Lie Algebra of Lorentz Group: Weird Notation?

Click For Summary

Discussion Overview

The discussion centers around the notation used in the commutation relations of the generators of the Lorentz group as presented in Srednicki's text on quantum field theory. Participants explore the meaning and standardization of the double-arrow notation in the context of linear algebra and quantum mechanics.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions the standardization of the double-arrow notation used in the commutation relations and seeks clarification on its meaning.
  • Another participant explains that the notation indicates a swapping of indices, specifically that \(\mu\) and \(\nu\) can be interchanged without changing the expression.
  • A third participant agrees, noting that the left-hand side should be antisymmetric in \(\mu\) and \(\nu\), as well as in \(\rho\) and \(\sigma\), which the notation reflects.
  • A fourth participant describes the notation as a typographical convenience, contrasting it with Weinberg's approach, which does not use such shorthand.

Areas of Agreement / Disagreement

Participants generally agree on the interpretation of the notation as a means of indicating index swapping, but there is a lack of consensus on its standardization and usage across different texts.

Contextual Notes

The discussion does not resolve whether the notation is widely accepted or if it varies significantly between different authors in the field.

Theage
Messages
11
Reaction score
1
In Srednicki's text on quantum field theory, he has a chapter on quantum Lorentz invariance. He presents the commutation relations between the generators of the Lorentz group (equation 2.16) as follows:
$$[M^{\mu\nu},M^{\rho\sigma}] = i\hbar\Big(g^{\mu\rho}M^{\nu\sigma}-(\mu\leftrightarrow\nu)\Big)-(\rho\leftrightarrow\sigma)$$
I have never seen the strange double-arrow notation in any linear algebra book before. Is this notation at all standard and if so, what does it mean?
 
Physics news on Phys.org
It just means "its the same thing as the last term, just put \mu instead of \nu and vice-versa." The same about the last parenthesis.
 
Indeed. The lhs should be antisymm. in mu and nu, and also in rho and sigma. That's what this notation says :)
 
A typographical convenience, nothing more. Weinberg for instance makes no LaTex shortenings.
 

Similar threads

  • · Replies 7 ·
Replies
7
Views
4K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 7 ·
Replies
7
Views
1K
  • · Replies 50 ·
2
Replies
50
Views
4K
  • · Replies 7 ·
Replies
7
Views
3K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K
Replies
27
Views
4K
  • · Replies 6 ·
Replies
6
Views
2K
  • · Replies 1 ·
Replies
1
Views
4K