Relativistic quantum kinematics in the Moyal representation

In this paper, we obtain the phase-space quantization for relativistic spinning particles. The main tool is what we call a "Stratonovich-Weyl quantizer" 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. Our phase spaces are coadjoint orbits of the restricted Poincaré group; we compute and explicitly coordinatize the orbits corresponding to massive particles, with or without spin. Some orbits correspond to unitary irreducible representations of the Poincaré group; we show that there is a unique Stratonovich-Weyl quantizer from each of these phase spaces to operators on the corresponding representation spaces, and compute it explicitly. We develop the formalism by computing relativistic Wigner functions and twisted products for Klein-Gordon particles; these Wigner functions are supported on the mass shell. We thereby obtain an expression for the position probability density which is local, that is, free from the difficulty of supraluminal propagation of the usual position probability density. It is shown explicitly how observables on phase space may be quantized; for example, we prove that the canonical position coordinate corresponds to the Newton-Wigner position operator, irrespective of spin. We show how relativistic phase-space quantization applies to particles governed by the Dirac equation. In effect, we construct a Stratonovich-Weyl quantizer whose associated Hilbert space is the space of positive-energy solutions of the Dirac equation.UCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Matemátic

Structural aspects of Hamilton–Jacobi theory

The final publication is available at Springer via http://dx.doi.org/10.1142/S0219887816500171In our previous papers [11, 13] we showed that the Hamilton–Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how constants of the motion help to solve the Hamilton–Jacobi equation. Here we want to delve into this interpretation by considering the most general case: a dynamical system on a manifold that is described in terms of a family of dynamics (‘slicing vector fields’) on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamilton– Jacobi theory, by considering special cases like fibred manifolds and Hamiltonian dynamics, in the symplectic framework and the Poisson one. We also show how a set of functions on a tangent bundle can determine a second-order dynamics for which they are constants of the motion.Peer ReviewedPostprint (author's final draft

Distinguished Hamiltonian theorem for homogeneous symplectic manifolds

A diffeomorphism of a finite-dimensional flat symplectic manifold which is canonoid with respect to all linear and quadratic Hamiltonians preserves the symplectic structure up to a factor: so runs the "quadratic Hamiltonian theorem". Here we show that the same conclusion holds for much smaller "sufficiency subsets" of quadratic Hamiltonians, and the theorem may thus be extended to homogeneous infinite-dimensional symplectic manifolds. In this way we identify the distinguished Hamiltonians for the Kähler manifold of equivalent quantizations of a Hilbertizable symplectic space.UCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Matemátic

Structural aspects of Hamilton–Jacobi theory

The final publication is available at Springer via http://dx.doi.org/10.1142/S0219887816500171In our previous papers [11, 13] we showed that the Hamilton–Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how constants of the motion help to solve the Hamilton–Jacobi equation. Here we want to delve into this interpretation by considering the most general case: a dynamical system on a manifold that is described in terms of a family of dynamics (‘slicing vector fields’) on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamilton– Jacobi theory, by considering special cases like fibred manifolds and Hamiltonian dynamics, in the symplectic framework and the Poisson one. We also show how a set of functions on a tangent bundle can determine a second-order dynamics for which they are constants of the motion.Peer Reviewe