Space-time-symmetric extension of nonrelativistic quantum mechanics

Abstract

In quantum theory we refer to the probability of finding a particle between positions x and x + dx at the instant t, although we have no capacity of predicting exactly when the detection occurs. In this work, first we present an extended non-relativistic quantum formalism where space and time play equivalent roles. It leads to the probability of finding a particle between x and x + dx during [t,t + dt]. Then, we find a Schrödinger-like equation for a "mirror" wave function φ(t, x) associated with the probability of measuring the system between t and t + dt, given that detection occurs at x. In this framework, it is shown that energy measurements of a stationary state display a non-zero 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. In Schrödinger quantum mechanics (QM) there is a clear asymmetry between time and space. Time is a continuous parameter that can be chosen with arbitrary precision and used to label the solution of the wave equation. In contrast, the position of a particle is seen as an operator, and therefore its value under a measurement is inherently probabilistic. It is common to hear that this asymmetry is due to the non-relativistic character of the Schrödinger equation (SE). Although partially correct, this argument is largely insufficient to justify all the disparity between space and time in the formalism of QM.A clear illustration is as follows. In a position measurement, ψ(x, t) = x|ψ(t) gives the probability amplitude of finding the particle within [x, x + dx], given that the time of detection is t. Would it not be equally reasonable, even in the non-relativistic domain, to ask about the probability of measuring the particle between x and x + dx, and t and t + dt? In this broader scenario, inquiring about the state of a particle at a given time t (as we often do), should make as much sense as asking about the state of that particle in a given position x (which we never do). In addition, if symmetry is to hold at this level, then there should exist a "mirror" wave function φ(t, x) = t|φ(x) , where x is a continuous parameter and t is the eigenvalue of an observable. If the location of particle becomes a physical reality only when a measurement is made, then it is a tenable position to expect that time should emerge in the same way. To earnestly consider these issues is the main goal of this manuscript.Time has been addressed in different contexts in QM . Common to several of these works is the attempt to remain within the borders of the standard theory. However, the solution to the arrival-time problem is considered by several authors to lay outside the framework of QM. It concerns the arrival of a particle in a spatially localized apparatus, where a time operator may be defined so that the relation [T ,Ĥ] = i is satisfied, and * corresponding author: eduardodias@df.ufpe.br † parisio@df.ufpe.br ...

A quantum formalism for events and how time can emerge from its foundations

“…Nevertheless, we believe that the physical connections between the results of the present manuscript and those of Ref. [6] deserve to be investigated in future works.…”

“…Since Eq. ( 10) has the same form as the PW's Hamiltonian (6), the solutions ( 7) and ( 8) still hold. Let us consider the timer's observable T such that T |t T = t|t T and [ T , ĤT ] = i , and the initial condition of R = ST D being…”

“…where |ψ(t) is the state of a system S at the instant t and α is an observable such that α|α = α|α . To predict experimental outcomes, the wave function ψ(α, t) = α|ψ(t) is interpreted as the probability amplitude of measuring the system in the state |α , given that the detection takes place at time t [2][3][4][5][6] . To avoid any confusion with other functions defined in the present work, from now on we will highlight the time-conditioned character of the Schrödinger wave function with the notation ψ(α|t).…”

“…In addition, it should be remarked that Refs. [6,25] propose the state of Eq. ( 4) for the particular case α = x.…”

History state formalism for scalar particles

“…The introduction of the concept of time in a quantum mechanical framework [1][2][3] has recently attracted renewed attention [4][5][6][7][8][9][10][11][12][13]. One persistent motivation is its connection with fundamental open problems, mainly related to the quantization of gravity [14][15][16][17][18][19][20][21][22][23][24][25][26], whose classical description is a general covariant theory [27].…”