The canonical formalism and path integrals in curved spaces

D’Olivo, J. C.; Torres, M. A. P.

doi:10.1088/0305-4470/21/16/012

Cited by 17 publications

(7 citation statements)

References 21 publications

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“…We only use the lattice formulation of (177) in this paper unless otherwise (and explicitly) stated. Of course, this lattice definition is completely equivalent with the midpoint definition [24,40,74,78, 791 and others (e.g. [l5, 21, 69, 72, 841).…”

Section: Gkosciie El Al Path Integral Discussion I1mentioning

confidence: 90%

Path Integral Discussion for Smorodinsky-Winternitz Potentials: II. The Two- and Three-Dimensional Sphere

1995

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Steps towards path integral formulations for Smorodinsky‐Winternitz potentials, respectively systems with accidental degeneracies, on the two‐ and three‐dimensional sphere, and a complete classification of super‐integrable systems on spaces of constant curvature are presented. We mention all coordinate systems which separate the Smorodinsky‐Winternitz potentials on a sphere, and state the corresponding path integral formulations. Whereas in many coordinate systems explicit path integral solutions are not possible, we list in all soluble cases the path integral in terms of the propagator, respectively the spectral expansion into the wave functions and the energy spectrum.

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Section: Gkosciie El Al Path Integral Discussion I1mentioning

confidence: 90%

Path Integral Discussion for Smorodinsky-Winternitz Potentials: II. The Two- and Three-Dimensional Sphere

1995

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“…Also, other ambiguities arise in the process of calculation of a Hamilton operator from the path integral (26). Instead of reviewing them, further I expose briefly main points of a special approach the initial idea of which is taken from paper by D'Olivo and Torres [16] but essentially modified in [19] and consists of the following steps: 1. Consider the Hamiltonian representation of G (F) (q, t|q 0 , t 0 ) as a fold of the short-time propagators in the configuration space representation [12]:…”

Section: Feynman Quantization Of Natural Systemsmentioning

confidence: 99%

Unfinished history and paradoxes of quantum potential. I. Non-relativistic origin, history and paradoxes

Tagirov

2013

Gravit. Cosmol.

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This is the first of two related papers analising and explaining the origin, manifestations and parodoxical features of the quantum potential (QP) from the non-relativistic and relativistic point of view. QP arises in the quantum Hamiltonian, under various procedures of quantization of the natural systems, i.e. the Hamilton functions of which are the positive-definite quadratic forms in momenta with coefficients depending on the coordinates in ( n -dimensional) configurational space V n endowed so by a Riemannian structure. The result of quantization may be considered as quantum mecanics (QM) of a particle in V n in the normal Gaussian system of reference in the globally-static spacetime V 1,n . Contradiction of QP to the Principles of General Covariance and Equivalence is discussed.It is found that actually the historically first Hilbert space based quantization by E. Schrödinger (1926), after revision in the modern framework of QM, also leads to QP in the form that B. DeWitt had been found 26 years later. Efforts to avoid QP or reduce its drawbacks are discussed. The general conclusion is that some form of QP and a violation of the principles of general relativity which it induces are inevitable in the non-relativistic quantum Hamiltonian. It is shown also that Feynman (path integration) quantization of natural systems singles out two versions of QP, which both determine two bi-scalar (indepedendent on choice of coordinates) propagators fixing two different algorithms of path integral calculation.In the accompanying paper under the same general title and the subtitle "The Relativistic Point of View", relation of the non-relativistic QP to the quantum theory of the scalar field non-minimally coupled to the curved space-time metric is considered.

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“…In order to set up our notation we proceed in the canonical way for path integrals on curved spaces (DeWitt [18], D'Olivio and Torres [20], Feynman [31], Gervais and Jevicki [36], [43,55], McLaughlin and Schulman [80], Mayes and Dowker [83], Mizrahi [85], and Omote [89]). In the following x denotes a D-dimensional cartesian coordinate, q a D-dimensional arbitrary coordinate, and x, y, z etc.…”

Section: Formulation Of the Path Integralmentioning

confidence: 99%

“…Actually, an expansion into the corresponding wave-functions in the coordinates µ and ν yields the Mathieu functions me ν (η, h 2 ) and Me (1) ν (ξ, h 2 ) (h 2 = mEd 2 /2h 2 ), respectively, as eigen-function of the Hamiltonian, a specific class of higher transcendental functions [84]. However, because we know on the one side the eigenfunctions of the Hamiltonian in terms of these functions [84], and on the other the kernel in E (2) in terms of the invariant distance d E (2) we can state the following path integral identity (note the implemented time-transformation) 20) and d E (2) (q ′′ , q ′ ) must be taken in elliptic coordinates. The functions me ν (η, h 2 ) and Me (1) ν (ξ, h 2 ), are mutually determined through the separation parameter λ = λ ν (h 2 ) yielding a countable set of numbers ν ∈ Λ.…”

Section: Cylindrical Coordinatesmentioning

confidence: 99%

Path Integration and Separation of Variables in Spaces of Constant Curvature in Two and Three Dimensions

Grosche

1994

Fortschr. Phys.

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In this paper path integration in two‐ and three‐dimensional spaces of constant curvature is discussed: i.e. the flat spaces R2 and R3, the two‐ and three‐dimensional sphere and the two‐ and three‐dimensional pseudosphere. We are going to discuss all coordinates systems where the Laplace operator admits separation of variables. In all of them the path integral formulation will be stated, however in most of them an explicit solution in terms of the spectral expansion can be given only on a formal level. What can be stated in all cases, are the propagator and the corresponding Green function, respectively, depending on the invariant distance which is a coordinate independent quantity. This property gives rise to numerous identities connecting the corresponding path integral representations and propagators in various coordinate systems with each other.

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The canonical formalism and path integrals in curved spaces

Cited by 17 publications

References 21 publications

Path Integral Discussion for Smorodinsky-Winternitz Potentials: II. The Two- and Three-Dimensional Sphere

Path Integral Discussion for Smorodinsky-Winternitz Potentials: II. The Two- and Three-Dimensional Sphere

Unfinished history and paradoxes of quantum potential. I. Non-relativistic origin, history and paradoxes

Path Integration and Separation of Variables in Spaces of Constant Curvature in Two and Three Dimensions

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