Casimir's trick / Evaluating trace

In summary, the person is working on a project and needs to evaluate a trace using Casimir's trick. They are unsure of how to simplify the form and are asking for help. The suggested solution is to expand the products and use the trace of a sum rule, as well as gamma matrix identities.
  • #1
BookWei
13
0
Hi all, I am working on a project at the moment, and I have to evaluate the trace by using the Casimir's trick.
The trace form is
$$Tr[(\displaystyle{\not} P +M_{0})\gamma^{\mu}(\displaystyle{\not} P^{'} +M^{'}_{0})(\displaystyle{\not} p^{'}_{1} +m^{'}_{1})\gamma^{\nu}(\displaystyle{\not} p_{1} +m_{1})]$$
I have no idea how can I simplify this form by using Casimir's trick.
Can somebody help me solve this problem?
Many thanks!
 
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  • #2
BookWei said:
Hi all, I am working on a project at the moment, and I have to evaluate the trace by using the Casimir's trick.
The trace form is
$$Tr[(\displaystyle{\not} P +M_{0})\gamma^{\mu}(\displaystyle{\not} P^{'} +M^{'}_{0})(\displaystyle{\not} p^{'}_{1} +m^{'}_{1})\gamma^{\nu}(\displaystyle{\not} p_{1} +m_{1})]$$
I have no idea how can I simplify this form by using Casimir's trick.
Can somebody help me solve this problem?
Many thanks!
Just expand the products and then use that the trace of a sum is the sum of the traces. Then you will have a bunch of traces of monomials. For each you may use the identities (see for example the wikipedia entry on gamma matrices)
 
  • #3
nrqed said:
Just expand the products and then use that the trace of a sum is the sum of the traces. Then you will have a bunch of traces of monomials. For each you may use the identities (see for example the wikipedia entry on gamma matrices)

Thank you so much.
 

1. What is Casimir's trick and how does it work?

Casimir's trick is a mathematical technique used to evaluate the trace of a matrix. It involves using the properties of the trace operator to manipulate the matrix into a more manageable form. This allows for a simpler and more efficient way to calculate the trace.

2. Why is evaluating the trace important in scientific research?

The trace of a matrix is a useful tool in many areas of science, including physics, chemistry, and engineering. It can provide information about the behavior and properties of a system, and is often used in calculations and equations to solve complex problems.

3. What are the limitations of Casimir's trick?

Casimir's trick is not applicable to all matrices, and it may not always provide the most accurate result. It also requires a good understanding of linear algebra and matrix operations, which can be challenging for those without a strong mathematical background.

4. How is Casimir's trick different from other methods of evaluating the trace?

Casimir's trick is unique in that it uses the properties of the trace operator to simplify the calculation, rather than directly summing the diagonal elements of the matrix. This can often lead to a more efficient and elegant solution compared to other methods.

5. Can Casimir's trick be applied to non-square matrices?

No, Casimir's trick is specifically designed for square matrices. If a non-square matrix needs to be evaluated, alternative methods such as the sum of diagonal elements or the use of eigenvalues may be more suitable.

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