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What is the physical constraint that gauge invariance is a required condition for electromagnetic fields? What would happen if the electromagnetic fields were not gauge invariant?

Qmechanic
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linuxfreebird
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1 Answers1

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By disposing of the $U(1)$ gauge symmetry, we also dispose of the global version of the $U(1)$ symmetry which gives rise to both a conserved current and conserved charge due to Noether's theorem.

Specifically, every continuous global symmetry, by Noether's theorem, gives rise to a conserved current $j^{\mu}$ which satisfies the continuity equation,

$$\partial_\mu j^{\mu} = \frac{\partial j^{0}}{\partial t} + \nabla \cdot \vec{j} = 0.$$

The conserved Noether charge is defined as,

$$Q = \int_V d^3 x \, \, j^{0},$$

and by demanding $\partial_t Q = 0$, we see this implies,

$$\frac{\partial Q}{\partial t} = -\int_V d^3x \, \, \nabla \cdot \vec{j} = -\int_{\partial V} \vec{j} \cdot ndS = 0,$$

or in other words the flux of $\vec{j}$ is zero, and hence $Q$ is conserved locally. By losing $U(1)$ gauge symmetry, we lose a redundancy in the description of the system, but also a conservation law.

JamalS
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  • Good answer. Where does the broken gauge invariance come into the math? Do we have to start with the Lagrangian? – linuxfreebird Mar 17 '14 at 14:44
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    If you want to break gauge invariance, you need to add a term to the Lagrangian, e.g. a mass term. The reason we need the Higgs mechanism in the standard model for certain fields is because we can't add a mass term that preserves symmetries for those fields. – JamalS Mar 17 '14 at 14:48
  • Whoa, the higgs field breaks charge conservation? – linuxfreebird Mar 17 '14 at 14:49
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    No, certainly not. We introduce the Higgs field to avoid introducing mass terms which break symmetries, in a nutshell – JamalS Mar 17 '14 at 14:52
  • It should be noted many symmetries are actually spontaneously (not explicitly) broken in Nature, for example $SU(2) \times U(1){L} \to U(1){EM}$. – JamalS Mar 17 '14 at 14:53