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Gamma matrix identity (QFT)

  • Thread starter Catria
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  • #1
149
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Homework Statement



Prove that [itex]\gamma^{a}\gamma^{b}\gamma^{c}\gamma^{d}\gamma^{e}\gamma_{a}[/itex] = [itex]2\left(\gamma^{e}\gamma^{b}\gamma^{c}\gamma^{d}+\gamma^{d}\gamma^{c} \gamma^{b}\gamma^{e}\right)[/itex]

Each of the [itex]\gamma^{i}[/itex]s are as used in the Dirac equation.

Homework Equations



[itex]\gamma^{a}\gamma^{b}\gamma^{c}\gamma^{d}\gamma_{a}[/itex] = [itex]-2\gamma^{d}\gamma^{c}\gamma^{b}[/itex]

[itex]\gamma^{a}\gamma^{b} + \gamma^{b}\gamma^{a} = 2g^{ab}[/itex]

The Attempt at a Solution



[itex]\gamma^{a}\gamma^{b}\gamma^{c}\gamma^{d}\gamma^{e}\gamma_{a}[/itex] = [itex]2g^{ab}\gamma^{c}\gamma^{d}\gamma^{e}\gamma_{a}[/itex] - [itex]\gamma^{b}\gamma^{a}\gamma^{c}\gamma^{d}\gamma^{e}\gamma_{a}[/itex]

= [itex]2\gamma^{c}\gamma^{d}\gamma^{e}\gamma^{b}[/itex] + [itex]2\gamma^{b}\gamma^{e}\gamma^{d}\gamma^{c}[/itex]

Perhaps I mixed up something or there is a typo...
 

Answers and Replies

  • #2
TSny
Homework Helper
Gold Member
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I think what you wrote so far is correct. But, as you can see, it is not yet very close to what you want.

Instead, start over and try commuting ##\gamma^{e}## with ##\gamma_{a}##. To do this, you will need to use the identity obtained by lowering ##a## in the identity [itex]\gamma^{a}\gamma^{b} + \gamma^{b}\gamma^{a} = 2g^{ab}[/itex]
 
  • #3
149
4
Thank you.
 

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