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:quant-ph/0612002 · AIP conference proceedings

Algebraic Quantum Mechanics and Pregeometry

David Böhm

Published Jan 1, 2006• 4 citations
arXivHTMLPDF

Abstract

We discuss the relation between the q‐number approach to quantum mechanics suggested by Dirac and the notion of “pregeometry” introduced by Wheeler. By associating the q‐numbers with the elements of an algebra and regarding the primitive idempotents as “generalized points”, we suggest an approach that may make it possible to dispense with an a priori given space‐time manifold. In this approach the algebra itself would carry the symmetries of translation, rotation, etc. Our suggestion is illustrated in a preliminary way by using a particular generalized Clifford algebra proposed originally by Weyl, which approaches the ordinary Heisenberg algebra a suitable limit. We thus obtain a certain insight into how quantum mechanics may be regarded as a purely algebraic theory, provided that we further introduce a new set of “neighbourhood operators”, which remove an important kind of arbitrariness that has thus far been present in the attempt to treat quantum mechanics solely in terms of a Heisenberg algebra.

DOI
10.1063/1.2158735

Related papers

Algebraic Quantum Mechanics and Pregeometry

David Böhm, Davies, Philip, Basil J. Hiley

We discuss the relation between the q-number approach to quantum mechanics suggested by Dirac and the notion of "pregeometry" introduced by Wheeler. By associating the q-numbers with the elements of an algebra and regarding the primitive idempotents as "generalized points" we suggest an approach that may make it possible to dispense with an a priori given space manifold. In this approach the algebra itself would carry the symmetries of translation, rotation, etc. Our suggestion is illustrated in a preliminary way by using a particular generalized Clifford algebra proposed originally by Weyl, which approaches the ordinary Heisenberg algebra a suitable limit. We thus obtain a certain insight into how quantum mechanics may be regarded as a purely algebraic theory, provided that we further introduce a new set of "neighbourhood operators", which remove an important kind of arbitrariness that has thus far been present in the attempt to treat quantum mechanics solely in terms of a Heisenberg algebra.

•Jan 1, 2006•11 citations

Relativity quantum mechanics with an application to compton scattering

P. A. M. Dirac

Abstract The new quantum mechanics, introduced by Heisenberg and since developed from different points of view by various authors, takes its simplest form if one assumes merely that the dynamical variables are numbers of a special type (called q-numbers to distinguish them from ordinary or c-numbers) that obey all the ordinary algebraic laws except the commutative law of multiplication, and satisfy instead of this the relations qrqs – qsqr =0, prps – pspr = 0 } qrqs – psqr = 0 (r ≠ s) or ih(r = s) where the p's and q's are a set of canonical variables and h is a c-number euqal to (2π)-1 times the usual Planck’s constant. Equations (1) may be regarded as replacing the commutative law of the classical theory, as one can, with their help, build up a complete algebraic theory of quantities that are analytic functions of a set of canonical variables. Further, it may easily be seen that the quantity [x, y] defined by xy – yz = ih [x, y] is completely analogous to the Poisson bracket of the classical theory. By means of this analogy the whole of the classical dynamical theory, in so far as it can be expressed in terms of P. B.’s instead of differential coefficients, may be taken over immediately into the quantum theory.

Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character•Jun 2, 1926•145 citations

Dirac, Bohm and the Algebraic Approach

Basil J. Hiley, Glen Dennis

In this paper we show how Dirac, in 1947, anticipated the Bohm approach using an argument based on what is now called the Heisenberg picture. From a detailed examination of these ideas, we show that the role played by the Dirac standard ket is equivalent to the introduction and use of the idempotent in the orthogonal and symplectic Clifford algebras. This formalism is then used to show that the so-called `Bohm trajectories' are the average of an ensemble of individual stochastic Feynman paths. Since the Bohm approach can be simply reduced to classical mechanics, the algebraic formalism presented here provides a natural way to relate quantum mechanics to classical mechanics without the need for decoherence. We show that this approach suggests an underlying fractal space-time of the type discussed by Nottale.

arXiv:1901.01979•Jan 5, 2019•0 citations

A Review of The Algebraic Approaches to Quantum Mechanics. Appraisals on Their Theoretical Relevance

Antonino Drago

I review the various algebraic foundations of quantum mechanics. They have been suggested since the birth of this theory till up to last year. They are the following ones: Heisenberg-Born-Jordan (1925), Weyl (1928), Dirac (1930), von Neumann (1936), Segal (1947), T.F. Jordan (1986), Morchio and Strocchi (2009) and Buchholz and Fregenhagen (2019). Three cases are stressed: 1) the misinterpretation of Dirac foundation; 2) von Neumann conversion from the analytic approach of Hilbert space to the algebraic approach of the rings of operators; 3) the recent foundation of quantum mechanics upon the algebra of perturbation Lagrangians. Moreover, historical considerations on the go-and-stop path performed by the algebraic approach in the history of QM are offered. The level of formalism has increased from the mere introduction of matrices till up to group theory and C*-algebras. But there was no progress in approaching closer the foundations of physics; therefore the problem of discovering an algebraic formulation of QM organized as a problem-based theory and making use of no more than constructive mathematics is open.

arXiv:2102.00861•Jan 29, 2021•0 citations

The algebraic foundations of Dirac notation and path integrals are examined. It is shown that a formulation of quantum mechanics on the basis of the theory of C* alge- bras permits these questions to be analyzed from a common standpoint.

O. G. Mishnev

In connection with the existence of irreducible representations of commutation relations which are not equivalent from the unitary standpoint, with a number of degrees of freedom 3N = ~, it is pertinent to study the algebraic foundations of quantum mechanics for the case 3N < ~, i.e., under the applicability conditions of von Neumann's theorem of the unitary equivalence of any representations of commutation relations to the direct sum of Schr~dinger relations. One reason for this situation is that the discussion often involves the limit N--~ V-+~ N/V==-!,so that Von Neumann's theorem does not apply. It is thus interesting to examine the process by which a Hilbert space arises from purely algebraic postulates, without direct appeal to the GNS construction. As we will show below, this question is intimately related to the basis for Dirac notation; we will also see that the well-known representation of the transition amplitude <xt x'l'L~ by a path integral and thus the Feynman construction of quantum mechanics can be given an algebraic footing. These aspects of our results are closely related to a study by Berezin [i], who does not work from the C* algebra of R generated by commutation relations, as we will do, but from representations of the algebras of functions defined in a phase space. We will not attempt to treat the problem in its most general form in this paper. The proofs will be given for some simple examples, but the necessary generalizations are easily made. i. We make a transition from the Weyl basis in the C* algebra of R, X(q p) to the coordiA hate basis D(~ y) and we write the latter out explicitly (h = i)

•Jan 1, 1982•0 citations

The Geometrical Meaning of Spinors Lights the Way to Make Sense of Quantum Mechanics

G. Coddens

This paper aims at explaining that a key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R3. In the three-dimensional rotation group, the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding that is acquired to derive the free-space Dirac equation from scratch, proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine’s statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further, we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.

Symmetry•Apr 12, 2021•4 citations
We discuss the relation between the q‐number approach to quantum mechanics suggested by Dirac and the notion of “pregeometry” introduced by Wheeler. By associating the q‐numbers with the elements of an algebra and regarding the primitive idempotents as “generalized points”, we suggest an approach that may make it possible to dispense with an a priori given space‐time manifold. In this approach the algebra itself would carry the symmetries of translation, rotation, etc. Our suggestion is illustrated in a preliminary way by using a particular generalized Clifford algebra proposed originally by Weyl, which approaches the ordinary Heisenberg algebra a suitable limit. We thus obtain a certain insight into how quantum mechanics may be regarded as a purely algebraic theory, provided that we further introduce a new set of “neighbourhood operators”, which remove an important kind of arbitrariness that has thus far been present in the attempt to treat quantum mechanics solely in terms of a Heisenberg algebra.
We discuss the relation between the q-number approach to quantum mechanics suggested by Dirac and the notion of "pregeometry" introduced by Wheeler. By associating the q-numbers with the elements of an algebra and regarding the primitive idempotents as "generalized points" we suggest an approach that may make it possible to dispense with an a priori given space manifold. In this approach the algebra itself would carry the symmetries of translation, rotation, etc. Our suggestion is illustrated in a preliminary way by using a particular generalized Clifford algebra proposed originally by Weyl, which approaches the ordinary Heisenberg algebra a suitable limit. We thus obtain a certain insight into how quantum mechanics may be regarded as a purely algebraic theory, provided that we further introduce a new set of "neighbourhood operators", which remove an important kind of arbitrariness that has thus far been present in the attempt to treat quantum mechanics solely in terms of a Heisenberg algebra.
Abstract The new quantum mechanics, introduced by Heisenberg and since developed from different points of view by various authors, takes its simplest form if one assumes merely that the dynamical variables are numbers of a special type (called q-numbers to distinguish them from ordinary or c-numbers) that obey all the ordinary algebraic laws except the commutative law of multiplication, and satisfy instead of this the relations qrqs – qsqr =0, prps – pspr = 0 } qrqs – psqr = 0 (r ≠ s) or ih(r = s) where the p's and q's are a set of canonical variables and h is a c-number euqal to (2π)-1 times the usual Planck’s constant. Equations (1) may be regarded as replacing the commutative law of the classical theory, as one can, with their help, build up a complete algebraic theory of quantities that are analytic functions of a set of canonical variables. Further, it may easily be seen that the quantity [x, y] defined by xy – yz = ih [x, y] is completely analogous to the Poisson bracket of the classical theory. By means of this analogy the whole of the classical dynamical theory, in so far as it can be expressed in terms of P. B.’s instead of differential coefficients, may be taken over immediately into the quantum theory.
In this paper we show how Dirac, in 1947, anticipated the Bohm approach using an argument based on what is now called the Heisenberg picture. From a detailed examination of these ideas, we show that the role played by the Dirac standard ket is equivalent to the introduction and use of the idempotent in the orthogonal and symplectic Clifford algebras. This formalism is then used to show that the so-called `Bohm trajectories' are the average of an ensemble of individual stochastic Feynman paths. Since the Bohm approach can be simply reduced to classical mechanics, the algebraic formalism presented here provides a natural way to relate quantum mechanics to classical mechanics without the need for decoherence. We show that this approach suggests an underlying fractal space-time of the type discussed by Nottale.
I review the various algebraic foundations of quantum mechanics. They have been suggested since the birth of this theory till up to last year. They are the following ones: Heisenberg-Born-Jordan (1925), Weyl (1928), Dirac (1930), von Neumann (1936), Segal (1947), T.F. Jordan (1986), Morchio and Strocchi (2009) and Buchholz and Fregenhagen (2019). Three cases are stressed: 1) the misinterpretation of Dirac foundation; 2) von Neumann conversion from the analytic approach of Hilbert space to the algebraic approach of the rings of operators; 3) the recent foundation of quantum mechanics upon the algebra of perturbation Lagrangians. Moreover, historical considerations on the go-and-stop path performed by the algebraic approach in the history of QM are offered. The level of formalism has increased from the mere introduction of matrices till up to group theory and C*-algebras. But there was no progress in approaching closer the foundations of physics; therefore the problem of discovering an algebraic formulation of QM organized as a problem-based theory and making use of no more than constructive mathematics is open.
In connection with the existence of irreducible representations of commutation relations which are not equivalent from the unitary standpoint, with a number of degrees of freedom 3N = ~, it is pertinent to study the algebraic foundations of quantum mechanics for the case 3N < ~, i.e., under the applicability conditions of von Neumann's theorem of the unitary equivalence of any representations of commutation relations to the direct sum of Schr~dinger relations. One reason for this situation is that the discussion often involves the limit N--~ V-+~ N/V==-!,so that Von Neumann's theorem does not apply. It is thus interesting to examine the process by which a Hilbert space arises from purely algebraic postulates, without direct appeal to the GNS construction. As we will show below, this question is intimately related to the basis for Dirac notation; we will also see that the well-known representation of the transition amplitude <xt x'l'L~ by a path integral and thus the Feynman construction of quantum mechanics can be given an algebraic footing. These aspects of our results are closely related to a study by Berezin [i], who does not work from the C* algebra of R generated by commutation relations, as we will do, but from representations of the algebras of functions defined in a phase space. We will not attempt to treat the problem in its most general form in this paper. The proofs will be given for some simple examples, but the necessary generalizations are easily made. i. We make a transition from the Weyl basis in the C* algebra of R, X(q p) to the coordiA hate basis D(~ y) and we write the latter out explicitly (h = i)
This paper aims at explaining that a key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R3. In the three-dimensional rotation group, the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding that is acquired to derive the free-space Dirac equation from scratch, proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine’s statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further, we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.