# Operator Product Expansion in Massless 2D QED

+ 3 like - 0 dislike
620 views

In Peskin & Schroeder chapter 19 page 656, where the axial current anomaly of massless 2D QED is discussed, the authors go from: $$\bar\psi(x+\varepsilon/2)\Gamma(x)\psi(x-\varepsilon/2)\tag{19.25}$$ (where $\Gamma(x)$ is some operator) to: $$\bar\psi(x+\varepsilon/2)\Gamma(x)\psi(x-\varepsilon/2) \tag{19.27}$$ (where now the two Fermionic fields are contracted) to: $$Tr\left[\Gamma(x)S_F(\varepsilon)\right] \tag{19.27}$$ (where $S_F(x)$ is the Fermion propagator between a spacetime interval $x$)

I really don't understand these transitions and would appreciate any help with how to do them. In particular:

1) Is it implicitly understood (from the very beginning of this derivation) that the axial current is in fact time-ordered (so that we can employ Wick's theorem) and always assume it operates on the vacuum (so that normal ordered terms vanish)?

2) Why does the contraction of the two Fermion fields over the $\Gamma$ operator lead to a trace, as if we had a loop?

This post imported from StackExchange Physics at 2014-07-14 07:37 (UCT), posted by SE-user PPR
asked Jul 13, 2014
retagged Jul 14, 2014
Observe that the trace is invariant under cyclic permutation, this should give you the answer to your second question.

This post imported from StackExchange Physics at 2014-07-14 07:37 (UCT), posted by SE-user ACuriousMind

 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$\varnothing$ysicsOverflowThen 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.