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

183 submissions , 143 unreviewed
4,703 questions , 1,912 unanswered
5,260 answers , 22,431 comments
1,470 users with positive rep
741 active unimported users
More ...

  Existences of Majorana spinors in $\rm Spin(4)$ and $\rm Spin(1,3)$

+ 2 like - 0 dislike
34 views

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$

We know that $$ \Spin(1,3)=\SL(2,\mathbb C) $$ and $$ \Spin(4)=\SU(2) \times \SU(2). $$ It is also said $\Spin(1,3)$ is the complexification of $\SU(2)$.

  • My question is that why there is a Majorana spinor (4 component Real spinor representation?) in $\Spin(1,3)$? But there is NO Majorana spinor (4 component Real spinor representation?) in $\Spin(4)$?

See this fact from the Table https://en.wikipedia.org/wiki/Spinor#Spinors_in_representation_theory

enter image description here

Refs from google search:

http://www-personal.umich.edu/~williams/notes/spinor.pdf http://scipp.ucsc.edu/~haber/ph251/Spinor_Shijun https://en.wikipedia.org/wiki/Complexification_(Lie_group)

This post imported from StackExchange Physics at 2020-12-03 13:04 (UTC), posted by SE-user annie marie heart
asked Jul 28, 2020 in Theoretical Physics by annie marie heart (1,155 points) [ no revision ]
Did I (more or less) answer your question?

This post imported from StackExchange Physics at 2020-12-03 13:04 (UTC), posted by SE-user Stijn

1 Answer

+ 3 like - 0 dislike

A Majorana spinor is a spinor for which the Dirac conjugate spinor $\psi^\dagger \gamma^0$ is equal to the Majorana conjugate spinor $\psi^T C$, i.e. $\psi$ satisfies the Majorana condition $$ (\gamma^0)^T \psi^* = C^T \psi. $$ Here $T$ stands for transposition, $*$ for (complex) conjugation, $\dagger$ for conjugate transposition, and $C$ is the charge conjugation matrix, which maps gamma matrices to their transposed cousins $$ C \gamma^\mu C^{-1} = \pm (\gamma^\mu)^T. $$ From this and the defining relations of the gamma matrices, it is possible to derive details on the form of the possible charge conjugation matrices and the complex conjugation properties of the gamma matrices. With that information we can then check whether the Majorana condition can have solutions. The upshot is that this derivation depends on the signature of the spacetime, and that the Majorana condition turns out to have a solution for signature $(-+++)$, but not for signature $(++++)$.

For the details of this derivation you can look at e.g. the notes in section 3 of https://arxiv.org/pdf/hep-th/9910030.pdf. For an intuitive hint why the existence of Majorana spinors depends on the signature, consider that if we have a set of gamma matrices $\{\gamma^0, \gamma^1, \gamma^2, \gamma^3\}$ for signature $(-+++)$, we can use $\tilde{\gamma}^0=i \gamma^0$ and $\tilde{\gamma}^j=\gamma^j, j=1,2,3$, as a set of gamma matrices for signature $(++++)$. While things look similar, the appearance of an extra $i$ certainly can (and does) have consequences when looking at an equation like the Majorana condition, involving complex conjugation.

This post imported from StackExchange Physics at 2020-12-03 13:04 (UTC), posted by SE-user Stijn
answered Jul 28, 2020 by Stijn (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$\hbar$ysicsOverf$\varnothing$ow
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
...