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Time in Quantum Mechanics and the Local Non-Conservation of the Probability Current

Giovanni Modanese

Published Jan 1, 2018• 9 citations
OpenAlex

Abstract

In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein–Gordon wavefunctions, as special cases; and in turn for non-relativistic quantum field theory and for the Schrödinger and Ginzburg–Landau equations, regarded as low energy limits. Quantum mechanics, however, is wider than quantum field theory, as an effective model of reality. For instance, fractional quantum mechanics and Schrödinger equations with non-local terms have been successfully employed in several applications. The non-locality of these formalisms is strictly related to the problem of time in quantum mechanics. We explicitly compute, for continuum wave packets, the terms of the fractional Schrödinger equation and the non-local Schrödinger equation by Lenzi et al. that break local current conservation. Additionally, we discuss the physical significance of these terms. The results are especially relevant for the electromagnetic coupling of these wavefunctions. A connection with the non-local Gorkov equation for superconductors and their proximity effect is also outlined.

Related papers

Time in Quantum Mechanics and the Local Non-Conservation of the Probability Current

Giovanni Modanese

In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein–Gordon wavefunctions, as special cases; and in turn for non-relativistic quantum field theory and for the Schrödinger and Ginzburg–Landau equations, regarded as low energy limits. Quantum mechanics, however, is wider than quantum field theory, as an effective model of reality. For instance, fractional quantum mechanics and Schrödinger equations with non-local terms have been successfully employed in several applications. The non-locality of these formalisms is strictly related to the problem of time in quantum mechanics. We explicitly compute, for continuum wave packets, the terms of the fractional Schrödinger equation and the non-local Schrödinger equation by Lenzi et al. that break local current conservation. Additionally, we discuss the physical significance of these terms. The results are especially relevant for the electromagnetic coupling of these wavefunctions. A connection with the non-local Gorkov equation for superconductors and their proximity effect is also outlined.

arXiv:1806.08222•Mathematics•Sep 4, 2018•18 citations

Uniqueness of conserved currents in quantum mechanics

Peter Holland

It is proved by a functional method that the conventional expression for the Dirac current is the only conserved 4-vector implied by the Dirac equation that is a function of just the quantum state. The demonstration is extended to derive the unique conserved currents implied by the coupled Maxwell-Dirac equations and the Klein-Gordon equation. The uniqueness of the usual Pauli and Schrödinger currents follows by regarding these as the non-relativistic limits of the Dirac and Klein-Gordon currents, respectively. The existence and properties of further conserved vectors that are not functions of just the state is examined.

arXiv:quant-ph/0305175•Annalen der Physik•Oct 14, 2003•29 citations

Uniqueness of conserved currents in quantum mechanics

Peter Holland

Abstract It is proved by a functional method that the conventional expression for the Dirac current is the only conserved 4‐vector implied by the Dirac equation that is a function of just the quantum state. The demonstration is extended to derive the unique conserved currents implied by the coupled Maxwell‐Dirac equations and the Klein‐Gordon equation. The uniqueness of the usual Pauli and Schrödinger currents follows by regarding these as the non‐relativistic limits of the Dirac and Klein‐Gordon currents, respectively. The existence and properties of further conserved vectors that are not functions of just the state is examined.

Annalen der Physik•Oct 13, 2003•5 citations

Towards a Relativistic Quantum Mechanics

J. Iliopoulos, Theodore N. Tomaras

Abstract The Klein–Gordon and the Dirac equations are studied as candidates for a relativistic generalisation of the Schrödinger equation. We show that the first is unacceptable because it admits solutions with arbitrarily large negative energy and has no conserved current with positive definite probability density. The Dirac equation on the other hand does have a physically acceptable conserved current, but it too suffers from the presence of negative energy solutions. We show that the latter can be interpreted as describing anti-particles. In either case there is no fully consistent interpretation as a single-particle wave equation and we are led to a formalism admitting an infinite number of degrees of freedom, that is a quantum field theory. We can still use the Dirac equation at low energies when the effects of anti-particles are negligible and we study applications in atomic physics.

•Oct 25, 2021•0 citations

Relativistic Quantum Mechanics

Tao Xiang

In special relativity, time and space are transformed according to the Lorentz transformation. Relativistic quantum mechanics is quantum mechanics that is consistent with special relativity. Nevertheless, relativistic quantum mechanics is only an approximation to a fully self-consistent relativistic theory of known particle interactions because it does not describe cases where the number of particles changes. In special relativity, mass can emerge from energy, which is not possible in quantum mechanics based on the conservation of probability. Dirac proposed the equation named after him to solve the difficulty encountered in the Klein-Gordon equation stemmed from the fact that it is a second-order differential equation in time.

CRC Press eBooks•Mar 31, 2022•6 citations

The time-dependent Schrodinger equation for a closed quantum system

Hitoshi Kitada, J. Jeknić-Dugić, M. Arsenijević +1 more

Ever since Schrodinger, the time-dependent Schrodinger equation has been regarded as the fundamental physical law of the unitary-only quantum theory. However, while the origin of time for non-conservative systems can be traced back to the environmental influence, this procedure is inapplicable for the closed (conservative) quantum systems. Hence the question of the origin of time for closed systems. In addressing this issue, in order to avoid circularity of the standard universal (global) time as well as assumptions that could be additional to the standard postulates of quantum theory, we stick to the so-called quantum-mechanical local time. The problems that inevitably appear in such minimalist rejection of the standard role of time in physics are solved with mathematical rigor while leading to the emergent time for every, at least approximately, closed (conservative) system as well as to establishing the time-dependent Schrodinger law as the fundamental law of the unitary non-relativistic as well as relativistic-field quantum theory.

•Jun 15, 2016•0 citations
In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein–Gordon wavefunctions, as special cases; and in turn for non-relativistic quantum field theory and for the Schrödinger and Ginzburg–Landau equations, regarded as low energy limits. Quantum mechanics, however, is wider than quantum field theory, as an effective model of reality. For instance, fractional quantum mechanics and Schrödinger equations with non-local terms have been successfully employed in several applications. The non-locality of these formalisms is strictly related to the problem of time in quantum mechanics. We explicitly compute, for continuum wave packets, the terms of the fractional Schrödinger equation and the non-local Schrödinger equation by Lenzi et al. that break local current conservation. Additionally, we discuss the physical significance of these terms. The results are especially relevant for the electromagnetic coupling of these wavefunctions. A connection with the non-local Gorkov equation for superconductors and their proximity effect is also outlined.
In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein–Gordon wavefunctions, as special cases; and in turn for non-relativistic quantum field theory and for the Schrödinger and Ginzburg–Landau equations, regarded as low energy limits. Quantum mechanics, however, is wider than quantum field theory, as an effective model of reality. For instance, fractional quantum mechanics and Schrödinger equations with non-local terms have been successfully employed in several applications. The non-locality of these formalisms is strictly related to the problem of time in quantum mechanics. We explicitly compute, for continuum wave packets, the terms of the fractional Schrödinger equation and the non-local Schrödinger equation by Lenzi et al. that break local current conservation. Additionally, we discuss the physical significance of these terms. The results are especially relevant for the electromagnetic coupling of these wavefunctions. A connection with the non-local Gorkov equation for superconductors and their proximity effect is also outlined.
It is proved by a functional method that the conventional expression for the Dirac current is the only conserved 4-vector implied by the Dirac equation that is a function of just the quantum state. The demonstration is extended to derive the unique conserved currents implied by the coupled Maxwell-Dirac equations and the Klein-Gordon equation. The uniqueness of the usual Pauli and Schrödinger currents follows by regarding these as the non-relativistic limits of the Dirac and Klein-Gordon currents, respectively. The existence and properties of further conserved vectors that are not functions of just the state is examined.
Abstract It is proved by a functional method that the conventional expression for the Dirac current is the only conserved 4‐vector implied by the Dirac equation that is a function of just the quantum state. The demonstration is extended to derive the unique conserved currents implied by the coupled Maxwell‐Dirac equations and the Klein‐Gordon equation. The uniqueness of the usual Pauli and Schrödinger currents follows by regarding these as the non‐relativistic limits of the Dirac and Klein‐Gordon currents, respectively. The existence and properties of further conserved vectors that are not functions of just the state is examined.
Abstract The Klein–Gordon and the Dirac equations are studied as candidates for a relativistic generalisation of the Schrödinger equation. We show that the first is unacceptable because it admits solutions with arbitrarily large negative energy and has no conserved current with positive definite probability density. The Dirac equation on the other hand does have a physically acceptable conserved current, but it too suffers from the presence of negative energy solutions. We show that the latter can be interpreted as describing anti-particles. In either case there is no fully consistent interpretation as a single-particle wave equation and we are led to a formalism admitting an infinite number of degrees of freedom, that is a quantum field theory. We can still use the Dirac equation at low energies when the effects of anti-particles are negligible and we study applications in atomic physics.
In special relativity, time and space are transformed according to the Lorentz transformation. Relativistic quantum mechanics is quantum mechanics that is consistent with special relativity. Nevertheless, relativistic quantum mechanics is only an approximation to a fully self-consistent relativistic theory of known particle interactions because it does not describe cases where the number of particles changes. In special relativity, mass can emerge from energy, which is not possible in quantum mechanics based on the conservation of probability. Dirac proposed the equation named after him to solve the difficulty encountered in the Klein-Gordon equation stemmed from the fact that it is a second-order differential equation in time.
Ever since Schrodinger, the time-dependent Schrodinger equation has been regarded as the fundamental physical law of the unitary-only quantum theory. However, while the origin of time for non-conservative systems can be traced back to the environmental influence, this procedure is inapplicable for the closed (conservative) quantum systems. Hence the question of the origin of time for closed systems. In addressing this issue, in order to avoid circularity of the standard universal (global) time as well as assumptions that could be additional to the standard postulates of quantum theory, we stick to the so-called quantum-mechanical local time. The problems that inevitably appear in such minimalist rejection of the standard role of time in physics are solved with mathematical rigor while leading to the emergent time for every, at least approximately, closed (conservative) system as well as to establishing the time-dependent Schrodinger law as the fundamental law of the unitary non-relativistic as well as relativistic-field quantum theory.