English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Multi-time wave functions are wave functions that have a time variable forevery particle, such as $\phi(t_1,x_1,\ldots,t_N,x_N)$. They arise as arelativistic analog of the wave functions of quantum mechanics but can beapplied also in quantum field theory. The evolution of a wave function with Ntime variables is governed by N Schr\"odinger equations, one for each timevariable. These Schr\"odinger equations can be inconsistent with each other,i.e., they can fail to possess a joint solution for every initial condition; infact, the N Hamiltonians need to satisfy a certain commutator condition inorder to be consistent. While this condition is automatically satisfied fornon-interacting particles, it is a challenge to set up consistent multi-timeequations with interaction. We prove for a wide class of multi-timeSchr\"odinger equations that the presence of interaction potentials (given bymultiplication operators) leads to inconsistency. We conclude that interactionhas to be implemented instead by creation and annihilation of particles, which,in fact, can be done consistently, as we show elsewhere [17]. We also prove thefollowing result: When a cut-off length $\delta>0$ is introduced (in the sensethat the multi-time wave function is defined only on a certain set of spacelikeconfigurations, thereby breaking Lorentz invariance), then the multi-timeSchr\"odinger equations with interaction potentials of range $\delta$ areconsistent; however, in the desired limit $\delta\to 0$ of removing thecut-off, the resulting multi-time equations are interaction-free, whichsupports the conclusion expressed in the title.

Multi-Time Schr\"odinger Equations Cannot Contain Interaction Potentials