Effective action from the background field

[physengineer]

Hello,

Assuming that I have a pure U(1) gauge theory. The partition function can be written as

$$Z=\int D(A) \exp (-F_{\mu\nu} F^{\mu\nu})$$

If I want to find the effective action in terms of an external classical field I can write it in terms of
$A\rightarrow A+B$ where $B$ is background and then integrate over $A$. Interestingly I get the same original $Z$ as in Gaussian integral the shift in integration does not change the result. So my final result will be independent of $B$! What does this mean? Does it mean that I can treat my pure U(1) gauge theory as classical? Or I probably misunderstood the background field method.

I appreciate any comments in this regard.

 

[eljose79]

Hello,

Thank you for your post. It is interesting that you have found that the final result is independent of the background field B. This means that the pure U(1) gauge theory can indeed be treated as classical, as the background field does not affect the partition function. This result is not surprising, as the background field method is commonly used in classical field theory to simplify calculations.

However, it is important to note that this result may not hold for more general gauge theories. In fact, for non-Abelian gauge theories, the background field method can lead to non-trivial results and can be a useful tool for calculating effective actions. So while your finding is valid for the pure U(1) gauge theory, it may not be applicable to other gauge theories.

I hope this helps clarify the use of the background field method in the context of U(1) gauge theory. Keep exploring and asking questions – this is how science progresses!