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arXiv:2308.04376

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

Eduardo O. Dias

Published Aug 8, 2023• 0 citations
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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 $ψ(x, y, z | t)$, we introduce space-conditional wave functions such as $ϕ(t, y, z | x)$, where $x$ plays the role of the evolution parameter. The function $ϕ(t, y, z | x)$ 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 $x^μ\in \{t, x, y, z\}$ 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, $\hat{P}^μ |ϕ^μ(x^μ)\rangle = -i\hbar \, η^{μν} \frac{d}{dx^ν} |ϕ^μ(x^μ)\rangle$. It reproduces standard QM when $x^μ= t$, with $|ϕ^0(x^0)\rangle = |ψ(t)\rangle$, and recovers the STS extension when $x^μ= x^i \in \{x, y, z\}$. For a free particle, we show that $ϕ(t, y, z | x) = \langle t, y, z | ϕ(x) \rangle$ 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 $|ϕ^μ(x^μ)\rangle$ can emerge by conditioning (i.e., projecting) a timeless and spaceless physical state onto the eigenstate $|x^μ\rangle$, leading to constraint equations of the form $\hat{\mathbb{P}}^μ|Φ^μ\rangle = 0$. 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.

DOI
10.48550/arxiv.2308.04376

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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 95, 032133 (2017)]. In this approach, the authors derived a space-time-symmetric (STS) extension of quantum mechanics (QM) where a new quantum state (intrinsic to the particle) $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ is defined at each point in space. The quantum state $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ obeys a space-conditional (SC) Schr\"odinger equation and its projection on $|t\ensuremath{\rangle}, \ensuremath{\langle}t|\ensuremath{\phi}(x)\ensuremath{\rangle}$, represents the probability amplitude of the particle's arrival time at xxx. In this work 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 $|\ensuremath{\phi}({x}_{0})\ensuremath{\rangle}$, which predicts any measurement on the particle performed by a detector localized at x0{x}_{0}x0​, the SC Schr\"odinger equation provides $|\ensuremath{\phi}(x)\ensuremath{\rangle}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbb{U}}(x,{x}_{0})|\ensuremath{\phi}({x}_{0})\ensuremath{\rangle}$, enabling us to predict measurements when the detector is at $x\ensuremath{\lessgtr}{x}_{0}$. We also verify that for space-dependent potentials, momentum eigenstates in the STS extension $|{P}_{b}(x)\ensuremath{\rangle}$ 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, $|\ensuremath{\psi}(t)\ensuremath{\rangle}={|P\ensuremath{\rangle}|}_{t}$, at time ttt, has momentum PPP (and indefinite position), the same particle in the state $|\ensuremath{\phi}(x)\ensuremath{\rangle}=|{P}_{b}(x)\ensuremath{\rangle}$ arrives at position xxx with momentum Pb(x){P}_{b}(x)Pb​(x) (and indefinite arrival time). By investigating the fact that $|\ensuremath{\psi}(t)\ensuremath{\rangle}$ and $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ describe experimental data of the same observables collected at ttt and xxx, respectively, we conclude that they provide complementary information about the same particle. Finally, we solve the SC Schr\"odinger equation for an arbitrary space-dependent potential. We apply this solution to a potential barrier and compare it with a generalized Kijowski distribution, showing that they can predict distinct traversal times.
In quantum theory we refer to the probability of finding a particle between positions xxx and x+dxx+dxx+dx at the instant ttt, although we have no capacity of predicting exactly when the detection occurs. In this work, we first present an extended nonrelativistic quantum formalism where space and time play equivalent roles. It leads to the probability of finding a particle between xxx and x+dxx+dxx+dx during [t,t+dtt,\phantom{\rule{0.16em}{0ex}}t+dtt,t+dt]. Then, we find a Schr\"odinger-like equation for a ``mirror'' wave function $\ensuremath{\phi}(t,x)$ associated with the probability of measuring the system between ttt and t+dtt+dtt+dt, given that detection occurs at xxx. In this framework, it is shown that energy measurements of a stationary state display a nonzero dispersion, and that energy-time uncertainty arises from first principles. We show that a central result on arrival time, obtained through approaches that resort to ad hoc assumptions, is a natural, built-in part of the formalism presented here.
Time-symmetric theory (TST) is formulated as a theoretical foundation of time-symmetric physics (TSP), classical, relativistic and quantum. TST is based on the widely used in particle physics fact that the description of positive energy antiparticles moving forward in space and time is equivalent to the description of negative energy particles moving backward in space and time (Zisman 1940, Stückelberg 1941, Feynman 1949). In TST, this fundamental fact is generalized by formulating the equivalence principle of particle physics and applying it to all physical phenomena in the rest frames of negative energy particles. As a result, the group of transformations TST also includes such inertial frames that move backward in space and time with coordinates related to the ordinary ones by 4-inversion. This led to the formulation of time-symmetric relativity theory and corresponding relativistic quantum mechanics. In the latter, the probabilities remain positive, but the bilinear forms of the wave functions are associated with probability flows, the signs of which depend on the direction of movement along the coordinate axes, including the time axis. In TST, therefore, the Klein-Gordon equation is a consistent equation for probability amplitudes, and the fermion theory contains the necessary corrections, which make it physically more consistent, unchanging its main observational consequences. Applications of TST to quantum fields will be considered in the second article.
A covariant generalization of a non-relativistic stochastic quantum mechanics introduced by de la Peña and Cetto is formulated. The analysis is done in space-time and avoids the use of a non-covariant time evolution parameter in order to search for Lorentz invariance. The covariant form of the set of iterative equations for the joint coordinate and momentum distribution function Q(x,p)Q(x,p)Q(x,p) is derived and expanded in power series of the charge of the particle. Then, solutions of the zeroth order in the charge of the iterative equations for Q(x,p)Q(x,p)Q(x,p) are considered. For them, it follows that the space-time probability density ρ(x)ρ(x)ρ(x) and the function S(x)S(x)S(x) which gradient defines the mean value of the momentum at the space time point xxx, define a complex function ψ(x)ψ(x)ψ(x) which exactly satisfies the Klein-Gordon (KG) equation. It is argued that when the KG solution is either of positive or negative energy, the total number of particles conserves in the random motion. It follows that the total number of particles conserves when ψ(x)ψ(x)ψ(x) composed alternatively of pure positive or negative energy modes. In addition, solutions for the joint distribution function in lowest order, satisfying the positive condition are also found. They suggests a possible link with the Couder's experiments on droplet movements over oscillating liquid surfaces.
Understanding whether the mathematical structure of quantum mechanics is fundamental or emergent remains a central question in the foundations of physics. In particular, the special role played by time in quantum theory, appearing as an external evolution parameter rather than a dynamical observable, suggests that the formalism itself may arise from deeper structural considerations. In this work, we investigate the emergence of quantum mechanical formalism from classical wave dynamics by adopting a dimensional framework in which time is treated as a +1 evolution parameter relative to the dimensions through which physical phenomena (fields or disturbances of a field) propagate and interact. Within this perspective, different fields may evolve with respect to different effective dimensions, while remaining embedded in a common higher-dimensional space, allowing time to acquire a relational and context-dependent role. This means that in our proposed model, time is not a fixed dimension which is experienced the same way for every field or field interaction of any dimensionality. In that sense, time for one physical phenomenon can behave as space for a higher dimensional physical phenomenon, whose time is a different +1 dimension. The central objective of this paper is to determine how a higher-dimensional deterministic field can be consistently represented by a lower-dimensional description that lacks direct access to its full set of evolution parameters and evolves through a spatial (for the higher-dimensional field) dimension. To this end, we introduce a general projection framework in which a higher-dimensional field is mapped to a reduced field through an interaction-based recording process. Crucially, we do not assume the form of this mapping a priori. Instead, we impose the requirement that it preserve the maximum amount of physically accessible information. In particular, we demand the faithful encoding of phase relations, interference structure, and spectral composition, including the relative contributions of different Fourier modes and their superposition. We first demonstrate, within a purely classical 3+1-dimensional wave framework that these constraints severely restrict the admissible form of the reduced description and naturally lead to complex amplitudes, linear superposition, Hilbert space structure, and canonical operator relations. This analysis provides an intuitive and mathematically explicit route to quantum-like descriptions without assuming quantum postulates. We then generalize the construction to a 4+1-dimensional framework, introducing an additional evolution parameter and showing that under the same information-preserving constraints, the Schrödinger equation appears as an effective low-energy description of the reduced dynamics, while a relativistic dispersion relation emerges simultaneously through the encoding of the hidden evolution parameter as an invariant frequency scale. In this way, both quantum mechanical and relativistic structures arise from the same underlying requirement: the consistent and information-preserving representation of higher-dimensional wave propagation in a lower-dimensional observational framework. The results suggest that the formal structure of quantum mechanics need not be postulated a priori, but may instead be understood as the unique mathematical language required to encode the observable remnant of a higher-dimensional deterministic dynamics under strict constraints of symmetry, invariance, and information preservation.
In non-relativistic quantum mechanics of N particles in three spatial dimensions, the wave function ψ ( q 1 , ..., q N , t ) is a function of 3 N position coordinates and one time coordinate. It is an obvious idea that in a relativistic setting, such functions should be replaced by ϕ (( t 1 , q 1 ), ..., ( t N , q N )), a function of N space-time points called a multi-time wave function because it involves N time variables. Its evolution is determined by N Schrödinger equations, one for each time variable; to ensure that simultaneous solutions to these N equations exist, the N Hamiltonians need to satisfy a consistency condition. This condition is automatically satisfied for non-interacting particles, but it is not obvious how to set up consistent multi-time equations with interaction. For example, interaction potentials (such as the Coulomb potential) make the equations inconsistent, except in very special cases. However, there have been recent successes in setting up consistent multi-time equations involving interaction, in two ways: either involving zero-range ( δ potential) interaction or involving particle creation and annihilation. The latter equations provide a multi-time formulation of a quantum field theory. The wave function in these equations is a multi-time Fock function, i.e., a family of functions consisting of, for every n = 0, 1, 2, ..., an n -particle wave function with n time variables. These wave functions are related to the Tomonaga–Schwinger approach and to quantum field operators, but, as we point out, they have several advantages.