Reduced Phase Space Quantization Research Articles

A gravitationally induced decoherence model using Ashtekar variables

We consider the coupling of a scalar field to linearised gravity and derive a relativistic gravitationally induced decoherence model using Ashtekar variables. The model is formulated at the gauge invariant level using suitable geometrical clocks in the relational formalism, broadening existing gauge invariant formulations of decoherence models. For the construction of the Dirac observables we extend the known observable map by a kind of dual map where the role of clocks and constraints is interchanged. We also discuss a second choice of geometrical clocks existing in the ADM literature. Then we apply a reduced phase space quantisation on Fock space and derive the final master equation choosing a Gibbs state for the gravitational environment and using the projection operator technique. The resulting master equation is not automatically of Lindblad type, a starting point sometimes assumed for phenomenological models, but still involves a residual time dependence at the level of the effective operators in the master equation due to the form of the correlation functions that we express in terms of thermal Wightman functions. Furthermore, we discuss why in the model analysed here the application of a second Markov approximation in order to obtain a set of time independent effective system operators is less straightforward than in some of the quantum mechanical models.

Causal diamonds in ( 2+1 )-dimensional quantum gravity

We develop the reduced phase space quantization of causal diamonds in pure ($2+1$)-dimensional gravity with a nonpositive cosmological constant. The system is defined as the domain of dependence of a topological disc with fixed boundary metric. By solving the initial value constraints in a constant-mean-curvature time gauge and removing all the spatial gauge redundancy, we find that the phase space is the cotangent bundle of ${\text{Diff}}^{+}({S}^{1})/\mathrm{PSL}(2,\mathbb{R})$. To quantize this phase space we apply Isham's group-theoretic quantization scheme, with respect to a ${\mathrm{BMS}}_{3}$ group, and find that the quantum theory can be realized by wave functions on some coadjoint orbit of the Virasoro group, with labels in irreducible unitary representations of the corresponding little group. We find that the twist of the diamond boundary loop is quantized in integer or half-integer multiples of the ratio of the Planck length to the boundary length.

Reduced phase space quantization of black holes: Path integrals and effective dynamics

We consider the loop quantum theory of the spherically symmetric model of gravity coupled to Gaussian dust fields, where the Gaussian dust fields provide a material reference frame of the space and time to deparametrize gravity. This theory, used to study the quantum features of the spherically symmetric BH, is constructed based on a 1-dimensional lattice $\ensuremath{\gamma}\ensuremath{\subset}\mathbb{R}$. Taking advantage of the path integral formulation, we investigate the quantum dynamics and obtain an effective action. With this action, we get an effective continuous description of this quantum lattice system which is not the same as the one described by the effective Hamiltonian used in [M. Han and H. Liu, Improved effective dynamics of loop-quantum-gravity black hole and Nariai limit], i.e., the classical Hamiltonian with the holonomy correction. It turns out that, the Hamiltonian derived in this paper can return to that used in [M. Han and H. Liu] only for macro black holes, since the lattice $\ensuremath{\gamma}$ is required to be sufficiently fine. Indeed, it is necessary to propose this fine-grained lattice structure in order to well describe the underlying lattice theory by the continuous description.

Reduced phase space approach to the U(1)3 model for Euclidean quantum gravity

If one replaces the constraints of the Ashtekar–Barbero SU(2) gauge theory formulation of Euclidean gravity by their U(1)3 version, one arrives at a consistent model which captures significant structure of its SU(2) version. In particular, it displays a non trivial realisation of the hypersurface deformation algebra which makes it an interesting testing ground for (Euclidean) quantum gravity as has been emphasised in a recent series of papers due to Varadarajan et al. In this paper we consider a reduced phase space approach to this model. This is especially attractive because, after a canonical transformation, the constraints are at most linear in the momenta. In suitable gauges, it is therefore possible to find a closed and explicit formula for the physical Hamiltonian which depends only on the physical observables. The corresponding reduced phase space quantisation can be confronted with the constraint quantisation due to Varadarajan et al to gain further insights into the quantum realisation of the hypersurface deformation algebra.

Hamiltonian Renormalization V: Free Vector Bosons

In a recent proposal we applied methods from constructive QFT to derive a Hamiltonian Renormalization Group in order to employ it ultimately for canonical quantum gravity. The proposal was successfully tested for free scalar fields and thus a natural next step is to test it for free gauge theories. This can be done in the framework of reduced phase space quantization which allows using techniques developed earlier for scalar field theories. In addition, in canonical quantum gravity one works in representations that support holonomy operators which are ill defined in the Fock representation of say Maxwell or Proca theory. Thus, we consider toy models that have both features, i.e. which employ Fock representations in which holonomy operators are well-defined. We adapt the coarse graining maps considered for scalar fields to those theories for free vector bosons. It turns out that the corresponding fixed pointed theories can be found analytically.

Towards a reduced phase space quantization in loop quantum cosmology with an inflationary potential

We explore a reduced phase space quantization of loop quantum cosmology (LQC) for a spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) universe filled with reference fields and an inflaton field in a Starobinsky inflationary potential. We consider three separate cases in which the reference fields are taken to be the Gaussian dust, the Brown-Kucha\ifmmode \check{r}\else \v{r}\fi{} dust, and massless Klein-Gordon scalar reference fields respectively. This is a ``two-fluid'' model in which reference fields act as global clocks providing a physical time in an inflationary spacetime, and allow bypassing various technical hurdles in conventional quantum cosmological models. The reduced phase space is obtained in terms of the Dirac observables of the gravitational as well as the inflaton degrees of freedom. The physical Hamiltonians of the two dust models take the same form but turn out to be quite different from that of the Klein-Gordon reference field which reflects an aspect of the multiple choice problem of time. Loop quantization is implemented using the so-called $\overline{\ensuremath{\mu}}$ scheme and the Schr\"odinger equations involving the physical Hamiltonian operators generating the evolution in the physical time in the dust and massless Klein-Gordon models are obtained. These turn out to be quantum difference equations with the same nonsingular structure as for other models in LQC. We study some phenomenological implications of the quantization using the effective dynamics resulting from the reduced phase space quantization including the resolution of the big bang singularity via a quantum bounce, and effects of the different reference fields on e-foldings in both the preinflationary and the slow-roll inflationary phases. We find that different clocks, even when starting with a small but same energy density, can leave tiny but different imprints on the inflationary dynamics. In addition, for Brown-Kucha\ifmmode \check{r}\else \v{r}\fi{} dust, the choice of a negative energy density can result in a cyclic evolution before the onset of inflation constraining certain values for ideal clocks.

Semiclassical limit of new path integral formulation from reduced phase space loop quantum gravity

Recently, a new path integral formulation of loop quantum gravity (LQG) has been derived in M. Han and H. Liu, Phys. Rev. D 101, 046003 (2020). from the reduced phase space formulation of the canonical LQG. This paper focuses on the semiclassical analysis of this path integral formulation. We show that dominant contributions of the path integral come from solutions of semiclassical equations of motion (EOMs), which reduce to Hamilton's equations of holonomies and fluxes $h(e)$, ${p}^{a}(e)$ in the reduced phase space ${\mathcal{P}}_{\ensuremath{\gamma}}$ of the cubic lattice $\ensuremath{\gamma}$: $\frac{\mathrm{d}h(e)}{\mathrm{d}\ensuremath{\tau}}={h(e),\mathbf{H}}$, $\frac{\mathrm{d}{p}^{a}(e)}{\mathrm{d}\ensuremath{\tau}}={{p}^{a}(e),\mathbf{H}}$, where $\mathbf{H}$ is the discrete physical Hamiltonian. The semiclassical dynamics from the path integral becomes an initial value problem of Hamiltonian time evolution in ${\mathcal{P}}_{\ensuremath{\gamma}}$. Moreover when we take the continuum limit of the lattice $\ensuremath{\gamma}$, these Hamilton's equations reproduce correctly classical reduced phase space EOMs of gravity coupled to dust fields in the continuum, as far as initial and final states are semiclassical. Our result proves that the new path integral formulation has the correct semiclassical limit and indicates that the reduced phase space quantization in LQG is semiclassically consistent. Based on these results, we compare this path integral formulation and the spin foam formulation, and show that this formulation has several advantages including the finiteness, the relation with canonical LQG, and the freedom from cosine and flatness problems.

Effective dynamics from coherent state path integral of full loop quantum gravity

A new routine is proposed to relate Loop Quant Cosmology (LQC) to Loop Quantum Gravity (LQG) from the perspective of effective dynamics. We derive the big-bang singularity resolution and big bounce from the first principle of full canonical LQG. Our results are obtained in the framework of the reduced phase space quantization of LQG. As a key step in our work, we derive with coherent states a new discrete path integral formula of the transition amplitude generated by the physical Hamiltonian. The semiclassical approximation of the path integral formula gives an interesting set of effective equations of motion (EOMs) for full LQG. When solving the EOMs with homogeneous and isotropic ansatz, we reproduce the LQC effective dynamics in $\mu_0$-scheme. The solution replaces the big-bang singularity by a big bounce. In the end, we comment on the possible relation between the $\bar{\mu}$-scheme of effective dynamics and the continuum limit of the path integral formula.

Reduced loop quantization with four Klein–Gordon scalar fields as reference matter

We perform a reduced phase space quantization of gravity using four Klein–Gordon scalar fields as reference matter as an alternative to the Brown–Kuchar dust model in Giesel and Thiemann (2010 Class. Quantum Grav. 27 175009), where dust scalar fields are used. We compare our results to an earlier model by Domagala et al (2010 Phys. Rev. D 82 104038) where only one Klein–Gordon scalar field is considered as reference matter for the Hamiltonian constraint but the spatial diffeomorphism constraints are quantized using Dirac quantization. As a result we find that the choice of four conventional Klein–Gordon scalar fields as reference matter leads to a reduced dynamical model that cannot be quantized using loop quantum gravity techniques. However, we further discuss a slight generalization of the action for the four Klein–Gordon scalar fields and show that this leads to a model which can be quantized in the framework of loop quantum gravity. By comparison of the physical Hamiltonian operators obtained from the model by Domagala et al (2010 Phys. Rev. D 82 104038) and the one introduced in this work we are able to make a first step towards comparing Dirac and reduced phase space quantization in the context of the spatial diffeomorphism constraints.

Gravitation in terms of observables

In the 1960s, Mandelstam proposed a new approach to gauge theories and gravity based on loops. The program was completed for Yang–Mills theories by Gambini and Trias in the 1980s. In this approach, gauge theories could be understood as representations of certain group: the group of loops. The same formalism could not be implemented at that time for the gravitational case. Here we would like to propose an extension to the case of gravity. The resulting theory is described in terms of loops and open paths and can provide the underpinning for a new quantum representation for gravity distinct from the one used in loop quantum gravity or string theory. In it, space-time points are emergent entities that would only have quasi-classical status. The formulation may be given entirely in terms of Dirac observables that form a set of gauge invariant functions that completely define the Riemannian geometry of the spacetime. At the quantum level this formulation will lead to a reduced phase space quantization free of any constraints.

Probabilistic interpretation of the wave function for the Bianchi I model

We compare two different approaches for quantization of the Bianchi I model: a reduced phase space quantization, in which the isotropic Misner variables is taken as time, and the Vilenkin proposal, in which a semiclassical approximation is performed for the same variable. We outline the technical and interpretative issues of these two methods and we demonstrate that they provide equivalent results only if the dynamics is essentially dictated by the isotropic matter contribution.

Scalar material reference systems and loop quantum gravity

In the past, the possibility to employ (scalar) material reference systems in order to describe classical and quantum gravity directly in terms of gauge invariant (Dirac) observables has been emphasized frequently. This idea has been picked up more recently in loop quantum gravity with the aim to perform a reduced phase space quantization of the theory, thus possibly avoiding problems with the (Dirac) operator constraint quantization method for a constrained system. In this work, we review the models that have been studied on the classical and/or the quantum level and parametrize the space of theories considered so far. We then describe the quantum theory of a model that, to the best of our knowledge, has only been considered classically so far. This model could arguably be called the optimal one in this class of models considered as it displays the simplest possible true Hamiltonian, while at the same time reducing all constraints of general relativity.

Dirac observables and boundary proposals in quantum cosmology

We study the reduced phase space quantization of a closed Friedmann Universe, where matter content is constituted by two (no-interacting) fluids, namely dust (or cold dark matter) and radiation. It is shown that, for this particular model, specific boundary conditions can be related to the algebra of Dirac observables.

On a partially reduced phase space quantization of general relativity conformally coupled to a scalar field

The purpose of this paper is twofold. On the one hand, after a thorough review of the matter free case, we supplement the derivations in our companion paper on ‘loop quantum gravity without the Hamiltonian constraint’ with calculational details and extend the results to standard model matter, a cosmological constant, and non-compact spatial slices. On the other hand, we provide a discussion on the role of observables, focused on the situation of a symmetry exchange, which is key to our derivation. Furthermore, we comment on the relation of our model to reduced phase space quantizations based on deparametrization.

Towards loop quantum supergravity (LQSG): II. p-form sector

In our companion paper, we focused on the quantization of the Rarita–Schwinger sector of supergravity theories in various dimensions by using an extension of loop quantum gravity to all spacetime dimensions. In this paper, we extend this analysis by considering the quantization of additional bosonic fields necessary to obtain a complete SUSY multiplet next to graviton and gravitino in various dimensions. As a generic example, we study concretely the quantization of the 3-index photon of minimal 11d SUGRA, but our methods easily extend to more general p-form fields. Due to the presence of a Chern–Simons term for the 3-index photon, which is due to local SUSY, the theory is self-interacting and its quantization far from straightforward. Nevertheless, we show that a reduced phase space quantization with respect to the 3-index photon Gauß constraint is possible. Specifically, the Weyl algebra of observables, which deviates from the usual CCR Weyl algebras by an interesting twist contribution proportional to the level of the Chern–Simons theory, admits a background independent state of the Narnhofer–Thirring type.

Gaussian state for the bouncing quantum cosmology

We present results concerning propagation of the Gaussian state across the cosmological quantum bounce. The reduced phase space quantization of loop quantum cosmology is applied to the Friedman-Robertson-Walker universe with a free massless scalar field. Evolution of quantum moments of the canonical variables is investigated. The covariance turns out to be a monotonic function so it may be used as an evolution parameter having quantum origin. We show that for the Gaussian state the Universe is least quantum at the bounce. We propose explanation of this counter-intuitive feature using the entropy of squeezing. The obtained time dependence of entropy is in agreement with qualitative predictions based on von Neumann entropy for mixed states. We show that, for the considered Gaussian state, semiclassicality is preserved across the bounce, so there is no cosmic forgetfulness.

Do gauge fields really contribute negatively to black hole entropy?

Quantum fluctuations of matter fields contribute to the thermal entropy of black holes. For free minimally coupled scalar and spinor fields, this contribution is precisely the entanglement entropy. For gauge fields, Kabat found an extra negative divergent ``contact term'' with no known statistical interpretation. We compare this contact term to a similar term which arises for nonminimally coupled scalar fields. Although both divergences may be interpreted as terms in the Wald entropy, we point out that the contact term for gauge fields comes from a gauge-dependent ambiguity in Wald's formula. Revisiting Kabat's derivation of the contact term, we show that it is sensitive to the treatment of infrared modes. To explore these infrared issues, we consider two-dimensional compact manifolds, such as Euclidean de Sitter space, and show that the contact term arises from an incorrect treatment of zero modes. In a manifestly gauge-invariant reduced phase space quantization, the gauge field contribution to the entropy is positive, finite, and equal to the entanglement entropy.

Shape dynamics in 2 + 1 dimensions

Shape Dynamics is a formulation of General Relativity where refoliation invariance is traded for local spatial conformal invariance. In this paper we explicitly construct Shape Dynamics for a torus universe in 2+1 dimensions through a linking gauge theory that ensures dynamical equivalence with General Relativity. The Hamiltonian we obtain is formally a reduced phase space Hamiltonian. The construction of the Shape Dynamics Hamiltonian on higher genus surfaces is not explicitly possible, but we give an explicit expansion of the Shape Dynamics Hamiltonian for large CMC volume. The fact that all local constraints are linear in momenta allows us to quantize these explicitly, and the quantization problem for Shape Dynamics turns out to be equivalent to reduced phase space quantization. We consider the large CMC-volume asymptotics of conformal transformations of the wave function. We then use the similarity of Shape Dynamics on the 2-torus with the explicitly constructible strong gravity (BKL) Shape Dynamics Hamiltonian in higher dimensions to suggest a quantization strategy for Shape Dynamics.

Reduced phase space approach to Kasner universe and the problem of time in quantum theory

We apply the reduced phase space quantization to the Kasner universe. We construct the kinematical phase space, find solutions to the Hamilton equations of motion, identify Dirac observables and arrive at physical solutions in terms of Dirac observables and an internal clock. We obtain the physical Hilbert space, which is the carrier space of the self-adjoint representation of the Dirac observables. Then, we discuss the problem of time. We demonstrate that the inclusion of evolution in a gravitational system, at classical level as well as at quantum level, leads respectively to canonically and unitarily inequivalent theories. The example of Hubble operator in two different clock variables and with two distinct spectra is given.

Evolution in bouncing quantum cosmology

We present the method of describing an evolution in quantum cosmology in the framework of the reduced phase space quantization of loop cosmology. We apply our method to the flat Friedmann–Robertson–Walker model coupled to a massless scalar field. We identify the physical quantum Hamiltonian that is positive-definite and generates globally a unitary evolution of the considered quantum system. We examine the properties of expectation values of physical observables in the process of the quantum big bounce transition. The dispersion of evolved observables is studied for the Gaussian state. Calculated relative fluctuations enable an examination of the semi-classicality conditions and possible occurrence of the cosmic forgetfulness. Preliminary estimations based on the cosmological data suggest that there was no cosmic amnesia. Presented results are analytical, and numerical computations are only used for the visualization purposes. Our method may be generalized to sophisticated cosmological models including the Bianchi-type universes.