Multi-Time Wave Functions versus Multiple Timelike Dimensions

TL;DR

This paper compares multi-time wave functions with wave equations in multiple timelike dimensions, highlighting their differences in mathematical well-posedness and physical relevance, and emphasizing the covariant nature of multi-time quantum states.

Contribution

It clarifies the distinction between multi-time wave functions and multiple timelike dimensions, demonstrating the well-posedness and physical applicability of the former.

Findings

Multi-time wave functions have unique solutions for initial data.

Wave equations in multiple timelike dimensions are generally ill-posed.

Multi-time wave functions provide a covariant quantum state representation.

Abstract

Multi-time wave functions are wave functions for multi-particle quantum systems that involve several time variables (one per particle). In this paper we contrast them with solutions of wave equations on a space-time with multiple timelike dimensions, i.e., on a pseudo-Riemannian manifold whose metric has signature such as ${+}{+}{-}{-}$ or ${+}{+}{-}{-}{-}{-}{-}{-}$, instead of ${+}{-}{-}{-}$. Despite the superficial similarity, the two behave very differently: Whereas wave equations in multiple timelike dimensions are typically mathematically ill-posed and presumably unphysical, relevant Schr\"odinger equations for multi-time wave functions possess for every initial datum a unique solution on the spacelike configurations and form a natural covariant representation of quantum states.