whereH+ 1 will be positive energy solutions withJC = +i andH− 1 will be
positive energy solutions withJC=−i. Taking as beforeα 1 (p),α 2 (p) for the
momentum space initial data for elements ofH 1 , we will define
α(p) =
1
√
2
(α 1 (p)−iα 2 (p))∈H 1 +, β(p) =
1
√
2
(α 1 (p) +iα 2 (p))∈H− 1
The negative energy solution space can be decomposed as
M−Jr=H 1 =H
+
1 ⊕H
−
1
and
α(p) =
1
√
2
(α 1 (p) +iα 2 (p))∈H 1 +, β(p) =
1
√
2
(α 1 (p)−iα 2 (p))∈H− 1
We will writeA(p),B(p) for the solutionsα,β with initial data delta-
functions atp,A(p),B(p) for their conjugates, and quantization will take
A(p)→a†(p) =
1
√
2
(a† 1 (p)−ia† 2 (p))
B(p)→b†(p) =
1
√
2
(a† 1 (p) +ia† 2 (p))
A(p)→a(p) =
1
√
2
(a 1 (p) +ia 2 (p))
B(p)→b(p) =
1
√
2
(a 1 (p)−ia 2 (p))
with the non-zero commutation relations between these operators given by
[a(p),a†(p′)] =δ(p−p′), [b(p),b†(p′)] =δ(p−p′)
The state space of this theory is a tensor product of two copies of the state
space of a real scalar field. The operatorsa†(p),a(p) act on the state space by
creating or annihilating a positively charged particle of momentump, whereas
theb†(p),b(p) create or annihilate antiparticles of negative charge. The vacuum
state will satisfy
a(p)| 0 〉=b(p)| 0 〉= 0
The Hamiltonian operator for this theory will be
Ĥ=
∫
R^3
ωp(a†(p)a(p) +b†(p)b(p))d^3 p
and the charge operator is
Q̂=
∫
R^3
(a†(p)a(p)−b†(p)b(p))d^3 p