and one can formally compute the commutators
[Ψ(̂x),Ψ(̂x′)] = [Ψ̂†(x),Ψ̂†(x′)] = 0[Ψ(̂x),Ψ̂†(x′)] =1
2 π∫∞
−∞∫∞
−∞eipxe−ip′x′
[a(p),a†(p′)]dpdp′=
1
2 π∫∞
−∞∫∞
−∞eipxe−ip′x′
δ(p−p′)dpdp′=
1
2 π∫∞
−∞eip(x−x′)
dp=δ(x−x′)getting results consistent with the interpretation of the field operator and its
adjoint as operators that annihilate and create particles at a pointx.
This sort of definition relies upon making sense of the operatorsa(p) and
a†(p) as distributional operators, using the action of elements ofH 1 andH 1 on
Fock space given by equations 36.12 and 36.13. We could instead more directly
proceed exactly as in section 36.3, but with solutions characterized by initial
data given by a functionψ(x) rather than its Fourier transformα(p). One gets
all the same objects and formulas, related by Fourier transform. Our notation
for these transformed objects will be
α(p)→ψ(x), α(p)→ψ(x)A(α)→Ψ(ψ), A(α)→Ψ(ψ)
A(p)→Ψ(x), A(p)→Ψ(x)
Ψ(ψ) will be the solution inH 1 with initial dataψ(x) andΨ(ψ) the conjugate
solution inH 1. Ψ(x) can be interpreted as the distributional solution equal to
δ(x−x′) att= 0 and we can write
Ψ(ψ) =∫
ψ(x)Ψ(x)dx, Ψ(ψ) =∫
ψ(x)Ψ(x)dxThese satisfy the Poisson bracket relations
{Ψ(ψ 1 ),Ψ(ψ 2 )}={Ψ(ψ 1 ),Ψ(ψ 2 )}= 0, {Ψ(ψ 1 ),Ψ(ψ 2 )}=i〈ψ 1 ,ψ 2 〉 (37.2){Ψ(x),Ψ(x′)}={Ψ(x),Ψ(x′}= 0, {Ψ(x),Ψ(x′)}=iδ(x−x′) (37.3)
Quantization then takes (note the perhaps confusing choice of notational
convention due to following the physicist’s convention thatΨ is the annihilation̂
operator)
Ψ(ψ)→−iΨ̂†(ψ), Ψ(ψ)→−iΨ(̂ψ)The quantum field operators can be defined in terms of tensor products using
the same generalization of the finite dimensional case of section 26.4 that we
used to definea(α) anda†(α). Here
Ψ̂†(ψ)P+(Ψ(ψ 1 )⊗···⊗Ψ(ψn)) =√n+ 1P+(Ψ(ψ)⊗Ψ(ψ 1 )⊗···⊗Ψ(ψn)) (37.4)