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Prove that dirac matrices have a vanishing trace

  • Thread starter elduderino
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  • #1
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Not a Homework problem, but I think it belongs here.

Homework Statement


Consider four dirac matrices that obey

[tex] M_i M_j + M_j M_i = 2 \delta_{ij} I [/tex]

knowing the property that [tex] Tr ABC = Tr CAB = Tr BCA [/tex] show that the matrices are traceless.


Homework Equations



[tex] Tr MN = Tr NM [/tex]

The Attempt at a Solution



The square of each dirac matrix is a unit matrix according to the definition above. For i,j unequal

[tex]M_i M_j = - M_j M_i [/tex]

Since these matrices are equal their traces should be equal

[tex] Tr M_i M_j = - Tr M_j M_i =- Tr M_i M_j [/tex]

implying [tex] Tr M_i M_j = Tr M_j M_i = 0 [/tex] for [tex] i \neq j [/tex]

So far, I have not been able to prove that each of these dirac matrices individually as a vanishing trace. I tried

[tex] Tr M_i M_j = \sum_k \sum_r (M_i)_{kr}(M_j)_{rk} = 0 [/tex]

but cant conclude anything. This is embarrasing, as this seems pretty basic. Can someone help?
 
Last edited:

Answers and Replies

  • #2
CompuChip
Science Advisor
Homework Helper
4,302
47
Try using the cyclic trace property to show that
[tex]\operatorname{Tr}(M_i M_j) = \frac12 \operatorname{Tr}\left( M_i M_j + M_j M_i \right)[/tex]
 

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