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

Proceedings of the Steklov Institute of Mathematics

Formulation of quantum mechanics with dynamical time

М. Г. Иванов

Published Jul 24, 2014• 4 citations
Publisher

Abstract

We consider quantum mechanics for which the system time is one of generalized coordinates. The generalized Hamiltonian has an unbounded spectrum, which allows us to introduce a Hermitian time operator. In the proposed formulation of quantum mechanics, a system time and observer’s time are introduced. The Schrödinger equation in the system time either does not hold or holds only approximately. The wave function is assumed to be square integrable with respect to all coordinates, including the system time. In some limit, this formalism reproduces standard quantum mechanics and the corresponding measurement theory.

DOI
10.1134/s0081543814040117

Related papers

The Time as an Observable in Quantum Mechanics

Taku Goto, S. Naka, K. Yamaguchi

Following homogeneous canonical formalism where the time is treated on an equal footing with the space coordinates, we propose an alternative formulation of quantum mechanics in which the time is considered as a dynamical variable. The basic equation corresponding to and being equivalent to the ordinary Schrödinger equation is obtained when we consider the space coordinates x as a parameter corresponding to the time t in the ordinary formulation. The formal solution of this equation is given and examined. By making use of this solution the scattering theory is formulated in our formalism. It is shown that observables corresponding to the time and space coordinates can be defined. A short comment on the energy-time uncertainty relation is also given.

Progress of Theoretical Physics•Jul 1, 1980•12 citations

On foundation of quantum physics

Е. А. Соловьев

Some aspects of the interpretation of quantum theory are discussed. It is emphasized that quantum theory is formulated in the Cartesian coordinate system; in other coordinates the result obtained with the help of the Hamiltonian formalism and commutator relations between “canonically conjugated” coordinate and momentum operators leads to a wrong version of quantum mechanics. The origin of time is analyzed by the example of atomic collision theory in detail; it is shown that the time-dependent Schrödinger equation is meaningless since in the high-impact-energy limit it transforms into an equation with two time-like variables. Following the Einstein-Rozen-Podolsky experiment and Bell’s inequality, the wave function is interpreted as an actual field of information in the elementary form. The concept “measurement” is also discussed.

arXiv:1001.2683•Physics of Atomic Nuclei•May 1, 2009•14 citations

The Foundations of Quantum Physics

Some aspects of the interpretation of quantum theory are discussed.It is emphasized that quantum theory is formulated in a Cartesian coordinate system; in other coordinates the result obtained with the help of the Hamiltonian formalism and commutator relations between 'canonically conjugated' coordinate and momentum operators leads to a wrong version of quantum mechanics.In this connection the Feynman integral formalism is also discussed.In this formalism the measure is not well-defined and there is no idea how to distinguish between the true version of quantum mechanics and an incorrect one; it is rather a mnemonic rule to generate perturbation series from an undefined zero order term.The origin of time is analyzed in detail by the example of atomic collisions.It is shown that the time-dependent Schrödinger equation for the closed three-body (two nuclei + electron) system has no physical meaning since in the high impact energy limit it transforms into an equation with two independent time-like variables; the time appears in the stationary Schrödinger equation as a result of extraction of a classical subsystem (two nuclei) from a closed three-body system.Following the Einstein-Rosen-Podolsky experiment and Bell's inequality the wave function is interpreted as an actual field of information in the elementary form.The relation between physics and mathematics is also discussed.

Taylor & Francis eBooks•Aug 9, 2006•1 citation

Lagrangian and phase-space formulations

Giampiero V. M. Esposito, Giuseppe Marmo, George Sudarshan

We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location x i at time t i and the final location x f at time t f . Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle. Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space. The Schwinger formulation of quantum dynamics In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the Lagrangian.

Cambridge University Press eBooks•Mar 11, 2004•0 citations

Hamiltonian formulation of quantum mechanics with semiclassical implications. II. Variational treatment

Kenneth G. Kay

It is shown that application of the time-dependent variational principle to trial wave functions of a particular form results in a description of the quantum system as a classical system consisting of two mutually interacting parts: the ordinary classical analog of the quantum system and a (generally finite) number of additional degrees of freedom. This holds out the prospect of simulating quantum dynamics by coupling the corresponding classical system to additional degrees of freedom, and it is pointed out that such calculations have, in effect, already been performed. Since the trial wave functions that qualify for this treatment may be arbitrarily accurate, there is no limitation to the accuracy of the resulting classical description. In a particular case, the canonical coordinates and momenta for all degrees of freedom of the overall classical system are identified and investigated. In general, certain parameters appearing in the trial wave functions treated in this work may become redundant at certain times, leading to singularities in the Hamiltonian equations of motion. It is shown, however, that in certain such cases, a classical formalism for the quantum dynamics remains valid if the Hamiltonian function is suitably modified. Calculations applying the present Hamiltonian formalism are carried out that demonstrate that it has better convergence properties than an alternative, nonvariational, Hamiltonian technique suggested by previous work. These numerical studies also suggest that the observed spreading of wave packets and development of complicated structure in wave functions may be interpreted classically in terms of energy transfer among the various degrees of freedom in the overall classical system.

Physical Review A•Aug 1, 1992•24 citations

Quantum Mechanics

Hitoshi Kitada

I consider in this book a formulation of Quantum Mechanics. Usually QM is formulated based on the notion of time and space, both of which are thought a priori given quantities or notions. However, when we try to define the notion of velocity or momentum, we encounter a difficulty as we will see in chapter 1. The problem is that if the notion of time is given a priori, the velocity is definitely determined when given a position, which contradicts the uncertainty principle of Heisenberg. We then set the basis of QM on the notion of position and momentum operators as in chapter 2. Time of a local system then is defined approximately as a ratio $|x|/|v|$ between the space coordinate $x$ and the velocity $v$. In this formulation of QM, we can keep the uncertainty principle, and time is a quantity that does not have precise values unlike the usually supposed notion of time has. The feature of local time is that it is a time proper to each local system, which is defined as a finite set of quantum mechanical particles. We now have an infinite number of local times that are unique and proper to each local system. Based on the notion of local time, the motion inside a local system is described by the usual Schrödinger equation. We investigate such motion in a given local system in part II. This is a usual quantum mechanics. After some excursion of the investigation of local motion, we consider in part III the relative relation or motion between plural local systems. In the final part IV, we will prove that there is at least one Universe wave function $ϕ$ in which all local systems have local motions and thus local times. This concludes our formulation of Quantum Mechanics.

arXiv:quant-ph/0410061•Oct 8, 2004•0 citations
We consider quantum mechanics for which the system time is one of generalized coordinates. The generalized Hamiltonian has an unbounded spectrum, which allows us to introduce a Hermitian time operator. In the proposed formulation of quantum mechanics, a system time and observer’s time are introduced. The Schrödinger equation in the system time either does not hold or holds only approximately. The wave function is assumed to be square integrable with respect to all coordinates, including the system time. In some limit, this formalism reproduces standard quantum mechanics and the corresponding measurement theory.
Following homogeneous canonical formalism where the time is treated on an equal footing with the space coordinates, we propose an alternative formulation of quantum mechanics in which the time is considered as a dynamical variable. The basic equation corresponding to and being equivalent to the ordinary Schrödinger equation is obtained when we consider the space coordinates x as a parameter corresponding to the time t in the ordinary formulation. The formal solution of this equation is given and examined. By making use of this solution the scattering theory is formulated in our formalism. It is shown that observables corresponding to the time and space coordinates can be defined. A short comment on the energy-time uncertainty relation is also given.
Some aspects of the interpretation of quantum theory are discussed. It is emphasized that quantum theory is formulated in the Cartesian coordinate system; in other coordinates the result obtained with the help of the Hamiltonian formalism and commutator relations between “canonically conjugated” coordinate and momentum operators leads to a wrong version of quantum mechanics. The origin of time is analyzed by the example of atomic collision theory in detail; it is shown that the time-dependent Schrödinger equation is meaningless since in the high-impact-energy limit it transforms into an equation with two time-like variables. Following the Einstein-Rozen-Podolsky experiment and Bell’s inequality, the wave function is interpreted as an actual field of information in the elementary form. The concept “measurement” is also discussed.
Some aspects of the interpretation of quantum theory are discussed.It is emphasized that quantum theory is formulated in a Cartesian coordinate system; in other coordinates the result obtained with the help of the Hamiltonian formalism and commutator relations between 'canonically conjugated' coordinate and momentum operators leads to a wrong version of quantum mechanics.In this connection the Feynman integral formalism is also discussed.In this formalism the measure is not well-defined and there is no idea how to distinguish between the true version of quantum mechanics and an incorrect one; it is rather a mnemonic rule to generate perturbation series from an undefined zero order term.The origin of time is analyzed in detail by the example of atomic collisions.It is shown that the time-dependent Schrödinger equation for the closed three-body (two nuclei + electron) system has no physical meaning since in the high impact energy limit it transforms into an equation with two independent time-like variables; the time appears in the stationary Schrödinger equation as a result of extraction of a classical subsystem (two nuclei) from a closed three-body system.Following the Einstein-Rosen-Podolsky experiment and Bell's inequality the wave function is interpreted as an actual field of information in the elementary form.The relation between physics and mathematics is also discussed.
We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location x i at time t i and the final location x f at time t f . Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle. Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space. The Schwinger formulation of quantum dynamics In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the Lagrangian.
It is shown that application of the time-dependent variational principle to trial wave functions of a particular form results in a description of the quantum system as a classical system consisting of two mutually interacting parts: the ordinary classical analog of the quantum system and a (generally finite) number of additional degrees of freedom. This holds out the prospect of simulating quantum dynamics by coupling the corresponding classical system to additional degrees of freedom, and it is pointed out that such calculations have, in effect, already been performed. Since the trial wave functions that qualify for this treatment may be arbitrarily accurate, there is no limitation to the accuracy of the resulting classical description. In a particular case, the canonical coordinates and momenta for all degrees of freedom of the overall classical system are identified and investigated. In general, certain parameters appearing in the trial wave functions treated in this work may become redundant at certain times, leading to singularities in the Hamiltonian equations of motion. It is shown, however, that in certain such cases, a classical formalism for the quantum dynamics remains valid if the Hamiltonian function is suitably modified. Calculations applying the present Hamiltonian formalism are carried out that demonstrate that it has better convergence properties than an alternative, nonvariational, Hamiltonian technique suggested by previous work. These numerical studies also suggest that the observed spreading of wave packets and development of complicated structure in wave functions may be interpreted classically in terms of energy transfer among the various degrees of freedom in the overall classical system.
I consider in this book a formulation of Quantum Mechanics. Usually QM is formulated based on the notion of time and space, both of which are thought a priori given quantities or notions. However, when we try to define the notion of velocity or momentum, we encounter a difficulty as we will see in chapter 1. The problem is that if the notion of time is given a priori, the velocity is definitely determined when given a position, which contradicts the uncertainty principle of Heisenberg. We then set the basis of QM on the notion of position and momentum operators as in chapter 2. Time of a local system then is defined approximately as a ratio ∣x∣/∣v∣|x|/|v|∣x∣/∣v∣ between the space coordinate xxx and the velocity vvv. In this formulation of QM, we can keep the uncertainty principle, and time is a quantity that does not have precise values unlike the usually supposed notion of time has. The feature of local time is that it is a time proper to each local system, which is defined as a finite set of quantum mechanical particles. We now have an infinite number of local times that are unique and proper to each local system. Based on the notion of local time, the motion inside a local system is described by the usual Schrödinger equation. We investigate such motion in a given local system in part II. This is a usual quantum mechanics. After some excursion of the investigation of local motion, we consider in part III the relative relation or motion between plural local systems. In the final part IV, we will prove that there is at least one Universe wave function ϕϕϕ in which all local systems have local motions and thus local times. This concludes our formulation of Quantum Mechanics.