Let us consider the functional integral: \begin{equation} \int \mathcal{D} A e^{iS[A]} \end{equation} where $S[A]$ is the action for $U(1)$ gauge field and \begin{equation} \mathcal{D}A\equiv \mathcal{D}A_0 \mathcal{D}A_1 \mathcal{D}A_2 \mathcal{D}A_3; \\ \mathcal{D}A_i = \prod_x dA_i(x). \end{equation}

Now I have two questions:

1. How to show that the integration measure $\mathcal{D} A $ is invariant under the gauge transformation: \begin{equation} A_\mu (x) \to A_\mu (x) + \frac{1}{e}\partial_\mu \alpha(x) \end{equation} 2. How to show that the integration measure $\mathcal{D} A $ is invariant under the Lorentz transformation?

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    $\begingroup$ Hints: 1. $\alpha(x)$ is a set function, it isn't being varied over like $A_\mu(x)$ is. 2. How does a measure change under a change of coordinates in regular integration? $\endgroup$ – Will Jun 30 '13 at 16:41
  • $\begingroup$ Doesn't the answer to this depend on the regulator? In particular, couldn't one concoct some terrible regulator that would render the integration measure non-gauge-invariant? $\endgroup$ – joshphysics Jun 30 '13 at 18:18
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    $\begingroup$ @joshphysics: You can certainly break symmetries with a non-covariant regulator, but it is almost universally assumed that one is using a gauge-covariant regulator since the consequence of gauge non-covariance are so dire. $\endgroup$ – BebopButUnsteady Jul 1 '13 at 22:29
  • $\begingroup$ @Will: Can u elaborate the hint for question no. 2 ? $\endgroup$ – layman Jul 2 '13 at 5:59
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    $\begingroup$ Lorentz transformations take $A_\mu \rightarrow A^\prime_{\nu} = \Lambda_{\nu}^{\mu}A_{\mu}$, right? Now, consider the process you go through when changing coordinates in regular integration of several variables. Hopefully this helps :) $\endgroup$ – Will Jul 2 '13 at 6:13

In quantum field theory, when manipulating the path integral, we naively assume the measures (or strictly speaking the product of the measures and integrand) are invariant under the gauge transformations. In a fundamental paper, Fujikawa demonstrated the flaw in this assumption (in certain cases), and how to rigorously compute the analogue of a Jacobian factor for the path integral, which he employed to derive the chiral anomaly of quantum electrodynamics. For a complete derivation, I recommend the sources:

  • Introduction to Quantum Field Theory, by Peskin and Schroeder, Chapter 19, pg. 651+
  • Beyond the Standard Model, Lecture 5 (13/14, Course by Prof. R. Mann), Perimeter Institute

I hope these may provide some clarification regarding the change of the path integral measure under a general transformation of the constituent fields.

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