[Paper Review] Trajectory-based Theory of Relativistic Quantum Particles

Bill Poirier|arXiv (Cornell University)|Aug 30, 2012

Quantum Mechanics and Applications10 references3 citations

TL;DR

This paper proposes a trajectory-based, wavefunction-free formulation of relativistic quantum mechanics for spin-zero, massive particles in Minkowski spacetime, using real-valued quantum trajectories that extremize an action and introduce a novel global simultaneity structure for accelerated observers. The key contribution is a generally covariant, single-particle relativistic theory that resolves the probability density and causality issues of the Klein-Gordon equation by replacing complex wavefunctions with ensembles of real trajectories governed by a new class of second-order-in-space, first-order-in-time PDEs.

ABSTRACT

Recently, a self-contained trajectory-based formulation of non-relativistic quantum mechanics was developed [Ann. Phys. 315, 505 (2005); Chem. Phys. 370, 4 (2010); J. Chem. Phys. 136, 031102 (2012)], that makes no use of wavefunctions or complex amplitudes of any kind. Quantum states are represented as ensembles of real-valued quantum trajectories that extremize a suitable action. Here, the trajectory-based approach is developed into a viable, generally covariant, relativistic quantum theory for single (spin-zero, massive) particles. Central to this development is the introduction of a new notion of global simultaneity for accelerated particles--together with basic postulates concerning probability conservation and causality. The latter postulate is found to be violated by the Klein-Gordon equation, leading to its well-known problems as a single-particle theory. Various examples are considered, including the time evolution of a relativistic Gaussian wavepacket.

Motivation & Objective

• To develop a generally covariant, single-particle relativistic quantum theory without relying on wavefunctions or complex amplitudes.
• To resolve the longstanding problems of negative probability densities and causality violations in the Klein-Gordon equation.
• To introduce a new notion of global simultaneity for accelerated particles, enabling consistent time evolution in relativistic quantum mechanics.
• To extend the trajectory-based non-relativistic quantum formalism to the relativistic domain for massive spin-zero particles.
• To establish a foundation for future extensions to curved spacetimes, external fields, and particles with spin or varying particle number.

Proposed method

• Represent quantum states as ensembles of real-valued trajectories, x(t,C), where C labels individual trajectories.
• Define a relativistic action functional whose extremization yields the equations of motion for the trajectories.
• Introduce a new global simultaneity structure for accelerated observers, based on the worldline geometry of each trajectory.
• Derive a system of partial differential equations (PDEs) for the trajectory density, with second-order spatial derivatives and first-order time derivatives.
• Enforce probability conservation and causality via postulates that constrain the form of the PDEs and the trajectory dynamics.
• Ensure Lorentz invariance of the PDEs despite the apparent asymmetry in time and space derivative orders.

Experimental results

Research questions

• RQ1 Can a consistent, generally covariant, single-particle relativistic quantum theory be formulated without wavefunctions or complex amplitudes?
• RQ2 How can global simultaneity be defined for accelerated observers in a way that preserves causality and probability conservation?
• RQ3 Why does the Klein-Gordon equation violate causality, and can this be resolved through a trajectory-based reformulation?
• RQ4 What is the structure of the PDEs governing the trajectory density, and how do they differ from standard relativistic wave equations?
• RQ5 Can the trajectory-based approach reproduce known results, such as the time evolution of a relativistic Gaussian wavepacket, without invoking wavefunctions?

Key findings

• The trajectory-based formulation successfully avoids the negative probability densities and causality violations inherent in the Klein-Gordon equation by replacing the wavefunction with a real-valued trajectory ensemble.
• The derived PDEs for the trajectory density are second-order in spatial coordinates (C) and first-order in time (T), yet remain invariant under Lorentz transformations.
• The theory introduces a new global simultaneity structure for accelerated particles, which is essential for defining consistent time evolution and probability conservation.
• The approach reduces seamlessly to the non-relativistic trajectory-based quantum mechanics in the low-velocity limit, confirming consistency with established non-relativistic results.
• The method provides a viable, self-contained, and generally covariant alternative to standard relativistic quantum mechanics, particularly for massive spin-zero particles.
• The framework suggests that the asymmetry in derivative orders (2 in space, 1 in time) is fundamental and not a flaw, contrasting with the symmetric second-order structure of the Klein-Gordon equation that leads to its problems.