative functions of the one galileon case.
Of course, there are other Lorentz invariant Lagrangians that are in Class I, but those donot obey the Galilean symmetry in φ.
Next, we generalize to Class II theories. The null-vector now is non-zero, but does notdepend on all derivatives. For one second-order eld, this can be written as
LφqβL−1qβ qα
= V α1
(φ, ∂µφ, q
β). (141)
The easiest way to accomplish this is to enforce that all components on the left-hand side donot depend on the higher-order derivatives, i.e.
Lφqβ = Lφqβ (φ, ∂µφ, qγ) ,
Lqβ qα = Lqβ qα (φ, ∂µφ, qγ) .
We can draw several conclusions from these lines:
• Any quadratic appearance of the second-order time derivative of the galileon is not coupledto the time derivative of the scalar.
• The second-order time derivative of the galileon is only linearly coupled to the rst-ordertime derivative of the scalars.
• The Lagrangians depends at most quadratically on the rst order derivatives of the scalar.
We are however interested in not just the form of these Class II theories, but we want tocheck their equivalence to a rst order theory. According to Crisostomi et al. [8], all Class IItheories can be mapped to a Class I theory via a eld redenition of the qα. The transformedelds qα are the solutions to the system of partial dierential equations [8]
∂u
∂φ+ V β
1 (∂φ, φ, qα)∂u
∂qβ= 0, (142)
and the transformed Lagrangian is given by
L (∂µ∂νφ, ∂µφ, φ, ∂µqα, qα) = LI (∂µ∂νφ, ∂µφ, φ, ∂µq
α, qα) . (143)
Lorentz invariant Class I theories are always a total divergence rst-order theory, but the ClassI theory we receive due to the eld redenition need not be Lorentz invariant. We can howeverstill end up with a st-order theory. Let L be the thus gained Class I theory. It can be rewritteninto the form of (139),
LI = φf (∂i∂µφ, ∂µφ, φ, ∂iqα, qα) + g (∂i∂µφ, ∂µφ, φ, ∂iqα, qα) .
Although this Lagrangian need not be Lorentz invariant, there is only one second order variable,thus the condition ∂fa
∂∂iφb= 0 still holds. Since we only have one second order variable, we
72
do not have the anti-symmetric requirement ∂fa∂φb
= ∂fb∂φa
. Thus there exists an F 0 such that
F 0φ
= f and L+ ddtF 0 is rst-order, and thus we conclude that all Class II One-Galileon, many
scalar Lagrangians are divergence equivalent to a regular rst-order Lagrangian, though thisLagrangian need not be Lorentz Invariant.
This leaves us with the Class III theories. For these theories, Crisostomi et al. [8] did not ndany relation between Class III theories and rst-order Lagrangians. As a result, our analysisof one-galileon theories stops at this point. The existence of a divergence equivalent rst-orderLagrangian can be tested using Cartan's equivalence algortihm discussed in chapters 10 and10.5.
9.4 Many Galileons
The next step is to promote the additional scalar elds to galileon elds. In other words, weare going to investigate multi-galileon theories. From section 5.1, and the work by Padilla andSivanesan [23] referenced therein, the Lagrangian for such a general Lagrangian is given by(72):
L =5∑
m=1
αa1...amφa1δµ2...µmν2...νm
∂µ2∂ν2φa2 · · · ∂µm∂νmφam ,
where the indexes aj count over the elds. Since there are no rst order scalar elds present,these theories are all in Class I. Since they are Lorentz invariant, we know that they aredivergence equivalent to rst-order Lagrangians by theorem 7. To arrive at the required totalderivative function F , we shall split up the sum in (72) into its component parts, and thenrewrite each term in the form of (128) individually. Linearity of dierentials then allows us toconsider each of these components separately.
L = L1 + L2 + L3 + L4 + L5
L1 = αaφa
L2 = αabφa∂µ∂µφ
b
= αabφaφb + αmnφ
a∂i∂iφb
L3 = αa1a2a3φa1δµ2µ3ν2ν3
∂µ2∂ν2φa2∂µ3∂
ν3φa3
= αa1a2a3φa1(φa2∂j∂
jφa3 + ∂j∂jφa2φa3 + ∂j∂
jφa2∂k∂kφa3 − 2∂jφ
a2∂jφa3 − ∂j∂kφa2∂j∂kφa3)
= φb(2αa1ba3φ
a1∂j∂jφa3
)+ g3
L4 = αa1a2a3a4φa1δµ2µ3µ4ν2ν3ν4
∂µ2∂ν2φa2∂µ3∂
ν3φa3∂µ4∂ν4φa4
= αa1a2a3a4φa1(
3∂j∂jφa2∂k∂kφ
a3φa4 + 6∂jφa2∂j∂kφ
a3∂kφa4 − 6∂j∂jφa2∂kφ
a3∂kφa4 − 3∂j∂kφa2∂j∂kφa3φa4
)= 3αa1a2a3a4φ
a1(∂j∂
jφa2∂k∂kφa3 − ∂j∂kφa2∂j∂kφa3
)φa4
+ 6αa1a2a3a4φa1(∂jφ
a2∂j∂kφa3 − ∂kφa2∂j∂jφa3
)∂kφa4
L5 = αa1a2a3a4a5φa1δµ2µ3µ4µ5ν2ν3ν4ν5
∂µ2∂ν2φa2∂µ3∂
ν3φa3∂µ4∂ν4φa4∂µ5∂
ν5φa5
= 4αa1a2a3a4a5φa1δj2j3j4k2k3k4
∂j2∂k2φa2∂j3∂
k3φa3∂j4∂k4φa4φa5 + g (∂i∂µφ
a, ∂µφa, φa)
In this split, we have that L1 is zero-th order, and thus naturally ghost free. For the moreinteresting cases have f2,a = αabφb and f3,b = 2αa1a2bφ
a1∂j∂jφa2 . For the fourth Lagrangian, we
have f4,b = 3αa1a2a3bφa1(∂j∂
jφa2∂k∂kφa3 − ∂j∂kφa2∂j∂kφa3
), and last for the fth Lagrangian
f5,b = 4αa1a2a3a4bφa1δj2j3j4k2k3k4
∂j2∂k2φa2∂j3∂
k3φa3∂j4∂k4φa4 . We could calculate which functions Fi
produce the functions fi, and thus calculate the rst order Lagrangians. This is left as anexercise for the reader.
73
Finally, we note that the Galilean invariance is not responsible for the equivalence. AllOstrogradsky free Lagrangians with only second order variables are in Class I, and thus aredivergence equivalent to a rst order theory by them 7, although this may not be a Lorentzinvariant theory.
9.5 Many Galileons, and many scalars
Essentially, this is the most general case. Our ghost free theories could fall in any of the threeclasses. We discuss the theories by class.
• The Class I theories are all equivalent to a rst-order theory, by theorem 7.
• The Class II theories are all equivalent to a Class I theory, by theorem 8. Unfortunately,the resulting Class I Lagrangian need not be Lorentz invariant, and neither is it genericallytrue that ∂fa
∂∂iφb= 0. As a result, we can not at this point state whether all Class II theories
are equivalent to a rst-order theory.
• Unfortunately, Crisostomi et al. [8] found no general transformation that turns a ClassIII theory into a Class I theory, and from there into a rst order one. As a result, we cannot state anything about them right now.
The Class II and Class III theories can be looked into, using the Cartan algortihm discussedin chapters 10 and 10.5.
9.6 Covariant Galileons
In the previous part of this section, only non-covariant galileons, and galileon like Lagrangians,were considered, i.e. theories on a at background gµν = ηµν . The reason for this is that, as faras I'm aware, no one has yet formulated general criteria for second order tensor theories to beOstrogradsky ghost free. Therefore is no class of theories to apply any equivalence calculationsor consideration to.
However, as Chogoya and Tasinato [19] found that several Galileon theories remain ghost-free upon minimal-coupling-covariantization, it is not illogical to assume that the theories fromthis section remain ghost-free upon covariantization.
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10 An explanation of Cartan's equivalence algorithm
Elie Cartan formulated an algorithm to solve equivalence problems on dierential forms. Healso formulated a procedure to cast several other equivalence problems in dierential form.This allows one to use his algorithm to solve these problems [6, 17]. In this section, I shallclosely follow the book by Olver [6, chapters 8 to 10] to explain Cartan's algorithm, and use hispaper together with Kamran [17] to show how to rewrite the Lagrangian equivalence problemin dierential form.
First, we have the strict version of Cartan's algorithm. It answers the question: Giventwo coframes θi an θi, when does there exist a dieomorphism Φ such that Φ∗
(θi)
= θi?Next, we concern ourselves with a more general question: Given two coframes θi and θi, whendoes there exist a dieomorphism Φ and a matrix Aij (subject to some constraints) such that
Φ∗(θi)
= Aijθj? We solve this more general question by reducing to the strict problem.
10.1 Strict equivalence of coframes
To answer the strict question, we rst need to dene coframes and the equivalence problem onthese coframes.
Denition 8 (Frames and coframes, [6]). Let M be a smooth, n-dimensional manifold. Aframe on M is a set of vector elds V = v1, . . . ,vn with the property that they form a basison the tangent space TxM at each point x ∈ M . Dually, a coframe on M is an ordered setof one-forms θ = θ1, . . . , θn which form a basis on the cotangent space T ∗xM at each pointx ∈M .
Given a coframe θ, its dual frame of vector elds is the frame of vector elds ∂/∂θ1 , . . . , ∂/∂θnthat satises ⟨
θi;∂
∂θj
⟩= δij.
Denition 9 (Equivalence of coframes [6]). Let θ = θ1, . . . , θn be a coframe on a manifoldMand θ =
θ1, . . . , θn
a coframe on a potentially dierent manifold M , with dimM = dim M .
We say that these two coframes are equivalent if and only if there exists a dieomorphismΦ : M → M such that
Φ∗θi = θi. (144)
The key observation made by Cartan in developing his algorithm was that the exteriorderivative operator d is invariant under dieomorphisms [6]. Thus, for any two equivalentcoframes, we also necessarily have
Φ∗ dθi = dθi (145)
Since θ is a coframe, we can write the two-forms dθi as a linear combination of wedgeproducts of the θi. This gives the fundamental structure equations
dθi = T ijkθj ∧ θk. (146)
We can also compute these for θ, this yields
dθi = T ijkθj ∧ θk.
Substituting these two equations into (145) gives
75
T ijk (x) θj ∧ θk
= dθi
=Φ∗ dθi
=Φ∗(T ijkθ
j ∧ θk)
=T ijk (Φ (x))Φ∗θj ∧ Φ∗θk
=T ijk (x) θj ∧ θk,
where in the last line the equivalence of the coframes (144) is used to write Φ∗θj∧Φ∗θk = θj∧θk.The above calculation implies the invariance of the structure equations:
T ijk (x) = T ijk (x) , when x = Φ (x) . (147)
This invariance already gives some necessary conditions for equivalence, for if T ijk for θ is a
constant for some indices i, j, k, then for any equivalent coframe θ, T ijk must assume the sameconstant value for the same indices i, j, k [6].
These structure equations appear not just in the derivatives of the coframe, if we computethe commutation relations for the dual-frame to our coframe, we nd the following:[
∂
∂θj,∂
∂θk
]= −T ijk
∂
∂θi. (148)
These equations will later allow us to rewrite the order of derivation of the structure equationsusing lower order derivatives.
We now have a set of 12n2(n− 1) invariant functions. However, we can easily compute more.
Let I (x) be any scalar invariant (T ijk is an example), which is mapped to I (x) under the actionof Φ, then the dierentials of these invariants must also agree: Φ∗ dI = dI. We can re-expressthese dierentials in terms of the coframes, using the dual frames. Thus, we nd
∂I (x)
∂θjθj = dI (x) = Φ∗ dI (x) =
∂I (Φ (x))
∂θjθj, (149)
or, since the θi are linearly independent,
∂I
∂θi(x) =
∂I
∂θi(x) . (150)
We can do this ad innitum (or at least ad nauseam), giving us an innite number of invariants.These invariants may not be all independent, there may be functional relations between them.These relations are also invariant. For example, assume that we have 3 invariants, I1, I2 andI3, with the relation I3 = H (I1, I2), for some function H(x1, x2). Then any equivalent coframemust have this same functional relation: I3 = H
(I1, I2
)for the same function H (x1, x2).
We can introduce these dependence relations by permuting the derivatives. The Lie brackets(148) allow us to do this, at the expense of introducing lower order derivatives, using the relation:
∂2I
∂θk∂θj− ∂2I
∂θj∂θk= T ijk
∂I
∂θi(151)
Before continuing, let us introduce some notation for the derivatives of the structure equa-tions:
Tσ =∂sT ijk
∂θls∂θls−1 · . . . · ∂θl1, where σ = (i, j, k, l1, . . . , ls) .
76
This is the most general structure invariant associated with the coframe [6]. σ is an orderedmulti-index, since the order of derivation matters. The integer s = order σ = |σ| − 3 is calledthe order of the derived invariant. According to the above calculations, for two equivalentcoframes, their structure equations and all their derivatives must agree, i.e.
Tσ (x) = Tσ (x) , with x = Φ (x) , ∀σ. (152)
Through constant derivation, we have an innite cascade of necessary conditions for equiva-lence, but we would like some sucient conditions. In order to reach these conditions, we needto introduce the classifying spaces and manifolds.
Denition 10 (Classifying spaces [6]). The sth order classifying space K(s) = K(s) (n) associ-ated with an n-dimensional manifold M is the Euclidean space of dimension qs(n) = 1
2n2(n −
1)( n+sn ), which has local coordinates given by z(s) = (. . . , zσ, . . . ). The entries of z
(s) are labelledby non-decreasing multi-indices σ = (i, j, k, l1, . . . , lr) , with j < k, 1 6 l1 6 l2 6 . . . 6 lr 6 n,and 0 6 r 6 s.
Denition 11 (Structure map [6]). The sth order structure map associated with a coframe θon M is the map T (s) : M → K(s) whose components are the structure constants: zσ = Tσ (x),for order (σ) 6 s.
Thus, the structure map is the set of all structure equations and their derivatives, andthe classifying space is where the values of the components of the structure map lie. Thesedenitions already take into account that we are allowed to permute the order of dierentiationin our invariants, thus only the qs(n) indices with non-decreasing indices are needed.
If two coframes θ and θ are equivalent, then the invariance equations (152) tell us that theirstructure maps must have the same image. We can give this image a name: the classifying set:
Denition 12 (Classifying set). The sth order classifying set C(s) = C(s) (θ, U) associated witha coframe θ on an open subset U ⊂M is dened as the image of the structure map T (s):
C(s) (θ, U) =T (s) (x) |x ∈ U
⊂ K(s).
This denition and (152) then immediately give rise to the following proposition
Proposition 1. Suppose θ and θ are equivalent coframes under Φ : M → M . Then for eachs > 0 the sth order classifying sets are the same, e.g. C(s)
(θ, U
)= C(s) (θ, U), where U ⊂M is
the domain and U ⊂ M is the range of the local equivalence map Φ.
Ergo, two coframes are equivalent if and only if all of their classifying sets are the same [6].From the denition of the classifying set, it should be clear that if C(s)
(θ, U
)= C(s) (θ, U)
for a certain s, then also C(r)(θ, U
)= C(r) (θ, U) for each r 6 s.
This proposition is not just a necessary condition, it is also sucient. However, it doesrequire us to compute innitely many classifying sets. The next paragraph aims to reduce theamount of classifying sets that need to be computed to determine equivalence.
Let ρs = rank(T (s)
)be the rank of the structure map. As long as the structure map is
regular, C(s) is a ρs dimensional sub-manifold of K(s). This rank ρs is equal to the number offunctionally independent structure invariants among the Tσ, and all structure invariants in C(s)
can be expressed as a function of a suitably chosen set of ρs invariants.It should be clear from the construction of the classifying set that its dimension is non-
decreasing. Furthermore, it is bounded by n, the dimension of M . Therefore, it must convergeto a constant value. These considerations lead to the following denition.
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Denition 13. Let θ be a coframe and let ρs denote the rank of the sth order structure mapT (s). The smallest s for which ρs = ρs+1 is called the order of the coframe. We also have
0 6 ρo < ρ1 < · · · < ρs = ρs+1 = ρs+2 = · · · = r 6 n.
This stabilizing rank r is called the rank of the coframe.
This proposition gives an upper bound to the order of C(s), and thus an upper bound to thesize of the classifying sets. Thus we can sumarize this section in the following theorem:
Theorem 9 (Fundamental Theorem of Equivalence of Coframes [6]). Let θ and θ be smooth,regular coframes dened, respectively, on n-dimensional manifolds M and M . There exists alocal dieomorphism Φ : M → M mapping the coframes to each other, Φ∗θ = θ, if and onlyif they have the same order, s = s, the same rank, r = r, and their (s+ 1)th order classifyingsets C(s+1)
(θ)and C(s+1) (θ) overlap.
Example 14. Let us discuss an example of the computation. Let M = R2\ x = 0, on whichwe dene the coframe
θ1 = dx, θ2 = x dy.
Its dual frame is given by
∂
∂θ1=
∂
∂x,
∂
∂θ2=
1
x
∂
∂y.
Compute the dierentials:
dθ1 = d dx = 0,
dθ2 = d (x dy)
= dx ∧ dy
=x
xdx ∧ dy
=1
xθ1 ∧ θ2.
Thus T 112 = 0 and T 2
12 = 1x. Our rank-0 structure map thus only has 1 independent structure
constant. To see if this is enough, compute the rank-1 structure map.
T2,1,2,1 =∂T 2
12
∂θ1=∂ 1x
∂x
= − 1
x2
T2,1,2,2 =∂T 2
12
∂θ2=
1
x
∂ 1x
∂y
= 0
T2,1,2,1 is a function of T 212: T2,1,2,1 = − (T 2
12)2and is thus not independent. Therefore, our
coframe has rank 1 and order 0. Its 1th order classifying set is given by
C1(dx, x dy,R2\ 0
)=T 1
12, T212, T1,1,2,1, T1,1,2,2, T2,1,2,1, T2,1,2,2
=
0,
1
x, 0, 0,
−1
x2, 0
.
78
For an equivalent coframe, consider
θ1 = dy, θ2 = y dx. (153)
Clearly, it is equivalent through the dieomorphism Φ given as
x = Φ1(x, y) = y,
y = Φ2(x, y) = x.
To prove the equivalence, calculate the pull-backs:
Φ∗(θ1)
= Φ∗ (dy)
= dx
= θ1,
Φ∗(θ2)
= Φ∗ (y dx)
= x dy
= θ2.
However, we can also compute the rank, order and classifying sets of θ. First, the dual frameis now given by
∂
∂θ1=
∂
∂y,
∂
∂θ2=
1
y
∂
∂x. (154)
The structure equations are
T 212 =
1
y,
T2,1,2,1 =−1
y2,
with all others equal to zero. The rst-order classifying set is now given as
C1(dy, y dx,R2\ 0
)=T 1
12, T212, T1,1,2,1, T1,1,2,2, T2,1,2,1, T2,1,2,2
=
0,
1
y, 0, 0,
−1
y2, 0
,
which is identical, concluding this example.
10.2 Equivalence with structure-groups
In the previous section, the equivalence problem for coframes was solved. However, for theLagrangians we do not require the strict equivalence. For example, we only require that thecontact ideal I as a whole is preserved. If θαI represents a contact form, then the relation
Φ∗θαI = θαI certainly conserves the contact ideal, but so does Φ∗θαI =(aαβ)JIθβJ , i.e. a linear
transformation mixing the contact forms themselves. Thus, we need to somehow include theseallowed transformations in the formulation of the equivalence problem. We do this by the useof structure groups, which are the groups of allowed transformations.
Denition 14 (G-equivalence problem [6]). Let G ⊆ GL(n) be a Lie group. Let ω and ωbe coframes dened respectively on the n-dimensional manifolds M and M . The G-valuedequivalence problem for these coframes is to determine if there exists a dieomorphism Φ :M → M and a G-valued function g : M → G with the property that
Φ∗ω = g (x)ω. (155)
The group G is called the structure group of the problem.
79
We use ω instead of θ to denote the coframes to dierentiate between the regular equivalenceproblem and the G-valued equivalence problem.
There are two special cases to look at. The rst is if G = e. In that case, (155) reducesto (144) and we have reduced the problem to the case described above, in section 10.1. Theother case is if G = GL(n). In that case, any two coframes are equivalent and the problem istrivial. Thus the interesting cases are those when e ⊂ G ⊂ GL(n). We want to reduce thisgroup G, through a series of invariant operations (to be dened later), to the trivial group e,which allows us to use the already developed machinery from subsection 10.1.
First, we make (155) more symmetric. Instead of just one G-valued function, we now searchfor two functions, g : M → G and g : M → G such that (155) becomes
Φ∗ [g (x) ω] = g (x)ω. (156)
If this can be done, then the two coframes dened by θ = g (x) ω and θ = g (x)ω areobviously invariant: Φ∗θ = θ. So, how to nd these functions? The key lies in nding functionsthat depend on the group parameters gij and maybe the coordinates xk that are invariant orunchanged by the coordinate transformations. We can then use such a function to x one ofthe group parameters25. More precisely, we are looking for scalar functions H(g, ω) that satisfy
H (g (x) , ω|x) = H (g (x) , ω|x) , (157)
whenever x = Φ (x) and Φ∗ (g (x) ω) = g (x)ω.For each such a function H, we can then normalize one of the group parameters and set it
equal to some constant or combination of the other group parameters. Of course, in doing so,the restrictions imposed by the group must be taken into account. E.g, if a group parametermust be non-zero, we can not normalize this parameter to zero. This reduces the diameter ofthe group G by one. If this is done often enough, the problem is reduced to the e-equivalenceproblem that was discussed in section 10.1 [6, Chapter 10].
The invariant combinations used are often the structure functions computed by taking thederivative of θ = g (x)ω:
θi = gij (x)ωj
dθi = d(gij (x)ωj
)= d
(gij (x)
)∧ ωj + gij (x) dωj.
The two forms dωj can be written as the wedge product of the one forms ωi, since ω forms acoframe on M by construction. Furthermore, we can rewrite these one forms ωi in terms of theone forms θi, resulting in
dθi = γijθj + T ijkθ
j ∧ θk. (158)
The functions T ijk are in general not invariants of the problem. In order to get to actualinvariants, the process of absorption is used.
First, one notes that the functions γij can be written in the Maurer-Cartan basis [6, chapter2] of the group G: γij = Aijκα
κ. Thus, (158) becomes
dθi = Aijκακ ∧ θj + T ijkθ
j ∧ θk, (159)
and for the equivalent frame:
dθi = Aijκακ ∧ θj + T ijkθ
j ∧ θk. (160)
25This xing of parameters is also called normalization, because often the structure constants are normalizedto zero or one.
80
Not all of the terms Aijκ are non-zero. We also do not care about the exact form of the ακ
forms. Thus, we can choose to absorb other terms into the ακ forms. The precise mathematicalfoundations can be found in [6, pages 307-310]. In particular, for a given ακ wedging θj (i.e.Aijκ is non-zero) we can redene it as
πκ = ακ − zκi θi, (161)
with the zκi satisfyingAijκz
κk − Aikκzκj = −T ijk. (162)
With this substitution, (159) becomes
dθi = Aijκπκ ∧ θj + U i
jkθj ∧ θk. (163)
The remaining non-zero terms U ijk are the essential torsion. These essential torsion coecients
may depend explicitly on the group-parameters. These are then normalized to convenientconstant values (usually ±1, 0) by solving for the group-parameters. After thus xing thegroup parameters, one substitutes the thus obtained formulae back into the formula for thecoframe and recompute. This continues until either all group parameters are normalized andwe have an explicit, invariant coframe, or we are left with some undetermined group parametersbut no essential torsion to x them. We shall deal with those cases in the next subsections.
To summarise, the procedure is as follows. One is given a coframe ω and an structure groupG. The invariant coframe is then given by θ = gω for a certain group element g ∈ G. Performthe process of normalization and absorption as described above to obtain the essential torsionelements and x the group-parameters to x g. Thus, the invariant coframe θ is now fullydetermined, and one can compute the order, rank and classifying sets for this invariant frame.
Example 15. As an example, consider M = R2\ x = 0 or y = 0. As coframe, take
ω1 = y dx, ω2 = x dy.
For the structure group, use
g =
(1 00 a
),
for some constant a 6= 0. Then (θ1, θ2)T
= g (ω1, ω2)T. Next, compute the dierentials. First
for θ1:
dθ1 = d (y dx)
= dy ∧ dx
=ω2 ∧ ω1
xy
= − 1
xyaθ1 ∧ θ2,
and then for θ2:
dθ2 = d (ax dy)
= x da ∧ dy + a dx ∧ dy
= da ∧ ω2 +a
xyω1 ∧ ω2
=da
a∧ θ2 +
1
xyθ1 ∧ θ2
= α ∧ θ2.
81
In the last line, I have absorbed θ1
xyinto α = da
a+ θ1
xy.
We are thus left with one essential torsion component U112 = −1
xya. This can be normalized to -1
by setting a = 1xy. After normalization, we repeat the procedure of computing the dierentials.
Again, rst we look at θ1:
dθ1 = d (y dx)
= dy ∧ dx
=ω2 ∧ ω1
xy
= −θ1 ∧ θ2.
The torsion component did not change, as expected. Next, look at θ2:
dθ2 = d
(1
ydy
)= − 1
y2dy ∧ dy
= 0,
which thus yields no additional torsion components.Thus, the nal invariant frame is given by
θ1 = y dx
θ2 =1
ydy.
With this invariant frame, we go through the strict algorithm. However, we have already donethe rst step, the dierentials of θi. The torsion components are T 1
12 = −1 and T 212 = 0. Since
both of these are constants, their derivatives are zero and we are left with a rank 0, order 1system and classifying set
C1(θ1, θ2,M
)=T 1
12, T212, T1,1,2,1, T1,1,2,2, T2,1,2,1, T2,1,2,2
= −1, 0, 0, 0, 0, 0
10.3 Involutive systems
It is possible that after a number of loops through the algorithm we have eliminated all essentialtorsion, but have not xed all group parameters. The reason for this is the existence of asymmetry group of the coframe, and the dimension of this symmetry group changes the solutionspace [6, Chapter 11]. There are two cases to distinguish, as the dimension of the symmetrygroup is either nite or innite, and each of these cases is dealt with through a dierent method.
First, we need to determine which case we're in. For this, we will use Cartan's test forinvolutivity. This test determines whether a system is in involution, which precisely means thatthe system has an innite dimensional symmetry group [6].
Before we can head to the test, we have to dene some new concepts.
Denition 15 (Degree of indeterminacy). The absorption equations (162) may not be com-pletely solvable, i.e. some of the functions zκi are not xed by solving these equations. Thedimension of the solution space of (162) is the degree of indeterminacy of the system, and islabelled by r(1) [6].
Aijκzκk − Aikκzκj = −T ijk. (162)
82
Next are the Cartan Characters of the system.
Denition 16 (Cartan Characters). Let v ∈ Rn, with n the dimension of the manifold M .Then dene L [v] ∈ Rn×r as the matrix with elements
Liκ [v] = Aijκvj.
From this matrix, we can determine the n reduced characters. The rst one is given by
s′1 = max rank(L [v]) | v ∈ Rm .
The next ones, up to and including the (m− 1)th one, are then subsequently given by
s′1 + s′2 + · · ·+ s′k = max
rank
L [v1]L [v2]...
L [vk]
∣∣∣∣∣∣∣∣∣ v1, . . .vk ∈ Rm
,
and the last one is dened by s′1 + s′2 + · · · + s′n = r, with r the dimension of the original,un-reduced structure group [6].
With these two concepts dened, we can now use Cartan's test for involution. According tohim, a system of coframes θ is involutive if [6]
s′1 + 2s′2 + · · ·+ ns′n = r(1). (164)
If the system at hand is both involutive, and has only constant essential torsion remaining,then it is called transitive. Furthermore, the following theorem about equivalence is true [6]:
Theorem 10 (Transitive Equivalence). Let θ and θ both be coframes onM and M respectively,with the same structure group G. If both θ and θ are in involution, have only constant essentialtorsion, and the torsion coecients are equal, then θ and θ are equivalent.
If the essential torsion is not constant, then we have the intransitive case. The next step isthen to look at the derivatives of the essential torsion coecients. It is possible that these dodepend on some group parameters. In this case, use this dependence to x the group parametersand normalize these derivative essential torsion and go through the algorithm again. Eventually,either all group parameters will be xed or all essential torsion derivatives will be independentof the group parameters. For the latter case, we have the following theorem [6]
Theorem 11. Let θ and θ both be lifted coframes on M and M respectively, with the samestructure group G. Furthermore, both θ and θ are involutive and neither the essential torsion,nor its derivatives, depend on the group parameters. Then θ and θ are equivalent only ifthey have the same order s = s and their (s+ 1)th order classifying manifolds C(s+1)
(θ)and
C(s+1) (θ) overlap.
10.4 Prolonged systems
In this subsection we will treat the non-involutive systems. These are systems with a nitesymmetry group with dimension greater than 0. These systems then are dened on a too smallbase manifold to handle this additional symmetry group. The solution to this is to prolong thesystem onto a larger base manifold, and try to solve the system on this larger manifold.
The prolongation takes us from the base manifold M to the manifold M ×G, i.e. the basemanifold appended with the reduced structure group. On this new, larger manifold, the coframe
83
will be formed by formed by the coframe θ on M , appended by the modied Maurer-Cartanforms πκ on G. These modied Maurer-Cartan forms are given by
πκ = ακ − zκi θi,
where ακ are the regular Maurer-Cartan forms on G, and the zκj are the solutions to (162).
If the degree of indeterminacy r(1) = 0, the zκi are uniquely determined and we have a fullydened coframe on M ×G. Thus, we can try to solve the system on M ×G.
However, maybe r(1) > 0, which is called the indeterminate case. Now, we have r(1) freevariables, which I denote by wu, on which the zκi depend. In this case, equivalence implies the
existence that there exists both a g = g(x) ∈ G and w = w(x) ∈ Rr(1) under which both θ andπ are invariant. The appended coframe now comprises of both θ = g ·ω and πκ = ακ + Sκj θ
j,where Sκj is the particular solution to (162) with wu = 0. In order to determine the functions
wu(x) that are part of the equivalence procedure, we have an additional structure group G(1),with elements given by
G(1) =
(In 0
K [w] Ir
) ∣∣∣∣ w ∈ Rr(1), (165)
with K [w] the homogeneous solution to (162), i.e. a solution to
Aijκzκk − Aikκzκj = 0. (166)
The nal coframe is then given by
Θ = g(1) ·(g ·ωπ
). (167)
Next, use the absorption and normalization procedure to determine the values for wu (x) anddetermine the remaining group parameters of g(x).
Should one again terminate to a solution where not all group parameters are determined,the procedure is to see if the new, reduced system, is in involution or not. If not, again prolongthe system by adding another copy of G. Keep doing this, until all group parameters aredetermined. This procedure will terminate (for non-involutive systems)26 [6].
Thus, all possible branches of the equivalence problem have now been solved. Next, we castthe Lagrangian equivalence problem into dierential form.
10.5 Application of Cartan's algorithm to Lagrangians
Cartan's algorithm allows us to check if two coframes attached to a manifold M are equivalent.We want to apply his algorithm to the problem of equivalence between Lagrangians. To dothat, we need to rewrite (127) into a dierential form, and set up the rest of the coframe to usein our equivalence problem. Last, we need to dene the structure group to use.
As a reminder, we call two Lagrangians divergence equivalent if there exists a map Φ :Jk (Rp,Rq)→ Jk (Rp,Rq) and a function F : Rp → R such that (127)
L(xi, u(r)
)=L(xi, u(r)
)+∇F
| det J |,
where J ij is the Jacobian associated with Φ. This relation opperates on the jet-bundle Jk (Rp,Rq),
and thus this is the manifold on which we need to dene our coframe.
26This non-termination is why involutive systems are treated dierently
84
The rst elements of our coframe we nd by considering the variational problem describedby L. It is given by the functional S =
∫L dx, with dx the volume form. It is the integrand
that is to be preserved, modulo a total divergence. From [42], the pull back of L is
Φ∗(L dx
)= L det J dx+ L dxi
duαθα ∧ dxi,
where dxi =∏
j∈1,...,p\i ∧ dxj. Since we want divergence equivalence, we can use (127) to
substitute L det J . We also consider the contact ideal I. The allowed transformations preservethis ideal. Furthermore, solutions to the variational problem live on the submanifold on whichthe contact forms vanish. Thus, we can remove dxi
duαθα ∧ dxi by working mod I.
Φ∗(L dx
)= L dx+∇F dx mod I. (168)
Dene the one-forms ωi = p√L dxi. These obey ω1 ∧ · · · ∧ ωp = L dx. We are interested
in how these one-forms transform under ber-preserving, point and contact transformations.Since we want (168) to hold for the wedge product of ωi, a general transformation can bewritten as [42]
Φ∗(ωi)
= (B0)iα θα + (B1)i,jα θαj + (B2)i,Jα θαJ + J ijω
j
= J ijωj mod I.
If Φ is a point transformation, we have B2 = 0, and if Φ is a ber-preserving transformation,we have B2 = 0 and B1 = 0, as a consequence of Bäcklund's theorem [42].
The transformation laws for the contact forms were already given in the proof of theorem 5,they are
Φ∗(θα)
=dψα
duβθβ, (169)
Φ∗(θαI)
=dψαIduβ
θβ +∑|J |6|I|
dψαIdpβJ
θβJ .
These transformations are all linear transformations, and thus we can follow Kamran andOlver [42]; we rescale the contact forms and place them into a column vector. Dene
θ =(θ0,L−1/pθ1,L−2/pθ2, . . . ,L−(r−1)/pθr−1
),
so that (169) becomesθ = Aθ, (170)
where A is a block-lower triangular matrix with the diagonal blocks satisfying Ak = A0⊗kJ−T[42]. With these two equations (168) and (170), we have given part of our coframe and denedthe structure group for this coframe. These one-forms however do not span a full co-frameon the jet-space. We are lacking coframe elements corresponding to the rth-order derivatives.Thus, append the coframe by adding elements [42]
παI = L−r/p dpαI ,
with |I| = r. The transformation law for these elements are governed by the fact that pαI hasto obey the chain rule.
Grouping this all together, we now have the following transformation matrix group of one-forms θω
π
=
A 0 0B J 0C D · J E
θωπ
.
85
The restrictions on these block-matrices can be found in [42, page 378]. We now need to insertthe divergence into this equation. We do this through considering the Jacobian J ij . Introducep additional coordinates wi, which transform according to wi = wi +F i. We are thus no longerworking on Jr, but on Jr × Rp and we need to extend our coframe accordingly. Like Kamranand Olver [42], append the co-frame with the one forms vi = L1/p−1 dwi. With this addition,the transformation law becomes
θωπv
=
A 0 0 0B J 0 0C D · J E 0Q T ·R T ·M T
θωπv
. (171)
The exact restrictions on the block-matrices can again be found in [42, page 378], but theimportant one is that we restrict det J = 1 + traceR.
Next, we show that this transformation law indeed transforms the Lagrangian as in (127).Let
(biµ)be the entries of B, J ij the entries of J and I the contact ideal generated by the contact
forms θ27. Under the transformation (171), the Lagrangian L becomes
L dx = ω1 ∧ ω2 ∧ · · · ∧ ωp
=(b1µθ
µ + J1i ω
i)∧(b2µθ
µ + J2i ω
i)∧ · · · ∧
(bpµθ
µ + Jpi ωi)
= det J ·ω1 ∧ ω2 ∧ · · · ∧ ωp mod I= (1 + traceR) ·ω1 ∧ ω2 ∧ · · · ∧ ωp mod I= L dx+ traceR · L dx mod I,
To nd the trace of R we look at the transformation law for v from two perspectives. First,using the transformation (171);
vi = qiµθµ +
1
det Jjikr
kjω
j +1
det Jjikm
kIα π
αI +
1
det Jjijv
j
= qiµθµ +
1
det Jjikr
kjω
j +1
det Jjikm
kIα π
αI +
1
det JjijL(1/p)−1 dwj,
and next, using the denition of v:
vi = L(1/p)−1 dwi
= L(1/p)−1 d(wi + F i
)= L(1/p)−1
(dwi + F i
xj dxj + Fθµθµ).
We compare these two transformation laws. Using the linear indepence of the one-forms dwi
and ωi, we see:
L(1/p)−1 dwm =1
det Jjmn L(1/p)−1 dwn, (172)
L(1/p)−1F ixjL−(1/p)ωj =
1
det Jjikr
kjω
j.
To distil the trace of R out of this, rst rewrite the second line,
δmj L(1/p)−1 =L(1/p)
det J
(F−1x
)mijikr
kj
27Here θµ runs over the entire θ. The index µ thus incorporates both the index α and the multi-indices I.
86
and then substitute this into (172)
L(1/p)
det J
(F−1x
)mijikr
kj dwj =
1
det Jjmj L(1/p)−1 dwj
F lxm
(F−1x
)mijikr
kj = L−1F l
xmjmj(
j−1)hljlkr
kj = L−1
(j−1)hlF lxmj
mj
rhj = L−1(j−1)hlF lxmj
mj
Thus the trace of R is given by
rhh = L−1(j−1)hlF lxmj
mh
= L−1F lxmδ
ml
= L−1divF
and the transformation for L dx becomes
L dx = L dx+ traceR · L dx mod I= L dx+ divF dx mod I,
as desired.To conclude, in this section we have written the Lagrangian equivalence problem (127) into
a form to which we can apply Cartan's equivalence algorithm. In the next section, we willapply the algorithm to the specic problem of determining if there is an equivalence betweensecond-order Ostrogradsky free and regular rst-order Lagrangians for the case of mechanics,i.e. only 1 independent variable.
87
11 The case for 1 independent, and q dependent variables,
all second order
In this section, we use the Cartan equivalence algorithm to determine whether second-order,Ostrogradsky-free Lagrangians are divergent equivalent to regular rst-order theories. As thetitle says, in this section we will deal with Lagrangians that are second order in all of theirdependent variables, i.e. ∂L
∂uα6= 0. Of course, we do want the Lagrangian as a whole to be
healthy, i.e. we enforce that (30) and (31) hold.Let us set-up the coframe needed. We are working on J2 (R,Rq), on which we dene the
coframe according to [42]:
θα = duα − uα dx
θαx = duα − uα dx
ω = L dx
πα = L−2 duα
v = dw
Including normalization on the contact forms, the coframe vector is Σ = (θ,L−1θx, ω,π, v).Since q > 1, according to Bäcklund's Theorem all contact transformations are prolongations of apoint transformation [42]. Therefore, we the most general transformation we use in this case arepoint transformations. In this case then, the structure group is given by the (q+q+1+q+1)2-block matrix
H =
(A1)αβ 0 0 0 0
(A2)αβ1j
(A1)αβ 0 0 0
(b1)β 0 j 0 0
(C1)αβ (C2)αβ (d1)α 1j2
(A1)αβ 0
(q1)β (q2)β j − 1 0 1
. (173)
However, we can be more restrictive than just that. In [8] it was already shown that p = 1,r = 2, q = arbitrary, all second-order, Ostrogradsky-free Lagrangians are divergence equiv-alent to rst order elds using only a total derivative, i.e. without a transformation to the(in)dependent coordinates. We have discussed this in section 9 using theorem 6. Since we wantto check this result using Cartan's algorithm, we can restrict the structure group to not includeany transformation on the dependent coordinates. A positive result in this restricted case, isstill valid for the unrestricted case. In our restriction, the only allowed transformations aretransformations purely on the independent coordinate and the addition of a total derivative.After this restriction, the structure group matrix is given by
H =
δαβ 0 0 0 0
0δαβj
0 0 0
0 0 j 0 0
0 0 0δαβj2
0
(q1)β (q2)β j − 1 0 1
. (174)
The inverse transformation is also required to calculate the dierentials in the proper co-frame.It is given by
H−1 =
δαβ 0 0 0 00 jδαβ 0 0 00 0 1
j0 0
0 0 0 j2δαβ 0
− (q1)β −j (q2)βj−1j
0 1
.
88
Next we need the dierentials. The total dierential of a function F is given by
dF (x, uα, uα, uα) = Fuα θα + jLFuα θαx +
DxF
jLω + j2L2Fuαπ
α
and the dierential dΣ is given by
dθα = −θαx ∧ ω1
Ldθαx = jω ∧ πα
dω =dLL∧ ω
=dLjL∧ ω
=LuαjL
θα ∧ ω + Luα θαx ∧ ω − jLLuαω ∧ πα
dπα = −2j2LuβLθβ ∧ πα − 2j3Luβ θβx ∧ πα − 2j
DxLL2
ω ∧ πα − 2j4LLuβ πβ ∧ πα
= −2j3 dLL∧ πα
The rst part of dΣ is relatively easy to compute:
dθα = δαβ dθβ = −θαx ∧ ω
dθαx = δαβ d1
jLθβx
= −1
jdj ∧ θαx −
1
LdL ∧ θαx + ω ∧ πα
= −ι ∧ θαx + ω ∧ πα
dω = djω
= dj ∧ ω + j dω
= ι ∧ ω
dπα = δαβ d1
j2πβ
= − 2
j3dj ∧ πα +
1
j2dπα
= −2ι ∧ πα,
where we use the substitution ι = djj
+ dLL . The last dierential is given by
dv = d(
(q1)β θβ)
+ d
((q2)β
1
Lθβx
)+ d ((j − 1)ω) + dv
= d (q1)β ∧ θβ + (q1)β dθβ + d (q2)β ∧
1
Lθβx + (q2)β d
(1
Lθβx
)+ dj ∧ ω + (j − 1) dω
= (ξ1)α ∧ θα + (ξ2)α ∧ θ
αx + ι ∧ ω + j [(q2)α + LLuα ] ω ∧ πα
These equations give 1 set of essential torsion coecients to be normalized: j [(q2)α + LLuα ] ω∧ πα.We normalize to zero by xing (q2)α = −LLuα
89
With this normalization, we enter the second round. Since this choice of group parame-ters does not aect the rst 4 groups of coframe elements, we only need to consider the lastcoecient. Again, we have
dv = d(
(q1)β θβ)− d
(Luβθβx
)+ d ((j − 1)ω) + dv
= d (q1)β ∧ θβ + (q1)β dθβ − dLuβ ∧ θβx − Luβ dθβx + dj ∧ ω + j dω − dω
= (ξ1)α ∧ θα + ι ∧ ω − [(q1)α + Luα −DxLuα ] θαx ∧ ω
Again, there is only one set of coecients to normalize, − [(q1)α + Luα −DxLuα ] θαx ∧ω. Thiscan be normalized to zero by choosing (q1)β = DxLuβ − Luβ
We now need to compute dv for the third time.
dv = d((DxLuβ − Luβ) θβ
)− d
(Luβθβx
)+ d ((j − 1)ω) + dv
= dDxLuβ ∧ θβ − dLuβ ∧ θβ + (DxLuβ − Luβ) dθβ − dLuβ ∧ θβx − Luβ dθβx + dj ∧ ω + j dω − dω
= ι ∧ ω + (DxLu[βuα] − Lu[βuα]) θα ∧ θβ + jL (Luαuβ − Luβuα −DxLuαuβ − Luαuβ) θα ∧ θβx
− DxDxLuβ −DxLuβ + LuβjL
θβ ∧ ω
This calculation leaves us with three sets of essential torsion coecients. (DxLu[βuα] − Lu[βuα]) θα∧θβ can not be normalized, since it does not contain any group parameters. As such, it isa fundamental invariant of the system. jL (Luαuβ − Luβuα −DxLuαuβ − Luαuβ) θα ∧ θβx canbe normalized. We can not normalize to 0, since j = 0 is not allowed. Normalization toδαβ is also not possible, since we have only 1 parameter to x. Thus, instead normalize to
(Luαuβ − Luβuα −DxLuαuβ − Luαuβ) θα ∧ θβx by xing j = L−1. The last torsion coecient isDxDxLuβ−DxLuβ+L
uβ
jL θβ∧ω = EL (L)uβ θβ∧ω28, i.e. the Euler-Lagrange equations for this system.
The Euler-Lagrange equations can thus also not be normalized, and thus are also fundamentalinvariants of the system.
With j thus chosen, all group parameters are xed and the nal structure group matrixlooks like
H =
δαβ 0 0 0 00 Lδαβ 0 0 00 0 L−1 0 00 0 0 L2δαβ 0
DxLuβ − Luβ −LLuα L−1 − 1 0 1
. (175)
Let us compute the dierentials of our invariant coframe.
dθα = dθα = −θαx ∧ ω,
dθαx = d
(L 1
Lθαx
)= dθαx = ω ∧ πα,
dω = d(L−1ω
)= d
(L−1L dx
)= 0,
dπα = d(L2πα
)= d
(L2L−2 duα
)= 0,
28In this step, I have both used the normalization j = L−1, and that for the E-L equations the truncateddierential and the regular total dierential match-up.
90
and the last dierential is
dv = d((DxLuβ − Luβ) θβ
)− d
(Luβθβx
)+ d
((L−1 − 1
)ω)
+ dv
= (DxLuβuα − Luβuα) θα ∧ θβ + (Luαuβ − Luβuα −DxLuαuβ − Luαuβ) θα ∧ θβx− (DxDxLuβ −DxLuβ + Luβ) θβ ∧ ω
We thus have two constant torsion coecients; T θα
θαxω= −1 and T
θαxωπα = 1; and three sets of
non-constant torsion coecients; T vθαθβ
= (DxLuβuα − Luβuα), T vθαθβx
= (Luαuβ − Luβuα −DxLuαuβ − Luαuβ)
and T vθαω = − (DxDxLuα −DxLuα + Luα); on our invariant coframe. Since the wedge prod-uct anti-commutes, the T v
θαθβonly yield 1
2q (q − 1) independent coecients. Similarly, due to
the symmetric nature of T vθαθβx
, this coecient only gives us another 12q (q + 1) coecients.
The Euler-Lagrange equations contribute another q invariants. Together, the coecients giveq2 + q + 1 independent coecients, and therefore the zeroth order classifying set has an orderof (at most) q2 + q + 1. Thus, the zeroth order classifying manifold is given by
T θ
α
θαxω, T vθαθβ , T
v
θαθβx, T vθαω
=
−1, (DxLuβuα − Luβuα) , (Luαuβ − Luβuα −DxLuαuβ − Luαuβ) ,− (DxDxLuα −DxLuα + Luα)
Since we work on J2 (R,Rq), there can be no more than dim J2 (R,Rq) = 3q+2 independentcoecients. Therefore, for q > 3, the zeroth classifying set is already 'large enough' to haveenough invariants. Still, we need to calculate the derivatives of our coframe elements and ndthe rst order classifying set. To calculate this set, rst we again need the total dierential:
dF (x, uα, uα, uα) = Fuα θα + Fuα θ
αx +DxFω + Fuαπ
α
Using this dierential, the derivatives of T vθαθβ
can be found to be
dT vθαθβ = d (DxLuβuα − Luβuα)
= (DxLuβuα − Luβuα)uγ θγ + (DxLuβuα − Luβuα)uγ θ
γx
+Dx (DxLuβuα − Luβuα) ω + (DxLuβuα − Luβuα)uγ πγ
= (DxLuβuαuγ − Luβuαuγ ) θγ + (DxLuβ uγuα + Luβuγuα − Luβ uγuα) θγx+ (DxDxLuβuα −DxLuβuα) ω + (Luβ uγuα − Luγ uβuα) πγ
= (DxLuβuαuγ − Luβuαuγ ) θγ + (DxLuβ uγuα + Luβuγuα − Luβ uγuα) θγx + (DxDxLuβuα −DxLuβuα) ω
=T vθαθβ ;θγ θγ + T vθαθβ ;θγx
θγx + T vθαθβ ;ωω + T vθαθβ ;πγ πγ + T vθαθβ ;vv
Similarly, the derivatives of T vθαθβx
can be computed as
dT vθαθβx
= d (Luαuβ −DxLuαuβ − Lu(αuβ))= (Luαuβ − Luβuα −DxLuαuβ − Luαuβ)uγ θ
γ + (Luαuβ − Luβuα −DxLuαuβ − Luαuβ)uγ θγx
+Dx (Luαuβ − Luβuα −DxLuαuβ − Luαuβ) ω + (Luαuβ − Luβuα −DxLuαuβ − Luαuβ)uγ πγ
= (Luαuβuγ − Luβuαuγ −DxLuαuβuγ − Luαuβuγ ) θγ
+ (Luαuβ uγ − Luβ uγuα −DxLuαuβ uγ − Luαuβuγ − Luαuγuβ) θγx+ (DxLuαuβ −DxLuβuα −DxDxLuαuβ −DxLuαuβ) ω + (Luγ uαuβ − Luαuβ uγ ) πγ
= (Luαuβuγ − Luβuαuγ −DxLuαuβuγ − Luαuβuγ ) θγ
+ (Luαuβ uγ −DxLuαuβ uγ − Luαuβuγ − Luβ uγuα − Luγ uαuβ) θγx+ (DxLuαuβ −DxLuβuα −DxDxLuαuβ −DxLuαuβ) ω
=T vθαθβx ;θγ
θγ + T vθαθβx ;θγx
θγx + T vθαθβx ;ω
ω + T vθαθβx ;πγ
πγ + T vθαθβx ;v
v
91
Last is the dierential of the Euler-Lagrange equations,
dT vθαω =− d (DxDxLuα −DxLuα + Luα)
=− (DxDxLuα −DxLuα + Luα)uγ θγ − (DxDxLuα −DxLuα + Luα)uγ θ
γx
−Dx (DxDxLuα −DxLuα + Luα) ω − (DxDxLuα −DxLuα + Luα)uγ πγ
=− (DxDxLuαuγ −DxLuαuγ + Luαuγ ) θγ − (DxDxLuαuγ + 2DxLuαuγ −DxLuαuγ − Lu[αuγ]) θγx− (DxDxDxLuα −DxDxLuα +DxLuα) ω − (DxLuαuγ − Luαuγ + Lu(αuγ)) πγ
=T vθαω;θγ θγ + T vθαω;θγx
θγx + T vθαω;ωω + T vθαω;πγ πγ + T vθαω;vv
The total number of non-zero rst-order torsion coecients is q (2q2 + 4q + 1). Combinedwith the zeroth order invariants, this is larger than 3q + 2. Therefore, there must be functionaldependency between these coecients. Since the functional dependence of the invariants is alsoinvariant [6, Chapter 8], any equivalent Lagrangian must have the same functional dependencebetween invariants.
First, we can see that T vθαθβx
= T vθαω;πβ
. A second relation is
T vθβθαx ;ω + T vθ[αθβ] − Tv
θαω;θβx= (DxLuβ uα −DxLuαuβ −DxDxLuβ uα −DxLuβuα) + (DxLuβuα − Luβuα)
− (DxLuαuβ − Luαuβ) + (DxDxLuαuβ + 2DxLuαuβ −DxLuαuβ − Lu[αuβ])=DxLuβ uα −DxLuαuβ −DxDxLuβ uα −DxLuβuα +DxLuβuα − Luβuα−DxLuαuβ + Luαuβ +DxDxLuαuβ + 2DxLuαuβ −DxLuαuβ − Lu[αuβ]
=0
The third relation is T vθαθβ ;θγx
= T vθβθγx ;θα
.From this we have the rst order classifying manifoldT θ
α
θαxω, T vθαθβ , T
v
θαθβx, T vθαω, T
vθαθβ ;θγ , T
vθαθβ ;θγx
, T vθαθβ ;ω, Tv
θαθβx ;θγx, T v
θαθβx ;ω, T vθαω;θγ , T
vθαω;ω
=−1, (DxLuβuα − Luβuα) , (Luαuβ − Luβuα −DxLuαuβ − Luαuβ) ,− (DxDxLuα −DxLuα + Luα) ,
(DxLuβuαuγ − Luβuαuγ ) , (DxLuβ uγuα + Luβuγuα − Luβ uγuα) , (DxDxLuβuα −DxLuβuα) ,
(Luαuβ uγ −DxLuαuβ uγ − Luαuβuγ − Luβ uγuα − Luγ uαuβ) , (DxLuαuβ −DxLuβuα −DxDxLuαuβ −DxLuαuβ) ,
− (DxDxLuαuγ −DxLuαuγ + Luαuγ ) ,− (DxDxDxLuα −DxDxLuα +DxLuα).
Of course, just having the invariant coframe and its invariants is not enough, we need tocompare it to the invariants that belong to a rst order Lagrangian. We set up the samecoframe and start with the same structure group.
H =
δαβ 0 0 0 0
0δαβj
0 0 0
0 0 j 0 0
0 0 0δαβj2
0
(q1)β (q2)β j − 1 0 1
. (176)
The total dierential of a function F not depending on uα is given by
dF (x, uα, uα) = Fuα θα + jLFuα θαx +
DxF
jLω
92
and the dierential dΣ is given by
dθα = −θαx ∧ ω1
Ldθαx = jω ∧ πα
dω =dLL∧ ω
=dLjL∧ ω
=LuαjL
θα ∧ ω + Luα θαx ∧ ω
dπα = −2dLLπα
= −2j2 dLLπα
= −2j2LuβLθβ ∧ πα − 2j3Luβ θβx ∧ πα − 2j
DxLL2
ω ∧ πα
The rst set of dierentials in dΣ remains unchanged:
dθα = δαβ dθβ = −θαx ∧ ω
dθαx = δαβ d1
jLθβx
= −1
jdj ∧ θαx −
1
LdL ∧ θαx + ω ∧ πα
= −ι ∧ θαx + ω ∧ πα
dω = djω
= dj ∧ ω + j dω
= ι ∧ ω
dπα = δαβ d1
j2πβ
= − 2
j3dj ∧ πα +
1
j2dπα
= −2ι ∧ πα
The last dierential is a dierent though:
dv = d(
(q1)β θβ)
+ d
((q2)β
1
Lθβx
)+ d ((j − 1)ω) + dv
= d (q1)β ∧ θβ + (q1)β dθβ + d (q2)β ∧
1
Lθβx − (q2)β
dLL2∧ θβx + (q2)β
1
Ldθβx + dj ∧ ω + j dω − dω
= (ξ1)α ∧ θα + (ξ2)α ∧ θ
αx + ι ∧ ω + j (q2)α ω ∧ π
α
The non-zero, non-constant torsion coecient T vωπβ
can be normalized to zero by choosing(q2)β = 0.
Since xing (q2)β does not aect the other dierentials, we only need to recompute dv.
dv = d(
(q1)β θβ)
+ d ((j − 1)ω) + dv
= d (q1)β ∧ θβ + (q1)β dθβ + dj ∧ ω + j dω − dω
= (ξ1)α ∧ θα + ι ∧ ω − ((q1)α + Luα) θαx ∧ ω
93
We again have 1 non-constant torsion coecient, T vθβxω
, and this is normalized to zero by choosing
(q1)β = −Luβ .Again, we only need to recompute dv:
dv = − d(Luβθβ
)+ d ((j − 1)ω) + dv
= − dLuβ ∧ θβ − Luβ dθβ + dj ∧ ω + j dω − dω
= ι ∧ ω − Luβuα θα ∧ θβ + jLLuβ uα θβ ∧ θαx +DxLuα − Luα
jLθα ∧ ω
We are in a similar situation as with the second-order Lagrangians. From these torsioncoecients, we choose the same normalization for j as we did for the second-order Lagrangians,i.e. j = L−1.
With this, we nd the following structure group
H =
δαβ 0 0 0 00 Lδαβ 0 0 00 0 L−1 0 00 0 0 L2δαβ 0−Luβ 0 L−1 − 1 0 1
. (177)
Let us compute the dierentials of our invariant coframe.
dθα = dθα = −θαx ∧ ω,
dθαx = d
(L 1
Lθαx
)= dθαx = ω ∧ πα,
dω = d(L−1ω
)= d
(L−1L dx
)= 0,
dπα = d(L2πα
)= d
(L2L−2 duα
)= 0,
and the last dierential is
dv = − d(Luβθβ
)+ d
((L−1 − 1
)ω)
+ dv
= − dLuβ ∧ θβ − Luβ dθβ − L−2 dL ∧ ω + L−1 dω − dω
= −Luβuα θα ∧ θβ + Luβ uα θβ ∧ θαx + (DxLuα − Luα) θα ∧ ω
In total, this gives us the same number of invariants as the second-order case did, from thesame sources. The zeroth order classifying manifold is thus given by:
T θα
θαxω, T vθαθβ , T
v
θαθβx, T vθαω
= −1,−Luβuα ,Luαuβ , DxLuα − Luα
To compare these rst-order with the second order Lagrangians, we also need to computethe rst order classifying manifold. For that, we rst need the dierential:
dF (x, uα, uα, uα) = Fuα θα + Fuα θ
αx +DxFω + Fuαπ
α
Using this dierential, the derivatives of T vθαθβ
can be found to be
dT vθαθβ = − dLuβuα= − (Luβuα)uγ θ
γ − (Luβuα)uγ θγx −DxLuβuαω − (Luβuα)uγ π
γ
= −Luβuαuγ θγ − Luβ uγuα θγx −DxLuβuαω= T vθαθβ ;θγ θ
γ + T vθαθβ ;θγxθγx + T vθαθβ ;ωω + T vθαθβ ;πγ π
γ + T vθαθβ ;vv
94
Similarly, the derivatives of T vθαθβx
can be computed as
dT vθαθβx
= dLuαuβ= (Luαuβ)uγ θ
γ + (Luαuβ)uγ θγx +DxLuαuβ ω + (Luαuβ)uγ π
γ
= Luαuβuγ θγ + Luαuβ uγ θγx +DxLuαuβ ω= T v
θαθβx ;θγθγ + T v
θαθβx ;θγxθγx + T v
θαθβx ;ωω + T v
θαθβx ;πγπγ + T v
θαθβx ;vv
Last is the dierential of the Euler-Lagrange equations,
dT vθαω = d (DxLuα − Luα)
= (DxLuα − Luα)uγ θγ + (DxLuα − Luα)uγ θ
γx +Dx (DxLuα − Luα) ω + (DxLuα − Luα)uγ π
γ
= (DxLuαuγ − Luαuγ ) θγ + (DxLuαuγ + Lu[αuγ]) θγx + (DxDxLuα −DxLuα) ω + Luαuγ πγ
= T vθαω;θγ θγ + T vθαω;θγx
θγx + T vθαω;ωω + T vθαω;πγ πγ + T vθαω;vv
Part of comparing these dierentials with those from the second order system, is to try andnd the same functional relations between them. Again, we have T v
θαω;πβ= T v
θαθβx, and also
T vθβθαx ;ω
+ T vθ[αθβ]
− T vθαω;θβx
= 0. The third relation is again T vθαθβ ;θγx
= T vθβθγx ;θα
, and these are all
relations.Again, we have the rst-order classifying manifold.
T θα
θαxω, T vθαθβ , T
v
θαθβx, T vθαω, T
vθαθβ ;θγ , T
vθαθβ ;θγx
, T vθαθβ ;ω, Tv
θαθβx ;θγx, T v
θαθβx ;ω, T vθαω;θγ , T
vθαω;ω
=−1,−Luβuα ,Luαuβ , DxLuα − Luα ,−Luβuαuγ ,−Luβ uγuα ,−DxLuβuα ,Luαuβ uγ , DxLuαuβ , (DxLuαuγ − Luαuγ ) , (DxDxLuα −DxLuα)
We can start comparing these coecients to those generated from the second order La-grangians. Referring to the rst-order Lagrangians with L from now on, we have the followingconditions for equivalence:
T vθαθβ = T vθαθβ ⇔ (178)
DxLuβuα − Luβuα = −Luβuα ,
T vθαθβx
= T vθαθβx⇔ (179)
Luαuβ −DxLuαuβ − Lu(αuβ) = Luβ uα ,
and
T vθαxω = T vθαxω ⇔ (180)
DxDxLuα −DxLuα + Luα = −DxLuα + Luα .
This last conditions states that the Euler-Lagrange equations must overlap, as we expectedand required from the start.
We want to see if a second-order, Ostrogradsky-free Lagrangian is divergent equivalent to aregular rst order Lagrangian. To check this, we calculated the invariants listed above, and nowwe need to see if both theories have the same invariants. Assume that we have such a secondorder Lagrangian L and a rst order Lagrangian L that are equivalent, i.e. ∃ F = F (x, uα, uα)s.t. L = L+DxF
29. Compute the left half of (178),
29Since F does not depend on uα, the truncated and full total derivatives are equivalent.
95
T vθαθβ = DxLuβuα − Luβuα= Dx
(L+DxF
)uβuα−(L+DxF
)uβuα
= Dx (DxF )uβuα − Luβuα − (DxF )uβuα
= Dx (Fuβuα)− (DxFuβ + Fuβ)uα − Luβuα= −Fuβuα − Luβuα= −Fuβuα + T vθαθβ .
This shows that the function F must satisfy Fuβuα = 0 for equivalence.Next, we compare (179),
T vθαθβx
= Luαuβ −DxLuαuβ − Lu(αuβ)=(L+DxF
)uαuβ−Dx
(L+DxF
)uαuβ−(L+DxF
)u(αuβ)
= Luαuβ + (DxF )uαuβ −Dx (DxF )uαuβ − (DxF )u(αuβ)
= Luαuβ + (DxFuα + Fuα)uβ −Dx (Fuα)uβ − (Fu(α)uβ)
= Luαuβ +DxFuαuβ + Fuαuβ + Fuβuα −DxFuαuβ − Fu(αuβ)= Luαuβ= T v
θαθβx
This calculation yields no new conditions on F .Last, we need to compare the Euler-Lagrange equations, i.e. equations (180):
T vθαxω = DxDxLuα −DxLuα + Luα= DxDx
(L+DxF
)uα−Dx
(L+DxF
)uα
+(L+DxF
)uα
= DxDxFuα −DxLuα −Dx (DxFuα + Fuα) + Luα +DxFuα
= −DxLuα + Luα= T vθαxω,
as expected.Since these invariants already match-up everywhere, there is no need to also compare the
derivatives of these invariants and we can instead conclude that all second-order, Ostrogradsky-free Lagrangians are equivalent to a rst-order Lagrangian plus a total derivative.
96
12 Conclusion
This goal of this thesis was to answer the question: Do the conditions that free a higher orderLagrangian of Ostrogradsky ghosts force such a Lagrangian to be divergence equivalent to arst-order Lagrangian? In trying to answer this question, we also discussed the subquestions
• What types of Ostrogradsky stable theories are there?
• What are the properties of these theories?
• What are the conditions to call two Lagrangians divergence equivalent?
• How can you determine such equivalence?
We found that there are seval types of Ostrogradsky stable theories. First, we can split themup into classical mechanical and eld theories. Second, the eld theories can be split up,depending on Lorentz (in)variance. Last, all theories can be split up depending on the presenceof rst-order helper elds.
Within the Lorentz invariant eld theories, Galileons are the most interesting theory. Es-pecially when combined with gravity, they allow for interesting modications to gravity. Wediscussed these couplings in chapter 7. A more general coupling of an Ostrogradsky-free second-order scalar to gravity leads to Beyond-Horndeski theories, which is an attempt to nd the mostgeneral Ostrogradsky-free tensor-scalar theory. Tensorial General Relativity itself is also an Os-trogradsky free theory, as we found in section 6.3.
After having discussed these types of Ostrogradsky free problems, we dened the conditionsunder which two Lagrangians are divergence equivalent. To be able to dene this equivalence,we discussed the theory of Jet spaces and contact transformations in chapter 8. In the end, weused the denition in section 8.4, which states that two Lagrangians are divergence equivalentif and only if there exists a contact transformation Φ : J2 (Rp,Rq)→ J2 (Rp,Rq) and an p-tupleF : J2 (Rp,Rq)→ Rp that relates the two Lagrangian under
L(xi, u(r)
)=L(xi, u(r)
)+∇F
| det J |, (127)
where (J)ij = dφi
dxj.
We applied this denition to several Galileon theories in section 9, using the work by Crisos-tomi et al. [8], before we expaned to try and nd a more general answer. To this end, we foundand explained the Cartan Equivalence algorithm in chapter 10, which can be used to answerthe main research question. In chapter 11 we applied Cartan's algorithm to classical mechanics,with an arbitrary number of second-order variables. I was unable to apply Cartan's algorithmto classical mechanics with helper elds, and eld theories altogether due to time constraintsand the additional complexity. As a result, the main research question can not be answeredin full generality. Using the results from chapters 11 and 9 we can formulate the followingconclusion:
Question Do the conditions that free a higher order Lagrangian of Ostrogradsky ghosts force sucha Lagrangian to be divergence equivalent to a rst-order Lagrangian?
Answer For theories with only second-order variables, the conditions that remove the Ostro-gradsky ghosts also force such a Lagrangian to be divergence equivalent to a rst-orderLagrangian. Field theories with only one second-order variable are also always equiva-lent, independent on the presence of rst-order elds. For theories with general numberof second-order and rst-order elds, we were unable to draw any conclusions.
97
As I was unable to fully answer the research question, there still is some further work tpdetermine equivalence for the unanswered cases. Furthermore, the conditions for Ostrogradsky-free theories have only been formulated for scalar elds. These conditions can be generalized tospinor, vector and tensor elds. With these conditions then known, we can then consider theequivalence of these Ostrogradsky-free theories to rst-order theories, be it through Cartan'salgorithm or through some other method.
98
A Notation
Throughout this document, several methods of notation are used. Below, I have tried to listthem all to give an overview.
A.1 Summation conventions
Throughouth this thesis, we use the Einstein summation convention, and thus summation overrepeated indices is implied:
aibi =
∑i
aibi.
Futhermore, we use [i, j] to refer to the anti-symmetric sum over the indices i and j, while (i, j)refers to the symmetric sum. for example
S[i,j] = Si,j − Sj,i
andS(i,j) = Si,j + Sj,i.
A.2 Jet Bundles
The notation in the table below refers to notation used in reference to jet bundles, and contactforms thereon. The notation is primarily taken from Ivey and Landsberg in [7] and Olver in [6].
xi The independent coordinates. In general, there are p of them [42, page371]. When discussing eld theories, the notation xµ is also used.
∂i The partial derivative w.r.t. xi.2 = ∂i∂
i The second order derivative.uα The dependent coordinates. In general, there are q of them [42, page
371]. In the physics chapters 2-9 also referred to as φa or qα.uαi = ∂uα
∂xiThe rst derivatives of the dependent coordinates to the independentcoordinates.
I = i1, i2, . . . , is A multi index of order s. Possibly un-ordered.uαI = ∂suα
∂xi1 · ∂xi2 · ... · ∂xis The |I| = sth order derivative of uα w.r.t. xI = xi1 , xi2 , . . . , xis [42,page 371].
us The set of all derivatives of all the functions uα of order exactly s.u(s) The set of all derivatives of all the functions uα of order up to and
including s, i.e. u(s) = u0, u1, . . . , us.Υ (X) The space of all curves over the domain X.
Jk (M,N) The kth jet-bundle from M to N . Sometimes shortened to Jk if it isobvious what M and N are.
I The contact ideal on a jet space Jk (M,N). Its exact form dependson k, M and N [7, Appendix B.4].
[ω] = ω + J The equivalence class of a dierential form ω modulo some dierentialideal J .
A.3 Geometry
The notation below refers to notation used in the sections dealing with gravity and geometry,and is the same as used by Horndeski in [40], unless otherwise specied.
99
δi1...ihj1...jh= det
∣∣∣∣∣∣∣δi1j1 . . . δi1jh...
. . ....
δihj1 . . . δihjh
∣∣∣∣∣∣∣ The generalised Kronecker delta.
gµν The metric tensor in regular (3+1)-dimensional space.g = | det gµν | The determinant of the metric.
ηµν The Minkowski metric in (3+1)-dimensions, I use amostly-positive signature.
GAB The metric tensor in (3 +N + 1) dimensions.G = | detGAB| The determinant of this metric.
ηAB The Minkowski metric in (3 +N + 1) dimensions.Γ ijk = 1
2gih (∂kgjh + ∂jgkh − ∂hgjk) The Christoel symbols for a general metric.
∇i The covariant derivative w.r.t xi and the Christoel sym-bols dened above.
R νµ ρσ The Riemann tensor.
Rµν = R ρµ νρ The Ricci-tensor, derived from the Riemann tensor.
R = Rµµ The Ricci-scalar, the trace of the Ricci-tensor.
Gµν = Rµν − 12gµνR The Einstein Tensor.
100
B Variational Problems
The calculus of variations is concerned with the extremals of functions, whose domain is theinnite-dimensional space of curves. Such functions are called functionals [10]. To explain thissome more, let us consider Rp. Then let Υ (Rp) denote the space of all curves over Rp. Afunctional Φ is a map Φ : Υ (Rp) → R. Note that the value of Φ only depends on the entirecurve f ∈ Υ (Rp), it does not depend on any local coordinates x ∈ Rp.
Example 16. One example of a functional is the length of a curve in the Euclidean plane. Letγ = (t, x, y) |x(t) = x, y(t) = y, ti < t < tf describe a curve30. Furthermore, let x = dx/dtand y = dy/dt. Then the functional that describes the length of this curve is given by
Φ (γ) =
∫ tf
ti
√x2 + y2 dt.
To consider extremes of a functional, like we do when we consider extremes of a regu-lar function, we need to know what happens to the value of the functional if we make asmall change to the input arguments. Let us consider an 'approximation' γ′ to γ, γ′ =(t, xi) : xi = xi(t) + hi(t). Let us call it γ′ = γ + h, with the requirement that h (ti) =h (tf ) = 0. Consider the change in Φ, ∆Φ = Φ (γ + h) − Φ (γ). We want this change to meetcertain conditions, if we want so speak of extremes of a functional. To be exact, we look atArnol'd [10]
Denition 17. A functional Φ is called dierentiable if
∆Φ = Φ (γ + h)− Φ (γ) = F +R,
where F depends linearly on h and R (h, γ) = O (h2) [10, page 56].
The function F can be called the dierential of Φ at γ, in the same way one denes thederivative of a regular function.
Next, we can dene what it means for a functional to have an extreme.
Denition 18. An extremal of a dierentiable functional Φ (γ) is a curve γ such that F (h) = 0for all functions h [10, page 57].
Again, this is similar to the denition of the extreme of a function, where we say a functionobtains an extremal point at those points where df
dx= 0.
For a functional of the form Φ (γ) =∫L (x, x, t) dt, the derivative is given by
F (h) =
∫ [∂L
∂x− d
dt
∂L
∂x
]h dt+
(∂L
∂xh
),
and an extremal is reached (F (h) = 0∀h) when [10, page 57]
∂L
∂x− d
dt
∂L
∂x= 0.
This is exactly the Euler-Lagrange equation (7) for the case p = q = r = 1. For derivation ofthe Euler-Lagrange equations for higher-order Lagrangians (r > 1), I refer to [11], or to chapter2 for a less rigorous derivation.
30This curve technically does not exist in the plane, but in 3-space. The curve in the xy-plane is the projectionof the above curve to the xy-plane. If γ behaves nice enough, the projection γ2 is itself a curve that can be
described as γ2 = (x, y) |y(x) = y, xi < x < xf and then Φ2 (γ2) =∫ xf
xi
√1 + (dy/dx)
2dx = Φ (γ).
101
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