Quantum Mechanics for Mathematicians

An important property ofN̂that can be straightforwardly checked is that

[N,̂Ĥ] =

[

N,̂

∫+∞

−∞

Ψ̂†(x)−^1

2 m

∂^2

∂x^2

Ψ(̂x)dx

]

= 0

This implies that particle number is a conserved quantity: if we start out with
a state with a definite particle number, this will remain constant. Note that
the origin of this conservation law comes from the fact thatN̂is the quantized
generator of theU(1) symmetry of phase transformations on complex-valued
fields Ψ. If we start with any Hamiltonian functionhonH 1 that is invariant
under theU(1) (i.e., built out of terms with an equal number of Ψs andΨs),
then for such a theoryN̂will commute withĤand particle number will be
conserved.

38.2.2 U(n) symmetry

By taking fields with values inCn, or, equivalently, ndifferent species of
complex-valued field Ψj, j= 1, 2 ,...,n, quantum field theories with larger in-
ternal symmetry groups thanU(1) can easily be constructed. Taking as Hamil-
tonian function

h=

∫+∞

−∞

∑n

j=1

Ψj(x)

− 1

2 m

∂^2

∂x^2

Ψj(x)dx (38.4)

gives a Hamiltonian that will be invariant not just underU(1) phase transfor-
mations, but also under transformations











Ψ 1

Ψ 2

..

.

Ψn











→U











Ψ 1

Ψ 2

..

.

Ψn











whereUis annbynunitary matrix. The Poisson brackets will be

{Ψj(x),Ψk(x′)}=iδ(x−x′)δjk

and are also invariant under such transformations byU∈U(n).
As in theU(1) case, we begin by considering the case of one particular value
ofpor ofx, for which the phase space isCn, with coordinateszj,zj. As we saw
in section 25.2, then^2 quadratic combinationszjzkforj= 1,...,n, k= 1,...,n
will generalize the role played byzzin then= 1 case, with their Poisson bracket
relations exactly the Lie bracket relations of the Lie algebrau(n) (or, considering
all complex linear combinations,gl(n,C)).
After quantization, these quadratic combinations become quadratic combi-
nations of annihilation and creation operatorsaj,a†jsatisfying

[aj,a†k] =δjk