Quantizing Time

I will sketch how the perspective-neutral approach to (quantum) frame covariance brings together some recent developments on dynamical reference frames in quantum foundations, gauge theories and gravity. The survey will touch on spatial frames, quantum clocks and the problem of time, edge modes, and the relativity of subsystems.

"We propose a time-of-arrival operator in quantum mechanics by conditioning on a quantum clock. This allows us to bypass some of the problems of previous proposals, and to obtain a Hermitian time of arrival operator whose probability distribution arises from the Born rule and which has a clear physical interpretation. The same procedure can be employed to measure the ""time at which some event happens"" for arbitrary events (and not just specifically for the arrival time of a particle). This talk is based on the paper: L. Maccone, K. Sacha, Quantum measurements of time, Phys. Rev. Lett. 124, 110402 (2020)."

Transformations between reference frames play a crucial role in our understanding of physical processes. In practice, reference frames are realised by physical systems, which are standardly treated as classical. However, assuming that every physical system is ultimately quantum, it is interesting to ask how a theory of transformations wrt quantum reference frames would look like, and what implications it would have for our description of spacetime. Recently, there has been a lot of effort towards developing a quantum generalisation of reference frame transformations, unveiling novel phenomena that are absent in the classical treatment of reference frames. Here, we develop a first-principles framework for quantum reference frame transformations which clarifies important conceptual issues of previous treatments. Based on the algebra of relative observables between a system and a reference frame, our operational perspective leads naturally to a mixed-state approach (incoherent twirling), in contrast to current pure-state approaches (coherent twirling). Within our framework, the full invariant quantum subsystem contains not only the algebra of relative observables between the system and the reference frame but also an “extra particle,” related to the invariant degrees of freedom of the reference frame itself. Importantly, this extra particle contains information about the “quantumness” of the reference frame and is essential to the unitarity of quantum reference frame transformations. Our approach is general, in the sense that it can be applied to a vast set of symmetry groups and to any type of system. We illustrate the physical meaning of the concepts developed by analysing quantum reference frame transformations with respect to the (centrally extended) Galilei group.

Time is absolute in quantum mechanics, whereas it is dynamical in general relativity. This is considered as one of the main obstacles towards unifying quantum theory and gravity. Relational quantum dynamics offers a possible solution by treating clocks as internal quantum systems, which promotes time to a dynamical quantity. This talk begins with a quick overview of time in relational quantum dynamics. We then explain that the inclusion of an interaction term coupling the clock and system causes the system dynamics to be governed by a time-nonlocal Schrödinger equation. Moreover, we demonstrate a quantum time dilation phenomena wherein we analyze the effect of non-classical states of quantum clocks on relativistic time dilation.

A possible solution of the problem of time in quantum gravitational systems is presented based on a relational description between the parameterized Dirac observables of the system under consideration and the clocks. The use of physical clocks required by a quantum gravitational description of time is shown to induce a loss of unitarity. The evolution is described by a Lindblad-type master equation unless it is possible to construct a perfect clock. I show that fundamental uncertainties in time measurements could arise due to quantum and gravitational effects, leading to the conclusion that there is always a loss of unitarity induced by the use of physical clocks. The extension of the analysis to physical reference frames in totally constrained systems is sketched.

"It is widely believed that the homogeneity of time is the symmetry related by Noether's (first) theorem to the conservation of energy, and indeed that it explains energy conservation. Both claims are questionable, and in particular seemingly hard to reconcile with the modern version of Noether's first theorem due independently to Martínes Alonso (1979) and Olver (1986). The talk is based on: 'Do symmetries ""explain"" conservation laws? ...' arXiv:2010.10909v1"

I argue that modern physics gives us good reason to take seriously the possibility of laws which are non-local, global, or in some other way non-dynamical. I set out a general framework for lawhood which does not presuppose the standard kinematical/dynamical split, and I apply it to the problem of giving a generalized definition of determinism for the non-dynamical context. Finally I make some suggestions about how to draw conclusions about the global structure of the laws of nature from the local observations we are able to make.

"Physics is formulated in terms of timeless axiomatic mathematics. However, time is essential in all our stories, in particular in physics. For example, to think of an event is to think of something in time. A formulation of physics based of intuitionism, a constructive form of mathematics built on time-evolving processes, would offer a perspective that is closer to our experience of physical reality and may help bridging the gap between static relativity and quantum indeterminacy. Historically, intuitionistic mathematics was introduced by L.E.J. Brouwer with a very subjectivist view where an idealized mathematician continually produces new information by solving conjectures. Here, in contrast, I’ll introduce intuitionism as an objective mathematics that incorporates a dynamical/creative time and an open future. Standard (classical) mathematics appears as the view from the “end of time” and the usual real numbers appear as the hidden variables of classical physics. Similarly, determinism appears as indeterminism seen from the “end of time”. Relativity is often presented as incompatible with indeterminism. Hence, at the end of this presentation I’ll argue that these incompatibility arguments are based on unjustified assumptions and present the “relativity of indeterminacy”. References: C. Posy, Mathematical Intuitionism, Cambridge Univ. Press, 2020. N. Gisin, Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real?, Erkenntnis (2019), https://doi.org/10.1007/s10670-019-00165-8 N. Gisin, Real Numbers are the Hidden Variables of Classical Mechanics, Quantum Studies: Mathematics and Foundations 7, 197-201 (2020). Flavio Del Santo and N. Gisin, Physics without determinism: Alternative interpretations of classical physics, Physical Review A 100.6 (2019). N. Gisin, Mathematical languages shape our understanding of time in physics, Nature Physics 16, 114-119 (2020). N. Gisin Indeterminism in Physics and Intuitionistic Mathematics, arXiv:2011.02348 Flavio Del Santo and N. Gisin, The Relativity of Indeterminacy, arXiv:2101.04134"

Time plays a fundamental role in our ability to make sense of the physical laws in the world around us. The nature of time has puzzled people –- from the ancient Greeks to the present day -– resulting in a long running debate between philosophers and physicists alike to whether time needs change to exist (the so-called relatival theory), or whether time flows regardless of change (the so-called substantival theory). One way to decide between the two is to attempt to measure the flow of time with a stationary clock, since if time were substantival, the flow of time would manifest itself in the experiment. Alas, conventional wisdom suggests that in order for a clock to function, it cannot be a static object, thus rendering this experiment seemingly impossible. We show that counter-intuitively, a quantum clock can measure the passage of time, even while being switched off, lending support for the substantival theory of time.

The concrete perspective of using interference to measure Gravity Induced Entanglement in the lab is a very exciting development for quantum gravity. While the measurements considered so far only test the nonrelativistic regime, the same technique might allow access to genuine relativistic quantum effects. Among these, there might be the possibility of direct detection of time quantum discreteness.