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

Reparametrization Invariance of Path Integrals

Kleinert, H, Chervyakov, A

Published Jan 1, 1999• 29 citations
OpenAlex

Abstract

We demonstrate the reparametrization invariance of perturbatively defined one-dimensional functional integrals up to the three-loop level for a path integral of a quantum-mechanical point particle in a box. We exhibit the origin of the failure of earlier authors to establish reparametrization invariance which led them to introduce, superfluously, a compensating potential depending on the connection of the coordinate system. We show that problems with invariance are absent by defining path integrals as the $\\epsilon \\to 0$ limits of $1+ \\epsilon$-dimensional functional integrals.

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 for two‐ and three‐dimensional δ‐function perturbations

Christian Grosche

Abstract The incorporation of two‐ and three‐dimensional Δ‐function perturbations into the path integral formalism is discussed. Contrary to the one‐dimensional case, a regularization prescription is needed due to the divergence of the Green function corresponding to a potential V, G(v) (x,y;E) (x, y ϵ ℝ2, ℝ3) for x→y. The known procedure to define proper self‐adjoint extensions for Hamiltonians with point‐interactions can be exploited to define the incorporation of δ‐function perturbations in the path integral. Several examples illustrate the formalism.

arXiv:hep-th/9308082•Annalen der Physik•Jan 1, 1994•13 citations

Renormalized path integral for the two-dimensional δ-function interaction

Horacio E. Camblong, CARLOS R. ORDÓÑEZ

A path-integral approach for \ensuremath{\delta}-function potentials is presented. Particular attention is paid to the two-dimensional case, which illustrates the realization of a quantum anomaly for a scale-invariant problem in quantum mechanics. Our treatment is based on an infinite summation of perturbation theory that captures the nonperturbative nature of the \ensuremath{\delta}-function bound state. The well-known singular character of the two-dimensional \ensuremath{\delta}-function potential is dealt with by considering the renormalized path integral resulting from a variety of schemes: dimensional, momentum-cutoff, and real-space regularization. Moreover, compatibility of the bound-state and scattering sectors is shown.

arXiv:hep-th/0110176•Physical Review A•May 17, 2002•30 citations

Reparametrization-invariant formulation of classical mechanics and the Schrödinger equation

Alexei A. Deriglazov, B. F. Rizzuti

Any classical-mechanics system can be formulated in reparametrization-invariant form. That is, we use the parametric representation for the trajectories, x=x(τ) and t=t(τ) instead of x=x(t). We discuss the quantization rules in this formulation and show that some of the rules become clearer. In particular, both the temporal and the spatial coordinates are subject to quantization, and the canonical Hamiltonian in the reparametrization-invariant formulation is proportional to H̃=pt+H, where H is the usual Hamiltonian and pt is the momentum conjugate to the variable t. Due to reparametrization invariance, H̃ vanishes for any solution, and hence the corresponding quantum-mechanical operator has the property H̃∧Ψ=0, which is the time-dependent Schrödinger equation, iℏ∂tΨ=H∧Ψ. We discuss the quantum mechanics of a relativistic particle as an example.

arXiv:1105.0313•American Journal of Physics•Jul 26, 2011•20 citations

Path Integral Invariance under Point Transformations

Andres Jordan, Matias Libedinsky

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.

arXiv:hep-th/9703173•Mar 25, 1997•52 citations

Hamiltonian path integrals

Прохоров Лев Васильевич, Sergei V. Shabanov

Introduction Preliminary remarks The path integral formalism is one of the most important tools in modern theoretical physics. Integration in functional spaces was considered for the first time by Norbert Wiener in his studies of diffusion processes [60, 61]. The corresponding integration measure is now known as the Wiener measure, i.e. the so-called Gaussian path integrals were studied. In quantum mechanics, functional integrals were introduced by R. Feynmann [62, 63] who was inspired by important observations due to P. A. M. Dirac concerning a relation between wave function and classical action [64, 65]. In modern physics the path integral formalism is applied not only to developing a quantum description for systems with infinitely many degrees of freedom as, for example, in quantum field theories or condensed matter physics, but also to ordinary quantum mechanical problems. Moreover there is a tendency to reformulate standard quantum mechanical problems (such as a particle in a “box” or potential well [66, 67], or a particle in the Coulomb field [68]) within the framework of the path integral method. The reason for doing this is rather obvious: physicists develop a new mathematical tool, a new language and, in order to comprehend all of its subtleties, it is necessary to investigate as many concrete examples as possible. Use of the path integral formalism is widespread in quantum field theory because it is the most adequate mathematical tool for a wide range of problems which allows, not only for elegant formulation of problems, but also for a great deal of simplification of their theoretical analysis.

Cambridge University Press eBooks•Sep 22, 2011•5 citations
We demonstrate the reparametrization invariance of perturbatively defined one-dimensional functional integrals up to the three-loop level for a path integral of a quantum-mechanical point particle in a box. We exhibit the origin of the failure of earlier authors to establish reparametrization invariance which led them to introduce, superfluously, a compensating potential depending on the connection of the coordinate system. We show that problems with invariance are absent by defining path integrals as the epsilonto0\\epsilon \\to 0epsilonto0 limits of 1+epsilon1+ \\epsilon1+epsilon-dimensional functional integrals.
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.
Abstract The incorporation of two‐ and three‐dimensional Δ‐function perturbations into the path integral formalism is discussed. Contrary to the one‐dimensional case, a regularization prescription is needed due to the divergence of the Green function corresponding to a potential V, G(v) (x,y;E) (x, y ϵ ℝ2, ℝ3) for x→y. The known procedure to define proper self‐adjoint extensions for Hamiltonians with point‐interactions can be exploited to define the incorporation of δ‐function perturbations in the path integral. Several examples illustrate the formalism.
A path-integral approach for \ensuremath{\delta}-function potentials is presented. Particular attention is paid to the two-dimensional case, which illustrates the realization of a quantum anomaly for a scale-invariant problem in quantum mechanics. Our treatment is based on an infinite summation of perturbation theory that captures the nonperturbative nature of the \ensuremath{\delta}-function bound state. The well-known singular character of the two-dimensional \ensuremath{\delta}-function potential is dealt with by considering the renormalized path integral resulting from a variety of schemes: dimensional, momentum-cutoff, and real-space regularization. Moreover, compatibility of the bound-state and scattering sectors is shown.
Any classical-mechanics system can be formulated in reparametrization-invariant form. That is, we use the parametric representation for the trajectories, x=x(τ) and t=t(τ) instead of x=x(t). We discuss the quantization rules in this formulation and show that some of the rules become clearer. In particular, both the temporal and the spatial coordinates are subject to quantization, and the canonical Hamiltonian in the reparametrization-invariant formulation is proportional to H̃=pt+H, where H is the usual Hamiltonian and pt is the momentum conjugate to the variable t. Due to reparametrization invariance, H̃ vanishes for any solution, and hence the corresponding quantum-mechanical operator has the property H̃∧Ψ=0, which is the time-dependent Schrödinger equation, iℏ∂tΨ=H∧Ψ. We discuss the quantum mechanics of a relativistic particle as an example.
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.
Introduction Preliminary remarks The path integral formalism is one of the most important tools in modern theoretical physics. Integration in functional spaces was considered for the first time by Norbert Wiener in his studies of diffusion processes [60, 61]. The corresponding integration measure is now known as the Wiener measure, i.e. the so-called Gaussian path integrals were studied. In quantum mechanics, functional integrals were introduced by R. Feynmann [62, 63] who was inspired by important observations due to P. A. M. Dirac concerning a relation between wave function and classical action [64, 65]. In modern physics the path integral formalism is applied not only to developing a quantum description for systems with infinitely many degrees of freedom as, for example, in quantum field theories or condensed matter physics, but also to ordinary quantum mechanical problems. Moreover there is a tendency to reformulate standard quantum mechanical problems (such as a particle in a “box” or potential well [66, 67], or a particle in the Coulomb field [68]) within the framework of the path integral method. The reason for doing this is rather obvious: physicists develop a new mathematical tool, a new language and, in order to comprehend all of its subtleties, it is necessary to investigate as many concrete examples as possible. Use of the path integral formalism is widespread in quantum field theory because it is the most adequate mathematical tool for a wide range of problems which allows, not only for elegant formulation of problems, but also for a great deal of simplification of their theoretical analysis.