Let me say the same thing very pedantically in order to alleviate some confusion. Let MC2 be the vector space of 2 by 2 complex matrices. Consider the smallest subspace C of MC2 which includes each {σ_{j}} for j = 1, 2, 3, and is closed under multiplication, addition and scalar multiplication by reals. C is isomorphic φ to a real rank 3 Clifford algebra where φ(σ_{j}) = γ_{j} for j = 0, 1, 2, 3.
P = {Σ_{j} a_{j}σ_{j} | for j = 0, 1, 2, 3; a_{j} is real} where σ_{0} = I.
There is an associative algebra P generated by {I, σ_{1}, σ_{2}, σ_{3}} which is strategic to quantum mechanics. P is the even part of C. C includes reflections; P does not. Note that V (the span of {iσ_{j}}) is not in P; only Hermitian matrices belong to P. You might get away with saying that iσ_{1} represents a reflection in the x direction but I doubt that this is warranted.
To be a Clifford algebra it suffices that γ_{j}γ_{j} = −1 and γ_{j}γ_{k} + γ_{k}γ_{j} = 0 when j≠k. These are all satisfied. The corresponding Clifford group provides the rigid rotations of R^{3}.
Perhaps an isomorphism table will help;
Pauli | Clifford |
0I | 0 |
I | 1 |
iσ_{j} | γ_{j} |
iI | γ_{1}γ_{2}γ_{3}(=k) |
σ_{j} | −kγ_{j} |
σ_{1} | γ_{3}γ_{2} |
σ_{2} | γ_{1}γ_{3} |
σ_{3} | γ_{2}γ_{1} |
This is a twisted isomorphism! Perhaps there is a better one. The two red entries are not even in the space P.
I subsequently found section II of this. This paper uses the rule γ_{i}γ_{j} + γ_{j}γ_{i} = 4δ_{ij} for the algebra Cl_{3,0}. This is contrary to all the other papers I have seen where γ_{1}γ_{1} = −1. His rule works for what I would call Cl_{0,3} except for a factor of two.
I want to do the same thing with the Dirac matrices but I can’t improve on this page which shows that the ‘gamma matrices’ already form a Clifford algebra for a vector space with quadratic form with signature (+ − − −). Even the convention of naming the basis elements γ carries over. Here are further gropings for Dirac.
P.S.
The original Clifford algebras require γ_{j}γ_{j} = −1 but Minkowski space has a non-definite signature for the metric and describing boosts for that space requires a ‘Clifford algebra’ where γ_{j}γ_{j} = ±1, depending on j. The definition has thus been stretched so that γ_{j}γ_{j} = 1 yields a new sort of Clifford algebra. I don’t know whether Clifford considered it. That algebra also yields the orthogonal rotations of R^{3}.