Nature of Time in Quantum Mechanics

We have an uncertain future continuously coming into existence relative to the instant absorption and emission of photon energy. The future is a blank canvas that we can interact with forming the possible into the actual. And the future we are dealing with is from our point of reference. The future is unfolding with each photonelectron interaction with the respective elements of the periodic table and the individual waves formed by the electromagnetic spectrum. It’s a process of energy exchange that forms reality. If we look at the quantum world, the ‘past’ is represented by anti-matter annihilation which maintains a balance between matter and anti-matter showing a balance between the future and the past. In the Delayed Choice Quantum Eraser Experiment, time is unfolding photon by photon within the experiment, within our frame of reference. But we have an infinite number of reference frames coming in and out of existence with a timeline for each object. What is our past might be the future to an observer from a different frame of reference. Everything is relative.

Time Observables in a Timeless Universe

Time in quantum mechanics is peculiar: it is an observable that cannot be associated to an Hermitian operator. As a consequence it is impossible to explain dynamics in an isolated system without invoking an external classical clock, a fact that becomes particularly problematic in the context of quantum gravity. An unconventional solution was pioneered by Page and Wootters (PaW) in 1983. PaW showed that dynamics can be an emergent property of the entanglement between two subsystems of a static Universe. In this work we first investigate the possibility to introduce in this framework a Hermitian time operator complement of a clock Hamiltonian having an equally-spaced energy spectrum. An Hermitian operator complement of such Hamiltonian was introduced by Pegg in 1998, who named it "Age". We show here that Age, when introduced in the PaW context, can be interpreted as a proper Hermitian time operator conjugate to a "good" clock Hamiltonian. We therefore show that, still following Pegg's formalism, it is possible to introduce in the PaW framework bounded clock Hamiltonians with an unequally-spaced energy spectrum with rational energy ratios. In this case time is described by a POVM and we demonstrate that Pegg's POVM states provide a consistent dynamical evolution of the system even if they are not orthogonal, and therefore partially undistinguishables.

Time Operator, Real Tunneling Time in Strong Field Interaction and the Attoclock

Attosecond science, beyond its importance from application point of view, is of a fundamental interest in physics. The measurement of tunneling time in attosecond experiments offers a fruitful opportunity to understand the role of time in quantum mechanics. In the present work, we show that our real T-time relation derived in earlier works can be derived from an observable or a time operator, which obeys an ordinary commutation relation. Moreover, we show that our real T-time can also be constructed, inter alia, from the well-known Aharonov–Bohm time operator. This shows that the specific form of the time operator is not decisive, and dynamical time operators relate identically to the intrinsic time of the system. It contrasts the famous Pauli theorem, and confirms the fact that time is an observable, i.e., the existence of time operator and that the time is not a parameter in quantum mechanics. Furthermore, we discuss the relations with different types of tunneling times, such as Eisenbud–Wigner time, dwell time, and the statistically or probabilistic defined tunneling time. We conclude with the hotly debated interpretation of the attoclock measurement and the advantage of the real T-time picture versus the imaginary one.

Time in Quantum Mechanics and the Local Non-Conservation of the Probability Current

In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein–Gordon wavefunctions, as special cases; and in turn for non-relativistic quantum field theory and for the Schrödinger and Ginzburg–Landau equations, regarded as low energy limits. Quantum mechanics, however, is wider than quantum field theory, as an effective model of reality. For instance, fractional quantum mechanics and Schrödinger equations with non-local terms have been successfully employed in several applications. The non-locality of these formalisms is strictly related to the problem of time in quantum mechanics. We explicitly compute, for continuum wave packets, the terms of the fractional Schrödinger equation and the non-local Schrödinger equation by Lenzi et al. that break local current conservation. Additionally, we discuss the physical significance of these terms. The results are especially relevant for the electromagnetic coupling of these wavefunctions. A connection with the non-local Gorkov equation for superconductors and their proximity effect is also outlined.

What is time in quantum mechanics?

Time of arrival in quantum mechanics is discussed in two versions: the classical axiomatic "time of arrival operator" introduced by Kijowski and the event enhanced quantum theory (EEQT) method. It is suggested that for free particles the two methods may lead to the same result. On the other hand, the EEQT method can be easily geometrized within the framework of Galilei–Newton general relativistic quantum mechanics developed by M. Modugno and collaborators, and it can be applied to non-free evolutions. The way of geometrization of irreversible quantum dynamics based on dissipative Liouville equation is suggested.

Emergence of Time in Quantum Gravity: Is Time Necessarily Flowing?

We discuss the emergence of time in quantum gravity and ask whether time is always “something that flows.” We first recall that this is indeed the case in both relativity and quantum mechanics, although in very different manners: time flows geometrically in relativity (i.e., as a flow of proper time in the four dimensional space-time), time flows abstractly in quantum mechanics (i.e., as a flow in the space of observables of the system). We then ask the same question in quantum gravity in the light of the thermal time hypothesis of Connes and Rovelli. The latter proposes to answer the question of time in quantum gravity (or at least one of its many aspects) by postulating that time is a state-dependent notion. This means that one is able to make a notion of time as an abstract flow—that we call the thermal time—emerge from the knowledge of both: the algebra of observables of the physical system under investigation; a state of thermal equilibrium of this system. Formally, the thermal time is similar to the abstract flow of time in quantum mechanics, but we show in various examples that it may have a concrete implementation either as a geometrical flow or as a geometrical flow combined with a non-geometric action. This indicates that in quantum gravity, time may well still be “something that flows” at some abstract algebraic level, but this does not necessarily imply that time is always and only “something that flows” at the geometric level.