Abstract We develop a relativistic framework of integral quantization applied to the motion of spinless particles in the four-dimensional Minkowski spacetime. The proposed scheme is based on coherent states generated by the action of the Heisenberg–Weyl group and has been motivated by the Hamiltonian description of the geodesic motion in General Relativity. We believe that this formulation should also allow for a generalization to the motion of test particles in curved spacetimes. A key element in our construction is the use of suitably defined positive operator-valued measures. We show that this approach can be used to quantize the one-dimensional nonrelativistic harmonic oscillator, recovering the standard Hamiltonian as obtained by the canonical quantization. A direct application of our model, including a computation of transition amplitudes between states characterized by fixed positions and momenta, is postponed to a forthcoming article.
Abstract We develop a relativistic framework of integral quantization applied to the motion of spinless particles in the four-dimensional Minkowski spacetime. The proposed scheme is based on coherent states generated by the action of the Heisenberg–Weyl group and has been motivated by the Hamiltonian description of the geodesic motion in General Relativity. We believe that this formulation should also allow for a generalization to the motion of test particles in curved spacetimes. A key element in our construction is the use of suitably defined positive operator-valued measures. We show that this approach can be used to quantize the one-dimensional nonrelativistic harmonic oscillator, recovering the standard Hamiltonian as obtained by the canonical quantization. A direct application of our model, including a computation of transition amplitudes between states characterized by fixed positions and momenta, is postponed to a forthcoming article.
Abstract We use an integral quantization model based on the Heisenberg–Weyl group to describe the motion of a spinless particle in the Minkowski background spacetime. This work is a sequel to a previous paper, devoted to mathematical aspects of our model: construction of the space of coherent states and properties of elementary observables. We compute transition amplitudes corresponding to the free motion of a particle between two coherent states. These amplitudes are then used to model quantum random walks of free relativistic particles. Our quantization scheme allows us to recover interference patterns occurring in a standard double-slit experiment, known from the classical approach. This result is obtained by modeling the slits in terms of eigenstates of the position operator and computing transition amplitudes between position and coherent states. We design our model in a way which allows for a future generalization to a semiclassical quantization of the geodesic motion in curved spacetimes.
We present the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg–Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group C*-algebra giving the algebra of observables. The key feature is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators. Galilean limit and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space.
The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space S q,p ≊ 2 . In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space S ϕ,I corresponds to the multiply connected punctured plane 2 – {0}, on which the 3-dimensional symplectic group Sp(2, ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U (1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator K o of U (l) represents the classical action variable I . It has the possible spectra n + k, n = 0,1,...; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK 0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K 0 etc. of Sp (2, ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.
The method of Weyl transforms is used to rigorously derive path integral forms for position and momentum transition amplitudes from the time-dependent Schrödinger equation for arbitrary Hermitian Hamiltonians. It is found that all paths in phase space contribute equally in magnitude, but that each path has a different phase, equal to 1/h/ times an ’’effective action’’ taken along it. The latter is the time integral of p⋅q̇−h (p,q), h (p,q) being the Weyl transform of the Hamiltonian operator H, which differs from the classical Hamiltonian function by terms of order h/2, vanishing in the classical limit. These terms, which can be explicitly computed, are zero for relatively simple Hamiltonians, such as (1/2M)[P−eA (Q)]2+V (Q), but appear when the coupling of the position and momentum operators is stronger, such as for a relativistic spinless particle in an electromagnetic field, or when configuration space is curved. They are always zero if one opts for Weyl’s rule for forming the quantum operator corresponding to a given classical Hamiltonian. The transition amplitude between two position states is found to be expressible as a path integral in configuration space alone only in very special cases, such as when the Hamiltonian is quadratic in the momenta.
In this paper we describe and apply integral quantization, a procedure based on operator-valued measures and the resolution of the identity.We explore its covariance properties in the context of group representation theory.Three applications based on group representations are carried out.The first one concerns the covariant integral quantization based on the spin one-half irreducible representation of SU(2).In this case we show that the quantization of both the quadratic Hopf variables and the Euler angles reproduces the SU(2) Lie algebra.The second example revisits integral quantization based on the Weyl-Heisenberg group.By completing previous results, we show the universality of the canonical commutation relation in such quantizations.In the last example we revisit and enrich the integral quantization based on the affine group of the real line and we give a short account of a relevant application.
Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalized positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is their ability to circumvent problems due to the presence of singularities in the classical models. In this paper, we implement covariant integral quantizations for systems whose phase space is Z×S1, i.e., for systems moving on the circle. The symmetry group of this phase space is the discrete & compact version of the Weyl–Heisenberg group, namely the central extension of the abelian group Z×SO(2). In this regard, the phase space is viewed as the right coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on L2(S1), is square integrable on the phase space. We show how to derive corresponding covariant integral quantizations from (weight) functions on the phase space and resulting resolution of the identity. As particular cases of the latter we recover quantizations with de Bièvre-del Olmo–Gonzales and Kowalski–Rembielevski–Papaloucas coherent states on the circle. Another straightforward outcome of our approach is the Mukunda Wigner transform. We also look at the specific cases of coherent states built from shifted gaussians, Von Mises, Poisson, and Fejér kernels. Applications to stellar representations are in progress.