Noether’s theorem for field theories

In this section, I will first discuss Noether’s theorem in classical field theories, before going into quantum physics. Noether’s theorem connects every continuous transformation that leaves the action integral invariant with a conserved quantity (cf. Greiner & Reinhardt 1996, ch. 2.4; Maggiore 2005, ch. 3.2.1). To show this, it is sufficient here to study only infinitesimal transformations, which can be defined as follows

xµ_!x0µ = xµ+ Aµ(x)

(x)! 0(x0) = (x) + F ( , @ ) . The transformations Aµ_{(x)and F}

i( , @ ) are defined as symmetry transformations,

that is, they leave the action integral S( ) invariant. This condition can be written as S( ) = ˆ V d4x0_L0(x0) ˆ V d4x_{L(x) = 0 .}

By calculating this expression, one finds the following continuity equation @

@xµ

fµ(x) = 0 (5.3.49)

with the current density fµ(x) =

@_L(x) @(@µ ₎(A

⌫_(x)@

⌫ F ( , @ )) Aµ(x)L .

16_{Throughout this section I was only concerned with particles for which we have empirical evidence.}

These particles, however, are not the only ones allowed by group structure. For example, besides Fermions and Bosons, it is at least theoretically possible that there are paraparticles, following parastatistics (cf. French & Krause 2006, sec. 3.8). To regard something as a brute fact of nature is not meant as an argument to exclude the possibility of there being more than these brute facts or the brute facts to be wrong.

Integrating 5.3.49 leads to 0 = d dx0 ˆ V d3x f0(x) + ˛ @V d~s ~f (x) . (5.3.50)

If it is assumed that the field falls off to infinity sufficiently fast, then the integral over the surface @V is zero and

G := ˆ

V

d3x f0(x)

is constant in time, that is, it is a conserved quantity.

It will be important for my further discussion to note that the continuity equation 5.3.49 defines a local conservation law. Which means that either the quantity G is conserved in the volume V or it is not conserved, but then there is a current through the surface element @V . It follows that a conserved quantity cannot simply vanish at one point in space and appear at another distant point. A process of the latter sort is only possible with global conservation laws, but not with local conservation laws (cf. Ryder 2002, p. 79). That said, it has to be kept in mind that the conserved current is not observable in field theories. It is possible to construct two theories, which have the exact same equations of motion for the fields and therefore are empirically equivalent, but nevertheless have different conserved currents. Only the conserved quantity G in volume V is uniquely defined. As an additional consequence, the conserved quantity cannot in general be exactly localised at points, because V has to be chosen such that the field falls to zero at the boundaries and the surface integral in 5.3.50 does not contribute.17

An example for a conserved quantity is energy-momentum, which follows from symmetry under spacetime translations. A translation is given by the transformation

xµ_{! x}0µ= xµ+ ✏µ

while the fields remain invariant, that is, 0_(x0_{) = (x). The conserved current is}

then the energy-momentum tensor ✓µ⌫ =

@_L(x)

@(@µ _{) @}⌫ ⌘µ⌫L

It follows that the conserved quantity is the energy-momentum four-vector P⌫ =

ˆ

V

d3x ✓0⌫ = (E, ~p) .

Going into QFT, the condition of energy-momentum conservation can be expressed as follows. As we know from 5.3.33 the basic expression one has to calculate is the

17_{Things might however be different if the gravitational field is included, which couples to the}

matrix element for the evolution of an initial into a final state. With the evolution operator in the interaction picture, ˆH, this can be expressed as

i ˆ

d4x_h out|H(x)| ini .

With the help of the operator ˆPµ _{for spacetime translations, the evolution operator}

can be written as H(x) = ei ˆP xH(0)e i ˆP x It follows that i ˆ d4x_h out|ei ˆP xH(0)e i ˆP x| ini = i ˆ d4x ei(Pout Pin)x_h out|H(0)| ini = i (2⇡)4 (4)(Pin Pout)h out|H(0)| ini

where the conservation is made explicit by the Dirac delta function (cf. Maggiore 2005, p. 205).

The influence of energy-momentum conservation also becomes obvious when the interaction between different fields is considered. In the discussion of QED, we have seen that the QED Lagrangian, 5.3.28, consists of three parts, namely, of the matter field the electromagnetic field and the interaction field. It can be shown that the continuity equation, 5.3.49, for the whole system then also has three parts, which are the energy-momentum tensors for the three fields

@µ✓µ⌫_Dirac+ @µ✓µ⌫e.m.+ @µ✓µ⌫_int.= 0 .

It follows that energy-momentum is conserved for the whole system and if, e.g., the energy-momentum of the Dirac field changes, it has to change for the other fields accordingly (cf. Greiner & Reinhardt 1996, exercise 6.2).

Finally, an example for another conserved quantity is the norm of the Dirac field 5.3.25, which follows from the symmetry under global phase transformations, so-called internal symmetries. Phase transformations can be expressed as ! ei

and †_! †_e i _{. The corresponding current destiny is then}

jµ= i ✓ @L(x) @(@µ ₎ @L(x) @(@µ †₎ † ◆ = ¯ µ

and the conserved charge is G =

ˆ

d3x f0(x) =

ˆ

d3x †

which is the norm of the Dirac field. This also can be regarded as implying the conservations of probability for expectation values (cf. Greiner & Reinhardt 1996, p. 120).

These short examples for conservation laws in QFT have merely heuristic and no general value. However, it can be rigorously shown that the above results hold for QFT. In the case of the path integral formulation, Noether’s theorem is then formulated in terms of the Ward-Takahashi identities (cf. Duncan 2012, p. 441 ff.). To be more precise, however, it has to be added that Noether’s theorem is in general not valid in QFT. So-called anomalies lead to non-vanishing divergencies for the currents, that is, 5.3.49 will be non-zero. Nevertheless, since these anomalies arise from local gauge symmetry, they do not impugn the conserved quantities, energy in particular, that I will rely on in my further discussion. I will therefore not go deeper into this topic (cf. Duncan 2012, ch. 15.5).

I want to close this chapter by following Brading & Brown (2003) and emphasise the empirical significance of Noether’s theorem. Spacetime symmetries can be interpreted actively, in that they tell us that if the laws of nature are valid in one system, they will also be valid in the transformed system. As Brading and Brown argue, however, this is not what gives Noether’s theorem its empirical significance. The reason for this is that Noether’s theorem is not restricted to spacetime symmetries, but also works with gauge symmetries, which cannot have the same active interpretation. Gauge transformations can in general not be connected to an empirical operation like going from one inertial frame into another. Instead, Brading and Brown make the point that the significance of the symmetries and hence Noether’s theorem is comprised in what follows for the equations of motion.

The imposition of a symmetry on a theory places a restriction on the possible form of the equations of motion of that theory, and insofar as this restriction has empirical significance then so too does the symmetry itself. This is the proper place to look when analysing the empirical significance of a given Noether symmetry. (Brading & Brown, 2003, p. 113)

On the one hand, we know from the validity of a symmetry that the equations of motion, that is the Euler-Lagrange equations, must transform according to these symmetries. On the other hand, the fields in the theory must be solutions to the Euler-Lagrange equations. So for example, if the equations of motion are symmetric under spacetime translations, then we know that the energy of the fields, solving the equations of motion, is conserved. This is where symmetries leave the realm of pure mathematics and gain empirical significance.18