Physical Hilbert Space Research Articles

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.

Discretisations, Constraints and Diffeomorphisms in Quantum Gravity

In this review we discuss the interplay between discretization, constraint implementation, and diffeomorphism symmetry in Loop Quantum Gravity and Spin Foam models. To this end we review the Consistent Discretizations approach, which is an application of the master constraint program to construct the physical Hilbert space of the canonical theory, as well as the Perfect Actions approach, which aims at finding a path integral measure with the correct symmetry behavior under diffeomorphisms.

Euclidean formulation of relativistic quantum mechanics

In this paper, we discuss a formulation of relativistic quantum mechanics that uses model Euclidean Green functions or their generating functional as input. This formalism has a close relation to quantum field theory, but as a theory of linear operators on a Hilbert space, it has the advantages of quantum mechanics. One interesting feature of this approach is that matrix elements of operators in normalizable states on the physical Hilbert space can be calculated directly using the Euclidean Green functions without performing an analytic continuation. The formalism is summarized in this paper. We discuss the motivation, advantages, and difficulties in using this formalism. We discuss how to compute bound states, scattering cross sections, and finite Poincar\'e transformations without using analytic continuation. A toy model is used to demonstrate how matrix elements of ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ in normalizable states can be used to construct sharp-momentum transition-matrix elements.

Formula omitted]-site-lattice analogues of [formula omitted

A discrete N-level alternative to the popular imaginary cubic oscillator is proposed and studied. As usual, the unitarity of evolution is guaranteed by the introduction of an ad hoc, Hamiltonian-dependent inner-product metric, the use of which defines the physical Hilbert space and renders the Hamiltonian (with real spectrum) observable. Due to the simplicity of our model of dynamics the construction of the set of eligible metrics is shown tractable by non-numerical means which combine the computer-assisted algebra with the extrapolation and/or perturbation techniques.

Constraint rescaling in refined algebraic quantisation: Momentum constraint

We investigate refined algebraic quantisation within a family of classically equivalent constrained Hamiltonian systems that are related to each other by rescaling a momentum-type constraint. The quantum constraint is implemented by a rigging map that is motivated by group averaging but has a resolution finer than what can be peeled off from the formally divergent contributions to the averaging integral. Three cases emerge, depending on the asymptotics of the rescaling function: (i) quantisation is equivalent to that with identity scaling; (ii) quantisation fails, owing to nonexistence of self-adjoint extensions of the constraint operator; (iii) a quantisation ambiguity arises from the self-adjoint extension of the constraint operator, and the resolution of this purely quantum mechanical ambiguity determines the superselection structure of the physical Hilbert space. Prospects of generalising the analysis to systems with several constraints are discussed.

Quantum theory of massless (p, 0)-forms

We describe the quantum theory of massless (p,0)-forms that satisfy a suitable holomorphic generalization of the free Maxwell equations on Kaehler spaces. These equations arise by first-quantizing a spinning particle with a U(1)-extended local supersymmetry on the worldline. Dirac quantization of the spinning particle produces a physical Hilbert space made up of (p,0)-forms that satisfy holomorphic Maxwell equations coupled to the background Kaehler geometry, containing in particular a charge that measures the amount of coupling to the U(1) part of the U(d) holonomy group of the d-dimensional Kaehler space. The relevant differential operators appearing in these equations are a twisted exterior holomorphic derivative and its hermitian conjugate (twisted Dolbeault operators with charge q). The particle model is used to obtain a worldline representation of the one-loop effective action of the (p,0)-forms. This representation allows to compute the first few heat kernel coefficients contained in the local expansion of the effective action and to derive duality relations between (p,0) and (d-p-2,0)-forms that include a topological mismatch appearing at one-loop.

Transition representations of quantum evolution with application to scattering resonances

A Lyapunov operator is a self-adjoint quantum observable whose expectation value varies monotonically as time increases and may serve as a marker for the flow of time in a quantum system. In this paper it is shown that the existence of a certain type of Lyapunov operator leads to representations of the quantum dynamics, termed transition representations, in which an evolving quantum state ψ(t) is decomposed into a sum ψ(t) = ψb(t) + ψf(t) of a backward asymptotic component and a forward asymptotic component such that the evolution process is represented as a transition from ψb(t) to ψf(t). When applied to the evolution of scattering resonances, such transition representations separate the process of decay of a scattering resonance from the evolution of outgoing waves corresponding to the probability “released” by the resonance and carried away to spatial infinity. This separation property clearly exhibits the spatial probability distribution profile of a resonance. Moreover, it leads to the definition of exact resonance states as elements of the physical Hilbert space corresponding to the scattering problem. These resonance states evolve naturally according to a semigroup law of evolution.

On knottings in the physical Hilbert space of LQG as given by the EPRL model

We consider the EPRL spin foam amplitude for arbitrary embedded two-complexes. Choosing a definition of the face- and edge amplitudes which lead to spin foam amplitudes invariant under trivial subdivisions, we investigate invariance properties of the amplitude under consistent deformations, which are deformations of the embedded two-complex where faces are allowed to pass through each other in a controlled way. Using this surprising invariance, we are able to show that the physical Hilbert space, as defined by the sum over all spin foams, contains no information about knotting classes of graphs anymore.

A COMPOSITE FERMION APPROACH TO THE ULTRACOLD DILUTE FERMI GAS

It is argued that the recently observed Fermi liquids in strongly interacting ultracold Fermi gases are adiabatically connected to a projected Fermi gas. This conclusion is reached by constructing a set of Jastrow wavefunctions, following Tan's observations on the structure of the physical Hilbert space [Annals of Physics 323, 2952 (2008)]. The Jastrow projection merely implements the Bethe–Peierls condition on the BCS and Fermi gas wavefunctions. This procedure provides a simple picture of the emergence of Fermi polarons as composite fermions in the normal state of the highly polarized gas. It is also shown that the projected BCS wavefunction can be written as a condensate of pairs of composite fermions (or Fermi polarons). A Hamiltonian for the composite fermions is derived. Within a mean-field theory, it is shown that the ground state and excitations of this Hamiltonian are those of a non-interacting Fermi gas although they are described by Jastrow–Slater wavefunctions.

No-Ghost Theorem for Neveu-Schwarz String in 0-Picture via Similarity Transformation

A proof of the no-ghost theorem for Neveu-Schwarz string in 0-picture is outlined in a manner using a similarity transformation. It is shown that a nontrivial metric consistent with the BRST cohomology is needed to define a positive semidefinite norm in the physical Hilbert space. The one-to-one correspondence between physical states in 0-picture and those in the conventional (−1)-picture are confirmed. As a by-product, we find a new inverse picturechanging operator, which is noncovariant but has nonsingular operator product with itself. A possibility to construct a new gauge-invariant superstring field theory is discussed.

No-Ghost Theorem for Neveu-Schwarz String in 0-Picture

The no-ghost theorem for Neveu-Schwarz string is directly proved in 0-picture. The one-to-one correspondence between physical states in 0-picture and those in the conventional (-1)-picture are confirmed. It is shown that a non-trivial metric consistent with the BRST cohomology is needed to define a positive semi-definite norm in the physical Hilbert space. As a by-product, we find a new inverse picture changing operator, which is non-covariant but has non-singular operator product with itself. A possibility to construct a new gauge invariant superstring field theory is discussed.

HIGGS MECHANISM FOR GRAVITONS

Just like the vector gauge bosons in the gauge theories, it is now known that gravitons acquire mass in the process of spontaneous symmetry breaking of diffeomorphisms through the condensation of scalar fields. The point is that we should find the gravitational Higgs mechanism such that it results in massive gravity in a flat Minkowski spacetime without non-unitary propagating modes. This is usually achieved by including higher-derivative terms in scalars and tuning the cosmological constant to be a negative value in a proper way. Recently, a similar but different gravitational Higgs mechanism has been advocated by Chamseddine and Mukhanov where one can relax the negative cosmological constant to zero or positive one. In this work, we investigate why the non-unitary ghost mode decouples from physical Hilbert space in a general spacetime dimension. Moreover, we generalize the model to possess an arbitrary potential and clarify under what conditions the general model exhibits the gravitational Higgs mechanism. By searching for solutions to the conditions, we arrive at two classes of potentials exhibiting gravitational Higgs mechanism. One class includes the model by Chamseddine and Mukhanov in a specific case while the other is a completely new model.

The physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory

A covariant spin-foam formulation of quantum gravity has been recently developed, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity. In this paper we reconsider the implementation of the constraints that defines the model. We define in a simple way the boundary Hilbert space of the theory, introducing a slight modification of the embedding of the SU(2) representations into the ones. We then show directly that all constraints vanish on this space in a weak sense. The vanishing is exact (and not just in the large quantum number limit). We also generalize the definition of the volume operator in the spin-foam model to the Lorentzian signature and show that it matches the one of loop quantum gravity, as in the Euclidean case.

Dirac quantization of membrane winding uniformly in time dependent orbifold

We present quantum theory of a membrane propagating in the vicinity of a time dependent orbifold singularity. The dynamics of a membrane, with the parameters space topology of a torus, winding uniformly around compact dimension of the embedding spacetime is mathematically equivalent to the dynamics of a closed string in a flat FRW spacetime. The construction of the physical Hilbert space of a membrane makes use of the kernel space of self-adjoint constraint operators. It is a subspace of the representation space of the constraints algebra. There exist non-trivial quantum states of a membrane evolving across the singularity.

Algebraic quantum gravity (AQG): IV. Reduced phase space quantization of loop quantum gravity

We perform a canonical, reduced phase space quantization of general relativity by loop quantum gravity (LQG) methods. The explicit construction of the reduced phase space is made possible by the combination of (a) the Brown–Kuchař mechanism in the presence of pressure-free dust fields which allows to deparametrize the theory and (b) Rovelli's relational formalism in the extended version developed by Dittrich to construct the algebra of gauge-invariant observables. Since the resulting algebra of observables is very simple, one can quantize it using the methods of LQG. Basically, the kinematical Hilbert space of non-reduced LQG now becomes a physical Hilbert space and the kinematical results of LQG such as discreteness of spectra of geometrical operators now have physical meaning. The constraints have disappeared; however, the dynamics of the observables is driven by a physical Hamiltonian which is related to the Hamiltonian of the standard model (without dust) and which we quantize in this paper.

2+1 quantum gravity with a Barbero–Immirzi-like parameter on toric spatial foliation

We consider gravity in 2+1 spacetime dimensions, with a negative cosmological constant and a ‘Barbero–Immirzi’ (B-I)-like parameter, when the spacetime topology is of the form . The phase space structure, both in covariant and canonical framework, is analyzed. Full quantization of the theory in the ‘constrain first’ approach reveals a finite-dimensional physical Hilbert space. An explicit construction of wavefunctions is presented. The dimension of the Hilbert space is found to depend on the ‘Barbero–Immirzi’-like parameter in an interesting fashion. A comparative study of this parameter in light of some of the recent findings in the literature for similar theories is presented.

The QCD nature of dark energy

The origin of the observed dark energy could be explained entirely within the standard model, with no new fields required. We show how the low-energy sector of the chiral QCD Lagrangian, once embedded in a non-trivial spacetime, gives rise to a cosmological vacuum energy density which can be presented entirely in terms of QCD parameters and the Hubble constant H as ρ Λ ≃ H ⋅ m q 〈 q ¯ q 〉 / m η ′ ∼ ( 4.3 ⋅ 10 − 3 eV ) 4 . In this work we focus on the dynamics of the ghost fields that are essential ingredients of the aforementioned Lagrangian. In particular, we argue that the Veneziano ghost, being unphysical in the usual Minkowski QFT, exhibits important physical effects if the universe is expanding. Such effects are naturally very small as they are proportional to the rate of expansion H / Λ QCD ∼ 10 − 41 . The co-existence of these two drastically different scales ( Λ QCD ∼ 100 MeV and H ∼ 10 − 33 eV ) is a direct consequence of the auxiliary conditions on the physical Hilbert space that are necessary to keep the theory unitary. The exact cancellation taking place in Minkowski space due to this auxiliary condition is slightly violated when the system is upgraded to an expanding background. Nevertheless, this “tiny” effect would in fact the driving force accelerating the universe today. We also derive the time-dependent equation of state w ( t ) for the dark energy component which tracks the dynamics of the Veneziano ghost in a FLRW universe.

The Hilbert space of the Chern–Simons theory on a cylinder: a loop quantum gravity approach

As a laboratory for loop quantum gravity, we consider the canonical quantization of the three-dimensional Chern–Simons theory on a noncompact space with the topology of a cylinder. Working within the loop quantization formalism, we define at the quantum level the constraints appearing in the canonical approach and completely solve them, thus constructing a gauge and diffeomorphism invariant physical Hilbert space for the theory. This space turns out to be infinite dimensional, but separable.

Consistent Histories in Quantum Cosmology

We illustrate the crucial role played by decoherence (consistency of quantum histories) in extracting consistent quantum probabilities for alternative histories in quantum cosmology. Specifically, within a Wheeler-DeWitt quantization of a flat Friedmann-Robertson-Walker cosmological model sourced with a free massless scalar field, we calculate the probability that the univese is singular in the sense that it assumes zero volume. Classical solutions of this model are a disjoint set of expanding and contracting singular branches. A naive assessment of the behavior of quantum states which are superpositions of expanding and contracting universes may suggest that a "quantum bounce" is possible i.e. that the wave function of the universe may remain peaked on a non-singular classical solution throughout its history. However, a more careful consistent histories analysis shows that for arbitrary states in the physical Hilbert space the probability of this Wheeler-DeWitt quantum universe encountering the big bang/crunch singularity is equal to unity. A quantum Wheeler-DeWitt universe is inevitably singular, and a "quantum bounce" is thus not possible in these models.

Loop quantum cosmology evolution operator of an FRW universe with a positive cosmological constant

The self-adjointness of an evolution operator $\Theta_{\Lambda}$ corresponding to the model of flat FRW universe with massless scalar field and cosmological constant quantized in the framework of Loop Quantum Cosmology is studied in the case $\Lambda>0$. It is shown, that for $\Lambda<\Lambda_c\approx 10.3 \lPl^{-2}$ the operator admits many self-adjoint extensions, each of the purely discrete spectrum. On the other hand for $\Lambda\geq\Lambda_c$ the operator is essentially self-adjoint, however the physical Hilbert space of the model does not contain any physically interesting states.