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 (Schr\"odinger) wave function $\psi(x, y, z | t)$, we introduce space-conditional wave functions such as $\phi(t, y, z | x)$, where $x$ plays the role of the evolution parameter. The function $\phi(t, y, z | x)$ 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 Schr\"odinger-type equation, $\hat{P}^{\mu} |\phi^\mu(x^\mu)\rangle = -i\hbar \, \eta^{\mu\nu} \frac{d}{dx^\nu} |\phi^\mu(x^\mu)\rangle$. It reproduces standard QM when $x^\mu = t$, with $|\phi^0(x^0)\rangle = |\psi(t)\rangle$, and recovers the STS extension when $x^\mu = x^i \in {x, y, z}$. For a free particle, we show that $\phi(t, y, z | x) = \langle t, y, z | \phi(x) \rangle$ 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 $|\phi^\mu(x^\mu)\rangle$ can emerge by conditioning (i.e., projecting) a timeless and spaceless physical state onto the eigenstate $|x^\mu\rangle$, leading to constraint equations of the form $\hat{\mathbb{P}}^\mu |\Phi^\mu\rangle = 0$. 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.