By Peskin and Schroeder
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Additional info for An Introduction to Quantum Field Theory
Generalize these results to the case of n identical complex scalar elds. 3 Evaluate the function Z 3 h0j (x) (y) j0i = D(x ; y) = (2d p)3 2E1 p e;ip (x;y) for (x ; y) spacelike so that (x ; y)2 = ;r2 , explicitly in terms of Bessel functions. Chapter 3 The Dirac Field Having exhaustively treated the simplest relativistic eld equation, we now move on to the second simplest, the Dirac equation. * In this chapter our viewpoint will be quite di erent. First we will rederive the Dirac equation as a classical relativistic eld equation, with special emphasis on its relativistic invariance.
In fact, we will soon nd that much worse problems arise when we impose commutation relations on the Dirac eld. But it is instructive to see how far we can get, in order to better understand the relation between spin and statistics. So let us press on just remember that the next few pages will eventually turn out to be a blind alley. Our rst task is to nd a representation of the commutation relations in terms of creation and annihilation operators that diagonalizes H . 84), it will clearly be helpful to expand (x) in a basis of eigenfunctions of hD .
111) does not contribute, and we can easily write the spin term in terms of ladder operators. 87) for (x): Jz = Z d3 x Z d3 p d3p0 p X (2 )6 1 2Ep 2Ep 0 ;ip0 x eip x e 0 3 arp0y ur0 y (p0 ) + brp0 0 vr0 y (;p0 ) 2 arpur (p) + brpy vr (;p) : r r0 We would like to apply this operator to the one-particle zero-momentum state as0y j0i. This is most easily done using a trick: Since Jz must annihilate the vacuum, Jz as0y j0i = Jz 0 as0y ] j0i. The only nonzero0term in this latter quantity has the structure arpy arp as0y ] = (2 )3 (3) (p)ar0y r s the other three terms in the commutator either vanish or annihilate the vacuum.