What is the significance of tau and sigma matrices in quantum theory?

[jhaber]

In Zee's quantum theory text, introducing the Dirac equation, he states the gamma matrices as direct products of Pauli matrices. The statements involve the identity matrix, sigma matrices, and tau matrices. It took me a bit to realize that the latter were identical. I hadn't seen the tau notation before; it's only sigma in my hardest quantum mechanics text, Messiah. I see that Wikipedia says that tau is used in context of isospin (not mentioned in this chapter in Zee).

Can I ask more about the convention, so that I understand why both notations are used in the same equation? They appear a few pages later in straightforward further manipulation as well. That is, I understand the math but not the notation. Thanks!

 

[juzzy]

In Zee's quantum theory text, introducing the Dirac equation, he states the gamma matrices as direct products of Pauli matrices. The statements involve the identity matrix, sigma matrices, and tau matrices. It took me a bit to realize that the latter were identical. I hadn't seen the tau notation before; it's only sigma in my hardest quantum mechanics text, Messiah. I see that Wikipedia says that tau is used in context of isospin (not mentioned in this chapter in Zee).

Can I ask more about the convention, so that I understand why both notations are used in the same equation? They appear a few pages later in straightforward further manipulation as well. That is, I understand the math but not the notation. Thanks!

The sigma and tau notation, as far as I know, is used to represent the possible spin states in a two state entangled system.

sigma: < u d | < d u |
Tau: | u d > | d u >


eg < u d | + < d u | (spin operator) | u d > + | d u >

The spin operator acts on sigma and tau leaving tau with it's sign interchanged

= < u d | + < d u | | u d > - | d u >

hope this helps

 

[jhaber]

I'll have to think about that one. The in-text equations are quite consistent with both notations referring to the same set of three matrices. For example, the third is

1 0
0 -1

where you'll excuse me if I don't know Latex, so that gammma-0 =

I 0
0 I

works out to the direct product of I and tau-3. Also there's no restriction here in mixing notations of one to any particular spin state, rather than to components of Dirac matrices that govern both spin states. It's more like they're being used for left-hand multiplier/super vs right-hand/sub, like co- and contravariant. But I'll look at it again from your angle and see if it gives added insight.

 

[juzzy]

I'll have to think about that one. The in-text equations are quite consistent with both notations referring to the same set of three matrices. For example, the third is

1 0
0 -1

where you'll excuse me if I don't know Latex, so that gammma-0 =

I 0
0 I

works out to the direct product of I and tau-3. Also there's no restriction here in mixing notations of one to any particular spin state, rather than to components of Dirac matrices that govern both spin states. It's more like they're being used for left-hand multiplier/super vs right-hand/sub, like co- and contravariant. But I'll look at it again from your angle and see if it gives added insight.

The sigma 3 matrix and the tau 3 matrix are both the same matrix, so are sigma 2 and tau 2, and sigma 1 and tau 1. It is called a sigma matrix when acting on sigma and a tau matrix when acting on tau. It doesn't matter which one you call sigma and which one you call tau, it works out the same, the inner product will be equal to zero...I think that's right, perhaps someone will correct me if wrong.

 

[jhaber]

That's fine, thanks much. I know I'll have to read up more this chapter, as my group theory background is limited. I took an algebra course in math out of Herstein, where a group was just a set with a single operation that didn't necessarily commute, and such things as symmetry, matrices, and types like SO(5) were not on the table. (Apologies: an earlier version of this referred to the text as Lang, my linear algebra text from two years before.)