Solving Tricky Pauli Matrices with Einstein Notation

In summary, the author is working on a problem where they need to expand a matrix in terms of a product of matrices. They are having trouble understanding the properties of a Pauli matrix and need help.
  • #1
33
0
Hello,
I am trying to recover the following calculation (where [tex]K,A[/tex] are 4x4 matrices in SL(2,C)):

--(start)--
"We expand [tex]K'=AKA^{\dagger}[/tex] in terms of [tex]k^a[/tex] and [tex]k'^{a}=(\delta_a^{b} + \lambda_a^{b} d\tau)k^b[/tex]. Multiplying by a general Pauli matrix and using the relation [tex]\frac{1}{2}tr(\sigma_{a}\sigma_{b})=\delta_{ab}[/tex] yields the expression:
[tex]
\lambda_b^{a} = \frac{1}{2}\eta^{ac}tr(\sigma_{b}\sigma_{c}A+\sigma_{c}\sigma_{b}A^{\dagger})
[/tex]."
--(end)--

I have been playing with the relations for a while but I guess I miss some knowledge on the properties of Pauli matrices because I don't manage to find the result. In particular, what would the "expansion" of [tex]AKA^{\dagger}[/tex] (which I guess is necessary here?) look like in Einstein summation notation ? Any help would be extremely appreciated!
 
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  • #2
the answer lies on wikipedia.
 
  • #3
ryuunoseika said:
the answer lies on wikipedia.

Well, give a link, then :rolleyes:which article? :smile:
 
  • #4
Well I had of course already checked beforehand the wikipedia page on Pauli matrices (http://en.wikipedia.org/wiki/Pauli_matrices) but had not found a relation to solve this problem... So Ryuunoseika, which article are you talking about ?
 
  • #5
Something here seems either not quite right or incomplete.

If [itex]k'^{a} = \left(\delta_b^{a} + \lambda_b^{a} d\tau \right) k^b[/itex] (careful with index placement) and [itex]d \tau[/itex] is infinitesimal (?), then [itex]k'[/itex] and [itex]k[/itex] differ by an infinitesimal amount, so the transformation is an infinitesimal version of [itex]K'=AKA^{\dagger}[/itex]. Then, the sum in final result could come from the product rule.

I'm just guessing. More context is needed.
 
  • #6
Dear George,

Thanks a lot for your answer. First, yes sorry I misplaced the indices in the first relation, the correct relation is:
[tex]k'_{a}=(\delta_a^{b} + \lambda_a^{b} d\tau)k_b[/tex]

Concerning [tex]d\tau[/tex], it is actually a [tex]\delta u[/tex], where [tex]u[/tex] is the affine parameter along the trajectory of a photon.

But what do you mean by "product rule" ? Do I have to develop explicitely [tex]AKA^{\dagger}[/tex] in indices notation and try to recover at the end the [tex]A[/tex] and [tex]A^{\dagger}[/tex] which appear in the trace ?
 

1. What are Pauli matrices?

Pauli matrices are a set of three 2x2 matrices named after physicist Wolfgang Pauli. They are used in quantum mechanics to represent spin states of particles, and they have important properties when it comes to solving mathematical equations in this field.

2. What is Einstein notation?

Einstein notation, also known as summation convention, is a mathematical notation used to condense and simplify equations involving repeated indices. It involves using Greek letters to represent indices and implies summation over all possible values of that index.

3. How are Pauli matrices and Einstein notation related?

Pauli matrices can be manipulated using Einstein notation to solve tricky equations in quantum mechanics. Using this notation, the properties of Pauli matrices, such as commutation and anti-commutation rules, can be easily expressed and used to simplify equations.

4. Can Einstein notation be used to solve other types of matrices?

Yes, Einstein notation can be used to solve any type of matrix equation involving repeated indices. It is commonly used in fields like physics and engineering to simplify calculations and make them more concise.

5. Are there any drawbacks to using Einstein notation to solve equations?

One potential drawback is that it may take some time to get used to this notation, as it differs from the traditional matrix notation. Additionally, it can become confusing when dealing with more complex equations with multiple indices. However, once mastered, it can be a powerful tool for solving tricky equations efficiently.

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