Sunday 9 February 2020

path integral - Symmetries of the theory that are not symmetries of the action nor of the measure?


Can anybody think of an example of any theory in which there is a transformation law that does not leave the action nor the path integral measure invariant, but such that the product of both is invariant so that the transformation is a symmetry?



Answer




The theory of a chiral fermion $\psi$ coupled to a Maxwell field in four dimensions has a famous anomaly in the chiral transformations $$ \psi \rightarrow \psi'=e^{i \gamma_5 \theta} \psi $$ which arises from the non-invariance of the fermion path integral measure $$ D\psi' D\bar\psi' = D\psi D\bar\psi \, {\rm\exp}\left( i\frac{\theta}{(4\pi)^2}\int F\wedge F\right)\,. $$ This can be fixed by adding a pseudoscalar (axion) $a(x)$ to the theory, that couples to the Maxwell field as follows $$ \mathcal{L}_a = (da + A)\wedge * (da + A) + a \frac{1}{(4\pi)^2}F \wedge F $$ and transforms under the symmetry with a shift $$ a \rightarrow a' = a - \theta. $$ The non-invanriace of the classical action cancels the quantum anomaly, giving an example of what you are looking for. This is the toy version of a more general story that goes by the name of "Green-Schwarz anomaly cancellation mechanism".


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...