# Symmetries in quantum field theory and anomalies

+ 2 like - 0 dislike
363 views

Suppose we have a lagrangian quantum field theory, thus a theory where we can write an action in the form

S = \displaystyle \int d^4 x \; \mathcal L \, \left( \partial_{\mu} \phi , \phi \right) \ .

In classical field theory, we define a symmetry as an action on the fields and/or on spacetime such that the lagrangian changes by a total divergence, which does not change the action $S$ (in this premise I made some simplifying assumptions: I take the lagrangian to be time independent and spacetime to be Minkowski $\mathcal M_4$).
Suppose we consider a quantum field theory with the same lagrangian as above. The symmetry found before (as I know) is said to be a classical symmetry of the system. However, at the quantum level, this symmetry may be spoiled. I know two mechanisms that may spoil this symmetry.

1. If we treat the QFT perturbatively, by means of renormalization loop diagrams may add additional terms to our lagrangian which may spoil the symmetry. This may happen also if under the flow of renormalization, some couplings of the initial lagrangian changes and break the symmetry.

2. If we look at the QFT via the partition function, it may happen that the functional measure is not invariant, and it changes by the determinant of the Jacobian of the transformation. This is usually called anomaly. This gives an additional term to the lagrangian, using the fact that

\text{det}(A) = e^{\text{tr}(\text{log}(A))} \ .

Notice also that the logarithm is not a local term in the lagrangian, and thus the anomaly introduce a non-local term.

- My first question is whether there are more mechanisms which can spoil a symmetry at the quantum level

- The second question is whether these two that I described are somehow related. For example, it may happen that expanding the log in the formula for the anomaly, we can recast the series of local terms that come from renormalization.
Personally, I think they are independent, because I think we may have an anomalous symmetry even in a renormalizable theory, where e.g. the form of the local terms is fixed and only couplings change.

- Last, to connect all the dots, is there a systematic approach to the question whether a quantum field theory spoils a symmetry or not? I am really interested in the subject but I find it confusing, probably because of this lack of systematicity (at least in my mind).

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOve$\varnothing$flowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.