Quantum Mechanics for Mathematicians

space does not completely resolve the problem. Using theα+(p),α−(p) allows
for an explicitly positive-definite inner product, which is just two copies of the
Klein-Gordon one for scalars, see equation 43.18. In the next section we will
quantize the theory using these. This inner product is not however manifestly
Lorentz invariant (due to the dependence on the choice of polarization vectors
u±(p)).

47.2.2 Quantization of the Majorana field

Quantization of the dual pseudo-classical phase spaceM, using the fermionic
Bargmann-Fock method, the relativistic complex structure, and the functions
α±(p) from equation 47.14 is given by annihilation and creation operators
a±(p),a†±(p) that anticommute, except for the relations

[a+(p),a†+(p′)]+=δ^3 (p−p′), [a−(p),a†−(p′)]+=δ^3 (p−p′)

The field operator is then constructed using these, giving

Definition(Majorana field operator).The Majorana field operator is given by

Ψ(̂x,t) =^1
(2π)^3 /^2

∫

R^3

∑

s=±

(as(p)us(p)e−iωpteip·x+a†s(p)us(p)eiωpte−ip·x)

d^3 p

√

2 ωp

If one uses commutation instead of anticommutation relations, the Hamilto-
nian operator will have eigenstates with arbitrarily negative energy, and there
will be problems with causality due to observable operators at space-like sep-
arated points not commuting. These two problems are resolved by the use of
anticommutation instead of commutation relations. The multi-particle state
space for the theory has occupation numbers 0 or 1 for each value ofpand for
each value ofs=±. Like the case of the real scalar field, the particles described
by these states are their own antiparticles. Unlike the case of the real scalar
field, each particle state has aC^2 degree of freedom corresponding to its spin^12
nature.
One can show that the Hamiltonian and momentum operators are given by

Ĥ=

∫

R^3

ωp(a†+(p)a+(p) +a†−(p)a−(p))d^3 p

and

P̂=

∫

R^3

p(a†+(p)a+(p) +a†−(p)a−(p))d^3 p

The angular momentum and boost operators are much more complicated to
describe, again due to the dependence of theα±(p) on a choice of polarization
vectorsu±(p).
Note that, as in the case of the real scalar field, the theory of a single
Majorana field has no internal symmetry group acting, so no way to introduce
a charge operator and couple the theory to electromagnetic fields.