- One can take elements ofH 1 to beC^4 -valued functionsα= (α 0 (p),α(p))
onR^3 with Lorentz invariant indefinite inner product
〈α,α′〉=
∫
R^3
(−α 0 (p)α′ 0 (p) +α(p)·α′(p))d^3 p (46.22)
Here eachαμ(p) is defined as in equation 43.7 for the single component
field case. The subspace satisfyingχ(A)+= 0 will be the subspaceH′ 1 ⊂
H 1 ofαμsatisfying
−p 0 α 0 (p) +
∑^3
j=1
pjαj(p) = 0
This subspace will in turn have a subspaceH′′ 1 ⊂H′ 1 corresponding toAμ
that are gauge transforms of 0, i.e., with Fourier coefficients satisfying
α 0 (p) =p 0 f(p), αj(p) =pjf(p)
for some functionf(p). Both of these subspaces carry an action of the
Lorentz group, and so does the quotient space
H′ 1 /H′′ 1
One can show that the indefinite inner product 46.22 is non-negative on
H′ 1 and null onH′′ 1 , so positive definite on the quotient space. Note that
this is an example of a symplectic reduction, although in the context of
an action of an infinite dimensional complex group (the positive energy
gauge transformations satisfying the massless wave equation). One can
construct the quantum theory by applying the Bargmann-Fock method to
this quotient space.
- One can instead implement the gauge condition after quantization, first
quantizing the four components ofAμas massless fields, getting a state
spaceH(which will not have a positive definite inner product), then defin-
ingH′⊂Hto be the subspace of states satisfying
(
−
∂ 0
∂t
+∇·Â
)+
|ψ〉= 0
where the positive energy part of the operator is taken. The state space
H′in turn has a subspaceH′′of states of zero norm, and one can define
Hphys=H′/H′′
Hphyswill have a positive definite Hermitian inner product, and carry a
unitary action of the Poincar ́e group. It can be shown to be isomorphic
to the physical state space of transverse photons constructed using the
Coulomb gauge.
