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

Q-Deformed Path Integral

Масуд Чайчиан, Demichev, A P

Published Jan 1, 1994• 22 citations
OpenAlex

Abstract

Using differential and integral calculi on the quantum plane which are invariant with respect to quantum inhomogeneous Euclidean group E(2)q , we construct path integral representation for the quantum mechanical evolution operator kernel of q-oscillator.

Related papers

Quantum and classical mechanics of q-deformed systems

Sergei V. Shabanov

Quantum and classical mechanics of a system of q-deformed bosonic oscillators are considered. The q-deformed Heisenberg-Weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables. The corresponding Hilbert space is constructed. In this approach, the deformation parameter turns out to be a function of the Planck constant, an oscillator frequency and a parameter with dimension of length. The Hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q-deformed classical theory. The phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory. The dimensional parameter introduced determines its radius. It is argued that the q-deformation can be associated with a special non-canonical transformation. The principle of least action for the classical q-deformed system is formulated. A representation of U q (m) in a space of functions on a phase space spanned by commutative coordinates is constructed.

Journal of Physics A Mathematical and General•Jun 7, 1993•21 citations

The Path Integral Formulation of Quantum Mechanics

Marcos Mariño

This chapter presents Feynman’s formulation of quantum mechanics, based on a path integral representation of the evolution operator. The chapter presents detailed examples which make it possible to understand clearly Feynman’s “sum over paths,” and it contains a complete discussion of how to calculate Gaussian path integrals. It also discusses the Euclidean version of the path integral, as well as Wick’s theorem and Feynman diagrams. Finally, it discusses instantons in quantum mechanics.

Cambridge University Press eBooks•Dec 2, 2021•0 citations

Constructing measures for path integrals

Ingrid C. Daubechies, John R. Klauder

The overcompleteness of the coherent states for the Heisenberg–Weyl group implies that many different integral kernels can be used to represent the same operator. Within such an equivalence class we construct an integral kernel to represent the quantum-mechanical evolution operator for certain dynamical systems in the form of a path integral that involves genuine (Wiener) measures on continuous phase-space paths. To achieve this goal it is necessary to employ an expression for the classical action different from the usual one.

Journal of Mathematical Physics•Oct 1, 1982•15 citations

The Feynman path integral, from the harmonic oscillator to quantum electrodynamics

Rafael Carlos Silva Tonhon

The Feynman path-integral formulation of quantum mechanics is developed. The math- ematical framework is worked out from Schrödinger's equation, illustrating the compati- bility of the two formulations. We introduce the sum over all paths, known as the kernel K, and discuss its implications for events occurring in succession. We compute the kernel for quadratic Lagrangians. Then, making use of imaginary time, we consider the discrete version of the kernel, known as the Euclidean path integral, and use it to solve the (quan- tum) harmonic and anharmonic oscillators through numerical Monte Carlo simulations. Lastly, we use the framework to quantize the electromagnetic field, as a motivation to discuss perturbation theory via Feynman diagrams.

•Jan 1, 2022•0 citations

Path Integrals in Quantum Mechanics

Dennis V. Perepelitsa

We present the path integral formulation of quantum mechanics and demon-strate its equivalence to the Schrödinger picture. We apply the method to the free particle and quantum harmonic oscillator, investigate the Euclidean path integral, and discuss other applications. 1

World Scientific lecture notes in physics•Dec 1, 1990•17 citations

Euclidean path integrals and quantum mechanics (QM)

Jean Zinn‐Justin

Abstract Functional integrals are basic tools to study first quantum mechanics (QM), and quantum field theory (QFT). The path integral formulation of QM is well suited to the study of systems with an arbitrary number of degrees of freedom. It makes a smooth transition between nonrelativistic QM and QFT possible. The Euclidean functional integral also emphasizes the deep connection between QFT and the statistical physics of systems with short-range interactions near a continuous phase transition. The path integral representation of the matrix elements of the quantum statistical operator e-β H for Hamiltonians of the simple separable form p2/2m +V(q) is derived. To the path integral corresponds a functional measure and expectation values called correlation functions, which are generalized moments, and related to quantum observables, after an analytic continuation in time. The path integral corresponding to the Euclidean action of a harmonic oscillator, to which is added a time-dependent external force, is calculated explicitly. The result is used to generate Gaussian correlation functions and also to reduce the evaluation of path integrals to perturbation theory. The path integral also provides a convenient tool to derive semi-classical approximations.

•Apr 15, 2021•0 citations
Quantum and classical mechanics of a system of q-deformed bosonic oscillators are considered. The q-deformed Heisenberg-Weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables. The corresponding Hilbert space is constructed. In this approach, the deformation parameter turns out to be a function of the Planck constant, an oscillator frequency and a parameter with dimension of length. The Hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q-deformed classical theory. The phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory. The dimensional parameter introduced determines its radius. It is argued that the q-deformation can be associated with a special non-canonical transformation. The principle of least action for the classical q-deformed system is formulated. A representation of U q (m) in a space of functions on a phase space spanned by commutative coordinates is constructed.
This chapter presents Feynman’s formulation of quantum mechanics, based on a path integral representation of the evolution operator. The chapter presents detailed examples which make it possible to understand clearly Feynman’s “sum over paths,” and it contains a complete discussion of how to calculate Gaussian path integrals. It also discusses the Euclidean version of the path integral, as well as Wick’s theorem and Feynman diagrams. Finally, it discusses instantons in quantum mechanics.
The overcompleteness of the coherent states for the Heisenberg–Weyl group implies that many different integral kernels can be used to represent the same operator. Within such an equivalence class we construct an integral kernel to represent the quantum-mechanical evolution operator for certain dynamical systems in the form of a path integral that involves genuine (Wiener) measures on continuous phase-space paths. To achieve this goal it is necessary to employ an expression for the classical action different from the usual one.
The Feynman path-integral formulation of quantum mechanics is developed. The math- ematical framework is worked out from Schrödinger's equation, illustrating the compati- bility of the two formulations. We introduce the sum over all paths, known as the kernel K, and discuss its implications for events occurring in succession. We compute the kernel for quadratic Lagrangians. Then, making use of imaginary time, we consider the discrete version of the kernel, known as the Euclidean path integral, and use it to solve the (quan- tum) harmonic and anharmonic oscillators through numerical Monte Carlo simulations. Lastly, we use the framework to quantize the electromagnetic field, as a motivation to discuss perturbation theory via Feynman diagrams.
Abstract Functional integrals are basic tools to study first quantum mechanics (QM), and quantum field theory (QFT). The path integral formulation of QM is well suited to the study of systems with an arbitrary number of degrees of freedom. It makes a smooth transition between nonrelativistic QM and QFT possible. The Euclidean functional integral also emphasizes the deep connection between QFT and the statistical physics of systems with short-range interactions near a continuous phase transition. The path integral representation of the matrix elements of the quantum statistical operator e-β H for Hamiltonians of the simple separable form p2/2m +V(q) is derived. To the path integral corresponds a functional measure and expectation values called correlation functions, which are generalized moments, and related to quantum observables, after an analytic continuation in time. The path integral corresponding to the Euclidean action of a harmonic oscillator, to which is added a time-dependent external force, is calculated explicitly. The result is used to generate Gaussian correlation functions and also to reduce the evaluation of path integrals to perturbation theory. The path integral also provides a convenient tool to derive semi-classical approximations.