QuantumPhysics.dvi

Evaluating the time derivative, we find,

i∂ 0 ψ=

∫

d^3 k

(2π)^3

1

√

2 k^0

∑

s=1, 2

{

k^0 us(k)bs(k)e−ik·x−k^0 vs(k)d†s(k)e+ik·x

}

(23.28)

and we find,

H=

∫ d (^3) k
(2π)^3
k^0
∑
s=1, 2
{
b†s(k)bs(k)−ds(k)d†s(k)
}
(23.29)

23.4 Quantization of fermion oscillators

We are now ready to proceed with quantization. Recall that wehad declaredb†andd†to be
the creation operators for particles and anti-particles, having charges +1 and−1 respectively, and
both with positive energy. The oscillatorsb,d,b†,d†have to be quantized as fermions. As we have
seen earlier, this is done in terms of anti-commutation relations. It will turn out that the correct
relations are,

{br(k),bs(k′)}={dr(k),ds(k′)} = 0

{br(k),ds(k′)}={br(k),d†s(k′)} = 0

{br(k),b†s(k′)}={dr(k),d†s(k′)} = (2π)^3 δr,sδ(3)(k−k′) (23.30)

as well as the adjoint relations of the first two lines. Normalordering now the expressions for
electric chargeQ, HamiltonianH, as well as the momentumP~, we get,

Q =

∫ d (^3) k
(2π)^3
∑
s=1, 2
{
b†s(k)bs(k)−d†s(k)ds(k)
}

H =

∫ d (^3) k
(2π)^3
k^0
∑
s=1, 2
{
b†s(k)bs(k) +d†s(k)ds(k)
}

P~ =

∫ d (^3) k
(2π)^3
~k
∑
s=1, 2
{
b†s(k)bs(k) +d†s(k)ds(k)
}
(23.31)

23.5 Canonical anti-commutation relations for the Dirac field

From the anti-commutation relations of the oscillatorsb,d,b†,d†, we deduce those of the Dirac fields
themselves. It is immediate that

{ψα(x),ψβ(y)}={ψ†α(x),ψβ†(y)}= 0 (23.32)

whereα,βrun over the spinor components of the Dirac fields. It remainsto compute

{ψα†(x),ψβ(y)} =

∫

d^3 k

(2π)^3

∫

d^3 k′

(2π)^3

∑

s,s′

1

2

√

k^0 k′^0

{

u†sα(k)b†s(k)eik·x+vsα†(k)ds(k)e−ik·x,

us′β(k′)bs′(k′)e−ik

′·y

+vs′β(k′)d†s′(k′)eik

′·y

}

(23.33)