Quantum Mechanics for Mathematicians

timet 1 , propagates for a timet 2 −t 1 , and is annihilated at positionx 2. Using
the solution for the time-dependent field operator given earlier we find

U(x 2 ,t 2 ,x 1 ,t 1 ) =

1

2 π

∫∫

R^2

〈 0 |eip^2 x^2 e−i

p^22

2 mta(p 2 )e−ip^1 x^1 ei

p 12

2 mt^1 a†(p 1 )| 0 〉dp 2 dp 1

=

1

2 π

∫∫

R^2

eip^2 x^2 e−i

p^22

2 mt^2 e−ip^1 x^1 ei

p 12

2 mt^1 δ(p 2 −p 1 )dp 2 dp 1

=

1

2 π

∫+∞

−∞

e−ip(x^1 −x^2 )e−i

p^2

2 m(t^2 −t^1 )dp

This is exactly the same calculation (see equation 12.5) already discussed in
detail in section 12.5. As described there, the result (equation 12.9) is

U(x 2 ,t 2 ,x 1 ,t 1 ) =U(t 2 −t 1 ,x 2 −x 1 ) =

(

m

i 2 π(t 2 −t 1 )

) (^12)
e
2 i(tm
2 −t 1 )(x^2 −x^1 )
2
which satisfies
lim
t→ 0 +
U(t,x 2 −x 1 ) =δ(x 2 −x 1 )
If we extend the definition ofU(t,x 2 −x 1 ) tot <0 by taking it to be zero there,
as in section we get the retarded propagatorU+(t,x 2 −x 1 ) and its Fourier
transformed version in frequency-momentum space of section 12.6 as well as the
relation to Green’s functions of section 12.7.

37.4 Interacting quantum fields

To describe an arbitrary number of particles moving in an external potential
V(x), the Hamiltonian can be taken to be

Ĥ=

∫∞

−∞

Ψ̂†(x)

(

−

1

2 m

d^2

dx^2

+V(x)

)

Ψ(̂x)dx

If a complete set of orthonormal solutionsψn(x) to the Schr ̈odinger equation
with potential can be found, they can be used to describe this quantum system
using similar techniques to those for the free particle, taking as basis forH 1 the
ψn(x) instead of plane waves of momentump. A creation-annihilation operator
pairan,a†nis associated to each eigenfunction, and quantum fields are defined
by
Ψ(̂x) =

∑

n

ψn(x)an, Ψ̂†(x) =

∑

n

ψn(x)a†n

For Hamiltonians quadratic in the quantum fields, quantum field theories
are relatively tractable objects. They are in some sense decoupled quantum os-
cillator systems, although with an infinite number of degrees of freedom. Higher
order terms in the Hamiltonian are what makes quantum field theory a difficult
and complicated subject, one that requires a year-long graduate level course to