Quantum Mechanics for Mathematicians

The quadratic functions we will consider will be “local”, multiplying elements
parametrized by the same points in position space. Often these will be differen-
tial operators. As a result, the generalization from the finite dimensional case
will take ∑

j,k

→

∫

dxor

∫

dp

and
Ajk→O(x) orO(p)

38.2 Internal symmetries

Since the phase spaceH 1 is a space of complex functions, there is an obvious
group that acts unitarily on this space: the groupU(1) of phase transformations
of the complex values of the function. Such a group action that acts trivially
on the spatial coordinates but non-trivially on the values ofψ(x) is called an
“internal symmetry”. If the fieldsψhave multiple components, taking values
inCn, there will be a unitary action of the larger groupU(n).

38.2.1 U(1) symmetry

In chapter 2 we saw that the fact that irreducible representations ofU(1) are
labeled by integers is responsible for the term “quantization”: since quantum
states are representations of this group, they break up into states characterized
by integers, with these integers counting the number of “quanta”. In the non-
relativistic quantum field theory, this integer will be the total particle number.
Such a theory can be thought of as a harmonic oscillator with an infinite number
of degrees of freedom, and the total particle number is the total occupation
number, summed over all degrees of freedom.
Consider theU(1) action on the fields Ψ(x),Ψ(x) given by

Ψ(x)→e−iθΨ(x), Ψ(x)→eiθΨ(x) (38.1)

This is an infinite dimensional generalization of the case worked out in section
24.2, where recall that the moment map wasμ=zzand

{zz,z}=iz {zz,z}=−iz

There were two possible choices for the unitary operator that will be the
quantization ofzz:

• zz→−

i

2

(a†a+aa†)

This will have eigenvalues−i(n+^12 ),n= 0, 1 , 2 ....