Squaring a Pauli matrix results in the identity matrix.
These bits of documentation (weakly, to a Mathematica-newbie like me) imply that some algebraic identities that link the Pauli matrices together are built into Mathematica:
- (other basic Pauli Matrix documentation on the Wolfram pages)
I enter in Mathematica:
PauliMatrix FullSimplify[PauliMatrix . PauliMatrix . PauliMatrix]
Is there a way of getting the second expression simplified to just
PauliMatrix . PauliMatrix is the identity matrix?
Mathematica simply prints out the resulting matrices for comparison rather than sticking to the algebra.
I see that someone else is trying to manipulate the Pauli matrices algebraically (for a similar purpose): http://homepage.cem.itesm.mx/lgomez/quantum/v7pauli.pdf (page 2. Setup Pauli Algebra)
... so I imagine that Mathematica doesn't have the behaviour built in?