Quantum Mechanics for Mathematicians

number and a particle number operatorN̂, but the lowest energy| 0 〉may not
have the properties

N̂| 0 〉= 0, e−iθN̂| 0 〉=| 0 〉

Instead the state| 0 〉gets taken bye−iθN̂to some other state, with

N̂| 0 〉6= 0, e−iθN̂| 0 〉≡|θ〉6=| 0 〉 (forθ 6 = 0)

and the vacuum state not an eigenstate ofN̂, so it does not have a well-defined
particle number. If [N,̂Ĥ] = 0, the states|θ〉will all have the same energy
as| 0 〉and there will be a multiplicity of different vacuum states, labeled by
θ. In such a case theU(1) symmetry is said to be “spontaneously broken”.
This phenomenon occurs when non-relativistic quantum field theory is used to
describe a superconductor. There the lowest energy state will be a state without
a definite particle number, with electrons pairing up in a way that allows them
to lower their energy, “condensing” in the lowest energy state.
When, as for the multi-component free particle (the Hamiltonian of equation
38.4), the Hamiltonian is invariant underU(n) transformations of the fieldsψj,
then we will have
[X,̂Ĥ] = 0

forX̂the operator giving the Lie algebra representation ofU(n) on the multi-
particle state space (see section 38.2.2). In this case, if| 0 〉is invariant under the
U(n) symmetry, then energy eigenstates of the quantum field theory will break
up into irreducible representations ofU(n) and can be labeled accordingly. As
in theU(1) case, theU(n) symmetry may be spontaneously broken, with

X̂| 0 〉6= 0

for some directionsXinu(n). When this happens, just as in theU(1) case states
did not have well-defined particle number, now they will not carry well-defined
irreducibleU(n) representation labels.

39.5 Higher order operators and renormaliza-

tion

We have generally restricted ourselves to considering only products of basis
elements of the Heisenberg Lie algebra (position and momentum in the finite
dimensional case, fields in the infinite dimensional case) of degree less than or
equal to two, since it is these that after quantization have an interpretation as
the operators of a Lie algebra representation. In the finite dimensional case
one can consider higher-order products of operators, for instance systems with
Hamiltonian operators of higher order than quadratic. Unlike the quadratic
case, typically no exact solution for eigenvectors and eigenvalues will exist, but
various approximation methods may be available. In particular, for Hamiltoni-
ans that are quadratic plus a term with a small parameter, perturbation theory