Affine Coherent States Research Articles

Affine density, von Neumann dimension and a problem of Perelomov

We provide a solution to Perelomov's 1972 problem concerning the existence of a phase transition (known in signal analysis as ‘Nyquist rate’) determining the basis properties of certain affine coherent states labelled by Fuchsian groups. As suggested by Perelomov, the transition is given according to the hyperbolic volume of the fundamental region. The solution is a more general form (in phase space) of the PSL(2,R) variant of a 1989 conjecture of Kristian Seip about wavelet frames, where the same value of ‘Nyquist rate’ is obtained as the trace of a certain localization operator. The proof consists of first connecting the problem to the theory of von Neumann algebras, by introducing a new class of projective representations of PSL(2,R) acting on non-analytic Bergman-type spaces. We then adapt to this setting a new method for computing von Neumann dimensions, due to Sir Vaughan Jones. Our solution contains necessary conditions in the form of a ‘Nyquist rate’ dividing frames from Riesz sequences of coherent states and sampling from interpolating sequences. They hold for an infinite sequence of spaces of polyanalytic functions containing the eigenspaces of the Maass operator and their orthogonal sums. Within mild boundaries, we show that our result is best possible, by characterizing our sequence of function spaces as the only invariant spaces under the non-analytic PSL(2,R)-representations.

Ascribing quantum system to Schwarzschild spacetime with naked singularity

We quantize the Schwarzschild spacetime with naked singularity using the affine coherent states quantization method. The novelty of our approach is quantization of both temporal and spatial coordinates. Quantization smears the gravitational singularity indicated by the Kretschmann invariant avoiding its localization in the configuration space. This way we resolve the singularity problem of considered spacetime at quantum level.

Dependence of the affine coherent states quantization on the parametrization of the affine group

The affine coherent states quantization is a promising integral quantization of Hamiltonian systems when the phase space includes at least one conjugate pair of variables which takes values from a half plane. Such a situation is common for gravitational systems which include singularities. The construction of the quantization map includes a one-to-one mapping of the half plane onto the affine group. Particular cases of this mapping define specific parametrizations of the group. Our aim is showing that different such parametrizations lead to unitarily inequivalent quantum theories. Depending on the Hamiltonian system under consideration, this dependence could potentially be used constructively.

Quantum Oppenheimer-Snyder model

We construct two reduced quantum theories for the Oppenheimer-Snyder model, respectively taking the point of view of the comoving and the exterior stationary observer, using affine coherent states quantization. Investigations of the quantum corrected dynamics reveal that both observers can see a bounce, although for the exterior observer certain quantization ambiguities have to be chosen correctly. The minimal radius for this bounce as seen from the stationary observer is then shown to always be outside of the photon sphere. Possible avenues to lower this minimal radius and reclaim black holes as an intermediate state in the collapse are discussed. We demonstrate further that switching between the observers at the level of the quantum theories can be achieved by modifying the commutation relations.

Robustness of the quantum BKL scenario

The quantum Belinski–Khalatnikov–Lifshitz scenario presents an unitary evolution of the system. However, the affine coherent states quantization applied to the quantization of the underlying classical scenario depends on the choice of the group parametrization. Using the two simplest parameterizations of the affine group, we show that qualitative features of our quantum system do not depend on the choice. It means that the quantum bounce replacing a singular classical scenario is expected to be a generic feature of the considered system. This paper complements our recent article (Góźdź et al. in Eur Phys J C 79:45, 2019).

Quantum Belinski\u2013Khalatnikov\u2013Lifshitz scenario

We present the quantum model of the asymptotic dynamics underlying the Belinski–Khalatnikov–Lifshitz (BKL) scenario. The symmetry of the physical phase space enables making use of the affine coherent states quantization. Our results show that quantum dynamics is regular in the sense that during quantum evolution the expectation values of all considered observables are finite. The classical singularity of the BKL scenario is replaced by the quantum bounce that presents a unitary evolution of considered gravitational system. Our results suggest that quantum general relativity has a good chance to be free from singularities.

Three examples of quantum dynamics on the half-line with smooth bouncing

This article is an introductory presentation of the quantization of the half-plane based on affine coherent states (ACS). The half-plane carries a natural affine symmetry, i.e. it is a homogeneous space for the 1d-affine group, and it is viewed as the phase space for the dynamics of a positive physical quantity evolving with time. Its affine symmetry is preserved due to the covariance of this type of quantization. We promote the interest of such a procedure for transforming a classical model into a quantum one, since the singularity at the origin is systematically removed, and the arbitrariness of boundary conditions for the Schrödinger operator can be easily overcome. We explain some important mathematical aspects of the method. Three elementary examples of applications are presented, the quantum breathing of a massive sphere, the quantum smooth bouncing of a charged sphere, and a smooth bouncing of “dust” sphere as a simple model of quantum Newtonian cosmology.

Smooth big bounce from affine quantization

We examine the possibility of dealing with gravitational singularities on a quantum level through the use of coherent state or wavelet quantization instead of canonical quantization. We consider the Robertson-Walker metric coupled to a perfect fluid. It is the simplest model of a gravitational collapse, and the results obtained here may serve as a useful starting point for more complex investigations in the future. We follow a quantization procedure based on affine coherent states or wavelets built from the unitary irreducible representation of the affine group of the real line with positive dilation. The main issue of our approach is the appearance of a quantum centrifugal potential allowing for regularization of the singularity, essential self-adjointness of the Hamiltonian, and unambiguous quantum dynamical evolution.

Affine coherent states and Toeplitz operators

We study a parameterized family of Toeplitz operators in the context of affine coherent states based on the Calderón reproducing formula (= resolution of unity on ) and the specific admissible wavelets (= affine coherent states in ) related to Laguerre functions. Symbols of such Calderón–Toeplitz operators as individual coordinates of the affine group (= upper half-plane with the hyperbolic geometry) are considered. In this case, a certain class of pseudo-differential operators, their properties and their operator algebras are investigated. As a result of this study, the Fredholm symbol algebras of the Calderón–Toeplitz operator algebras for these particular cases of symbols are described.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

New Affine Coherent States Based on Elements of Nonrenormalizable Scalar Field Models

Recent proposals for a nontrivial quantization of covariant, nonrenormalizable, self‐interacting, scalar quantum fields have emphasized the importance of quantum fields that obey affine commutation relations rather than canonical commutation relations. When formulated on a spacetime lattice, such models have a lattice version of the associated ground state, and this vector is used as the fiducial vector for the definition of the associated affine coherent states, thus ensuring that in the continuum limit, the affine field operators are compatible with the system Hamiltonian. In this article, we define and analyze the associated affine coherent states as well as briefly review the author′s approach to nontrivial formulations of such nonrenormalizable models.

Fundamentals of quantum gravity

The outline of a recent approach to quantum gravity is presented. Novel ingredients include: (1) Affine kinematical variables; (2) Affine coherent states; (3) Projection operator approach toward quantum constraints; (4) Continuous-time regularized functional integral representation without/with constraints; and (5) Hard core picture of nonrenormalizability. The "diagonal representation" for operator representations, introduced by Sudarshan into quantum optics, arises naturally within this program.

Coherent states and annihilation–creation operators associated with the irreducible unitary representations of 𝔰𝔲(1,1)

We construct a kind of annihilation–creation operators related to the affine coherent states. Next, we re-interpret them as annihilation–creation operators associated with the irreducible unitary representation of the algebra 𝔰𝔲(1,1), by adding another generator to the two generators of the unitary representation of the one-dimensional affine group.

Generalized affine coherent states: A natural framework for the quantization of metric-like variables

Affine variables, which have the virtue of preserving the positive-definite character of matrix-like objects, have been suggested as replacements for the canonical variables of standard quantization schemes, especially in the context of quantum gravity. We develop the kinematics of such variables, discussing suitable coherent states, their associated resolution of unity, polarizations, and finally the realization of the coherent-state overlap function in terms of suitable path-integral formulations.

Subnormal operators regarded as generalized observables and compound-system-type normal extension related to 𝔰𝔲(1,1)

In this paper, subnormal operators, not necessarily bounded, are discussed as generalized observables. In order to describe not only the information about the probability distribution of the output data of their measurement but also a framework of their implementations, we introduce a new concept compound-system-type normal extension, and we derive the compound-system-type normal extension of a subnormal operator, which is defined from an irreducible unitary representation of the algebra su(1,1). The squeezed states are characterized as the eigenvectors of an operator from this viewpoint, and the squeezed states in multi-particle systems are shown to be the eigenvectors of the adjoints of these subnormal operators under a representation. The affine coherent states are discussed in the same context, as well.

The universal propagator for spin [or SU (2)] coherent states

Following constructions in the case of canonical and affine coherent states, a universal propagator for a general Hamiltonian appropriate to a single spin degree of freedom is introduced. The universal propagator is a single function, independent of any particular choice of fiducial vector, which, nonetheless, propagates all coherent state Hilbert space representatives correctly. Furthermore, we explicitly construct the universal propagator for several examples.

Continuous and Discrete Wavelet Transforms

This paper is an expository survey of results on integral representations and discrete sum expansions of functions in $L^2 ({\bf R})$ in terms of coherent states. Two types of coherent states are considered: Weyl–Heisenberg coherent states, which arise from translations and modulations of a single function, and affine coherent states, called ’wavelets,’ which arise as translations and dilations of a single function. In each case it is shown how to represent any function in $L^2 ({\bf R})$ as a sum or integral of these states. Most of the paper is a survey of literature, most notably the work of I. Daubechies, A. Grossmann, and J. Morlet. A few results of the authors are included.

Painless nonorthogonal expansions

In a Hilbert space ℋ, discrete families of vectors {hj} with the property that f=∑j〈hj‖ f〉hj for every f in ℋ are considered. This expansion formula is obviously true if the family is an orthonormal basis of ℋ, but also can hold in situations where the hj are not mutually orthogonal and are ‘‘overcomplete.’’ The two classes of examples studied here are (i) appropriate sets of Weyl–Heisenberg coherent states, based on certain (non-Gaussian) fiducial vectors, and (ii) analogous families of affine coherent states. It is believed, that such ‘‘quasiorthogonal expansions’’ will be a useful tool in many areas of theoretical physics and applied mathematics.