Quantum Mechanics for Mathematicians

Quantization should give unitary operatorsU(a,Λ), which act on field operators
by
Φ̂→U(a,Λ)Φ(̂x)U(a,Λ)−^1 =Φ(Λ̂ x+a) (44.6)

TheU(a,Λ) will provide a unitary representation of the Poincar ́e group on the
quantum field theory state space, acting by intertwining operators of the sort
discussed in the finite dimensional context in chapter 20. We would like to con-
struct these operators by the usual method: using the moment map to get a
quadratic polynomial on phase space, quantizing to get Lie algebra representa-
tion operators, and then exponentiating to get theU(a,Λ).
This will require that the symplectic structure on the phase spaceH 1 be
Poincar ́e invariant. The Poisson bracket relations on the position space fields

{Φ(x),Π(x′)}=δ^3 (x−x′), {Φ(x),Φ(x′)}={Π(x),Π(x′)}= 0

are easily seen to be invariant under the action of the Euclidean group of spatial
translations and rotations by

Φ(x)→Φ(Rx+a), Π(x)→Π(Rx+a)

(since the delta-function is). Things are not so simple for the rest of the Poincar ́e
group, since the definition of the Φ(x),Π(x) is based on a choice of the distin-
guishedt= 0 hyperplane. In addition, the complicated form of the relativistic
complex structureJrin these coordinates (see equation 43.15) makes it difficult
to see if this is invariant under Poincar ́e transformations.
Taking Fourier transforms, recall that solutions to the Klein-Gordon equa-
tion can be written as (see 43.3)

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

so these are given by functions (actually distributions)f(p) on the positive and
negative energy hyperboloids. The complex structureJris +ion functions on
the negative energy hyperboloid,−ion functions on the positive energy hyper-
boloid. The action of the Poincar ́e group preservesJr, since it acts separately on
the negative and positive energy hyperboloids. It also preserves the Hermitian
inner product (see equations 43.17 and 43.18), and thus gives a unitary action
onM+Jr=H 1.
Just as for the finite dimensional case in chapter 25 and the non-relativistic
quantum field theory case in section 38.3, we can find for each elementLof the
Lie algebra of the group acting (here the Poincar ́e groupP) a quadratic expres-
sion in theA(p),A(p) (this is the moment mapμL). Quantization then gives
a corresponding normal ordered quadratic operator in terms of the operators
a†(p),a(p).