Multi-Time Equations, Classical and Quantum

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Abstract

Multi-time equations are evolution equations involving several time variables, one for each particle. Such equations have been considered for the purpose of making theories manifestly Lorentz invariant. We compare their status and significance in classical and quantum physics.

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Relativistic Invariance and Hamiltonian Theories of Interacting Particles

D Currie, T. Jordan, E. C. G. Sudarshan (1963)

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A Relativistic Version of the Ghirardi-Rimini-Weber Model

Roderich Tumulka (2004)

Carrying out a research program outlined by John S. Bell in 1987, we arrive at a relativistic version of the Ghirardi-Rimini-Weber (GRW) model of spontaneous wavefunction collapse. The GRW model was proposed as a solution of the measurement problem of quantum mechanics and involves a stochastic and nonlinear modification of the Schr\"odinger equation. It deviates very little from the Schr\"odinger equation for microscopic systems but efficiently suppresses, for macroscopic systems, superpositions of macroscopically different states. As suggested by Bell, we take the primitive ontology, or local beables, of our model to be a discrete set of space-time points, at which the collapses are centered. This set is random with distribution determined by the initial wavefunction. Our model is nonlocal and violates Bell's inequality though it does not make use of a preferred slicing of space-time or any other sort of synchronization of spacelike separated points. Like the GRW model, it reproduces the quantum probabilities in all cases presently testable, though it entails deviations from the quantum formalism that are in principle testable. Our model works in Minkowski space-time as well as in (well-behaved) curved background space-times.