Hamiltonian Constraint Research Articles

Canonical loop quantization of the lowest-order projectable Horava gravity

The Hamiltonian formulation of the lowest-order projectable Horava gravity, namely the so-called $\lambda$-$R$ gravity, is studied. Since a preferred foliation has been chosen in projectable Horava gravity, there is no local Hamiltonian constraint in the theory. In contrast to general relativity, the constraint algebra of $\lambda$-$R$ gravity forms a Lie algebra. By canonical transformations, we further obtain the connection-dynamical formalism of the $\lambda$-R gravity theories with real $su(2)$-connections as configuration variables. This formalism enables us to extend the scheme of non-perturbative loop quantum gravity to the $\lambda$-$R$ gravity. While the quantum kinematical framework is the same as that for general relativity, the Hamiltonian constraint operator of loop quantum $\lambda$-$R$ gravity can be well defined in the diffeomorphism-invariant Hilbert space. Moreover, by introducing a global dust degree of freedom to represent a dynamical time, a physical Hamiltonian operator with respect to the dust can be defined and the physical states satisfying all the constraints are obtained.

Cosmological perturbation theory with matter time

We study cosmological perturbation theory with scalar field and pressureless dust in the Hamiltonian formulation, with the dust field chosen as a matter-time gauge. The corresponding canonical action describes the dynamics of the scalar field and metric degrees of freedom with a non-vanishing physical Hamiltonian and spatial diffeomorphism constraint. We construct a momentum space Hamiltonian that describes linear perturbations, and show that the constraints to this order form a first class system. We then write the Hamiltonian as a function of certain gauge invariant canonical variables and show that it takes the form of an oscillator with time dependent mass and frequency coupled to an ultralocal field. We compare our analysis with other Hamiltonian approaches to cosmological perturbation theory that do not use dust time.

Primordial scalar power spectrum from the hybrid approach in loop cosmologies

We compare the primordial scalar power spectra in the loop cosmological models using the effective dynamics of the hybrid approach to cosmological perturbations in which the background is loop quantized, but the perturbations are Fock quantized. The three loop cosmological models under consideration are the standard loop quantum cosmology (LQC), the modified LQC-I (mLQC-I) and the modified LQC-II (mLQC-II) in the spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker universe with a Starobinsky potential. These models arise from different regularizations of the classical Hamiltonian constraint in the symmetry reduced spacetimes and aim to capture certain features of quantization in loop quantum gravity. When applying the techniques in the hybrid approach to mLQC-I/II, we find the effective Mukhanov-Sasaki equations take the same form as in LQC. The difference among the three models is encoded in the unique expressions of the effective masses in each model. We find that the relative difference in the amplitude of power spectrum between LQC and mLQC-II is approximately 50% in the infrared and the oscillatory regimes, whereas this difference can be as large as 100% between mLQC-I and LQC/mLQC-II. Interestingly, in the infrared and the oscillatory regimes of mLQC-I, we obtain a suppressed power spectrum from the hybrid approach which is far below the Planck scale. This result is in a striking contrast to the one obtained from the dressed metric approach to perturbations where the corresponding amplitude in this regime is extremely large. Our analysis shows that while the phenomenological predictions are in agreement between the two approaches for LQC and mLQC-II, for mLQC-I the differences between the dressed and hybrid approaches can be quite significant. Our result provides the first robust evidence of difference in predictions between the dressed and hybrid approaches due to respective underlying constructions.

Hamiltonian constraints and unfree gauge symmetry

We study Hamiltonian form of unfree gauge symmetry where the gauge parameters have to obey differential equations. We consider the general case such that the Dirac-Bergmann algorithm does not necessarily terminate at secondary constraints, and tertiary and higher order constraints may arise. Given the involution relations for the first-class constraints of all generations, we provide explicit formulas for unfree gauge transformations in the Hamiltonian form, including the differential equations constraining gauge parameters. All the field theories with unfree gauge symmetry share the common feature: they admit sort of "global constants of motion" such that do not depend on the local degrees of freedom. The simplest example is the cosmological constant in the unimodular gravity. We consider these constants as modular parameters rather than conserved quantities. We provide a systematic way of identifying all the modular parameters. We demonstrate that the modular parameters contribute to the Hamiltonian constraints, while they are not explicitly involved in the action. The Hamiltonian analysis of the unfree gauge symmetry is precessed by a brief exposition for the Lagrangian analogue, including explicitly covariant formula for degrees of freedom number count. We also adjust the BFV-BRST Hamiltonian quantization method for the case of unfree gauge symmetry. The main distinction is in the content of the non-minimal sector and gauge fixing procedure. The general formalism is exemplified by traceless tensor fields of irreducible spin $s$ with the gauge symmetry parameters obeying transversality equations.

How to switch between relational quantum clocks

Every clock is a physical system and thereby ultimately quantum. A naturally arising question is thus how to describe time evolution relative to quantum clocks and, specifically, how the dynamics relative to different quantum clocks are related. This is a particularly pressing issue in view of the multiple choice facet of the problem of time in quantum gravity, which posits that there is no distinguished choice of internal clock in generic general relativistic systems and that different choices lead to inequivalent quantum theories. Exploiting a recent unifying approach to switching quantum reference systems [Vanrietvelde et al 2020 Quantum 4 225; Vanrietvelde et al 2018 arXiv:1809.05093[quant-ph])], we exhibit a systematic method for switching between different clock choices in the quantum theory. We illustrate it by means of the parametrized particle, which, like gravity, features a Hamiltonian constraint. We explicitly switch between the quantum evolution relative to the non-relativistic time variable and that relative to the particle’s position, which requires carefully regularizing the zero-modes in the so-called time-of-arrival observable. While this toy model is simple, our approach is general and, in particular, directly amenable to quantum cosmology. It proceeds by systematically linking the reduced quantum theories relative to different clock choices via the clock-choice-neutral Dirac quantized theory, in analogy to coordinate changes on a manifold. This method suggests a new perspective on the multiple choice problem, indicating that it is rather a multiple choice feature of the complete relational quantum theory, taken as the conjunction of Dirac quantized and quantum deparametrized theories. Precisely this conjunction permits one to consistently switch between different temporal reference systems, which is a prerequisite for a quantum notion of general covariance. Finally, we show that quantum uncertainties generically lead to a discontinuity in the relational dynamics when switching clocks, in contrast to the classical case.

Effective loop quantum gravity framework for vacuum spherically symmetric spacetimes

We develop an effective framework for the $\bar\mu$ scheme of holonomy corrections motivated by loop quantum gravity for vacuum spherically symmetric space-times. This is done by imposing the areal gauge in the classical theory, and then expressing the remaining components of the Ashtekar-Barbero connection in the Hamiltonian constraint in terms of holonomies of physical length $\ell_{\rm Pl}$. The stationary solutions to the effective Hamiltonian constraint can be found exactly, and we give the explicit form of the effective metric in Painlev\'e-Gullstrand coordinates. This solution has the correct classical limit, the quantum gravity corrections decay rapidly at large distances, and curvature scalars are bounded by the Planck scale, independently of the black hole mass $M$. In addition, the solution is valid for radii $x \ge x_{\rm min} \sim (\ell_{\rm Pl}^2 M)^{1/3}$ indicating the need for a matter field, with an energy density bounded by the Planck scale, to provide a source for the curvature in the space-time. Finally, for $M \gg m_{\rm Pl}$, the space-time has an outer and also an inner horizon, within which the expansion for outgoing radial null geodesics becomes positive again. On the other hand, for sufficiently small $M \sim m_{\rm Pl}$, there are no horizons at all in the effective metric.

Physical Hamiltonian for mimetic gravity

Starting from a local action for mimetic gravity that includes higher derivatives of a scalar field $\phi$, we derive a gauge-fixed canonical action of the theory in the ADM canonical formalism in the time gauge $\phi=t$. This reduced action reveals (i) a non-vanishing conserved physical Hamiltonian that is a sum of two terms, the expression for the Hamiltonian constraint of general relativity and a function of the expansion scalar, and (ii) a reduced symplectic structure that geometrically provides the Dirac brackets. As applications of our general analysis, we compute the physical Hamiltonians and canonical equations for perturbations around Minkowski spacetime, homogeneous cosmologies, and spherically symmetric spacetimes.

(2+1)-dimensional loop quantum cosmology of Bianchi I models

We study the anisotropic Bianchi I loop quantum cosmology in [Formula: see text] dimensions. The [Formula: see text] scheme is considered in the present paper and the following expected results are established: (i) the massless scalar field again play the role of emergent time variables and serves as an internal clock; (ii) by imposing the fundamental discreteness of length operator, the total Hamiltonian constraint is obtained and gives rise the evolution as a difference equation; and (iii) the exact solutions of Friedmann equation are constructed rigorously for both classical and effective level. The investigation extends the domain of validity of loop quantum cosmology to beyond the four dimensions.

Primordial perturbations in the Dapor-Liegener model of hybrid loop quantum cosmology

In this work, we extend the formalism of hybrid loop quantum cosmology for primordial perturbations around a flat, homogeneous, and isotropic universe to the new treatment of Friedmann-Lema\^{\i}tre-Robertson-Walker geometries proposed recently by Dapor and Liegener, based on an alternative regularization of the Hamiltonian constraint. In fact, our discussion is applicable also to other possible regularization schemes for loop quantum cosmology, although we specialize our analysis to the Dapor-Liegener proposal and construct explicitly all involved quantum operators for that case.

Loop quantum cosmology from an alternative Hamiltonian. II. Including the Lorentzian term

The scheme of using the Chern-Simons action to regularize the gravitational Hamiltonian constraint is extended to including the Lorentzian term in the $k=0$ cosmological model. The Euclidean term and the Lorenzian term are thus regularized separately mimicking the treatment of full loop quantum gravity. The new quantum dynamics for the spatially flat Friedmann-Robertson-Walker model with a massless scalar field as an emergent time is studied. By semiclassical analysis, the effective Hamiltonian constraint is obtained, which indicates that the new quantum dynamics has the correct classical limit. The classical big-bang singularity is again replaced by a quantum bounce. Similar to the case of quantizing only the Euclidean term, the backward evolution of the cosmological model determined by the new effective Hamiltonian will be bounced to an asymptotic de Sitter universe coupled to a massless scalar field, while the problem of negative energy density of matter in the former case is resolved.

DC conductivities and Stokes flows in Dirac semimetals influenced by hidden sector

In the holographic model of Dirac semimetals, the Einstein–Maxwell scalar gravity with the auxiliary U(1)-gauge field, coupled to the ordinary Maxwell one by a kinetic mixing term, the black brane response to the electric fields and temperature gradient has been elaborated. Using the foliation by hypersurfaces of constant radial coordinate we derive the exact form of the Hamiltonian and equations of motion in the phase space considered. Examination of the Hamiltonian constraints enables us, to the leading order expansion of the linearised perturbations at the black brane event horizon, to derive the Stokes equations for an incompressible doubly charged fluid. Solving the aforementioned equations, one arrives at the DC conductivities for the holographic Dirac semimetals.

Space-time collocation method: Loop quantum Hamiltonian constraints

A space-time collocation method (STCM) using asymptotically-constant basis functions is proposed and applied to the quantum Hamiltonian constraint for a loop-quantized treatment of the Schwarzschild interior. Canonically, these descriptions take the form of a partial difference equation (PDE). The space-time collocation approach presents a computationally efficient, convergent, and easily parallelizable method for solving this class of equations, which is the main novelty of this study. Results of the numerical simulations will demonstrate the benefit from a parallel computing approach; and show general flexibility of the framework to handle arbitrarily-sized domains. Computed solutions will be compared, when applicable, to a solution computed in the conventional method via iteratively stepping through a predefined grid of discrete values, computing the solution via a recursive relationship.

Quantum geometry from higher gauge theory

Higher gauge theories play a prominent role in the construction of 4D topological invariants and have been long ago proposed as a tool for 4D quantum gravity. The Yetter lattice model and its continuum counterpart, the BFCG theory, generalize BF theory to 2-gauge groups and—when specialized to 4D and the Poincaré 2-group—they provide an exactly solvable topologically-flat version of 4D general relativity. The 2-Poincaré Yetter model was conjectured to be equivalent to a state sum model of quantum flat spacetime developed by Baratin and Freidel after work by Korepanov (KBF model). This conjecture was motivated by the origin of the KBF model in the theory of two-representations of the Poincaré 2-group. Its proof, however, has remained elusive due to the lack of a generalized Peter–Weyl theorem for 2-groups. In this work we prove this conjecture. Our proof avoids the Peter–Weyl theorem and rather leverages the geometrical content of the Yetter model. Key for the proof is the introduction of a kinematical boundary Hilbert space on which 1- and two-Lorentz invariance is imposed. Geometrically this allows the identification of (quantum) tetrad variables and of the associated (quantum) Levi-Civita connection. States in this Hilbert space are labelled by quantum numbers that match the two-group representation labels. Our results open exciting opportunities for the construction of new representations of quantum geometries. Compared to loop quantum gravity, the higher gauge theory framework provides a quantum representation of the ADM—Regge initial data, including an identification of the intrinsic and extrinsic curvature. Furthermore, it leads to a version of the diffeomorphism and Hamiltonian constraints that acts on the vertices of the discretization, thus providing a prospect for a quantum realization of the hypersurface deformation algebra in 4D.

Equivalence of the Chern-Simons state and the Hartle-Hawking and Vilenkin wave functions

We show that the Chern-Simons (CS) state when reduced to minisuperspace is the Fourier dual of the Hartle-Hawking (HH) and Vilenkin (V) wave functions of the Universe. This is to be expected, given that the former and latter solve the same constraint equation, written in terms of conjugate variables (loosely, the expansion factor and the Hubble parameter). A number of subtleties in the mapping, related to the contour of integration of the connection, shed light on the issue of boundary conditions in quantum cosmology. If we insist on a real Hubble parameter, then only the HH wave function can be represented by the CS state, with the Hubble parameter covering the whole real line. For the V (or tunneling) wave function, the Hubble parameter is restricted to the positive real line (which makes sense, since the state only admits outgoing waves), but the contour also covers the whole negative imaginary axis. Hence, the state is not admissible if reality conditions are imposed upon the connection. Modifications of the V state, requiring the addition of source terms to the Hamiltonian constraint, are examined and found to be more palatable. In the dual picture, the HH state predicts a uniform distribution for the Hubble parameter over the whole real line; the modified V state a uniform distribution over the positive real line.

Loop quantum Schwarzschild interior and black hole remnant

The interior of Schwarzschild black hole is quantized by the method of loop quantum gravity. The Hamiltonian constraint is solved and the physical Hilbert space is obtained in the model. The properties of a Dirac observable corresponding to the ADM mass of the Schwarzschild black hole are studied by both analytical and numerical techniques. It turns out that zero is not in the spectrum of this Dirac observable. This supports the existence of a stable remnant after the evaporation of a black hole. Our conclusion is valid for a general class of schemes adopted for loop quantization of the model.

Minimally modified gravity fitting Planck data better than Lambda CDM

We study the phenomenology of a class of minimally modified gravity theories called f(mathcal {H}) theories, in which the usual general relativistic Hamiltonian constraint is replaced by a free function of it. After reviewing the construction of the theory and a consistent matter coupling, we analyze the dynamics of cosmology at the levels of both background and perturbations, and present a concrete example of the theory with a 3-parameter family of the function f. Finally, we compare this example model to Planck data as well as some later-time probes, showing that such a realization of f(mathcal {H}) theories fits the data significantly better than the standard Lambda CDM model, in particular by modifying gravity at intermediate redshifts, zsimeq 743.

Alternative dynamics in loop quantum Brans-Dicke cosmology

To inherit more features of full loop quantum Brans-Dicke theory, the Euclidean and Lorentzian terms of the Hamiltonian constraint are quantized independently in loop quantum Brans-Dicke cosmology. An alternative Hamiltonian constraint operator and its effective expression are obtained in the cosmological model. A residual quantum correction term is found in the effective Hamiltonian constraint, which has no analog in the effective Hamiltonian of the loop quantum cosmology from general relativity. The dynamics driven by this effective Hamiltonian constraint is analyzed in detail. For the physically interesting case of $\omega\gg 1$, this effective Hamiltonian drives a bouncing evolution which evolves from a de Sitter universe to a classical Brans-Dicke solution.

The Hamiltonian dynamics of Ho\u0159ava gravity

We consider the Hamiltonian formulation of Hořava gravity in arbitrary dimensions, which has been proposed as a renormalizable gravity model for quantum gravity without the ghost problem. We study the full constraint analysis of the non-projectable Hořava gravity whose potential, mathcal{V}(R), is an arbitrary function of the (intrinsic) Ricci scalar R but without the extension terms which depend on the proper acceleration a_i. We find that there exist generally three distinct cases of this theory, A, B, and C, depending on (i) whether the Hamiltonian constraint generates new (second-class) constraints or just fixes the associated Lagrange multipliers, or (ii) whether the IR Lorentz-deformation parameter {lambda } is at the conformal point or not. It is found that, for Cases A and C, the dynamical degrees of freedom are the same as in general relativity, while, for Case B, there is one additional phase-space degree of freedom, representing an extra (odd) scalar graviton mode. This would achieve the dynamical consistency of a restricted model at the fully non-linear level and be a positive result in resolving the long-standing debates about the extra graviton modes of the Hořava gravity. Several exact solutions are also studied as some explicit examples of the new constraints. The structure of the newly obtained, “extended” constraint algebra seems to be generic to Hořava gravity and its general proof would be a challenging problem. Some other challenging problems, which include the path integral quantization and the Dirac bracket quantization are discussed also.

Constrained dynamics: generalized Lie symmetries, singular Lagrangians, and the passage to Hamiltonian mechanics

Guided by the symmetries of the Euler–Lagrange equations of motion, a study of the constrained dynamics of singular Lagrangians is presented. We find that these equations of motion admit a generalized Lie symmetry, and on the Lagrangian phase space the generators of this symmetry lie in the kernel of the Lagrangian two-form. Solutions of the energy equation—called second-order, Euler–Lagrange vector fields (SOELVFs)—with integral flows that have this symmetry are determined. Importantly, while second-order, Lagrangian vector fields are not such a solution, it is always possible to construct from them a SOELVF that is. We find that all SOELVFs are projectable to the Hamiltonian phase space, as are all the dynamical structures in the Lagrangian phase space needed for their evolution. In particular, the primary Hamiltonian constraints can be constructed from vectors that lie in the kernel of the Lagrangian two-form, and with this construction, we show that the Lagrangian constraint algorithm for the SOELVF is equivalent to the stability analysis of the total Hamiltonian. Importantly, the end result of this stability analysis gives a Hamiltonian vector field that is the projection of the SOELVF obtained from the Lagrangian constraint algorithm. The Lagrangian and Hamiltonian formulations of mechanics for singular Lagrangians are in this way equivalent.

Cosmological spinor

We build upon previous investigation of the one-dimensional conformal symmetry of the Friedman-Lema\^ itre-Robertson-Walker (FLRW) cosmology of a free scalar field and make it explicit through a reformulation of the theory at the classical level in terms of a manifestly $\textrm{SL}(2,\mathbb{R})$-invariant action principle. The new tool is a canonical transformation of the cosmological phase space to write it in terms of a spinor, i.e. a pair of complex variables that transform under the fundamental representation of $\textrm{SU}(1,1)\sim\textrm{SL}(2,\mathbb{R})$. The resulting FLRW Hamiltonian constraint is simply quadratic in the spinor and FLRW cosmology is written as a Schr\"odinger-like action principle. Conformal transformations can then be written as proper-time dependent $\textrm{SL}(2,\mathbb{R})$ transformations. We conclude with possible generalizations of FLRW to arbitrary quadratic Hamiltonian and discuss the interpretation of the spinor as a gravitationally-dressed matter field or matter-dressed geometry observable.