Canonical Commutation Relation Explained?

[ice109]

what is it? i need to know everything about it. i know it encompasses a lot of different stuff but yea if someone could point me to a book or webpage that explains it thoroughly.

additionally what does this equal

$$\sigma_{\mu}\sigma_{\alpha}\sigma_{\alpha}\sigma_{\mu}$$

those are pauli matrices btw. alpha is just an arbitrary index to differentiate it from the $\mu$ index

 

[ice109]

anyone?

 

[olgranpappy]

...
additionally what does this equal

$$\sigma_{\mu}\sigma_{\alpha}\sigma_{\alpha}\sigma_{\mu}$$

some four by four matrix that depends on all the indices. The expression doesn't get any simpler than that... are you sure you don't maybe want the value of the trace? that is easily simplified.

 

[olgranpappy]

what is it? i need to know everything about it. i know it encompasses a lot of different stuff but yea if someone could point me to a book or webpage that explains it thoroughly.

...probably any QM book. Try Messiah's book.

 

[ice109]

some four by four matrix that depends on all the indices. The expression doesn't get any simpler than that... are you sure you don't maybe want the value of the trace? that is easily simplified.

what's the identity?

 

[olgranpappy]

what's the identity?

You mean, you want to know the RHS of the equation
$$Tr(\sigma_\alpha\sigma_\beta\sigma_\gamma\sigma_\delta)=?$$

You can figure it out by commuting one of the sigma matrices on the far left all the way to the right (using the *anti*commutation relations for sigma matrices) and then using the cyclic property of the trace to get it back. This gives you an equation for the trace of four sigma matrices in terms of the trace of two sigma matrices. The trace of two sigma matrices can be then figured out in the same way. E.g.
$$Tr(\sigma_i \sigma_j)=Tr(-\sigma_i\sigma_j+2\delta_{ij})=-Tr(\sigma_i\sigma_j)+4$$