Quantum Field Theory

So what are we to make of this? We have a theory with an infinite number ofsymmetries, one for each function λ(x). Previously we only encountered symmetrieswhich act the same at all points in spacetime, for example ψ → e iα ψ for acomplex scalar field. Noether’s theorem told us that these symmetries give riseto conservation laws. Do we now have an infinite number of conservation laws?The answer is no! Gauge symmetries have a very different interpretation thanthe global symmetries that we make use of in Noether’s theorem. While thelatter take a physical state to another physical state with the same properties,the gauge symmetry is to be viewed as a redundancy in our description. That is,two states related by a gauge symmetry are to be identified: they are the samephysical state. (There is a small caveat to this statement which is explained inSection 6.3.1). One way to see that this interpretation is necessary is to noticethat Maxwell’s equations are not sufficient to specify the evolution of A µ . Theequations read,[η µν (∂ ρ ∂ ρ ) − ∂ µ ∂ ν ] A ν = 0 (6.13)But the operator [η µν (∂ ρ ∂ ρ )−∂ µ ∂ ν ] is not invertible: it annihilates any function ofthe form ∂ µ λ. This means that given any initial data, we have no way to uniquelydetermine A µ at a later time since we can’t distinguish between A µ and A µ +∂ µ λ.This would be problematic if we thought that A µ is a physical object. However,if we’re happy to identify A µ and A µ +∂ µ λ as corresponding to the same physicalstate, then our problems disappear.Since gauge invariance is a redundancy of the system,GaugeGauge Orbitswe might try to formulate the theory purely in terms of Fixingthe local, physical, gauge invariant objects E ⃗ and B. ⃗ Thisis fine for the free classical theory: Maxwell’s equationswere, after all, first written in terms of E ⃗ and B. ⃗ But it isnot possible to describe certain quantum phenomena, suchas the Aharonov-Bohm effect, without using the gaugepotential A µ . We will see shortly that we also require theFigure 29:gauge potential to describe classically charged fields. Todescribe Nature, it appears that we have to introduce quantities A µ that we can nevermeasure.The picture that emerges for the theory of electromagnetism is of an enlarged phasespace, foliated by gauge orbits as shown in the figure. All states that lie along a given– 126 –