A Wiener-regularized path integral is presented as an alternative way to formulate Berezin-Toeplitz quantization on a toroidal phase space. Essential to the result is that this quantization prescription for the torus can be constructed as an induced representation from anti-Wick quantization on its covering space, the plane. When this construction is expressed in the form of a Wiener-regularized path integral, symmetrization prescriptions for the propagator emerge similar to earlier path-integral formulas on multiply-connected configuration spaces.
We present a new method for the consistent construction of time-continuous coherent-state path integrals using the theory of half-form quantization. Through the inversion of the quantization procedure we construct a de-quantization map taking first order operators to their corresponding path integrals. We generalize our results using functional techniques, allowing for the consistent path integral study of more general operators, including higher orders and interactions.
Using a scheme proposed earlier we set up Hamiltonian path integral quantization for a particle in two dimensions in plane polar coordinates.This scheme uses the classical Hamiltonian, without any $O(\hbar^2)$ terms, in the polar varivables. We show that the propagator satisfies the correct Schrödinger equation.
Systems with constraints pose problems when they are quantized. Moreover, the Dirac procedure of quantization prior to reduction is preferred. The projection operator method of quantization, which can be most conveniently described by coherent state path integrals, enables one to directly impose a regularized form of the quantum constraints. This procedure also overcomes conventional difficulties with normalization and second class constraints that invalidate conventional Dirac quantization procedures.
Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalised positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is to allow 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 $\mathbb{Z}\times\,\mathbb{S}^1$, i.e., for systems moving on the circle. The symmetry group of this phase space is the discrete \& com…
We give analytical expressions for the eigenvalues and generalized eigenfunctions of $\hat{T}_3$, the $z$-axis projection of the toroidal dipole operator, in a system consisting of a particle confined in a thin film bent into a torus shape. We find the quantization rules for the eigenvalues, which are essential for describing measurements of $\hat{T}_3$. The eigenfunctions are not square-integrable, so they do not belong to the Hilbert space of wave functions, but they can be interpreted in the formalism of rigged Hilbert spaces as kernels of distributions. While these kernels appear to be pro…
The contour integrals, occurring in the arbitrary-order phase-integral quantization conditions given in a previous paper, are in the first- and third-order approximations expressed in terms of complete elliptic integrals in the case that the charges of the Coulomb centres are equal. The evaluation of the integrals is facilitated by the knowledge of quasiclassical dynamics. The resulting quantization conditions involving complete elliptic integrals are solved numerically to obtain the energy eigenvalues and the separation constants of the $1sσ$ and $2pσ$ states of the hydrogen molecule ion for …
A path-integral quantization on a homogeneous space G/H is proposed based on the guiding principle `first lift to G and then project to G/H'. It is then shown that this principle gives a simple procedure to obtain the inequivalent quantizations (superselection sectors) along with the holonomy factor (induced gauge field) found earlier by algebraic approaches. We also prove that the resulting matrix-valued path-integral is physically equivalent to the scalar-valued path-integral derived in the Dirac approach, and thereby present a unified viewpoint to discuss the basic features of quantizing on…
In this paper we take up the quantal two-centre problem where the Coulomb centres have arbitrary positive charges. In analogy with the symmetric case, treated in the second paper of this series of papers, we use the knowledge on the quasiclassical dynamics to express the contour integrals in the first- and third-order approximations of the phase-integral quantization conditions, given in the first paper of this series of papers, in terms of complete elliptic integrals. For various values of the distance between these charges the accuracy of the formulas obtained is illustrated by comparison wi…
We present a list of formulae useful for Weyl-Heisenberg integral quantizations, with arbitrary weight, of functions or distributions on the plane. Most of these formulae are known, others are original. The list encompasses particular cases like Weyl-Wigner quantization (constant weight) and coherent states (CS) or Berezin quantization (Gaussian weight). The formulae are given with implicit assumptions on their validity on appropriate space(s) of functions (or distributions). One of the aims of the document is to accompany a work in progress on Weyl-Heisenberg integral quantization of dynamics…
The Berezin--Simon (BS) quantization is a rigorous version of the ``operator formalism'' of quantization procedure. The goal of the paper is to present a rigorous real-time (not imaginary-time) path-integral formalism corresponding to the BS operator formalism of quantization; Here we consider the classical systems whose phase space $M$ is a (possibly non-compact) Kähler manifold which satisfies some conditions, with a Hamiltonian $H:M\rightarrow\mathbb{R}$. For technical reasons, we consider only the cases where $H$ is smooth and bounded. We use Güneysu's extended version of the Feynman--Kac …
We implement a SU(1,1) covariant integral quantization of functions or distributions on the unit disk. The latter can be viewed as the phase space for the motion of a test "massive" particle on 1+1 Anti de Sitter space-time, and the relevant unitary irreducible representations of SU(1,1) corresponding to the quantum version of such motions are found in the discrete series and its lower limits. Our quantization method depends on a weight function on the phase space, and it includes Perelomov coherent states quantization. Semi-classical portraits or lower symbols of main physically relevant oper…
We develop a mathematically well-defined path integral formalism for general symplectic manifolds. We argue that in order to make a path integral quantization covariant under general coordinate transformations on the phase space and involve a genuine functional measure that is both finite and countably additive, the phase space manifold should be equipped with a Riemannian structure (metric). A suitable method to calculate the metric is also proposed.
The method for quantization of constrained theories that was suggested originally by Faddeev and Jackiw along with later modifications is discussed. The particular emphasis of this paper is to show how it is simple to implement their method within the path integral framework using the natural geometric structure that their method utilizes. The procedure is exemplified with the analysis of two models: a quantum mechanical particle constrained to a surface (of which the hypersphere is a special case), and a quantized Schrödinger field interacting with a quantized vector field for both the massiv…
Covariant affine integral quantization of the half-plane is studied and applied to the motion of a particle on the half-line. We examine the consequences of different quantizer operators built from weight functions on the half-plane. To illustrate the procedure, we examine two particular choices of the weight function, yielding thermal density operators and affine inversion respectively. The former gives rise to a temperature-dependent probability distribution on the half-plane whereas the later yields the usual canonical quantization and a quasi-probability distribution (affine Wigner functio…
The conversion of second-class constraints into first-class constraints is used to extend the coordinate-free path integral quantization, achieved by a flat-space Brownian motion regularization of the coherent-state path integral measure, to systems with second-class constraints.
The Feynman-Garrod path integral representation for time evolution is extended to arbitrary one-parameter continuous canonical transformations. One thereupon obtains a generalized Kerner-Sutcliffe formula for the unique quantum representation of the transformation generator, which can be an arbitrary classical dynamical variable. This closed-form quantization procedure is shown to be equivalent to a natural extension of the standard canonical quantization recipe -- an extension that resolves the operator-ordering ambiguity in favor of the Born-Jordan rule.
The path integral quantization method is applied to a relativistically covariant version of the Hopfield model, which represents a very interesting mesoscopic framework for the description of the interaction between quantum light and dielectric quantum matter, with particular reference to the context of analogue gravity. In order to take into account the constraints occurring in the model, we adopt the Faddeev-Jackiw approach to constrained quantization in the path integral formalism. In particular we demonstrate that the propagator obtained with the Faddeev-Jackiw approach is equivalent to th…
The paper is devoted to integral quantization, a procedure based on operator-valued measure and resolution of the identity. We insist on covariance properties in the important case where group representation theory is involved. We also insist on the inherent probabilistic aspects of this classical-quantum map. The approach includes and generalizes coherent state quantization. Two applications based on group representation are carried out. The first one concerns the Weyl-Heisenberg group and the euclidean plane viewed as the corresponding phase space. We show that there exists a world of quanti…
Covariant affine integral quantization is studied and applied to the motion of a particle in a punctured plane R^2_\ast=R^2\{0}, for which the phase space is R^2_\ast=R^2\{0}X R^2. We examine the consequences of different quantizer operators built from weight functions on this phase space. To illustrate the procedure, we examine two examples of weights. The first one corresponds to 2-D coherent state families, while the second one corresponds to the affine inversion in the punctured plane. The later yields the usual canonical quantization and a quasi-probability distribution (2-D affine Wigner…
The coordinate-free formulation of canonical quantization, achieved by a flat-space Brownian motion regularization of phase-space path integrals, is extended to a special class of closed first-class constrained systems that is broad enough to include Yang-Mills type theories with an arbitrary compact gauge group. Central to this extension are the use of coherent state path integrals and of Lagrange multiplier integrations that engender projection operators onto the subspace of gauge invariant states.
The quantization of systems with first- and second-class constraints within the coherent-state path-integral approach is extended to quantum systems with fermionic degrees of freedom. As in the bosonic case the importance of path-integral measures for Lagrange multipliers, which in this case are in general expected to be elements of a Grassmann algebra, is emphasized. Several examples with first- and second-class constraints are discussed.
The Hamiltonian of the harmonic oscillator is usually defined as a differential operator, but an integral representation can be obtained by using the coherent state quantization. The finite frame quantization is a finite counterpart of the coherent state quantization and it allows us to define a finite oscillator by starting from the integral representation of the harmonic oscillator. Our purpose is to investigate the oscillator obtained in this way, and to present a possible application to the discrete fractional Fourier transform.
We introduce, for the first time, bicoherent-state path integration as a method for quantizing non-hermitian systems. Bicoherent-state path integrals arise as a natural generalization of ordinary coherent-state path integrals, familiar from hermitian quantum physics. We do all this by working out a concrete example, namely, computation of the propagator of a certain quasi-hermitian variant of Swanson's model, which is not invariant under conventional $PT$-transformation. The resulting propagator coincides with that of the propagator of the standard harmonic oscillator, which is isospectral wit…
