Proving the Gamma Matrix Identity in QFT: Is There a Mistake in My Attempt?

In summary, the task is to prove that \gamma^{a}\gamma^{b}\gamma^{c}\gamma^{d}\gamma^{e}\gamma_{a} = 2\left(\gamma^{e}\gamma^{b}\gamma^{c}\gamma^{d}+\gamma^{d}\gamma^{c}\gamma^{b}\gamma^{e}\right). The equations provided are \gamma^{a}\gamma^{b}\gamma^{c}\gamma^{d}\gamma_{a} = -2\gamma^{d}\gamma^{c}\gamma^{b} and \gamma^{a}\gamma^{b} + \gamma^{b}\gamma^{a} = 2g^{ab}. The attempt at a solution involves
  • #1
Catria
152
4

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...
 
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  • #2
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
Thank you.
 

1. What is the Gamma matrix identity?

The Gamma matrix identity is a mathematical relationship that arises in quantum field theory (QFT). It relates the Dirac gamma matrices, which are mathematical objects used to represent spin and other properties of particles, to the identity matrix. It is an important tool in understanding and solving problems in QFT.

2. Why is the Gamma matrix identity important in quantum field theory?

The Gamma matrix identity is crucial in QFT because it helps us to simplify complex mathematical equations and expressions. It also allows us to make connections between different physical quantities, making it easier to study the behavior of particles and fields in quantum systems.

3. How is the Gamma matrix identity derived?

The Gamma matrix identity is derived from the Dirac equation, which describes the behavior of fermions (particles with half-integer spin). By manipulating the Dirac equation using mathematical tools such as the Feynman slash notation, we can arrive at the Gamma matrix identity.

4. Can the Gamma matrix identity be used in other areas of physics?

Yes, the Gamma matrix identity has applications in other fields of physics, such as condensed matter physics and string theory. In these areas, the identity is used to describe the behavior of fermions in different systems and to study the properties of higher-dimensional objects.

5. Are there different versions of the Gamma matrix identity?

Yes, there are multiple versions of the Gamma matrix identity, each corresponding to different dimensions and numbers of gamma matrices. The most commonly used version is the four-dimensional identity, but there are also versions for higher dimensions, such as 10 or 11 dimensions in string theory.

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