Discrete Time in Quantum Gravity

Carlo Rovelli (Aix-Marseille) gives a talk at the Workshop on the Problem of Time in Perspective (3-4 July, 2015) titled "Physics without Fundamental Time". Abstract: Rivers of ink have flown on the basic conceptual structure of quantum gravity -a theory where we expect the notions of classical spacetime, particles, fields, energy and momentum to require substantial revision-. I discuss a specific solution to these questions and apply concretely it to a physical calculation, the tunneling time of a Hajicek-Kiefer black-to-white hole transition. This is a quantum gravitational effect that might have some chance to be actually observable, or could have even been already observed in the "Fast Radio Bursts" observed by the Arecibo and Parkes radio-telescopes.

Kurt Sundermeyer (FU Berlin/MPIWG) gives a talk at the Workshop on the Problem of Time in Perspective (3-4 July, 2015) titled "Facets of Time in Physics". Abstract: In an attempt to understand slogans such as "The End Of Time", "Forget About Time", "Time Reborn", "Time Remains", I started to write an essay about the various notions of time in physics (with a glimpse to philosophy) from classical mechanics to quantum gravity. By this curiosity-driven motivation I realized that there are not only various notions of time, but that there are diverse problems with time and problems of time. In my talk I will present the structure and inputs of this still unfinished treatise, and I expect to receive valuable feedback from the competent participants of this workshop.

Tim A. Koslowski (Brunswick) gives a talk at the Workshop on the Problem of Time in Perspective (3-4 July, 2015) titled "The Gravitational Arrow of Time". Abstract: The arrow of time appears to always to point in one direction (i.e. we can clearly tell whether a movie is played forward or backward) although the underlying physics is time-reversal symmetric. The most widely accepted explanation for this is that the experienced arrow is the thermodynamic arrow of time (i.e. the direction of entropy growth). This scenario requires the past hypothesis, i.e. an atypical initial condition. We propose an alternative mechanism: The arrow of time is the direction in which the complexity of the universe grows. Gravity generates this arrow of time and creates subsystems with low entropy initial conditions spontaneously. I show this in detail in the Newtonian limit and discuss the extension to cosmological models in GR.

Oliver Pooley (Oxford) gives a talk at the Workshop on the Problem of Time in Perspective (3-4 July, 2015) titled "First-Class Constraints, Gauge, and the Wheeler-DeWitt Equation". Abstract: Recently, Pitts (2014) has argued that the claim that first-class constraints generate gauge transformations (hereafter “orthodoxy”) fails even in electromagnetism, which is standardly taken to illustrate its correctness. Independently, Barbour and Foster (2008) have argued that a key presupposition of the primary argument for orthodoxy (due to Dirac, 1964) is not satisfied in the important case of reparameterization-invariant theories. In assessing these claims, one needs to distinguish between, (i) transformations that relate points of phase space that represent the same instantaneous state and (ii) transformations that map curves on phase space to curves that represent the same history. Pitts shows that arbitrary first-class constraints fail to generate transformations of type (ii), but leaves untouched the orthodox position concerning (i). Barbour and Foster show that we have no reason to regard transformations generated by Hamiltonian constraints as type (i), but that leaves open that they may be regarded as transformations of type (ii). I will discuss whether the latter possibility allows one to reconcile applying Dirac’s constrained quantization procedure to the Hamiltonian constraint and interpreting the Wheeler–DeWitt wavefunction as representing genuine change.

J. Brian Pitts (Cambridge) gives a talk at the Workshop on the Problem of Time in Perspective (3-4 July, 2015) titled "Changing Observables in Canonical General Relativity from Hamiltonian-Lagrangian Equivalence". Abstract: Is change missing in classical canonical General Relativity? If one insists on Hamiltonian-Lagrangian equivalence, then there is Hamiltonian change just when there is no time-like Killing vector field. Change has seemed missing partly due to Dirac’s belief that a first-class constraint, especially a primary, generates a gauge transformation. Pons showed that Dirac’s argument stops too soon: working to second order in time brings in first-class secondaries and hence the gauge generator G, a tuned sum of first-class constraints used by Anderson and Bergmann (1951) and recovered by Mukunda, Castellani et al. from the 1980s. I observe that trouble happens immediately: a first-class primary constraint generates an illegal change of initial data in GR, Maxwell and Yang-Mills. Dirac’s subtractive derivation misses it by cancellation; confusion between the electric field E(dA) and canonical momenta p (auxiliary fields in the canonical action \int dt (p \dot{q}-H) also obscures the problem. Dirac’s conjecture that a first-class secondary constraint generates a gauge transformation rests on a false assumption. Looking for gauge symmetries of the canonical action, one finds that the gauge generator G changes the action by at most a boundary term, but an isolated first-class constraint does not. The gauge generator G generates spatio-temporal coordinate transformations (not just spatial ones) for the space-time metric (not just the spatial metric). But are there locally varying _observables_ in canonical General Relativity? Hamiltonian-Lagrangian equivalence guarantees that Hamiltonian observables are equivalent on-shell to Lagrangian observables. (Historically, Lagrangian-inequivalent observables may have arisen within Bergmann’s school due to novel postulation in Bergmann-Schiller 1953.) With first-class constraints exposed as not generating gauge transformations, observables’ Poisson brackets should be taken with the gauge generator G, as noted by Pons, Salisbury and Sundermeyer. Heeding Einstein’s point-coincidence argument excludes primitive point individuation and thus active diffeomorphisms in favor of (4-d) tensor calculus. Kuchař’s unsystematic waiver of the vanishing Poisson brackets condition to permit change has a more principled extension: observables should be internally gauge _invariant_ (0 Poisson bracket with G for Maxwell, Yang-Mills, etc.) but externally gauge _covariant_. Hence the Poisson bracket with the coordinate-changing G should be the Lie derivative, indeed the Lie derivative of a geometric object (on-shell). For GR with no matter gauge group, observables are (on-shell) space-time geometric objects (components in coordinates with a transformation law). Hence the space-time metric and its concomitants (connection, curvature, etc.) are locally varying observables. Questions regarding Legendre projectability when an internal gauge group is also present and regarding the mixed supergravity transformations are noted. Velocity-dependent gauge transformations call for phase space extended by time---“phase space-time”; GR’s Lie derivative is an example. Vacuum GR’s phase space-time has 20 infinity^3 + 1 dimensions and 8 infinity^3 first-class constraints; one should not have expected a reduced phase _space_ description of a theory with many-fingered time. Classical clarity might be of some use in quantization.

Philipp Hoehn (Perimeter Institute) gives a talk at the Workshop on the Problem of Time in Perspective (3-4 July, 2015) titled "Constraints, Dirac Observables and Chaos". Abstract: I will discuss fundamental challenges to the standard relational paradigm arising from chaotic dynamics.

Bianca Dittrich (Perimeter Institute) gives a talk at the Workshop on the Problem of Time in Perspective (3-4 July, 2015) titled "The Consistent Boundary Formulation". Abstract: I will introduce the consistent boundary formulation which allows to express renormalization flow in a background independent context. I will discuss consequences of this formulation for the Hamiltonian framework and explore in which sense Hamiltonian constraints do actually exists in this context and how this influences the notion of time.

Sean Gryb (Radboud) gives a talk at the Workshop on the Problem of Time in Perspective (3-4 July, 2015) titled "In Favour of a Schrödinger Evolution of the Universe". Abstract: In the canonical formulation of reparametrization invariant systems, time evolution on phase space is generated by a fully constrained Hamiltonian. On the orthodoxy view, the quantum formalism for such systems is constructed through Dirac quantization, which leads to a real, time-independent constraint on the quantum state. The question then remains how to extract a notion of time evolution from this frozen formalism. On one predominant view, the system is to be split in terms of ''partial observables'' -- which may be used as internal clocks -- and a set of ''complete observables'' -- which are understood to evolve in terms of the former. This has led to a controversy around the interpretation of the partial observables within the formalism. In this talk, we will provide a negative argument against the orthodoxy view that clarifies the role that should be played by the partial observables. We then present a proposal for the canonical quantization of reparametrization invariant systems that naturally encodes a genuine notion of time evolution and illustrate how this proposal can be applied to gravity.

Henrique Gomes (Perimeter Institute) gives a talk at the Workshop on the Problem of Time in Perspective (3-4 July, 2015) titled "Timeless Quantum Mechanics in Configuration Space: an Outsider View". Abstract: In this talk, I will explore a timeless interpretation of quantum mechanics of closed systems, solely in terms of path integrals in non-relativistic timeless configuration space. What prompts a fresh look at the foundational problems in this context, is the advent of multiple gravitational models in which Lorentz symmetry is only emergent. In this setting, I propose a new understanding of records as certain relations between two configurations, the recorded one and the record-holding one. These relations are formalized through a factorization of the amplitude kernel, which forbids unwanted 'recoherence' of branches. On this basis, I show that in simple cases the Born rule is consistent with counting the relative density of observers with the same records. Furthermore, unlike what occurs in consistent histories, in this context there is indeed a preferred notion of coarse-grainings: those centered around piece-wise classical paths in configuration space (with a certain radius). Thus, this new understanding claims to resolve aspects of the measurement problem which are still deemed controversial in the standard approaches (but which probably leaves others open...).

Francesca Vidotto (Radboud) gives a talk at the Workshop on the Problem of Time in Perspective (3-4 July, 2015) titled "Discrete Time in Quantum Gravity". Abstract: We study the quantization of geometry in the presence of a cosmological constant, using a discretiza- tion with constant-curvature simplices. Phase space turns out to be compact and the Hilbert space finite dimensional for each link. Not only the intrinsic, but also the extrinsic geometry turns out to be discrete, pointing to discreetness of time, in addition to space. We work in 2+1 dimensions, but these results may be relevant also for the physical 3+1 case.