Quantum Mechanics for Mathematicians

∂

∂t

Π = [̂ Π̂,−iĤ] = (∆−m^2 )̂Φ

which have as solution the following equation for the time-dependent field op-
erator:

Φ(̂t,x) =^1

(2π)^3 /^2

∫

R^3

(a(p)e−iωpteip·x+a†(p)eiωpte−ip·x)

d^3 p

√

2 ωp

(43.23)

Unlike the non-relativistic case, where fields are non-self-adjoint operators, here
the field operator is self-adjoint (and thus an observable), and has both an an-
nihilation operator component and a creation operator component. This gives a
theory of positive energy quanta that can be interpreted as either particles mov-
ing forward in time or antiparticles of opposite momentum moving backwards
in time.

43.5 The scalar field propagator

As for any quantum field theory, a fundamental quantity to calculate is the
propagator, which for a free quantum field theory actually can be used to calcu-
late all amplitudes between multiparticle states. For the free relativistic scalar
field theory, we have

Definition(Propagator, Klein-Gordon theory).The propagator for the rela-
tivistic scalar field theory is the amplitude

U(t 2 ,x 2 ,t 1 ,x 1 ) =〈 0 |Φ(̂t 2 ,x 2 )Φ(̂t 1 ,x 1 )| 0 〉

By translation invariance, the propagator will only depend ont 2 −t 1 and
x 2 −x 1 , so we can just evaluate the case (t 1 ,x 1 ) = (0, 0 ),(t 2 ,x 2 ) = (t,x), using
the formula 43.23 for the time-dependent quantum field to get

U(t,x, 0 , 0 ) =

1

(2π)^3

∫

R^3 ×R^3

〈 0 |(a(p)e−iωpteip·x+a†(p)eiωpte−ip·x)

(a(p′) +a†(p′))| 0 〉

d^3 p

√

2 ωp

d^3 p′

√

2 ωp′

=

1

(2π)^3

∫

R^3 ×R^3

δ^3 (p−p′)e−iωpteip·x

d^3 p

√

2 ωp

d^3 p′

√

2 ωp′

=

1

(2π)^3

∫

R^3

e−iωpteip·x

d^3 p

2 ωp

=

1

(2π)^3

∫

R^4

θ(p 0 )δ(p^2 +m^2 )e−ip^0 teip·xd^3 pdp 0

The last line shows that this distribution is the Minkowski space Fourier trans-
form of the delta-function distribution on the positive energy hyperboloid

p^2 =−p^20 +|p|^2 =−m^2