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 ...

  Strange factor multiplying the fermionic part in the NS mass-squared operator?

+ 1 like - 0 dislike
522 views

In the Neveu-Schwarz sector, the worldsheet fermions can be expanded as

$$ \psi^I(\tau,\sigma) \sim \sum\limits_{r\in Z+1/2}b_r^Ie^{-ir(\tau-\sigma)} $$

and the total mass squared operator can then be written as

$$ M^2 = \frac{1}{\alpha'}\left( \frac{1}{2} \sum\limits_{p\neq 0} \alpha_{-p}^I\alpha_p^I + \frac{1}{2}\sum\limits_{r\in Z+1/2} r \, b_{-r}^I b_r^I \right) $$

The first sum gives the contribution of the bosons, the second one the contributions of the fermions.

Why are the summands in in the fermionic part multiplied by $r$, how does this factor come in mathematically? Does this have something to do with the Pauli exclusion principle?

The same thing issue appears with the fermionic part mass operator in the Ramond sector, where I dont understand it either ...

asked Oct 27, 2013 in Theoretical Physics by Dilaton (6,240 points) [ revision history ]
retagged Mar 25, 2014 by dimension10
Funny, the proof of this has not even been given in BBS, maybe I should check my trustable lecture notes. The "lecture" is the trustiest/.

This post imported from StackExchange Mathematics at 2014-03-09 15:47 (UCT), posted by SE-user dimensio1n0
@DImension10AbhimanyuPS if you find the answer, you could post it here :-). When trying to look at the lecture notes, the nasty firewall we have at work (that should not exist as two young bright colleagues of Lumo nicely proved ...) intervenes, so I'll look at it at home.

This post imported from StackExchange Mathematics at 2014-03-09 15:47 (UCT), posted by SE-user Dilaton
Unfortunately, all three lecture notes betrayed me, as did BBS and Mc Mohan : ( However, I think I found the answer myself, and will be posting it here soon.

This post imported from StackExchange Mathematics at 2014-03-09 15:47 (UCT), posted by SE-user dimensio1n0

1 Answer

+ 1 like - 0 dislike
If one solves the field equations for the bosonic field, with the Newmann/Dirchilet/Closed String Boundary conditions, one can see that the mode expansion is something like: $$X^\mu=...+i\sqrt{2\alpha'} \sum_{n\neq0 }^{ } \frac{\alpha^\mu}{n}\exp\left(in\sigma^0\right)\cos\left(n\sigma^1\right)$$ On the other hand, the fermionic field mode expansion goes like: $$\psi^\mu_\pm = \frac1{\sqrt2}\sum_{n\in\mathbb Z \ \mathrm{or} \ \mathbb{Z}+\frac12 }b_r^\mu \exp\left(-ir\sigma^\pm\right) $$ Notice that there is a missing factor of $\frac1r$ in the second equation. $[N,b_{-r}^\mu]=rb_{-r}^\mu$ and the conclusion follows.
answered Oct 28, 2013 by dimension10 (1,985 points) [ revision history ]
Gosh, I'll have to check if the factor 1/r and maybe a factor r in the commutator is missing in my Zweibach book too or if it is just missing in my question ... Your answer will then solve the issue, thanks :-). In fact, thnking about it I should have seen that it is missing in the fermion expansion myself, darn ...!

This post imported from StackExchange Mathematics at 2014-03-09 15:47 (UCT), posted by SE-user Dilaton

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$\hbar$ysicsOverfl$\varnothing$w
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
...