Space-time-symmetric extension of quantum mechanics: Interpretation and arrival-time predictions

Abstract

An alternative quantization rule, in which time becomes a self-adjoint operator and position is a parameter, was proposed by Dias and Parisio [Phys. Rev. A {\bf 95}, 032133 (2017)]. In this approach, the authors derive a space-time-symmetric (STS) extension of quantum mechanics (QM) where a new quantum state (intrinsic to the particle), $|{\phi}(x)\rangle$, is defined at each point in space. $|\phi(x)\rangle$ obeys a space-conditional (SC) Schr\"odinger equation and its projection on $|t\rangle$, $\langle t|\phi(x)\rangle$, represents the probability amplitude of the particle's arrival time at $x$. In this work, first we provide an interpretation of the SC Schr\"odinger equation and the eigenstates of observables in the STS extension. Analogous to the usual QM, we propose that by knowing the "initial" state $|\phi(x_0)\rangle$ -- which predicts any measurement on the particle performed by a detector localized at $x_0$ -- the SC Schr\"odinger equation provides $|\phi(x)\rangle={\hat U}(x,x_0)|\phi(x_0)\rangle$, enabling us to predict measurements when the detector is at $x \lessgtr x_0$. We also verify that for space-dependent potentials, momentum eigenstates in the STS extension, $|P_b(x)\rangle$, depend on position just as energy eigenstates in the usual QM depend on time for time-dependent potentials. In this context, whereas a particle in the momentum eigenstate in the standard QM, $|\psi(t)\rangle=|P\rangle|_t$, at time $t$, has momentum $P$ (and indefinite position), the same particle in the state $|\phi(x)\rangle=|P_b(x)\rangle$ arrives at position $x$ with momentum $P_b(x)$ (and indefinite arrival time). By investigating the fact that $|\psi(t)\rangle$ and $|{\phi}(x)\rangle$ describe experimental data of the same observables collected at $t$ and $x$, respectively, we conclude that they provide complementary information about the same particle...