where one should keep in mind that each degree of freedom can be rescaled
separately, allowing different parametersωjfor the different degrees of freedom.
The energy and number operator eigenstates will be written
|n 1 ,...,nd〉where
a†jaj|n 1 ,...,nd〉=Nj|n 1 ,...,nd〉=nj|n 1 ,...,nd〉
Ford= 3 the harmonic oscillator problem is an example of the central po-
tential problem described in chapter 21, and will be discussed in more detail
in section 25.4.2. It has anSO(3) symmetry, with angular momentum opera-
tors that commute with the Hamiltonian, and spaces of energy eigenstates that
can be organized into irreducibleSO(3) representations. In the Schr ̈odinger
representation states are inH=L^2 (R^3 ), described by wavefunctions that can
be written in rectangular or spherical coordinates, and the Hamiltonian is a
second-order differential operator. In the Bargmann-Fock representation, states
inF 3 are described by holomorphic functions of 3 complex variables, with op-
erators given in terms of products of annihilation and creation operators. The
Hamiltonian is, up to a constant, just the number operator, with energy eigen-
states homogeneous polynomials (with eigenvalue of the number operator their
degree).
Either thePj,Qjor theaj,a†jtogether with the identity operator will give a
representation of the Heisenberg Lie algebrah 2 d+1onH, and by exponentiation
a representation of the Heisenberg groupH 2 d+1. Quadratic combinations of
these operators will give a representation of the Lie algebrasp(2d,R), one that
exponentiates to the metaplectic representation of a double cover ofSp(2d,R).
25.2 Complex coordinates on phase space andU(d)⊂Sp(2d,R)
As in thed= 1 case, annihilation and creation operators can be thought of
as the quantization of complexified coordinateszj,zjon phase space, with the
standard choice given by
zj=1
√
2
(qj−ipj), zj=1
√
2
(qj+ipj)Such a choice ofzj,zj gives a decomposition of the complexified Lie algebra
sp(2d,C) (as usual, the Lie bracket is the Poisson bracket) into three Lie sub-
algebras as follows:
- A Lie subalgebra with basis elementszjzk. There are^12 (d^2 +d) distinct
such basis elements. This is a commutative Lie subalgebra, since the
Poisson bracket of any two basis elements is zero. - A Lie subalgebra with basis elementszjzk. Again, this has dimension
1
2 (d
(^2) +d) and is a commutative Lie subalgebra.