Space-time-symmetric non-relativistic quantum mechanics: Time and position of arrival and an extension of a Wheeler-DeWitt-type equation

Abstract

We generalize a space-time-symmetric (STS) extension of non-relativistic quantum mechanics (QM) to describe a particle moving in three spatial dimensions. In addition to the conventional time-conditional (Schrödinger) wave function $\psi (x,y,z|t)$, we introduce space-conditional wave functions such as $\varphi (t,y,z|x)$, where $x$ plays the role of the evolution parameter. The function $\varphi (t,y,z|x)$ represents the probability amplitude for the particle to arrive on the plane $x=\text{constant}$ at time $t$ and transverse position $(y,z)$. Within this framework, the coordinate $x^{\mu }\in \{t,x,y,z\}$ can be conveniently chosen as the evolution parameter, depending on the experimental context under consideration. This leads to a unified formalism governed by a generalized Schrödinger-type equation, ${\hat{P}}^{\mu }|{\varphi }^{\mu }(x^{\mu })⟩=-i\hslash {\eta }^{\mu \nu }\frac{d}{d{x}^{\nu }}|{\varphi }^{\mu }(x^{\mu })⟩$. It reproduces standard QM when $x^{\mu }=t$, with $|{\varphi }^0(x^0)⟩=|\psi (t)⟩$, and recovers the STS extension when $x^{\mu }=x^i\in \{x,y,z\}$. For a free particle, we show that $\varphi (t,y,z|x)=⟨t,y,z|\varphi (x)⟩$ naturally reproduces the same dependence on the momentum wave function as the axiomatic Kijowski distribution. Possible experimental tests of these predictions are discussed. Finally, we demonstrate that the different states $|{\varphi }^{\mu }(x^{\mu })⟩$ can emerge by conditioning (i.e., projecting) a timeless and spaceless physical state onto the eigenstate $|x^{\mu }⟩$, leading to constraint equations of the form ${\hat{\mathbb{P}}}^{\mu }|{\Phi }^{\mu }⟩=0$. This formulation generalizes the spirit of the Wheeler-DeWitt-type equation: instead of privileging time as the sole evolution parameter, it treats all coordinates on equal footing.

12 pages