so thezjare a basis ofM+J 0 , thezjofM−J 0. They have Poisson brackets
{zj,zk}=iδjkIn the Bargmann-Fock quantization, the state space is taken to be polyno-
mials in thezj, with annihilation and creation operators
aj=∂
∂zj, a†j=zjA.5 Special relativity
The “mostly plus” convention for the Minkowski inner product is used, so four-
vectorsx= (x 0 ,x 1 ,x 2 ,x 3 ) satisfy
(x,x) =||x||^2 =−x^20 +x^21 +x^22 +x^23The relativistic energy-momentum relation is then
p^2 =−E^2 +||p||^2 =−m^2A.6 Clifford algebras and spinors
The Clifford algebra associated to an inner product (·,·) satisfies the relation
uv+vu= 2(u,v)With the choice of signature for the Minkowski inner product above, the Clifford
algebra is isomorphic to a real matrix algebra
Cliff(3,1) =M(4,R)Under this isomorphism, basis element of Minkowski space correspond toγ
matrices, which satisfy
γ 02 =− 1 , γ^21 =γ 22 =γ^23 = 1Explicit choices of these matrices are described in sections 41.2, 47.2 and 47.3.
The Dirac equation is taken to be
(∂/−m)ψ(x) = 0see section 47.1.