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

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

185 submissions , 145 unreviewed
4,725 questions , 1,928 unanswered
5,268 answers , 22,442 comments
1,470 users with positive rep
743 active unimported users
More ...

  Lorentz spinor in Lorentz $\rm Spin(3,1)$ signature and the real structure?

+ 5 like - 0 dislike
57 views

In this paper:

it says the following in p.2,

enter image description here

It says for $3+1$ dimensional spacetime, the Weyl spinor with $SU(2)$ isospin $1/2$ , "Lorentz signature always carry a real structure; if a fermion field appears in the Lagrangian, so does its hermitian adjoint"

What does it mean to be real ? If :

  • Weyl spinor is complex in $\text {Spin}(3,1)$, and
  • its $SU(2)$ isospin $1/2$ is pseudoreal in $SU(2) = \text {Spin} (3)$,

why do we get a Lorentz signature always carry a real structure (instead of just complex or pseudoreal)? Does it mean the whole Weyl spinor is in a real representation (4 component) of $\text {Spin} (3,1)$ and $SU(2) $ together?

What is the emphasis to say "In the Euclidean signature, nothing like that is true in general; what happens depends on the spacetime dimension"? Isnt that the Lorentz signature : real, pseudoreal, or complex also depends on the spacetime dimension ?


This post imported from StackExchange Physics at 2020-12-03 13:06 (UTC), posted by SE-user annie marie heart

asked Aug 3, 2020 in Theoretical Physics by annie marie heart (1,205 points) [ revision history ]
edited Apr 27 by annie marie heart

1 Answer

+ 2 like - 0 dislike

The highlighted statement is quite gnomic. There are many like it the paper. The claim that an an odd number of zero modes makes the path integral measure change sign under a gauge transformation by the central element $-1\in {\rm SU}(2)$ is likewise a mystery. The measure has both a $d\bar \psi_0$ and $d\psi_0$ for each zero mode, and so is invariant under the simultaneous transformation $\psi\to -\psi$ and $\bar\psi \to -\bar \psi$. Further, while it is clear that the mod 2 index of the five dimensional Dirac operator connects to the five dimensional operator $(-1)^F$, (which comes from the mapping torus periodic boundary condition in the $S^1$ direction) and is given by the ordinary four dimensional index reduced mod 2, I do not see how they can claim that this relates to the four dimensional $(-1)^F$ which is not well defined on $S^4$. I look forward to some interesting answers to your question.

This post imported from StackExchange Physics at 2020-12-03 13:06 (UTC), posted by SE-user mike stone
answered Aug 6, 2020 by mike stone (30 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$\hbar$ysic$\varnothing$Overflow
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).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...