Quantizing Time

In classical mechanics, the representations of dynamical evolutions of a system and those of interactions the system can have with its environment are different vector fields on the space of states: evolutions and interactions are conceptually, physically and mathematically different in classical physics, and those differences arise from the generic structure of the very dynamics of classical systems ("Newton's Second Law"). Correlatively, there is a clean separation of the system's degrees of freedom from those of its environment, in a sense one can make precise. I present a theorem showing that these features allow one to reconstruct the entire flat affine 4-dimensional geometry of Newtonian spacetime---the dynamics is inextricably tied to the underlying spacetime structure. In quantum theory (QT), contrarily, the representations of possible evolutions and interactions with the environment are exactly the same vector fields on the space of states ("add another self-adjoint operator to the Hamiltonian and exponentiate"): there is no difference between "evolution" and "interaction" in QT, at least none imposed by the structure of the dynamics itself. Correlatively, in a sense one can make precise, there is no clean separation of the system's degrees of freedom from those of the environment. Finally, there is no intrinsic connection between the dynamics and the underlying spacetime structure: one has to reach in and attach the dynamics to the spacetime geometry by hand, a la Wigner (e.g.). How we distinguish interaction from evolution in QT and how we attach the dynamics to a fixed underlying spacetime structure come from imposing classical concepts foreign to the theory. Trying to hold on to such a distinction is based on classical preconceptions, which we must jettison if we are to finally come to a satisfying understanding of QT. These observatons offer a way to motivate and make sense of, inter alia, the idea of indefinite causal structures.

"The Causaloid framework [1] is useful to study Theories with Indefinite Causality; since Quantum Gravity is expected to marry the radical aspects of General Relativity (dynamic causality) and Quantum Theory (probabilistic-ness). To operationally study physical theories one finds the minimum set of quantities required to perform any calculation through physical compression. In this framework, there are three levels of compression: 1) Tomographic Compression, 2) Compositional Compression and 3) Meta Compression. We present a diagrammatic representation of the Causaloid framework to facilitate exposition and study Meta compression. We show that there is a hierarchy of theories with respect to Meta compression and characterise its general form. Next, we populate the hierarchy. The theory of circuits forms the simplest case, which we express diagrammatically through Duotensors, following which we construct Triotensors using hyper3wires (hyperedges connecting three operations) for the next rung in the hierarchy. Finally, we discuss the implications for the field of Indefinite Causality. [1] Journal of Physics A: Mathematical and Theoretical, 40(12), 3081"

It has been previously discussed how events (interactions) in quantum mechanics are time-symmetric and an arrow of time is only due to the arrow of inference in the paper “Quantum information and the arrow of time”, arXiv:2010.05734 by Andrea Di Biagio, Pietro Dona, and Carlo Rovelli. In the relational interpretation of Quantum Mechanics, these interactions are relative facts. Stable facts result from relative facts through the process of decoherence as shown in the paper "Di Biagio, A., Rovelli, C., Foundations of Physics 51, 30 (2021)". They are separate from observed facts in laboratories due to the reason that they do not depend on a decision-making agent for their creation. In my talk, I will discuss my work with Carlo Rovelli and Andrea Di Biagio where we show that the process of decoherence and the notion of stability of facts is indeed time-symmetric. This is in contrast to the observed facts of our everyday world where an arrow of time emerges due to the presence of agents and traces.

"Making progress in quantum gravity requires resolving possible tensions between quantum mechanics and relativity.  One such tension is revealed by Bell's Theorem, but this relies on relativistic Local Causality, not merely the time-reversal symmetric aspects of relativity.  Specifically, it depends on an arrow-of-time condition, taken for granted by Bell, which we call No Future-Input Dependence.  One may replace this condition by the weaker Signal Causality arrow-of-time requirement -- only the latter is necessary, both for empirical viability and in order to avoid paradoxical causal loops.  There is then no longer any ground to require Local Causality, and Bell's tension disappears.  The locality condition which is pertinent in this context instead is called Continuous Action, in analogy with Einstein's ""no action at a distance,"" and the corresponding ""local beables"" are ""spacetime-local"" rather than ""local in space and causal in time.""    That such locally mediated mathematical descriptions of quantum entanglement are possible not only in principle but also in practice is demonstrated by a simple toy-model -- a ""local"" description of Bell correlations.  Describing general physical phenomena in this manner, including both quantum systems and gravitation, is a grand challenge for the future. [K.B. Wharton and N. Argaman, ""Colloquium: Bell's Theorem and Locally-Mediated Reformulations of Quantum Mechanics,"" Rev. Mod. Phys. 92, 21002 (2020).]"

What does it mean to say that a curve in state space describes change with respect to time, as opposed to space or any other parameter? What does it mean to say it's time is asymmetric? Inspired by the Wigner-Bargmann analysis of the Poincaré group, I discuss a general framework for understanding the meaning of time evolution and temporal symmetry in terms of the representation of a semigroup that includes "time translations", amongst the automorphisms of a state space. I discuss the structuralist and functionalist philosophical underpinnings of this view, and show how time reversal, parity, matter-antimatter exchange, and CPT are best viewed as extensions of a representation of continuous symmetries, whose existence is sensitive to the underlying structure of state space. I conclude with some comments on how an arrow of time can be defined in this framework, as well as prospects for such an arrow in the context of gravitation.

To this date no empirical evidence contradicts general relativity. In particular, there is no experimental proof a quantum theory of gravity is needed. Surprisingly, it appears likely that the first such evidence would come from experiments that involve non relativistic matter and extremely weak gravitational fields. The conceptual key for this is the Planck mass, a mesoscopic mass scale, and how it relates with what remains of general relativity in the Newtonian limit: time dilation. Indeed, current technological capabilities can amplify differences in time dilation superposition that are much smaller than the smallest time interval that can be measured by an atomic clock. Inspired from recent proposals to detect non--classicality of the gravitational field, we devise and examine the feasibility of an experiment that could detect a granularity of time at the Planck scale.

"Even though path-integral formulations of quantum theory are thought to be equivalent to state-based approaches, path-integrals are rarely used to motivate answers to foundational questions. This talk will summarize a number of implications concerning time and time-symmetry which result from the path-integral viewpoint. Such a perspective sheds serious doubt on dynamical collapse theories, and also pushes against efforts to extend configuration space to include multiple time dimensions. A recently-developed map between all possible two-qubit entangled states and spacetime-based path-integrals sheds further doubt on any need to extend spacetime to a large ontological configuration space. (References include arXiv:2103.02425, 1512.00740, 1103.2492 .)"

"We revisit the arguments underlying two well-known arrival-time distributions in quantum mechanics, viz., the Aharonov-Bohm and Kijowski (ABK) distribution, applicable for freely moving particles, and the quantum flux (QF) distribution. An inconsistency in the original axiomatic derivation of Kijowski’s result is pointed out, along with an inescapable consequence of the “negative arrival times” inherent to this proposal (and generalizations thereof). The ABK free-particle restriction is lifted in a discussion of an explicit arrival-time setup featuring a charged particle moving in a constant magnetic field. A natural generalization of the ABK distribution is in this case shown to be critically gauge-dependent. A direct comparison to the QF distribution, which does not exhibit this flaw, is drawn (its acknowledged drawback concerning the quantum backflow effect notwithstanding). Based on a recent paper (https://arxiv.org/abs/2102.02661), to be published in Proceedings of the Royal Society A."

Quantum cosmology faces the problem of time: the Universe has no background time, only interacting dynamical degrees of freedom within it. The relational view is to use one degree of freedom (which can be matter or geometry) as a clock for the others. In this talk we discuss a cosmological model of a flat FLRW universe filled with a massless scalar field and a perfect fluid. We study three quantum theories based on three different choices of (relational) clock and show that, if we require the dynamics to be unitary, all three make drastically different predictions regarding resolution of the classical (Big Bang) singularity or a possible quantum recollapse at large volume. The talk is based on [arXiv:2005.05357] and a second paper to appear on arXiv in May 2021. We plan to give two talks: one covering the foundations and general properties of the model, and one showing detailed results and physical interpretation. (We will merge these talks into one if the organisers decide to accept only one talk.)

I discuss the new dimension that the relational approach to the problem of time takes in quantum gravity contexts in which spacetime and geometry are understood as emergent. I argue that, in this case, the relational strategy is best realized at an approximate and effective level, after suitable coarse graining and only in terms of special quantum states. I then show a concrete realization of such effective relational dynamics in the context of a cosmological application of the tensorial group field theory formalism for quantum gravity.

"Candidate theories of quantum gravity must answer the questions: how can the dynamics of quantum states of matter and geometry be defined in a diffeomorphism-invariant way? How are the quantum states related to probabilities in the absence of a preferred time? To address these issues, we discuss the construction and interpretation of relational observables in quantum theories with worldline diffeomorphism invariance, which serve as toy models of quantum gravity. In this context, we present a method of construction of quantum relational observables which is analogous to the construction of gauge-invariant extensions of noninvariant quantities in usual gauge (Yang-Mills) theories. Furthermore, we discuss how the notion of a physical propagator may be used to define a unitary evolution in the quantum theory, which is to be understood in terms of a generalized clock, as is the classical theory.  We also discuss under which circumstances this formalism can be related to the use of conditional probabilities in a generalization of the Page-Wootters approach. Finally, we also examine how our formalism can be adapted to calculations of quantum-gravitational effects in the early Universe. Refs.: L. Chataignier, Phys. Rev. D 101, 086001 (2020); 103, 026013 (2021); 103, 066005 (2021)"