Quantum Mechanics for Mathematicians

Besides specifying the functionψ(x,t), elements of the single-particle spaceH 1
could be uniquely characterized in two other ways by a function onR^3 : either
the initial valueψ(x,0) or its Fourier transformψ ̃(p,0).
In the relativistic case, since the Klein-Gordon equation is second order in
time, solutionsφ(t,x) will be parametrized by initial data which, unlike the
non-relativistic case, now requires the specification att= 0 of not one, but two
functions:

φ(x) =φ(0,x), φ ̇(x) =

∂

∂t

φ(t,x)|t=0≡π(x)

the values of the field and its first time derivative.
In the relativistic case a continuous basis of solutions of the Klein-Gordon
equation will be given by the functions

e±iωpteip·x

and a general solution can be written

φ(t,x) =

1

(2π)^3 /^2

∫

R^4

δ(p^20 −ω^2 p)f(p)ei(−p^0 t+p·x)d^4 p (43.3)

forf(p) a complex function satisfyingf(p) =f(−p) (so thatφwill be real). The
solution will only depend on the valuesftakes on the hyperboloidsp 0 =±ωp.
The integral 43.3 is expressed in a four dimensional, Lorentz invariant man-
ner using the delta-function, but this is really an integral over the two-component
hyperboloid. This can be rewritten as a three dimensional integral overR^3 by
the following argument. For eachp, applying equation 11.9 to the case of the
function ofp 0 given by
g(p 0 ) =p^20 −ω^2 p

onR^4 , and using

d

dp 0

(p^20 −ω^2 p)|p 0 =±ωp= 2p (^0) |p 0 =±ωp=± 2 ωp
gives
δ(p^20 −ω^2 p) =

1

2 ωp

(δ(p 0 −ωp) +δ(p 0 +ωp))

We will often in the future use the above to provide a Lorentz invariant measure
on the hyperboloidsp^20 =ωp^2 , which we’ll write

d^3 p

2 ωp

For a functionφ(p) on these hyperboloids the integral over the hyperboloids
can be written in two equivalent ways

∫

R^4

δ(p^20 −ωp^2 )φ(p)d^4 p=

∫

R^3

(φ(ωp,p) +φ(−ωp,p))

d^3 p

2 ωp

(43.4)