An extension of standard quantum mechanics is proposed in which the Newtonian time appearing as a parameter in the unitary evolution operator is replaced with the time shown by a `quantum clock'. Such a clock is defined by the following properties: (a) the time that the clock shows is nondecreasing, (b) the clock ticks at random Newtonian times with random tick sizes, and (c) on average the clock shows the Newtonian time. We show that the leading term in the evolution equation for the density matrix associated with any quantum clock gives the von Neumann equation. The leading correction to the von Neumann equation is given by the Lindblad equation generated by the Hamiltonian, but there are higher-order terms that generalize the von Neumann equation and the Lindblad terms. Modifications to the von Neumann equation are worked out in detail in a parametric family of models for which the tick sizes are gamma distributed. Lower bounds on the parameters of these quantum clock models are derived using the precision limit of an atomic clock. An anomalous term in the Ehrenfest theorem for a free particle is derived, which in principle can be used as a basis such models.
Comments
14 pages, 3 figures, additional material on the correction to the Ehrenfest theorem included
An extension of standard quantum mechanics is proposed in which the Newtonian time appearing as a parameter in the unitary evolution operator is replaced with the time shown by a `quantum clock'. Such a clock is defined by the following properties: (a) the time that the clock shows is nondecreasing, (b) the clock ticks at random Newtonian times with random tick sizes, and (c) on average the clock shows the Newtonian time. We show that the leading term in the evolution equation for the density matrix associated with any quantum clock gives the von Neumann equation. The leading correction to the von Neumann equation is given by the Lindblad equation generated by the Hamiltonian, but there are higher-order terms that generalize the von Neumann equation and the Lindblad terms. Modifications to the von Neumann equation are worked out in detail in a parametric family of models for which the tick sizes are gamma distributed. Lower bounds on the parameters of these quantum clock models are derived using the precision limit of an atomic clock. An anomalous term in the Ehrenfest theorem for a free particle is derived, which in principle can be used as a basis such models.
Does Time Tick in the Schrödinger Equation? is a short conceptual note about a distinction that is entirely standard in quantum mechanics and yet remarkably easy to lose in ordinary explanatory language. The time-dependent Schrödinger equation contains a temporal parameter t, but t is not itself a clock, an observable, a physical substance, or a causal agent. It orders and metrically distinguishes members of a parameterized state family and becomes operationally temporal through correlation with physical clocks. The Hamiltonian generates the evolution; the parameter indexes it. This paper is emphatically not an attempt to teach physicists how to suck eggs. Physicists already know that t is conventionally a parameter. The problem is that, when quantum mechanics is discussed informally, even technically sophisticated accounts repeatedly slip from “the state is parameterized by time” into language suggesting that time itself flows, acts, drives the state, produces phase change, or creates physical effects. The note uses reparameterization, stationary energy eigenstates, relative phase, entanglement, and the distinction between ordering, metric parameter, and physical clock process to expose this category shift. No modification of quantum mechanics is proposed. The aim is simply to keep the ontology no less disciplined than the mathematics: clocks tick, Hamiltonians generate, and the symbol t indexes their correlation. For further information about the ENSO Framework, please contact Eric Needham:ensotheory1@gmail.com
The problem of time is a deep paradox in our physical description of the world. According to Aristotle's relational theory, time is a measure of change and does not exist on its own. In contrast, quantum mechanics, just like Newtonian mechanics, is equipped with a master clock that dictates the evolution of a system. This clock is infinitely precise and tacitly supplied free of charge from outside physics. Not only does this absolute time make it notoriously difficult to make a consistent theory of quantum gravity, it is also the underlying problem in establishing the second law. Indeed, contrary to our experience, the Wheeler-deWitt equation --a canonical quantization of general relativity-- predicts a static universe. Similarly, when simply concerned with the dynamics of a closed quantum system, there is no second law because the Von Neumann entropy is invariant under unitary transformations. Here we are mainly concerned with the latter problem and we show that it can be resolved by attributing a minimal amount of resources to the measurement of time. Although there is an absolute time in quantum mechanics, an observer can only establish a time by measuring a clock. For a local measurement, the minimal entropy production is equal to the number of ticks. This lower bound is attained by a black hole.
The local conservation of a physical quantity whose distribution changes with time is mathematically described by the continuity equation. The corresponding time parameter, however, is defined with respect to an idealized classical clock. We consider what happens when this classical time is replaced by a nonrelativistic quantum-mechanical description of the clock. From the clock-dependent Schr\"odinger equation (as an analog of the time-dependent Schr\"odinger equation) we derive a continuity equation, where, instead of a time derivative, an operator occurs that depends on the flux (probability current) density of the clock. This clock-dependent continuity equation can be used to analyze the dynamics of a quantum system and to study degrees of freedom that may be used as internal clocks for an approximate description of the dynamics of the remaining degrees of freedom. As an illustration, we study a simple model for coupled electron-nuclear dynamics and interpret the nuclei as quantum clock for the electronic motion. We find that whenever the Born-Oppenheimer approximation is valid, the continuity equation shows that the nuclei are the only relevant clock for the electrons.
Quantum dynamics is usually written relative to a smooth external time parameter, while actual control schedules often contain waits, pulses, gates, resets, and event-like interventions. This paper develops the first unitary part of a sigma-time quantum dynamics framework. We define a bounded-Hamiltonian quantum evolution over a finite-horizon bounded-variation sigma clock with a continuous clock component, flat clock intervals, and a locally finite set of atom times carrying unitary gates. Under these hypotheses the sigma-chronological product exists, is unitary, composes as an evolution family, conserves wave-function norm, is strongly continuous away from atom times, and obeys a sharp atom/gate insertion law. We also prove a fixed-clock perturbation estimate and a bounded isolated-pulse corridor in which shrinking finite-width pulses converge in operator norm to declared atomic gates. The result is deliberately scoped: it does not claim an unbounded Schroedinger-operator theorem, a measurement theorem, an open-system theorem, or a Zeno limit theorem. A finite qubit benchmark with two bounded waits and one atomic gate is included as a reproducibility appendix and verifies unitary defect, trace preservation, Choi positivity, and output-state validity to machine precision
The meaning of time asymmetry in quantum physics is discussed.On the basis of a mathematical theorem, the Stone-von Neumann theorem, the solutions of the dynamical equations, the Schrödinger equation (1) for states or the Heisenberg equation (6a) for observables are given by a unitary group.Dirac kets require the concept of a RHS (rigged Hilbert space) of Schwartz functions; for this kind of RHS a mathematical theorem also leads to time symmetric group evolution.Scattering theory suggests to distinguish mathematically between states (defined by a preparation apparatus) and observables (defined by a registration apparatus (detector)).If one requires that scattering resonances of width Γ and exponentially decaying states of lifetime τ = Γ should be the same physical entities (for which there is sufficient evidence) one is led to a pair of RHS's of Hardy functions and connected with it, to a semigroup time evolution t 0 ≤ t < ∞, with the puzzling result that there is a quantum mechanical beginning of time, just like the big bang time for the universe, when it was a quantum system.The decay of quasi-stable particles is used to illustrate this quantum mechanical time asymmetry.From the analysis of these processes, we show that the properties of rigged Hilbert spaces of Hardy functions are suitable for a formulation of time asymmetry in quantum mechanics.
Quantum mechanics rests on the assumption that time is a classical variable. As such, classical time is assumed to be measurable with infinite accuracy. However, all real clocks are subject to quantum fluctuations, which leads to the existence of a nonzero uncertainty in the time variable. The existence of a quantum of time modifies the Heisenberg evolution equation for observables. Here we propose and analyse a generalisation of Heisenberg's equation for observables evolving in real time (the time variable measured by real clocks), that takes the existence of a quantum of time into account. This generalisation of Heisenberg's equation turns out to be a delay-differential equation.