# Spin angular momentum

Spin angular momentum is defined as $\Sigma^{i}= 1/2 e^{ijk} \sigma_{jk}$ . Thus, I can write $\Sigma^{1}$ as $[\gamma_{2},\gamma_{3}]$ . I want some insights on this definition. Also, Can anybody explain the difference in spin and spin angular momentum?

$\gamma_{i}$ are Dirac matrices appearing in dirac equation. And, $\sigma_{jk}= e_{ijk} \sigma^{i}$ where $\sigma_{i}$ are Pauli matrices.

• What are the $\sigma_{jk}$? What are the $\gamma_i$? Please define your notation and give references which treat "spin angular momentum" as different from "spin". – ACuriousMind Aug 13 '15 at 12:57
• Write out your $\sigma_{jk}$ and use a Levi-Civita identity. Then, $\Sigma^i = \delta^i_j \sigma_j$, so there's no difference between $\Sigma$ and $\sigma$. – ACuriousMind Aug 13 '15 at 13:10

If you define spin angular momentum as $\Sigma^{i}= 1/2 e^{ijk} \sigma_{jk}$ and $\sigma_{jk}= e^{ajk} \sigma_{a}$ then there is some inconsistency about when you place upper versus lower indices but you get:

$$\Sigma^{i}=\frac{ 1}{2 }e^{ijk} \sigma_{jk}= \frac{ 1}{2 }e^{ijk} e^{ajk} \sigma_{a} =\frac{ 2\delta^i_a}{2}\sigma_{a} =\sigma_i$$

It really comes down to whether you prefer to write things in terms of $\gamma_\mu$s or $\sigma_i$s or $\sigma_{ij}$s.

Myself, I to think of the $\gamma_\mu$ as vectors in spacetime and the $\sigma_i$ as spacetime bivectors with a fixed timelike vector factor which themselves have a vector space structure as a euclidean 3space ($\gamma_0\gamma_1,$ $\gamma_0\gamma_2,$ and $\gamma_0\gamma_3$ all square to +1 and all anticommute but are clearly just one split of the full space of bivectors like $\gamma_\mu\gamma_\nu$ into two 3d subspaces). So I like to build everything out of spacetime vectors that have a clear geometric interpretation. And I'd rather have the geometric interpretation of everything being linear combinations of products of the spacetime vectors $\gamma_\mu$ so I know what everything is both geometrically and algebraically.

I'm not even impressed with all the lower and upper indexices because I want to distinguish between the vectors which just are what they are and their products which again just are what they are and the linear combinations which needs scalars to combine them.

I'd rather see $1,$ $\gamma_\mu,$ $\gamma_\mu\gamma_\nu,$ $\gamma_\mu\gamma_\nu\gamma_\alpha,$ and $\gamma_0\gamma_1\gamma_2\gamma_3$ as a basis so I can just look at it as an algebraic basis of geometric entities. Breaking the bivectors $\gamma_\mu\gamma_\nu$ into a subcollection that looks like their own geometric algebra for a euclidean 3 space is clearly an arbitrary frame dependent choice, exactly the way we break an electromagnetic field into electric and magnetic fields.

But if you want lots of upper and lower indexes and and you prefer to write things in terms of $sigma_i$ or in terms of $\sigma_{ij}$ you are free to do that too. Your basis doesn't really matter. You don't even have to always pick a basis or even a representation sometimes you can just use the algebra as an algebra just like you can use a vector space as a vector space (just some things with an addition and some product(s)).

As for the terms spin and spin angular momentum. You might want to be careful to distinguish between an operator with integer eigenvalues and one with eigenvalues that are multiples of $\hbar/2$ only the latter has units of angular momentum.

Again, since you asked for insight. When I write a $\gamma_\mu$ I'm not writing a matrix I'm just writing an element of an algebra (with a multiplicative identity) with an addition and multiplication that has $\gamma_\mu\gamma_\nu+\gamma_\nu\gamma_\mu$ equal twice the multiplicative identity of the algebra if $\mu=\nu$ and gives the zero of the algebra if $\mu\neq\nu.$

In many situations a representation or even a basis is irrelevant, the algebra is what you are using and you can interpret the $\gamma_mu$ as vectors in spacetime and their products as rotations of each other or their antisymmetric products as planes or higher dimensional subspaces spanned by the vectors. So you can just interpret everything geometrically.

Leave a basis or coordinates or a representation for a particular situation if it comes up, but not part of the general understanding. For instance your whole question was about what basis to use for your algebra. The $\sigma_i$ can't generate the whole algebra by itself neither can a single rank three tensor, but the four basis spacetime vectors can. And how much time and confusion is it worth to worry about what basis to use if you don't even need a basis?