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## The U(1) gauge invariance of standard electrodynamics

The presence of a gauge symmetry implies the presence of redundancy in the description of the degrees of freedom of the theory.   Take for example the Lagrangian of the gauge part of electrodynamics which is given in terms of the gauge field $A_\mu$ by

$L = -\frac{1}{2} \partial^\mu A^\nu \partial^\mu A_\nu + \frac{1}{2} \partial^\mu A^\nu \partial_{\nu} A_\mu + J^\mu A_\mu$

contains no time derivative as can be can be seen by writing out the contractions.  As a consequence there is no canonically conjugate momentum and therefore the field has no dynamics.  This is the gauge invariance of electromagnetism.

So how would one realize that an S4 universe would not have this $U(1)$ gauge invariance?  When the electromagnetic potential is written as a 1-form, the usual choice is:

$A = A_0 dt + A_1 dx^1 + A_2 dx^2 + A_3 dx^3$

In an S4 universe, one considers for electromagnetism the form

$A = A_0 e_4 + A_1 e_1 + A_2 e_2 + A_3 e_3$

for a coframe adapted to a hypersurface with $e_4$ normal to it in $S^4(1/h)$.  Assuming one accepts that form of the Lagrangian is as above, and more importantly that this form of the electromagnetic potential is consistent with Maxwell’s equations, we can see that the gauge invariance is not achieved for $A_0$.