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Why this expression is timereversal odd?by Chenkb
Tags: expression, gamma matrix., particle physics, quantum field theory, time reversal odd, timereversal 
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#1
Dec1313, 02:22 AM

P: 27

P and k are fourmomentum of two particles.
I read in a paper which said that [itex][\slashed{P},\slashed{k}]=\slashed{P}\slashed{k}  \slashed{k}\slashed{P}[/itex] is timereversal odd. Why? 


#2
Dec1313, 01:16 PM

P: 855

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


#3
Dec1313, 01:47 PM

Emeritus
Sci Advisor
PF Gold
P: 29,238

... and please make a full citation of the paper.
Zz. 


#4
Dec1313, 02:05 PM

Mentor
P: 6,239

Why this expression is timereversal odd?
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##. [edit]Or just use (the possible more useful in this case) ##\gamma^\mu P_\mu## and ##\gamma^\nu k_\nu##.[/edit] 


#5
Dec1313, 04:02 PM

Thanks
P: 1,948

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


#6
Dec1313, 08:07 PM

P: 27



#7
Dec1313, 08:08 PM

P: 27




#8
Dec1313, 08:15 PM

P: 855

*wrong answer*
Well, the reason why there is Toddness I guess is because there is a [itex]i[/itex] in front, making that term complex.. 


#9
Dec1313, 08:31 PM

P: 27

Actually, I think it is ##[P, k]## be Todd that infers ##A_4## be Todd, not the opposite, because ##A_4## is an unknown coefficient function. So I wonder why. Thanks a lot! 


#10
Dec1413, 05:12 AM

P: 855

lol, I changed my initial post because it was wrong, I guess that:
"Well, the reason why there is Toddness I guess is because there is a [itex]i[/itex] in front, making that term complex.." is your answer... 


#11
Dec1413, 06:10 AM

P: 27

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
Dec1413, 09:32 AM

Sci Advisor
Thanks
P: 4,160




#13
Dec1413, 09:50 AM

P: 27

Regards! 


#14
Dec1413, 09:58 AM

Sci Advisor
Thanks
P: 4,160




#15
Dec1413, 10:06 AM

P: 27




#16
Dec1413, 11:39 AM

Sci Advisor
Thanks
P: 4,160

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 γ^{1}γ^{3}, but the value actually depends on your choice of representation for the gamma matrices. He uses the Weyl representation Eq.(3.25), in which γ^{2} 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^{1}γ^{μ}D = γ'^{μ} where γ'^{0} =  γ^{0} and γ'^{j} = γ^{j}. 


#17
Dec1413, 12:07 PM

P: 27




#18
Dec3113, 01:54 PM

P: 184

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



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