Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:hep-th/9703173

Path Integral Invariance under Point Transformations

Andres Jordan, Matias Libedinsky

Published Mar 25, 1997• 52 citations
arXivHTMLPDF

Abstract

We give here a covariant definition of the path integral formalism for the Lagrangian, which leaves a freedom to choose anyone of many possible quantum systems that correspond to the same classical limit without adding new potential terms nor searching for a strange measure, but using as a framework the geometry of the spaces considered. We focus our attention on the set of paths used to join succesive points in the discretization if the time-slicing definition is used to calculate the integral.If this set of paths is not preserved when performing a point transformation, the integral may change. The reasons for this are geometrically explained. Explicit calculation of the Kernel in polar coordinates is made, yielding the same system as in Cartesian coordinates.

DOI
10.48550/arxiv.hep-th/9703173

Related papers

Path integral invariance under point canonical transformations

Andres Jordan, Matias Libedinsky

It is often stated in the literature on path integrals that naive changes of coordinates might in general give inequivalent theories. The discrepancy is presumably connected to subtleties in the discretization, to the stochastic nature of quantum paths, or to operator order ambiguities in the canonical quantization. Here we argue that in order to define a path integral one needs not only a Lagrangian but also a set of paths that join succesive points in the discretized paths. If the set of paths is maintained when performing a point canonical transformation the path integral does not change. We explicitly show this with the calculation of the free particle kernel in polar coordinates.

•Mar 25, 1997•1 citation

Path integrals from classical momentum paths

John J. Hegseth

The path integral formulation of quantum mechanics constructs the propagator by evaluating the action S for all classical paths in coordinate space. A corresponding momentum path integral may also be defined through Fourier transforms in the endpoints. Although these momentum path integrals are especially simple for several special cases, no one has, to my knowledge, ever formally constructed them from all classical paths in momentum space. I show that this is possible because there exists another classical mechanics based on an alternate classical action R. Hamilton's Canonical equations result from a variational principle in both S and R. S uses fixed beginning and ending spatial points while R uses fixed beginning and ending momentum points. This alternative action's classical mechanics also includes a Hamilton-Jacobi equation. I also present some important points concerning the beginning and ending conditions on the action necessary to apply a Canonical transformation. These properties explain the failure of the Canonical transformation in the phase space path integral. It follows that a path integral may be constructed from classical position paths using S in the coordinate representation or from classical momentum paths using R in the momentum representation. Several example calculations are presented that illustrate the simplifications and practical advantages made possible by this broader view of the path integral. In particular, the normalized amplitude for a free particle is found without using the Schrodinger equation, the internal spin degree of freedom is simply and naturally derived, and the simple harmonic oscillator is calculated.

arXiv:quant-ph/0403005•Feb 29, 2004•3 citations

Time-Slicing Path-integral in Curved Space

Mingnan Ding, Xiangjun Xing

Path integrals constitute powerful representations for both quantum and stochastic dynamics. Yet despite many decades of intensive studies, there is no consensus on how to formulate them for dynamics in curved space, or how to make them covariant with respect to nonlinear transform of variables (NTV). In this work, we construct a rigorous and covariant formulation of time-slicing path integrals for dynamics in curved space. We first establish a rigorous criterion for equivalence of time-slice Green's function (TSGF) in the continuum limit (Lemma 1). This implies the existence of infinitely many equivalent representations for time-slicing path integral. We then show that, for any dynamics with second order generator, all time-slice actions are equivalent to a Gaussian (Lemma 2). We further construct a continuous family of equivalent path-integral actions parameterized by an interpolation parameter α∈[0,1] (Lemma 3). The action generically contains term linear in Δx , whose concrete form depends on α . Finally we also establish the covariance of our path-integral formalism, by demonstrating how the action transforms under NTV. The α=0 representation of time-slice action is particularly convenient because it is Gaussian and transforms as a scalar, as long as Δx transforms according to Ito's formula .

arXiv:2107.14562•Quantum•Apr 21, 2022•8 citations

Path Integral Quantization and Riemannian-Symplectic Manifolds

Sergei V. Shabanov, John R. Klauder

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. 1. Most quantization formulations lead to correct results only when the system in question is expressed in one of a family of special (Cartesian) coordinate systems [1]. However, for Hamiltonian systems on a generic symplectic manifold, the canonical variables do not admit such special coordinates. Moreover classical Hamiltonian dynamics exhibits covariance under general coordinate transformations on the symplectic manifold, and, in this sense, is coordinate-free. This covariance is lost upon a formal phase-space path integral quantization. The reason is that the Hamiltonian action involves terms linear in the time derivatives and, therefore, its exponential (even in Euclidean time) does not induce a

•Jan 1, 1998•6 citations

Selecta from a Life-Long Obsession with Path Integrals

John R. Klauder, Christopher Casenas Bernido, M. Victoria Carpio-Bernido

The definition and interpretation of canonical, phase space path integrals has evolved over many years to achieve a form that now admits a correct and rigorous formulation, which is also covariant under canonical coordinate transformations. Such formulations involve coherent state representations, which, in their modern version, were originally introduced as an alternative tool to construct phase space path integrals. Moreover, coherent state representations lead to physical interpretations that are more natural than those afforded by more traditional representations. Suitable continuous time regularization procedures lead to a covariant phase space path integral formulation that greatly clarifies the vague phrase that canonical quantization requires Cartesian coordinates.

AIP conference proceedings•Jan 1, 2008•0 citations

The unitary transformation of the path-integral measure

J. Manjavidze

The aim of the article is to show how a coordinate transformation can be applied to the path-integral formalism. For this purpose the unitary definition of the quantum measure, which guarantees the conservation of total probability, is offered. As the examples, the phase space transformation to the canonically conjugate pare $(energy, time)$ and the transformation to the cylindrical coordinates are shown. The transformations of the path-integral measure looks classically but they can not be deduced from naive transformations of quantum trajectories.

arXiv:quant-ph/9507003•Jul 6, 1995•1 citation
We give here a covariant definition of the path integral formalism for the Lagrangian, which leaves a freedom to choose anyone of many possible quantum systems that correspond to the same classical limit without adding new potential terms nor searching for a strange measure, but using as a framework the geometry of the spaces considered. We focus our attention on the set of paths used to join succesive points in the discretization if the time-slicing definition is used to calculate the integral.If this set of paths is not preserved when performing a point transformation, the integral may change. The reasons for this are geometrically explained. Explicit calculation of the Kernel in polar coordinates is made, yielding the same system as in Cartesian coordinates.
It is often stated in the literature on path integrals that naive changes of coordinates might in general give inequivalent theories. The discrepancy is presumably connected to subtleties in the discretization, to the stochastic nature of quantum paths, or to operator order ambiguities in the canonical quantization. Here we argue that in order to define a path integral one needs not only a Lagrangian but also a set of paths that join succesive points in the discretized paths. If the set of paths is maintained when performing a point canonical transformation the path integral does not change. We explicitly show this with the calculation of the free particle kernel in polar coordinates.
The path integral formulation of quantum mechanics constructs the propagator by evaluating the action S for all classical paths in coordinate space. A corresponding momentum path integral may also be defined through Fourier transforms in the endpoints. Although these momentum path integrals are especially simple for several special cases, no one has, to my knowledge, ever formally constructed them from all classical paths in momentum space. I show that this is possible because there exists another classical mechanics based on an alternate classical action R. Hamilton's Canonical equations result from a variational principle in both S and R. S uses fixed beginning and ending spatial points while R uses fixed beginning and ending momentum points. This alternative action's classical mechanics also includes a Hamilton-Jacobi equation. I also present some important points concerning the beginning and ending conditions on the action necessary to apply a Canonical transformation. These properties explain the failure of the Canonical transformation in the phase space path integral. It follows that a path integral may be constructed from classical position paths using S in the coordinate representation or from classical momentum paths using R in the momentum representation. Several example calculations are presented that illustrate the simplifications and practical advantages made possible by this broader view of the path integral. In particular, the normalized amplitude for a free particle is found without using the Schrodinger equation, the internal spin degree of freedom is simply and naturally derived, and the simple harmonic oscillator is calculated.
Path integrals constitute powerful representations for both quantum and stochastic dynamics. Yet despite many decades of intensive studies, there is no consensus on how to formulate them for dynamics in curved space, or how to make them covariant with respect to nonlinear transform of variables (NTV). In this work, we construct a rigorous and covariant formulation of time-slicing path integrals for dynamics in curved space. We first establish a rigorous criterion for equivalence of time-slice Green's function (TSGF) in the continuum limit (Lemma 1). This implies the existence of infinitely many equivalent representations for time-slicing path integral. We then show that, for any dynamics with second order generator, all time-slice actions are equivalent to a Gaussian (Lemma 2). We further construct a continuous family of equivalent path-integral actions parameterized by an interpolation parameter α∈[0,1] (Lemma 3). The action generically contains term linear in Δx , whose concrete form depends on α . Finally we also establish the covariance of our path-integral formalism, by demonstrating how the action transforms under NTV. The α=0 representation of time-slice action is particularly convenient because it is Gaussian and transforms as a scalar, as long as Δx transforms according to Ito's formula .
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. 1. Most quantization formulations lead to correct results only when the system in question is expressed in one of a family of special (Cartesian) coordinate systems [1]. However, for Hamiltonian systems on a generic symplectic manifold, the canonical variables do not admit such special coordinates. Moreover classical Hamiltonian dynamics exhibits covariance under general coordinate transformations on the symplectic manifold, and, in this sense, is coordinate-free. This covariance is lost upon a formal phase-space path integral quantization. The reason is that the Hamiltonian action involves terms linear in the time derivatives and, therefore, its exponential (even in Euclidean time) does not induce a
The definition and interpretation of canonical, phase space path integrals has evolved over many years to achieve a form that now admits a correct and rigorous formulation, which is also covariant under canonical coordinate transformations. Such formulations involve coherent state representations, which, in their modern version, were originally introduced as an alternative tool to construct phase space path integrals. Moreover, coherent state representations lead to physical interpretations that are more natural than those afforded by more traditional representations. Suitable continuous time regularization procedures lead to a covariant phase space path integral formulation that greatly clarifies the vague phrase that canonical quantization requires Cartesian coordinates.
The aim of the article is to show how a coordinate transformation can be applied to the path-integral formalism. For this purpose the unitary definition of the quantum measure, which guarantees the conservation of total probability, is offered. As the examples, the phase space transformation to the canonically conjugate pare (energy,time)(energy, time)(energy,time) and the transformation to the cylindrical coordinates are shown. The transformations of the path-integral measure looks classically but they can not be deduced from naive transformations of quantum trajectories.