Classical mechanics in phase space involves the concepts of Poisson brackets and canonical transformations. In (nonrelativistic) quantum mechanics, based on fundamental Cartesian variables, the kinematics is expressed via the Heisenberg canonical commutation relations (CCR’s). One now has unitary transformations on wave functions in Hilbert space. In both cases, Hamiltonians quadratic in the canonical variables lead to linear equations of motion whose solutions are linear canonical transformations. In all this, the key role is played by the family of real symplectic groups Sp (2n ,ℝ). They are the ‘noncompact’ versions of the classical compact Lie groups Cl = USp (2l) , and their defining representations are in real even dimensional vector spaces. In addition to their significance in the canonical formulation of classical mechanics and its counterpart in quantum mechanics, these groups are important in several specific quantum mechanical problems as well as in classical and quantum optics.

We look at some aspects of these groups in this chapter. While we describe the main features, we will not present complete proofs of all statements made, but a motivated reader should be able to supply the details and proceed to make practical uses of these groups.

In both classical and quantum mechanics we have in mind only Cartesian fundamental variables. So for n degrees of freedom, or n canonical pairs, the classical phase space is R2n, while in quantum mechanics we have the Hilbert space H = L 2(ℝn). Since the symplectic groups are somewhat unfamiliar to most students of physics, we study the case n = 1 to begin with, and later go on to general n.

The GroupSp (2,ℝ)

For one degree of freedom with Cartesian variables, in classical mechanics we have real canonical variables q and p , varying from −∞ to ∞, and the Poisson brackets (PB)

The phase space is the two dimensional plane ℝ2. For the corresponding system in quantum mechanics we have an irreducible pair of hermitian operators q , p obeying the Heisenberg CCR’s

In the usual Schrödinger solution to these relations we have a Hilbert space H = L2 (ℝ) with q , p acting as follows:

(Strictly speaking, since q and p are unbounded operators, their domains should be carefully specified; we assume such precautions are implicitly observed.)

Now consider linear homogeneous transformations on q , p which in the classical case preserve the PB's (8.1) and in the quantum case the CCR's (8.2).