QuantumPhysics.dvi

Using the anti-commutation relations andbanddoscillators, we obtain contributions of either 0
orδs,s′(2π)^3 δ(~k−~k′). The expression thus simplifies to

{ψ†α(x),ψβ(y)}=

∫

d^3 k

(2π)^3

1

2 k^0

∑

s

{

u†sα(k)usβ(k)eik·(x−y)+vsα†(k)vsβ(k)e+ik·(x−y)

}

(23.34)

It remains to obtain the final spinor relation we need,
∑
s

u†sαusβ = (γμkμ+m)αβ

∑

s

vsα†vsβ = (γμkμ−m)αβ (23.35)

Putting all together, we have

{ψ†α(x),ψβ(y)}=

∫

d^3 k

(2π)^3

1

2 k^0

{

(γμkμ+m)γ^0 eik·(x−y)+ (γμkμ−m)γ^0 e+ik·(x−y)

}

αβ

(23.36)

Now, this is the expression for free fermions. Setting the time components equal to one another,
we have a relation which is independent of dynamics, and giving the canonical anti-commutation
relations of the fields,

{ψ†α(x),ψβ(y)}δ(x^0 −y^0 ) =δαβδ(4)(x−y) (23.37)

23.6 The fermion propagator

The free fermion propagator is the “inverse of the Dirac operator”. A useful way to look at this
quantity is from how it arises from the Dirac field. The two pieces of information we need are
summarized as follows,

(γμ∂μ−m)ψ(x) = 0

{ψα†(x),ψβ(y)}δ(x^0 −y^0 ) = δαβδ(x−y) (23.38)

where the derivative operator acts as∂μ=∂/∂xμ. So, the Dirac field may be viewed as a “homoge-
neous” solution to the Dirac equation. The Dirac propagatorshould thus satisfy the inhomogeneous
equation,

(γμ∂μ−m)S(x,y) =I 4 δ(4)(x−y) (23.39)

whereI 4 is the identity matrix in the 4-component Dirac spinor space. On the one hand, this
equation may be solved (formally) by Fourier transform, andwe find,

∫

d^4 k

(2π)^4

eik·(x−y)

iγμkμ−m

(23.40)

This is formal because the denominator will have a vanishingeigenvalue for a whole family of values
ofk, namely those that obeykμkμ+m^2 = 0. This problem is familiar from the Lippmann-Schwinger