The Physics of Scales - uni-heidelberg.de · The Physics of Scales Björn M. Schäfer, Matthias Bartelmann Zentrum für Astronomie ... 6.1.3 Static, spherically-symmetric gravitational

The Physics of Scales

Björn M. Schäfer, Matthias BartelmannZentrum für AstronomieUniversität Heidelberg

ii

Contents

1 Lagrangian classical mechanics 3

1.1 classical mechanics - universality . . . . . . . . . . . . 3

1.2 scales of modern physics . . . . . . . . . . . . . . . . 4

1.3 Lagrange formalism . . . . . . . . . . . . . . . . . . . 5

1.4 Constants of motion . . . . . . . . . . . . . . . . . . . 5

1.5 Lagrangian invariants . . . . . . . . . . . . . . . . . . 6

1.5.1 mechanical similarity . . . . . . . . . . . . . . 6

1.5.2 Lagrangian gauge transformation . . . . . . . 6

1.6 Kepler’s problem . . . . . . . . . . . . . . . . . . . . 7

1.6.1 Kepler’s laws . . . . . . . . . . . . . . . . . . 7

1.6.2 gravitationally bound systems . . . . . . . . . 7

1.6.3 planetary orbits . . . . . . . . . . . . . . . . . 8

1.6.4 mechanical similarity in the Kepler-problem . . 10

1.7 summary . . . . . . . . . . . . . . . . . . . . . . . . 11

2 The Speed of Light and the Lorentz Transform 13

2.1 The space-time structure of classical mechanics . . . . 13

2.1.1 The Galilei group . . . . . . . . . . . . . . . . 13

2.1.2 Affine spaces and Galilei space-time . . . . . . 14

2.2 The space-time structure of special relativity . . . . . . 16

2.2.1 Electrodynamics and classical mechanics . . . 16

2.2.2 The Lorentz transform . . . . . . . . . . . . . 17

2.2.3 Some properties of the Lorentz group . . . . . 20

iii

iv CONTENTS

2.2.4 Minkowski space-time . . . . . . . . . . . . . 21

3 Electrodynamics, Gauge Theories, and Scales 23

3.1 Electrodynamics . . . . . . . . . . . . . . . . . . . . 23

3.1.1 The field tensor . . . . . . . . . . . . . . . . . 23

3.1.2 The Lagrange density of electrodynamics andMaxwell’s equations . . . . . . . . . . . . . . 24

3.1.3 Scales in electrodynamics . . . . . . . . . . . 25

3.2 Gauge theory . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 Local gauge invariance . . . . . . . . . . . . . 28

3.2.2 Mass-less gauge fields and the Higgs mechanism 31

4 Quantum Mechanics I 33

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 The Photoelectric effect . . . . . . . . . . . . . . . . . 34

4.3 de Broglie waves . . . . . . . . . . . . . . . . . . . . 35

4.3.1 Velocity of a wave . . . . . . . . . . . . . . . 35

4.3.2 Dispersion . . . . . . . . . . . . . . . . . . . 36

4.3.3 Wave packets . . . . . . . . . . . . . . . . . . 36

4.3.4 Dispersion of a Gaussian classical wave packet 37

4.4 Hamiltonian mechanics . . . . . . . . . . . . . . . . . 39

4.4.1 Lagrangian equations of motion . . . . . . . . 39

4.4.2 Hamiltonian equations of motion . . . . . . . 39

4.4.3 Poisson brackets . . . . . . . . . . . . . . . . 40

4.4.4 Poisson brackets and commutators . . . . . . . 40

4.4.5 Constants of motion . . . . . . . . . . . . . . 41

4.4.6 Ehrenfest theorem . . . . . . . . . . . . . . . 41

5 Quantum Mechanics II 43

5.1 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . 43

5.1.1 General uncertainty relation . . . . . . . . . . 43

5.1.2 Position and momentum . . . . . . . . . . . . 45

CONTENTS v

5.1.3 Energy and time . . . . . . . . . . . . . . . . 45

5.2 Path integrals . . . . . . . . . . . . . . . . . . . . . . 45

5.2.1 Propagators . . . . . . . . . . . . . . . . . . . 46

5.2.2 Marginalisation and the propagator group . . . 47

5.2.3 Phase space path integrals . . . . . . . . . . . 48

5.2.4 Recovery of Lagrange . . . . . . . . . . . . . 49

5.2.5 configuration space path integrals . . . . . . . 49

5.2.6 Huygens principle and probability waves . . . 50

5.3 summary . . . . . . . . . . . . . . . . . . . . . . . . 50

6 The gravitational field of a point mass 53

6.1 Fundamental ideas of general relativity . . . . . . . . . 53

6.1.1 The meaning of the metric . . . . . . . . . . . 53

6.1.2 The space-time structure of general relativity . 54

6.1.3 Static, spherically-symmetric gravitational fields 55

6.1.4 Newtonian gravity as a limit to general relativity 56

6.2 Relativistic dynamics . . . . . . . . . . . . . . . . . . 57

6.2.1 Derivation from the effective relativistic action 57

6.2.2 Comparison with the Newtonian case . . . . . 59

6.2.3 Perihelion shift . . . . . . . . . . . . . . . . . 60

7 Hydrostatics and stability of self-gravitating systems 63

7.1 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . 63

7.1.1 Hydrostatics in Newtonian gravity . . . . . . . 63

7.1.2 Hydrostatics in general relativity . . . . . . . . 65

7.2 Stability and scales in self-gravitating systems . . . . . 66

7.2.1 The Tolman-Oppenheimer-Volkoff equation . . 66

7.2.2 Occurrence of a scale . . . . . . . . . . . . . . 68

7.2.3 Scales and the linear approximation . . . . . . 68

8 Thermodynamics 73

vi CONTENTS

8.1 thermal energy, heat and temperature . . . . . . . . . . 73

8.1.1 heat and thermal energy . . . . . . . . . . . . 73

8.1.2 dissipation in Lagrangian systems . . . . . . . 73

8.1.3 first law of thermodynamics . . . . . . . . . . 74

8.2 measurements of temperature . . . . . . . . . . . . . . 75

8.2.1 empirical temperatures . . . . . . . . . . . . . 75

8.2.2 Carnot’s engine . . . . . . . . . . . . . . . . . 75

8.2.3 Probabilities of thermal fluctuations . . . . . . 76

8.2.4 second law of thermodynamics . . . . . . . . . 77

8.2.5 Engines as thermometers . . . . . . . . . . . . 77

9 Statistical Mechanics 79

9.1 Boltzmann-factor . . . . . . . . . . . . . . . . . . . . 79

9.2 continuum mechanics and statistics . . . . . . . . . . . 80

9.3 thermal wavelength . . . . . . . . . . . . . . . . . . . 80

9.4 particle exchange symmetry . . . . . . . . . . . . . . 81

9.5 occupation number statistics . . . . . . . . . . . . . . 82

9.6 photon number statistics . . . . . . . . . . . . . . . . 82

9.7 Planck-spectrum . . . . . . . . . . . . . . . . . . . . 84

10 Renormalisation 87

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 87

10.1.1 Partition functions in statistical physics . . . . 87

10.1.2 Block spins in the Ising model . . . . . . . . . 88

10.1.3 The renormalisation group and its flow . . . . 89

10.2 Application to path integrals . . . . . . . . . . . . . . 90

10.2.1 Path integrals in quantum mechanics . . . . . . 90

10.2.2 Path-integral formulation of field theory . . . . 92

10.2.3 The effective action . . . . . . . . . . . . . . . 94

10.2.4 Functional renormalisation . . . . . . . . . . . 95

10.2.5 The saddle-point approximation . . . . . . . . 96

CONTENTS vii

10.2.6 Wetterich’s equation . . . . . . . . . . . . . . 96

11 The Planck scales 99

11.1 values of natural constants . . . . . . . . . . . . . . . 99

11.2 Planck scales as natural units . . . . . . . . . . . . . . 100

12 Problems 103

12.1 Problem Sheet 1: Lagrangian classical mechanics . . . 104

12.2 Problem Sheet 2: The Speed of Light and the LorentzTransform . . . . . . . . . . . . . . . . . . . . . . . . 105

12.3 Problem Sheet 3: Quantum Mechanics I . . . . . . . . 106

12.4 Problem Sheet 4: Quantum Mechanics II . . . . . . . . 108

12.5 Problem Sheet 5: The gravitational field of a point mass 110

12.6 Problem Sheet 6: Hydrostatics and stability of self-gravitating systems . . . . . . . . . . . . . . . . . . . 112

12.7 Problem Sheet 7: Thermodynamics . . . . . . . . . . . 113

12.8 Problem Sheet 8: Statistical Mechanics . . . . . . . . 114

12.9 Problem Sheet 9: Renormalisation . . . . . . . . . . . 115

viii CONTENTS

CONTENTS 1

Abstract

The script Concepts and scales in modern physics summarizes the con-tent of a lecture given at the University of Heidelberg in the summer term2013. The motivation for the lecture was the observation that many stu-dents think that the essence of e.g. electrodynamics was vector calculusand that of quantum mechanics was Hilbert space algebra, while only fewpeople grasped the concept of Lorentz covariance in the construction of alinear field theory, or the emergence of probability effects on small scales.Therefore, we wanted to present a lecture on the fundamental conceptsof theoretical physics and the scales c, ~, G and kB involved, as thosescales are, in contrast to classical mechanics, essential to modern physics.Starting from scale-free classical mechanics with the Kepler problem asthe primary example, we introduce Maxwellian electrodynamics fromits construction of a Lorentz-covariant linear field theory, quantum me-chanics as a concept for describing the motion of particles if the actionis close to the scale ~, the description of gravity by general relativityand the Schwarzschild scale rS = 2GM/c2 of the gravitational field andfinally the concept of thermal energy kBT of an object of temperature T .We conclude with an explanation of renormalisation.

2 CONTENTS

Chapter 1

Lagrangian classical mechanics

1.1 classical mechanics - universality

Classical mechanics is a description the motion of particles under theinfluence of forces. It is based on Galilean kinematics for expressingthe state of motion in different systems of reference (so-called inertialframes) and Newtonian dynamics for connecting changes in the stateof motion when forces are acting on the particle. Classical mechanicsderives the equations of motion of a system from an extremal principle,the Hamiltonian principle. It does not have a scale associated with itand at its inception it was thought to be universal, meaning valid on allscales: The motion of apples and moons is governed by the same forcesand the trajectories are scaled versions of each other. For the scientistsat the time universality was a desirable feature of a theory, as it was anexpression of fundamentality.

In classical physics, constants are there to fix units, for instance inHooke’s law F = −Dr with the spring constant D in the same way as inNewton’s law F = −GmM/r2 with the gravitational constant G. Modernphysics, on the contrary, has certain principles which are based on scales,and one recovers classical universality far away from that scale. In theabove example, one would have a fundamental understanding of G fromgeneral relativity, recovering Newtonian gravity in the weak field limit.

A nice example illustrating the emergence of a scale is the Yukawapotential as opposed to the Newtonian potential. The latter follows fromthe solution of the Poisson equation

∆ϕ = 4πGρ (1.1)

for a point charge, and the factor 4π has been added for convenienceby redefining the coupling constant G (the convenience being anotherfactor of 1/(4π) in the Green’s theorem, which cancel each other). In 3d

3

4 CHAPTER 1. LAGRANGIAN CLASSICAL MECHANICS

theory scale concept modelelectrodynamics c Lorentz-covariance massless linear gauge theoryquantum mechanics ~ probability wave mechanicsgeneral relativity G varying metric nonlinear gauge theorythermodynamics kB thermal energy statistical theory of heat

Table 1.1: Summary of the four theories of modern physics with theirscale, their fundamental concept and their model formulation.

spherical coordinates we have

∆ϕ =1r2

∂

∂r

(r2∂ϕ

∂r

)= 0 (1.2)

for the Newtonian potential ϕ ∝ 1/r. It makes a lot of sense that thesolution for ϕ is a power law with no scale associated to it, because thePoisson equation itself is scale-free: Transforming r → r′ = αr definesa new Laplace operator,

∆′ =∂2

∂(r′)2 =1α2

∂2

∂r2 =1α2 ∆ (1.3)

which implies that a redefinition of the length scale can be absorbed by aredefinition of the charge/mass or by the coupling constant. In contrastthe Klein-Gordon equation (

∆ + λ2)ϕ = 0 (1.4)

does not show this behavior, as it is solved by the Yukawa-potential ϕ ∝exp(−λr)/r, which has a specific scale 1/λ, breaking the invariance r →αr. Of course, the Newtonian behavior is recoverd on small scales λr 1: Expanding the exponential exp(−λr) ' 1 − λr yields exp(−λr)/r '(1 − λr)/r ' 1/r to first order in 1/r.

1.2 scales of modern physics

Modern physics is not universal, the four cornerstones of modern physicshave scales associated with them, and we aim to explain the conceptshow these scales are built into the theories. G for instance, was knownwell before Einstein in the context of a universal theory, namely Newto-nian gravity, but was reinterpreted in general relativity as the couplingconstant of a metric field theory. Likewise, c was known as the speed oflight and showed up in Maxwell’s equations, but the Lorentz-covariantformulation of electrodynamics as a relativistic massless gauge theorycame much later. It is worth noting the difference of a concept suchas a velocity scale in transformations between inertial frames and theformulation of a field theory, for which one might have some degree of

1.3. LAGRANGE FORMALISM 5

freedom and which does not need to be unambiguously constraint by theconcepts on which it is based.

Often, a scale might have different aspects to it: Most notably, in quantummechanics ~ is the magnitude of uncertainty in simultaneous measure-ments, i.e. the scale of probability effects becoming important, but itis also the scale of dispersion of wave packets, the natural scale for theaction, the unit for angular momenta and the relation between wavenumber and momentum.

1.3 Lagrange formalism

We will review the Lagrange-description of classical mechanics anddiscuss the Kepler-problem because in this case classical universality isnicely encapsulated. The Hamiltonian principle asserts that a particlefollows the trajectory that minimizes the action function(al) S =

∫dt L,

where the Lagrange function L is a function of the position and velocityof the particle, L(x, x). Variation of the action gives

δS = δ

∫dt L(x, x) =

∫dt

(∂L∂xδx +

∂L∂xδx

)(1.5)

Noting that δx = dδxdt by interchanging variation and taking the timederivative makes it possible to carry out a partial integration:

δS =

∫dt

(∂L∂x−

ddt∂L∂x

)︸︷︷︸

=0

= 0 (1.6)

and by application of the Hamiltonian principle we identify the Euler-Lagrange equation:

∂L∂x−

ddt∂L∂x

(1.7)

If the Lagrange function L is chosen to be the difference between kineticand potential energy,

L =m2

x2 − mΦ(x) (1.8)

with the kinetic energy mx2/2 and the potential Φ variation yields exactlythe Newtonian equation of motion x = −∇Φ which is necessarily ofsecond order due to the Euler-Lagrange equation.

1.4 Constants of motion

If the time-derivative of a coordinate appears in the Lagrange functionbut not the coordinate itself, the generalized momentum

∂L∂x≡ p (1.9)


is conserved, because the time derivative in the Euler-Lagrange equationis zero. Of course, this implies momentum conservation in the absenceof potentials and is known as the Noether-theorem.

1.5 Lagrangian invariants

There are two transformations of the Lagrange function that leave theequations of motions invariant: mechanical similarity and Lagrangiangauge transformations.

1.5.1 mechanical similarity

Mechanical similarity refers to the fact that one can multiply the La-grange function with an arbitrary constant without changing the equa-tion of motion. Clearly, the equation of motion

ddt∂L∂x

=∂L∂x

(1.10)

is invariant under the transformation L→ const · L due to the linearityof the derivatives.

In fact this is the way how the equivalence principle is effectively builtinto the Lagrange density: In the expression (1.8) the mass plays twodifferent roles in the two terms making up the Lagrange density: Them in the kinetic term is expressing inertia, whereas in the second termit is a consequence of coupling to the gravitational field. Accordingto the equivalence principle the two need to be the same (well, at leastproportional to each other, but the constant of proportionality can beabsorbed in the units) or bodies of different mass would acceleratedifferently, in contradiction with Galilei’s experiments. The fact that wecan divide out m in the Lagrange function is a consequence of mechanicalsimilarity.

But mechanical similarity goes much beyond that: It shows us that thechoice of units does not matter and that there is no scale associated witha classical system. Solutions to the Euler-Lagrange equation are simplyscaled versions of each other, hence the similarity of solutions.

1.5.2 Lagrangian gauge transformation

The second invariant of the Lagrangian formalism is the addition of atotal time derivative to the Lagrange function. Transforming the action

1.6. KEPLER’S PROBLEM 7

S by addition of a total time derivative dg/dt yields

S ′ =

∫dt

(L +

dgdt

)=

∫dt L + g|t2t1 = S + ∆g (1.11)

The additional term ∆g = g(t2) − g(t1) drops out in carrying out thevariation and is of no relevance to the equation of motion. It is veryuseful, however, for describing the choice of new coordinates, whichare function of the old coordinates, and of course this redefinition of thecoordinates, known as canonical transformations, should not alter themotion of the system. One can introduce g as the generating function forthe coordinate transformation, and writing it as a total time derivative inthe action ensures that the equation of motion are unchanged.

1.6 Kepler’s problem

1.6.1 Kepler’s laws

The Kepler problem, i.e. an explanation of the orbits of planets aroundthe Sun was the decisive test of Newtonian gravity which needed toprovide a simple explanation of the three Kepler laws,

1. Planets orbit the Sun in elliptical orbits

2. The connecting line between the Sun and the planet traces outconstant areas in each time interval

3. The cubes of the semi-major axes are proportional to the squaresof the orbital period

which Kepler extracted from Brahe’s precise astrometric (well, actuallyplanetometric) data.

1.6.2 gravitationally bound systems

The gravitational potential Φ only depends on the radial distance r andnecessarily gives rise to a radial force ∇Φ, ~r×∇Φ = 0. The motion takesplace in a plane, and choosing polar coordinates (r, ϕ) for the system, theLagrange function reads:

L =m2

(r2 + r2ϕ2

)+ mΦ(r) (1.12)

Because of the spherical symmetry, the azimuthal angle (or phase angleof the orbit) ϕ is a cyclic coordinate in the Lagrange function and angular


momentum conservation applies,

~L =∂L∂ϕ

= m~r ×~v = mr2ϕ (1.13)

Furthermore, the gravitational potential is static and there is no dissipa-tion, such that the time t is the second cyclic variable and the associatedenergy ε is conserved,

ε =m2

(r2 + r2ϕ2

)+ mΦ(r) =

m2

r2 +L2

2mr2 + mΦ(r) (1.14)

where in the last step ϕ2 was replaced by the angular momentum, whichreduces the dimensionality of the problem to 1. At the same time, angularmomentum conservation as a consequence of the central potential alreadyprovides an explanation for the second law, because the area covered bythe connecting line between planet and central body is tracing out anarea A = 1

2r × v × ∆t, and if L = mvr we see that the product r × v mustbe constant, if L is conserved.

1.6.3 planetary orbits

The energy balance can be solved for the radial velocity r,

r =

√2m

[ε − mΦ(r)] −L2

m2r2 (1.15)

which can be solved by separation of variables and integration,

t =

∫dr√

2m [ε − mΦ(r)] − L2

m2r2

+ const (1.16)

Substitution of the angular momentum L = mr2ϕ → dt = mr2/L dϕgives

ϕ =

∫ Lr2 dr√

2m [ε − mΦ(r)] − L2

r2

(1.17)

from which we identify the effective potential

mΦeff(r) = mΦ(r) +L2

2mr2 (1.18)

The second term ∝ L2 is caused by the centrifugal force and allows theorbits of bound planets to be stable, as it keeps them from falling into thecentral body: At small distances it overcomes the gravitational attractionand repulses the planet if the angular momentum is not zero.

1.6. KEPLER’S PROBLEM 9

The interpretation of the motion is that the planet (if it’s bound and iftherefore ε < 0) performs oscillations in the radial distance in the effec-tive potential while orbiting the central body under angular momentumconservation. The turning points are determined by r = 0, meaning

ε = mΦeff(r) (1.19)

which has a solution in the 1/r2-dominated regime called perihel rmin,and one in the 1/r-regime referred to as aphel rmax.

Are the orbits closed? And are they ellipses?

If the planet moves in radial direction from rmin to rmax and back to rmin,the phase angle ϕ needs to change by 2π (or a rational fraction) for theorbit to be periodic:

∆ϕ2∫ rmax

rmin

Lr2 dr√

2m [E − mΦ(r)] − L2

r2

= 2πpq

(1.20)

with two integers p and q. Substituting now the Newton-potential

Φ(r) = −GM

r(1.21)

yields the solution

cosϕ =

Lr −

GmML√

2mε −(

GmML

)2(1.22)

or written differentlypr

= 1 + e cosϕ (1.23)

with the conic section parameters

p =L2

m2GMand e =

√1 + 2

ε

m

(L

(GmM)

)2

(1.24)

In summary the solution orbits to the Kepler-problem are conic sections,and therefore, the orbits of different planets need to be scaled versionsof each other. Furthermore, the solutions are closed orbits because theconic sections are circles or ellipses, if ε < 0. It is worth noting thatthe product GM never gets separated in the calculation and defines anequivalent length scale of the potential r0 = GM, such that Φ(r) = r0/r,which reflects the mass of the central body. The gravitational constant Gis there for fixing the units and of course the scale r0 does not stand outin gravitational potential. Any length scale in the problem such as rmin

and rmax involve the angular momentum L of the orbiting planet and cannot be consequences of the potential alone.


Kepler’s third law

The second law implies that 2mA = L∆t for the area A covered in time∆t by the radius vector, and the area of the ellipse would be given byA = πab with the semi-axes a and b. Those are given by

a =p

1 − e2 =GmM

2εand b =

p√

1 − e2=

L√

2mε(1.25)

(check that they’re both length scales!) implying that

∆t =2mA

L=

2π√

GMa3/2 (1.26)

i.e. ∆t2 ∝ a3 as a consequence of the 1/r-shape of the potential and itsspherical symmetry.

1.6.4 mechanical similarity in the Kepler-problem

Mechanical similarity states that you can multiply the Lagrange functionof the problem with an arbitrary constant without changing the equationof motion:

L→ const × L (1.27)

If we rescale r → αr and t → βt (or, equivalently, r ∝ 1/α and t ∝ 1/β),we see that the kinetic energy scales with α2/β2 whereas the potentialΦ ∝ 1/rn scales ∝ α−n. Applying mechanical similarity now yields thatthe two scaling factors should be equal, α2/β2 = α−n, or β−2 = α−n−2,implying that t2 ∝ rn+2, i.e. Kepler’s third law for the Newtonian casen = 1.

The remarkable thing here is that of course this is much faster than anexplicit solution of the Kepler-problem, and that we obtained this scalingrelation without knowing at all what the trajectory of the planet is. Infact, the third Kepler law also applies to parabolic (ε = 0) and hyperbolic(ε > 0) orbits. From this point of view one really needs all three Kepler-laws, as you can not conclude the first from the second and third law,and conversely, the third Kepler law also applies to non-periodic motion.

The only assumption made was that of a scale-free potential and ofcourse the similarity solution would not work if scales are involved, i.e.potentials other than power laws. A nice example of this which we willcover later is the perihelion shift of the planet Mercury, where deviationsfrom the 1/r-potential due to general relativity become important.

1.7. SUMMARY 11

1.7 summary

We have seen that classical mechanics was thought to be universal: Be-cause of mechanical similarity, solutions to the Euler-Lagrange equationsof a problem are scaled versions of each other. This made perfect sense atthe time because the two force laws known (Hooke’s law and Newton’slaw) are scale-free. The Lagrange-formalism allows

• the derivation of second-order equations of motion in dissipation-less systems,

• the identification of conserved quantities, and

• the transition to other coordinates which might be better suited.

It is a bit of a surprise that the two force laws mentioned always give riseto closed orbits (Bertrand’s law) which are the Lissajous figures in thefirst and the Kepler-orbits in the second case.


Chapter 2

The Speed of Light and theLorentz Transform

2.1 The space-time structure of classical me-chanics

2.1.1 The Galilei group

Central for classical mechanics are Newton’s axioms. The first axiomor inertial law states that bodies move along straight lines at constantvelocity unless forces act upon them. The second axiom is the equationof motion, which states that the mass of a body times its accelerationequals the force acting on the body, or

m~x = ~F . (2.1)

The third law or reaction law states that the action of one body on anotherequals a reaction of opposite sign.

The first law holds in inertial frames, with inertial frames being definedas reference frames in which no forces are felt. More precisely, aninertial frame can be defined as a frame in which three bodies, drivenapart on trajectories which are not co-planar, are all observed to move onstraight lines. The inertial time is then defined as a measure of time inwhich bodies moving with constant velocity move by a constant amountper unit time.

In classical mechanics, inertial frames are related by Galilei transforms.Such transforms are characterised by a constant rotation matrix R, aconstant velocity ~v, a constant vector ~a and a constant time interval b,

g = g(R,~v,~a, b) , (2.2)

such that the coordinates (t′, ~x ′) in the new inertial frame are related to

13

14CHAPTER 2. THE SPEED OF LIGHT AND THE LORENTZ TRANSFORM

the coordinates (t, ~x ) in the old inertial frame via

t′ = t + b , ~x ′ = R~x +~vt + ~a . (2.3)

The Galilei transforms constitute a ten-parameter group with the compo-sition or multiplication rule

g1 g2 = g(R1,~v1, ~a1, t1)g(R2,~v2, ~a2, b2)= g(R1R2,R1~v2 +~v1,R1~a2 + ~a1 +~v1b2, b1 + b2) , (2.4)

the neutral elemente = g(I, 0, 0, 0) (2.5)

with I being the unit matrix in three dimensions, and the inverse element

g−1(R,~v,~a, b) = g(R−1,−R−1~v,−R−1~a + R−1~vb,−b) . (2.6)

If the matrix R is orthogonal with determinant det R = +1, the group iscalled the orthochronous Galilei group G↑+. The complete Galilei groupcombines G↑+ with the discrete time-inversion and parity transforms,

T : (t, ~x) 7→ (−t, ~x) , P : (t, ~x) 7→ (t,−~x) . (2.7)

Subgroups of the Galilei group are the Euclidean group of motion,

g(R, 0, ~a, 0) , (2.8)

and the special Galilei transforms

g(~v) = (I,~v, 0, 0) . (2.9)

Of particular importance for us is that, according to the group multipli-cation rule (2.4),

g(~v1)g(~v2) = g(~v1 +~v2) , (2.10)

which states the Galilean addition rule for velocities.

Solutions of the equation of motion given by Newton’s second law (2.1)are indeed invariant under Galilei transforms. Galilei invariance is thus adefining property of classical mechanics.

2.1.2 Affine spaces and Galilei space-time

With the mathematical concept of an affine space, the space-time struc-ture of classical mechanics can be further specified. Generally, an affinespace is a triple (M,V,+) of a set M, a vector space V and an addititiveoperation

+ : M × V → M , (p, v) 7→ p + v (2.11)

with the following properties:

2.1. THE SPACE-TIME STRUCTURE OF CLASSICAL MECHANICS15

1. (v1 + v2) + p = v1 + (v2 + p)

2. v + p = p ⇔ v = 0 (for all p ∈ M)

3. For any two points p, q ∈ M a vector v ∈ V exists such thatq = p + v.

For our purposes, the set M is the set of space-time points, while thevector space V provides the position vectors −→pq between any two pointsof M. Property (3) ensures that such position vectors exist, while property(2) guarantees that they are unique. The position vectors are also writtenas −→pq = q − p.

A homomorphism between two affine spaces (M1,V1,+) and (M2,V2,+)is a map f : M1 → M2 preserving the affine structure on M, i.e.

1. if −→pq =−−→p′q′ then

−−−−−−−→f (p) f (q) =

−−−−−−−−→f (p′) f (q′) and

2. the map f : V1 → V2 defined by −→pq 7→ f (−→pq) =−−−−−−−→f (p) f (q) is linear.

Galilean space-time can now be defined as a four-dimensional affinespace (M,V,+) with the following additional properties:

1. A linear form τ exists on the difference space V which allowsdefining the absolute and objective time difference between anytwo points p, q ∈ M,

τ : M × M → R , (p, q) 7→ τ(−→pq

)= τ(q − p) . (2.12)

2. The subspace E := v ∈ V | τ(v) = 0 of all simultaneous points(defined as those points whose position vectors have no time dif-ference) is Euclidean, i.e. it is equipped with a positive-definitebilinear form (a metric) 〈·, ·〉.

Finally, a Galilei system is an affine coordinate frame whose basis vectorseµ with µ = 0, 1, 2, 3 satisfy

1. τ(ei) = 0 for i = 1, 2, 3,

2. 〈ei, e j〉 = δi j for i, j = 1, 2, 3, and

3. τ(e0) = ±1.

In other words, Galilei systems define an orthonormal, three-dimensionalcoordinate frame on the simultaneous subspace and a unit vector parallelor antiparallel to the time direction.

Equipped with these mathematical specifications, we state without proofthat the Galilei transforms are the group of automorphisms of a Galileispace-time, and that the Galilei systems are the mathematical equivalentof inertial frames in physics. This should clarify the space-time structureof classical mechanics sufficiently.


2.2 The space-time structure of special rela-tivity

2.2.1 Electrodynamics and classical mechanics

This space-time structure is questioned by classical electrodynamics. Letus recall Maxwell’s equations in vacuum,

∂t~B = −c~∇ × ~E , ∂t ~E = c~∇ × ~B , (2.13)~∇ · ~B = 0 , ~∇ · ~E = 0 . (2.14)

Taking the curl of the first and the time derivative of the second equation(2.13) and eliminating ∂t~∇× ~B between the two resulting equations gives

∂2t~E − c2~∇2 ~E = 0 , (2.15)

where we have used that ~E is divergence-free in vacuum. This is ad’Alembert equation with the solutions

~E = ~E0 f (~x ± ct~e) (2.16)

for any sufficiently smooth function f , where ~e is a unit vector pointinginto the arbitrary propagation direction of the electric field. In otherwords, electric fields in vacuum propagate with a unique and absolutevelocity c. The same holds true for magnetic fields.

This is clearly in contradiction with the Galilei invariance of classicalmechanics. According to the Galilean law for adding velocities, novelocity should be absolute. The velocity of electromagnetic waves seenby an observer should be the sum of the light speed and the velocity ofthe source emitting the light.

At least one of classical mechanics and classical electrodynamics haveto be modified to remove this contradiction. The solution proposed byEinstein led in 1905 to the theory of special relativity, which has sincebeen thoroughly confirmed experimentally. Einstein suggested to acceptthe absolute light speed in Maxwell’s equations as a principle and toaugment it by the principle of relativity:

1. The laws of nature are the same in all inertial frames, irrespectiveof their state of motion with respect to each other.

2. The speed of light is a universal constant independent of the motionof the source relative to the observer.

If one is willing to accept these two principles, Maxwell’s theory canremain unchanged, but the Galilei transform between inertial framesneeds to be replaced. Let us see now which part of the Galilean space-time structure we can save.

2.2. THE SPACE-TIME STRUCTURE OF SPECIAL RELATIVITY17

2.2.2 The Lorentz transform

Even though we need to replace the Galilean space-time structure bysomething else, we should not give up that inertial motion traces straightlines in all inertial frames. This immediately requires that any newtransform between inertial frames needs to be affine. Moreover, thetransformation should respect the homogeneity and the isotropy of spaceas well as the homogeneity of time. In other words, the transform shouldneither single out any preferred spatial directions nor should it introducepreferred points in space or time. Rather, the transform should dependonly on the relative velocity of any two inertial frames, and it should doso in a universal way. Let us now work out the consequences of theseassumptions.

For simplicity of notation, we bundle the four coordinates introduced inthe definition of Galilean space-time into a vector x,

x = (t, x1, x2, x3) . (2.17)

Since the transformation needs to be affine, new coordinates x′ have tobe related to old coordinates x by

x′ = Λx + a , (2.18)

where the vector a now has four components and Λ is a 4 × 4 matrix.By homogeneity, the vector a must not be essential for the transform,allowing us to restrict ourselves to special transforms, x′ = Λx. It is nowcrucial to realise that, given the isotropy requirements stated before, Λ

must be of the form

Λ =

(b c~v>

d~v R

), (2.19)

where R is a 3 × 3 matrix to be defined, and b is a function dependingon the modulus v = |~v | of the relative velocity between the two inertialframes, b = b(v). If b depended on the direction of ~v, the universalityof the transform was broken. Likewise, the c and d must depend onv only. Given the affine nature of the transform and the requirementof spatial isotropy, the matrix R can only be linearly composed of twocontributions,

R = eI +fv2~v ⊗~v , (2.20)

one given by the unit matrix and the other by the tensor product of theonly vector available, which is the relative velocity ~v. Again, e and f canonly be functions of v. Dividing f by v2 is merely for convenience.

Given the form (2.20) for R, equations (2.18) turn into

t′ = bt + c〈~v, ~x 〉 ,

~x ′ = d~v t + e~x +fv2 〈~v, ~x 〉~v . (2.21)


By the physical meaning of the relative velocity~v between the two inertialframes, the motion ~x = ~v t in the unprimed frame must be mapped to~x ′ = 0 in the primed frame if the clocks are set such that the origins ofthe two frames coincide at t = 0. This allows us to conclude

d + e + f = 0 . (2.22)

The transform with velocity −~v should invert the transform with velocity~v. We thus have from (2.21)

t = bt′ − c〈~v, ~x ′〉 ,

~x = −d~v t′ + e~x ′ +fv2 〈~v, ~x

′〉~v . (2.23)

Inserting t′ and ~x ′ from (2.21) gives

t = b(bt + c〈~v, ~x〉

)− c

⟨~v,

(d~v t + e~x +

fv2 〈~v, ~x〉~v

)⟩~x = −d~v

(bt + c〈~v, ~x〉

)+ e

(d~v t + e~x +

fv2 〈~v, ~x〉~v

)+

fv2

⟨~v,

(d~v t + e~x +

fv2 〈~v, ~x〉~v

)⟩~v . (2.24)

Sorting terms and using (2.22) leads to

t =(b2 − cdv2

)t + c (b + d) 〈~v, ~x 〉 ,

~x = e2~x − d (b + d)~v t −((e + d)(e − d)

v2 + cd)〈~v, ~x 〉~v . (2.25)

Satisfying these equations identically requires

b2 − cdv2 = 1 , c(b + d) = 0 , e2 = 1 ,

d(b + d) = 0 , d(d − cv2) = 1 , (2.26)

where e2 = 1 was already used for the last equation. We can excludee = −1 and thus set e = 1. By the last equation d , 0, so b = −d fromthe fourth equation, satisfying also the second equation. The first andthe last equation then also agree. We are thus left with

f = b − 1 , e = 1 , d = −b , c =1 − b2

bv2 , (2.27)

simplifying the transform (2.21) to

t′ = bt +1 − b2

bv2 〈~v, ~x 〉 ,

~x ′ = ~x − b~v t +b − 1v2 〈~v, ~x 〉~v . (2.28)


We can now without loss of generality rotate our coordinate frame suchthat ~v points into the positive x1 direction. Then 〈~v, ~x 〉 = vx1, and (2.28)reduces to (

t′

x′1

)=

(b 1−b2

bv−bv b

) (tx1

)=: A(v)

(tx1

). (2.29)

Applying the matrix A twice with two different velocities v and v′ mustpreserve its form. In particular, its two diagonal elements must remainthe same. This requires the expression

κ :=b2 − 1(bv)2 (2.30)

to be a universal constant, independent of v. Solving (2.30) for b gives

b =1

√1 − κv2

(2.31)

and the transformation (2.29) to be(t′

x′1

)= b

(1 −κv−v 1

) (tx1

). (2.32)

Quite obviously, κ must have the dimension of an inverse squared veloc-ity. If we set this velocity to infinity and thus κ to zero, b = 1 and (2.32)reproduces the special Galilei transform.

If κ is not zero, however, we can verify quite easily that the transform(2.32) then leaves the quadratic form

Q =t2

κ− x2

1 (2.33)

invariant. Setting κ = c−2 will thus specify the transform (2.29) suchthat signals propagating with the speed of light in one inertial frame willdo so in all inertial frames, regardless of the relative velocity v of theframes.

We have thus achieved the following remarkable result: Beginning withthe requirement that the transform between inertial frames should leavestraight trajectories straight, leave space homogeneous and isotropic anddepend on the relative velocity of the inertial frames only, we have foundthat such transforms are necessarily characterised by a characteristicvelocity κ−1/2. Setting this velocity to infinity returns the Galilei trans-form. However, setting κ−1/2 = c returns a new transform, the Lorentztransform, which leaves the quadratic form Q = c2t2 − x2 invariant and isthus compatible with the principle that the speed of light be a universalconstant.


2.2.3 Some properties of the Lorentz group

We introduce x0 = ct instead of t as the zeroth coordinate and expressthe special Lorentz transform by the matrix

Λ =

γ −βγ 0 0−βγ γ 0 0

0 0 1 00 0 0 1

(2.34)

with the usual abbreviations

β =v

cand γ =

1√1 − β2

. (2.35)

Aiming at a slightly more general mathematical characterisation ofLorentz transforms, we first express the quadratic form Q from (2.33) as

Q = xTGx with G = diag(−1, 1, 1, 1) . (2.36)

Then, the homogeneous Lorentz group L is defined as the set of all real,4 × 4 matrices Λ that are orthonormal with respect to G,

L := Λ|ΛTGΛ = G . (2.37)

The inhomogeneous Lorentz transforms x′ = Λx + a are also calledPoincaré transforms.

Since det G = −1 , 0, the defining equation ΛTGΛ = G impliesdet Λ = ±1. The proper orthochronous Lorentz group L↑+ is the subgroupof L defined by

L↑+ := Λ ∈ L| det L = 1 . (2.38)

Those transformations from L↑+ that leave x2 and x3 invariant form animportant subgroup of L↑+. They are represented by the matrices

Λ =

a bc d 0

01 00 1

(2.39)

which, by the condition ΛTGΛ = G, need to satisfy(a cb d

) (−1 00 1

) (a bc d

)=

(−1 00 1

)(2.40)

ora2 − c2 = 1 , ab − cd = 0 , b2 − d2 = 1 . (2.41)

The first and third of these equations suggest introducing parameters ψ1

and ψ2 such that

a = coshψ1 , c = − sinhψ1 , b = − sinhψ2 , d = − coshψ2 .(2.42)


The second equation (2.41) then demands ψ1 = ψ2 =: ψ, allowing us towrite (2.39) as

Λ =

coshψ − sinhψ− sinhψ coshψ 0

01 00 1

(2.43)

Settingtanhψ = β =

v

c, (2.44)

we can identify (2.43) with (2.34), since

γ =1√

1 − β2=

coshψ√cosh2 ψ − sinh2 ψ

= coshψ

βγ = tanhψ coshψ = sinhψ . (2.45)

It is easily seen that, in terms of ψ,

Λ(ψ1)Λ(ψ2) = Λ(ψ1 + ψ2) , (2.46)

replacing the Galilean law (2.10) for the addition of velocities. Fromψ = ψ1 + ψ2, we can conclude according to (2.44)

tanhψ = β = tanh(ψ1 + ψ2) =tanhψ1 + tanhψ2

1 + tanhψ1 tanhψ2

=β1 + β2

1 + β1β2, (2.47)

thus the velocities v1 and v2 add to

v =v1 + v2

1 + v1v2/c2 . (2.48)

For β1 → 1 or β2 → 1, (2.47) implies β→ 1: Velocities are bounded byc from above.

2.2.4 Minkowski space-time

The special Lorentz group can be augmented by spatial rotations in astraightforward way, but this is not our point of concern here. Rather,we now replace the Galilei space-time by the Minkowski space-time anddefine:

A Minkowski space is a four-dimensional affine space (M,V,+) in whichV is equipped with a non-degenerate, symmetric bilinear form 〈·, ·〉 withthe signature (+,−,−,−). This turns V into a pseudo-Euclidean vectorspace. The group of inhomogeneous Lorentz transforms is isomorphicto the group of automorphisms on the Minkowski space.

The space-time manifold of the theory of special relativity has the struc-ture of a Minkowski space. The Galilean space-time structure of classicalmechanics thus needs to be replaced by Minkowski space-time.


Chapter 3

Electrodynamics, GaugeTheories, and Scales

3.1 Electrodynamics

3.1.1 The field tensor

Electromagnetism can now be characterised a classical field theory withsix degrees of freedom, namely the three components each of the electricand magnetic fields ~E and ~B. Fields are functions of space and time. Anyfield theory must be explicitly constructed to agree with the space-timestructure of special relativity. The electromagnetic field must thus beexpressed as a four-vector or a tensor field. Obviously, a four vector isnot sufficient to describe six degrees of freedom. The simplest objectavailable is a rank-2 tensor, which offers 16 independent components inits most general form. A symmetric rank-2 tensor in four dimensionsstill has ten independent components, while an antisymmetric rank-2tensor has exactly the required six degrees of freedom. The simplestpossibility to describe six degrees of freedom with a Lorentz-covariantobject in four dimensions is thus provided by an antisymmetric fieldtensor F of rank two, whose components must satisfy

Fµν = −Fνµ , Fµν = −Fνµ . (3.1)

The antisymmetry is most conveniently ensured expressing the compo-nents of F as derivatives of a four-potential A with components

Aµ =

(Φ~A

), (3.2)

where Φ is the ordinary scalar potential and ~A is the three-dimensionalvector potential. The components of the rank-(2, 0) field tensor are thenwritten in the manifestly antisymmetric form

Fµν = ∂µAν − ∂νAµ . (3.3)

23

24CHAPTER 3. ELECTRODYNAMICS, GAUGE THEORIES, AND SCALES

They can be conveniently summarised as

(Fµν) =

(0 ~E−~E B

)(3.4)

where the matrixBi j = εi jaBa (3.5)

is formed from the components of the magnetic field. The fields them-selves are thus given by

~E = −1c~A − ~∇Φ , ~B = ~∇ × ~A . (3.6)

Given our signature (−,+,+,+) of the Minkowski metric, the associatedrank-(0, 2) tensor has the components

(Fµν) =

(0 −~E~E B

). (3.7)

The source of the electromagnetic field is the four-current density jwhich has the components

( jµ) =

(ρc~j

), (3.8)

where ρ is the charge density and ~j is the three-dimensional current den-sity. Charge conservation is expressed by the vanishing four-divergenceof the four-current,

∂µ jµ =∂ρ

∂t+ ~∇ · ~j = 0 . (3.9)

3.1.2 The Lagrange density of electrodynamics and Maxwell’sequations

The dynamical equations of a field theory are the Euler-Lagrange equa-tions applied to a Lagrange density which, for a linear theory like elec-trodynamics, must satisfy three conditions: It must be Lorentz invariant,it must contain at most quadratic terms in the field quantities to ensure alinear theory, and it must reproduce the Coulomb force law in the caseof electrodynamics. The only Lagrangian that satisfies these criteria is

L = −1

16πFµνFµν −

1c

Aµ jµ , (3.10)

where the constants must be chosen such as to reproduce the measuredcoupling strength of the electromagnetic field to matter. The otherwiseperfectly legitimate term AµAµ is excluded because it would give the

3.1. ELECTRODYNAMICS 25

electromagnetic field an effective mass and thus violate the Coulombforce law.

Since the field tensor depends on Aµ only through derivatives, it isinvariant under the gauge transformation

Aµ → Aµ + ∂µχ , (3.11)

where χ is an arbitrary function of all four coordinates xµ. At first sight,the Lagrangian (3.10) appears to violate gauge invariance, but Gauss’law applied to the action

S =

∫d4xL (3.12)

shows that charge conservation (3.9) ensures gauge invariance.

Maxwell’s equations are now the Euler-Lagrange equations

∂ν∂L

∂(∂νAµ)−∂L

∂Aµ= 0 (3.13)

of the Lagrangian (3.10). They turn out to be

∂νFµν =4πc

jµ , (3.14)

which are four inhomogeneous equations. Since the field tensor isantisymmetric, it identically satisfies the equation

∂[αFβγ] = 0 , (3.15)

which represents the homogeneous Maxwell equations.

With the definition (3.3) of the field tensor in terms of the four-potentialand with the Lorenz gauge condition ∂µAµ = 0, the inhomogeneousequations (3.14) can be cast into the form

Aµ = −4πc

jµ , (3.16)

where = −∂20 + ~∇2 is the d’Alembert operator.

3.1.3 Scales in electrodynamics

Let us now consider the solution of (3.16) in a simple case, e.g. in anelectrostatic situation. Then, since no currents are flowing, the spatialcomponents of jµ vanish, and j0 = ρc is the only remaining component.Only one of the four equations (3.16) is then relevant, which reads

~∇2Φ = −4π ρ . (3.17)


This Poisson equation is most easily solved by the well-known Green’sfunction of the Laplacian operator,

G(~x, ~x ′) =1

|~x − ~x ′|, (3.18)

which gives immediately

Φ(~x) =

∫d2x′

ρ(~x ′)|~x − ~x ′|

. (3.19)

For a point charge q at the origin, ρ(~x ′) = qδD(~x ′), the electrostaticpotential turns into

Φ(~x) = −qr, r := |~x | . (3.20)

This is an important result, as it clearly shows that electrodynamics asconstructed above is scale-free: The scalar potential, as an example for asimple solution of Maxwell’s equations (3.16), is a power law. Of course,we could have seen this directly in (3.17) or (3.18): The Laplacian is scalein variant, and so is its Green’s function. No scale appears in the problem.Since electrodynamics is linear, solutions of Maxwell’s equations formore complicated charge distributions can simply be constructed bylinear superposition. Since the d’Alembert operator is also scale-free, noscale appears in classical electrodynamics.

What aspect of the Lagrange density (3.10) prevented the occurence of ascale? Could we add a Lorentz invariant term giving rise to a scale? Anobvious suspect is the squared four-potential AµAµ, which is not only aLorentz scalar, but would also appear linearly in the field equations. Letus therefore augment the Lagrange density by such a term,

L → L −κ2

8πAµAµ , (3.21)

and work out the consequences. By the Euler-Lagrange equation, thefield equations would acquire a single extra term and read

Aµ − κ2Aµ = −4πc

jµ . (3.22)

Note that we still had to assume Lorentz gauge for arriving at (3.22),which is now required by charge conservation. To see this, take the fieldequations in the form

∂µFµν − κ2Aν = −4πc

jν (3.23)

and take the derivative with respect to xν,

∂µ∂νFµν − κ2∂νAν = −4πc∂ν jν . (3.24)

3.1. ELECTRODYNAMICS 27

The first term on the left-hand side vanishes because it is a contraction ofthe symmetric tensor ∂µ∂ν with the antisymmetric tensor Fµν. The right-hand side has to vanish for charge conservation, hence (3.24) requiresthe Lorentz gauge condition ∂νAν = 0.

Let us apply (3.22) again to the simple electrostatic case, further assum-ing a single point charge at the coordinate origin as a source. Then,

~∇2Φ − κ2Φ = −4πqδD(~x ) . (3.25)

In Fourier space, this equation reads(k2 + κ2

)Φ = 4πq . (3.26)

Solving for the Fourier-transformed electrostatic potential Φ and trans-forming back to configuration space gives

Φ(~x ) = 4πq∫

d3k(2π)3

ei~k·~x

k2 + κ2 =qπ

∫ ∞

0k2dk

∫ 1

−1dµ

eikxµ

k2 + κ2

=2qπx

∫ ∞

0kdk

sin kxk2 + κ2 . (3.27)

The remaining integral is of the type∫ ∞

0dx

x sinαxβ2 + x2 =

π

2e−αβ , (3.28)

and the potential turns out to be

Φ(~x ) =qr

e−κr . (3.29)

Now, a characteristic scale appears: The potential of a point charge isnow exponentially cut off at the scale λ = κ−1. Since this is clearly inconflict with measurements telling us that the Coulomb force ~F = −~∇Φ

falls off like r−2, we see that the otherwise perfectly legitimate term AµAµ

must not appear in the Lagrange density of electrodynamics.

Nonetheless, for our further purposes, it is worthwhile discussing theso-called Proca term κ2/2 AµAµ in a little more detail. Suppose we insertthe plane-wave ansatz

Aµ = aµei(~k·~x−ωt) (3.30)

into the field equation (3.22), where aµ is an amplitude constant in spaceand time. Then, in vacuum, (3.22) gives

ω2

c2 −~k 2 = κ2 . (3.31)

Multiplying with ~2c2 returns

E2 = c2~p 2 + ~2c2κ2 , (3.32)


where E = ~ω is the energy and ~p = ~~k the momentum carried by theplane wave (3.30). Comparing (3.32) to the relativistic relation

E2 = c2~p 2 + m2c4 (3.33)

reveals that the inverse length scale κ is related to a mass m by

m2c2 = ~2κ2 ⇒ κ =mc~. (3.34)

A spatial scale κ−1 in electrodynamics would thus immediately imply arest mass m = ~κ/c for the photon. This suggests that spatial scales andmass scales are closely related. It also shows that the elementary resultthat the Coulomb force depends on the inverse square of the distanceindicates that the photon is mass-less.

3.2 Gauge theory

3.2.1 Local gauge invariance

Clearly, the Proca term AµAµ violates gauge invariance: The gaugetransformation Aµ → Aµ + ∂µχ transforms the Proca term as

AµAµ → (Aµ + ∂µχ)(Aµ + ∂µχ) = AµAµ + 2Aµ∂µχ + ∂µχ∂

µχ . (3.35)

This also suggests that there is a fundamental connection between thescale-free character of electrodynamics and gauge invariance. This,however, is potentially troubling since we saw in (3.24) that we need theLorentz gauge for charge conservation.

Let us now sketch a completely different approach to electrodynamics.We begin with a simple, relativistic theory for a complex-valued scalarfield φ with a mass term, coupling to nothing but itself. We choose theLagrange density

L = −12

(∂µφ)∗∂µφ −κ2

2φ∗φ (3.36)

which implies the field equations

φ − κ2φ = 0 . (3.37)

This equation can be interpreted as the Klein-Gordon equation for arelativistic quantum field: Expand the d’Alembert operator, multiply theequation by ~ and insert (3.34) for κ. This directly gives

(i~∂t)2 φ =(−i~~∇

)2φ + m2c2φ , (3.38)

which is already the Klein-Gordon equation. We can thus imaginebeginning with a simple example of relativistic quantum mechanics. We

3.2. GAUGE THEORY 29

emphasise that the Klein-Gordon equation (3.38) simply expresses therelativistic energy-momentum relation (3.33). Indeed, replacing energyand momentum by their respective operators,

E → i~∂t , pi → −i~∂i , (3.39)

directly leads from (3.33) to (3.38). The Dirac equation, which is linearrather than quadratic in the operators, can also be derived directly from(3.33).

Now, let us change the phase of the quantum field φ by some amount qα,

φ→ eiqαφ , (3.40)

where q is a constant introduced here for later convenience. Clearly, thiswould leave the Lagrange density (3.36) invariant. We call this a globalgauge transformation of the quantum field φ and find that the simpletheory given by (3.36) is globally gauge invariant. What happens if welocally gauge-transform the field φ,

φ→ eiqα(x)φ ? (3.41)

The mass term in (3.36) is invariant, but the kinetic term is not, since

∂µφ→ ∂µ(eiqα(x)φ

)= eiqα(x) (∂µφ + iq∂µα(x)φ) . (3.42)

Under a local gauge transformation φ→ eiqα(x)φ, the Lagrange densitythus changes to

L = −12

(∂µ − iq∂µα)φ∗(∂µ + iq∂µα)φ −κ2

2φ∗φ . (3.43)

This seems to be the end of the discussion. However, local gaugeinvariance can be restored if the derivatives

Dµ = ∂µ + iq∂µα (3.44)

are used instead of the partial derivatives. Such derivatives resemble thecovariant derivatives which may be familiar from differential geometry.Let us abbreviate the derivatives of the phase by the vector field

Aµ := ∂µα , (3.45)

which is called the gauge field. Then,

Dµ = ∂µ + iqAµ. (3.46)

Upon a further change of gauge,

φ→ eiqα(x)φ→ eiq(α(x)+β(x))φ , (3.47)


the gauge field changes as

Aµ → Aµ + ∂µβ(x) . (3.48)

Having introduced the gauge field, we need to extend the Lagrange den-sity to account for the dynamics of the gauge field itself. In fact, since thegauge field was required to depend on space and time, this dependenceneeds to be described by field equations which are expected to be theEuler-Lagrange equations of a suitable Lagrange density. Clearly, thisextension must contain a kinetic term, containing all derivative ∂νAµ

of the gauge field, it must respect the local gauge symmetry, and itshould return a linear term after variation of the action. Clearly, thesimplest kinetic term ∂νAµ would break the local gauge invariance, butits antisymmetric form

∂νAµ − ∂µAν =: Fνµ (3.49)

would be compatible with it. The simplest possible extension of theLagrange density (3.36) satisfying the above criteria is thus

L = −12

(Dµφ)∗(Dµφ) −κ2

2φ∗φ − ηFαβFαβ , (3.50)

where η is some constant to be fixed by measurement.

The kinetic term for the gauge field Aµ in this Lagrange density resem-bles the kinetic term for the electromagnetic field in the Lagrange density(3.10) of electrodynamics. The gauge field allows the gauge transforma-tions (3.37), just like the four-potential of the electromagnetic field. Thegauge-covariant derivative Dµ from (3.35) can be interpreted as replacingthe momentum operator pµ ∝ ∂µ of the original quantum theory for thefield φ by the momentum operator pµ+qAµ, expressing how the quantumfield φ couples to the gauge field.

Thus, we have achieved the following remarkable result: Beginningwith what could be considered the simplest relativistic quantum theoryexpressed by the Lagrange density (3.36), we have constructed a theoryincluding electrodynamics simply by the requirement that the theory beinvariant under local gauge transformations of the form (3.30). Sincethese transformations are operations of the group U(1) on φ, we have thusrecovered electrodynamics as the theory of the gauge field expressingthe U(1) symmetry, or as the U(1) gauge theory for short.

Thus, the following picture emerges: The fundamental equations ofmotion for quantum fields directly follow from the relativistic relation(3.33) between energy and momentum. All other interactions except forgravity can then be derived from there as interactions with gauge fields,following from invariance requirements under the operation of certainsymmetry groups. We have shown this for the simplest example here,revealing electrodynamics as the gauge interaction following from the

3.2. GAUGE THEORY 31

requirement of U(1) symmetry, but the weak and the strong interactionscan be constructed in a formally equivalent way.

This is a remarkable insight: Relativistic energy-momentum conservationplus local gauge symmetries are sufficient to construct all interactionsknown in physics, excluding gravity.

3.2.2 Mass-less gauge fields and the Higgs mechanism

The gauge field Aµ just introduced by the requirement of local gaugeinvariance has to be mass-less. As we have seen before, a mass would beintroduced by a term in the Lagrangian quadratic in the gauge field itself,but such a term would not be invariant under the gauge transformations(3.37) of the gauge field itself. Evidently, this would introduce aninconsistency into the whole idea of local gauge theories: A gaugefield would be introduced by a symmetry requirement that would laterbe violated by the gauge field itself. Gauge fields thus seem to benecessarily mass-less, thus interactions mediated by gauge fields, forwhich electrodynamics is an example, would have to be scale-free.

However, this is not observed. While the photon, which is the gaugeparticle of electrodynamics, turns out to be mass-less, the gauge bosonsof the weak and strong interactions are not. The weak interaction, forexample, can be constructed as the S U(2) gauge theory. Its gauge fieldsshould be mass-less, as seen above, but they are not: The masses of theW± and Z0 bosons are 80.835±0.015 GeVc−2 and 91.188±0.002 GeVc−2,respectively, and definitely not zero. How can the beautifully elegantconcept of local gauge theories be saved?

The answer is provided by the Higgs mechanism, the principle of whichcan straightforwardly be demonstrated. Suppose we augment the La-grange density of the scalar field φ by a self-interaction term W(φ),

L → L −W(φ) , (3.51)

satisfying the following criteria: It should be invariant under a globaloperation from some symmetry group called the structure group, and itshould have a global minimum at a field value φ0 which is degenerate inthe sense that it is not invariant under operations of the structure group.

For example, W(φ) could be the Mexican-hat potential

W(φ) = λ(φ∗φ)2 − µφ∗φ + C , (3.52)

with positive constants λ, µ and an irrelevant constant C. This potentialis globally invariant under rotations about the origin at φ = 0. However,it has minima φ0 where

φ∗0φ0 =µ

2λ. (3.53)


Only the modulus of the minimum is constrained rather than the fielditself, hence the minimum is degenerate.

Beginning from a symmetric state at high energy, the field φ will movetowards φ0 at low energy and thus spontaneously break the symmetry ofthe structure group. The dynamically relevant field is then

ϕ = φ − φ0 (3.54)

rather than φ itself. Inserting ϕ instead of φ into the kinetic term of theLagrange density (3.50) now produces a term

(Dµφ0)∗(Dµφ0) = q2AµAµ(φ∗0φ0) =q2µ

2λAµAµ , (3.55)

where the partial derivatives disappear because φ0 is a constant. Sincethe additional self-interaction term W(φ) is gauge invariant, its introduc-tion does not break the local gauge invariance. Breaking the structuralsymmetry of W, however, introduces the desired mass term, with a massgiven by the constants λ and µ appearing in the self-interaction potentialW(φ) and by the charge q. This is a simple model for the essentialfeatures of the Higgs mechanism, by which mass and length scales canbe introduced into a gauge theory.

Chapter 4

Quantum Mechanics I

4.1 Introduction

Quantum mechanics is a description of motion on microscopic scalesdefined by the action S =

∫dt L being small, of the order ~. Classical

mechanics is recovered from quantum mechanics in the limit S ~. Itis based on Hamiltonian classical mechanics which implies that

1. it is invariant under Galilean transformations

2. there is a global time as a parameter

New concepts in quantum mechanics are the

1. particle-wave duality

2. complex wave function, motivated by de Broglie waves

3. probability

4. Schrödinger equation

5. uncertainty relation

We will illustrate how the concepts of matter waves was constructed(chapter 4.2), investigate the properties of de Boglie-waves (chapter 4.3)and how quantum mechanics arises from Hamiltonian classical me-chanics (chapter 4.4). We derive the motion of expectation values, theuncertainty principle (chapter 5.1) and finally the path integral formalism(chapter 5.2), bridging to variation principles in classical mechanics.

At the beginning it might be quite confusing that the scale ~ appears indifferent roles, but we hope to make clear that those roles are mutualconsequences. Clearly, ~ is the scale of quantum mechanical uncertainty

33

34 CHAPTER 4. QUANTUM MECHANICS I

∆p∆x ' ~ and the scale at which probability effects become important,but it is also the scale of dispersion of wave packets and the scaleof time evolution in quantum mechanics. Finally, it gives a naturalscale for the time integral of the Lagrange function, i.e. the action S .Perhaps surprisingly, we will try to do as many derivations as possiblewithout using the Schrödinger equation, which is in fact the part ofquantum mechanics which can be replaced easiest with something morecomplicated.

4.2 The Photoelectric effect

If a metal is irradiated by light, it releases electrons by the photoelectriceffect. Surprisingly, this effect does not depend on the light intensity buton the photon frequency. Let’s start with the classical description usingMaxwell electrodynamics and see why this is so puzzling:

The electromagnetic field performs work on charges with the electricpart of the Lorentz-force because the magnetic term ~v × ~b is alwaysperpendicular to the direction of motion. The density of mechanicalpower P/V would be:

PV

= q~v · ~e

V= ρ~v · ~e = ~j · ~e (4.1)

The mechanical work performed by the electric field can be rewritten

~j · ~e = ~e · ~∇ × ~b − ~e · ∂t~e (4.2)

substituting Ampère’s law ~∇×~b = ~j+∂t~e. Using ÷(~e×~b) = ~b·~∇×~e−~e·~∇×~byields

~j · ~e = ~b · ~∇ × ~e − ÷(~e × ~b) − ~e · ∂t~e. (4.3)

Substituting the induction law ~∇ × ~e = −∂t~b then suggests

~j · ~e = −~b · ∂t~b − ~e · ∂t~e − ÷(~e × ~b) (4.4)

where we identify the first two terms on the right side as the time deriva-tive of the energy density ε = (~e2 + ~b2)/2 and the energy flux density~S = ~e × ~b:

~j · ~e = − ÷ ~S − ∂tε (4.5)

which is Poynting’s law and reads in integral notation:∫∂V

d~A · ~S + ∂t

∫V

dV ε = −

∫V

dV ~j · ~e (4.6)

i.e. the amount of mechanical work performed per time interval insidethe volume is equal to the energy flux through a surface and the changein field energy.

4.3. DE BROGLIE WAVES 35

In this interpretation, an atom collects energy S ∆A∆T from the Poyntingflux and ejects an electron if the energy exceeds the binding energy.Because classically, the energy is proportional to the squared field am-plitudes, you’d just have to wait when the illumination is dim, but inreality the photoelectric effect sets in instantaneously provided that thefrequency of the photons is high enough, i.e. if they carry enough energy~ω to overcome the binding energy.

4.3 de Broglie waves

The photoelectric effect suggests that energy in quantum mechanicalwaves is associated with angular frequency, E = ~ω. Similarly, fromdiffraction experiments we know that particles can have wave proper-ties, that they’re described by de Broglie-waves and that they carry amomentum p = ~k associated with wave number k:

ψ(x, t) = ψ0 exp(i(kx − ωt)) (4.7)

with the definition of the de Broglie-wave length λ = 2π/k = h/p, where~ = h/(2π). This is surprising because classically, the energy is relatedto the squared amplitudes and this suggests now that the wave amplitudeshould be related to the number density of particles.

4.3.1 Velocity of a wave

Any wave can be characterized by two velocities: the phase velocity atwhich the individual Fourier components travel and the group velocity atwhich any feature of the wave travels. In general, the two velocities arenot identical, which gives rise to dispersion.

Phase velocity

The phase velocity u is derived from the phase φ(x, t) = kx − ωt of thewave ψ(x, t) = ψ0 exp(iφ(x, t)). A point of constant phase is defined by

dφdt

= 0 = kx − ω = ku − ω → u =ω

k(4.8)

and this feature travels at the velocity u.

Group velocity

But waves representing particles must show dispersion even in vacuum(in contrast to electromagnetic waves). The energy-momentum relation


reads:E2 = m2c4 + p2c2 = (mc2)2 (4.9)

which can be Taylor expanded for small momenta,

E = mc2 =√

m2c4 + p2c2 ' m0c2 +p2

2m0(4.10)

to recover the classical energy-momentum relation. With that, the dis-persion relation reads:

ω(k) =m0c2

~+~k2

2m0, (4.11)

which implies that waves with different wave number travel at differentphase velocities u = ω(k)/k. But the phase velocity u can never beassociated with the particle velocity v because

u =ω

k=~ω

~k=

Ep

=mc2

mv=

c2

v, (4.12)

i.e. for a particle, where v < c the phase velocity u must be > c, becausethe relation uv = c2 holds. Obviously, relativistic particles such asphotons are fine, they are not subjected to dispersion due to the linearenergy-momentum relation.

4.3.2 Dispersion

A purely Galilean argumentation would identify v = dω(k)/dk as thegroup velocity of the particle: The variation of the energy E under theforce ~F along the displacement d~s is dE = ~F · d~s by the work performed.Newton’s third law implies ~F = d~p/dt such that dE = d~p/dt · ~s = d~p ·~vand therefore

dEdp

= v (4.13)

if ~p and ~v are parallel, and this means that the mechanical energy travelswith v = dE/dp.

4.3.3 Wave packets

The above argumentation can be understood in more detail if we con-struct a wave packet from superposition of individual harmonic wavesthat travel at the phase velocity u and see how fast a feature of this wavepacket travels. Synthesizing a wave packet ψ(x, t) from Fourier-modesinside the interval k0 − ∆k . . . k0 + ∆k:

ψ(x, t) =

∫ k0+∆k

k0−∆kdk ψ(k) exp(−i(kx − ω(k)t)) (4.14)

4.3. DE BROGLIE WAVES 37

ω(k) is the dispersion relation of the medium. If the spread in wavenumber ∆k is small compared to the centre wave number k0 we can applya Taylor-expansion of the dispersion relation at k0:

ω(k) − ω(k0) =dωdk

∣∣∣∣∣k0

(k − k0) +12

d2ω

dk2

∣∣∣∣∣∣k0

(k − k0)2 + . . . (4.15)

and drop terms ∝ κ2 on if κ = k − k0 is small enough in comparison tothe variation scale of the dispersion relation. Let’s choose a particularlysimple amplitude function ψ(k), which should be constant inside theinterval and zero outside. Then:

ψ(x, t) = exp(−i(k0x − ω(k0)t))∫ +∆k

−∆kdκ exp(i(x − vt)κ) (4.16)

with the group velocity v,

v ≡dω(k)

dk

∣∣∣∣∣k0

(4.17)

Carrying out the last integration yields

ψ(x, t) = 2sin(∆k(x − vt))

x − vtexp(−i(k0x − ω(k0)t)) (4.18)

which is a wave consisting of an envelope traveling at the group velocityv and individual Fourier components at the phase velocity u = ω(k0)/k0,constantly overtaking the wave packet.

It is very interesting to see that in the definition of the the phase velocitythe Planck constant ~ drops out

u =ω(k)

k

∣∣∣∣∣k0

(4.19)

whereas the group velocity is proportional to ~:

v =dω(k)

dk

∣∣∣∣∣k0

=dEdp

∣∣∣∣∣p0

=p0

m=~k0

m(4.20)

Therefore, we conclude that ~ sets the scale at which dispersion of wavepackets matters.

4.3.4 Dispersion of a Gaussian classical wave packet

The difference between phase velocity and group velocity is very nicelyillustrated by a Gaussian wave packet for a particle obeying a quadraticdispersion relation. As before, we construct a wave packet from thesuperposition of Fourier-modes, but this time with a Gaussian envelope


in Fourier space at time t = 0 with variance σ2. At time t = 0 this wouldbe

ψ(x, 0) = ψ0 exp(−

x2

2σ2

)exp(ik0x) (4.21)

in position space and consequently

ψ(k, 0) = ψ0

∫dxψ(x, 0) exp(−ikx) = ψ0

∫dx exp

(−

x2

2σ2 + i(k0 − k)x)

(4.22)in Fourier- or momentum space, using the de Broglie’s idea about matterwaves. The integral can be solved by completing the square of thequadratic form in the exponent

ψ(k, 0) =

∫dx exp

(−

(x√

2σ− i

σ(k − k0)√

2

))exp

(−σ2(k − k0)

2

)(4.23)

which yields

ψ(k, 0) =√

2πσ ψ0 exp(−σ2(k − k0)2

2

)(4.24)

i.e. a wave packet with inverse width. If we use this expression forthe amplitude distribution in Fourier space at t = 0 and reassemblethe wave packet in position space allowing a time evolution of the el-ementary waves with a classical quadratic energy-momentum relationω = ~k2/(2m) we would get:

ψ(x, t) = ψ0

∫dk2π

ψ(k, 0) exp(i(kx − ω(k)t)) (4.25)

which can be solved, again by completing the square,

ψ(x, t) =ψ0√

2π

∫dk exp

(−σ2(k − k0)2

2+ ikx − i

~k2m

t)

(4.26)

to yield

ψ(x, t) =ψ√

1iαtexp

(x2 − 2iσ2k0x + iσ4α2t

2σ2(1 + iαt)

)(4.27)

This corresponds to a wave packet that increases its width ∝ t, i.e. isslowly dispersing. The behavior of the wave packet becomes a littlemore apparent if we compute the probability density,

|ψ(x, t)|2 =ψ2

0

1 + (αt)2 exp(−

(x − σ2αk0t)2

σ2(1 + (αt)2)

)(4.28)

which is interpreted as a wave packet that shifts with the group velocityv = σ2αk0 = ~k0/m = p/m and widens ∝

√1 + (αt)2. If t 1/α the

width is σ, but at early times t 1/α, the width would be σ(1 + αt/2)with the diffusion scale 1/α = mσ2/~. In summary, there is a quadraticaddition of the width σ of the initial condition of the wave packet andthe dynamical width acquired by dispersion while propagating:

variance = σ2 +

(m~

t)2

(4.29)

the latter being given by the scale ~.

4.4. HAMILTONIAN MECHANICS 39

4.4 Hamiltonian mechanics

4.4.1 Lagrangian equations of motion

Lagrangian mechanics, based on the variation of S =∫

dt L necessarilyleads to a second order equation of motion of the type x = −∇Φ. Ina numerical solution, one would rewrite this second order differentialequation to a set of two coupled first order differential equations by intro-ducing a velocity v = x as the first and v = −∇Φ as the second equation.This is generally possibly as can be seen from this consideration: Writingthe total derivative of the Lagrange function L yields:

dL =∂L∂x

dx +∂L∂x

dx (4.30)

which suggests that the second term ∂L/∂x = mx would be such avelocity, and in fact it corresponds to a generalized momentum as itallows us to identify a constant of motion if the first term L/∂x vanishes.

4.4.2 Hamiltonian equations of motion

Starting from the differential of the Lagrange function

dL =∂L∂x

dx − pdx (4.31)

with the substitution ∂L/∂x = p carried out allows us to write

d(px − L) = −pdx + xdp (4.32)

with the differential d(px) = pdx + xdp. Now, the left side is thedifferential of a new function, the Hamiltonian H ≡ px − L and we canread off the equation of motions of H from the above equation:

∂H∂p

= −x and∂H∂x

= p (4.33)

which are now two first order differential equations instead of a singlesecond order differential equation following from the Euler-Lagrangeequation.

The minus sign in the first equation of motion ensures energy conser-vation: If the Hamiltonian does not depend explicitly on time we’vegot:

dHdt

=∂H∂p

p +∂H∂x

x = 0 (4.34)

which vanishes if the Hamilton equations of motion are substituted.


4.4.3 Poisson brackets

Suppose there is a phase space function g(p, x, t) that depends on thephase space coordinates p and x and, optionally, on the time t. Forexample, you might want to use that function for computing statisticalproperties of the system, so it is important how that function varies withtime. Constructing the total time derivative of g(p, x, t) yields

dgdt

=∂g

∂xdxdt

+∂g

∂pdpdt

+∂g

∂t. (4.35)

We can substitute the Hamiltonian equations of motion for the timederivatives of x and p:

dgdt

=∂g

∂x∂H∂p−∂g

∂p∂H∂x−∂g

∂t(4.36)

where we introduce the Poisson bracket as a shorthand notation:

dgdt

= H, g +∂g

∂t. (4.37)

Therefore, the time evolution of the phase space function g is given bya very simple prescription: it is equal to the gradients of g with respectto the coordinates x times the gradient of the Hamilton function H withrespect to the momenta p minus the same expression with the roles of gand H reversed.

Obvious properties of the Poisson bracket are:

1. antisymmetry: H, g = − g,H

2. The Poisson bracket of the Hamiltonian H is trivially fulfilled,H,H such that the total energy H is conserved, dH/dt = 0

3. Poisson brackets with the positions themselves give g, x = ∂g/∂pbecause ∂x/∂p = 0, and

4. Poisson brackets with the momenta are g, p = −∂g/∂x because∂p/∂x = 0.

5. The Poisson bracket between position and momentum given byp, x = 1, which will have important implications.

4.4.4 Poisson brackets and commutators

Of course probability effects play an important role in quantum mechan-ics. Suppose there is a property A of which you’d like to measure astatistical average. Born’s postulate says that one needs to weight the

4.4. HAMILTONIAN MECHANICS 41

property A with the modulus squared of the wave function ψ(x) (which it-self is normalized to unity) Born’s postulate: weight statistical quantitieswith |ψ|2 for constructing the estimates, and that this statistical propertycan be measured:

〈A〉 (t) =

∫dV prob(x, t) A(x, t) with prob(x, t) ≡ |ψ(x, t)|2 (4.38)

The time evolution of 〈A〉 would follow from

d 〈A〉dt

=

∫dV ψ∗

∂A∂tψ +

∫dV

(∂ψ∗

∂tAψ + ψ∗A

∂ψ

∂t

)(4.39)

If we now were to substitute the Schrödinger equation −i~∂tψ = Hψ forthe two time derivatives, assuming that H is Hermitean, H+ = H, oneobtains:

d 〈A〉dt

=∂ 〈A〉∂t

+i~〈[H, A]〉 (4.40)

with the commutator [H, A] = HA − AH. Of course we would haveexpected this result from the Poisson bracket, but now the scale ~ appearsthat puts the commutator in relation to the time dependence of A. In thissense, ~ sets the scale of probability effects being important.

4.4.5 Constants of motion

Using this result we can conclude that 〈A〉 is a constant of motion if∂t 〈A〉 = 0 and 〈[H, A]〉 = 0, and we recover energy conservation because∂tH = 0 and 〈[H,H]〉 = 0 just as in the classical case.

4.4.6 Ehrenfest theorem

If we choose for A the position x or the momentum p of the quantummechanical system, they obey statistically the relations

dxdt

=i~

[H, x] (4.41)

dpdt

=i~

[H, p

]. (4.42)

The commutators assume for classical Hamiltonians of the type H =

p2/(2m) + Φ the shapes

[H, x] =1

2m

[p2, x

]=~

ipm

(4.43)[H, p

]=

[Φ, p

]= −~

i∇Φ (4.44)


such that one obtains classical Newtonian equations of motion

dxdt

=pm

(4.45)

dpdt

= −∇Φ (4.46)

for the statistical, |ψ|2-weighted averages. This implies that the centreof gravity of wave packets travels along classical trajectories and thatquantum effects can only concern the width of the wave packets, i.e. theuncertainty in position and momentum, which will be the topic of thenext chapter.

In summary we have recoverd exactly the classical behaviour for aver-ages, and ~ as a quantum mechanical scale always cancels. The motionof averages is perfectly regular, and can be predicted using constants ofmotion, the Newtonian equations of motion and the virial theorem. Wewill need ~ for making statements about uncertainty of measurements,in much the same way as the width of a wave packet (i.e. the uncertaintyin position) is governed by ~.

Chapter 5

Quantum Mechanics II

5.1 Uncertainty

In the previous chapter we have seen that the motion of the centresof wave packets is completely regular, they propagate proportional tothe group velocity and in general the classical equations of motion(Ehrenfest’s theorem) and even the virial theorem applies. But withBorn’s interpretation of the wave function, we would think that theseequations describe motion in a probabilistic way: Averaged over manymeasurements we would find the particle at the classical position. Themeasurement will however, differ on average from that average position,so it makes sense to quantify the dispersion of the measurement bycomputing the variance. For the variance

⟨x2

⟩of position in case of

the Gaussian wave packet we have already seen that it increases withtime ∝ t. A genuinely new result in quantum mechanics is now that thesimultaneous measurement of different observables is only possible to abound given by ~, if those observables have different eigensystems.

5.1.1 General uncertainty relation

Let’s have a look at the properties of statistical measurements: Theexpectation value or the average of an observable A would be given by

〈A〉 =

∫dx |ψ|2 A(x) (5.1)

and the variance would be⟨A2

⟩=

∫dx |ψ|2 A2(x) (5.2)

where we used Born’s postulate that |ψ|2 has a probability interpretation.

The standard deviation ∆A =

√⟨A2⟩ − 〈A〉2 is the uncertainty of the

43

44 CHAPTER 5. QUANTUM MECHANICS II

measurement and we will assume for simplicity that the mean value〈A〉 = 0 is zero, such that ∆A =

√⟨A2⟩.

If we estimate two statistical properties of the observables A and B withthe same weight given by the wave function, we can constrain the valueof their error with the Cauchy-Schwarz inequality:(∫

dx |ψ|2 AB∗)2

≤

(∫dx |ψ|2 A2

) (∫dx |ψ|2 B2

)(5.3)

which can be written 〈AB〉2 ≤⟨A2

⟩ ⟨B2

⟩.

The Cauchy-Schwarz inequality can be proved in the following way:Define two numbers α =

⟨B2

⟩≥ 0 and β = − 〈AB〉. We can construct

the expectation variances⟨(αA + βB)2

⟩and

⟨(αA − βB)2

⟩, which both

are larger than zero, and substitute α and β, which yields:

2αβ 〈AB〉 ≥ −2α2β2 (5.4)2αβ 〈AB〉 ≤ +2α2β2 (5.5)

which can be divided by 2αβ and concatenated to

−

√⟨A2⟩√⟨

B2⟩ ≤ 〈AB〉 ≤ +

√⟨A2⟩√⟨

B2⟩ (5.6)

and thus

|〈AB〉| ≤√⟨

A2⟩ ⟨B2⟩ (5.7)

If one now applies the Cauchy-Schwarz inequality for using 〈AB〉 as alower bound on ∆A∆B one can construct:√⟨

A2⟩√⟨B2⟩ ≥ 〈AB〉 ≥

∣∣∣∣∣ 12i〈AB〉 − 〈AB〉∗

∣∣∣∣∣ = |〈AB − BA〉| = |〈[A, B]〉|

(5.8)where we have used that a complex number is always larger or equalto its complex part, and that the operators are Hermitean, such that weobtain in summary

⟨A2

⟩ ⟨B2

⟩≥

14|〈[A, B]〉|2 (5.9)

and the relation

∆A∆B ≥12〈[A, B]〉 (5.10)

taking advantage of the monotonicity of the square root. This means thatthere is a fundamental limit to the uncertainty of two observables, if theydo not share a common eigensystem.

5.2. PATH INTEGRALS 45

5.1.2 Position and momentum

The commutator[p, x

]between position −i~d/dx and position x is given

by i~, in accordance with the Poisson bracket p, x = 1, such that oneobtains the vintage Heisenberg uncertainty relation

∆p∆x ≥~

2(5.11)

which, as we have seen before, is a consequence of the dispersion ofthe estimators for x and p, and the factor 1/2 is a consequence of thecomplex valued expectation values.

5.1.3 Energy and time

Please take great care in the energy-time uncertainty. It might be quitetempting to think that there’s the Hamiltonian operator H = −i~∂t thathas a non-vanishing commutator with the time t, [H, t] = i~, so that youmight conclude that ∆E∆t ≥ ~/2 just as in the case of the momentum-position uncertainty, because 〈H〉 = E. While the result is correct, thereasoning behind it is not: There is no time-operator in non-relativisticquantum mechanics, because time is not an observable but rather a(global Galilean) parameter describing motion and it would not makesense to define expectation values like 〈t〉 =

∫dx |ψ|2 t.

Instead, the energy-time uncertainty follows consistently from the momentum-position uncertainty: Writing E = p2/(2m) for a classical particle, theuncertainty would scale as ∆E = ∂E/∂p∆p = p/m∆p. Position and timeare related via the velocity p/m, such that x = p/m t and ∆x = p/m ∆t.Collecting the results yields ∆E∆t = ∆p∆x ≥ ~/2 (to first order).

5.2 Path integrals

As the last subject, we will link ideas like propagation and dispersion ofwaves to variational principles in classical point mechanics by means ofpath integrals. Path integrals give a very intuitive interpretation of propa-gation in the microscopic world and they help to understand variationalprinciples in the classical limit. Specifically we cover propagators, thephase-space path integral and the configuration space path integral forquadratic energy-momentum relations.

“As far as I am aware, path integrals give us no dramatic new resultsin the quantum mechanics of a single particle. Indeed, most if not allcalculations in quantum which can be done by path integrals can bedone with considerably greater ease using the standard formulations ofquantum” (Richard MacKenzie)


5.2.1 Propagators

Propagators are conditional probabilities, describing the transition am-plitude for a particle originating at (xi, ti) to move to (x f , t f ) in the timeinterval t f − ti. If we start with the time evolution operator acting on astate ψ at ti∣∣∣ψ(t f )

⟩= U(ti → t f ) |ψ(ti)〉 with U(ti → t f ) = exp

(−

i~

H(t f − ti))

(5.12)we can introduce the position representation of the wave function

ψ(x f , t f ) ≡ 〈x′|ψ(t f )〉 = 〈x f | exp(−

i~

H(t f − ti))|ψ(t f )〉 (5.13)

and a basis for decomposing the initial state,⟨x f |ψ(t f )

⟩=

∫dxi 〈x f | exp

(−

i~

H(t f − ti))|xi〉︸︷︷︸

≡K(xi,ti→x f ,t f )

〈xi|ψ(ti)〉 (5.14)

which suggests that the propagator or Green-function K time evolves thewave function from the initial to the final state by performing a weightedsummation over individual modes of the wave initial wave function suchthat the synthesis of the individually propagated modes is exactly thewave function in the final state:

ψ(x f , t f ) =

∫dxi K(xi, ti → x f , t f )ψ(xi, ti) (5.15)

with the transition probability

K(xi, ti → x f , t f ) = 〈x f | exp(−

i~

H(t f − ti))|xi〉 (5.16)

The expression for the propagator K can be simplified if the particlemotion is described by a quadratic dispersion H = p2/(2m), which canbe seen by working in the momentum representation:

K(xi, ti → x f , t f ) =

∫dpi

∫dp f 〈x f |p f 〉〈p f | exp

(−

i~

p2

2m(t − t0)

)|pi〉〈pi|xi〉

(5.17)Substitution of the plane waves 〈x f |p f 〉 and 〈pi|xi〉 yields the propagatorintegral

K(xi, ti → x f , t f ) = exp(

i~

[p(x f − xi) −

p2i

2m(t f − ti)

])(5.18)

We can complete square, and carry out substitution

P = pi − mx f − xi

t − t0and λ2 =

i~

t f − ti

2m(5.19)


and finally perform integration over dpi

K(xi, ti → x f , t f ) =

√m~

2πi(t f − ti)exp

(im2~

(x f − xi)2

t f − ti

)(5.20)

which again illustrates the diffusive motion we encountered in the wavepacket, only now we can time-propagate arbitrary wave forms with theabove Green-function by convolution.

5.2.2 Marginalisation and the propagator group

The way multiple propagators for different time intervals can be linkedtogether is nicely related to the maginalisation process in probabilitytheory. Imagine there is a conditional random process p(c|a) that returnsa random number c under the condition a. If one links two such randomprocesses assuming mutual independence, p(c|b)p(b|a) would be theprobability that three specific numbers a, b, c are returned, and we’reusing the outcome b from the first process as a condition for the second.In the case that we average over the intermediate outcome b, we shouldrecover the process p(c|a):

p(c|a) =

∫db p(c|b)p(b|a) (5.21)

which is called marginalisation.

Propagators can be thought of conditional transition probabilities, K(xi, ti →

x f , t f ) = K(x f , t f |xi, ti), i.e. the probability that an amplitude is trans-ported to (x f , t f ) under the condition that it has been at (xi, ti). In fact, wewill see that marginalisation is the correct rule for linking up propagators,and that this is a consequence of the time evolution operators to form agroup. Let’s start with two time evolutions, from ti to t1 and further onfrom t1 to t f :

U(ti → t f ) = U(t1 → t f )U(ti → t1) (5.22)

which follows from the calculation rules of the exponential. Buildingthe expectation value 〈x f U(ti → t f )xi〉 yields the propagator K(xi, ti →

x f , t f ). The product on the right side can be resolved by introducing acomplete set of position eigenfunctions:

K(xi, ti → x f , t f ) =

∫dx1 〈x f U(t1 → t f )x1〉〈x1U(ti → t1)xi〉 (5.23)

which gives the composition rule for propagators:

K(xi, ti → x f , t f ) =

∫dx1 K(x1, t1 → x f , t f )K(xi, ti → x1, t1) (5.24)


Let’s take this idea a bit further by subdividing the time interval t f − ti

in n steps of size δ = (t f − ti)/n, and by using the group property of thetime evolution operators again:

K(xi, ti → x f , t f ) = 〈x f | exp(−

i~

Hδ)n

|xi〉 (5.25)

which is obviously equal to

K(xi, ti → x f , t f ) = 〈x f | exp(−

i~

Hδ)× . . . × exp

(−

i~

Hδ)

︸︷︷︸n times

|xi〉 (5.26)

There are n − 1 gaps between the n differential time evolution operators,and at those gaps we introduce n − 1 complete basis sets:

K(xi, ti → x f , t f ) =

∫ n−1∏j=0

dx j K(x j, t j → x j+1, t j + δ) (5.27)

with x0 = xi and xn = x f .

5.2.3 Phase space path integrals

If the time interval δ is chosen to be small, the time evolution operatorcan be Taylor-expanded:

K(x j, t j → x j+1, t j + δ) = 〈x j+1| exp(−

i~

Hδ)

x j〉 ' 〈x j+1|1 −i~

Hδ|x j〉

(5.28)such that the propagator becomes

K(x j, t j → x j+1, t j + δ) = 〈x j+1|x j〉 −i~δ〈x j+1|H|x j〉 (5.29)

The first term yields a δD-function which can be expanded in momentumeigenstates⟨

x j+1|x j

⟩= δD(x j+1 − x j) =

∫dp j

2πexp(ip j(x j+1 − x j)) (5.30)

The second term can be rewritten by inserting again a complete set ofmomentum eigenstates

− iδ⟨x j+1|H|x j

⟩= −iδ

∫dp j

2πH exp(ip j(x j+1 − x j)) (5.31)

If we combine both equations and write the velocity x = (x j+1 − x j)/δand finally apply this reasoning recursively to all terms of the Taylorexpansion of the time-evolution operator we get:

K(x j, t j → x j+1, t j + δ) =

∫ n−1∏j=0

dp j

2πexp

i~δ

n−1∑j=0

(p jx j − H

) (5.32)

when replacing the series by the exponential again.


5.2.4 Recovery of Lagrange

The last result can be used in the expression for the complete propagatorby substituting it into every differential time step:

K =

∫ n−1∏j=0

dx j

2π

∫ n−1∏j=0

dp j

2πexp

i~δ

n−1∑j=0

(p jx j − H

) (5.33)

where the term in the exponential is the classical Lagrange-function L,made dimensionless by ~:

K =

∫ n−1∏j=0

dx j

2π

∫ n−1∏j=0

dp j

2πexp

i~δ

n−1∑j=0

L(p j, x j, t)

(5.34)

In the continuum limit where n→ ∞ the sum can be approximated withan integral yielding the action s

δ

n−1∑j=0

L→∫

dt : L = S (5.35)

such that the propagator is given by

K =

∫Dx

∫Dp exp

(i~

∫dt L(p, x, t)

)=

∫Dx

∫Dp exp

(i~

S)

(5.36)In this equation we see that ~ is a natural unit for the action S . TheGerman word Planck’sches Wirkungsquantum is reminiscent of this fact.The latter expression is referred to as the phase space path integral.

5.2.5 configuration space path integrals

In the case of standard forms of the Hamiltonian function, H = p2/(2m)+

Φ we can carry out the momentum integrations of the phase space pathintegral such that the result only comprises position space integrations:

K =

∫ n−1∏j=0

dx j exp

− i~δ

n−1∑j=0

Φ(x j)

∫ n−1∏j=0

dp j

2πexp

− i~δ

n−1∑j=0

p j x j −p2

j

2m

(5.37)

Due to the quadratic dispersion relation, the dp j-integrals are Gaussianand can be solved by completing the square (yet again!):

K =

(m~

2πiδ

) n2∫ n−1∏

j=0

dx j exp

− i~δ

n−1∑j=0

m2

x2 − Φ(x j)

(5.38)

which results in the configuration space path integral

K =

∫Dx exp

(i~

∫dt L

)=

∫Dx exp

(i~

S)

(5.39)

where the classical action enters in units of the Planck-constant ~.


5.2.6 Huygens principle and probability waves

The path integral formalism suggests a very intuitive interpretation ofprobabilities in microscopic propagation. A wave packet is transportedby the propagators during a time interval by convolution, and the propa-gator ca be written as a position (and momentum integral) on the kernelexp(iS/~) with the classical action S . The path integral formalism iseven more than that: it is a microscopic explanation of how particlescarry out the variation of L in order to find their trajectory: Assume thereis a variation x′(t) = x(t) + η(t) with a small η(t), we would get

S [x′] = S [x + η] = S [x] +

∫dt η(t)

δS [x]δx

(5.40)

for the action. If η is large compared to ~, there will be a large phasedifference (∝ η

~) and destructive interference in the propagators, making

the path unlikely. Close to classical path however, S is extremised, andthe derivative δS [x]

δx is small, even in units of ~, such that the classicalpath becomes likely. You can think of ~ as the size of an allowed tubearound the classical trajectory in which a quantum mechanical particleis likely to be found.

Microscopic propagation would now work like this: a quantum me-chanical particle sends out Huygens/de Broglie-waves, that interfereconstructively along the classical path predicted by the Lagrangian for-malism and destructively elsewhere, making the classical path the mostlikely trajectory.

5.3 summary

We have seen that in quantum mechanics the scale ~ occurs in threeplaces: it is the scale for the action S , it sets the scale of uncertainty andof probability effects, and it is the dispersion scale for classical wavepackets. The basic concept of using |ψ|2 as a probability for weightingexpectation values and their variances is a generalisation of phase spacefunctions, whose time evolution is set by the Poisson-brackets. Theseget replaced by commutators in quantum mechanics, which bring in ~ inthe uncertainty relation.

Wave packets are described as carrying momentum and the width ofwave packets (i.e. the variance of their position expectation value) isincreasing with time on the scale ~, which can be easily understood in thepropagator formalism. Bridging back to classical Lagrangian mechanicswe introduced the path integral formalism, that compares the classicalaction ~.

In summary we would like to add that all these results have been derived

5.3. SUMMARY 51

by using the concepts of probability and de Broglie waves, and withoutmaking use of the Schrödinger equation.


Chapter 6

The gravitational field of a pointmass

6.1 Fundamental ideas of general relativity

6.1.1 The meaning of the metric

In special relativity, we have defined a scalar product between any twofour-vectors x and y by means of a matrix G,

〈x, y〉 = xTGy = −x0y0 + x1y1 + x2y2 + x3y3 . (6.1)

With this Minkowskian scalar product, we can write the squared distancebetween two infinitesimally close points x and x + dx as

ds2 = −(dx0)2 + (dx1)2 + (dx2)2 + (dx3)2 = −(dx0)2 + (d~x )2 . (6.2)

In this way, the Minkowskian scalar product induces the Minkowskianmetric η, represented by the matrix (ηµν) = diag(−1, 1, 1, 1). The metricgains its physical meaning by identifying its line element ds with theproper time dτ passing for an observer moving in Minkowski space fromx to x + dx. More precisely,

dτ =(−ds2

)1/2=

(−ηµνdxµdxν

)1/2. (6.3)

General relativity replaces the global Minkowskian metric η by a metricg depending on space and time. Inertial frames are replaced by freely-falling frames of reference. By the equivalence principle, gravity isabsent in freely-falling frames, which means that the metric in suchframes must be locally Minkowskian.

53

54CHAPTER 6. THE GRAVITATIONAL FIELD OF A POINT MASS

6.1.2 The space-time structure of general relativity

This subsection is inserted for completeness. If this is your first exposureto the ideas of general relativity, it can be skipped.

The space-time of general relativity is a four-dimensional differentiablemanifold M. At any point p ∈ M, a tangent vector space TpM can beconstructed. Together, the tangent spaces at all points of the manifoldform the tangent bundle T M. The space-time manifold M is furthersupplied with a Riemannian metric g and a covariant derivative ∇, alsocalled connection. The metric is a symmetric, non-degenerate, rank-2tensor field on M,

g : T M × T M → R , (x, y) 7→ g(x, y) = 〈x, y〉 (6.4)

with

〈x, y〉 = 〈y, x〉 and 〈x, y〉 = 0 ∀x ⇒ y = 0 . (6.5)

The connection is defined by

∇ : T M × T M → T M , (v, x) 7→ ∇vx (6.6)

with∇v( f v) = d f (x) + f∇vx and ∇ f vx = f∇vx (6.7)

for each scalar function f : M → R on M. The covariant derivative ∇vxgeneralises the ordinary partial derivative and defines how the vector xchanges upon infinitesimal transport along the direction v.

If a set eµ of basis vectors is given on T M, the covariant derivative isdetermined either by the Christoffel symbols or, more elegantly, by theconnection 1-forms ωα

β ,

∇vx = ∇v(xµeµ) = dxµ(v)eµ + xνωµν(v)eµ =

⟨dxµ + xνωµ

ν , v⟩

eµ . (6.8)

Given the basis eµ, an orthonormal dual basis can be defined such that

θµ(eν) = δµν . (6.9)

In this dual basis, the metric can be expressed as

g = gµνθµ ⊗ θν . (6.10)

By means of the covariant derivative, the curvature is defined by

R : T M × T M × T M → T M ,

(u, v, x) 7→ R(u, v)x =(∇u∇v − ∇v∇u − ∇[u,v]

)x . (6.11)

This operation has an intuitive meaning: It first moves the vector xalong u, then along v, and subtracts the result of moving it in the reverse

6.1. FUNDAMENTAL IDEAS OF GENERAL RELATIVITY 55

order. The misalignment intrinsically quantifies the local curvature ofthe manifold. While the connection is directly related to the gravitationalfield and can thus be transformed away in freely-falling reference frames,the curvature is related to the gravitational tidal field.

The curvature is not a tensor yet since its result is a vector. In a basiseµ, the components of the Riemann curvature tensor are given by

Rαβγδ = θα

[R(eγ, eδ)eβ

]. (6.12)

The components of the Ricci tensor are given by the contraction

Rαβ = Rµαµβ . (6.13)

The Ricci tensor is symmetric and thus has ten independent components.The Ricci tensor defines the Einstein tensor by

Gαβ = Rαβ −gαβ

2R , (6.14)

where the Ricci scalar R = Rµµ is the contraction of the Ricci tensor. Ein-

steins field equations relate the Einstein tensor to the energy-momentumtensor T ,

Gαβ =8πGc4 Tαβ , (6.15)

where we ignore the cosmological constant since it will not be rele-vant for our purposes. By the second contracted Bianchi identity, thedivergence of the Riemann tensor is

∇µRµν =

12∂νR =

12∇µ

(δµνR

), (6.16)

hence the divergence of the Einstein tensor vanishes identically. This isnecessary to ensure local energy-momentum conservation.

6.1.3 Static, spherically-symmetric gravitational fields

Static space-times allow identifying a unique time direction. The orthog-onal three-space can then be required to be spherically symmetric. Thismeans that angular coordinates (ϑ, ϕ) can be introduced on sphericalshells defined by a radius r. The most general metric of such space-timeshas the form

ds2 = −e2adt + e2bdr + r2dΩ2 , dΩ2 = r2dϑ2 + r2 sin2 ϑdϕ2 , (6.17)

where a and b are functions of r only, to be determined from Einstein’sfield equations. The Einstein tensor of the metric (6.18) has the tworelevant components

G00 =1r2 +

(2b′

r−

1r2

)e−2b ,

G11 = −1r2 +

(2a′

r+

1r2

)e−2b . (6.18)


It requires a lengthy, but straightforward calculation to see this, whichwe omit here.

We first wish to construct a vacuum solution to Einstein’s field equations,Tµν = 0. Then, by Einstein’s field equations, G00 = 0 = G11. The sumand the difference of G00 and G11 must then also vanish, which gives

a′ + b′ = 0 and(re−2b

)′= 1 . (6.19)

We require that our vacuum solution asymptotically turns into the Minkowskispace-time for r → ∞. This implies limr→∞ a = 0 = limr→∞ b. Sincea′ = −b′, this is possible only if a = −b. The second equation (6.19)requires

re−2b = r − 2m , (6.20)

where −2m simply appears as an integration constant. Thus, we arrive at

e2a = 1 −2mr, e2b =

(1 −

2mr

)−1

(6.21)

and the Schwarzschild metric

ds2 = −

(1 −

2mr

)dt2 +

(1 −

2mr

)−1

dr2 + r2dΩ2 . (6.22)

Evidently, for r = 2m, the radial component of the line element divergesand the time component vanishes. Does this result indicate a radial scalefor gravity?

6.1.4 Newtonian gravity as a limit to general relativity

Identifying the Ricci tensor for weak fields with the tidal field in Newto-nian gravity, it can be shown that the line element of the metric is

ds2 = −(1 + 2φ)dt2 + (1 − 2φ)d~x 2 (6.23)

in the Newtonian limit of general relativity, where

φ =Φ

c2 (6.24)

is the dimension-less Newtonian gravitational potential. Notice that(1 + 2φ)−1 ≈ (1− 2φ) for small φ 1 to first order of a Taylor expansion.For a point mass M,

φ = −GMrc2 . (6.25)

Comparing with (6.22), we see that we need to identify

m =GMc2 . (6.26)

6.2. RELATIVISTIC DYNAMICS 57

The quantity 2m =: rs is the Schwarzschild radius. This indicates that theradial scale r = rs is not set by general relativity, but is already present inNewtonian gravity. In fact, the velocity v of a test particle on a circularorbit of radius r around a point mass M is set by

v2 =GM

r. (6.27)

The light speed thus immediately sets the radial scale rs also in New-tonian gravity. In order to see how general relativity breaks the scaleinvariance of Newtonian gravity, we therefore have to look deeper.

6.2 Relativistic dynamics

6.2.1 Derivation from the effective relativistic action

For a free point particle in special relativity, the only relativistic invariantwhich the action can be constructed from is the proper time itself. Thus,its action needs to be proportional to

S ∝∫

dτ =

∫ √−ηµνdxµdxν . (6.28)

Inserting the four-velocity

uµ =dxµ

dτ, (6.29)

we recognise in (6.3) the normalisation condition

− ηµνuµuν = 1 (6.30)

for the four-velocity. Since the proportionality constant is irrelevant here,we write the action as

S =

∫ √−ηµνuµuνdτ (6.31)

and identify the Lagrange function

L =√−ηµνuµuν . (6.32)

Since its variation is

δL = −δ(ηµνuµuν

)2√−ηµνuµuν

= −12δ(ηµνuµuν

), (6.33)

it is convenient to introduce the effective action L by

2L = −ηµνuµuν = 1 = −〈u, u〉 . (6.34)


We adopt the last expression directly for general relativity.

In the Schwarzschild metric, therefore,

2L = 1 =

(1 −

2mr

)t2 −

(1 −

2mr

)−1

r2 − r2ϑ2 − r2 sin2 ϑϕ2 . (6.35)

Since the metric is spherically symmetric, angular momentum is con-served. We can thus rotate the spatial coordinate frame such that themotion is confined to the equatorial plane, ϑ = π/2 and ϑ = 0. Theazimuthal angle ϕ is cyclic, hence

∂L

∂ϕ= r2ϕ = const =: L . (6.36)

The time is also cyclic, hence

∂L

∂t=

(1 −

2mr

)t = const =:

√E . (6.37)

Inserting (6.36) and (6.37) into (6.35), we find

1 =E − r2

1 − 2mr

−L2

r2 , (6.38)

which can immediately be turned into

r2 + Veff(r) = E − 1 , Veff(r) = −2mr

+L2

r2 −2mL2

r3 , (6.39)

where Veff(r) is an effective potential which contains a repulsive contri-bution from the angular momentum, known from Newtonian dynamics.

We replace the derivative with respect to the proper time by a derivativewith respect to the azimuthal angle,

r = r′ϕ =Lr′

r2 (6.40)

and substitute the reciprocal radius u = r−1 to find

r = Lu2(1u

)′= −Lu′ . (6.41)

Then, the equation of motion (6.39) simplifies to

(Lu′)2 − 2mu + L2u2 − 2mL2u3 = E − 1 . (6.42)

A further derivation with respect to ϕ gives

u′u′′ −mu′

L2 + uu′ − 3mu2u′ = 0 . (6.43)

The trivial solution is u′ = 0, corresponding to a circular orbit. Assumingu′ , 0 gives

u′′ + u =mL2 + 3mu2 (6.44)

for the orbital equation of a test mass near a Schwarzschild black hole.


6.2.2 Comparison with the Newtonian case

In Newtonian gravity, the Lagrangian can be written as

L =12

(r2 + r2ϕ2

)+

GMr

, (6.45)

confining the motion to the equatorial plane from the start because ofangular-momentum conservation. Again, since ϕ is cyclic,

r2ϕ = const = L . (6.46)

The Euler-Lagrange equation for the only relevant coordinate is

r −L2

r3 +GMr2 = 0 . (6.47)

For this equation, r is an integrating factor. After multiplication with r,we can integrate the equation once, getting

r2 + Veff(r) = E − 1 , Veff(r) = −2GM

r+

L2

r2 (6.48)

where we have chosen E − 1 as in integration constant to emphasise thesimilarity with the relativistic calculation. Here, an important differenceappears: There is one term less in the effective potential, which in the rel-ativistic case appears by multiplying the angular-momentum contributionwith the ordinary Newtonian potential.

Now consider the order of magnitude of the terms involved. The New-tonian potential, in dimensionless form, is of order (v/c)2, where v is atypical velocity for orbital motion in the potential. Even for a galaxycluster where v ≈ 1000 km s−1, this is of order 300−2 ≈ 10−5. SinceL = r2ϕ ≈ rv in our notation, the angular-momentum term is of order v2

as well, which also has to be compared to the velocity of light. The prod-uct between the gravitational-potential and angular-momentum termsis thus of order (v/c)4 and thus extremely small. It does not show up inNewtonian gravity just for that reason.

Let us return to (6.47) and substitute u = r−1 again. By (6.41), we seethat

r = −Lu′′ϕ = −L2u2u′′ . (6.49)

Back in (6.47), this implies

u′′ + u =GML2 . (6.50)

Compared to the Schwarzschild case (6.44), just the term 3mu2 is missingon the right-hand side. By (6.49), this corresponds to an effective forcecontribution per unit mass of

3mu2 · (−L2u2) = −3GML2

r4 . (6.51)


Here it is: The difference between the Newtonian and the relativisticorbital equation is a force term proportional to r−4, which belongs toa potential contribution ∝ r−3. Apart from the repulsive centrifugalpotential, the complete potential in the Schwarzschild metric has twocontributions, one with the Newtonian form ∝ r−1, and another ∝ r−3.The latter, as we have seen, is due to a coupling between the angularmomentum and the ordinary Newtonian potential, which is absent inNewtonian gravity.

To clarify the issue further, let us look at the dynamics of a point par-ticle in the Newtonian gravitational field of a point mass, but this timebeginning with the metric (6.23) and following the path sketched for theSchwarzschild metric. We have

2L = 1 = (1 + 2φ)t2 − (1 − 2φ)(r2 + r2ϕ2

)(6.52)

if we again confine the motion to the equatorial plane from the start.We proceed in much the same way as for the Schwarzschild metric, i.e.we identify the two constants of the motion

√E and L implied by the

Lagrane function’s being cyclic in t and ϕ, respectively, and transformthe effective Lagrange function into

1 =E

1 + 2φ− (1 − 2φ)

(r2 +

L2

r2

). (6.53)

Since φ 1, we can write

E − r2 −L2

r2 = 1 + 2φ (6.54)

to first order in φ, or

r2 + Veff(r) = E − 1 , Veff = 2φ +L2

r2 = −2GMrc2 +

L2

r2 . (6.55)

This equation is identical to (6.48), derived from ordinary Newtoniandynamics in the gravitational field. Notice, however, that the approxima-tion to first order in φ was necessary to achieve this. Had we kept termsof higher order, the coupling term between the Newtonian potential andthe angular momentum would have reappeared.

6.2.3 Perihelion shift

Let us return to equation (6.44) to investigate its solutions. Withoutthe term quadratic in u on the right-hand side, the equation resembles aconstantly driven, harmonic oscillator. Its general solution is

u = u(ϕ) = A cos(ϕ) + B sin(ϕ) +mL2 , (6.56)


where the constants A and B are to be set by initial conditions. Let usrequire that u is extremal at ϕ = 0, then u′ = 0 implies B = 0. Wefurther set A such that the extremum at ϕ = 0 is a maximum of u, hencea minimum in r = u−1. Denoting the closest distance of the orbitingobject from its central body with rmin and umax = r−1

min, we must have

A = umax −mL2 (6.57)

oru(ϕ) = umax +

mL2

(1 − cosϕ) . (6.58)

Defining the orbital parameter p = L2/m and the eccentricity e = pumax−

1 gives the usual equation for the radius of a Newtonian orbit,

r(ϕ) =p

1 + e cosϕ. (6.59)

We now evaluate the term in u on the right-hand side of (6.44) by insert-ing the Newtonian orbit, assuming that the perturbation of this orbit canbe calculated along the unperturbed orbit, denoted u0. We then have tosolve

u′′ + u =mL2 +

3mp2

(1 + e cosϕ)2 . (6.60)

This is quite straightforward because equations of the form

u′′ + u =

AB cosϕC cos2 ϕ

(6.61)

have the particular solutions

u1 = A , u2 =B2ϕ sinϕ , u3 =

C2

(1 −

13

cos 2ϕ). (6.62)

Beginning with the Keplerian orbit u0, (6.60) thus has the completesolution

u =1 + e cosϕ

p+

3mp2

[1 + eϕ sinϕ +

e2

2

(1 −

13

cos 2ϕ)]. (6.63)

Beginning with a perihelion transit at ϕ = 0, the next will be reached atϕ = 2π + δϕ. At a perihelion, u′ = 0, hence

0 = u′ = −e sinϕ

p+

3mep2

[sinϕ + ϕ cosϕ +

e3

sin 2ϕ]. (6.64)

Taken at ϕ = 2π + δϕ and to linear approximation in δϕ, this gives

δϕ =6mp

(δϕ + π +

e3δϕ

)(6.65)


or, isolating δϕ,

δϕ =6πm

p

[1 −

6mp

(1 +

e3

)]−1

≈6πm

p. (6.66)

This is the lowest-order approximation to the perihelion shift. Withp = a(1 − e2), we arrive at the perhaps more familiar form

δϕ

2π=

3ma(1 − e2)

=3rs

2a(1 − e2)≈

3rs

2a, (6.67)

where we have assumed e 1 in the final step. For the Sun, rs ≈

2.95 km; for Mercury, a = 5.79 × 107 km. Per Mercurian orbit, theperihelion shifts by

δϕ =(2.75 × 10−5

)= 0.1′′ (6.68)

Since an orbit takes Mercury 87.97 d, this amounts to the famous valueof

δϕ = 42′′ per century , (6.69)

measured by Urbain Le Verrier in the 19th century.

It is quite intuitive that the perihelion shift should depend on the ratiobetween rs and a because these are the only scales in the problem. As aperturbative effect, the perihelion shift should thus be linear in rs/a.

Chapter 7

Hydrostatics and stability ofself-gravitating systems

7.1 Hydrostatics

7.1.1 Hydrostatics in Newtonian gravity

The fundamental equations of hydrodynamics can be written as conser-vation equations: They state the conservation of mass, momentum andenergy. In non-relativistic hydrodynamics, the matter current is given bythe four-vector

jµ =ρ

c

(c~v

). (7.1)

Its vanishing four-divergence,

∂µ jµ = 0 , (7.2)

gives the continuity equation,

∂tρ + ∂i(ρvi) = ∂tρ + ~∇ · (ρ~v) = 0 . (7.3)

The conservation equations for momentum and energy follow from thevanishing four-divergence of the energy-momentum tensor T . For anideal fluid, the energy-momentum tensor is

T =(ρc2 + P

)u ⊗ u + Pη , T µν =

(ρc2 + P

)uµuν + Pηµν , (7.4)

with the pressure P and the matter density ρ. For non-relativistic matter,the pressure can be neglected compared to the energy density ρc2, andwe have

T µν = ρc2uµuν + Pηµν . (7.5)

Its vanishing four-divergence,

∂µT µν = 0 , (7.6)

63

64CHAPTER 7. HYDROSTATICS AND STABILITY OF SELF-GRAVITATING SYSTEMS

states momentum conservation for ν = i,

∂µT µi = ∂t

(ρvi

)+ ∂ j

(ρv jvi

)+ ∂ j

(Pη ji

)= vi

[∂tρ + ∂ j

(ρv j

)]+ ρ

(∂tv

i + v j∂ jvi)

+ ∂iP

= ρ(∂tv

i + v j∂ jvi)

+ ∂iP = 0 , (7.7)

where we have used the continuity equation (7.3) in the last step. Invector notation, the latter equation is

ρ[∂t~v +

(~v · ~∇

)~v]

= −~∇P , (7.8)

which is the familiar form of Euler’s equation. Since the left-hand sidecan be re-written as the total time derivative of the velocity, this equationmanifestly states that the acceleration of the fluid times the matter densityis given by the force density. For an ideal fluid without gravitationalfields, only the pressure-gradient force appears. If the fluid flows ina gravitational field, the gravitational force density −ρ~∇Φ needs to beadded to the right-hand side,

ρdt~v = −~∇P − ρ~∇Φ . (7.9)

In a static situation, the acceleration vanishes, dt~v = 0, and the hydrostaticequation follows,

~∇Pρ

= −~∇Φ . (7.10)

If we augment this equation by the gravitational field equation,

~∇2Φ = 4πGρ , (7.11)

it describes a non-relativistic, self-gravitating body in equilibrium. For aspherically-symmetric body, these equations can be cast into the form

Φ′ = −P′

ρ, Φ′ =

GM(r)r2 with M(r) = 4π

∫ r

0x2ρdx . (7.12)

Together, these two equations determine the pressure gradient in a New-tonian, self-gravitating body,

P′ = −GM(r)ρ(r)

r2 . (7.13)

This equation can be turned into the famous Lane-Emden equation if itis assumed that the pressure scales with the density in a polytropic way,

P = P0

(ρ

ρ0

)κ, (7.14)

where κ is the polytropic index and P0 and ρ0 are arbitrary, fiducialvalues for the pressure and the density, respectively.

7.1. HYDROSTATICS 65

7.1.2 Hydrostatics in general relativity

Replacing the partial divergence in (7.6) by the covariant divergence, weimmediately arrive at the equations for relativistic hydrodynamics,

∇µT µν = 0 . (7.15)

Aiming at the relativistic analog to the hydrostatic equation (7.10), weneed to identify the spatial components of (7.15) first. Due to localLorentz invariance, identifying the time direction is less straightforwardin general relativity than in Newtonian physics.

We proceed as follows. For an observer moving with four-velocity u, thelocal time direction is defined by u. If we thus define projection operatorby

h = g + u ⊗ u , hµν = gµν + uµuν , (7.16)

it maps the four-velocity to zero,

hµνuν = uµ − uµ = 0 (7.17)

because uµuµ = −1. Any four-vector that h is applied to is thus projectedonto the three-space perpendicular to the local time direction. Thus, weexpect the Euler equation to result from

hµν(∇λT λν

)= 0 , (7.18)

where we now have to insert the fully relativistic energy-momentumtensor (7.4) with the Minkowski metric η replaced by the metric g. Theresult of this operation is the generally-relativistic Euler equation(

ρc2 + P)∇uu + grad P + ∇uP = 0 . (7.19)

The three terms may require some explanation. First, ∇uu is the covariantderivative of the four-velocity field along itself,

∇uu =⟨duµ + uνωµ

ν , u⟩

eµ =[duµ(u) + uνωµ

ν(u)]eµ . (7.20)

For a static situation, u = u0e0 = e0 since the four-velocity can onlypoint into the time direction and has to be normalised. Then, du = 0, and(7.20) simplifies to

∇uu = ωi0(e0)ei . (7.21)

The term grad P is the four-vector associated to the four-gradient of thepressure. If we restrict our consideration to the hydrostatic equilibrium ofa spherical body, the pressure P can only depend on the radial coordinater, and

grad P = (∂rP) dr[ = P′dr[ , (7.22)


where dr[ is the vector associated to the dual vector dr. Since the pressuregradient can only point in radial direction, the covariant derivative ∇uumust also have a radial component only,

∇uu = ω10(e0)e1 . (7.23)

Finally, in a static situation, the derivative ∇uP of the pressure along thefour-velocity field must vanish, ∇uP = 0. Our relativistic hydrostaticequation thus reads (

ρc2 + P)ω1

0(e0) + P′dr[ = 0 . (7.24)

The metric for a static, spherically-symmetric body is generally givenby (6.17), with radius-dependent functions a and b to be determinedaccording to the matter model given by the energy-momentum tensor.For this metric, the dual basis

θ0 = eadt , θ1 = ebdr , θ2 = rdϑ , θ3 = r sinϑdϕ (7.25)

suggests itself, in which the connection form ω10 is

ω10 = a′e−bθ0 , ω1

0(e0) = a′e−b . (7.26)

In the same basis,

grad P = P′dr[ = P′e−b(θ1)[ = P′e−be1 . (7.27)

Thus, the hydrostatic equation (7.24) turns into

a′ = −P′

ρc2 + P. (7.28)

7.2 Stability and scales in self-gravitating sys-tems

7.2.1 The Tolman-Oppenheimer-Volkoff equation

The components G00 and G11 derived earlier for the metric (6.17) remainof course the same, but now the relevant field equations read

G00 =1r2 +

(2b′

r−

1r2

)e−2b =

8πGc2 ρ ,

G11 = −1r2 +

(2a′

r+

1r2

)e−2b =

8πGc4 P . (7.29)

After multiplying with r2, we see that the first of these equations can bewritten as

1 −(re−2b

)′=

8πGc2 ρr2 (7.30)

7.2. STABILITY AND SCALES IN SELF-GRAVITATING SYSTEMS67

and integrated to give

e−2b = 1 −2GM(r)

rc2 . (7.31)

Equipped with this result, the second equation (7.29) gives

a′ =

(12r

+4πGc4 Pr

)e2b −

12r

=

(12r

(1 − e−2b

)+

4πGc4 Pr

)e2b

=

(GM(r)

r2c2 +4πGc4 Pr

) (1 −

2GM(r)rc2

)−1

. (7.32)

Now, in order to understand the changes due to general relativity com-pared to the Newtonian equations, identify the potential gradient φ′ =

Φ′/c2 with a′ in the Newtonian equations (7.12),

a′ = −P′

ρc2 , a′ =GM(r)

r2c2 , (7.33)

while we have found

a′ = −P′

ρc2 + P, a′ =

(GM(r)

r2c2 +4πGc4 Pr

) (1 −

2GM(r)rc2

)−1

(7.34)

in general relativity. The first equation replaces the hydrostatic equation,showing that the gravitational acceleration of the total equivalent massdensity

ρ = ρ +Pc2 (7.35)

needs to be balanced by the pressure gradient rather than the mass densityρ only. The second equation shows that not only the matter density is thesource of the gravitational field. Instead, the total mass is replaced by

M(r)→ M(r) = M(r) +4πPr3

c2 . (7.36)

With these replacements, the relativistic equations (7.34) read

a′ = −P′

ρc2 , a′ =GM(r)

r2c2

(1 −

2GM(r)rc2

)−1

. (7.37)

Combining these two equations gives the Tolman-Oppenheimer-Volkoff

equation, often written as the condition

P′ = −GM(r)ρ

r2

(1 −

2GM(r)rc2

)−1

. (7.38)

for the radial pressure gradient inside a relativistic, static, spherically-symmetric, self-gravitating body.


7.2.2 Occurrence of a scale

Writing the relativistic hydrostatic equations as in (7.37) emphasises thatall forms of energy have inertia and contribute to the gravitational fieldin general relativity. However, compared to the Newtonian case, theradial scale

2GM(r)c2 (7.39)

appears in the field equation, i.e. the second equation (7.37), wheregradient of a formally diverges. In a realistic, physical situation, it mustnot diverge, however, hence the condition

2GM(r)rc2 < 1 (7.40)

needs to be satisfied. If we approximate the density inside our spherically-symmetric object by a constant, ρ = ρc, the condition (7.40) turns into

8πGρcr2

3c2 < 1 (7.41)

everywhere inside the object. Let R be the radius of the object, thenr ≤ R and

8πGρcr2

3c2 ≤8πGρcR2

3c2 < 1 (7.42)

and thus the size of the object must satisfy

R < Rmax =

√3c2

8πGρc. (7.43)

Therefore, its mass is limited from above by

M < Mmax =4πρc

3R3

max =4πρc

3

(3c2

8πGρc

)3/2

=

√3

4√

2π

(c2

Gρ1/3c

)3/2

.

(7.44)If one inserts nuclear densities here, ρc ≈ 1015 g cm−3, the mass limitturns out to be close to a solar mass: This is the maximum mass of astatic, spherically-symmetric object that can be stabilised against its owngravity if it is compressed to the density of nuclear matter. Neutron starscan have at most approximately this mass before they collapse to a blackhole.

7.2.3 Scales and the linear approximation

To further clarify the origin of the radial scale in the equations forrelativistic, self-gravitating objects, let us return to the first field equation(7.29) and assume that

GM(r)rc2 1 . (7.45)


Then,

− 2b(r) = ln e−2b = ln(1 −

2GM(r)rc2

)≈ −

2GM(r)rc2 , (7.46)

henceb(r) ≈

GM(r)rc2 = rφ′ , (7.47)

where (7.12) was used in the last step. Therefore,

2b′

r=

2r(rφ′

)′= 2

(φ′′ +

φ′

r

). (7.48)

At the same time, the first field equation (7.29) must reproduce thePoisson equation

~∇2φ = φ′′ +2φ′

r=

4πGc2 ρ (7.49)

in the Newtonian limit. In order to achieve this, we need to approximate

G00 =1r2 +

(2b′

r−

1r2

)e−2b

≈1r2 +

[2(φ′′ +

φ′

r

)−

1r2

] (1 − 2rφ′

)≈ 2

(φ′′ +

2φ′

r

), (7.50)

dropping terms of higher than first order in φ. Again, this is the keypoint: As soon as we leave the linear approximation in the gravitationalpotential, the scale-invariance of gravity is broken.

However, the Poisson equation of Newtonian gravity is linear in thepotential, which implies that Newtonian gravity, like electrodynamics,obeys the superposition principle. Newtonian gravity is a linear theory,while General Relativity is not. This closes our argument: Analysing thedynamics of point particles in a static gravitational field as well as thestructure of static, self-gravitating bodies, we have found that Newtoniangravity is scale-free because it is a linear theory. Scales, either in lengthor mass, enter the theory of gravity as soon as the assumption of linearityis given up. Deviations from the scale-free behaviour of Newtoniangravity arise as soon as the fundamental assumption

Φ c2 (7.51)

underlying Newtonian gravity has to be given up. Again, this showsthat the essential reason for both the non-linearity of gravity as well asthe appearance of absolute scales is a consequence of the finite lightspeed. Only the finite value of c sets an absolute scale c2 to compare thegravitational potential to. Linear gravity, and thus the absence of scales,can be maintained only as long as the gravitational potential is smallcompared to c2.


Appendix

For the interested reader, we summarise here the calculation leadingfrom the ansatz (6.17) for the metric to the Einstein equations (6.18). Weuse Cartan’s formalism for this purpose.

We begin by defining the dual basis vectors

θ0 = eadt , θ1 = ebdr , θ2 = rdϑ , θ3 = r sinϑdϕ , (7.52)

in terms of which the metric coefficients become Minkowskian. Theexterior derivatives of these dual basis vectors are

dθ0 = a′e−bθ1 ∧ θ0 , dθ1 = 0 ,

dθ2 =e−b

rθ1 ∧ θ2 , dθ3 =

e−b

rθ1 ∧ θ3 +

cotϑr

θ2 ∧ θ3 . (7.53)

Cartan’s first structure equation for constant metric coefficients,

dθα + ωαβ ∧ θ

β = 0 , (7.54)

then implies the connection forms

ω01 = ω1

0 = a′e−bθ0 , ω02 = ω2

0 = 0 = ω03 = ω3

0 ,

ω21 = −ω1

2 =e−b

rθ2 , ω3

1 = −ω13 =

e−b

rθ3 ,

ω23 = −ω3

2 =cotϑ

rθ3 . (7.55)

Their exterior derivatives, together with Cartan’s second structure equa-tion

Ωαβ = dωα

β + ωαλ ∧ ω

λβ (7.56)

give the curvature forms

Ω01 = Ω1

0 = −a′′ − a′b′ + a2

e2b θ0 ∧ θ1 ,

Ω02 = Ω2

0 = −a′

re2b θ0 ∧ θ2 ,

Ω03 = Ω3

0 = −a′

re2b θ0 ∧ θ3 ,

Ω12 = −Ω2

1 =b′

re2b θ1 ∧ θ2 ,

Ω13 = −Ω3

1 =b′

re2b θ1 ∧ θ3 ,

Ω23 = −Ω3

2 =1 − e−2b

r2 θ2 ∧ θ3 . (7.57)

By the relationRµν = Ωλ

µ(eλ, eν) , (7.58)


these curvature forms immediately give the diagonal components of theRicci tensor,

R00 =

(a′′ − a′b′ + a′ 2 +

2a′

r

)e−2b ,

R11 = −

(a′′ − a′b′ + a′ 2 −

2b′

r

)e−2b ,

R22 =

(b′ − a′

r+

e2b − 1r2

)e−2b ,

R33 = R22 . (7.59)

With the Ricci scalar,

R = −2(a′′ − a′b′ + a′ 2 + 2

a′ − b′

r+

1 − e2b

r2

)e−2b , (7.60)

the diagonal components of the Einstein tensor now follow immediatelyfrom

Gµν = Rµν −gµν

2R . (7.61)


Chapter 8

Thermodynamics

8.1 thermal energy, heat and temperature

8.1.1 heat and thermal energy

The fundamental concept in thermodynamics is thermal energy, i.e. thenotion that there is energy associated with temperature. A body of massm at temperature T carries the thermal energy Q = c(T )mT where c(T )is the specific heat and can be measured as the slope of the Q-T -relation.Is is very important that heat is not a substance (as people used to think:it was called caloricum or phlogiston) but a property of objects. Heatalways flows from objects of larger to objects of smaller temperatureuntil a thermal equilibrium is established, at which the temperatures areidentical.

The generation of heat is easy and takes place in every dissipative me-chanical system by frictional forces, but the converse is very difficult:The generation of mechanical energy from heat can never be achievedcompletely and the best one can do is employing a Carnot-engine.

8.1.2 dissipation in Lagrangian systems

Dissipation is not contained in the Lagrangian description of motionin classical mechanics. Typically, the equation of motion would needto include a frictional term −Dx (think of Stokes’ law) and would readx − Dx = −∇Φ. This frictional term can not follow from the Lagrangiandescription and needs to be put in by hand as a generalised force:

ddt∂L∂x−∂L∂x

= −Dx (8.1)

and could be derived from the Rayleigh dissipation function R = Dx2/2by differentiation with respect to x.

73

74 CHAPTER 8. THERMODYNAMICS

Now, Boltzmann’s revolutionary idea was that heat is in fact the kineticenergy of microscopic particles. In principle a Lagrangian descriptionholds even for dissipational systems (in fact it would not make sense froma dynamical point of view to distinguish dissipation and dissipationlesssystems), and what would be needed is a description of how motion istransferred from macroscopic to microscopic scales.

8.1.3 first law of thermodynamics

If Boltzmann was right and if heat Q was a form of kinetic energy,one would need to include heat into the energy-conservation equationbecause it should be conserved in sum with the mechanical energy W:

U = W + Q = const → dU = dW + δQ (8.2)

This statement of energy conservation is known as the first law of ther-modynamics. The sum of both forms of energy is called internal energyU. A natural question would be whether it was possible to increase theenergy content of a system in a way other than increasing W: The answeris yes, there are numerous ways of achieving this:

dU = TdS − pdV + ΦdQ + ~Bd~µ + µdN + . . . (8.3)

For instance, one can work against

1. the chemical potential µ when increasing the particle number N

2. an external magnetic field ~B by orienting magnetic moments ~µ

3. an external electrical potential Φ by introducing charges Q

4. the pressure p when changing the volume V of the system

5. the temperature T of the system by changing the entropy S

In this formula, there’s always a pairing between an extensive quantity(proportional to the amount of matter: entropy S , volume V , electricalcharge Q, magnetic moment ~µ, particle number N) and an intensivequantity (independent on the amount of matter: temperature T , pressurep, external electrical potential Φ, external magnetic field ~B, chemicalpotential µ). It is important that the fields involved are external and notgenerated by the extensive quantity itself, because this is the reason forthe strange thermodynamic behavior of e.g. self-gravitating systems.The minus sign of the pressure-term in the above equation is chosen byconvention.

Perhaps the best way for imaging what entropy is would be that of amagic thing related to the exchanged heat where the temperature provides

8.2. MEASUREMENTS OF TEMPERATURE 75

the resistance against increasing the internal energy of a body: dU = δQif there’s no work performed, which at the same time is dU = TdS ,and therefore ∆S =

∫dQ/T . Processes in which there’s no exchange

in thermal energy Q = 0 are called adiabatic and the entropy change iszero.

8.2 measurements of temperature

There are two ways of measuring temperatures:

• One can stay on the thermodynamics side of the dU-relation andrelate the temperature to an observable quantity such as the pres-sure p or the volume V by means of an empirical equation of state,or

• one can have the thermal energy perform mechanical work andrelate the TdS -term in the first law to dW.

Let’s have a look at both of them in the following.

8.2.1 empirical temperatures

A thermometer can be constructed by using Gay-Lussac’s law for dilutegases, where temperature and volume are proportional to each other,V ∝ T , by inferring the temperature from observing volumes:

V0

T0=

V1

T1→ T1 =

T0

V0V1 (8.4)

meaning that with a measurement of V1 of a system that occupies avolume V0 at the reference temperature T0 one can determine T1. Ofcourse one needs to be sure of the proportionality T ∝ V (real van-der-Waals gases do have a different relation). In addition, the temperaturemeasurement has to be calibrated with a reference volume V0 at referencetemperature T0. This leads us to the question if it would be possible todetermine a temperature in a calibration-free measurement: In fact thisis possible by using a thermodynamic engine that transforms thermalenergy into mechanical work.

8.2.2 Carnot’s engine

Carnot’s brilliant idea was to invent an engine that transforms heat intomechanical energy and see at what efficiency this can be done. In a sensethe engine is supposed to invert the dissipation process, which is perfectly


efficient. The ultimate goal is to find a link between thermodynamicquantities and mechanics, and to measure temperatures by mechanicalmeans.

The classic construction works with an ideal gas in 4 steps: isothermalexpansion at T1 absorbing the thermal energy Q1, an adiabatic jumpwithout exchanging heat to the temperature T2, isothermal compressionat T2 squeezing out the thermal energy Q2 and a second adiabatic jumpback to the temperature T1.

In the isothermal phases heat is exchanged reversibly with the reservoirsand mechanical work is performed, for instance by the pressure termpdV: dU = 0→ δQ = TdS = pdV , but the sum of the work is not zerobecause the processes take place at different temperatures:

• the isothermal expansion at T1 sucks in heat Q1,

Q1 =

∫dS T = T1∆S > 0 (8.5)

• the isothermal compression at T2 squeezes out heat Q2,

Q2 =

∫dS T = T2∆S < 0 (8.6)

In contrast, the work needed for the two adiabatic phases cancels exactly.

After a cycle the system is back to its initial state and the total change ofinternal energy must be zero, dU = 0: dU = dW + δQ can be integratedto W = −Q = −(Q1 + Q2) = (T1 − T2)∆S , and the efficiency of theengine is the mechanical work generated from the heat intake Q1,

η =WQ1

=T1 − T2

T1< 1 (8.7)

The efficiency only depends on the temperature difference in relation tothe starting temperature.

8.2.3 Probabilities of thermal fluctuations

Another type of thermodynamic engine (and in fact a Carnot engine) wasinvented by Feynman himself: He thought of a propeller with a ratched-mechanism and a coil for a string lifting a weight on the same axis. Thepropeller is in contact with a gas at temperature T1, and occasionallythere’s a thermal fluctuation ε strong enough to disengage the ratchet andto turn the axle, lifting the weight and performing work W. The ratchetis in contact with a gas at temperature T2, and thermal fluctuations candisengage the ratchet too, allowing the machine to turn backwards todrop the weight and to lose energy.

8.2. MEASUREMENTS OF TEMPERATURE 77

For forward motion an energy W + ε is needed, which is taken from thepropeller at temperature T1, and after lifting the weight, the energy εis dissipated by the ratchet. This process occurs at the rate exp(−(ε +

W)/(kBT )). When the machine is moving backwards, the ratchet isdisengaged by a thermal fluctuation ε at temperature T2, and the energyε + W is dissipated by the first reservoir. This second process occurs atthe rate exp(−ε/(kBT )). If the machine is reversible, the two probabilitiesmust be equal, leading to the efficiency of the engine,

ε + WT1

=ε

T2→

Wε

= η =T1 − T2

T1(8.8)

which is equal to that of a Carnot engine. This thought experimentsuggests that the Boltzmann-factor describes the occurrence of a thermalfluctuation, and that the Boltzmann-constant sets the scale for comparingenergies with temperatures.

8.2.4 second law of thermodynamics

The incomplete conversion from thermal energy back to mechanicalenergy is referred to as the second law of thermodynamics, which statesthat this conversion can only be accomplished at the efficiency of aCarnot engine.

8.2.5 Engines as thermometers

With the Carnot-engine we can determine ∆T relative to a referencebody at temperature T just by measuring the amount of mechanicalwork the engine produces. This mechanical energy can be measuredin a calibration free way: Imagine you make the Carnot-engine lift akilogram weight in the Earth’s gravitational field by one meter, then themechanical work would be exactly 9.81 Joules of energy.

For the reference temperature used as the cold side of a Carnot engine onehas agreed to use the triple point of water. This is a special point in thephase diagram of water at a temperature of T = 273.16 K and a pressureof p = 0.006 bar, where liquid water, ice and vapor simultaneously exist,because this phase is comparatively easy to observe.


Chapter 9

Statistical Mechanics

A source of confusion is the fact that thermodynamics is a continuumtheory, whereas the microscopic theory behind it, statistical mechanics,is not. A nice way of illustrating the way in which the Boltzmann-factor describes the probability of a fluctuation to happen is given by thebarometric formula, which, derived from a continuum picture, requiresan equation of state for arriving result. We shall have a look at thesethings in the first sections before developing a statistical description forthe Planck blackbody and connecting it to thermodynamic quantities.

9.1 Boltzmann-factor

The Boltzmann-factor states the probability of a thermal fluctuation inenergy ε in a system that is at a temperature T :

p(ε,T ) = exp(−

ε

kBT

)(9.1)

where the Boltzmann-constant kB provides the unit in which the energyfluctuation ε is compared to the temperature T . The specific functionalform follows from requiring that probability ratios should be proportionalto energy differences:

p(ε2,T )p(ε1,T )

= g(ε2 − ε1) (9.2)

to ensure transitivity:

g(ε3 − ε1) =p(ε3)p(ε1)

=p(ε3)p(ε2)

p(ε2)p(ε1)

= g(ε3 − ε2)g(ε2 − ε1) (9.3)

which is uniquely solved by g(ε) = exp(−βε), and from the compari-son with the partition sum of the ideal gas one obtains β = 1/(kBT ).Heuristically, β ∝ 1/T would make sense as the ratio of probabilities

79

80 CHAPTER 9. STATISTICAL MECHANICS

should decrease with larger temperature, and the minus-sign is chosento ensure stability because almost all systems are energetically boundedfrom below.

9.2 continuum mechanics and statistics

For illustrating the Boltzmann-factor at work let’s have a look at thebarometric formula. The Euler-equation as the continuum version ofNewton’s second law describes the acceleration of a fluid in the presenceof forces,

∂tυ + υ∇υ = −∇pρ− ∇Φ (9.4)

In a static (υ = 0) and stationary ∂tυ = 0 situation one obtains a forcebalance between pressure and gravity,

∇pρ

= −∇Φ (9.5)

For ideal gases we can substitute p = ρT as a relation between pressureand density, and assuming that the temperature is constant yields:

∇ ln ρ = −∇Φ

T→ ρ ∝ exp

(−

Φ

T

)(9.6)

In a homogeneous gravitational field the potential would be given byΦ = mgh, we can insert kB in the denominator for fixing the units,

ρ ∝ exp(−

hh0

)with h0 =

kBTmg

(9.7)

and identify the scale height h0. In the microscopic picture we wouldinterpret the situation quite differently: Individual molecules would tryto gain height against the gravitational field and use the energy they gotfrom a thermal fluctuation. The probability that a thermal fluctuationprovides them with Φ is equal to the Boltzmann-factor exp(−Φ/(kBT )).The number density of molecules that succeeded in rising in the gravita-tional field by an energy difference Φ is proportional to the actual densityρ of molecules.

9.3 thermal wavelength

The thermal wavelength is the de Broglie-wavelength of a particle inthermal motion and provides a scale for judging if a thermodynamicsystem needs to be described by quantum mechanics or not. Assumingequipartition one would equate

p2

2m=

32

kBT (9.8)

9.4. PARTICLE EXCHANGE SYMMETRY 81

if there are 3 translational degrees of freedom for a particle, each degreecarrying kBT/2 of thermal energy. Substituting p = h/λ = ~k yields thethermal wavelength

λ =h

√3mkBT

(9.9)

Now, if the typical particle separation is much larger than the thermalwavelength, r0 λ, one would recover classical behavior as interferenceand superposition of individual wave packets is irrelevant. In contrast,if λ ' r0 these effects become important and a quantum mechanicaldescription is applicable.

9.4 particle exchange symmetry

Quantum mechanics comes up with a new type of symmetry, the symme-try of a wave function under particle exchange: The particle exchange isdescribed by a permutation operator P that operates on a 2-particle wavefunction in this way,

Pψ(x1, x2) = ψ(x2, x1) (9.10)

The permutation operator has the obvious property P2 = id and com-mutes with the Hamilton-operator H,

P−1HP = H → [H, P] = 0 (9.11)

If ψ is an eigenfunction of H, so is Pψ, which can be seen from theargument,

HPψ = PHP−1Pψ = PHψ = EPψ → Hψ = Eψ (9.12)

so ψ and Pψ describe the same state and they can differ at most by afactor. They have to be normalized,∫

dx P2 |ψ|2 =

∫dx |ψ|2 (9.13)

so this factor has to be ±1. Nature uses both possibilities for describingparticles,

Pψ(x1, x2) = +ψ(x1, x2) (9.14)

for bosons andPψ(x1, x2) = −ψ(x1, x2) (9.15)

for fermions.


9.5 occupation number statistics

We can generalize the above result to n particles and construct a wavefunction ψ(x1, . . . , xn), which needs to be either even or odd under ex-change of any pair of coordinates: Starting with 1-particle wave func-tions φi(x) that are required to form an orthonormal and complete set ofeigenfunctions, ∫

dx φi(x)φ j(x) = δi j (9.16)

we form a product wave function, which is motivated as a separationansatz for coping with the system’s Hamilton-operator, that necessarilycontains

∑i ∆i as the kinetic term and

∑i Φ(xi) as potentials. Assembling

the wave function then yields

ψ(x1, . . . , xn) =1√

n!

∑permutations

(sign

)P

[φα1(x1) · . . . · φαn(xn)

](9.17)

where the permutation operator P interchanges particle positions xi atfixed labels αi. The sign is either +1 for bosons or −1 for fermions. Then-particle wave function for fermions can be written in a very compactform with the help of the Slater-determinant,

ψ(x1, . . . , xn) =1√

n!det

ψα1(x1) · · · ψα1(xn)

......

ψαn(x1) · · · ψαn(x1)

(9.18)

using the properties of the determinant, which is the only multilinearform with alternating sign and norm 1. If a wave function has fermioniccharacter, we see that it must vanish due to the antisymmetry if bothparticles are at the same position,

φ1(x) = φ2(x) → ψ = φ1(x)φ2(x) − φ2(x)φ1(x) = 0, (9.19)

showing that statistics with fermions must work differently as there canonly be a single fermion in a given state.

9.6 photon number statistics

For constructing a statistical ensemble we start with a collection of statesi with weights gi and define a set ni of occupation numbers. We needto find the number of ways in which ni can be realized by distributingn particles in the states i by obeying the Pauli-exclusion principle, ifapplicable, ∏

i

w(ni, gi) (9.20)

9.6. PHOTON NUMBER STATISTICS 83

when w(ni, g) is the number of ways to put ni particles in the state i. ForBose-statistics there can be arbitrarily many particles in a single state,so we can write down as the number of ways for putting n particles in asingle state,

w(n, 1) = 1 (9.21)

, and the number of putting n particles in 2 states,

w(n, 2) = n+!1 =(n + 1)!

n!1!(9.22)

and for the number of putting n particles in 3 states,

w(n, 3) = w(n, 2)+. . .+w(0, 2) =∑

k

w(n−k, 2) =∑

k

(n − k + 1)!(n − k)!1!

=(n + 2)!

n!2!(9.23)

and finally,

w(n, g) =(n + g − 1)!n!(g − 1)!

'(n + g)!

n!g!(9.24)

by induction. Combining the combinations for all states by multiplicationyields,

Ω() =∏

i

(ni + gi)!ni!gi!

(9.25)

where we take for convenience the logarithm, which is monotonic andwill have its maximum at the same position as Ω itself,

ln Ω '∑

i

(ni + gi) ln(ni + gi) − ni ln ni − gi ln gi (9.26)

From that expression we can find the maximum under the conditions

1. total particle number:∑

i ni = N

2. total energy: E =∑

i niεi

by using two Lagrange-multipliers α and β

X ≡ ln Ω + α

N −∑

i

ni

+ β

E −∑

i

niεi

(9.27)

for taking care of the boundary conditions. Maximisation yields

dXdni

= 0 → ni = gi ×1

exp(α + βεi) + 1(9.28)

as a solution for the occupation numbers ni, and comparison with e.g.the result for the ideal gas gives the fugacity α = µ/(kBT ) with chemicalpotential µ and the well-known Boltzmann factor β = 1/(kBT ).


9.7 Planck-spectrum

Photons are ultra relativistic bosons with two spin states, and they followthe dispersion relation ε = cp with the momentum p = ~k = ~ω/c. Fromthe average occupation number we can derive the number density ofphotons by integrating over all momenta,

n = 2∫

d3

h3

1exp(βcp) − 1

=1π2c3

∫ ∞

0dω

ω2

exp(β~ω) − 1= κ

(kBT~c

)3

(9.29)with the constant κ = 4ζ(3) ' 0.23, showing that the photon density isproportional to the third power of the temperature. The energy densitycan be calculated quite similarly,

u =~

π2c3

∫ ∞

0dω

ω3

exp(β~ω) − 1= σT 4 (9.30)

giving the Stefan-Boltzmann law with the constant σ,

σ =πk4

B

15(~c)3 (9.31)

joining 3 of the 4 constants we have encountered in the lecture. ThePlanck-spectrum can be read off from the expression of the energydensity,

u =

∫dω S (ω) → S (ω) =

~

π2c3

ω3

exp(β~ω) − 1(9.32)

For the photon pressure we can consider the momentum transfer per unittime by reflection of photons by the walls: The momentum transfer of asingle photon would be 2p cos θ if the photon is reflected under an angleθ and the flux of photons is simply c cos θ, yielding the compact result

p⊥ = 2∫

0≤θ≤π/2

d3 p~3 2cp cos2 θ n(p) =

13

u (9.33)

Here, the momentum integral was written in spherical coordinates, dp =

p2dpd cos θdφ, the dφ-integration gives a pre factor of 2π and the dθ-integration with the specific boundary conditions is simply 1/3 cos3 θ.

The relation between the integrals that appear in the context of thePlanck-spectrum and the ζ-function is the following one:∫ ∞

0dω

ωn−1

exp(ω) − 1=

∫ ∞

0dωωn−1 1

exp(ω) − 1︸︷︷︸geometric series

=

∫ ∞

0dωωn−1

∑m

exp(−mω)

(9.34)which can be treated by substitution y ≡ mω,

=∑

m

m−n∫ ∞

0dy yn−1 exp(−y) = ζ(n)Γ(n) (9.35)

9.7. PLANCK-SPECTRUM 85

giving rise to the ζ-function and a factorial Γ(n) = (n − 1)!.

Naturally, the Planck-spectrum is approximated by the Rayleigh-Jeanslimit in the case ~ω kBT ,

S (ω) ∝ω3

exp(~ω/(kBT )) − 1→

ω3

1 + ~ωkBT − 1

∝ ω2 (9.36)

and by the Wien-law for ~ω kBT .

S (ω) ∝ω3

exp(~ω/(kBT )) − 1→ ω3 exp

(−~ω

kBT

)(9.37)

because kBT provides a scale against which the photon energy ~ω ismeasured. Let’s have a more detailed look at this from the perspectiveof the Wien-displacement law: The most likely photon energy followsfrom the spectrum S (ω) by differentiation,

dS (ω)dω

= 0 → (3 − x)1

exp(x) − 1= 0 (9.38)

with x = ~ω/(kBT ). This relation is solved by x ' 2.821. Comingback to the initial argument with thermal wavelengths, we would writecp = kBT/2 for equilibration of energies and set cp = c~k = ~ω for aphoton with ω = ck as its de Broglie-wave length: ~ω = kBT/2 wouldthen define the boundary between classical and quantum mechanicalbehavior. This scale related to the maximum of the Planck-curve as pre-dicted by Wien’s displacement law by a constant factor, and suggests todivide the spectrum in a quantum-effect dominated low-energy part anda classical high energy part. Of course Wien’s derivation of the radiationspectrum by having photons occupy the states with statistical weight4πk2dk weighted by the Boltzmann factor exp(−~ω/(kBT )) provides thecorrect approximation from purely classical arguments but we wouldexpect this formula to work only at ~ω kBT , which is to the right ofthe maximum.


Chapter 10

Renormalisation

10.1 Introduction

10.1.1 Partition functions in statistical physics

Renormalisation deals with the behaviour of macroscopic systems withvery many microscopic degrees of freedom. The question renormalisa-tion addresses is: How do the interactions of the microscopic degrees offreedom determine the macroscopic behaviour of the system?

Expressed in a more quantitative way, renormalisation asks the question:If a macroscopic system is studied at a certain scale L, how can the detailsof microscopic physics at a scale λ L be given up in a controlled waywithout losing its essentials?

This can be seen as a refinement of the procedure applied in thermo-dynamics. In thermodynamics, the details of microscopic physics areignored. According to the usual procedure, the partition function is setup,

Z =

∫dΓ(x) exp

[−βH(x)

], (10.1)

integrating over the state space of the system, and all thermodynamicalproperties follow from it. In (10.1), x = (q, p) are the phase-spacecoordinates, β = (kBT )−1 is the inverse thermal energy, H(x) is theHamilton function of the system under consideration, and

dΓ(x) =1

h f0

dq1 . . . dq f dp1 . . . dp f (10.2)

defines the integral measure on phase space, where f is the number ofmicroscopic degrees of freedom.

For simplicity only, we imagine a classical rather than a quantal systemhere. Hence, the classical phase space is our state space, and the constant

87

88 CHAPTER 10. RENORMALISATION

h0 introduced as a fundamental volume of a phase-space cell is arbitrary.In quantum mechanics, h0 has to be replaced by Planck’s constant, statespace will turn into a suitable Hilbert space, and integral measureson phase space are replaced by traces over density operators, but theformalism remains essentially unchanged.

If the partition function Z is canonical, its logarithm gives the Helmholtzfree energy,

F = −kBT ln Z , (10.3)

from which all other thermodynamical potentials can be derived byLegendre transform. All thermodynamical information on the system,including fluctuations, is contained in the partition function Z.

We can now phrase the central question of renormalisation in a morerefined manner as, given the partition function of one particular system,how can information on microscopic degrees of freedom be droppedin a controlled way while retaining the essential information on thesystem? Let us consider the two-dimensional Ising model as one specificexample.

10.1.2 Block spins in the Ising model

The two-dimensional Ising model consists of N spins si on a regulartriangular lattice. It has the Hamiltonian

H = −J∑i, j

sis j − B∑

i

si , (10.4)

where the summation over i, j extends over nearest neighbours only.The constants J and B control the coupling strength between spins andof the spins with an external field. The (canonical) partition function is

Z =∑si

exp (−βH) , (10.5)

with the sum extending over all spin configurations.

Let us now combine three neighbouring spins each to form block spinsS I enumerated with a capital index I, and define

S I = sign(sI1 + sI

2 + sI3) = ±1 . (10.6)

There are four different combinations σα+I of the spins sI

1,2,3 giving S I =

+1, and four different combinations σα−I giving S I = −1, with 1 ≤ α ≤ 4.

We can thus rewrite the partition function as

Z =∑S I

∑σα±I

exp (−βH) . (10.7)

10.1. INTRODUCTION 89

Summing or “integrating out” the internal block-spin configurationsσα±

I , we can define a new Hamiltonian H′ in the following manner,

Z =∑S I

∑σα±I

exp (−βH)

=∑S I

exp(−βH′

)with

exp(−βH′

)=

∑σα±I

exp (−βH) . (10.8)

This new Hamiltonian defines a new physical system. The introduction ofblock spins has thus mapped the original two-dimensional Ising model toa new model in which the smallest-scale information has been integratedout.

One can see quite easily that the new Hamiltonian H′ now contains notonly nearest-neighbour interactions as the original Ising model did, butinteractions between all blocks spins S I. These interactions must bedescribed by new coupling constants that arise only by integrating outsome of the microscopic degrees of freedom. To formalise the procedurea bit better, let us modify the original Hamiltonian (10.4) of the Isingmodel to account for the occurrence of these coupling constants,

H = −J1

∑i, j

sis j − J2

∑i, j

sis j − J3

∑i, j,k,l

sis jsksl − . . .− B∑

i

si , (10.9)

where all the Ji with i ≥ 2 are originally zero. Then, we can write

exp[−βH

(~J ′, S I

)]=

∑σα±I

exp[−βH

(~J, S I , σ

α±I

)]. (10.10)

This procedure can now be repeated by grouping the block spins into newblock spins, gradually removing one level of microscopic physics afterthe other. With each level of block-spin transformation, the original gridscale a increases by a factor of

√3. After each block-spin transformation,

the effective coupling constants ~J will have changed.

10.1.3 The renormalisation group and its flow

Let now be ~T (·, p) the map of the original coupling constants ~J = ~J(0) tothe effective coupling constants ~J(p) after p iterations of the block-spintransform,

~J(p) = ~T ( ~J(0), p) . (10.11)

The map ~T must satisfy the self-similarity property

~J(p) = ~T ( ~J(r), p − r) = ~T[~T ( ~J(0), r), p − r

](10.12)


or~T (·, p) = ~T

[~T (·, r), p − r

]. (10.13)

The key issue of the renormalisation procedure is now that the originalpartition sum often contains much more information than needed forstudying macroscopic or universal properties of a system. Then, weexpect that more or less elaborate approximation schemes can be intro-duced for the effective Hamiltonians. Moreover, the so-called flow ofthe coupling constants with the renormalisation-group transformations~T can be calculated under these approximations and is often sufficient tostudy the macroscopic behaviour of the system.

Let us move one step further by imagining continuous renormalisation-group transforms parameterised by λ. Then,

~Jλ = ~T ( ~J, λ) . (10.14)

Now, let us begin at some λ and move an infinitesimal step forward toλ(1 + ε). We can then write

~Jλ(1+ε) = ~T(~Jλ, 1 + ε

), ~Jλ = ~T ( ~Jλ, 1) (10.15)

and write~Jλ(1+ε) − ~Jλ = ~T

(~Jλ, 1 + ε

)− ~T

(~Jλ, 1

), (10.16)

allowing us to describe the renormalisation flow of the coupling constantsby the differential equation

λ∂ ~J∂λ

=∂~T ( ~Jλ, x)

∂x

∣∣∣∣∣∣x=1

. (10.17)

We thus arrive at a scheme for the controlled study of physical systemsand their effective coupling constants on arbitrary scales above the scalesof the microscopic degrees of freedom. Let us now see how this schemecan be ported to field theory.

10.2 Application to path integrals

10.2.1 Path integrals in quantum mechanics

We had seen in the discussion of quantum mechanics that the transitionamplitude for a quantum-mechanical system to move from the positionq′ at time t′ to the position q′′ at the time t′′ > t′ can be expressed by thepath integral

〈q′′, t′′|q′, t′〉 =

∫Dq

∫Dp eiS /~ , (10.18)

10.2. APPLICATION TO PATH INTEGRALS 91

where L(q, q, t) is the Lagrange function and

S =S~

=1~

∫ t′′

t′dt L(q, q, t) (10.19)

is the action into which we have absorbed the scaling by ~−1 for simplicityof notation.

The average of an operator Q(t1) is given by

〈q′′, t′′|Q(t1)|q′, t′〉 =

∫Dq

∫Dp q(t1)eiS (10.20)

or, more generally,

〈q′′, t′′|TQ(t1)Q(t2)|q′, t′〉 =

∫Dq

∫Dp q(t1)q(t2)eiS , (10.21)

with the operator T ensuring proper time ordering.

Now, we introduce functional derivatives. We first insert source functionsf and h to formally extend the Hamiltonian as

H → (1 − iε)H − f q − hp . (10.22)

The multiplication by (1 − iε) with an infinitesimally small, positive pa-rameter ε is a formal step greatly simplifying the subsequent calculations.For ε → 0 and f = 0 = h, the original theory is recovered.

Leaving ε , 0 in the Hamiltonian allows a substantial simplificationas one can show that then only transitions from the ground state to theground state remain with non-vanishing transition probabilities,

〈0|0〉 f ,h

=

∫Dq

∫Dp exp

i∫ t′′

t′dt

[pq − (1 − iε)H(p, q) + f q + hp

].

(10.23)

This expression is already very similar to a partition function in statisticalphysics, suggesting to write

〈0|0〉 f ,h = Z0[ f , h] . (10.24)

The partition function Z0 is now a functional of the sources f and h,while the partition function of thermodynamics is typically a functionof external macroscopic quantities such as the volume V or the temper-ature T . In thermodynamics, average quantities are obtained by takingderivatives of the partition function. For example, let

Z =

∫Γ

dΓ(x) exp

−β f∑i=1

εi

(10.25)


be the partition function of a simple thermodynamical system whose fdegrees of freedom have the energies εi, 1 ≤ i ≤ f . Then, the averageinternal energy is

U =1Z

∫dΓ

f∑i=1

εi exp

−β f∑i=1

εi

= −

1Z∂Z∂β

= −∂ ln Z∂β

, (10.26)

where the division by Z and thus the logarithm in the last expression isnecessary for normalisation only. We now proceed by applying the sameformalism to quantum mechanics and field theory.

In order to obtain averages, we now use functional rather than ordinaryderivatives to bring averages like (10.20) or (10.21) into the form

〈Q(t)〉 =1i

δ

δ f (t)〈0|0〉 f ,h

∣∣∣∣∣f =0=h

=1i

δ

δ f (t)ln Z0[ f , h] ,

〈Q(t1)P(t2)〉 =1i

δ

δ f (t1)1i

δ

δh(t2)ln Z0[ f , h] , (10.27)

where it is understood that the functional derivatives are to be evaluatedat f = 0 = h and in the limit ε → 0.

With this result, we have now established a complete analogy betweenstatistical physics and quantum mechanics: Expectation values for ob-servables are to be obtained by (functional) derivatives of suitable parti-tion functions. Let us now move on to quantum field theory.

10.2.2 Path-integral formulation of field theory

Let ϕ be a scalar quantum field, i.e. ϕ is a field operator consistingof creation and annihilation operators. Its Lagrange density could forexample be

L0 = −∂µϕ∂µϕ −

m2

2ϕ2 (10.28)

for the free field. Then, in complete analogy to quantum mechanics, wefind the transition amplitude

〈0|0〉J =: Z0[J] =

∫Dϕ exp

[i∫

d4x (L0 + Jϕ)], (10.29)

where |0〉 is again the ground or vacuum state of the field.

For an interacting field theory, an interaction term L1 needs to be addedto the Lagrange density, hence

L0 → L0 +L1 , S →∫

d4x (L0 +L1 + Jϕ) . (10.30)


The partition functional then changes accordingly,

Z0[J]→ Z[J] =

∫Dϕ exp

[i∫

d4x (L0 +L1 + Jϕ)]

= exp[i∫

d4xL1

(1i

δ

δJ(x)

)] ∫Dϕ exp

[i∫

d4x (L0 + Jϕ)],

(10.31)

where each occurrence of the field ϕ in the interaction Lagrangian hasbeen replaced by a functional derivative with respect to the source Jassociated with ϕ. The partition functional can thus be written as ininteraction operator applied to the generating functional Z0 of the freetheory,

Z[J] = exp[i∫

d4xL1

(1i

δ

δJ(x)

)]Z0[J] . (10.32)

For example, for a theory with L1 = gϕ3, the generating functional forthe interacting theory is

Z[J] = expi ∫ d4x g

(1i

δ

δJ(x)

)3 Z0[J] . (10.33)

The exponentials in both the interaction operator and the generatingfunctional for the free theory can be expanded,

Z[J] =

∞∑V=0

1V!

ig∫d4x

(1i

δ

δJ(x)

)3V

×

∞∑P=0

1P!

∫Dϕ

[i∫

d4x (L0 + Jϕ)]P

. (10.34)

The first factor contains the vertex terms, the second the propagators.Equation (10.34) can be expressed by a hierarchy of Feynman diagrams.For a diagram with V vertices and P propagators, the number E ofremaining, external sources is

E = 2P − 3V (10.35)

because two sources appear in each propagator term, while each vertexreduces the number of sources by three in this ϕ3 theory.

While this perturbative approach to quantum field theory has been andstill is most successful, increasing demand for precision requires thecalculation of a rapidly increasing number of perturbation terms, orFeynman diagrams. It depends on the structure of these terms and onthe strength of the coupling constants how many of these terms arenecessary for the precision aimed at, and it is not clear at all whether andunder which conditions the perturbation series converges. The controlled


neglect or subtraction of diverging terms, called renormalisation, hasoften raised mathematical and conceptual suspicion.

The block-spin model discussed in the beginning of this section offersa controlled way out, first gone by Wilson and Polchinski. Like there,one can coarse-grain the original Hamiltonian by integrating it over allscales larger than a certain cut-off scale and thus suppressing small-scale fluctuations in a controlled way. As in the block-spin model,effective Hamiltonians arise this way whose physical meaning is difficultto interpret.

10.2.3 The effective action

It has proven much more convenient to replace the external source J inthe partition sum by the average field ϕ using a Legendre transform andrather integrate out the fluctuations of the field ϕ about its mean. Wethus begin with the partition sum Z[J] from (10.31) and convert it firstto the thermodynamical potential

W[J] = ln Z[J] (10.36)

corresponding to the Helmholtz free energy in thermodynamics. Thefunctional derivatives of W with respect to the source J(x) now generatethe average field value ϕ and all its correlation functions,

1iδW[J]δJ(x)

=1Z

1iδZδJ(x)

= 〈ϕ〉 =: ϕ ,

1i

δ

δJ(x)1i

δ

δJ(y)W[J] = 〈ϕ(x)ϕ(y)〉 − ϕ(x)ϕ(y) (10.37)

and so on to correlations of arbitrary order.

Now, we carry out a Legendre transform replacing J by ϕ,

Γ[ϕ] =

∫d4x J(x)ϕ(x) −W[J] . (10.38)

The functional Γ now corresponds to the Gibbs free energy of thermo-dynamics and is called the effective action of the theory. Its functionalderivative with respect to ϕ is the source J,

δΓ[ϕ]δϕ(x)

= J(x) , (10.39)

which is the field equation for the average field in presence of the sourceJ(x).


10.2.4 Functional renormalisation

By (10.31), the effective action is related to the original partition sum by

exp −iΓ[ϕ] = Z[J] exp−

∫d4xJ(x)ϕ(x)

=

∫Dϕ exp

i∫

d4x[L + J (ϕ − ϕ)

]=

∫Dϕ exp

i∫

d4x[L +

δΓ

δϕ(ϕ − ϕ)

], (10.40)

where we have used (10.39) in the last step. Now, we rewrite the field interms of its average ϕ and its fluctuations χ,

ϕ = ϕ + χ , (10.41)

to transform the path integral to an integral over the fluctuations,

exp −iΓ[ϕ] =

∫Dχ exp

i∫

d4x(L[ϕ + χ] +

δΓ

δϕχ

). (10.42)

Now, we introduce a scale-dependent regularisation term where

∆S k[χ] =12

∫d4xd4y χ(x)Rk(x − y)χ(y) (10.43)

into the effective action chosen such as to suppress the action on scalesto be ignored, and to leave it unchanged on the scales of interest. Thisleads to the definition

exp −iΓk[ϕ] =

∫Dχ exp

i∆S k[χ] + i

∫d4x

(L[ϕ + χ] +

δΓ

δϕχ

)(10.44)

of the scale-dependent effective action. The shape of the regularisationterm Rk(x − y) needs to be chosen appropriately. As a function of thescale k, it can be chosen to remove either short (ultraviolet) or long(infrared) scales and thus to suppress possible corresponding infinities.If Rk is zero for all scales, the original theory is recovered.

The substantial advantage of this approach is that it is in principle non-perturbative, in contrast to summing up Feynman diagrams. Moreover,it allows the smooth transition from the complete theory including alldegrees of freedom to an effective theory from which microscopic de-grees of freedom can successively and continuously be removed. Diver-gences, which tend to give rise to tentatively obscure renormalisationand resumming procedures with Feynman diagrams, do not occur in thiseffective-action, so-called functional renormalisation approach. The foreach scale, the effective average action describes a different effectivetheory which would lead to different perturbation series and Feynmandiagrams.


The disadvantage is that the effective action of the theory is generallyunknown. If it was known, the theory would be solved. Finding suitableapproximation schemes for the effective action is thus essential for thesuccess of the approach. The following consideration, however, providessome guidance.

10.2.5 The saddle-point approximation

Suppose the source term J and the fluctuations χ about the mean field ϕare both small. Then, we can expand the action into a functional Taylorseries

S [ϕ] = S [ϕ + χ] = S [ϕ] +12χ(x)

δ2S [ϕ]δϕ(x)δϕ(y)

∣∣∣∣∣∣ϕ

χ(y)

=: S [ϕ] +i2χ(x)S (2)(x, y)χ(y) (10.45)

where the gradient does not appear since the action needs to be extrem-ised for the mean field ϕ to be a solution. Inserted into the definition(10.42) of the effective action, this shows

exp −Γ[ϕ] = exp iS [ϕ]∫Dχ exp

−

12χ(x)S (2)(x, y)χ(y)

.

(10.46)The remaining Gaussian functional integral can immediately be solvedand gives∫

Dχ exp−

12χ(x)S (2)(x, y)χ(y)

= det

(S (2)

)−1/2+ const . (10.47)

Thus, with (10.46)

Γ[ϕ] = S [ϕ] − ln det(S (2)

)−1/2

= S [ϕ] +12

tr ln(S (2)

). (10.48)

In hindsight, this approximate result justifies calling Γ the effectiveaction: In the limit of vanishing perturbations, the effective action equalsthe action itself.

10.2.6 Wetterich’s equation

In the course of this lecture, we could only briefly touch the main ideasof functional renormalisation. Let us close by mentioning without proofthat an exact flow equation exists for the effective action, which was first


discovered by Christof Wetterich and is sometimes called the Wetterichequation. It says

∂Γk[ϕ]∂k

=12

tr(

Γ(2)k [ϕ] + Rk

)−1 ∂Rk

∂k

, (10.49)

where Γ(2) is the matrix of second functional derivatives of the effectiveaction. This equation, extremely complicated as it is, has now turned intoa universal starting point for analysing a multitude of theories for theirscaling behaviour. The regularisation term can be chosen to suppressesany divergences that might otherwise arise in the infrared or ultravioletlimits of the theory.


Chapter 11

The Planck scales

The purpose of theis lecture was a look at the four grand theories ofmodern physics, i.e. electrodynamics, quantum mechanics, relativity andthermodynamics, from the scales c, ~, G and kB encapsulated by them.Those four fundamental constants are of very different type: a velocity,an action, a coupling constant (rather G/c2 is relevant than just G) and aratio between an energy and a temperature. When Planck realized that~ is a fundamental constant along with G, c and kB which have beenknown before, he immediately noticed that the four constants provide anatural system of units and that they can be combined to yield a lengthscale, a time scale, a mass scale and a temperature scale. These unitswould be "free of anthropocentric arbitrariness" present in choices likethe SI-units or the Gauss-units.

11.1 values of natural constants

The natural constants are known at an accuracy of 10−8 for h and kB andonly 10−4 for G. The value of c, however, is defined as a fundamental unit,and determines together with the definition of the second the fundamentalunit of length. Therefore, there is no uncertainty on c.

c = 299792458 m/s velocity exacth = 6.62606957 × 10−34 Js action 10−8

G/c2 = 7.524 × 10−28 m/kg coupling constant 10−4

kB = 1.3806488 × 10−23 J/K thermal energy 10−8

Table 11.1: Numerical values of the fundamental constants of nature andtheir uncertainties

99

100 CHAPTER 11. THE PLANCK SCALES

11.2 Planck scales as natural units

Naturally the question arises if it would be sensible to do this for allconstants, i.e. to define a basic system of units for length, time, mass andtemperature from the natural constants. This construction can be doneby dimensional analysis: Requesting that the combination of naturalconstants gives a mass unit mp,

mp = cαGβ~γ (11.1)

lets us to construct a linear system of equations: With the units [c] = L/T ,[G] = L3/M/T 2 and [~] = ML2/T the above requirement becomes

[cαGβ~γ] = Lα+3β+2γM−β+γT−α−2β−γ = M (11.2)

from which we isolate the linear system:

0 = α + 3β + 2γ (11.3)1 = −β + γ (11.4)0 = −α − 2β − γ (11.5)

which is solved by α = −β = γ = 1/2, so the Planck mass is given by

mp =

√c~G' 2 × 10−8kg ' 1016GeV/c2 (11.6)

At the same time this result could have been obtained by equating the deBroglie-wave length for a relativistic particle with momentum mc andthe Schwarzschild-radius Gm/c2:

~

mc= λ = rs =

Gmc2 → mp =

√c~G

(11.7)

where theorists suspect that both quantum field theory and general rela-tivity break down and need to make space for a unified theory of quantumgravity.

In analogy to the construction for the Planck-mass, scales for length,

lp =

√~Gc3 ' 10−35m (11.8)

from λ = ~/(mpc), time,

tp =lp

c' 10−44s (11.9)

and temperature,

kBTp =

√c3~

G' 10+32K (11.10)

11.2. PLANCK SCALES AS NATURAL UNITS 101

from KBT − p = mpc2. Building on the fundamental units, many sec-ondary Planck scales can be constructed, for instance a Planck-chargeqp,

qp =√

4πε0

√~c ' 11.7e (11.11)

by equating the rest mass energy to the electrostatic self energy of acharge concentrated within lp, mpc2 = q)p2/(4πε0lp).

If one express all lengths, times, masses and temperatures with thePlanck-scales as units, all unit conversions cancel by setting c = ~ =

G = kB = 1 which gives all equations a particularly clean appearance butthe importance of the concepts behind the scales is not as apparent.

The Planck-units look very far away from the scales we are used toand on which the SI- and Gauss-units are based. But even the Universelooks strange from the point of view of the Planck units. Cosmologistsimagine that the Universe evolved through an inflationary stage whereits size increased very rapidly by 30 orders of magnitude, followed byanother 24 orders of magnitude of cosmic expansion in 10 billion years.The age of the Universe is given by the inverse Hubble constant and istH ∼ 1060tp in units of the Planck time. The size of the universe χH =

c/H0 = 1060lp (because size and age are related by χH = ctH = c/H0)and the temperature would be TH = 10−32Tp (which can be understood bycooling down photons in adiabatic expansion by 1060 orders of volumeincrease). So, the Universe has evolved to a very unnatural state fromthe beginning of inflation where the size of the Universe was 1lp, the agewas 1tp and the temperature was 1Tp.

Another way of looking at the problem is to note that the Friedmann-equations, which provides a description of the expansion dynamics ofthe Universe, defines a density scale, the critical density:

ρcrit =3H2

0

8πG(11.12)

which is very difficult to reconcile with the Planck density

ρp =mp

l3p

=c5

~G2 = 10+122ρcrit (11.13)

using the value of the Hubble constant H0 = 10−60/tp. But we do in factobserve that the Universe is close to the critical density (and that thegeometry is flat), such that ρcrit is a valid density estimate rather than ρp.

Using Eddington’s number that there are 1080 protons in the Universesuggests that either there is far to little matter around, or that the particlesare far too light, or that gravity is very weak in order to reconcile ρcrit

and ρp. This issue arises for all forms of gravitating matter includingdark energy.

102 CHAPTER 11. THE PLANCK SCALES

Chapter 12

Problems

103

104 CHAPTER 12. PROBLEMS

12.1 Problem Sheet 1: Lagrangian classicalmechanics

1. Suppose that you’re a renaissance scientist and a friend proposesthat the planets are held in place by gigantic invisible springsobeying Hooke’s law. Can you disprove him? Which of the threeKepler laws would still apply if the potential was Φ ∝ r2 insteadof ∝ 1/r? Are you surprised by the equivalent third law?

2. Please show that the Lagrangian formalism is invariant underGalilei-transformations.

3. Show that the Galilei-transformations form a group. How manyparameters are there?

4. Show that you can construct all elements of the Galilei group withthe Greek geometry rules, i.e. only using a compass and a ruler(with no markings), whereas this is not possible with the Lorentzgroup (why?).

5. Please write down the Lagrange-function and derive the equationsof motions for a pendulum with an elastic string: it can do left-right oscillations under gravity and up-down oscillations becauseof Hooke’s law. What can you say about the solution?

6. Please formulate the Lagrange function for a central potential andderive the equations of motion in polar coordinates (r, ϕ). Canyou transform the second order equations of motion to a coupledsystem of first order equations by defining the velocities ω = ϕand v = r?

7. If you could observe a distant planetary system, could you distin-guish a weaker gravitational constant from a slower passage oftime? What would you in particular need to measure absolutely?

8. Please show that the Klein-Gordon equation (∆ + λ2)ϕ = 0 issolved by the Yukawa potential ϕ ∝ exp(−λr)/r in 3d sphericalcoordinates.

9. Can you, by using Newton’s axioms, derive the expression for thekinetic energy ε = p2/(2m)? Which of the axioms do you need?

10. Can you think of a proof why the Poisson-equation has an isotropicsolution, i.e. that it is invariant under orthogonal transformations?

11. What’s the orbit of the tightest bound planet?

12.2. PROBLEM SHEET 2: THE SPEED OF LIGHT AND THE LORENTZ TRANSFORM105

12.2 Problem Sheet 2: The Speed of Lightand the Lorentz Transform

1. Galilei group:

(a) Show that the Galilei transforms form indeed a group.

(b) Also show that the special Galilei transforms and the Eu-clidean transforms form subgroups of the Galilei group.

(c) Show that the Galilei and the Lorentz groups have ten param-eters each.

2. Galilei invariance:

(a) Prove that Newtons 2nd axiom, the law of motion ~x = ~F, isinvariant under Galilei transforms. Hint: You are likely toneed Newton’s third law for doing so.

(b) Can you prove Galilei invariance of classical mechanics be-ginning with the Lagrange function instead of Newton’s 2ndlaw?

3. Lorentz transforms:

(a) Show that the Lorentz transforms leave the quadratic form

Q =t2

κ− ~x 2 (12.1)

invariant.

(b) The generators of a (Lie) group are the derivatives of thematrices representing the group elements with respect to theircontinuous parameters, taken at the neutral element of thegroup. Find the generators of the special Lorentz transformscharacterised by the velocities in the three coordinate axes.Hint: It is easiest to use the representation by means ofhyperbolic functions.

(c) Calculate the commutators [A, B] = AB − BA

[Λ1,Λ2] , [Λ2,Λ3] , [Λ3,Λ1] . (12.2)

They define the Lie algebra of the special Lorentz group.


12.3 Problem Sheet 3: Quantum Mechanics I

1. Please show starting from the Schrödinger equation i~∂tψ = Hψthat the time evolution can be written as ψ(t) = exp(−iHt/~)ψ(t =

0) and that the operator U = exp(−iHt/~) is unitary, U−1 = U+.

2. Please show that probability is conserved if the time evolutionoperator U is unitary, and that this unitarity is a consequence ofthe Hermiticity H = H+ of the Hamiltonian H (hint: use the seriesexpansion of the exponential). What would happen in the case ofa non-Hermitean Hamiltonian of the type H + iΓ?

3. Please show that the time evolution operator U in quantum me-chanics fulfills the group axioms. Does the Hamiltonian have tobe Hermitean for this?

4. Can you show that Schrödinger quantum mechanics is Galilei-covariant?

5. Under what circumstances would there be no dispersion in thepropagation of a particle? (hint: there is an obvious and a far lessobvious solution)

6. Can you derive a continuity equation of the shape ρ + div~j = 0 forthe probability density ρ ≡ |ψ|2? what’s the form of the probabilitycurrent ~j? What’s ~j for a plane wave?

7. Do the Hermitean matrices form a group? Show that the productof two Hermitean operators A and B is only Hermitean if the twooperators commute [A, B] = AB − BA = 0.

8. Please show (using the series expansion again) that the operatorexp(iap/~) with the momentum operator p = −i~∇ and a fixeddistance a shifts a wave function ψ(x) by the distance a.

9. Show that the Pauli matrices σα together with the unit matrixσ0 ≡ id are a basis of all complex 2 × 2-matrices and that anymatrix A can be written as

A =12

4∑α=0

tr (Aσα)σα (12.3)

with (possibly complex) coefficients tr(Aσα). What is the keyproperty for this decomposition and where does the factor 1/2come from?

10. In time evolution with the operator U = exp(−iHt/~), could youdistinguish higher energy of the system from time passing fasteror a smaller Planck constant ~? Likewise, in the shifting operatorexp(−ipx/~) could you separate a higher momentum from shorterlength scales or a smaller value of ~?

12.3. PROBLEM SHEET 3: QUANTUM MECHANICS I 107

11. Consider the product exp(Lα) exp(Lβ) of two matrix exponentialsfor two non-commuting matrices Lα and Lβ, which leads to theBaker-Hausdorff formula. Can you show how that formula reducesdramatically if the commutator has the property

[Lα, Lβ

]= εαβγLγ

as we saw for some Lie-algebras? (NB: The matrices could beangular momenta, and perhaps you find it easier to imagine this.)

12. Can you formulate the uncertainty relation for non-Hermiteanoperators? what about anti-Hermitean operators? (Of course thisis not realized in Nature.)

13. Please show using the Cauchy-Schwarz inequality that⟨A2

⟩ ⟨B2

⟩≥

14

(⟨[A, B]2

⟩+

⟨A, B2

⟩)(12.4)

with the commutator [A, B] = AB − BA and the anticommutatorA, B = AB + BA.


12.4 Problem Sheet 4: Quantum MechanicsII

1. The covariance matrix Ci j =⟨xix j

⟩of a multivariate Gaussian

probability density p(xi)dxi,

p(xi)dxi =1

√(2π)ndet(C)

exp

−12

∑i j

xi(C−1)i jx j

dxi (12.5)

is positive definite. Why does this have to be the case and whatensures positive definiteness?

2. Please show that the trace of a commutator is always zero.

3. Please show that det(exp(A)) = exp(tr(A)).

4. Can you show that the numbers tr(An) are invariants under unitarytransformations? how many of those invariants can you construct,and how many are sensible? how is the determinant det(A), whichis invariant under unitary transformations as well (why?) relatedto the set of tr(An)?

5. Please show that quantum mechanical momentum conservationholds, d 〈p〉 /dt = 0 if the potential does not have a gradient,∇Φ = 0.

6. Would the quantum mechanical uncertainty evolve in time?

7. Can you show that∫dnx exp

(−

12

xiCi jx j

)=

√(2π)n

detC(12.6)

for a positive symmetric matrix C?

8. Suppose that two given Hermitean (symmetric) matrices A and Bare related by a unitary (orthogonal) transformation, B = U+AU.Can you construct U from A and B?

9. Please show that the determinant of a skew-symmetric matrix A,At = −A, is zero if the dimension is odd.

10. What’s the reason why the propagators K(xi, ti → xi+1, ti+1) forma group?

11. Please derive the path integral for the harmonic oscillator. Whichof the integrations can be carried out?

12. What happens to the path integral under a mechanical similaritytransformation?

12.4. PROBLEM SHEET 4: QUANTUM MECHANICS II 109

13. What would be the equivalent Bohr-radius for a system bound bygravity?

14. Why is the Bohr-radius a sensible quantity in quantum mechanicsand appears in the hydrogen wave functions that follow fromsolving the Schrödinger equation when it was originally derivedusing a balance between the Coulomb- and centrifugal forces whileimposing an ad-hoc quantization of angular momentum?


12.5 Problem Sheet 5: The gravitational fieldof a point mass

1. Dynamics in the Schwarzschild spacetime

(a) Light rays follow null geodesics in General Relativity, i.e.their tangent vectors k satisfy 〈k, k〉 = 0 = gµνkµkν. Thisimplies that their effective Lagrangian vanishes, Leff = 0.Given this piece of information, show that light rays in theSchwarzschild metric are determined by the equation

u′′ + u = 3mu2 , (12.7)

where u = r−1 and m = GMc−2 have their usual meaning andthe prime denotes the derivative with respect to the azimuthalangle ϕ.

(b) Solve this equation with the boundary condition that theminimal distance b between the light ray and the centre ofthe Schwarzschild metric is passed at ϕ = 0.

(c) Calculate the deflection angle of the light ray, which is twicethe angle reached at r → ∞. (Why twice?)

2. The Laplace-Lenz-Runge vector of an orbit ~x(t) is given by

~Q =1α

(~x × ~L

)−~xr, (12.8)

where α quantifies the gravitational potential,

Φ = −α

rn , n ∈ N , (12.9)

~L is the angular momentum of the orbiting test particle, and r = |~x |.

(a) Show that, if the potential is Newtonian, the Laplace-Lenz-Runge vector falls into the orbital plane and points into theperihelion direction.

(b) Determine n such that ~Q is constant in time. What does theresult mean?

3. For two versions of the Newtonian limit of General Relativity,

ds2 = −(1+2φ)dt2 +d~r 2 and ds2 = −(1+2φ)dt2 +(1−2φ)d~r 2 ,(12.10)

determine

(a) whether the dynamics of particles and

(b) the dynamics of light

12.5. PROBLEM SHEET 5: THE GRAVITATIONAL FIELD OF A POINT MASS111

differ between the two approximations. As usual in this approxi-mation,

φ =Φ

c2 = −GMrc2 1 , (12.11)

which entails t ≈ 1.


12.6 Problem Sheet 6: Hydrostatics and sta-bility of self-gravitating systems

1. Hydrostatics

(a) Solve the Newtonian hydrostatic equation assuming that thedensity is constant, ρ = ρ0, for 0 ≤ r ≤ R and zero outside.Adopt P = 0 at r = R as a boundary condition.

(b) Can the Tolman-Oppenheimer-Volkoff equation admit power-law solutions for the pressure?

2. The Lane-Emden equation

(a) Assuming that the pressure P is related to the density ρ bythe polytropic relation

P = P0

(ρ

ρ0

)γ(12.12)

with a constant polytropic index γ, turn the Newtonian hy-drostatic equation into a second-order differential equationfor the density.

(b) Introduce the polytropic index n by

γ − 1 =1n, (12.13)

define a dimension-less density θ by

ρ

ρ0= θn (12.14)

identify a suitable radial scale r0 and thus cast the hydrostaticequation into the form

1x2∂x

(x2∂xθ

)= −θn . (12.15)

(c) Solve this Lane-Emden equation for n = 0 and

(d) show that the spherical Bessel function

j0(x) =sin x

x(12.16)

is a solution for n = 1.

12.7. PROBLEM SHEET 7: THERMODYNAMICS 113

12.7 Problem Sheet 7: Thermodynamics

1. Imagine an electric circuit with a resistor at a certain temperatureand a diode. You could argue that the diode lets through half ofthe thermal voltage fluctuations on the resistor, and that there’sa net electrical current, that could potentially be used to performmechanical work. Where’s the flaw?

2. In a laser one has to establish an inversion of the Boltzmann-statistics of states: Would that imply the system has a negativeabsolute temperature?

3. Could you construct a Carnot engine with a photon gas as a work-ing substance? What property of the working substance is relevant?How would the adiabatic and isothermal parts of the cycle scale inthe p-V diagram?

4. Why do Carnot-engines always operate at the same efficiency η?

5. Could you set up a system that would be stable at negative absolutetemperatures? Would heat flow from a system with positive tem-perature to that system? What would happen to the temperatures?

6. What would happen to the efficiency η if you used the system fromexercise 5 as the cold side of a Carnot-engine?

7. Can you derive the barometric law for a polytropic equation ofstate p ∝ ργ with a fixed γ and with a uniform temperature? Whatabout a van-der-Waals-equation of state?

8. Please invent a machine for empirical (but reproducible!) tempera-ture measurements.

9. The shape of the Planck-spectrum depends only on temperature:How would you construct a thermometer using a blackbody cavity?What constants would you need to know?


12.8 Problem Sheet 8: Statistical Mechanics

1. Can you derive the intensity-frequency-relation for a Planck-blackbodyin n dimensions? Are there equivalents of the Wien-displacementlaw and of the Stefan-Boltzmann law?

2. Please derive entropy density, energy density and pressure fora Planck blackbody. How would the relations generalize to ndimensions?

3. Can you derive the equivalents of the laws of Gay-Lussac, Boyle-Mariotte and Charles for the Planck blackbody? Where are differ-ences due to the linear energy-momentum relation of photons?

4. Please show that in the case of photon wavelengths longer thanthe thermal wavelength the Rayleigh-Jeans spectrum is recoveredfrom the Planck-spectrum.

5. Why do we observe to good approximation a blackbody spectrumfrom the Sun? Shouldn’t we see emission lines from the elementsin the photosphere? Why are there absorption lines (the Fraunhoferlines)?

6. Why is it possible to touch a neon tube without burning your hand?

7. The Sun’s mass is 2 × 1030 kg, the Sun’s radius is 7 × 108 m, andthe solar energy flux on Earth is S = 1.4 kW/m2. What’s thetemperature on the Sun’s surface?

12.9. PROBLEM SHEET 9: RENORMALISATION 115

12.9 Problem Sheet 9: Renormalisation

1. Functional derivatives. Functionals are functions of functions.Functional derivatives are derivatives of functionals with respect totheir function arguments. They obey the usual rules for derivatives,in particular the Leibniz (product) and chain rules. In addition, thefunctional derivative of a function with respect to itself is a Diracdelta distribution,

δ f (x)δ f (x′)

= δD(x − x′) . (12.17)

(a) Determine the functional derivatives of the following func-tionals:

F[ f ] =

∫ b

af α(x)dx , F[ f ] =

∫ b

a

f α(x)g(x)

dx , F[ f ] =

∫ b

a

[f 2(x)

x+

1f (x)

]dx

(12.18)

(b) By functional derivative, find the function ρ(x) which extrem-ises the Shannon entropy

S [ρ] = −

∫ρ(x) ln ρ(x)dx . (12.19)

(c) Using partial integration, show that the functional derivativeof the action

S [q, q] =

∫ t1

t0L[q(t), q(t)

]dt (12.20)

with respect to q(t′) leads to the Euler-Lagrange equations ifset to zero.

2. Saddle-point expansion. Beginning with the simple action

S [ϕ] =

∫d4x′

[12∂µϕ∂

µϕ +m2

2ϕ2 +

λ

12ϕ4

], (12.21)

(a) show that the second functional derivative of this action withrespect to the field ϕ is given by

δ2S [ϕ]δϕ(x)δϕ(y)

=(−∂µ∂

µ + m2 + λϕ2)δD(x − y) . (12.22)

What does it mean that the Dirac delta distribution remains?

(b) If the action is augmented by a scale-dependent regulatorterm

∆S k[ϕ] =12

∫d4x′

∫d4y′ ϕ(x′)Rk(x′ − y′)ϕ(y′) , (12.23)

how does the second-order derivative of the action change?What does this tell about the physical meaning of the regula-tor term?


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The Physics of Scales - uni-heidelberg.de · The Physics of Scales Björn M. Schäfer, Matthias Bartelmann Zentrum für Astronomie ... 6.1.3 Static, spherically-symmetric gravitational