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Journal of Physics A Mathematical Nuclear and General

Coordinate free canonical commutation relations

F. J. Bloore, J. Underhill

Published Jul 1, 1973• 4 citations
Publisher

Abstract

The authors present an algebraic argument which shows that the coordinate free canonical commutation relations of Segal are the natural extension of the coordinate-velocity commutation relations. They do not invoke the Schrodinger representation. They define an algebra of quantum observables for any manifold and show it becomes the usual algebra when the manifold is riemannian.

DOI
10.1088/0305-4470/6/7/002

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The authors present an algebraic argument which shows that the coordinate free canonical commutation relations of Segal are the natural extension of the coordinate-velocity commutation relations. They do not invoke the Schrodinger representation. They define an algebra of quantum observables for any manifold and show it becomes the usual algebra when the manifold is riemannian.
Starting with Segal's postulates (1960) for quantum mechanics, augmented by the postulate that the commutators of the free Hamiltonian with position observables are canonical, the authors prove that the quantum mechanical observable Q(X), which corresponds to the function C(X) identical to X i (q)p i on phase space in classical mechanics, is equal to the anticommutator 1 / 2 (Q(X i ),Q(p i )). In coordinate free language, this means the Q( phi X)= 1 / 2 (Q( phi ),Q(X)) for any scalar field phi and any vector field X on the configuration space.
A formalism of quantum kinematics which rests on the regular representation of a Lie group is proposed. This representation leads to explicit canonical commutation relations for non-Abelian dynamical variables which, together with the Lie algebra, define the kinematic algebra. Generalised equations of motion for the group-parameter-dependent operators, as well as generalised wave equations, are introduced over the group manifold considered as a background arena. The formalism proposed affords a method of geometric quantisation stemming directly from the observed symmetries of a system.
This paper studies a class of representations (called quadratic) of the canonical commutation relations over symplectic spaces of arbitrary dimension, which naturally generalizes coherent and symplectic (i.e. quasifree) representations and which has previously been heuristically employed in the special case of finite degrees of freedom in the physics literature. An explicit characterization of canonical quadratic transformations in terms of a 'standard form' is given, and it is shown that they can be exponentiated to give representations of the Weyl algebra. Necessary and sufficient conditions are presented for the unitary equivalence of these representations with the Fock representation. Possible applications to quantum optics and quantum field theory are briefly indicated.
The classical system is usually quantized using the canonical formalism, i.e., in the language of coordinates and momenta. In addition to this “purely algebraic” ambiguity, there is also an ambiguity in the choice of realization of the operators p^ and q^: the commutation relations do not determine it uniquely, but up to a unitary equivalence (the well known uniqueness theorem of von Neumann pertains only to the Weyl form of the commutation relations). The realization must be chosen on the basis of additional considerations regarding the nature of the canonical variables p and q. In this book the term “theory” is used in various ways: in a broad sense (quantum field theory, relativistic theory) and in a narrow sense, to refer to a particular system or class of systems with certain dynamics. In quantum theory there are also systems whose operators corresponding to canonical variables satisfy not commutation relations, but anticommutation relations.
We consider Hilbert space representations of a generalization of canonical commutation relations , where 's are the elements of an algebra with identity , is the imaginary unit, and is a real number with antisymmetry . Some basic aspects on Hilbert space representations of the generalized CCR (GCCR) are discussed. We define a Schrödinger-type representation of the GCCR by an analogy with the usual Schrödinger representation of the CCR with degrees of freedom. Also, we introduce a Weyl-type representation of the GCCR. The main result of the present paper is a uniqueness theorem on Weyl representations of the GCCR.