Quantum Mechanics for Mathematicians

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