- #1
flybyme
- 20
- 0
Hello,
While reading up some about the Pauli matrices (which the author indexes by [itex]\sigma_{00}, \sigma_{01}, \sigma_{10}, \sigma_{11}[/itex]) I stumbled across an expression for the multiplication rules:
[itex] \sigma_{ab} \sigma_{a'b'} = i^{\tilde {w} ( a, b ; a', b' ) } \sigma_{a \oplus a', b \oplus b'}[/itex]
where [itex]\tilde {w} (a, b ; a', b') \in \mathbb{Z}_4[/itex].
Later an explicit formula for [itex]\tilde {w}[/itex] is given, and it is equal to [itex]ab + a'b' - (a \oplus a')(b \oplus b') + 2a' b \text{ mod } 4[/itex], which apparently is the same thing as [itex]a^2 b^2 + (a')^2 (b')^2 - (a + a')^2 (b + b')^2 + 2a'b[/itex].
Anyway, what I got to understand first of all, is what it means by having the two last parameters for [itex]\tilde {w}[/itex] separated by [itex];[/itex] instead of an ordinary [itex],[/itex].
That is: why [itex]\tilde{w}(a, b ; a', b')[/itex] instead of [itex]\tidle{w}(a, b, a', b')[/itex]?
(Excuse me for the non-existant TeX markup btw, I couldn't seem to find any info about it in the post form.)
Edit: Updated the post with marked up equations, thanks for the pointer.
While reading up some about the Pauli matrices (which the author indexes by [itex]\sigma_{00}, \sigma_{01}, \sigma_{10}, \sigma_{11}[/itex]) I stumbled across an expression for the multiplication rules:
[itex] \sigma_{ab} \sigma_{a'b'} = i^{\tilde {w} ( a, b ; a', b' ) } \sigma_{a \oplus a', b \oplus b'}[/itex]
where [itex]\tilde {w} (a, b ; a', b') \in \mathbb{Z}_4[/itex].
Later an explicit formula for [itex]\tilde {w}[/itex] is given, and it is equal to [itex]ab + a'b' - (a \oplus a')(b \oplus b') + 2a' b \text{ mod } 4[/itex], which apparently is the same thing as [itex]a^2 b^2 + (a')^2 (b')^2 - (a + a')^2 (b + b')^2 + 2a'b[/itex].
Anyway, what I got to understand first of all, is what it means by having the two last parameters for [itex]\tilde {w}[/itex] separated by [itex];[/itex] instead of an ordinary [itex],[/itex].
That is: why [itex]\tilde{w}(a, b ; a', b')[/itex] instead of [itex]\tidle{w}(a, b, a', b')[/itex]?
(Excuse me for the non-existant TeX markup btw, I couldn't seem to find any info about it in the post form.)
Edit: Updated the post with marked up equations, thanks for the pointer.
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