Quantum Mechanics for Mathematicians

(see equation 43.7) are related by this non-standard conjugation (only on real
solutions isα(p) the complex conjugate ofα(p)).
Also worth keeping in mind is that, while in terms of theα(p) the com-
plex structureJris just multiplication byi, for the basis of field variables
(φ(x),π(x)),Jr is not multiplication byi, but something much more com-
plicated. From equation 43.11 one sees that multiplication byion theα(p)
corresponds to

(φ(x),π(x))→

(

1

ωp

π(x),−ωpφ(x)

)

on theφ(x),π(x) coordinates. This transformation is compatible with the sym-
plectic structure (preserves the Poisson bracket relations 43.13). As a transfor-
mation on position space solutions the momentum is the differential operator
p=−i∇, soJrneeds to be thought of as a non-local operation that can be
written as

(φ(x),π(x))→

(

1

√

−∇^2 +m^2

π(x),−

√

−∇^2 +m^2 φ(x)

)

(43.15)

Poisson brackets of the continuous basis elementsA(p),A(p) are given by

{A(p),A(p′)}={A(p),A(p′)}= 0, {A(p),A(p′)}=iδ^3 (p−p′) (43.16)

Recall from section 26.2 that given a symplectic structure Ω and positive,
compatible complex structureJonM, we can define a Hermitian inner product
onM+J by
〈u,v〉=iΩ(u,v)

foru,v∈M+J. For the case ofMreal solutions to the Klein-Gordon equation
we have on basis elementsA(p) ofM+Jr

〈A(p),A(p′)〉=iΩ(A(p),A(p′)) =i{A(p),A(p′)}=δ^3 (p−p′) (43.17)

As a Hermitian inner product on elementsα(p)∈M+Jr, which is our single-
particle state spaceH 1 , equation 43.17 implies that

〈α 1 (p),α 2 (p)〉=

∫

R^3

α 1 (p)α 2 (p)d^3 p (43.18)

This inner product onH 1 will be positive definite and Lorentz invariant.
Note the difference with the non-relativistic case, where one has the same
E(3) invariant inner product on the position space fields or their momentum
space Fourier transforms. In the relativistic case the Hermitian inner product
is only the simple one (43.18) on the momentum space initial data for solutions
α(p) but another quite complicated one on the position space dataφ(x),π(x)
(due to the complicated expression 43.15 forJrthere). Unlike the Hermitian
inner product andJr, the symplectic form is simple in both position and mo-
mentum space versions, see the Poisson bracket relations 43.13 and 43.16.