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Journal of Physics A Mathematical and General

Relativistic quantum kinematics in the Moyal representation

José F. Cariñena, JoséM. Garcia-Bondia, Joseph C. Várilly

Published Mar 21, 1990• 79 citations
Publisher

Abstract

The authors obtain the phase-space quantisation for relativistic spinning particles. The main tool is what they call a 'Stratonovich-Weyl quantiser' which relates functions on phase space to operators on a suitable Hilbert space, and has the essential properties of covariance (under a group representation) and traciality. Their phase spaces are coadjoint orbits of the restricted Poincare group; they compute and explicitly coordinatise the orbits corresponding to massive particles, with or without spin. Some orbits correspond to unitary irreducible representations of the Poincare group; they show that there is a unique Stratonovich-Weyl quantiser from each of these phase spaces to operators on the corresponding representation spaces, and compute it explicitly.

DOI
10.1088/0305-4470/23/6/015

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The authors obtain the phase-space quantisation for relativistic spinning particles. The main tool is what they call a 'Stratonovich-Weyl quantiser' which relates functions on phase space to operators on a suitable Hilbert space, and has the essential properties of covariance (under a group representation) and traciality. Their phase spaces are coadjoint orbits of the restricted Poincare group; they compute and explicitly coordinatise the orbits corresponding to massive particles, with or without spin. Some orbits correspond to unitary irreducible representations of the Poincare group; they show that there is a unique Stratonovich-Weyl quantiser from each of these phase spaces to operators on the corresponding representation spaces, and compute it explicitly.
The Stratonovich–Weyl correspondence is a restatement of the Moyal quantization where the phase space is a manifold and where a group of transformations acts on it transitively. The first and most important step is to define a mapping from the manifold into the set of self-adjoint operators on a Hilbert space, under suitable conditions. This mapping is called a Stratonovich–Weyl kernel. The construction of this mapping is discussed on coadjoint orbits of the one-dimensional Galilei, Poincaré, and Newton–Hooke groups as well as the two-dimensional Euclidean group.
The building blocks of Hudson-Parthasarathy quantum stochastic calculus start with Weyl operators on a symmetric Fock space. To realize a relativistically covariant version of the calculus we construct representations of Poincare group in terms of Weyl operators on suitably constructed, Bosonic or Fermionic based on the mass and spin of the fundamental particle, Fock spaces. We proceed by describing the orbits of homogeneous Lorentz group on R4 and build fiber bundle representations of Poincaré group induced from the stabilizer subgroups (little groups) and build the Boson Fock space of the Hilbert space formed from the sections of the bundle. Our Weyl operators are constructed on symmetric Fock space of this space and the corresponding annihilation, creation, and conservation operators are synthesized in the usual fashion in relativistic theories for space-like, time-like, and light-like fields. We achieve this by constructing transitive systems of imprimitivity (second-quantized SI), which are dynamical systems with trajectories dense in the configuration space, by induced representations. We provide the details of the field operators for the case of massive Bosons as the rest are similar in construction and indicate the ways to construct adapted processes paving way for building covariant quantum stochastic calculus.
Spinor structure is usually introduced as part of relativistic quantum theory. We show that its underlying algebra can instead emerge without quantisation from a covariant statistical description of a massive relativistic particle subject to an explicit finite-dimensional linear factorisation of the complete massive mass-shell constraint motivated by requiring the theory retain both sheets of the massive mass shell. A finite-dimensional factorisation of the complete mass-shell constraint, linear in all four momentum components, forces a Clifford algebra. In (3+1) dimensions its minimal complex representation is four-dimensional, while each on-shell factor selects a rank-two sector; in this representation statistical completeness requires an arbitrary 2×22\times22×2 matrix distribution within that sector before quantisation. Projecting the Weyl-ordered matrix Liouvillian gives relativistic transport while preserving all internal populations and coherences. For universal deformations on flat canonical phase space satisfying the stated covariance, matrix-neutrality, homogeneity and associativity assumptions, the resulting product is the matrix Weyl--Moyal product. Its parameter has dimensions of action and, when identified with ℏ\hbarℏ, supplies the scale of physical spin-12\tfrac1221​. Requiring all-order constraint preservation gives a vanishing star commutator, while preservation of the selected star-spectral sector yields the two-sided Dirac--Wigner equations. Our central conclusion is therefore that relativistic mass-shell factorisation supplies spinor structure, while completion of the constraint-preservation hierarchy within the stated deformation class leads to the Weyl--Moyal phase-space formulation of quantum mechanics.
It is shown that to every Galilei-covariant nonrelativistic stochastic phase space representation of a system of massive particles, whose generator is rotationally invariant, corresponds a Poincaré-covariant relativistic representation sharing the same generator. The stochastic phase space probability densities of the two representations overlap in the limit of nonrelativistic velocities in the laboratory frame. The relativistic representations give rise to covariant and conserved probability currents at stochastic space–time points, in complete analogy with their nonrelativistic counterparts. This parallelism extends to the existence of a global representation of the proper Poincaré group in L2(Γ), which is reduced by each subspace of L2(Γ) spanned by the set of phase-space wavefunctions generated by some stochastic phase space representation of all pure states of the system.
This is a review paper on the formulation of nonrelativistic Quantum Mechanics on Phase Space. In an overlook to the traditional approach, we present the basic concepts including the Weyl mapping, the Wigner function, the Moyal propagator and the Spectral projections. We discuss the advantages and the disadvantages of this formulation. We apply it to the study of quadratic Hamiltonians. The Stratonovich-Weyl correspondence is presented as the natural framework in which the theory can be extended to include spin, relativity and constraints. The theory of identical particles is also considered.