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Tensor Product of Pauli Matrices

  1. Mar 30, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose that [tex][\sigma_a]_{ij}[/tex] and [tex][\eta_a]_{xy}[/tex] are Pauli matrices in two different two dimensional spaces. In the four dimensional tensor product space, define the basis:
    [tex]|1\rangle=|i=1\rangle|x=1\rangle[/tex]
    [tex]|2\rangle=|i=1\rangle|x=2\rangle[/tex]
    [tex]|3\rangle=|i=2\rangle|x=1\rangle[/tex]
    [tex]|4\rangle=|i=2\rangle|x=2\rangle[/tex]
    Write out the matrix elements of [tex]\sigma_2\otimes\eta_1[/tex]

    2. Relevant equations
    [tex]\sigma_a\sigma_b=\delta_{ab} + i\epsilon_{abc}\sigma_c[/tex]

    3. The attempt at a solution
    I know that that [tex]\sigma_2\otimes\sigma_1=\begin{bmatrix}0&0&0&-i\\0&0&-i&0\\0&i&0&0\\i&0&0&0\end{bmatrix}[/tex]
    And [tex]\langle i,x| \sigma_2\otimes\eta_1|j,y\rangle = \langle i| \sigma_2|j\rangle \langle x| \eta_1|y\rangle[/tex]
    I'm just confused about the matrices being in different spaces, how do I use the defined basis to calculate the matrix elements? I suspect I need the formula given as a relevant equation, but how can I use it with matrices in different spaces.
    I'm doing self study with Georgi - Lie Algebras.
     
  2. jcsd
  3. Mar 30, 2009 #2
    Here's what I'm thinking: We have two matrices A and B that represent linear transformations f and g in two spaces U and V with basis (u, u') and (v, v'), respectively. The tensor product space U otimes V will have as a basis (uv, uv', u'v, u'v') and A otimes B will be the matrix representation of f otimes g with the aforementioned basis.

    So represenet sigma_2 with respect to (|i=1>, |i=2>) and then eta_1 with respect to (|x=1>, |x=2>) and take their tensor product.

    I'm no expert on this though. Caveat emptor.
     
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