Quantum Mechanics for Mathematicians

that such irreducible representations are characterized in part by the scalar
value that the Casimir operatorP^2 takes on the representation. When this is
negative, we have the operator equation

P^2 =−P 02 +P 12 +P 22 +P 32 =−m^2 (43.1)

and we would like to find a state space with momentum operators satisfying
this relation. We can use wavefunctionsφ(x) on Minkowski space to get such a
state space.
Just as in the non-relativistic case, where we could represent momentum
operators as either multiplication operators on functions of the momenta, or
differentiation operators on functions of the positions, here we can do the same,
with functions now depending on the four space-time coordinates. TakingP 0 =
H=i∂t∂ as well as the conventional momentum operatorsPj=−i∂x∂j, equation

43.1 becomes:

Definition(Klein-Gordon equation).The Klein-Gordon equation is the second-
order partial differential equation

(

−

∂^2

∂t^2

+

∂^2

∂x^21

+

∂^2

∂x^22

+

∂^2

∂x^23

)

φ=m^2 φ

or (

−

∂^2

∂t^2

+ ∆−m^2

)

φ= 0 (43.2)

for functionsφ(x)on Minkowski space (these functions may be real or complex-
valued).

This equation is the simplest Lorentz invariant (the Lorentz group acting on
functions takes solutions to solutions) wave equation, and historically was the
one first tried by Schr ̈odinger. He soon realized it could not account for known
facts about atomic spectra and instead used the non-relativistic equation that
bears his name. In this chapter we will consider the quantization of the space of
real-valued solutions of this equation, with the case of complex-valued solutions
appearing later in section 44.1.2.
Taking Fourier transforms by

φ ̃(p) =^1

(2π)^2

∫

R^4

e−i(−p^0 x^0 +p·x)φ(x)d^4 x

the momentum operators become multiplication operators, and the Klein-Gordon
equation is now

(p^20 −p^21 −p^22 −p^23 −m^2 )φ ̃= (p^20 −ω^2 p)φ ̃= 0

where

ωp=

√

p^21 +p^22 +p^23 +m^2