We challenge the common belief that if there is no absolute time parameter in physics, a quantum system described without reference to an external clock can be assumed to be in a stationary state of its Hamiltonian. We present a time-reparameterisation invariant quantum evolution law, which for a given initial condition predicts the same trajectory in state space as the Schrödinger equation, except that for nontrivial trajectories it does not predict the speed at which the trajectory is traversed. The solutions of this evolution law are all time-reparameterised solutions of the Schrödinger equation. We show how the predictions of the Schrödinger equation are recovered relative to an internal clock in this framework. In contrast to the Page-Wootters formalism or Dirac's quantisation of the Hamiltonian constraint, here the global sate is not stationary. We discuss the assumptions leading to the common conclusion that the state can be taken stationary and suggest that they need to be revisited.
What would it be like to be in a superposition of yesterday, today, and tomorrow? This question may seem at best entertaining, but it is necessary, and exploring it allows us to understand how exact irreversible clocks and change are possible, despite the Unruh-Wald and Hegerfeldt-Ruijsenaars no-go theorems forbidding them.
Unruh and Wald (1989) proved that if energy is bounded from below, no observable can increase monotonically with the Schrödinger time parameter t. Perfectly monotonic clocks and irreversible observable changes (Hegerfeldt-Ruijsenaars, 1980) seem impossible. From the perspective of the Schrödinger time, the world appears in a superposition of different intrinsic clock states indicating different times and opposite time directions. This seems to directly contradict our daily experiences of time and change.
I show that there is no contradiction: from an intrinsic perspective of the world, sharp irreversible changes do happen, because the macroscopic pointer states resolve the superposition of different times. Large-scale time-reversing or discontinuous transitions are not internally observable in the records. From the intrinsic perspective, an unbounded intrinsic-time translation generator plays the role of the Hamiltonian, generating only forward time evolution with respect to the intrinsic time, but not to the Schrödinger parameter t, which is thus not justified to play the role of time. This allows sharp time observables even if the external Hamiltonian is bounded from below. In addition, this leads to a stationary wavefunction of the universe satisfying a Wheeler-DeWitt-type equation, without assuming gravity.
In quantum theory it is possible to explain time, and dynamics, in terms of entanglement. This is the timeless approach to time, which assumes that the universe is in a stationary state, where two noninteracting subsystems, the ``clock'' and the ``rest,'' are entangled. As a consequence, by choosing a suitable observable of the clock, the relative state of the rest of the universe evolves unitarily with respect to the variable labeling the clock observable's eigenstates, which is then interpreted as time. This model for an ``evolution without evolution'' (Page and Wootters, 1983), albeit elegant, has never been developed further, because it was criticized for generating severe ambiguities in the dynamics of the rest of the universe. In this paper we show that there are no such ambiguities; we also update the model, making it amenable to possible new applications.
Relativistically, time
t is an observable just like position
r. In quantum theory,
t is a parameter, in contrast to the observable
r. This discrepancy suggests that there exists a more elaborate formalization of time, which encapsulates both perspectives. Such a formalization is proposed in this paper. The evolution is described in terms of sequential time
n∈N, which is updated each time an event occurs. Sequential time
n is separated from relational time
t, which describes distances between events in space-time. There is a space-time associated with each
n, in which
t represents the knowledge at time
n about temporal relations. The evolution of the wave function is described in terms of the parameter
σ that interpolates between sequential times
n. For a free object we obtain a Stueckelberg equation
dσdΨ(r4,σ)=2⟨ε⟩ic2ℏ□Ψ(r4,σ), where
r4=(r,ict). Here
σ describes the time
m passed between the start of the experiment at time
n and the observation at time
n+m. The parametrization is assumed to be natural, meaning that
dσd⟨t⟩=1, where
⟨t⟩ is the expected temporal distance between the events that define
n and
n+m. The squared rest energy
ε02 is proportional to the eigenvalue
σ~ that describes a 'stationary state'
Ψ(r4,σ)=ψ(r4,σ~)eiσ~σ. The Dirac equation follows as a `square root' of the stationary state equation from the condition that
σ~>0, which follows from the directed nature of
n. The formalism thus implies that all observable objects have non-zero rest mass, including elementary fermions. The introduction of
n releases
t, so that it can be treated as an observable with uncertainty
Δt.
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 basic tenet of the present work is the assumption of the lack of external and fixed time in the Universe. This assumption is best embodied by general relativity, which replaces the fixed space-time structure with the gravitational field, which is subject to dynamics. The lack of time does not imply the lack of evolution but rather brings to the forefront the role of internal clocks which are some largely arbitrary internal degrees of freedom with respect to which the evolution of timeless systems can be described. We take this idea seriously and try to understand what it implies for quantum mechanics when the fixed external time is replaced by an arbitrary internal clock. We put the issue in a solid, mathematically rigorous framework. We find that the dynamical interpretation of a quantum state of a timeless system depends on the employed internal clock. In particular, we find that the continuous spectra of well-known dynamical observables like the position of a free particle on the real line may turn discrete if measured in unusual clocks. We discuss the meaning of our result for attempts at quantization of global gravitational degrees of freedom.