Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Chiral Anomaly in Massless QED

+ 2 like - 0 dislike
1082 views

Classical massless QED has axial current conservation. When quantizing the theory, we expect that suddenly $\partial_\mu \hat{j}^{\mu5}\neq0$ (as an operator equality).

I have two questions regarding this:

  1. Why is it then, that when trying to prove indeed that $\partial_\mu \hat{j}^{\mu5}\neq0$ in the quantum version, we go on to try to prove that $$\partial_\mu\langle\Omega\left|\hat{j}^{5 mu} (x) \hat{j}^\nu (y) \hat{j}^\lambda (z)\right|\Omega\rangle\neq0$$? I understand that the above relation would imply the violation of a naive Ward-identity, but what's the connection between axial current conservation and Ward-identities? Is it the case that if the Ward identity is violated then necessarily the current cannot be conserved?

  2. When calculating $\langle\Omega\left|\hat{j}^{5 mu} (x) \hat{j}^\nu (y) \hat{j}^\lambda (z)\right|\Omega\rangle$, it appears that the $\hat{j}$'s are treated as external photon lines, as in this image: diagrams in the bottom why is that? I would naively calculate such a transition amplitude by plugging in functional derivatives w.r.t. to the Fermion and anti-Fermion sources, for example, like so: $$\langle\Omega\left|\hat{j}^{5 mu} (z) \hat{j}^\nu (x) \hat{j}^\lambda (y)\right|\Omega\rangle = \gamma_{\mu\alpha_{1}\alpha_{2}}\gamma_{5\alpha_{2}\alpha_{3}}\gamma_{\nu\alpha_{4}\alpha_{5}}\gamma_{\lambda\alpha_{6}\alpha_{6}}\left.\left(\frac{\left(-\frac{1}{i}\frac{\delta}{\delta\eta_{\alpha_{1}}\left(z\right)}\right)\left(\frac{1}{i}\frac{\delta}{\delta\bar{\eta}_{\alpha_{3}}\left(z\right)}\right)\left(-\frac{1}{i}\frac{\delta}{\delta\eta_{\alpha_{4}}\left(x\right)}\right)\left(\frac{1}{i}\frac{\delta}{\delta\bar{\eta}_{\alpha_{5}}\left(x\right)}\right)\left(-\frac{1}{i}\frac{\delta}{\delta\eta_{\alpha_{6}}\left(y\right)}\right)\left(\frac{1}{i}\frac{\delta}{\delta\bar{\eta}_{\alpha_{7}}\left(y\right)}\right)\mathcal{Z}_{QED}}{\mathcal{Z}_{QED}}\right)\right|_{\mbox{sources}=0}$$

This post imported from StackExchange Physics at 2014-08-07 15:35 (UCT), posted by SE-user PPR
asked Aug 6, 2014 in Theoretical Physics by PPR (135 points) [ no revision ]
Question 1 seems to be related to page 17 here: www-personal.umich.edu/~larsenf/Lecture11.pdf

This post imported from StackExchange Physics at 2014-08-07 15:35 (UCT), posted by SE-user PPR

Your answer

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):
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$\hbar$ysi$\varnothing$sOverflow
Then 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).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...