Lie Algebra of Lorentz Group: Weird Notation?

[Theage]

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?

 

[ShayanJ]

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.

 

[haushofer]

Indeed. The lhs should be antisymm. in mu and nu, and also in rho and sigma. That's what this notation says :)

 

[dextercioby]

A typographical convenience, nothing more. Weinberg for instance makes no LaTex shortenings.

 

[sardar]

The double-arrow notation used in the commutation relations for the generators of the Lorentz group is a standard notation in the theory of Lie algebras. It is used to indicate that the terms inside the parentheses are to be swapped or interchanged. In this specific case, the notation $(\mu\leftrightarrow\nu)$ means that the terms $\mu$ and $\nu$ are to be swapped, while $(\rho\leftrightarrow\sigma)$ means that the terms $\rho$ and $\sigma$ are to be swapped. This notation is commonly used in the study of Lie algebras, which are mathematical structures that are closely related to the symmetries of physical systems. Therefore, it is a natural choice to use this notation in the context of Lorentz invariance, which is a fundamental symmetry in physics. While it may seem unfamiliar at first, the double-arrow notation is a standard and useful tool in the study of Lie algebras and their applications in physics.