ContinuousSymmetriesandConservationLaws. Noether…users.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-7-conslaw.pdf · side of (8.7) is called Noether charge. ... The existence

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8Continuous Symmetries and Conservation Laws.Noether’s Theorem

In many physical systems, the action is invariant under some continuous set oftransformations. In such systems, there exist local and global conservation laws

analogous to current and charge conservation in electrodynamics. The analogs ofthe charges can be used to generate the symmetry transformation, from which theywere derived, with the help of Poisson brackets, or after quantization, with the helpof commutators.

8.1 Point Mechanics

Consider a simple mechanical system with a generic action

A =∫ tb

ta

dt L(q(t), q(t), t). (8.1)

8.1.1 Continuous Symmetries and Conservation Law

Suppose A is invariant under a continuous set of transformations of the dynamicalvariables:

q(t) → q′(t) = f(q(t), q(t)), (8.2)

where f(q(t), q(t)) is some functional of q(t). Such transformations are called sym-metry transformations. Thereby it is important that the equations of motion arenot used when establishing the invariance of the action under (8.2).

If the action is subjected successively to two symmetry transformations , the re-sult is again a symmetry transformation. Thus, symmetry transformations form agroup called the symmetry group of the system. For infinitesimal symmetry trans-formations (8.2), the difference

δsq(t) ≡ q′(t)− q(t) (8.3)

will be called a symmetry variation. It has the general form

δsq(t) = ǫ∆(q(t), q(t), t). (8.4)

619

620 8 Continuous Symmetries and Conservation Laws. Noether’s Theorem

Symmetry variations must not be confused with ordinary variations δq(t) used inSection 1.1 to derive the Euler-Lagrange equations (1.8). While the ordinary vari-ations δq(t) vanish at initial and final times, δq(tb) = δq(ta) = 0 [recall (1.4)], thesymmetry variations δsq(t) are usually nonzero at the ends.

Let us calculate the change of the action under a symmetry variation (8.4). Usingthe chain rule of differentiation and an integration by parts, we obtain

δsA =∫ tb

ta

dt

[

∂L

∂q(t)− ∂t

∂L

∂q(t)

]

δsq(t) +∂L

∂q(t)δsq(t)

∣

∣

∣

∣

∣

tb

ta

. (8.5)

For orbits q(t) that satisfy the Euler-Lagrange equations (1.8), only boundary termssurvive, and we are left with

δsA = ǫ∂L

∂q∆(q, q, t)

∣

∣

∣

∣

∣

ta

tb

. (8.6)

Under the symmetry assumption, δsA vanishes for any orbit q(t), implying that thequantity

Q(t) ≡∂L

∂q∆(q, q, t) (8.7)

is the same at times t = ta and t = tb. Since tb is arbitrary, Q(t) is independent ofthe time t, i.e., it satisfies

Q(t) ≡ Q. (8.8)

It is a conserved quantity , a constant of motion. The expression on the right-handside of (8.7) is called Noether charge.

The statement can be generalized to transformations δsq(t) for which the actionis not directly invariant but its symmetry variation is equal to an arbitrary boundaryterm:

δsA = ǫΛ(q, q, t)∣

∣

∣

tb

ta. (8.9)

In this case,

Q(t) =∂L

∂q∆(q, q, t)− Λ(q, q, t) (8.10)

is a conserved Noether charge.It is also possible to derive the constant of motion (8.10) without invoking the

action, but starting from the Lagrangian. For it we evaluate the symmetry variationas follows:

δsL≡L (q+δsq, q+δsq)−L(q, q)=

[

∂L

∂q(t)−∂t

∂L

∂q(t)

]

δsq(t)+d

dt

[

∂L

∂q(t)δsq(t)

]

. (8.11)

8.1 Point Mechanics 621

On account of the Euler-Lagrange equations (1.8), the first term on the right-handside vanishes as before, and only the last term survives. The assumption of invarianceof the action up to a possible surface term in Eq. (8.9) is equivalent to assumingthat the symmetry variation of the Lagrangian is a total time derivative of somefunction Λ(q, q, t):

δsL(q, q, t) = ǫd

dtΛ(q, q, t). (8.12)

Inserting this into the left-hand side of (8.11), we find

ǫd

dt

[

∂L

∂q∆(q, q, t)− Λ(q, q, t)

]

= 0, (8.13)

thus recovering again the conserved Noether charge (8.8).The existence of a conserved quantity for every continuous symmetry is the

content of Noether’s theorem [1].

8.1.2 Alternative Derivation

Let us do the substantial variation in Eq. (8.5) explicitly, and change a classical orbitqc(t), that extremizes the action, by an arbitrary variation δaq(t). If this does not

vanish at the boundaries, the action changes by a pure boundary term that followsdirectly from (8.5):

δaA =∂L

∂qδaq

∣

∣

∣

∣

∣

tb

ta

. (8.14)

From this equation we can derive Noether’s theorem in yet another way. Supposewe subject a classical orbit to a new type of symmetry variation, to be called local

symmetry transformations , which generalizes the previous symmetry variations (8.4)by making the parameter ǫ time-dependent:

δtsq(t) = ǫ(t)∆(q(t), q(t), t). (8.15)

The superscript t of δtsq(t) indicates the new time dependence in the parameter ǫ(t).These variations may be considered as a special set of the general variations δaq(t)introduced above. Thus also δtsA must be a pure boundary term of the type (8.14).For the subsequent discussion it is useful to introduce the infinitesimally transformedorbit

qǫ(t) ≡ q(t) + δtsq(t) = q(t) + ǫ(t)∆(q(t), q(t), t), (8.16)

and the associated Lagrangian:

Lǫ ≡ L(qǫ(t), qǫ(t)). (8.17)

Using the time-dependent parameter ǫ(t), the local symmetry variation of the actioncan be written as

δtsA =∫ tb

ta

dt

[

∂Lǫ

∂ǫ(t)−

d

dt

∂Lǫ

∂ǫ(t)

]

ǫ(t) +d

dt

[

∂Lǫ

∂ǫ

]

ǫ(t)

∣

∣

∣

∣

∣

tb

ta

. (8.18)


Along the classical orbits, the action is extremal and satisfies the equation

δA

δǫ(t)= 0, (8.19)

which translates for a local action to an Euler-Lagrange type of equation:

∂Lǫ

∂ǫ(t)−

d

dt

∂Lǫ

∂ǫ(t)= 0. (8.20)

This can also be checked explicitly by differentiating (8.17) according to the chainrule of differentiation:

∂Lǫ

∂ǫ(t)=

∂Lǫ

∂q(t)∆(q, q, t) +

∂Lǫ

∂q(t)∆(q, q, t); (8.21)

∂Lǫ

∂ǫ(t)=

∂Lǫ

∂q(t)∆(q, q, t), (8.22)

and inserting on the right-hand side the ordinary Euler-Lagrange equations (1.8).We now invoke the symmetry assumption that the action is a pure surface term

under the time-independent transformations (8.15). This implies that

∂Lǫ

∂ǫ=

d

dtΛ. (8.23)

Combining this with (8.20), we derive a conservation law for the charge:

Q =∂Lǫ

∂ǫ− Λ. (8.24)

Inserting here Eq. (8.22), we find that this is the same charge as that derived by theprevious method.

8.2 Displacement and Energy Conservation

As a simple but physically important example consider the case that the Lagrangiandoes not depend explicitly on time, i.e., that L(q, q, t) ≡ L(q, q). Let us perform atime translation on the coordinate frame:

t′ = t− ǫ. (8.25)

In the new coordinate frame, the same orbit has the new description

q(t′) = q(t), (8.26)

i.e., the orbit q(t) at the translated time t′ is precisely the same as the orbit q(t) atthe original time t. If we replace the argument of q(t) in (8.26) by t′, we describe a

8.2 Displacement and Energy Conservation 623

time-translated orbit in terms of the original coordinates. This implies the symmetryvariation of the form (8.4):

δsq(t) = q′(t)− q(t) = q(t′ + ǫ)− q(t)

= q(t′) + ǫq(t′)− q(t) = ǫq(t). (8.27)

The symmetry variation of the Lagrangian is in general

δsL = L(q′(t), q′(t))− L(q(t), q(t)) =∂L

∂qδsq(t) +

∂L

∂qδsq(t). (8.28)

Inserting δsq(t) from (8.27) we find, without using the Euler-Lagrange equation,

δsL = ǫ

(

∂L

∂qq +

∂L

∂qq

)

= ǫd

dtL. (8.29)

This has precisely the form of Eq. (8.12), with Λ = L as expected, since timetranslations are symmetry transformations. Here the function Λ in (8.12) happensto coincide with the Lagrangian.

According to Eq. (8.10), we find the Noether charge

Q =∂L

∂qq − L(q, q) (8.30)

to be a constant of motion. This is recognized as the Legendre transform of theLagrangian which is, of course, the Hamiltonian of the system. s

Let us briefly check how this Noether charge is obtained from the alternativeformula (8.10). The time-dependent symmetry variation is here

δtsq(t) = ǫ(t)q(t), (8.31)

under which the Lagrangian is changed by

δtsL =∂L

∂qǫq +

∂L

∂q(ǫq + ǫq) =

∂Lǫ

∂ǫǫ+

∂Lǫ

∂ǫǫ, (8.32)

with∂Lǫ

∂ǫ=∂L

∂qq (8.33)

and∂Lǫ

∂ǫ=∂L

∂qq +

∂L

∂qǫq =

d

dtL. (8.34)

This shows that time translations fulfill the symmetry condition (8.23), and thatthe Noether charge (8.24) coincides with the Hamiltonian found in Eq. (8.10).


8.3 Momentum and Angular Momentum

While the conservation law of energy follows from the symmetry of the action undertime translations, conservation laws of momentum and angular momentum are foundif the action is invariant under translations and rotations.

Consider a Lagrangian of a point particle in a euclidean space

L = L(xi(t), xi(t), t). (8.35)

In contrast to the previous discussion of time translation invariance, which wasapplicable to systems with arbitrary Lagrange coordinates q(t), we denote the co-ordinates here by xi to emphasize that we now consider cartesian coordinates. Ifthe Lagrangian does depend only on the velocities xi and not on the coordinates xi

themselves, the system is translationally invariant . If it depends, in addition, onlyon x

2 = xixi, it is also rotationally invariant.The simplest example is the Lagrangian of a point particle of massm in euclidean

space:

L =m

2x2. (8.36)

It exhibits both invariances, leading to conserved Noether charges of momentumand angular momentum, as we now demonstrate.

8.3.1 Translational Invariance in Space

Under a spatial translation, the coordinates xi change to

x′i = xi + ǫi, (8.37)

where ǫi are small numbers. The infinitesimal translations of a particle path are[compare (8.4)]

δsxi(t) = ǫi. (8.38)

Under these, the Lagrangian changes by

δsL = L(x′i(t), x′i(t), t)− L(xi(t), xi(t), t)

=∂L

∂xiδsx

i =∂L

∂xiǫi = 0. (8.39)

By assumption, the Lagrangian is independent of xi, so that the right-hand sidevanishes. This has to be compared with the symmetry variation of the Lagrangianaround the classical orbit, calculated via the chain rule, and using the Euler-Lagrange equation:

δsL =

(

∂L

∂xi−

d

dt

∂L

∂xi

)

δsxi +

d

dt

[

∂L

∂xiδsx

i

]

=d

dt

[

∂L

∂xi

]

ǫi. (8.40)

8.3 Momentum and Angular Momentum 625

This has the form (8.6), from which we extract a conserved Noether charge (8.7) foreach coordinate xi:

pi =∂L

∂xi. (8.41)

These are simply the canonical momenta of the system.

8.3.2 Rotational Invariance

Under rotations, the coordinates xi change to

x′i = Rijx

j , (8.42)

where Rij is an orthogonal 3× 3 -matrix. Infinitesimally, this can be written as

Rij = δij − ωkǫkij, (8.43)

where ! is an infinitesimal rotation vector. The corresponding rotation of a particlepath is

δsxi(t) = x′i(t)− xi(t) = −ωkǫkijx

j(τ). (8.44)

It is useful to introduce the antisymmetric infinitesimal rotation tensor

ωij ≡ ωkǫkij , (8.45)

in terms of which

δsxi = −ωijx

j . (8.46)

Then we can write the change of the Lagrangian under δsxi,

δsL = L(x′i(t), x′i(t), t)− L(xi(t), xi(t), t)

=∂L

∂xiδsx

i +∂L

∂xiδsx

i, (8.47)

as

δsL = −

(

∂L

∂xixj +

∂L

∂xixj)

ωij = 0. (8.48)

If the Lagrangian depends only on the rotational invariants x2, x2,x · x, and on

powers thereof, the right-hand side vanishes on account of the antisymmetry of ωij.This ensures the rotational symmetry.

We now calculate once more the symmetry variation of the Lagrangian via thechain rule and find, using the Euler-Lagrange equations,

δsL =

(

∂L

∂xi−

d

dt

∂L

∂xi

)

δsxi +

d

dt

[

∂L

∂xiδsx

i

]

= −d

dt

[

∂L

∂xixj]

ωij =1

2

d

dt

[

xi∂L

∂xj− (i↔ j)

]

ωij. (8.49)


The right-hand side yields the conserved Noether charges of type (8.7), one for eachantisymmetric pair i, j:

Lij = xi∂L

∂xj− xj

∂L

∂xi≡ xipj − xjpi. (8.50)

These are the antisymmetric components of angular momentum.Had we worked with the original vector form of the rotation angles ωk, we would

have found the angular momentum in the more common form:

Lk =1

2ǫkijL

ij = (x× p)k. (8.51)

The quantum-mechanical operators associated with these, after replacing pi →−i∂/∂xi, have the well-known commutation rules

[Li, Lj ] = iǫijkLk. (8.52)

In the tensor notation (8.50), these become

[Lij , Lkl] = −i(

δikLjl − δilLjk + δjlLik − δjkLil

)

. (8.53)

8.3.3 Center-of-Mass Theorem

Consider now the transformations corresponding to a uniform motion of the coor-dinate system. We shall study the behavior of a set of free massive point particlesin euclidean space described by the Lagrangian

L(xi) =∑

n

mn

2x2n. (8.54)

Under Galilei transformations, the spatial coordinates and the time are changedto

xi(t) = xi(t)− vit, (8.55)

t′ = t,

where vi is the relative velocity along the ith axis. The infinitesimal symmetryvariations are

δsxi(t) = xi(t)− xi(t) = −vit, (8.56)

which change the Lagrangian by

δsL = L(xi − vit, xi − vi)− L(xi, xi). (8.57)

Inserting the explicit form (8.54), we find

δsL =∑

n

mn

2

[

(xin − vi)2 − (xni)2]

. (8.58)


This can be written as a total time derivative:

δsL =d

dtΛ =

d

dt

∑

n

mn

[

−xinvi +

v2

2t

]

, (8.59)

proving that Galilei transformations are symmetry transformations in the Noethersense. By assumption, the velocities vi in (8.55) are infinitesimal, so that the secondterm can be ignored.

By calculating δsL once more via the chain rule with the help of the Euler-Lagrange equations, and by equating the result with (8.59), we find the conservedNoether charge

Q =∑

n

∂L

∂xiδsx

i − Λ

=

(

−∑

n

mnxint+

∑

n

mnxin

)

vi. (8.60)

Since the direction of the velocity vi is arbitrary, each component is separately aconstant of motion:

N i = −∑

n

mnxit+

∑

n

mnxni = const. (8.61)

This is the well-known center-of-mass theorem [2]. Indeed, introducing the center-of-mass coordinates

xiCM ≡

∑

nmnxni

∑

nmn

, (8.62)

and the associated velocities

viCM =

∑

nmnxni

∑

nmn

, (8.63)

the conserved charge (8.61) can be written as

N i =∑

n

mn(−viCMt + xiCM). (8.64)

The time-independence of N i implies that the center-of-mass moves with uniformvelocity according to the law

xiCM(t) = xi0CM + viCMt, (8.65)

where

xi0CM =N i

∑

nmn

(8.66)

is the position of the center of mass at t = 0.Note that in non-relativistic physics, the center-of-mass theorem is a consequence

of momentum conservation since momentum ≡ mass × velocity. In relativisticphysics, this is no longer true.


8.3.4 Conservation Laws Resulting from Lorentz Invariance

In relativistic physics, particle orbits are described by functions in spacetime

xµ(τ), (8.67)

where τ is an arbitrary Lorentz-invariant parameter. The action is an integral oversome Lagrangian:

A =∫

dτL (xµ(τ), xµ(τ), τ) , (8.68)

where xµ(τ) denotes the derivative with respect to the parameter τ . If the La-grangian depends only on invariant scalar products xµxµ, x

µxµ, xµxµ, then it is

invariant under Lorentz transformations

xµ → xµ = Λµν x

ν , (8.69)

where Λµν is a 4× 4 matrix satisfying

ΛgΛT = g, (8.70)

with the Minkowski metric

gµν =

1−1

−1−1

. (8.71)

For a free massive point particle in spacetime, the Lagrangian is

L(x(τ)) = −Mc√

gµν xµxν . (8.72)

It is reparametrization invariant under τ → f(τ), with an arbitrary function f(τ).Under translations

δsxµ(τ) = xµ(τ)− ǫµ(τ), (8.73)

the Lagrangian is obviously invariant, satisfying δsL=0. Calculating this variationonce more via the chain rule with the help of the Euler-Lagrange equations, we find

0 =∫ τν

τµ

dτ

(

∂L

∂xµδsx

µ +∂L

∂xµδsx

µ

)

= −ǫµ∫ τν

τµ

dτd

dτ

(

∂L

∂xµ

)

. (8.74)

From this we obtain the Noether charges

pµ ≡ −∂L

∂xµ=Mc

xµ(τ)√

gµν xµxν=Mcuµ, (8.75)


which satisfy the conservation law

d

dτpµ(t) = 0. (8.76)

They are the conserved four-momenta of a free relativistic particle. The quantity

uµ ≡xµ

√

gµν xµxν(8.77)

is the dimensionless relativistic four-velocity of the particle. It has the propertyuµuµ = 1, and it is reparametrization invariant. By choosing for τ the physical timet = x0/c, we can express uµ in terms of the physical velocities vi = dxi/dt as

uµ = γ(1, vi/c), with γ ≡√

1− v2/c2. (8.78)

Note the minus sign in the definition (8.75) of the canonical momentum withrespect to the nonrelativistic case. It is necessary to write Eq. (8.75) covariantly.The derivative with respect to xµ transforms like a covariant vector with a subscriptµ, whereas the physical momenta are pµ.

For small Lorentz transformations near the identity we write

Λµν = δµν + ωµ

ν , (8.79)

whereωµ

ν = gµλωλν (8.80)

is an arbitrary infinitesimal antisymmetric matrix. An infinitesimal Lorentz trans-formation of the particle path is

δsxµ(τ) = xµ(τ)− xµ(τ)

= ωµνx

ν(τ). (8.81)

Under it, the symmetry variation of a Lorentz-invariant Lagrangian vanishes:

δsL =

(

∂L

∂xµxν +

∂L

∂xµxν)

ωµν = 0. (8.82)

This has to be compared with the symmetry variation of the Lagrangian calculatedvia the chain rule with the help of the Euler-Lagrange equation

δsL =

(

∂L

∂xµ−

d

dτ

∂L

∂xµ

)

δsxµ +

d

dτ

[

∂L

∂xµδsx

µ

]

=d

dτ

[

∂L

∂xµxν]

ωµν

=1

2ωµ

ν d

dτ

(

xµ∂L

∂xν− xν

∂L

∂xµ

)

. (8.83)


By equating this with (8.82), we obtain the conserved rotational Noether charges[containing again a minus sign as in (8.75)]:

Lµν = −xµ∂L

∂xν+ xν

∂L

∂xµ= xµpν − xνpµ. (8.84)

They are four-dimensional generalizations of the angular momenta (8.50). Thequantum-mechanical operators

Lµν ≡ i(xµ∂ν − xν∂µ) (8.85)

obtained after the replacement pµ → i∂/∂xµ satisfy the four-dimensional spacetimegeneralization of the commutation relations (8.53):

[Lµν , Lκλ] = i(

gµκLνλ − gµλLνκ + gνλLµκ − gνκLµλ)

. (8.86)

The quantities Lij coincide with the earlier-introduced angular momenta (8.50).The conserved components

L0i = x0pi − xip0 ≡Mi (8.87)

yield the relativistic generalization of the center-of-mass theorem (8.61):

Mi = const. (8.88)

8.4 Generating the Symmetry Transformations

As mentioned in the introduction to this chapter, the relation between invariancesand conservation laws has a second aspect. With the help of Poisson brackets,the charges associated with continuous symmetry transformations can be used togenerate the symmetry transformation from which they were derived. Explicitly,

δsx = −iǫ[Q, x(t)]. (8.89)

The charge derived in Section 7.2 from the invariance of the system under timedisplacement is the most famous example for this property. The charge (8.30) is bydefinition the Hamiltonian,

Q ≡ H,

whose operator version generates infinitesimal time displacements by the Heisenbergequation of motion:

˙x(t) = −i[H, x(t)]. (8.90)

This equation is obviously the same as (8.89).To quantize the system canonically, we may assume the Lagrangian to have the

standard form

L(x, x) =M

2x2 − V (x), (8.91)

8.4 Generating the Symmetry Transformations 631

so that the Hamiltonian operator becomes, with the canonical momentum p ≡ x:

H =p2

2M+ V (x). (8.92)

Equation (8.90) is then a direct consequence of the canonical equal-time commuta-tion rules

[p(t), x(t)] = −i, [p(t), p(t)] = 0, [x(t), x(t)] = 0. (8.93)

The charges (8.41), derived in Section 7.3 from translational symmetry, are an-other famous example. After quantization, the commutation rule (8.89) becomes,with (8.38),

ǫj = iǫi[pi(t), xj(t)]. (8.94)

This coincides with one of the canonical commutation relations (here it appears onlyfor time-independent momenta, since the system is translationally invariant).

The relativistic charges (8.75) of spacetime generate translations via

δsxµ = ǫµ = −iǫν [pν(t), x

µ(τ)], (8.95)

in agreement with the relativistic canonical commutation rules (29.27).Similarly we find that the quantized versions of the conserved charges Li in

Eq. (8.51) generate infinitesimal rotations:

δsxj = −ωiǫijkx

k(t) = iωi[Li, xj(t)], (8.96)

whereas the quantized conserved charges N i of Eq. (8.61) generate infinitesimalGalilei transformations, and that the charges Mi of Eq. (8.87) generate pure rota-tional Lorentz transformations:

δsxj = ǫix

0 = iǫi[Mi, xj ],

δsx0 = ǫix

i = iǫi[Mi, x0]. (8.97)

Since the quantized charges generate the rotational symmetry transformations,they form a representation of the generators of the symmetry group. When com-muted with each other, they obey the same commutation rules as the generatorsof the symmetry group. The charges (8.51) associated with rotations, for example,have the commutation rules

[Li, Lj ] = iǫijkLj, (8.98)

which are the same as those between the 3 × 3 generators of the three-dimensionalrotations (Li)jk = −iǫijk.

The quantized charges of the generators (8.84) of the Lorentz group satisfy thecommutation rules (8.86) of the 4× 4 generators (8.85)

[Lµν , Lµλ] = −igµµLνλ. (8.99)

This follows directly from the canonical commutation rules (8.95) [i.e., (29.27)].


8.5 Field Theory

A similar relation between continuous symmetries and constants of motion holds infield theory.

8.5.1 Continuous Symmetry and Conserved Currents

Let A be the action of an arbitrary field ϕ(x),

A =∫

d4xL(ϕ, ∂ϕ, x), (8.100)

and suppose that a transformation of the field

δsϕ(x) = ǫ∆(ϕ, ∂ϕ, x) (8.101)

changes the Lagrangian density L merely by a total derivative

δsL = ǫ∂µΛµ, (8.102)

or equivalently, that it changes the action A by a surface term

δsA = ǫ∫

d4x ∂µΛµ. (8.103)

Then δsL is called a symmetry transformation.Given such a symmetry transformation, we can find a current four-vector

jµ =∂L

∂∂µϕ∆− Λµ (8.104)

that has no four-divergence

∂µjµ(x) = 0. (8.105)

The expression on the right-hand side of (8.104) is called a Noether current ,and (8.105) is referred to as the associated current conservation law . It is a local

conservation law .The proof of (8.105) is just as simple as that of the time-independence of the

charge (8.10) associated with the corresponding symmetry of the mechanical action(8.1) in Section 8.1. We calculate the symmetry variation of L under the symmetrytransformation in a similar way as in Eq. (8.11), and find

δsL =

(

∂L

∂ϕ− ∂µ

∂L

∂∂µϕ

)

δsϕ+ ∂µ

(

∂L

∂∂µϕδsϕ

)

= ǫ

(

∂L

∂ϕ− ∂µ

∂L

∂∂µϕ

)

∆+ ∂µ

(

∂L

∂∂µϕ∆

)

. (8.106)

8.5 Field Theory 633

Then we invoke the Euler-Lagrange equation to remove the first term. Equating thesecond term with (8.102), we obtain

∂µjµ ≡ ∂µ

(

∂L

∂∂µϕ∆− Λµ

)

= 0. (8.107)

The relation between continuous symmetries and conservation is called Noether’s

theorem [1].Assuming all fields to vanish at spatial infinity, we can derive from the local law

(8.107) a global conservation law for the charge that is obtained from the spatialintegral over the charge density j0:

Q(t) =∫

d3x j0(x, t). (8.108)

Indeed, we may write the time derivative of the charge as an integral

d

dtQ(t) =

∫

d3x ∂0j0(x, t) (8.109)

and adding on the right-hand side a spatial integral over a total three-divergence,which vanishes due to the boundary conditions, we find

d

dtQ(t) =

∫

d3x ∂0j0(x, t) =

∫

d3x [∂0j0(x, t) + ∂ij

i(x, t)] = 0. (8.110)

Thus the charge is conserved:

d

dtQ(t) = 0. (8.111)

8.5.2 Alternative Derivation

There is again an alternative derivation of the conserved current that is analogousto Eqs. (8.15)–(8.24). It is based on a variation of the fields under symmetry trans-formations whose parameter ǫ is made artificially spacetime-dependent ǫ → ǫ(x),thus extending (8.15) to

δxs ϕ(x) = ǫ(x)∆(ϕ(x), ∂µϕ(x), x). (8.112)

As before in Eq. (8.17), we calculate the Lagrangian density for a slightly trans-formed field

ϕǫ(x) ≡ ϕ(x) + δxs ϕ(x), (8.113)

calling itLǫ ≡ L(ϕǫ(t), ∂ϕǫ(t)). (8.114)

The corresponding action differs from the original one by

δxsA =∫

dx

{[

∂Lǫ

∂ǫ(x)− ∂µ

∂Lǫ

∂∂µǫ(x)

]

δǫ(x) + ∂µ

[

∂Lǫ

∂∂µǫ(x)δǫ(x)

]}

. (8.115)


From this we obtain the Euler-Lagrange-like equation

∂Lǫ

∂ǫ(x)− ∂µ

∂Lǫ

∂∂µǫ(x)= 0. (8.116)

By assumption, the action is a pure surface term under x-independent transforma-tions, implying that

∂Lǫ

∂ǫ(x)= ∂µΛ

µ. (8.117)

Together with (8.116), we see that

jµ =∂δxs L

∂∂µǫ(x)− Λµ (8.118)

has no four-divergence. By the chain rule of differentiation we calculate

δtsL =∂L(x)

∂ϕǫ∆+

∂L

∂∂νϕ∂νǫ∆, (8.119)

and see that

∂Lǫ

∂∂µǫ(x)=

∂L

∂∂µϕ∆(ϕ, ∂ϕ, x), (8.120)

so that the current (8.118) coincides with (8.104).

8.5.3 Local Symmetries

If we apply the alternative derivation of a conserved current to a local symmetry,such as a local gauge symmetry, the current density (8.118) vanishes identically.Let us illuminate the symmetry origin of this phenomenon.

To be specific, we consider directly the field theory of electrodynamics. Thetheory does have a conserved charge resulting from the global U(1)-symmetry of thematter Lagrangian. There is a conserved current which is the source of a masslessparticle, the photon. This is described by a gauge field which is minimally coupledto the conserved current. A similar structure exists for many internal symmetriesgiving rise to nonabelian versions of the photon, such as gluons, whose exchangecauses the strong interactions, and W - and Z-vector mesons, which mediate theweak interactions. It is useful to reconsider Noether’s derivation of conservationlaws in such theories.

The conserved matter current in a locally gauge-invariant theory cannot be foundany more by the rule (8.118), which was so useful in the globally invariant theory. Forthe gauge transformation of quantum electrodynamics, the derivative with respectto the local field transformation ǫ(x) would simply be given by

jµ =δL

∂∂µΛ. (8.121)

8.5 Field Theory 635

This would be identically equal to zero, due to local gauge invariance. We may,however, subject just the matter field to a local gauge transformation at fixed gaugefields. Then we obtain the correct current

jµ ≡∂L

∂∂µΛ

∣

∣

∣

∣

∣

em

. (8.122)

Since the complete change under local gauge transformations δxsL vanishes identi-cally, we can alternatively vary only the gauge fields and keep the particle orbitfixed:

jµ = −∂L

∂∂µΛ

∣

∣

∣

∣

∣

m

. (8.123)

This is done most simply by forming the functional derivative with respect to the

gauge field, thereby omitting the contribution ofem

L :

jµ = −∂m

L

∂∂µΛ. (8.124)

An interesting consequence of local gauge invariance can be found for the gaugefield itself. If we form the variation of the pure gauge field action

δsem

A =∫

d4x tr

δxsAµ

em

A

δAµ

, (8.125)

and insert for δxsA an infinitesimal pure gauge field configuration

δxsAµ = −i∂µΛ(x), (8.126)

the right-hand side must vanish for all Λ(x). After a partial integration this impliesthe local conservation law ∂µj

µ(x) = 0 for the current:

jµ(x) = −iδem

A

δAµ

. (8.127)

In contrast to the earlier conservation laws derived for matter fields, which were validonly if the matter fields obey the Euler-Lagrange equations, the current conservationlaw for gauge fields is valid for all field configurations. It is an identity which wemay call Bianchi identity due to its close analogy with the Bianchi identities inRiemannian geometry.

To verify the conservation of (12.63), we insert the Lagrangian (12.3) into (12.63)and find jν = ∂µF

µν/2. This current is trivially conserved for any field configurationdue to the antisymmetry of F µν .


8.6 Canonical Energy-Momentum Tensor

As an important example for the field theoretic version of the theorem, consider theusual case that the Lagrangian density does not depend explicitly on the spacetimecoordinates x:

L = L(ϕ, ∂ϕ). (8.128)

We then perform a translation along an arbitrary direction ν = 0, 1, 2, 3 of spacetime

x′µ = xµ − ǫµ, (8.129)

under which field ϕ(x) transforms as

ϕ′(x′) = ϕ(x). (8.130)

This equation expresses the fact that the field has the same value at the sameabsolute point in space and time, which in one coordinate system is labeled by thecoordinates xµ and in the other by x′µ.

Under an infinitesimal translation of the field configuration coordinate, the La-grangian density undergoes the following symmetry variation

δsL ≡ L(ϕ′(x), ∂ϕ′(x))− L(ϕ(x), ∂ϕ(x))

=∂L

∂ϕ(x)δsϕ(x) +

∂L

∂∂µϕ∂µδsϕ(x), (8.131)

where

δsϕ(x) = ϕ′(x)− ϕ(x) (8.132)

is the symmetry variation of the fields. For the particular transformation (8.130)the symmetry variation becomes simply

δsϕ(x) = ǫν∂νϕ(x). (8.133)

The Lagrangian density (8.128) changes by

δsL(x) = ǫν∂νL(x). (8.134)

Hence the requirement (8.103) is satisfied and δsϕ(x) is a symmetry transformation.The function Λ happens to coincide with the Lagrangian density

Λ = L. (8.135)

We can now define a set of currents jνµ, one for each ǫν . In the particular case at

hand, the currents jνµ are denoted by Θν

µ, and read:

Θνµ =

∂L

∂∂µϕ∂νϕ− δν

µL. (8.136)

8.6 Canonical Energy-Momentum Tensor 637

They have no four-divergence

∂µΘνµ(x) = 0. (8.137)

As a consequence, the total four-momentum of the system, defined by

P µ =∫

d3xΘµ0(x), (8.138)

is independent of time.The alternative derivation of the currents goes as follows. Introducing

δxs ϕ(x) = ǫν(x)∂νϕ(x), (8.139)

we see thatδxs ϕ(x) = ϕν(x)∂νϕ(x). (8.140)

On the other hand, the chain rule of differentiation yields

δxsL =∂L

∂ϕ(x)ǫν(x)∂νϕ(x) +

∂L

∂∂µϕ(x){[∂µǫ

ν(x)]∂νϕ+ ǫν∂µ∂νϕ(x)} . (8.141)

Hence∂Lǫ

∂∂µǫν(x)=

∂L

∂∂µϕ∂νϕ, (8.142)

and we obtain once more the energy-momentum tensor (8.136).Note that (8.142) can also be written as

∂Lǫ

∂∂µǫν(x)=

∂L

∂∂µϕ

∂δxs ϕ

∂ǫν(x). (8.143)

Since ν is a contravariant vector index, the set of currents Θνµ forms a Lorentz

tensor called the canonical energy-momentum tensor . The component

Θ00 =

∂L

∂∂0ϕ∂0ϕ−L (8.144)

is recognized to be the Hamiltonian density in the canonical formalism.

8.6.1 Electromagnetism

As an important physical application of the field theoretic Noether theorem, considerthe free electromagnetic field with the action

L = −1

4cFλκF

λκ, (8.145)

where Fλκ are the components of the field strength Fλκ ≡ ∂λAκ − ∂κAλ. Under atranslation in space and time from xµ to xµ − ǫδµν , the vector potential undergoes asimilar change as in (8.130):

A′µ(x′) = Λµ(x). (8.146)


As before, this equation expresses the fact that at the same absolute spacetimepoint, which in the two coordinate frames is labeled once by x′ and once by x, thefield components have the same numerical values. The equation transformation law(8.146) can be rewritten in an infinitesimal form as

δsAλ(xµ) ≡ A′λ(xµ)− Aλ(xµ)

= A′λ(x′µ + ǫδνµ)− Aλ(xµ) (8.147)

= ǫ∂νAλ(xµ). (8.148)

Under it, the field tensor changes as follows

δsFλκ = ǫ∂νF

λκ, (8.149)

so that the Lagrangian density is a total four-divergence:

δsL = −ǫ1

2cFλκ∂νF

λκ = ǫ∂νL (8.150)

Thus, the spacetime translations (8.148) are symmetry transformations, and thecurrents

Θνµ =

∂L

∂∂µAλ∂νA

λ − δνµL (8.151)

are conserved:

∂µΘνµ(x) = 0. (8.152)

Using ∂L/∂∂µAλ = −F µ

λ, the currents (8.151) become more explicitly

Θνµ = −

1

c

(

F µλ∂νA

λ −1

4δν

µF λκFλκ

)

. (8.153)

They form the canonical energy-momentum tensor of the electromagnetic field.

8.6.2 Dirac Field

We now turn to the Dirac field which has the well-known action

A =∫

d4xL(x) =∫

d4xψ(x)(iγµ↔

∂ µ −M)ψ(x), (8.154)

where γµ are the Dirac matrices

γµ =

(

0 σµ

σµ 0

)

. (8.155)

Here σµ, σµ are four 2× 2 matrices

σµ ≡ (σ0, σi).σµ ≡ (σ0,−σi), (8.156)

8.6 Canonical Energy-Momentum Tensor 639

whose zeroth component is the unit matrix

σ0 =

(

1 00 1

)

, (8.157)

and whose spatial components consist of the Pauli spin matrices

σ1 =

(

0 11 0

)

, σ2 =

(

0 −ii 0

)

, σ3 =

(

1 00 −1

)

. (8.158)

On behalf of the algebraic properties of the Pauli matrices

σiσj = δij + iǫijkσk, (8.159)

the Dirac matrices (8.155) satisfy the anticommutation rules

{γµ, γν} = 2gµν. (8.160)

Under spacetime translations

x′µ = xµ − ǫµ, (8.161)

the Dirac field transforms in the same way as the previous scalar and vector fields:

ψ′(x′) = ψ(x), (8.162)

or infinitesimally:δsψ(x) = ǫµ∂µψ(x). (8.163)

The same is true for the Lagrangian density, where

L′(x′) = L(x), (8.164)

andδsL(x) = ǫµ∂µL(x). (8.165)

Thus we obtain the Noether current

Θνµ =

∂L

∂∂µψλ∂νψ

λ + c.c.− δνµL, (8.166)

with the local conservation law

∂µΘνµ(x) = 0. (8.167)

From (8.154), we see that∂L

∂∂µψλ=

1

2ψγµ, (8.168)

so that we obtain the canonical energy-momentum tensor of the Dirac field:

Θνµ =

1

2ψγµ∂νψ

λ + c.c.− δνµL (8.169)


8.7 Angular Momentum

Let us now turn to angular momentum in field theory. Consider first the case of ascalar field ϕ(x). Under a rotation of the coordinates,

x′i = Rijx

j , (8.170)

the field does not change, if considered at the same space point, i.e.,

ϕ′(x′i) = ϕ(xi). (8.171)

The infinitesimal symmetry variation is:

δsϕ(x) = ϕ′(x)− ϕ(x). (8.172)

Using the infinitesimal form (8.46) of (8.170),

δxi = −ωijxj , (8.173)

we see that

δsϕ(x) = ϕ′(x0, x′i − δxi)− ϕ(x)

= ∂iϕ(x)xjωij. (8.174)

Suppose we are dealing with a Lorentz-invariant Lagrangian density that has noexplicit x-dependence:

L = L(ϕ(x), ∂ϕ(x)). (8.175)

Then the symmetry variation is

δsL = L(ϕ′(x), ∂ϕ′(x))− ϕ(ϕ(x), ∂ϕ(x))

=∂L

∂ϕ(x)δsϕ(x) +

∂ϕ

∂∂ϕ(x)∂µδsϕ(x). (8.176)

For a Lorentz-invariant L, the derivative ∂L/∂∂µϕ is a vector proportional to ∂µϕ.For the Lagrangian density, the rotational symmetry variation Eq. (8.174) be-

comes

δsL =

[

∂L

∂ϕ∂iϕx

j +∂L

∂µϕ∂µ(∂iLx

j)

]

ωij

=

[

(∂iL)xj +

∂L

∂∂jϕ∂iϕ

]

ωij = ∂i(

Lxjωij

)

. (8.177)

The right-hand side is a total derivative. In arriving at this result, the antisymmetryof ϕij has been used twice: first for dropping the second term in the brackets, which

8.8 Four-Dimensional Angular Momentum 641

is possible since ∂L/∂∂iϕ is proportional to ∂iϕ as a consequence of the assumedrotational invariance1 of L. Second it is used to pull xj inside the last parentheses.

Calculating δsL once more with the help of the Euler-Lagrange equations gives

δsL =∂L

∂Lδsϕ+

∂L

∂∂µϕ∂µδsϕ (8.178)

=

(

∂L

∂ϕ− ∂µ

∂L

∂∂µϕ

)

δsϕ+ ∂µ

(

∂L

∂∂µϕδsϕ

)

= ∂µ

(

∂L

∂∂µϕ∂iϕ xj

)

ωij.

Thus the Noether charges

Lij,µ =

(

∂L

∂∂µϕ∂iϕx

j − δiµL xj

)

− (i↔ j) (8.179)

have no four-divergence

∂µLij,µ = 0. (8.180)

The associated charges

Lij =∫

d3xLij,µ (8.181)

are called the total angular momenta of the field system. In terms of the canonicalenergy-momentum tensor

Θνµ =

∂L

∂∂µϕ∂νϕ− δν

µL, (8.182)

the current density Lij,µ can also be rewritten as

Lij,µ = xiΘjµ − xjΘiµ. (8.183)

8.8 Four-Dimensional Angular Momentum

A similar procedure can be applied to pure Lorentz transformations. An infinitesimalboost to rapidity ζ i produces a coordinate change

x′µ = Λµνx

ν = xµ + δµiζixν + δµ0ζ

ixi. (8.184)

This can be written as

δxµ = ωµνx

ν , (8.185)

1Recall the similar argument after Eq. (8.48)


where ωij = 0,

ω0i = −ωi0 = ζ i. (8.186)

With the tensor ωµν , the restricted Lorentz transformations and the infinitesimal

rotations can be treated on the same footing. The rotations have the form (8.185)for the particular choice

ωij = ǫijkωk,

ω0i = ωi0 = 0. (8.187)

We can now identify the symmetry variations of the field as being

δsϕ(x) = ϕ′(x′µ − δxµ)− ϕ(x)

= −∂µϕ(x)xνωµ

ν . (8.188)

Just as in (8.177), the Lagrangian density transforms as the total derivative

δsϕ = −∂µ(Lxν)ωµ

ν , (8.189)

and we obtain the Noether currents

Lµν,λ = −

(

∂L

∂∂λϕ∂λϕxν − δµλL xν

)

+ (µ↔ ν). (8.190)

The right-hand side can be expressed in terms of the canonical energy-momentumtensor (8.136), so that we find

Lµν,λ = −

(

∂L

∂∂λϕ∂λϕxν − δµλLxν

)

+ (µ ↔ ν)

= xµΘνλ − xνΘµλ. (8.191)

These currents have no four-divergence

∂λLµν,λ = 0. (8.192)

The associated charges

Lµν ≡∫

d3xLµν,0 (8.193)

are independent of time.For the particular form of ωµν in (8.186), we find time-independent components

Li0. The components Lij coincide with the previously-derived angular momenta.The constancy of Li0 is the relativistic version of the center-of-mass theorem

(8.65). Indeed, since

Li0 =∫

d3x (xiΘ00 − x0Θi0), (8.194)

8.9 Spin Current 643

we can define the relativistic center of mass

xiCM =

∫

d3xΘ00xi∫

d3xΘ00, (8.195)

and the average velocity

viCM = cd3xΘi0

∫

d3xΘ00= c

P i

P 0. (8.196)

Since∫

d3xΘi0 = P i is the constant momentum of the system, also viCM is a con-stant. Thus, the constancy of L0i implies the center-of-mass moves with the constantvelocity

xiCM(t) = xi0CM + vi0CMt, (8.197)

with xi0CM = L0i/P 0. The quantities Lµν are referred to as four-dimensional orbitalangular momenta.

It is important to point out that the vanishing divergence of Lµν,λ makes Θνµ

symmetric:

∂λLµν,λ = ∂λ(x

µΘνλ − xνΘµλ)

= Θνµ −Θνµ = 0. (8.198)

Thus, translationally invariant field theories whose orbital angular momentum isconserved have always a symmetric canonical energy-momentum tensor.

Θµν = Θνµ. (8.199)

8.9 Spin Current

If the field ϕ(x) is no longer a scalar but carries spin degrees of freedom, the deriva-tion of the four-dimensional angular momentum becomes slightly more involved.

8.9.1 Electromagnetic Fields

Consider first the case of electromagnetism where the relevant field is the four-vectorpotential Aµ(x). When going to a new coordinate frame

x′µ = Λµνx

ν , (8.200)

the vector field at the same point remains unchanged in absolute spacetime. How-ever, since the components Aµ refer to two different basic vectors in the differentframes, they must be transformed accordingly. Indeed, since Aµ is a vector andtransforms like xµ, it must satisfy the relation characterizing a vector field:

A′µ(x′) = ΛµνA

ν(x). (8.201)


For an infinitesimal transformation

δsxµ = ωµ

νxν , (8.202)

this implies a symmetry variation

δsAµ(x) = A′µ(x)− Aµ(x) = A′µ(x− δx)− Aµ(x)

= ωµνA

ν(x)− ωλνx

ν∂λAµ. (8.203)

The first term is a spin transformation, the other an orbital transformation. Theorbital transformation can also be written in terms of the generators Lµν of theLorentz group defined in (8.84) as

δorbs Aµ(x) = −iωµνLµνA(x). (8.204)

It is convenient to introduce 4×4 spin transformation matrices Lµν with the matrixelements:

(Lµν)λκ ≡ i (gµλgνκ − gµκgνλ) . (8.205)

They satisfy the same commutation relations (8.86) as the differential operators Lµν

defined in Eq. (8.85). By adding together the two generators Lµν and Lµν , we formthe operator of total four-dimensional angular momentum

Jµν ≡ Lµν + Lµν , (8.206)

and can write the symmetry variation (8.203) as

δorbs Aµ(x) = −iωµν JµνA(x). (8.207)

If the Lagrangian density involves only scalar combinations of four-vectors Aµ,and if it has no explicit x-dependence, it changes under Lorentz transformations likea scalar field:

L′(x′) ≡ L(A′(x′), ∂′A′(x′)) = L(A(x), ∂A(x)) ≡ L(x). (8.208)

Infinitesimally, this makes the symmetry variation a pure gradient term:

δsL = −(∂µLxν)ωµ

ν . (8.209)

Thus Lorentz transformations are symmetry transformations in the Noether sense.Following Noether’s construction (8.179), we calculate the current of total four-

dimensional angular momentum:

Jµν,λ =∂L

∂∂λAµ

Aν −

(

∂L

∂∂λAκ∂µAκxν − δµλLxν

)

− (µ↔ ν). (8.210)


The last two terms have the same form as the current Lµν,λ of the four-dimensionalangular momentum of the scalar field. Here they are the currents of the four-

dimensional orbital angular momentum:.

Lµν,λ = −

(

∂L

∂∂λAκ∂µAκxν − δµλLxν

)

+ (µ ↔ ν). (8.211)

Note that this current has the form

Lµν,λ = −i∂L

∂∂λAκLµνAκ +

[

δµλLxν − (µ↔ ν)]

, (8.212)

where Lµν are the differential operators of four-dimensional angular momentum inthe commutation rules (8.86).

Just as the scalar case (8.191), the currents (8.211) can be expressed in terms ofthe canonical energy-momentum tensor as

Lµν,λ = xµΘνλ − xνΘµλ. (8.213)

The first term in (8.210),

Σµν,λ =

[

∂L

∂∂λAν

Aν − (µ↔ ν)

]

, (8.214)

is referred to as the spin current . It can be written in terms of the 4× 4-generators(8.205) of the Lorentz group as

Σµν,λ = −i∂L

∂∂λAκ(Lµν)κσA

σ. (8.215)

The two currents together,

Jµν,λ(x) ≡ Lµν,λ(x) + Σµν,λ(x), (8.216)

are conserved, satisfying ∂λJµν,λ(x) = 0. Individually, they are not conserved.

The total angular momentum is given by the charge

Jµν =∫

d3x Jµν,0(x). (8.217)

It is a constant of motion. Using the conservation law of the energy-momentumtensor we find, just as in (8.198), that the orbital angular momentum satisfies

∂λLµν,λ(x) = − [Θµν(x)−Θνµ(x)] . (8.218)

From this we find the divergence of the spin current

∂λΣµν,λ(x) = − [Θµν(x)−Θνµ(x)] . (8.219)


For the charges associated with orbital and spin currents

Lµν(t) ≡∫

d3xLµν,0(x), Σµν(t) ≡∫

d3xΣµν,0(x), (8.220)

this implies the following time dependence:

Lµν(t) = −∫

d3x [Θµν(x)−Θνµ(x)] ,

Σµν(t) =∫

d3x [Θµν(x)−Θνµ(x)] . (8.221)

Thus fields with a nonzero spin density have always a non-symmetric energy mo-mentum tensor.

In general, the current density Jµν,λ of total angular momentum reads

Jµν,λ =

(

∂δxsL

∂∂λωµν(x)− δµλLxν

)

− (µ↔ ν). (8.222)

By the chain rule of differentiation, the derivative with respect to ∂λωµν(x) can comeonly from field derivatives, for a scalar field

∂δxs L

∂∂λωµν(x)=

∂L

∂∂λϕ

∂δxs ϕ

∂ωµν(x), (8.223)

and for a vector field∂δxsL

∂∂λωµν(x)=

∂L

∂∂λAκ

∂δxsAκ

∂ωµν

. (8.224)

The alternative rule of calculating angular momenta is to introduce spacetime-dependent transformations

δxx = ωµν(x)x

ν , (8.225)

under which the scalar fields transform as

δsϕ = −∂λϕωλν(x)x

ν , (8.226)

and the Lagrangian density as

δxs ϕ = −∂λLωλν(x)x

ν = −∂λ(xνL)ωλ

ν(x). (8.227)

By separating spin and orbital transformations of δxsAκ, we find the two contributions

σµν,λ and Lµν,λ to the current Jµν,λ of the total angular momentum, the latterreceiving a contribution from the second term in (8.222).

8.9.2 Dirac Field

We now turn to the Dirac field. Under a Lorentz transformation (8.200), this trans-forms according to the law

ψ(x′)Λ

−−−→ ψ′Λ(x) =D(Λ)ψ(x), (8.228)


where D(Λ) are the 4×4 spinor representation matrices of the Lorentz group. Theirmatrix elements can most easily be specified for infinitesimal transformations. Foran infinitesimal Lorentz transformation

Λµν = δµ

ν + ωµν , (8.229)

under which the coordinates are changed by

δsxµ = ωµ

νxν , (8.230)

the spin components transform under the representation matrix

D(δµν + ωµ

ν) =(

1− i1

2ωµνσ

µν

)

, (8.231)

where σµν are the 4× 4 matrices acting on the spinor space

σµν =i

2[γµ, γν ]. (8.232)

From the anticommutation rules (8.160), it is easy to verify that the spin matricesSµν ≡ σµν/2 satisfy the same commutation rules (8.86) as the previous orbital and

spin-1 generators Lµνµ and Lµν of Lorentz transformations.The field has the symmetry variation [compare (8.203)]:

δsψ(x) = ψ′(x)− ψ(x) = D(δµν + ωµ

ν)ψ(x− δx)− ψ(x)

= −i1

2ωµνσ

µνψ(x)− ωλνx

ν∂λψ(x)

= −i1

2ωµν

(

Sµν + Lµν)

ψ(x) ≡ −i1

2ωµν J

µνψ(x), (8.233)

the last line showing the separation into spin and orbital transformation for a Diracparticle.

Since the Dirac Lagrangian is Lorentz-invariant, it changes under Lorentz trans-formations like a scalar field [compare (8.208)]:

L′(x′) = L(x). (8.234)

Infinitesimally, this amounts to

δsL = −(∂µLxν)ωµ

ν . (8.235)

With the Lorentz transformations being symmetry transformations in the Noethersense, we calculate the current of total four-dimensional angular momentum extend-ing the formulas (8.191) and (8.210) for scalar field and vector potential. The resultis

Jµν,λ =

(

−i∂L

∂∂λψσµνψ − i

∂L

∂∂λψLµνψ + c.c.

)

+[

δµλLxν − (µ↔ ν)]

. (8.236)


As before in (8.211) and (8.191), the orbital part of (8.236) can be expressed interms of the canonical energy-momentum tensor as

Lµν,λ = xµΘνλ − xνΘµλ. (8.237)

The first term in (8.236) is the spin current

Σµν,λ =1

2

(

−i∂L

∂∂λψσµνψ + c.c.

)

. (8.238)

Inserting (8.168), this becomes explicitly

Σµν,λ = −i

2ψγλσµνψ =

1

2ψγ[µγνγλ}]ψ =

1

2ǫµνλκψγκψ. (8.239)

The spin density is completely antisymmetric in the three indices [3].The conservation properties of the three currents are the same as in Eqs. (8.217)–

(8.221).Due to the presence of spin, the energy-momentum tensor is nonsymmetric.

8.10 Symmetric Energy-Momentum Tensor

Since the presence of spin is the cause of asymmetry of the canonical energy-momentum tensor, it is suggestive that an appropriate use of the spin current shouldhelp to construct a new modified momentum tensor

T µν = Θµν +∆Θνµ, (8.240)

that is symmetric, while still having the fundamental property of Θµν that its spatialintegral P µ =

∫

d3xT µ0 yields the total energy-momentum vector of the system.This is ensured by the fact that ∆Θµ0 being a three-divergence of a spatial vector.Such a construction was found in 1939 by Belinfante [4]. He introduced the tensor

T µν = Θµν −1

2∂λ(Σ

µν,λ − Σνλ,µ + Σλµ,ν), (8.241)

whose symmetry is manifest, due to (8.219) and the symmetry of the last two termsunder the exchange µ ↔ ν. Moreover, the relation (8.241) for the µ0-componentsof (8.241),

T µ0 = Θµ0 −1

2∂λ(Σ

µ0,λ − Σ0λ,µ + Σλµ,0), (8.242)

ensures that the spatial integral over Jµν,0 ≡ xµT ν0 − xνT µ0 leads to the same totalangular momentum

Jµν =∫

d3x Jµν,0 (8.243)

8.10 Symmetric Energy-Momentum Tensor 649

as the canonical expression (8.216). Indeed, the zeroth component of (8.242) is

xµΘν0 − xνΘµ0 −1

2

[

∂k(Σµ0,k − Σ0k,µ + Σkµ,0)xν − (µ↔ ν)

]

. (8.244)

Integrating the second term over d3x and performing a partial integration gives, forµ = 0, ν = i:

−1

2

∫

d3x[

x0∂k(Σi0,k − Σ0k,i + Σki,0)− xi∂k(Σ

00,k − Σ0k,0 + Σk0,0)]

=∫

d3xΣ0i,0,

(8.245)

and for µ = i, ν = j:

−1

2

∫

d3x[

xi∂k(Σj0,k − Σ0k,j + Σkj,0)− (i↔ j)

]

=∫

d3xΣij,0. (8.246)

The right-hand sides are the contributions of spin to the total angular momentum.For the electromagnetic field, the spin current (8.214) reads explicitly

Σµν,λ = −1

c

[

F λµAν − (µ ↔ ν)]

. (8.247)

From this we calculate the Belinfante correction

∆Θµν =1

2c[∂λ(F

λµAν − F λνAµ)− ∂λ(FµνAλ − F µλAν) + ∂λ(F

νλAµ − F νµAλ)]

=1

c∂λ(F

νλAµ). (8.248)

Adding this to the canonical energy-momentum tensor (8.153)

Θµν = −1

c(F ν

λ∂µAλ −

1

4gµνF λκFλκ), (8.249)

we find the symmetric energy-momentum tensor

T µν = −1

c(F ν

λFµλ −

1

4gµνF λκFλκ) +

1

c(∂λF

νλ)Aµ. (8.250)

The last term vanishes due to the free Maxwell field equations, ∂λFµν = 0. Therefore

it can be dropped. Note that the proof of the symmetry of T µν involves the fieldequations via the divergence equation (8.219).

It is useful to see what happens to Belinfante’s energy-momentum tensor in thepresence of an external current, i.e., if T µν is calculated from the Lagrangian

L = −1

4cFµνF

µν −1

c2jµAµ, (8.251)

with an external current. The energy-momentum tensor is

Θµν =1

c

(

F νλ∂

µAλ −1

4gµνF λκFλκ

)

+1

c2gµνjλAλ, (8.252)


generalizing (8.25).The spin current is the same as before, and we find Belinfante’s energy-

momentum tensor [4]:

T µν = Θµν +1

c∂λ(F

νλAµ) (8.253)

= −1

c(F ν

λFµλ −

1

4gµνF λκFλκ) +

1

c2gµνjλAλ +

1

c(∂λF

νλ)Aµ.

Using Maxwell’s equations ∂λFνλ = −jλ, the last term can also be rewritten as

−1

cjνAµ. (8.254)

This term prevents T µν from being symmetric, unless the current jλ vanishes.

8.10.1 Gravitational Field

The derivation of the canonical energy-momentum tensor Θµν for the gravitationalfield is similar to that for the electromagnetic field in Subsection 8.9.1. We startfrom the quadratic action of the gravitational field (4.372),

f

A = −1

8κ

∫

d4x(∂µhνλ∂µhνλ − 2∂λhµν∂

µhνλ + 2∂µhµν∂νh− ∂µh∂

µh), (8.255)

and identify the canonically conjugate field πλµν ,

πλµν ≡∂

f

L

∂∂λhµν, (8.256)

as being

πλµν=1

8κ[(∂λhµν−∂µhλν) (ηλν∂µh− ηµν∂λh)−ηλν∂

σhσµ+ηµν∂σhσλ]+(µ↔ ν).(8.257)

It is antisymmetric under the exchange λ↔ µ, and symmetric under µ↔ ν. Fromthe integrand in (8.255) we calculate, according to the general expression (8.136),

f

Θµν =

∂L

∂∂νhλκ∂µhλκ − ηµνL = πνλκ∂

µhλκ − ηµνL

=1

2κ(∂νhλκ − ∂κhδλ + ηνκ∂λh− ηνλ∂κh− ηνκ∂

σhσλ + ηνκ∂σhσν)∂

µhλκ

−ηµν8κ

(∂κhσλ∂κhσλ − 2∂λhσν∂

σhνλ + 2∂σhσν∂νh− ∂σh∂

σh). (8.258)

In order to find the symmetric energy-momentum tensorf

T µν , we follow Belinfante’sconstruction rule. The spin current density is calculated as in Subsection 8.9.1,starting from the substantial derivative of the tensor field

δshµν = ωµ

κhκν + ων

κhµκ. (8.259)

8.11 Internal Symmetries 651

Following the Noether rules, we find, as in (8.215),

Σµν,λ = 2∂φ

∂∂λhµκ− (µ↔ ν) = 2 [πλµκh

νκ − (µ↔ ν)] . (8.260)

Combining the two results according to Belinfante’s formula (4.57), we obtain thesymmetric energy-momentum tensor

f

Tµν= πνλκ∂µhλκ−∂c(π

λµκhνκ−πλνκhµκ−π

µνκhcd+ πµλκhνκπνλκhµκ−π

νµκhλκ)−ηµνL.

(8.261)

Using the field equation ∂µπµνλ = 0 and the Hilbert gauge (4.399) with ∂µφ

µν = 0,this takes the simple form in φµν :

f

Tµν =

1

8κ

[

2∂µφλκ∂νφλκ − ∂µφ∂νφ− ηµν(

∂λφκσ∂λφκσ −

1

2∂λφ∂

λφ)]

. (8.262)

8.11 Internal Symmetries

In quantum field theory, an important role is played by internal symmetries. Theydo not involve any change in the spacetime coordinate of the fields, whose symmetrytransformations have the simple form

φ′(x) = e−iαGφ(x), (8.263)

where G are the generators of some Lie group and α the associated transformationparameters. The field φ may have several indices on which the generators G act asa matrix. The symmetry variation associated with (8.263) is obviously

δsφ′(x) = −iαGφ(x). (8.264)

The most important example is that of a complex field φ and a generator G = 1,where (8.263) is simply a multiplication by a constant phase factor. One also speaksof U(1)-symmetry. Other important examples are those of a triplet or an octet offields φi with G being the generators of an SU(2) vector representation or an SU(3)octet representation (the adjoint representations of these groups). The first case isassociated with charge conservation in electromagnetic interactions, the other twowith isospin and SU(3) invariance in strong interactions. The latter symmetries are,however, not exact.

8.11.1 U(1)-Symmetry and Charge Conservation

Suppose that a Lagrangian density L(x) = L(φ(x), ∂φ(x), x) depends only on the ab-solute squares |φ|2, |∂φ|2, |φ∂φ|. Then L(x) is invariant under U(1)-transformations

δsφ(x) = −iφ(x). (8.265)


Indeed:

δsL = 0. (8.266)

On the other hand, we find by the chain rule of differentiation:

δsL =

(

∂L

∂φ− ∂µ

∂L

∂µφ

)

δsφ+

[

∂µ∂L

∂∂µφ

]

δsφ = 0. (8.267)

The Euler-Lagrange equation removes the first part of this, and inserting (8.265) wefind by comparison with (8.266) that

jµ = −∂L

∂∂µφφ (8.268)

is a conserved current.

For a free relativistic complex scalar field with a Lagrangian density

L(x) = ∂µϕ∗∂µϕ−m2ϕ∗ϕ (8.269)

we have to add the contributions of real and imaginary parts of the field φ in formula(8.268). Then we obtain the conserved current

jµ = −iϕ∗↔

∂µϕ (8.270)

where ϕ∗↔

∂µϕ denotes the left-minus-right derivative:

ϕ∗↔

∂µϕ ≡ ϕ∗∂µϕ− (∂µϕ∗)ϕ. (8.271)

For a free Dirac field, we find from (8.268) the conserved current

jµ(x) = ψ(x)γµψ(x). (8.272)

8.11.2 SU(N)-Symmetry

For more general internal symmetry groups, the symmetry variations have the form

δsϕ = −iαiGiϕ, (8.273)

and the conserved currents are

jµi = −i∂L

∂∂µϕGiϕ. (8.274)

8.12 Generating the Symmetry Transformations of Quantum Fields 653

8.11.3 Broken Internal Symmetries

The physically important symmetries SU(2) of isospin and SU(3) are not exact. TheLagrange density is not strictly zero. In this case we remember the alternative deriva-tion of the conservation law from (8.116). We introduce the spacetime-dependentparameters α(x) and conclude from the extremality property of the action that

∂µ∂Lǫ

∂∂µαi(x)=

∂Lǫ

∂αi(x). (8.275)

This implies the divergence law for the above derived current

∂µjµi (x) =

∂Lǫ

∂αi

. (8.276)

8.12 Generating the Symmetry Transformationsof Quantum Fields

As in quantum mechanical systems, the charges associated with the conserved cur-rents of the previous section can be used to generate the transformations of thefields from which they were derived. One merely has to invoke the canonical fieldcommutation rules.

As an important example, consider the currents (8.274) of an internal U(N)-symmetry. Their charges

Qi = −i∫

d3x∂L

∂∂µϕGiϕ (8.277)

can be written asQi = −i

∫

d3xπGiϕ, (8.278)

where π(x) ≡ ∂L(x)/∂∂µϕ(x) is the canonical momentum of the field ϕ(x). Afterquantization, these fields satisfy the canonical commutation rules:

[π(x, t), ϕ(x′, t)] = −iδ(3)(x− x′),

[ϕ(x, t), ϕ(x′, t)] = 0, (8.279)

[π(x, t), π(x′, t)] = 0.

From this we derive directly the commutation rule between the quantized charges(8.278) and the field ϕ(x):

[Qi, ϕ(x)] = −αiGiϕ(x). (8.280)

We also find that the commutation rules among the quantized charges are

[Qi, Qj] = [Gi, Gj]. (8.281)

Since these coincide with those of the matrices Gi, the operators Qi are seen to form

a representation of the generators of the symmetry group in the Fock space.


It is important to realize that the commutation relations (8.280) and (8.281)remain also valid in the presence of symmetry breaking terms, as long as these donot contribute to the canonical momentum of the theory. Such terms are called soft

symmetry breaking terms. The charges are no longer conserved, so that we mustattach a time argument to the commutation relations (8.280) and (8.281). All timesin these relations must be the same, in order to invoke the equal-time canonicalcommutation rules.

The most important example is the canonical commutation relation (8.95) itself,which holds also in the presence of any potential V (q) in the Hamiltonian. Thisbreaks translational symmetry, but does not contribute to the canonical momentump = ∂L/∂q. In this case, the relation generalizes to

ǫj = iǫi[pi(t), xj(t)], (8.282)

which is correct thanks to the validity of the canonical commutation relations (8.93)at arbitrary equal times, also in the presence of a potential.

Other important examples are the commutation rules of the conserved chargesassociated with the Lorentz generators (8.237):

Jµν ≡∫

d3xJµν,0(x), (8.283)

which are the same as those of the 4×4-matrices (8.205), and those of the quantummechanical generators (8.85):

[Jµν , Jµλ] = −igµµJνλ. (8.284)

The generators Jµν ≡∫

d3xJµν,0(x) are sums Jµν = Lµν(t)+Σµν(t) of charges (8.220)associated with orbital and spin rotations. According to (8.221), the individualcharges are time-dependent. Only their sum is conserved. Nevertheless, they bothgenerate Lorentz transformations: Lµν(t) on the spacetime argument of the fields,and Σµν(t) on the spin indices. As a consequence, they both satisfy the commutationrelations (8.284):

[Lµν , Lµλ] = −igµµLνλ,

[Σµν , Σµλ] = −igµµΣνλ. (8.285)

The commutators (8.281) have played an important role in developing a theoryof strong interactions, where they first appeared in the form of a charge algebra ofthe broken symmetry SU(3) × SU(3) of weak and electromagnetic charges. Thissymmetry will be discussed in more detail in Chapter 10.

8.13 Energy Momentum Tensor of a Relativistic Massive Point Particle 655

8.13 Energy Momentum Tensor of aRelativistic Massive Point Particle

If we want to study energy and momentum of charged relativistic point particles inan electromagnetic field, it is useful to consider the action (8.68) with (8.72) as aspacetime integral over a Lagrangian density:

A =∫

d4xL(x), with L(x) =∫ τb

τµ

L(xµ(τ))δ(4)(x− x(τ)). (8.286)

We can then derive for point particles local conservation laws that look very similarto those for fields. Instead of doing this from scratch, however, we shall simply takethe already known global conservation laws and convert them into the local ones byinserting appropriate δ-functions with the help of the trivial identity

∫

d4x δ(4)(x− x(τ)) = 1. (8.287)

Consider for example the conservation law (8.74) for the momentum (8.75). Withthe help of (8.287) this becomes

0 = −∫

d4x∫ ∞

−∞dτ

[

d

dτpλ(τ)

]

δ(4)(x− x(τ)). (8.288)

Note that the boundaries of the four volume in this expression contain the infor-mation on initial and final times. We now perform a partial integration in τ , andrewrite (8.288) as

0 = −∫

d4x∫ ∞

−∞dτ

d

dτ

[

pλ(τ)δ(4)(x− x(τ))

]

+∫

d4x∫ ∞

−∞dτ pλ(τ)∂τδ

(4)(x− x(τ)).

(8.289)

The first term vanishes if the orbits come from and disappear into infinity. Thesecond term can be rewritten as

0 = −∫

d4x ∂ν

[∫ ∞

−∞dτ pλ(τ)x

ν(τ)δ(4)(x− x(τ))]

. (8.290)

This shows that the tensor

Θλν(x) ≡∫ ∞

−∞dτ pλ(τ)xν(τ)δ(4)(x− x(τ)) (8.291)

satisfies the local conservation law

∂νΘλν(x) = 0. (8.292)

This is the conservation law of the energy-momentum tensor of a massive pointparticle.


The total momenta are obtained from the spatial integrals over Θλ0:

P µ(t) ≡∫

d3xΘλ0(x). (8.293)

For point particles, they coincide with the canonical momenta pµ(t). If the La-grangian depends only on the velocity xµ(t) and not on the position xµ(t), themomenta pµ(t) are constants of motion: pµ(t) ≡ pµ.

The Lorentz invariant quantity

M2 = P 2 = gµνPµP ν (8.294)

is called the total mass of the system. For a single particle it coincides with themass of the particle.

Subjecting the orbits xµ(τ) to Lorentz transformations according to the rules ofthe last section we find the currents of the total angular momentum

Lµν,λ ≡ xµΘνλ − xνΘµλ, (8.295)

to satisfy the conservation law:

∂λLµν,λ = 0. (8.296)

A spatial integral over the zeroth component of the current Lµν,λ yields the conservedcharges:

Lµν(t) ≡∫

d3xLµν,0(x) = xµpν(t)− xνpµ(t). (8.297)

8.14 Energy Momentum Tensor of a Massive ChargedParticle in a Maxwell Field

Let us consider an important combination of a charged point particle and an elec-tromagnetic field Lagrangian

A = −mc∫ τν

τµ

dτ√

gµν xµ(τ)xν(τ)−1

4c

∫

d4xFµνFµν −

e

c2

∫ τν

τµ

dτxµ(τ)Aµ(x(τ)).

(8.298)

By varying the action in the particle orbits, we obtain the Lorentz equation of motion

dpµ

dτ=e

cF µ

ν xν(τ). (8.299)

We now vary the action in the vector potential, and find the Maxwell-Lorentz equa-tion

−∂νFµν =

e

cxν(τ). (8.300)

The action (8.298) is invariant under translations of the particle orbits and theelectromagnetic fields. The first term is obviously invariant, since it depends only

8.14 Energy Momentum Tensor of a Massive Charged Particle in a Maxwell Field 657

on the derivatives of the orbital variables xµ(τ). The second term changes undertranslations by a pure divergence [recall (8.134)]. The interaction term also changesby a pure divergence, which is seen as follows: Since the symmetry variation changesthe coordinates as xν(τ) → xν(τ)−ǫν , and the field Aµ(x

ν) is transformed as follows:

Aµ(xν) → A′

µ(xν) = Aµ(x

ν + ǫν) = Aµ(xν) + ǫν∂µAµ(x

ν), (8.301)

we have altogether the symmetry variation

δsL = ǫν∂νm

L . (8.302)

We now calculate the same variation once more using the equations of motion.This gives

δsA =∫

dτd

dτ

∂Lm

∂x′µδsx

µ +∫

d4x∂

em

L

∂∂λAµδsA

µ. (8.303)

The first term can be treated as in (8.289)–(8.290), after which it acquires the form

−∫ τν

τµ

dτd

dτ

(

pµ +e

cAµ

)

= −∫

d4x∫ ∞

−∞dτ

d

dτ

[(

pµ +e

cAµ

)

δ(4)(x− x(τ)]

+∫

d4x∫ ∞

−∞dτ(

pµ +e

cAµ

)

d

dτδ(4)(x− x(τ)), (8.304)

and thus, after dropping boundary terms,

−∫ τν

τµ

dτd

dτ(pµ +

e

cAµ) = ∂λ

∫

d4x∫ ∞

−∞dτ(

pµ +e

cAµ

)

dxλ

dτδ(4)(x− x(τ)). (8.305)

The electromagnetic part is the same as before, since the interaction contains noderivative of the gauge field. In this way we find the canonical energy-momentumtensor

Θµν(x) =∫

dτ(

pµ +e

cAµ

)

xν(τ)δ(4)(x− x(τ))

−1

c

(

F νλ∂

µAλ −1

4gµνF λκFλκ

)

. (8.306)

Let us check its conservation by calculating the divergence:

∂νΘµν(x) =

∫

dτ(

p+e

cAµ

)

xν(τ)∂νδ(4)(x− x(τ))

−1

c∂νF

νλ∂

µAλ −1

c

[

F νλ∂ν∂

µAλ −1

4∂µ(F λκFλκ)

]

. (8.307)

The first term is, up to a boundary term, equal to

−∫

dτ(

pµ+e

τAµ

)

d

dτδ(4)(x− x(τ))=

∫

dτ

[

d

dτ

(

pµ+e

cAµ

)

]

δ(4)(x− x(τ)).(8.308)


Using the Lorentz equation of motion (8.299), this becomes

e

c

∫ ∞

−∞dτ

(

F µν x

ν(τ) +d

dτAµ

)

δ(4)(x− x(τ)). (8.309)

Inserting the Maxwell equation

∂νFµν = −e

∫

dτ(dxµ/dτ)δ(4)(x− x(τ)), (8.310)

the second term in Eq. (8.307) can be rewritten as

−e

c

∫ ∞

−∞dτdxλdτ

∂µAλδ(4)(x− x(τ)), (8.311)

which is the same as

−e

c

∫

dτ

(

dxµdτ

F µλ +dxλdτ

∂λAµ

)

δ(4)(x− x(τ)), (8.312)

thus canceling (8.309). The third term in (8.307) is finally equal to

−1

c

[

F νλ∂

µFνλ −

1

4∂µ(F λκFλκ)

]

, (8.313)

due to the antisymmetry of F νλ. With the help of the homogeneous Maxwell equa-tion we verify the Bianchi identity

∂λFµν + ∂µFνλ + ∂νFλµ = 0. (8.314)

is identically guaranteed.It is easy to construct from (8.306) Belinfante’s symmetric energy-momentum

tensor. We merely observe that the spin density comes entirely from the vectorpotential, and is hence the same as before in (8.247). Hence the additional piece tobe added to the canonical energy-momentum tensor is again [see (8.248)]

∆Θµν =1

c∂λ(F

µνAµ)

=1

2(∂λF

νλAµ + F νλ∂λAµ). (8.315)

The second term in this expression serves to symmetrize the electromagnetic partof the canonical energy-momentum tensor and brings it to the Belinfante form:

em

Tµν = −

1

c

(

F νλF

µλ −1

4gµνF λκFλκ

)

. (8.316)

The first term in (8.315), which in the absence of charges vanishes, is now just whatis needed to symmetrize the matter part of Θµν . Indeed, using once more Maxwell’sequation, it becomes

−e

c

∫

dτxν(τ)Aµδ(4)(x− x(τ)), (8.317)

Notes and References 659

thus canceling the corresponding term in (8.306). In this way we find that the totalenergy-momentum tensor of charged particles plus electromagnetic fields is simplythe sum of the two symmetric energy-momentum tensors:

T µν =m

Tµν+

em

Tµν

=1

m

∫ ∞

−∞dτ uµuνδ(4)(x− x(τ))−

1

c

(

F νλF

µλ −1

4gµνF λκFλκ

)

. (8.318)

For completeness, let us also cross-check its conservation:

∂νTµν = 0. (8.319)

Indeed, forming the divergence of the first term gives [in contrast to (8.309)]

e

c

∫

dτ xν(τ)F µν(x(τ)), (8.320)

which is canceled by the divergence in the second term [in contrast to (8.312)]

−1

c∂νF

νλF

µλ = −e

c

∫

dτ xλ(τ)Fµλ(x(τ)). (8.321)

Notes and References

For more details seeL.D. Landau and E.M. Lifshitz, The Classical Theory of Fields , Addison-Wesley, Reading, Mass.,1951;S. Schweber, Relativistic Quantum Fields , Harper and Row, New Yoerk, N.Y., 1961;A.O. Barut, Electrodynamics and Classical Theory of Fields and Particles , MacMillan, New York,N.Y. 1964;J.D. Jackson, Classical Electrodynamics , John Wiley & Sons, New York, N.Y., 1975;H. Ohanian, Classical Electrodynamics , Allyn and Bacon, Boston, Mass., 1988.

The individual citations refer to:

[1] E. Noether, Nachr. d. vgl. Ges. d. Wiss. Gottingen, Math-Phys. Klasse, 2, 235 (1918);See alsoE. Bessel-Hagen, Math. Ann. 84, 258 (1926);L. Rosenfeld, Me. Acad. Roy. Belg. 18, 2 (1938);F. Belinfante, Physica 6, 887 (1939).

[2] S. Coleman and J.H. VanVleck, Phys. Rev. 171, 1370 (1968).

[3] This property is important for being able to construct a consistent quantum mechanics inspacetime with torsion. See the textbookH. Kleinert, Path Integrals in Quantum Mechanics Statistics, Polymer Physics, and Finan-

cial Markets, World Scientific, Singapore 2008 (http://klnrt.de/b5).

[4] The Belinfante energy-momentum tensor is discussed further inH. Kleinert, Gauge Fields in Condensed Matter , Vol. II Stresses and Defects , World Sci-entific Publishing, Singapore 1989, pp. 744–1443 (http://klnrt.de/b2).H. Kleinert, Multivalued Fields in Condensed Matter, Electromagnetism, and Gravitation,World Scientific, Singapore 2009, pp. 1–497 (http://klnrt.de/b11).

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ContinuousSymmetriesandConservationLaws. Noether…users.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-7-conslaw.pdf · side of (8.7) is called Noether charge. ... The existence