[Paper Review] Relativistic probability waves

M. Grigorescu|ArXiv.org|May 21, 2008

Quantum Mechanics and Applications15 references3 citations

TL;DR

This paper proposes a Lorentz-covariant framework for relativistic quantum mechanics by extending phase space to include energy and time as conjugate variables, enabling a canonical structure compatible with the Lorentz group. It introduces relativistic probability waves via coherent states of an extended Liouville equation, derives action waves from momentum-space localized distributions, and shows that discretizing spacetime leads to relativistic Wigner functions and a relativistic Schrödinger equation, recovering nonrelativistic quantum mechanics in the appropriate limit.

ABSTRACT

A canonical structure compatible with the action of the Lorentz group can be obtained considering the energy and time as conjugate variables of an extended phase space. Scalar probability waves, describing free relativistic particles, are associated with functional coherent states for an extended Liouville equation. Relativistic action waves are provided by distributions localized in the momentum space, evolving according to the continuity and Hamilton-Jacobi equations. Presuming the existence of minimum space and time intervals, the action distributions take the form of relativistic Wigner functions. The nonrelativistic quantum dynamics is retrieved approximating the time distribution function by a Gaussian wave packet.

Motivation & Objective

• To develop a Lorentz-covariant formulation of classical and quantum mechanics by extending phase space to include energy and time as conjugate variables.
• To establish a canonical structure compatible with the Lorentz group, enabling a consistent relativistic statistical mechanics framework.
• To derive relativistic probability waves as coherent solutions of the extended Liouville equation in momentum space.
• To show that discretizing spacetime with a minimum length interval leads to relativistic Wigner functions and a relativistic Schrödinger equation.
• To recover nonrelativistic quantum mechanics as a limit by approximating the time distribution as a Gaussian wave packet.

Proposed method

• Extends classical phase space to include energy $p_0 = -E/c$ and time $q_0 = ct$ as canonical conjugate variables, forming an 8-dimensional extended phase space $M^e$.
• Defines an extended Hamiltonian $H^e_0 = -c\sqrt{p_0^2 - \mathbf{p}^2}$ for a free relativistic particle, yielding Lorentz-invariant equations of motion via the Lie derivative $\mathcal{L}_{H^e}$.
• Derives the relativistic Liouville equation for the distribution function on $M^e$, reducing to coupled continuity and Hamilton-Jacobi equations for momentum-localized coherent states.
• Introduces action waves as distributions localized in momentum space, evolving via continuity and Hamilton-Jacobi equations, with universal time $u$ as the evolution parameter.
• Applies a discretization scheme assuming a minimum length $\ell$ (independent of Planck scale), leading to a discrete representation of the wave function in an extended Hilbert space.
• Defines the Wigner function directly over the extended phase space as a quasiprobability distribution, avoiding trajectory-based constructions.

Experimental results

Research questions

• RQ1 How can a canonical structure compatible with the Lorentz group be constructed in classical mechanics by extending phase space to include energy and time?
• RQ2 What is the relativistic generalization of coherent states and probability waves in the extended phase space framework?
• RQ3 How do action waves—distributions localized in momentum space—evolve under the extended Liouville dynamics?
• RQ4 What is the role of spacetime discretization in recovering relativistic quantum mechanics and Wigner functions?
• RQ5 How does the nonrelativistic limit emerge from the relativistic framework, particularly in terms of wave packet evolution and the Schrödinger equation?

Key findings

• The extended phase space with $q_0 = ct$ and $p_0 = -E/c$ provides a canonical structure compatible with Lorentz transformations, enabling a unified description of relativity and statistical mechanics.
• Relativistic probability waves arise as coherent solutions of the extended Liouville equation, with universal time $u$ as the evolution parameter, and the time distribution function approximates a Gaussian wave packet in the nonrelativistic limit.
• Action waves are described by distributions localized in momentum space, evolving via the continuity and Hamilton-Jacobi equations, and are shown to be invariant under Lorentz boosts via the parameter $\rho$.
• Discretization of spacetime with a minimum length $\ell$ leads to a relativistic Wigner function defined directly on the extended phase space, replacing trajectory-based definitions.
• The wave function in the extended Hilbert space evolves according to a relativistic Schrödinger equation, reducing to the Klein-Gordon equation in the stationary case.
• The nonrelativistic limit is recovered when the time distribution is approximated as a Gaussian, yielding a nonstationary Schrödinger-type evolution, consistent with standard quantum mechanics.