Quantum Mechanics for Mathematicians

As in the non-relativistic case (see section 12.5), this is a distribution that
can be defined as a boundary value of an analytic function of complex time.
The integral can be evaluated in terms of Bessel functions, and its properties
are discussed in all standard quantum field theory textbooks. These include:

• Forx= 0, the amplitude is oscillatory in time, a superposition of terms
  with positive frequency.


• Fort= 0, the amplitude falls off exponentially ase−m|x|.

The resolution of the potential causality problem caused by the non-zero
amplitude at space-like separations between (t 1 ,x 1 ) and (t 2 ,x 2 ) (e.g., fort 1 =
t 2 ) is that the condition really needed on observable operatorsOlocalized at
points in space time is that

[O(t 2 ,x 2 ),O(t 1 ,x 1 )] = 0

for (t 1 ,x 1 ) and (t 2 ,x 2 ) space-like separated (this condition is known as “micro-
causality”). This will ensure that measurement of the observableOat a point
will not affect its measurement at a space-like separated point, avoiding potential
conflicts with causality. For the field operator Φ(t,x) one can calculate the
commutator by a similar calculation to the one above forU(t,x, 0 , 0 ), with
result

[Φ(̂t,x),Φ(0̂ , 0 )] =

1

(2π)^3

∫

R^4

δ(p^2 +m^2 )(θ(p 0 )eip·x−θ(−p 0 )e−ip·x)d^3 pdp 0

=U(t,x, 0 , 0 )−U(−t,−x, 0 , 0 )

This will be zero for space-like (t,x), something one can see by noting that the
result is Lorentz invariant, is equal to 0 att= 0 by the canonical commutation
relation
[Φ(̂x 1 ),Φ(̂x 2 )] = 0

and any two space-like vectors are related by a Lorentz transformation. Note
that the vanishing of this commutator for space-like separations is achieved by
cancellation of propagation amplitudes for a particle and antiparticle, showing
that the relativistic choice of complex structure for quantization is needed to
ensure causality.
One can also study the propagator using Green’s function methods as in the
single-particle case of section 12.7, now with

D=−

∂^2

∂t^2

+∇^2 −m^2

D̂(p 0 ,p) =p^20 −(|p|^2 +m^2 )

and

Ĝ(p 0 ,p) =^1

p^20 −(|p|^2 +m^2 )

=

1

(p 0 −ωp)(p 0 +ωp)