Simplifying gamma matrices

In summary, gamma matrices are mathematical objects used to represent the behavior of particles with spin in quantum mechanics. Simplifying them is important as it makes calculations and equations involving them more manageable. This can be done using techniques such as matrix algebra, properties and identities, and specific rules. Common mistakes include not properly applying these techniques and not carefully performing algebraic steps, which can lead to incorrect simplifications and results. It is crucial to follow correct procedures and double-check the simplification to ensure accuracy.
  • #1
ryanwilk
57
0

Homework Statement



I'm trying to find [itex]P_L \displaystyle{\not}p P_L[/itex] for a left-handed particle.

(I think the answer is zero...)

Homework Equations



[itex]P_L = \frac{1}{2} (1-\gamma_5)[/itex] (the left-handed projection operator)
[itex]\displaystyle{\not} p = \gamma_\mu p^\mu[/itex] (pμ is the 4-momentum)

μ, γ5 are the gamma matrices)

The Attempt at a Solution



So, [itex]P_L \displaystyle{\not} p P_L =
\frac{1}{4} (1-\gamma_5) \gamma_\mu p^\mu (1-\gamma_5) =
\frac{1}{4} [ \gamma_\mu p^\mu - \gamma_5 \gamma_\mu p^\mu - \gamma_\mu p^\mu \gamma_5 + \gamma_5 \gamma_\mu p^\mu \gamma_5 ] =
\frac{1}{4} \gamma_\mu [ p^\mu + \gamma_5 p^\mu - p^\mu \gamma_5 - \gamma_5 p^\mu \gamma_5] [/itex] (*)
(using the anticommutation relation {γ5μ} = 0).

Can this be simplified further? Since pμ is a 4x1 matrix, I'm guessing you can't just swap the order of [itex]\gamma_5[/itex] and [itex]p^\mu[/itex].

However, you can rewrite (*) as [itex]\gamma_\mu P_R p^\mu P_L[/itex], and then because it's a left-handed particle, P_R acting on the momentum gives zero?

Any help would be appreciated,

Thanks.
 
Last edited:
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  • #2


Hello! Your reasoning is correct, the answer is indeed zero. Let me explain why.

First, let's rewrite the expression using the definition of P_L and \displaystyle{\not} p:

P_L \displaystyle{\not} p P_L = \frac{1}{2} (1-\gamma_5) \gamma_\mu p^\mu (1-\gamma_5) = \frac{1}{2} (1-\gamma_5) \gamma_\mu p^\mu - \frac{1}{2} (1-\gamma_5) \gamma_5 \gamma_\mu p^\mu

Using the anticommutation relation {γ5,γμ} = 0, we can simplify the second term to:

\frac{1}{2} (1-\gamma_5) \gamma_5 \gamma_\mu p^\mu = -\frac{1}{2} \gamma_\mu p^\mu

Therefore, the expression becomes:

P_L \displaystyle{\not} p P_L = \frac{1}{2} (1-\gamma_5) \gamma_\mu p^\mu - \frac{1}{2} \gamma_\mu p^\mu

Now, as you mentioned, since pμ is a 4x1 matrix, we cannot just swap the order of \gamma_5 and p^\mu. However, we can rewrite the expression as:

\frac{1}{2} \gamma_\mu (1-\gamma_5) p^\mu - \frac{1}{2} \gamma_\mu p^\mu

And using the definition of P_L, we can rewrite it as:

\frac{1}{2} \gamma_\mu P_R p^\mu - \frac{1}{2} \gamma_\mu p^\mu

Finally, since we are dealing with a left-handed particle, P_R acting on the momentum gives zero, as you correctly stated. Therefore, the final expression becomes:

P_L \displaystyle{\not} p P_L = 0

I hope this helps! Let me know if you have any other questions.
 

1. What are gamma matrices?

Gamma matrices are a set of mathematical objects that are used to represent the Dirac equation in quantum mechanics. They are used to describe the behavior of particles with spin and are an essential tool in understanding the behavior of subatomic particles.

2. Why is it important to simplify gamma matrices?

Gamma matrices can be quite complex and difficult to work with. Simplifying them can make calculations and equations involving them much more manageable and easier to understand.

3. How can gamma matrices be simplified?

There are various techniques for simplifying gamma matrices, such as using matrix algebra, using the properties of gamma matrices, and using identities and rules specific to gamma matrices. It often involves breaking down the matrices into simpler components and manipulating them to simplify the overall expression.

4. What are some common mistakes when simplifying gamma matrices?

One common mistake is not properly applying the properties and identities of gamma matrices, leading to incorrect simplifications. Another mistake is not carefully performing the necessary algebraic steps, such as multiplying or factoring, which can also result in incorrect simplifications.

5. Can simplifying gamma matrices lead to incorrect results?

Yes, if done incorrectly, simplifying gamma matrices can result in incorrect or invalid results. It is important to carefully follow the correct procedures and double-check the simplification to ensure the accuracy of the final result.

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