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 ...

  Exact calculations with Moyal product by "Bopp Shift"

+ 2 like - 0 dislike
259 views

I'm now working on my Phd thesis on the area of deformation quantization and field theory. After doing all the "ground work" (definitions, motivations, basics of the theory etc) I have now to do some "dirty work" with many and many Moyal Products. So I have searched for some tools to make life easier. In particular I found this MO question with a nice comment by user @Comaszachos saying that we can "Bopp shift away". I look at the pdf linked in the comment and on page 27 we have a lemma stating:

$$f(x,p)*g(x,p)=f(\hat x,\hat p)g(x,p)=f(x+\frac{ih}{2} \partial_p , p-\frac{ih}{2} \partial_x)g(x,p)$$

So I tried to apply this to some simple examples and that is what I found: take $f(x,p)=g(x,p)=e^x e^p$. Than

$$f(x,p)*g(x,p)=e^{x+\frac{ih}{2} \partial_p}e^{p-\frac{ih}{2} \partial_x}e^xe^p$$

as $[x,\frac{ih}{2} \partial_p]=[p,\frac{ih}{2} \partial_x]=0$, we have

$$ f(x,p)*g(x,p)= (e^x e^{ \frac{ih}{2} \partial_p} e^p e^{-\frac{ih}{2} \partial_x}) (e^x e^p) $$

using Lagrange's formula for the shift operator

$$f(x,p)*g(x,p)=e^xe^{\frac{ih}{2} \partial_p}(e^pe^{x-\frac{ih}{2}}e^p) =e^xe^{p+\frac{ih}{2}}e^{x-\frac{ih}{2}}e^{p+\frac{ih}{2}} =e^{2x+2p+\frac{ih}{2}}$$

and this is wrong! We can see it by the fact that $f*g=fg+\frac{ih}{2}\{f,g\}+O(h^2)$ and $\{e^xe^p,e^xe^p\}=0$, but $$e^{2x+2p+\frac{ih}{2}}=e^{2x+2p}+e^{2x+2p}\frac{ih}{2}+O(h^2)$$

As a comment, the exact correct answer is $$f(x,p)*g(x,p)=e^{2x+2p}$$ where I used $$f(x,p)*g(x,p)=\lim_{(x',p')\to (x,p)}e^{\frac{ih}{2}(\partial_x \partial_{p'}-\partial_{x'} \partial_p)}f(x,p)g(x',p')$$

with the fact $e^{\frac{ih}{2}(\partial_x \partial_{p'}-\partial_{x'} \partial_p)}=e^{\frac{ih}{2}\partial_x \partial_{p'}}e^{-\frac{ih}{2}\partial_{x'} \partial_p}$ and expanding and collecting terms.

My question is: Where is the error? How do I properly apply the Bop shift? Is there any restriction to use it?

This post imported from StackExchange MathOverflow at 2023-11-30 17:41 (UTC), posted by SE-user Diego Santos
asked Nov 29, 2023 in Theoretical Physics by Diego Santos (10 points) [ no revision ]
retagged Nov 30, 2023

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$\hbar$ysicsOve$\varnothing$flow
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
...