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 ...

  Counting degrees of freedom in spinor-helicity formalism

+ 1 like - 0 dislike
599 views

Just a couple of quick questions about the spinor-helicity formalism.

We start with $p^\mu$ and $\epsilon^\mu$, so we have eight degrees of freedom. Then we have that $p^\mu p_\mu = 0$ and that $\epsilon^\mu p_\mu = 0$, which reduce this to six. Also there is a freedom to choose the gauge, which means that $\epsilon^\mu \rightarrow \epsilon^\mu + c p^\mu$ for $c \in \mathbb{R}$ can't affect the physics, which I think reduces the system by one degree of freedom more. So this gives us five remaining degrees of freedom.

Then we go to the spinor-helicity notation to reduce this redundancy, and instead we use $P_{a\dot{a}} = p_\mu \sigma^\mu_{a\dot{a}} = \lambda_a \tilde{\lambda}_\dot{a}$. Because the lambdas are constructed from $p^\mu$, I conclude that they encode the three degrees of freedom of our null momentum vector.
We now construct two different helicity polarisation vectors, using the lambdas
$\epsilon^{-}_{a\dot{a}} = -\sqrt{2}\frac{\lambda_a \tilde{\mu}_\dot{a}}{[\tilde{\lambda}\tilde{\mu}]}$
and
$\epsilon^{+}_{a\dot{a}} = -\sqrt{2}\frac{\mu_a \tilde{\lambda}_\dot{a}}{\langle\lambda\mu\rangle}$

Where $\mu$ and $\tilde{\mu}$ are aribitary reference spinors, representing the choice of gauge. Two questions:
1) What happened to the two degrees of freedom of the original polisation vector? We appear to have constructed our new polarisation vectors completely from the three degrees of freedom of our momentum
2) I'm happy to accept that the choice of $\mu$ corresponds to the choice of gauge as written for the polarisation vector, but I can't see it explicitly. Can someone help to show me why that is?

This post imported from StackExchange Physics at 2015-06-03 19:31 (UTC), posted by SE-user Joe
asked Jun 2, 2015 in Theoretical Physics by Joe (40 points) [ no revision ]

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$\hbar$ysicsOverflo$\varnothing$
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
...