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 Schrodinger representation approaches and observe that the boundary term prepares the system in the state of positive energy.
Systems invariant under the reparametrization of time were treated as constrained systems within Hamilton-Jacobi formalism. After imposing the integrability conditions the time-dependent Schrödinger equation was obtained. Three examples are investigated in details.
Reparametrization invariance being treated as a gauge symmetry shows some specific peculiarities. We study these peculiarities both from a general point of view and on concrete examples. We consider the canonical treatment of reparametrization invariant systems in which one fixes the gauge on the classical level by means of time-dependent gauge conditions. In such an approach one can interpret different gauges as different reference frames. We discuss the relations between different gauges and the problem of gauge invariance in this case. Finally, we establish a general structure of reparametr…
We study the fate of reparametrization invariance of Wilson loops, also known as 'zig-zag' symmetry, under the RG flow using some simple cases as guidance. We restrict our analysis to large-$N$, strongly coupled CFTs and use the holographic dual description of a Wilson loop as a fundamental string embedded in asymptotically AdS spaces, at zero and nonzero temperature. We then introduce a cutoff in the holographic radial direction and integrate out the the section of the string closer to the AdS boundary in the spirit of holographic Wilsonian renormalization. We make explicit the map between Wi…
In the present work we consider a time-dependent Schrödinger equation for systems invariant under the reparametrization of time. We develop the two-stage procedure of construction such systems from a given initial ones, which is not invariant under the time reparametrization. One of the first-class constraints of the systems in such description becomes the time-dependent Schrödinger equation. The procedure is applicable in the supersymmetric theories as well. The $n=2$ supersymmetric quantum mechanics is coupled to world-line supergravity, and the local supersymmetric action is constructed lea…
The superdiffeomorphisms invariant description of $N$ - extended spinning particle is constructed in the framework of nonlinear realizations approach. The action is universal for all values of $N$ and describes the time evolution of $D+2$ different group elements of the superdiffeomorphisms group of the $(1,N)$ superspace. The form of this action coincides with the one-dimensional version of the gravity action, analogous to Trautman's one.
We demonstrate the reparametrization invariance of perturbatively defined one-dimensional functional integrals up to the three-loop level for a path integral of a quantum-mechanical point particle in a box. We exhibit the origin of the failure of earlier authors to establish reparametrization invariance which led them to introduce, superfluously, a compensating potential depending on the connection of the coordinate system. We show that problems with invariance are absent by defining path integrals as the epsilon-> 0 -limit of 1+ epsilon -dimensional functional integrals.
We extend existing work on reparametrization invariance (RPI) of the heavy-quark expansion. We discuss the total rates of inclusive processes and obtain results which have a manifest RPI and can be expressed through matrix elements of operators and states defined in full QCD. This approach leads to a partial re-summation of higher-order terms in the heavy-quark expansion and has the advantage that the number of independent parameters is reduced.
The construction of physical models with local time-reparametrization invariance is reviewed. Negative-energy contributions to the hamiltonian are shown to be crucial for the realization of this reparametrization symmetry. The covariant formulation of the dynamics is used to develop a time and gauge invariant Hamilton-Jacobi theory. This formalism is applied to solve for the cosmology of a homogeneous universe of the Friedmann-Lemaitre-Robertson-Walker type. After a discussion of empty universes, the FLRW theory is extended with homogeneous scalar fields generically described by a $\sg$-model …
We show how to perform integrals over products of distributions in coordinate space such as to reproduce the results of momentum space Feynman integrals in dimensional regularization. This ensures the invariance of path integrals under coordinate transformations. The integrals are uniquely defined by expressing the propagators in 1- epsilon dimensions in terms of modified Bessel functions.
The Parisi scheme for equilibrium and the corresponding slow dynamics with multithermalization - same temperature common to all observables, different temperatures only possible at widely separated timescales -- imply one another. Consistency requires that two systems brought into infinitesimal coupling be able to rearrange their timescales in order that all their temperatures match: this time reorganisation is only possible because the systems have a set of time-reparametrization invariances, that are thus seen to be an essential component of the scenario.
Two topics in soft collinear effective theory (SCET) for gravitational interactions are explored. First, the collinear Wilson lines---necessary building blocks for maintaining multiple copies of diffeomorphism invariance in gravity SCET---are extended to all orders in the SCET expansion parameter $λ$, where it has only been known to $O(λ)$ in the literature. Second, implications of reparametrization invariance (RPI) for the structure of gravity SCET lagrangians are studied. The utility of RPI is illustrated by an explicit example in which $O(λ^2)$ hard interactions of a collinear graviton are …
I address some recently raised issues regarding the time-parametrization dependence in stochastic descriptions of eternal inflation. To clarify the role of the choice of the time gauge, I show examples of gauge-dependent as well as gauge-independent statements about physical observables in eternally inflating spacetimes. In particular, the relative abundance of thermalized and inflating regions is highly gauge-dependent. The unbounded growth of the 3-volume of the inflating regions is found in certain time gauges, such as the proper time or the scale factor gauge. Yet in the same spacetimes th…
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.
We consider the Hartle-Hawking wavefunction of the universe defined as a Euclidean path integral that satisfies the "no-boundary proposal." We focus on the simplest minisuperspace model that comprises a single scale factor degree of freedom and a positive cosmological constant. The model can be seen as a non-linear $σ$-model with a line-segment base. We reduce the path integral over the lapse function to an integral over the proper length of the base and use diffeomorphism-invariant measures for the ghosts and the scale factor. As a result, the gauge-fixed path integral is independent of the g…
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 Horava-in which the Hamiltonian constraint is deformed by arbitrary spatial diffeomorphism invariant terms, where some subtleties are found to arise.
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 Friedman 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…
The quantization of General Relativity invariant with respect to time-reparametrizations is considered. We construct the Faddeev-Popov generating functional for the unitary perturbation theory in terms of invariants of the kinemetric group of diffeomorphisms of a frame of reference as a set of Einstein's observers with the equivalent Hamiltonian description ($t'=t'(t)$, $x'^i=x'^i(t,x^1,x^2,x^3)$). The algebra of the kinemetric group has other dimensions than the constraint algebra in the conventional Dirac-Faddeev-Popov (DFP) approach to quantization. To restore the reparametrization invarian…
We discuss the constraints induced by the algebra of the Poincare' generators on non-relativistic effective field theories. In the first part we derive some relations among the matching coefficients of the HQET (and NRQCD), which have been formerly obtained by use of reparametrization invariance. In the second part we obtain new constraints on the matching coefficients of pNRQCD.
In this paper we discuss gravity in the light-front formulation (light-cone gauge) and show how possible counterterms arise. We find that Poincare invariance is not enough to find the three-point counterterms uniquely. Higher-spin fields can intrude and mimic three-point higher derivative gravity terms. To select the correct term we have to use the remaining reparametrization invariance that exists after the gauge choice. We finally sketch how the corresponding programme for N=8 Supergravity should work.
We study the space-time invariances of the bosonic relativistic particle and bosonic relativistic string using general formulations obtained by incorporating the Hamiltonian constraints into the formalism. We point out that massless particles and tensionless strings have a larger set of space-time invariances than their massive and tensionful partners, respectively. We also show that it is possible to use the reparametrization invariance of the string formulation we present to reach the classical conformal equations of motion without the use of two-dimensional Weyl scalings of the string world…
Approximation only by derivative (or more generally momentum) expansions, combined with reparametrization invariance, turns the continuous renormalization group for quantum field theory into a set of partial differential equations which at fixed points become non-linear eigenvalue equations for the anomalous scaling dimension $η$. We review how these equations provide a powerful and robust means of discovering and approximating non-perturbative continuum limits. Gauge fields are briefly discussed. Particular emphasis is placed on the rôle of reparametrization invariance, and the convergence of…
We study two-dimensional Einstein-aether (or equivalently Hořava-Lifshitz) gravity, which has an $AdS_2$ solution. We examine various properties of this solution in the context of holography. We first show that the asymptotic symmetry group is the full set of time reparametrizations, the one-dimensional conformal group. At the same time there are configurations with finite energy and temperature, which indicate a violation of the Ward identity associated with one-dimensional conformal invariance. These solutions are characterized by a universal causal horizon and we show that the associated en…
We study a modified bosonic string theory that has a pressureless ``dust'' field on the string worldsheet. The dust is a real scalar field with unit gradient which breaks conformal invariance. Hamiltonian analysis reveals a time reparametrization constraint linear in the dust field momentum and a spatial diffeomorphism constraint. This feature provides a natural ``dust time" gauge in analogy with the parametrized particle. In this gauge we give a Fock quantization of the theory, which is complete and self-consistent in d<26. The Hamiltonian of the theory is not a constraint; as a consequence t…
We study the possibility that the approximate time shift symmetry during inflation is promoted to the full invariance under time reparametrization t \to \tilde t(t), or equivalently under field redefinition of the inflaton ϕ\to \tildeϕ(ϕ). The symmetry allows only two operators at leading order in derivatives, so that all n-point functions of scalar perturbations are fixed in terms of the power spectrum normalization and the speed of sound. During inflation the decaying mode only decays as 1/a and this opens up the possibility to violate some of the consistency relations in the squeezed limit,…
We study the Neveu-Schwarz-Ramond (NSR) open super-string theory in the proper-time gauge. The string field action is obtained by evaluating the Polyakov string path integral. In this study, we focus on the open-string free-field action, which corresponds to the string path integral on a strip. Depending on the periodicity of the fermion fields, the open super-string has two sectors: The Neveu-Schwarz (NS) and Ramond (R) sectors. We can impose the gauge conditions to fix the (super-) reparametrization invariance on the two-dimensional metric and its super-partner on the string world sheet to s…
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 con…
Following a minisuperspace approach to the dynamics of a spherically symmetric shell, a reduced Lagrangian for the radial degree of freedom is derived directly from the Einstein-Hilbert action. The key feature of this new Lagrangian is its invariance under time reparametrization. Indeed, all classical and quantum dynamics is encoded in the Hamiltonian constraint that follows from that invariance. Thus, at the classical level, we show that the Hamiltonian constraint reproduces, in a simple gauge, Israel's matching condition which governs the evolution of the shell. In the quantum case, the vani…
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 diffeomorph…
The high degree of symmetry renders the dynamics of cosmological as well as some black hole spacetimes describable by a system of finite degrees of freedom. These systems are generally known as minisuperspace models. One of their important key features is the invariance of the corresponding reduced actions under reparametrizations of the independent variable, a fact that can be seen as the remnant of the general covariance of the full theory. In the case of a system of $n$ degrees of freedom, described by a Lagrangian quadratic in velocities, one can use the lapse by either gauge fixing it or …
We propose that the gauge principle of d-dimensional Euclidean quantum gravity is Weyl invariance in its stochastic (d+1)-dimensional bulk. Observables are defined as depending only on conformal classes of d-dimensional metrics. We work with the second order stochastic quantization of Einstein equations in a (d+1)-dimensional bulk. There, the evolution is governed by the stochastic time, which foliates the bulk into Euclidean d-dimensional leaves. The internal metric of each leaf can be parametrized by its unimodular part and conformal factor. Additional bulk metric components are the ADM stoc…
