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:2109.05756 · Modern Physics Letters A

Feynman’s propagator in Schwinger’s picture of Quantum Mechanics

Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, G. Marmo, Luca Schiavone, A. Zampini

Published Aug 30, 2021• 9 citations
arXivHTMLPDF

Abstract

A novel derivation of Feynman’s sum-over-histories construction of the quantum propagator using the groupoidal description of Schwinger picture of Quantum Mechanics is presented. It is shown that such construction corresponds to the GNS representation of a natural family of states called Dirac–Feynman–Schwinger (DFS) states. Such states are obtained from a q-Lagrangian function [Formula: see text] on the groupoid of configurations of the system. The groupoid of histories of the system is constructed and the q-Lagrangian [Formula: see text] allows us to define a DFS state on the algebra of the groupoid. The particular instance of the groupoid of pairs of a Riemannian manifold serves to illustrate Feynman’s original derivation of the propagator for a point particle described by a classical Lagrangian L.

DOI
10.1142/s021773232150187x

Related papers

A quantum route to the classical Lagrangian formalism

Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort +3 more

Using the recently developed groupoidal description of Schwinger’s picture of Quantum Mechanics, a new approach to Dirac’s fundamental question on the role of the Lagrangian in Quantum Mechanics is provided. It is shown that a function [Formula: see text] on the groupoid of configurations (or kinematical groupoid) of a quantum system determines a state on the von Neumann algebra of the histories of the system. This function, which we call q-Lagrangian, can be described in terms of a new function [Formula: see text] on the Lie algebroid of the theory. When the kinematical groupoid is the pair groupoid of a smooth manifold M, the quadratic expansion of [Formula: see text] will reproduce the standard Lagrangians on TM used to describe the classical dynamics of particles.

arXiv:2106.00998•Modern Physics Letters A•Apr 30, 2021•9 citations

A Lagrangian path integral approach to the qubit

Alberto Ibort, María Jiménez-Vázquez

Abstract A Lagrangian description of the qubit based on Schwinger’s picture of Quantum Mechanics that allows for a Feynman-like computation of its probability amplitudes is presented. The Lagrangian is a function on the groupoid that describes the qubit and at the same time determines a self-adjoint element on its associated algebra. Feynman’s paths are replaced by histories on the groupoid which form a groupoid again, and a simple method to compute the sum over all histories is discussed. The unitarity of the theory described in this way imposes quantization conditions on the parameters determining the Lagrangian, and some particular instances are solved completely.

arXiv:2401.13763•The European Physical Journal Plus•Apr 3, 2024•0 citations

A gentle introduction to Schwinger’s formulation of quantum mechanics: The groupoid picture

Florio M. Ciaglia, Alberto Ibort, Giuseppe Marmo

In this paper, we review Schwinger’s formulation of Quantum Mechanics and argue that the mathematical structure behind Schwinger’s “Symbolism of Atomic Measurements” is that of a groupoid. In this framework, both the Hilbert space (Schrödinger picture) and the [Formula: see text]-algebra (Heisenberg picture) of the system turn out to be derived concepts, that is, they arise from the underlying groupoid structure.

arXiv:1807.00519•Modern Physics Letters A•Jun 21, 2018•35 citations

The groupoidal picture of quantum mechanics

Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort +1 more

The basic ingredients of the groupoid interpretation of Quantum Mechanics are discussed. Starting with the motivation to consider measure groupoids as the appropriate statistical and kinematical setting to describe quantum mechanical systems, the algebra of observables of the system is constructed as the von Neumann algebra of the groupoid. The interpretation of the theory for classical systems is briefly analysed. As a final application of these ideas, Feynman's sum-over-histories interpretation of the theory is discussed in the groupoid picture.

Journal of Geometry and Physics•Jan 3, 2024•2 citations

Schwinger’s picture of quantum mechanics II: Algebras and observables

Florio M. Ciaglia, A. Ibort, G. Marmo

The kinematical foundations of Schwinger’s algebra of selective measurements were discussed in [F. M. Ciaglia, A. Ibort and G. Marmo, Schwinger’s picture of quantum mechanics I: Groupoids, To appear in IJGMMP (2019)] and, as a consequence of this, a new picture of quantum mechanics based on groupoids was proposed. In this paper, the dynamical aspects of the theory are analyzed. For that, the algebra generated by the observables, as well as the notion of state, are discussed, and the structure of the transition functions, that plays an instrumental role in Schwinger’s picture, is elucidated. A Hamiltonian picture of dynamical evolution emerges naturally, and the formalism offers a simple way to discuss the quantum-to-classical transition. Some basic examples, the qubit and the harmonic oscillator, are examined, and the relation with the standard Dirac–Schrödinger and Born–Jordan–Heisenberg pictures is discussed.

arXiv:1907.03883•International Journal of Geometric Methods in Modern Physics•Jul 4, 2019•29 citations

Schwinger’s picture of quantum mechanics I: Groupoids

Florio M. Ciaglia, A. Ibort, G. Marmo

A new picture of Quantum Mechanics based on the theory of groupoids is presented. This picture provides the mathematical background for Schwinger’s algebra of selective measurements and helps to understand its scope and eventual applications. In this first paper, the kinematical background is described using elementary notions from category theory, in particular the notion of 2-groupoids as well as their representations. Some basic results are presented, and the relation with the standard Dirac–Schrödinger and Born–Jordan–Heisenberg pictures are succinctly discussed.

arXiv:1905.12274•International Journal of Geometric Methods in Modern Physics•May 23, 2019•33 citations
A novel derivation of Feynman’s sum-over-histories construction of the quantum propagator using the groupoidal description of Schwinger picture of Quantum Mechanics is presented. It is shown that such construction corresponds to the GNS representation of a natural family of states called Dirac–Feynman–Schwinger (DFS) states. Such states are obtained from a q-Lagrangian function [Formula: see text] on the groupoid of configurations of the system. The groupoid of histories of the system is constructed and the q-Lagrangian [Formula: see text] allows us to define a DFS state on the algebra of the groupoid. The particular instance of the groupoid of pairs of a Riemannian manifold serves to illustrate Feynman’s original derivation of the propagator for a point particle described by a classical Lagrangian L.
Using the recently developed groupoidal description of Schwinger’s picture of Quantum Mechanics, a new approach to Dirac’s fundamental question on the role of the Lagrangian in Quantum Mechanics is provided. It is shown that a function [Formula: see text] on the groupoid of configurations (or kinematical groupoid) of a quantum system determines a state on the von Neumann algebra of the histories of the system. This function, which we call q-Lagrangian, can be described in terms of a new function [Formula: see text] on the Lie algebroid of the theory. When the kinematical groupoid is the pair groupoid of a smooth manifold M, the quadratic expansion of [Formula: see text] will reproduce the standard Lagrangians on TM used to describe the classical dynamics of particles.
Abstract A Lagrangian description of the qubit based on Schwinger’s picture of Quantum Mechanics that allows for a Feynman-like computation of its probability amplitudes is presented. The Lagrangian is a function on the groupoid that describes the qubit and at the same time determines a self-adjoint element on its associated algebra. Feynman’s paths are replaced by histories on the groupoid which form a groupoid again, and a simple method to compute the sum over all histories is discussed. The unitarity of the theory described in this way imposes quantization conditions on the parameters determining the Lagrangian, and some particular instances are solved completely.
The basic ingredients of the groupoid interpretation of Quantum Mechanics are discussed. Starting with the motivation to consider measure groupoids as the appropriate statistical and kinematical setting to describe quantum mechanical systems, the algebra of observables of the system is constructed as the von Neumann algebra of the groupoid. The interpretation of the theory for classical systems is briefly analysed. As a final application of these ideas, Feynman's sum-over-histories interpretation of the theory is discussed in the groupoid picture.
The kinematical foundations of Schwinger’s algebra of selective measurements were discussed in [F. M. Ciaglia, A. Ibort and G. Marmo, Schwinger’s picture of quantum mechanics I: Groupoids, To appear in IJGMMP (2019)] and, as a consequence of this, a new picture of quantum mechanics based on groupoids was proposed. In this paper, the dynamical aspects of the theory are analyzed. For that, the algebra generated by the observables, as well as the notion of state, are discussed, and the structure of the transition functions, that plays an instrumental role in Schwinger’s picture, is elucidated. A Hamiltonian picture of dynamical evolution emerges naturally, and the formalism offers a simple way to discuss the quantum-to-classical transition. Some basic examples, the qubit and the harmonic oscillator, are examined, and the relation with the standard Dirac–Schrödinger and Born–Jordan–Heisenberg pictures is discussed.
A new picture of Quantum Mechanics based on the theory of groupoids is presented. This picture provides the mathematical background for Schwinger’s algebra of selective measurements and helps to understand its scope and eventual applications. In this first paper, the kinematical background is described using elementary notions from category theory, in particular the notion of 2-groupoids as well as their representations. Some basic results are presented, and the relation with the standard Dirac–Schrödinger and Born–Jordan–Heisenberg pictures are succinctly discussed.