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:1112.5321 · Progress of Theoretical Physics

Salpeter Equation and Causality

Piotr Kosiński

Published Jul 1, 2012• 3 citations
arXivHTMLPDF

Abstract

Acausal behaviour of solutions to the free Salpeter equation is considered. It is shown that the formal properties of solutions suggest the acausal propagation of quantum phenomena. On the other hand, the same properties of solutions applied to macroscopic phenomena can be explained without appealing to the notion of acausality.
DOI
10.1143/ptp.128.59

Related papers

Causality in quantum electrodynamics

M. I. Shirokov

Causality, in the sense of finiteness of the velocity of propagation of signals, is discussed for quantum electrodynamics. The paper proposes an appropriate quantum-mechanical criterion for causality. According to this criterion local observables like the field strengths E(x), H(x) and the current density j/sub ..mu../(x) behave causally. However, the theory contains quantities (like the electron number density) which exhibit macroscopic acausality. The reason for this result is discussed in a theory where the usual causality condition, i.e., local commutativity of the operators A/sub ..mu../ and psi, is satisfied. (AIP)

•May 1, 1976•2 citations

On the Notions of Causality and Complementarity

Niels Henrik David Bohr

A short exposition is given of the foundation of the causal description in classical physics and the failure of the principle of causality in coping with atomic phenomena. It is emphasized that the individuality of the quantum processes excludes a separation between a behaviour of the atomic objects and their interaction with the measuring instruments denning the conditions under which the phenomena appear. This circumstance forces us to recognize a novel relationship, conveniently termed complementarity, between empirical evidence obtained under different experimental conditions. An appropriate tool for a complementary mode of description is provided by the quantum-mechanical formalism which allows us to account for regularities of definite or statistical character beyond the grasp of classical physical explanation. – N. B.

dialectica•Jan 1, 1948•340 citations

Time‐Dependent Free‐Particle Salpeter Equation: Numerical and Asymptotic Analysis in the Light of the Fundamental Solution

Amalia Torre, Ambra Lattanzi, Decio Levi B

Abstract A detailed study of the spinless (1+1)D free‐particle Salpeter equation is presented. It involves several aspects of the topic: from the analysis of the behavior of solutions of the equation, both numerically evaluated and asymptotically approximated for definite initial conditions, to the comparison with the behavior of the corresponding solutions of the Schrödinger equation in order to both highlight the differences and to possibly understand how the latter “flow” in the former. Interesting analogies with other fields of physics, in particular with optics, are suggested.

Annalen der Physik•Dec 19, 2016•9 citations

On the violation of causality in relativistic quantum mechanics

Piotr Kosiński, Paweł Maślanka

The estimates on causality violating part of the wave function are given. The problem of acausal behavior of charge and current densities is discussed.

Journal of Mathematical Physics•Jul 1, 1990•5 citations

Beyond the Light-Cone Propagation of Relativistic Wavefunctions: Numerical Results

X. Gutiérrez de la Cal, A. Matzkin

It is known that relativistic wavefunctions formally propagate beyond the light cone when the propagator is limited to the positive energy sector. By construction, this is the case for solutions of the Salpeter (or relativistic Schrödinger) equation or for Klein–Gordon and Dirac wavefunctions defined in the Foldy–Wouthuysen representation. In this work, we quantitatively investigate the degree of non-causality for free propagation for different types of wavepackets that all initially have a compact spatial support. In the studied examples, we find that non-causality appears as a small transient effect that can in most cases be neglected. We display several numerical results and discuss the fundamental and practical consequences of our findings concerning this peculiar dynamical feature.

arXiv:2212.08400•Dynamics•Feb 6, 2023•5 citations

Tunneling dynamics of the relativistic Schrödinger/Salpeter equation

F. Daem, A. Matzkin

Abstract We investigate potential scattering and tunneling dynamics of a particle wavepacket evolving according to the relativistic Schrödinger equation (also known as the Salpeter equation). The tunneling properties of the Salpeter equation differ from those of the standard relativistic wave equations (such as the Klein–Gordon or Dirac equations). In particular, the tunneling solutions must be found by working in momentum space, given that the equation in configuration space contains a pseudo-differential operator. The resulting integral equations are derived and solved numerically for wavepackets scattering on model potential barriers. The solutions are characterized by the absence of Klein tunneling and an effect of the potential on the fraction of the transmitted wavepacket that propagates outside the light cone, a feature that has in the past been well-studied only for free propagation.

arXiv:2406.16644•Physica Scripta•Nov 21, 2024•1 citation
Causality, in the sense of finiteness of the velocity of propagation of signals, is discussed for quantum electrodynamics. The paper proposes an appropriate quantum-mechanical criterion for causality. According to this criterion local observables like the field strengths E(x), H(x) and the current density j/sub ..mu../(x) behave causally. However, the theory contains quantities (like the electron number density) which exhibit macroscopic acausality. The reason for this result is discussed in a theory where the usual causality condition, i.e., local commutativity of the operators A/sub ..mu../ and psi, is satisfied. (AIP)
A short exposition is given of the foundation of the causal description in classical physics and the failure of the principle of causality in coping with atomic phenomena. It is emphasized that the individuality of the quantum processes excludes a separation between a behaviour of the atomic objects and their interaction with the measuring instruments denning the conditions under which the phenomena appear. This circumstance forces us to recognize a novel relationship, conveniently termed complementarity, between empirical evidence obtained under different experimental conditions. An appropriate tool for a complementary mode of description is provided by the quantum-mechanical formalism which allows us to account for regularities of definite or statistical character beyond the grasp of classical physical explanation. – N. B.
Abstract A detailed study of the spinless (1+1)D free‐particle Salpeter equation is presented. It involves several aspects of the topic: from the analysis of the behavior of solutions of the equation, both numerically evaluated and asymptotically approximated for definite initial conditions, to the comparison with the behavior of the corresponding solutions of the Schrödinger equation in order to both highlight the differences and to possibly understand how the latter “flow” in the former. Interesting analogies with other fields of physics, in particular with optics, are suggested.
It is known that relativistic wavefunctions formally propagate beyond the light cone when the propagator is limited to the positive energy sector. By construction, this is the case for solutions of the Salpeter (or relativistic Schrödinger) equation or for Klein–Gordon and Dirac wavefunctions defined in the Foldy–Wouthuysen representation. In this work, we quantitatively investigate the degree of non-causality for free propagation for different types of wavepackets that all initially have a compact spatial support. In the studied examples, we find that non-causality appears as a small transient effect that can in most cases be neglected. We display several numerical results and discuss the fundamental and practical consequences of our findings concerning this peculiar dynamical feature.
Abstract We investigate potential scattering and tunneling dynamics of a particle wavepacket evolving according to the relativistic Schrödinger equation (also known as the Salpeter equation). The tunneling properties of the Salpeter equation differ from those of the standard relativistic wave equations (such as the Klein–Gordon or Dirac equations). In particular, the tunneling solutions must be found by working in momentum space, given that the equation in configuration space contains a pseudo-differential operator. The resulting integral equations are derived and solved numerically for wavepackets scattering on model potential barriers. The solutions are characterized by the absence of Klein tunneling and an effect of the potential on the fraction of the transmitted wavepacket that propagates outside the light cone, a feature that has in the past been well-studied only for free propagation.