Physical and mathematical properties of the space-time-symmetric formalism

TL;DR

This paper explores a space-time-symmetric formalism in nonrelativistic quantum mechanics, introducing a new framework where time is an operator and position is a parameter, revealing novel symmetries and properties.

Contribution

It develops the mathematical and physical properties of the space-time-symmetric formalism, including solutions, symmetries, and gauge invariance, extending the theory's scope.

Findings

Symmetry between Hamilton-Jacobi and space-conditional equations

General solution for time-independent potentials

Gauge invariance of the space-conditional equation with electromagnetic fields

Abstract

It is well known that nonrelativistic quantum mechanics presents a clear asymmetry between space and time. Much of this asymmetry is attributed to the lack of Lorentz invariance of the theory. Nonetheless, a recent work [Phys. Rev. A \textbf{95}, 032133 (2017)] showed that even though this is partially true, there is a broader physical scenario in which space and time can be handled in nonrelativistic quantum theory in a more symmetric way. In this space-time-symmetric formalism, an additional Hilbert space is defined so that time is raised to the status of operator and position becomes a parameter. As a consequence, the Hilbert space now requires a space-conditional quantum state governed by a new quantum dynamics. In this manuscript, we reveal some physical and mathematical properties of the space-time-symmetric formalism such as: symmetries between the Hamilton-Jacobi and the space-conditional equation; the general solution for a time-independent potential; and a new Lagrangian for a spinless particle in one dimensional. Finally, we present the space-conditional equation for a particle under the effect of an electromagnetic field, and the gauge invariance of this equation is proved.