Quantum Mechanics for Mathematicians

operatorHwill make up a Lie algebra of symmetries of the quantum system,
and will take energy eigenstates to energy eigenstates of the same energy. Some
examples for the physical case ofd= 3 are:

• The groupR^3 of translations in coordinate space is a subgroup of the
  Heisenberg group and has a Lie algebra representation as linear combina-
  tions of the operators−iPj. If the Hamiltonian is position-independent,
  for instance the free particle case of

H=

1

2 m

(P 12 +P 22 +P 32 )

then the momentum operators correspond to symmetries. Note that the

position operatorsQjdo not commute with this Hamiltonian, and so do

not correspond to a symmetry of the dynamics.

• The groupSO(3) of spatial rotations is a subgroup of Sp(6,R), with
  so(3)⊂sp(6,R) given by the quadratic polynomials in equation 16.14 for
  Aan antisymmetric matrix. Quantizing, the operators

−i(Q 2 P 3 −Q 3 P 2 ), −i(Q 3 P 1 −Q 1 P 3 ), −i(Q 1 P 2 −Q 2 P 1 )

provide a basis for a Lie algebra representation ofso(3). This phenomenon

will be studied in detail in chapter 19.2 where we will find that for the

Schr ̈odinger representation on position-space wavefunctions, these are the

same operators that were studied in chapter 8 under the nameρ′(lj). They

will be symmetries of rotationally invariant Hamiltonians, for instance the

free particle as above, or the particle in a potential

H=

1

2 m

(P 12 +P 22 +P 32 ) +V(Q 1 ,Q 2 ,Q 3 )

when the potential only depends on the combinationQ^21 +Q^22 +Q^23.

17.5 More general notions of quantization

The definition given here of quantization using the Schr ̈odinger representation
ofh 2 d+1only allows the construction of a quantum system based on a classical
phase space for the linear case ofM=R^2 d. For other sorts of classical systems
one needs other methods to get a corresponding quantum system. One possible
approach is the path integral method, which starts with a choice of configuration
space and Lagrangian, and will be discussed in chapter 35.

Digression.The name “geometric quantization” refers to attempt to generalize
quantization to the case of any symplectic manifoldM, starting with the idea
of prequantization (see equation 15.8). This gives a representation of the Lie
algebra of functions onMon a space of sections of a line bundle with connection
∇, with∇a connection with curvatureω, whereωis the symplectic form on
M. One then has to deal with two problems: