The Moyal-Lie Theory of Phase Space Quantum Mechanics

A Lie algebraic approach to the unitary transformations in Weyl quantization is discussed. This approach, being formally equivalent to the ⋆-quantization is an extension of the classical Poisson-Lie formalism which can be used as an efficient tool in the quantum phase space transformation theory. The purpose of this paper is to show that the Weyl correspondence in the quantum phase space can be presented in a quantum Lie algebraic perspective which in some sense derives analogies from the Lie algebraic approach to classical transformations and Hamiltonian vector fields. In the classical case transformations of functions f(z) of the phase space z = (p, q) are generated by the generating functions Gμ(z). If the generating functions are elements of a set, then this set usually contains a subset closed under the action the Poisson Bracket (PB), so defining an algebra. To be more precise one talks about different representations of the generators and of the phase space algebras. The second well known representation is the so called adjoint representation of the Poisson algebra of the Gμ’s in terms of the classical Lie generators LGμ over the Lie algebra. These are the Hamiltonian vector fields with the Hamiltonians Gμ. By Liouville’s theorem, they characterize an incompressible and covariant phase space flow. The quantum Lie approach can be based on a parallelism with the classical one above. In this case the unitary transformations are represented by a set of quantum generating functions Aμ(z) which are the quantum partners of the Gμ’s. The non commutativity is encoded in a new multiplication rule, i.e. ⋆-product in z. The quantum Lie generators V̂Aμ over the Lie bracket are the adjoint representations of the generating functions Aμ(z) over the Moyal bracket and they correspond to quantum counterparts of the classical Hamiltonian vector fields LGμ . They are represented by an infinite number of phase space derivatives. The only exception is the generators of linear canonical transformations defining the sp2(R) sub algebra. In this case the generators are given by the classical Hamiltonian vector fields generated by quadratic Hamiltonians and the classical and quantum generators are identical [see (46) below]. The classical Lie approach is sometimes referred to as the Poisson-Lie theory (PLT). This terminology is quite appropriate for generalizations as the algebraic representations that are relevant are written by using the Poisson and Lie product rules. For a similar reason we suggest that the quantum approach be referred to as the Moyal-Lie theory (MLT). In section 1 we start with a brief textbook discussion of the PLT. The Lie algebraic treatment of the quantum transformation theory starts in section 2 with a discussion of Weyl correspondence (section 2.1) and the ⋆-product. The main results of this paper, the MLT and the ⋆-covariance are discussed at length in section 2.2. The associativity property is discussed comparatively with respect to Poisson-Lie and the Moyal-Lie algebras in section 3. In the light of the classical covariance versus ⋆-covariance of the phase space trajectories under canonical transformations the time evolution deserves a specific attention. The time evolution in the MLT and its extended ⋆-covariance property is examined in section 4. Finally, we discuss the equivalence of the MLT to the standard ⋆-quantization through the ⋆-exponentiation in section 5. A. The Poisson-Lie Theory and the classical phase space The classical Lie generators LGμ are given by LGμ = (∂zGμ) T J ∂z , (1) where Gμ = Gμ(z)’s are in the set of phase space functions (generating functions) where μ = 1, . . . , N describes the generator index, N being the total number of generators, z denotes the 2n dimensional phase space row vector z = (z1, . . . , z2n) = (q1, . . . , qn; p1, . . . , pn), ∂z = (∂z1 , . . . , ∂z2n), T is the transpose of a row vector, and J is the 2n × 2n symplectic matrix with elements (J)i j , i, j = (1, . . . , 2n)