Quantum Mechanics for Mathematicians

theγ 0 γare symmetric, and the derivative is antisymmetric. Applying the finite
dimensional theorem 30.1 in this infinite dimensional context, one finds that the
pseudo-classical equation of motion is

∂

∂t

Ψ(x) ={Ψ(x),h}+=γ 0 (γ·∇−m)Ψ(x)

which is the Dirac equation in Hamiltonian form (see equations 47.4 and 47.5).
The antisymmetry of the operatorγ 0 (γ·∇−m) that generates time evolution
corresponds to the fact that time evolution gives for eachtan (infinite dimen-
sional) orthogonal group action on the space of solutionsV, preserving the inner
product 47.11.
Corresponding to the Poincar ́e group action 47.9 on solutions to the Dirac
equation, at least for translations byaand rotations Λ, one has a corresponding
action on the fields, written

Ψ(x)→u(a,Λ)Ψ(x) =S(Λ)−^1 Ψ(Λ·x+a)

The quadratic pseudo-classical moment map that generates the action of
spatial translations on solutions is the momentum

P=−

1

2

∫

R^3

ΨT(x)∇Ψ(x)d^3 x

since it satisfies (generalizing equation 30.1)

{P,Ψ(x)}+=∇Ψ(x)

For rotations, the moment map is the angular momentum

J=−

1

2

∫

R^3

ΨT(x)(x×∇−s)Ψ(x)d^3 x

which satisfies
{J,Ψ(x)}+= (x×∇−s)Ψ(x)

Here the componentssjofsare the matrices

sj=

1

2

jklγkγl

Our use here of the fixed-time fields Ψ(x) as continuous basis elements on
the phase spaceVcomes with two problematic features:

• One cannot easily implement Lorentz transformations that are boosts,
  since these change the fixed-time hypersurface used to define the Ψ(x).


• The relativistic complex structure onVneeded for a consistent quantiza-
  tion is defined by a splitting ofV ⊗Cinto positive and negative energy
  solutions, but this decomposition is only easily made in momentum space,
  not position space.