Time Reparametrization Invariance Research Articles

Beyond semiclassical time: dynamics in quantum cosmology

We review two approaches to the definition of the Hilbert space and evolution in mechanical theories with local time-reparametrization invariance, which are often used as toy models of quantum gravity. The first approach is based on the definition of invariant relational observables, whereas the second formalism consists of a perturbative construction of the Hilbert space and a weak-coupling expansion of the Hamiltonian constraint, which is frequently performed as part of the Born-Oppenheimer treatment in quantum cosmology. We discuss in which sense both approaches exhibit an inner product that is gauge-fixed via an operator version of the usual Faddeev-Popov procedure, and, in the second approach, how the unitarity of the effective Schrödinger evolution is established perturbatively. We note that a conditional probability interpretation of the physical states is possible, so that both formalisms are examples of quantum mechanics with a relational dynamics.

Pseudoclassical system with gauge and time-reparametrization invariance

We present a pseudoclassical mechanics model which exhibits gauge symmetry and time-reparametrization invariance. As such, first- and second-class constraints restrict the phase space, and the Hamiltonian weakly vanishes. We show that the Dirac conjecture does not hold---the secondary first-class constraint is not a symmetry generator---and only the gauge fixing condition associated with the primary first-class constraint is needed to remove the gauge ambiguities. The gauge fixed theory is equivalent to the Fermi harmonic oscillator extended by a boundary term. We quantize in the deformation quantization and in the Schr\"odinger representation approaches and observe that the boundary term prepares the system in the state of positive energy.

The constraints of post-quantum classical gravity

We study a class of theories in which space-time is treated classically, while interacting with quantum fields. These circumvent various no-go theorems and the pathologies of semi-classical gravity, by being linear in the density matrix and phase-space density. The theory can either be considered fundamental or as an effective theory where the classical limit is taken of space-time. The theories have the dynamics of general relativity as their classical limit and provide a way to study the back-action of quantum fields on the space-time metric. The theory is invariant under spatial diffeomorphisms, and here, we provide a methodology to derive the constraint equations of such a theory by imposing invariance of the dynamics under time-reparametrization invariance. This leads to generalisations of the Hamiltonian and momentum constraints. We compute the constraint algebra for a wide class of realisations of the theory (the “discrete class”) in the case of a quantum scalar field interacting with gravity. We find that the algebra doesn’t close without additional constraints, although these do not necessarily reduce the number of local degrees of freedom.

Proper time reparametrization in cosmology: Möbius symmetry and Kodama charges

It has been noticed that for a large class of cosmological models, the gauge fixing of the time-reparametrization invariance does not completely fix the clock. Instead, the system enjoys a surprising residual Noether symmetry under a Möbius reparametrization of the proper time, which maps gauge-inequivalent solutions to the Friedmann equations onto each other. In this work, we provide a unified treatment of this hidden conformal symmetry and its realization in the homogeneous and isotropic sector of the Einstein-Scalar-Λ system. We consider the flat Friedmann-Robertson-Walker (FRW) model, the (A)dS cosmology and provide a first treatment of the model with spatial constant curvature.We derive the general condition relating the choice of proper time and the conformal weight of the scale factor, and give a detailed analysis of the conserved Noether charges generating this physical symmetry. Our approach allows us to identify new realizations of this symmetry while recovering previous results in a unified manner. We also present the general mapping onto the conformal particle and discuss the solution-generating nature of the transformations beyond the Möbius symmetry. Finally, we show that, at least in a restricted context, this hidden conformal symmetry is intimately related to the Kodama charges of spherically symmetric gravity. This new connection suggests that the Möbius invariance of cosmology is only the corner of a larger symmetry structure which could be relevant beyond cosmological models.

Time parametrizations in long-range interacting Bose-Einstein condensates

Time-reparametrization invariance in general relativistic space-time does not allow us to single out a time in quantum mechanics in a mechanical way of measurement. Motivated by this problem, we examine this gauge invariance in the ground state of the quasi-stationary coarse-grained state of a long-range interacting closed system of identical or identified, macroscopic, and spatiotemporally inhomogeneous Bose–Einstein condensates in the thermodynamic and Newtonian limits. As a result, we find that it is a theoretical counterexample of this gauge invariance, except for proper-time translational invariance, at a coarse-grained level.

Classical glasses, black holes, and strange quantum liquids

From the dynamics of a broad class of classical mean-field glass models one may obtain a quantum model with finite zero-temperature entropy, a quantum transition at zero temperature, and a time-reparametrization (quasi-)invariance in the dynamical equations for correlations. The low eigenvalue spectrum of the resulting quantum model is directly related to the structure and exploration of metastable states in the landscape of the original classical glass model. This mapping reveals deep connections between classical glasses and the properties of SYK-like models.

Discreteness of time in the evolution of the universe

In this paper, we will first derive the Wheeler–DeWitt equation for the generalized geometry which occurs in M-theory. Then we will observe that M2-branes act as probes for this generalized geometry, and as M2-branes have an extended structure, their extended structure will limits the resolution to which this generalized geometry can be defined. We will demonstrate that this will deform the Wheeler–DeWitt equation for the generalized geometry. We analyze such a deformed Wheeler–DeWitt equation in the minisuperspace approximation, and observe that this deformation can be used as a solution to the problem of time. This is because this deformation gives rise to time crystals in our universe due to the spontaneous breaking of time reparametrization invariance.

Constraints from conformal symmetry on the three point scalar correlator in inflation

Using symmetry considerations, we derive Ward identities which relate the three point function of scalar perturbations produced during inflation to the scalar four point function, in a particular limit. The derivation assumes approximate conformal invariance, and the conditions for the slow roll approximation, but is otherwise model independent. The Ward identities allow us to deduce that the three point function must be suppressed in general, being of the same order of magnitude as in the slow roll model. They also fix the three point function in terms of the four point function, upto one constant which we argue is generically suppressed. Our approach is based on analyzing the wave function of the universe, and the Ward identities arise by imposing the requirements of spatial and time reparametrization invariance on it.

Lorentz invariance in shape dynamics

Shape dynamics is a reframing of canonical general relativity in which time reparametrization invariance is ‘traded’ for a local conformal invariance. We explore the emergence of Lorentz invariance in this model in three contexts: as a maximal symmetry, an asymptotic symmetry and a local invariance.

Remarks on gauge invariance and first-class constraints

Gauge symmetries lead to first-class constraints. This assertion is, of course, true only for non-trivial gauge symmetries, i.e., gauge symmetries that act non-trivially on-shell on the dynamical variables. We illustrate this well-appreciated fact for time reparametrization invariance in the context of modifications of gravity-suggested in a recent proposal by Hořava-in which the Hamiltonian constraint is deformed by arbitrary spatial diffeomorphism invariant terms, where some subtleties are found to arise.

A dynamical inconsistency of Hořava gravity

The dynamical consistency of the non-projectable version of Horava gravity is investigated by focusing on the asymptotically flat case. It is argued that for generic solutions of the constraint equations the lapse must vanish asymptotically. We then consider particular values of the coupling constants for which the equations are tractable and in that case we prove that the lapse must vanish everywhere -- and not only at infinity. Put differently, the Hamiltonian constraints are generically all second-class. We then argue that the same feature holds for generic values of the couplings, thus revealing a physical inconsistency of the theory. In order to cure this pathology, one might want to introduce further constraints but the resulting theory would then lose much of the appeal of the original proposal by Horava. We also show that there is no contradiction with the time reparametrization invariance of the action, as this invariance is shown to be a so-called "trivial gauge symmetry" in Horava gravity, hence with no associated first-class constraints.

Glassy dynamics and coarsening

In this paper I present a short review of the present theoretical understanding of the dynamics, global and fluctuating, in the glassy out of equilibrium regime. The first part of the talks dealt with a summary of the results of mean-field models and served to introduce the conjecture that time-reparametrization invariance controls the fluctuating dynamics. Finally, I briefly discussed very recent results on the domain size fluctuations in coarsening problems and how these and similar studies of the dynamics of elastic lines could help us in understanding the evolution of more complex glassy systems.

Fluctuations in the coarsening dynamics of the O(N) model with N → ∞: are they similar to those in glassy systems?

We study spatio-temporal fluctuations in the non-equilibrium dynamics of thed-dimensionalO(N) model inthe large N limit. We analyse the invariance of the dynamic equations for the global correlation andresponse in the slow ageing regime under transformations of time. We find that theseequations are invariant under scale transformations. We extend this study to the action inthe dynamic generating functional, finding similar results. This model therefore fallsinto a different category to glassy problems in which full time reparametrizationinvariance, a larger symmetry that encompasses timescale invariance, is expected to berealized asymptotically. Consequently, the spatio-temporal fluctuations of the largeN O(N) model should follow a pattern different from that for glassy systems. We compute thefluctuations of local, as well as spatially separated, two-field composite operators andresponses, and we confront our results with the ones found numerically for the 3DEdwards–Anderson model and kinetically constrained lattice gases. We analyse thedependence of the fluctuations of the composite operators on the growing domain lengthand we compare to what has been found for supercooled liquids and glasses. Finally, weshow that the development of time reparametrization invariance in glassy systems isintimately related to a well-defined and finite effective temperature, specified from themodification of the fluctuation-dissipation theorem out of equilibrium. We then conjecturethat the global asymptotic time reparametrization invariance is broken down to timescaleinvariance in all coarsening systems.

A sigma-model approach to glassy dynamics

In this contribution we review recent progress in understanding fluctuations in the aging process of macroscopic systems, and we propose further tests of these ideas. We discuss how the emergence of a symmetry in aging systems, global timereparametrization invariance, could be responsible for the observed ‘universal’ behavior of local and mesoscopic non-equilibrium fluctuations. We discuss (i) the two-time scaling and functional form of the distribution of local correlations and responses; (ii) the scaling of multi-time correlations and susceptibilities; (iii) how the above can be derived from a random surface effective action; (iv) the behavior of a diverging two-time dependent correlation length; (v) how these ideas apply to off-lattice particle systems.

Chern-Simons state for the nondiagonal Bianchi type IX model

The Bianchi IX mixmaster model is quantized in its non-diagonal form, imposing spatial diffeomorphism, time reparametrization and Lorentz invariance as constraints on physical state vectors before gauge-fixing. The result turns out to be different from quantizing the diagonal model obtained by gauge-fixing already on the classical level. For the non-diagonal model a generalized 9-dimensional Fourier transformation over a suitably chosen manifold connects the representations in metric variables and in Ashtekar variables. A space of five states in the metric representation is generated from the single physical Chern-Simons state in Ashtekar variables by choosing five different integration manifolds, which cannot be deformed into each other. For the case of a positive cosmological constant $\Lambda$ we extend our previous study of these five states for the diagonal Bianchi IX model to the non-diagonal case. It is shown that additional discrete (permutation) symmetries of physical states arise in the quantization of the non-diagonal model, which are satisfied by two of the five states connected to the Chern-Simons state. These have the characteristics of a wormhole groundstate and a Hartle-Hawking `no-boundary' state, respectively. We also exhibit a special gauge-fixing of the time reparametrization invariance of the quantized system and define an associated manifestly positive scalar product. Then the wormhole ground state is left as the only normalizable physical state connected to the Chern-Simons state.

A SCHRÖDINGER EQUATION FOR MINIUNIVERSES

We discuss how to fix the gauge in the canonical treatment of Lagrangians, with a finite number of degrees of freedom, endowed with time reparametrization invariance. The motion can be described by an effective Hamiltonian acting on the gauge shell canonical space. The system is then suited for quantization. We apply this treatment to the case of a Robertson–Walker metric interacting with zero modes of bosonic fields and write a Schrödinger equation for the on-shell wave function.

Supersymmetric general Bianchi type IX cosmology with a cosmological term

The covariant action of the supersymmetrically extended spatially homogeneous general Bianchi type IX cosmology is given including time reparametrization invariance, inducing the hamiltonian constraint, and the constraints on the supercharges. A supersymmetric cosmological term is also included. A quantization respecting the supersymmetry and the conformal invariance of the superspace of geometrodynamics is given. The problem of the initial condition is naturally solved in this description by demanding that the wave function does not diverge if the anisotropy variables are taken to infinity for fixed scale parameter.

Two-dimensional quantum gravity in Minkowski space

We study exactly soluble models of gravity in two-dimensional Minkowski space. It is argued that an interpretation of these models as causal unitary quantum mechanics is possible only if we work in a Fock space of two-dimensional universes. The argument is general and should be valid for any generally covariant Minkowski field theory which contains the metric as a dynamical variable. It depends only on global time reparametrization invariance. We conjecture that the local time reparametrization invariance of theories of extended objects actually requires that the second quantized theory of such objects be an interacting Universal Field Theory, i.e. that it must contain wormhole processes.