Quantum Mechanics for Mathematicians

with the left-hand side an explicitly Lorentz invariant measure (sincep^20 −ωp^2 =
−p^2 −m^2 is Lorentz invariant).
An arbitrary solution of the Klein-Gordon equation (see equation 43.3) can
thus be written

φ(t,x) =

1

(2π)^3 /^2

∫

R^4

1

2 ωp

(δ(p 0 −ωp) +δ(p 0 +ωp))f(p)ei(−p^0 t+p·x)dp 0 d^3 p

=

1

(2π)^3 /^2

∫

R^3

(f(ωp,p)e−iωpteip·x+f(−ωp,−p)eiωpte−ip·x)

d^3 p

2 ωp

(43.5)

which will be real when

f(−ωp,−p) =f(ωp,p) (43.6)

Instead of the functionsf, we will usually instead use

α(p) =

f(ωp,p)

√

2 ωp

, α(p) =

f(−ωp,−p)

√

2 ωp

(43.7)

Other choices of normalization of these complex functions are often used, the
motivation for this one is that we will see that it will give simple Poisson bracket
relations. With this choice, the Klein-Gordon solutions are

φ(t,x) =

1

(2π)^3 /^2

∫

R^3

(α(p)e−iωpteip·x+α(p)eiωpte−ip·x)

d^3 p

√

2 ωp

(43.8)

Such solutions can be specified in terms of their initial data by either the
pair of real-valued functionsφ(x),π(x), or their Fourier transforms. We will
however find it much more convenient to characterize the momentum space
initial data by the single complex-valued functionα(p). The equations relating
these choices of initial data are

φ(x) =

1

(2π)^3 /^2

∫

R^3

(α(p)eip·x+α(p)e−ip·x)

d^3 p

√

2 ωp

(43.9)

π(x) =

∂

∂t

φ(x,t)|t=0=

1

(2π)^3 /^2

∫

R^3

(−iωp)(α(p)eip·x−α(p)e−ip·x)

d^3 p
√
2 ωp
(43.10)
and

α(p) =

1

(2π)^3 /^2

∫

R^3

1

√

2

(

√

ωpφ(x) +i

1

√

ωp

π(x)

)

e−ip·xd^3 x (43.11)

To construct a relativistic quantum field theory, we would like to proceed
as in the non-relativistic case of chapters 36 and 37, but taking the dual phase
spaceMto be the space of solutions of the Klein-Gordon equation rather than
of the Schr ̈odinger equation. As in the non-relativistic case, we have various
ways of specifying an element ofM: