• 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.


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


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,347 answers , 22,720 comments
1,470 users with positive rep
818 active unimported users
More ...

  Commutation and Anticommutation relations in lattice QCD

+ 3 like - 0 dislike

The article "Construction of a selfadjoint, strictly positive transfer matrix for euclidean lattice gauge theories" (Lüscher 1977), about lattice QCD, says the following:

> The fermion Hilbert space $\mathscr{H}_F$ is the Fock space built from an operator spinor field $\hat{\chi}_n$ which satisfies the usual canonical anticommutation relations:
> $$\{\hat{\chi}_{n,\alpha},\hat{\chi}^\dagger_{m,\beta} \}=\delta_{n,m} \delta_{\alpha,\beta}\qquad[...]\tag{11}$$
> The field operator $\hat{\psi}$ acts in $\mathscr{H}_F \otimes \mathscr{H}_G$.

(This is the hilbert space of fermions tensor product the hilbert space of the gauge fields.)

> It does _not_ have a canonical anticommutator, but:
> $$\{\hat{\psi}_{n,\alpha},\hat{\psi}^\dagger_{m,\beta} \}= B^{-1}_{n\alpha,m\beta}\qquad [...]\tag{12}$$
> The matrix $B_{n\alpha,m\beta}$ depends on the gauge field and is given by
> $$B_{n\alpha,m\beta} = \delta_{n,m}\delta_{\alpha,\beta} - K \sum_{j=1,2,3} U(n,j)_{\alpha,\beta}\delta_{n+\hat{j},m}+U(m,j)^{\dagger}_{\alpha,\beta}\delta_{m+\hat{j},n}\tag{13}$$

Then the article says in equation (14) that the relation between the field $\psi$ and the canonical field $\chi$ is
$$\psi_{n\alpha} = \sum_{m\beta} (B^{-1/2})_{n\alpha,m\beta} \chi_{m,\beta}$$
where $n$ and $m$ are the lattice sites, $U$ are the links variables (the gauge fields), $K$ is the Wilson hopping parameter, and the action is supposed to be the improved Wilson action:
$$S_F= \sum_{n} \biggl\{\bar{\psi}(n)\psi(n)-K \sum_{j=1,2,3}\bar{\psi}(n) U(n,j)(1-\gamma_j)\psi(n+\hat{j})+\bar{\psi}(n+\hat{j})U(n,j)^{\dagger}(1+\gamma_j)\psi(n) \biggr\}$$

How can these equations be justified? I have tried for days (without any success) to come up with an explanation nor I have found any demonstration from other sources for these assertions.
Especially on why the anticommutator of the fields is $B^{-1}$.

asked Mar 10, 2016 in Theoretical Physics by practical matter (50 points) [ revision history ]
reshown Oct 1, 2017 by practical matter

Could you add a reference?

I edited the post

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:
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