Proving the Trace of Gamma Matrices with (Anti-)Communtation Rules

In summary, by using the anti-commutation rules and the cyclic property of trace, it can be shown that tr(\gamma^{\mu}\gamma^{\nu}\gamma^{5}) = 0. This is because the expression is equal to minus itself, and the only number with this property is zero. To prove this, the (gamma^rho)^2 term can be inserted into the trace and one of the gamma^rho's can be moved around by commutation.
  • #1
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Homework Statement


Show that [tex]tr(\gamma^{\mu}\gamma^{\nu}\gamma^{5}) = 0[/tex]


Homework Equations


(anti-)commutation rules for the gammas, trace is cyclic


The Attempt at a Solution


I can do
[tex] tr(\gamma^{\mu}\gamma^{\nu}\gamma^{5}) = -tr(\gamma^{\mu}\gamma^{5}\gamma^{\nu}) = - tr(\gamma^{\nu}\gamma^{\mu}\gamma^{5}) [/tex]
and so on, but I don't see how that helps me. Any suggestions?
 
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  • #2
You've used the first of your "relevant equations", now use the second.
 
  • #3
If I continue like this
[tex] tr(\gamma^{\mu}\gamma^{\nu}\gamma^{5}) = - tr(\gamma^{\nu}\gamma^{\mu}\gamma^{5}) = - tr( (2\eta^{\nu \mu} - \gamma^{\mu}\gamma^{\nu} )\gamma^{5}) = tr(\gamma^{\mu}\gamma^{\nu}\gamma^{5}) - tr(2\eta^{\nu \mu} \gamma^{5}) [/tex]
which I guess means that [tex]tr(\gamma^{5})=0[/tex]
 
  • #4
Oops, I didn't notice that you've already gotten as far as you need to. You've shown that your original expression is equal to minus itself. What is the only number with this property?
 
  • #5
But, I have [tex]\gamma^{\mu}[/tex] and [tex]\gamma^{\nu}[/tex] in a different order, and the entire point is that they don't commute?
 
Last edited:
  • #6
Oops, sorry, I screwed up again.

To do this one, I think you need to insert (gamma^rho)^2 (which is either one or minus one, depending on your conventions and on whether or not rho=0) into the trace,
with rho not equal to mu or nu, and then move one of the gamma^rho's around by commutation.
 

1. What are gamma matrices?

Gamma matrices are a set of mathematical objects used in quantum mechanics to represent spin and other properties of subatomic particles.

2. How many gamma matrices are there?

There are typically 4 gamma matrices, denoted as γ0, γ1, γ2, and γ3. However, in some theories, there can be more than 4.

3. What is the purpose of gamma matrices?

Gamma matrices are used to describe the behavior of fermions, such as electrons and quarks, in quantum field theory. They are essential in understanding the properties and interactions of these particles.

4. How do gamma matrices relate to the Dirac equation?

The Dirac equation is an equation that describes the behavior of fermions in quantum mechanics. Gamma matrices appear in the equation as operators that act on the wavefunction of the particle, allowing us to solve for its energy and other properties.

5. Are there any applications of gamma matrices outside of physics?

Gamma matrices are also used in mathematics, specifically in the field of linear algebra. They have applications in computer graphics, where they are used to rotate and transform objects.

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