- #1
ismaili
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I'm considering a physical system of [tex]n[/tex]-species of fermions.
The creation and annihilation operators satisfy
[tex] \{b_i^-,b_j^-\} = 0 = \{ b_i^+ , b_j^+ \}, \quad \{ b_i^- , b_j^+ \} = \delta_{ij} [/tex]
I want to construct the matrix representations of the state vectors and operators.
I do it recursively, but, I can't find a way to systematically construct the representation, what I did is the following.
Consider only one species of fermion, with one fermion state represented by [tex] \left(\begin{array}{c} 1\\ 0 \end{array}\right) [/tex], and empty state represented by [tex] \left(\begin{array}{c} 0\\ 1 \end{array}\right) [/tex]. We can construct the creation operator [tex]B^+[/tex] as [tex] B^+ = \left(\begin{array}{cc} 0 &1\\ 0 &0 \end{array}\right) [/tex] and annihilation operator [tex] B^- = \left( \begin{array}{cc} 0 &0 \\ 1 &0 \end{array}\right) [/tex].
However, for two species fermion system, if the states should be the direct product of each single species states, i.e.
[tex]|\frac{1}{2},\frac{1}{2}\rangle = \left(\begin{array}{c} 1\\0 \end{array}\right) \otimes \left(\begin{array}{c} 1\\0 \end{array}\right) [/tex] represents two-fermion states, likewise we have other states [tex] |\frac{1}{2},-\frac{1}{2}\rangle, |-\frac{1}{2},\frac{1}{2}\rangle, |-\frac{1}{2},-\frac{1}{2} \rangle [/tex] represents two possible 1-fermion states, and zero-fermion state.
Now, what I was surprised is, the creation and annihilation operators cannot be constructed naively as the following direct products:
[tex] b_1^+ = B^+ \otimes \mathbf{1}, b_1^- = B^- \otimes \mathbf{1} [/tex] and
[tex] b_2^+ = \mathbf{1} \otimes B^+, b_2^- = \mathbf{1} \otimes B^- [/tex]
The creation operators and annihilation operators constructed in this way would not satisfy the correct anti-commutation relation, for example, [tex] \{ b_1^+ , b_2^+ \} \neq 0 [/tex]
I don't quite understand why this construction doesn't work?
The creation and annihilation operators satisfy
[tex] \{b_i^-,b_j^-\} = 0 = \{ b_i^+ , b_j^+ \}, \quad \{ b_i^- , b_j^+ \} = \delta_{ij} [/tex]
I want to construct the matrix representations of the state vectors and operators.
I do it recursively, but, I can't find a way to systematically construct the representation, what I did is the following.
Consider only one species of fermion, with one fermion state represented by [tex] \left(\begin{array}{c} 1\\ 0 \end{array}\right) [/tex], and empty state represented by [tex] \left(\begin{array}{c} 0\\ 1 \end{array}\right) [/tex]. We can construct the creation operator [tex]B^+[/tex] as [tex] B^+ = \left(\begin{array}{cc} 0 &1\\ 0 &0 \end{array}\right) [/tex] and annihilation operator [tex] B^- = \left( \begin{array}{cc} 0 &0 \\ 1 &0 \end{array}\right) [/tex].
However, for two species fermion system, if the states should be the direct product of each single species states, i.e.
[tex]|\frac{1}{2},\frac{1}{2}\rangle = \left(\begin{array}{c} 1\\0 \end{array}\right) \otimes \left(\begin{array}{c} 1\\0 \end{array}\right) [/tex] represents two-fermion states, likewise we have other states [tex] |\frac{1}{2},-\frac{1}{2}\rangle, |-\frac{1}{2},\frac{1}{2}\rangle, |-\frac{1}{2},-\frac{1}{2} \rangle [/tex] represents two possible 1-fermion states, and zero-fermion state.
Now, what I was surprised is, the creation and annihilation operators cannot be constructed naively as the following direct products:
[tex] b_1^+ = B^+ \otimes \mathbf{1}, b_1^- = B^- \otimes \mathbf{1} [/tex] and
[tex] b_2^+ = \mathbf{1} \otimes B^+, b_2^- = \mathbf{1} \otimes B^- [/tex]
The creation operators and annihilation operators constructed in this way would not satisfy the correct anti-commutation relation, for example, [tex] \{ b_1^+ , b_2^+ \} \neq 0 [/tex]
I don't quite understand why this construction doesn't work?