# Why this expression is time-reversal odd?

1. Dec 13, 2013

### Chenkb

P and k are four-momentum of two particles.
I read in a paper which said that
$[\slashed{P},\slashed{k}]=\slashed{P}\slashed{k} - \slashed{k}\slashed{P}$
is time-reversal odd.
Why?

2. Dec 13, 2013

### ChrisVer

could you please fix the Latex?
What is P and k? momenta?

3. Dec 13, 2013

### ZapperZ

Staff Emeritus
... and please make a full citation of the paper.

Zz.

4. Dec 13, 2013

### George Jones

Staff Emeritus
Unfortunately, MathJax does not support the "slashed" command from the amsmath package, which is used to implement Feynman's slash notation. Try

$$\left[\gamma \cdot P , \gamma \cdot k \right] = \gamma \cdot P \ \gamma \cdot k − \gamma \cdot k \ \gamma \cdot P,$$

where $\gamma \cdot P = \gamma^\mu P_\mu$.

Or just use (the possible more useful in this case) $\gamma^\mu P_\mu$ and $\gamma^\nu k_\nu$.[/edit]

Last edited: Dec 13, 2013
5. Dec 13, 2013

### dauto

Try [t e x]/ \!\!\!\! P[/ t e x]: $/ \!\!\!\! P$
and [t e x]/ \!\!\! k[/ t e x]: $/ \!\!\! k$

6. Dec 13, 2013

### Chenkb

7. Dec 13, 2013

### Chenkb

Thanks a lot for the texing.

8. Dec 13, 2013

### ChrisVer

Well, the reason why there is T-oddness I guess is because there is a $i$ in front, making that term complex..

Last edited: Dec 13, 2013
9. Dec 13, 2013

### Chenkb

But $\Phi(P,k,S|n)$ on the left of the Eq. must be T-even according to it's physical meaning, so, if $A_4$ is T-odd, $[\gamma \cdot P, \gamma \cdot k]$ must be T-odd.
Actually, I think it is $[P, k]$ be T-odd that infers $A_4$ be T-odd, not the opposite, because $A_4$ is an unknown coefficient function.
So I wonder why.
Thanks a lot!

10. Dec 14, 2013

### ChrisVer

lol, I changed my initial post because it was wrong, I guess that:
"Well, the reason why there is T-oddness I guess is because there is a $i$ in front, making that term complex.."

11. Dec 14, 2013

### Chenkb

Well, I think the $i$ in front is just for convenience. And the gamma matrices are not all real, for example, in Dirac representation, $\gamma^2$ is complex.
So, I think the problem is how does $\left[\gamma \cdot P , \gamma \cdot k \right]$change under time reversal operation.

I know that, under time reversal, $P=(P_0,\vec{P})$becomes $(P_0,-\vec{P})$, and in QFT course I've learned that $\gamma^{\mu}$becomes $(-1)^{\mu}\gamma^{\mu}$(Peskin's book, Page71). But what about $\left[\gamma \cdot P , \gamma \cdot k \right]$?

12. Dec 14, 2013

### Bill_K

No, it's not there just for convenience. It's there because i[γ·P, γ·K] = 2 σμν Pμ Kν. And if you look at the table on p.71 of Peskin, you'll see how σμν transforms.

13. Dec 14, 2013

### Chenkb

Year, but for the terms in $\sigma_{\mu\nu}P^{\mu}k^{\nu}$ like $\sigma_{02}P_0k_2$, is T-even, because $\sigma_{02}$ odd, $P_0$ even, and $k_2$ odd.
Regards!

14. Dec 14, 2013

### Bill_K

No, σ02 is even. In the sum, σ0i is even, σjk is odd, P0 and K0 are even, while Pj and Kj are odd. All the terms in the sum σμνPμKν wind up transforming the same way, namely they are all odd.

15. Dec 14, 2013

### Chenkb

Could you please explain why $\sigma_{0i}$ is T-even? isn't it $-(-1)^{\mu}(-1)^{\nu}$?

16. Dec 14, 2013

### Bill_K

Peskin uses a very strange notation here! He says "(-1)μ" is supposed to mean 1 for μ=0 and -1 for μ=1, 2, 3. In the table, for T we're given - (-1)0(-1)i, which is interpreted to mean (-1)(1)(-1) which is +1.

His derivation of the results in this table could use a little cleanup work too. In addition to complex conjugation ψ → ψ*, the time reversal operation includes a matrix acting on ψ. He says this matrix is γ1γ3, but the value actually depends on your choice of representation for the gamma matrices. He uses the Weyl representation Eq.(3.25), in which γ2 is imaginary and all the others real. In general, time reversal can be written ψ → Dψ* where D is a matrix having the property D-1γμD = γ'μ where γ'0 = - γ0 and γ'j = γj.

17. Dec 14, 2013

### Chenkb

WOW!:thumbs: Many thanks, I will check it.

18. Dec 31, 2013

### M. Bachmeier

Your looking for a transformation of the equation... yes? or, an explanation?

19. Dec 31, 2013

### Bill_K

Which do you think has not already been given?

20. Dec 31, 2013

### Chenkb

The transformation, why it's T-odd. And I think it is solved already.