By Peskin and Schroeder

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95) would indeed cancel if A = ;B , but this is impossible since A and B must both be positive. The solution, however, is now at hand. By setting A = B = 1, it is easy to obtain (outside the light-cone) h0j a (x) b (y) j0i = ; h0j b (y) a (x) j0i : That is, the spinor elds anti commute at spacelike separation. This is enough to preserve causality, since all reasonable observables (such as energy, charge, and particle number) are built out of an even number of spinor elds for any such observables O1 and O2 , we still have O1 (x) O2 (y)] = 0 for (x ; y)2 < 0.

F. Streater and A. S. , 1964). 5 Quantization of the Dirac Field 59 Thus both aspy and bspy create particles with energy +Ep and momentum p. We will refer to the particles created by aspy as fermions and to those created by bspy as antifermions. The one-particle states p jp si 2Ep aspy j0i (3:106) are de ned so that their inner product hp rjq si = 2Ep (2 )3 (3) (p ; q) rs (3:107) is Lorentz invariant. This implies that the operator U ( ) that implements Lorentz transformations on the states of the Hilbert space is unitary, even though for boosts, 21 is not unitary.

17). This computation goes through in any dimensionality, with Lorentz or Euclidean metric. 2 The Dirac Equation 41 space, and in fact we can simply write j i j (Pauli sigma matrices) i j = ;2 ij : so that The factor of i in the rst line and the minus sign in the second line are purely conventional. The matrices representing the Lorentz algebra are then (3:24) S ij = 21 ijk k which we recognize as the two-dimensional representation of the rotation group. Now let us nd Dirac matrices for four-dimensional Minkowski space.