Quantum Mechanics for Mathematicians

Note that the vacuum state| 0 〉is an eigenvector forQ̂andĤwith both eigen-
values 0: it has zero energy and zero charge. Statesa† 1 (p)| 0 〉anda† 2 (p)| 0 〉
are eigenvectors ofĤwith eigenvalue and thus energyωp, but these are not

eigenvectors ofQ̂, so do not have a well-defined charge.
All of this can be generalized to the case ofm >2 real scalar fields, with a
larger groupSO(m) now acting instead of the groupSO(2). The Lie algebra is
now multi-dimensional, with a basis the elementary antisymmetric matricesjk,
withj,k= 1, 2 ,···,mandj < k, which correspond to infinitesimal rotations in
thejkplanes. Group elements can be constructed by multiplying rotationseθjk
in different planes. Instead of a single operatorQ̂, we get multiple operators

−iQ̂jk=−i

∫

R^3

(̂Πk(x)Φ̂j(x)−Π̂j(x)̂Φk(x))d^3 x

and conjugation by

Ujk(θ) =e−iθ

Q̂jk

rotates the field operators in thejkplane. These also provide unitary operators
on the state space, and, taking appropriate products of them, a unitary repre-
sentation of the full groupSO(m) on the state space. TheQ̂jkcommute with the
Hamiltonian (generalized to them-component case) so the energy eigenstates
of the theory break up into irreducible representations ofSO(m) (a subject we
haven’t discussed form >3).

44.1.2 U(1) symmetry and complex scalar fields

Instead of describing a scalar field system withSO(2) symmetry using a pair
Φ 1 ,Φ 2 of real fields, it is sometimes more convenient to work with complex
scalar fields and aU(1) symmetry. This will also allow the use of field operators
and annihilation and creation operators for states with a definite value of the
charge observable. Taking asMthe complex vector space of complex solutions
to the Klein-Gordon equation however is confusing, since the Bargmann-Fock
quantization method requires that we complexifyM, and the complexification
of a complex vector space is a notion that requires some care. More simply, here
one can think of the spaceMof solutions to the Klein-Gordon equation for a
pair of real fields as having two different complex structures:

• The relativistic complex structureJr, which is +ion positive energy so-
  lutions inM⊗Cand−ion negative energy solutions inM⊗C.


• The “charge” complex structureJC, which is +ion positive charge solu-
  tions and−ion negative charge solutions.

The operatorsJrandJCwill commute, so we can simultaneously diagonalize
them onM ⊗C, and decompose the positive energy solution space into±i
eigenspaces ofJC, so
H 1 =M+Jr=H+ 1 ⊕H− 1