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It is known that the momentum operator canonically conjugated to the position operator for a particle moving in some bounded interval of the line {(with Dirichlet boundary conditions) is not essentially self-adjoint}: it has a continuous set of self-adjoint extensions. We prove that essential self-adjointness can be recovered by symmetrically weighting the momentum operator with a positive bounded function approximating the indicator function of the considered interval. This weighted momentum operator is consistently obtained from a similarly weighted classical momentum through the so-called Weyl-Heisenberg covariant integral quantization of functions or distributions., Comment: in press in Journal of Mathematical Analysis and Applications

We revisit the concept of `tachyonic' scalar fields, characterized by a real negative squared `mass', in de Sitter (dS) spacetime. Through a rigorous group-theoretical analysis, we argue that labeling these fields as `tachyonic' lacks physical relevance, often leading to misinterpretations in scholarly discourse, as their `mass' parameter holds no genuine physical significance., Comment: 5 pages, no figure

Background: Occupational safety and health institutions report that caregivers areparticularly at risk of developing work-related musculoskeletal disorders (WRMSDs) andpatient handling is often pointed out as one of the main causes. While lots of studiesaddressed the use of assistive devices in order to protect caregivers, it seems that motorlessstand-up lifts have not been studied yet.Objectives: The aim of this work is to provide quantitative data about the loads in thelow back area, as well as qualitative data about subjects perceptions, resulting from the useof a motorless stand-up lift and to compare them to those resulting from manual patienthandling.Methods: Nine caregivers participated to motion capture and ground reaction forcesmeasurement sessions. These recordings were performed in three cases of handling:manual handling with one caregiver, manual handling with two caregivers, motorless deviceassisted handling. Forces and torques at the L5/S1 joint were then estimated throughInverse Dynamics process. A questionnaire about manual and motorless device assistedhandling was also submitted.Results: Motorless device assisted handling involved the smallest loads whereasmanual handling with one caregiver involved the biggest loads.Conclusions: Our findings suggest that, if the situation allows it, caregivers should behelped by another caregiver or use a motorless stand-up lift when handling a patient fromsitting to standing position or from standing to sitting position considering the reduced loadsthese aids involve.

We have discovered a class of dynamically stable coherent states for motion on the half-line. The regularization of the half-line boundary and the consequent quantum motion are expounded within the framework of covariant affine quantization, although alternative approaches are also feasible. The former approach is rooted in affine coherent states and offers a consistent semiclassical representation of quantum motion. However, this method has been known to possess two shortcomings: (a) the dependence of affine coherent states on the choice of a vector, denoted as 'fiducial vector' (which remains unspecified), introduces significant arbitrariness in boundary regularization, and (b) regardless of the choice of 'fiducial vector,' affine coherent states fail to evolve parametrically under the Schr\"odinger equation, thus limiting the accuracy of the semiclassical description. This limitation, in particular, hampers their suitability for approximating the evolution of compound observables. We demonstrate that a distinct and more refined definition of affine coherent states can simultaneously address both of these issues. In other words, these new affine coherent states exhibit parametric evolution only when the 'fiducial vector,' denoted as $|\psi_0>$, possesses a highly specific character, such as being an eigenstate of a well-defined Hamiltonian. Our discovery holds significant relevance in the field of quantum cosmology, particularly in scenarios where the positive variable is the scale factor of the universe, and its regularized motion plays a crucial role in avoiding the big-bang singularity., Comment: 15 pages, 3 figures

This manuscript introduces a novel holographic correspondence in $d$-dimensional de Sitter (dS$_d$) spacetime, connecting bulk dS$_d$ scalar unitary irreducible representations (UIRs) with their counterparts at the dS$_d$ boundary ${\cal{I}}^\pm$, all while preserving reflection positivity. The proposed approach, with potential applicability to diverse dS$_d$ UIRs, is rooted in the geometry of the complex dS$_d$ spacetime and leverages the inherent properties of the (global) dS$_d$ plane waves, as defined within their designated tube domains., Comment: 6 pages, no figure, version accepted for publication in Physics Letters B

We elaborate the definition and properties of ''massive" elementary systems in the (1+3)-dimensional Anti-de Sitter (AdS$_4$) spacetime, on both classical and quantum levels. We fully exploit the symmetry group Sp$(4,\mathbb R)$, that is, the two-fold covering of SO$_0(2,3)$ (Sp$(4,\mathbb R) \sim$ SO$_0(2,3)\times \mathbb Z_2$), recognized as the relativity/kinematical group of motions in AdS$_4$ spacetime. In particular, we discuss that the group coset Sp$(4,\mathbb R)$/S(U(1)x SU(2)), as one of the Cartan classical domains, can be interpreted as a phase space for the set of free motions of a test massive particle on AdS$_4$ spacetime; technically, in order to facilitate the computations, the whole process is carried out in terms of complex quaternions. The (projective) unitary irreducible representations (UIRs) of the Sp$(4,\mathbb R)$ group, describing the quantum version of such motions, are found in the discrete series of the Sp$(4,\mathbb R)$ UIRs. We also describe the null-curvature (Poincar\'{e}) and non-relativistic (Newton-Hooke) contraction limits of such systems, on both classical and quantum levels. On this basis, we unveil the dual nature of ''massive" elementary systems living in AdS$_4$ spacetime, as each being a combination of a Minkowskian-like massive elementary system with an isotropic harmonic oscillator arising from the AdS$_4$ curvature and viewed as a Newton-Hooke elementary system. This matter-vibration duality will take its whole importance in the quantum regime (in the context of the validity of the equipartition theorem) in view of its possible r\^{o}le in the explanation of the current existence of dark matter., Comment: 49 pages, 1 figure

We present a covariant quantization scheme for the so-called "partially massless" graviton field in de Sitter spacetime. Our approach is founded on the principles of the de Sitter group representation theory (in the sense given by Wigner), the Wightman-G\"{a}rding axioms for gauge-invariant fields (Gupta-Bleuler scheme), and the essential analyticity prerequisites in the complexified pseudo-Riemannian manifold. To implement the quantization process effectively, we adopt coordinate-independent (global) de Sitter plane waves. These plane waves, defined in the appropriate tube domains of complex de Sitter spacetime, serve as the de Sitter counterparts to the standard Minkowskian plane waves. By employing these analytical plane waves, we enable a spectral analysis of the corresponding two-point function that closely resembles the Fourier analysis typically employed in the flat Minkowskian case. Within this framework, we present the Wightman two-point function for the partially massless graviton field, which satisfies the essential criteria of locality, covariance, and normal analyticity. Furthermore, we provide insights into the underlying Hilbert space structure and the corresponding unsmeared field operator. A direct consequence of this quantization construction confirms the widely accepted notion of light-cone propagation for the de Sitter partially massless graviton field., Comment: 24 pages, no figure, version accepted for publication in Physical Review D

We revisit the Perelomov SU(1,1) displaced coherent states states as possible quantum states of light. We disclose interesting statistical aspects of these states in relation with photon counting and squeezing. In the non-displaced case we discuss the efficiency of the photodetector as inversely proportional to the parameter k of the discrete series of unitary irreducible representations of SU(1,1). In the displaced case, we study the counting and squeezing properties of the states in terms of k and the number of photons in the original displaced state. We finally examine the quantization of a classical radiation field which is based on these families of coherent states. The procedure yields displacement operators which might allow to prepare such states in the way proposed by Glauber for the standard coherent states., Comment: 18 pages, 9 figures

We study a semiclassical model of the mixmaster universe. We first derive the quantum model and then introduce its semiclassical approximation. We employ a general integral quantization method that respects the symmetries of the model given by the affine and the Weyl-Heisenberg groups, and can produce a wide class of quantum models. The semiclassical approximation is based on the coherent states. The semiclassical dynamics is complex and cannot be solved by analytical methods. We focus on a key qualitative feature of the dynamics, namely, we investigate whether the primordial anisotropic universe can undergo a spontaneous inflationary phase driven by the anisotropic energy combined with semiclassical corrections. The answer to this question provides a useful perspective on the inflationary paradigm as well as on alternative bouncing models., Comment: 15 pages, 6 figures

Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalised positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is to allow to circumvent problems due to the presence of singularities in the classical models. In this paper we implement covariant integral quantizations for systems whose phase space is $\mathbb{Z}\times\,\mathbb{S}^1$, i.e., for systems moving on the circle. The symmetry group of this phase space is the discrete \& compact version of the Weyl-Heisenberg group, namely the central extension of the abelian group $\mathbb{Z}\times\,\mathrm{SO}(2)$. In this regard, the phase space is viewed as the right coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on $L^2(\mathbb{S}^1)$, is square integrable on the phase space. We show how to derive corresponding covariant integral quantizations from (weight) functions on the phase space {and resulting resolution of the identity}. {As particular cases of the latter} we recover quantizations with de Bi\`evre-del Olmo-Gonzales and Kowalski-Rembielevski-Papaloucas coherent states on the circle. Another straightforward outcome of our approach is the Mukunda Wigner transform. We also look at the specific cases of coherent states built from shifted gaussians, Von Mises, Poisson, and Fej\'er kernels. Applications to stellar representations are in progress., Comment: 21 pages, Special Issue "Continuous and Discrete Phase-Space Methods and Their Applications" of Quantum Reports

As an extension of Gabor signal processing, the covariant Weyl-Heisenberg integral quantization is implemented to transform functions on the eight-dimensional phase space $\left(x,k\right)$ into Hilbertian operators. The $x=\left(x^{\mu}\right)$'s are space-time variables and the $k=\left(k^{\mu}\right)$'s are As an extension of Gabor signal processing, the covariant Weyl-Heisenberg integral quantization is implemented to transform functions on the eight-dimensional phase space $\left(x,k\right)$ into Hilbertian operators. The $x=\left(x^{\mu}\right)$'s are space-time variables and the $k=\left(k^{\mu}\right)$'s are their conjugate wave vector-frequency variables. The procedure is first applied to the variables $\left(x,k\right)$ and produces canonically conjugate essentially self-adjoint operators. It is next applied to the metric field $g_{\mu\nu}(x)$ of general relativity and yields regularised semi-classical phase space portraits $\check{g}_{\mu\nu}(x)$. The latter give rise to modified tensor energy density. Examples are given with the uniformly accelerated reference system and the Schwarzschild metric. Interesting probabilistic aspects are discussed., Comment: 22 pages

We review the construction of ("free") elementary systems in de Sitter (dS) spacetime, in the Wigner sense, as associated with unitary irreducible representations (UIR's) of the dS (relativity) group. This study emphasizes the conceptual issues arising in the formulation of such systems and discusses known results in a mathematically rigorous way. Particular attention is paid to: "smooth" transition from classical to quantum theory; physical content under vanishing curvature, from the point of view of a local ("tangent") Minkowskian observer; and thermal interpretation (on the quantum level), in the sense of the Gibbons-Hawking temperature. We review three decompositions of the dS group physically relevant for the description of dS spacetime and classical phase spaces of elementary systems living on it. We review the construction of (projective) dS UIR's issued from these group decompositions. (Projective) Hilbert spaces carrying the UIR's (in some restricted sense) identify quantum ("one-particle") states spaces of dS elementary systems. Adopting a well-established Fock procedure, based on the Wightman-G\"{a}rding axioms and on analyticity requirements in the complexified Riemannian manifold, we proceed with a consistent quantum field theory (QFT) formulation of elementary systems in dS spacetime. This dS QFT formulation closely parallels the corresponding Minkowskian one, while the usual spectral condition is replaced by a certain geometric Kubo-Martin-Schwinger (KMS) condition equivalent to a precise thermal manifestation of the associated "vacuum" states., Comment: This draft is an early version of both the 1st and subsequent 2nd editions of the book, officially released by Springer Nature in 2022, albeit with a slightly modified title: 'The de Sitter (dS) Group and its Representations: An Introduction to Elementary Systems and Modeling the Dark Energy Universe'. Book DOI: 10.1007/978-3-031-56552-6

Quantisation with Gaussian type states offers certain advantages over other quantisation schemes, in particular, they can serve to regularise formally discontinuous classical functions leading to well defined quantum operators. In this work we define a squeezed state quantisation in two dimensions using several families of squeezed states for one- and two-mode configurations. The completeness relations of the squeezed states are exploited in order to tackle the quantisation and semiclassical analysis of a constrained position dependent mass model with harmonic potential. The effects of the squeezing parameters on the resulting operators and phase space functions are studied, and configuration space trajectories are compared between the classical and semiclassical models., Comment: 28 pages, 5 figures

The real plane with its set of orientations or angles in $[0,\pi)$ is the simplest non trivial example of a (projective) Hilbert space and provides nice illustrations of quantum formalism. We present some of them, namely covariant integral quantization, linear polarisation of light as a quantum measurement, interpretation of entanglement leading to the violation of Bell inequalities, and spin one-half coherent states viewed as two entangled angles., Comment: 8 pages. Matches the published version in the AAMP Special Issue of the diamond Open-Access journal Acta Polytechnica

In the same way as the realization of some of the famous gedanken experiments imagined by the founding fathers of quantum mechanics has recently led to the current renewal of the interpretation of quantum physics, it seems that the most recent progresses of observational astrophysics can be interpreted as the realization of some cosmological gedanken experiments such as the removal from the universe of the whole visible matter or the cosmic time travel leading to a new cosmological standard model. This standard model involves two dark components of the universe, dark energy and dark matter. Whereas dark energy is usually associated with the cosmological constant, we propose to explain dark matter as a pure QCD effect, namely a gluonic Bose Einstein condensate, following the transition from the quark gluon plasma phase to the colorless hadronic phase. Our approach not only allows us to assume a ratio Dark/Visible equal to 11/2 but also provides gluons (and di-gluons, viewed as quasi-particles) with an extra mass of vibrational nature. Such an interpretation would comfort the idea that, apart from the violation of the matter/antimatter symmetry satisfying the Sakharov's conditions, the reconciliation of particle physics and cosmology needs not the recourse to any ad hoc fields, particles or hidden variables., Comment: This article belongs to the Special Issue Selected Papers from the 1st International Electronic Conference on Universe (ECU 2021)

We investigate two aspects of the elementary example of POVMs on the Euclidean plane, namely their status as quantum observables and their role as quantizers in the integral quantization procedure. The compatibility of POVMs in the ensuing quantum formalism is discussed, and a Naimark dilation is found for the quantum operators. The relation with Toeplitz quantization is explained. A physical situation is discussed, where we describe the linear polarization of the light with the use of Stokes parameters. In particular, the case of sequential measurements in a real bidimensional Hilbert space is addressed. An interpretation of the Stokes parameters in the framework of unsharp or fuzzy observables is given. Finally, a necessary condition for the compatibility of two dichotomic fuzzy observables which provides a condition for the approximate joint measurement of two incompatible sharp observables is found., Comment: 43 pages, 2 figures. Title changed. Matches the version published in Annals of Physics

Susskind-Glogower coherent states, whose Fock expansion coefficients include Bessel functions, have recently attracted considerable attention for their optical properties. Nevertheless, identity resolution is still an open question, which is an essential mathematical property that defines an overcomplete basis in the Fock space and allows a coherent state quantization map. In this regard, the modified Susskind-Glogower coherent states have been introduced as an alternative family of states that resolve the identity resolution. In the present manuscript, the quantization map related to the modified Susskind-Glogower coherent states is exploited, which naturally leads to a particular representation of the $\mathfrak{su}(1,1)$ Lie algebra in its discrete series. The latter provides evidence about further generalizations of coherent states, built from the Susskind-Glogower ones by extending the indexes of the Bessel functions of the first kind and, alternatively, by employing the modified Bessel functions of the second kind. In this form, the new families of Susskind-Glogower-I and Susskind-Glogower-II coherent states are introduced. The corresponding quantization maps are constructed so that they lead to general representations of elements of the $\mathfrak{su}(1,1)$ and $\mathfrak{su}(2)$ Lie algebras as generators of the SU$(1,1)$ and SU$(2)$ unitary irreducible representations respectively. For completeness, the optical properties related to the new families of coherent states are explored and compared with respect to some well-known optical states.

In quantum information processing, {using a receiver device to differentiate between two nonorthogonal states leads to a quantum error probability. The minimum possible error is} known as the Helstrom bound. In this work we study and compare quantum limits for states which generalize the Glauber-Sudarshan coherent states, like non-linear, Perelomov, Barut-Girardello, and (modified) Susskind-Glogower coherent states. For some of these, we show that the Helstrom bound can be significantly lowered and even vanish in specific regimes., Comment: 31 pages, 25 figures

We explore the quantization of classical models with position-dependent mass (PDM) terms constrained to a bounded interval in the canonical position. This is achieved through the Weyl-Heisenberg covariant integral quantization by properly choosing a regularizing function $\Pi(q,p)$ on the phase space that smooths the discontinuities present in the classical model. We thus obtain well-defined operators without requiring the construction of self-adjoint extensions. Simultaneously, the quantization mechanism leads naturally to a semi-classical system, that is, a classical-like model with a well-defined Hamiltonian structure in which the effects of the Planck's constant are not negligible. Interestingly, for a non-separable function $\Pi(q,p)$, a purely quantum minimal-coupling term arises in the form of a vector potential for both the quantum and semi-classical models.

Signal analysis is built upon various resolutions of the identity in signal vector spaces, e.g. Fourier, Gabor, wavelets, etc. Similar resolutions are used as quantizers of functions or distributions, paving the way to a time-frequency or time-scale quantum formalism and revealing interesting or unexpected features. Extensions to classical electromagnetism viewed as a quantum theory for waves and not for photons are mentioned., Comment: Submitted to Proceedings of ATFA19

The Mixmaster solution to Einstein field equations was examined by C. Misner in an effort to better understand the dynamics of the early universe. We highlight the importance of the quantum version of this model for early universe. This quantum version and its semi-classical portraits are yielded through affine and standard coherent state quantizations and more generally affine and Weyl-Heisenberg covariant integral quantizations. The adiabatic and vibronic approximations widely used in molecular physics can be employed to qualitatively study the dynamics of the model on both quantum and semi-classical levels. Moreover, the semi-classical approach with the exact anisotropy potential can be effective in numerical integration of some solutions. Some promising physical features such as the singularity resolution, smooth bouncing, the excitation of anisotropic oscillations and a substantial amount of post-bounce inflation as the backreaction to the latter are pointed out. Finally, a realistic cosmological scenario based on the quantum mixmaster model, which includes the formation and evolution of local structures is outlined., Comment: 9 figures, Key words and phrases. quantum cosmology; mixmaster; coherent states; quantization; semi-classical portraits; smooth bouncing

Covariant affine integral quantization is studied and applied to the motion of a particle in a punctured plane R^2_\ast=R^2\{0}, for which the phase space is R^2_\ast=R^2\{0}X R^2. We examine the consequences of different quantizer operators built from weight functions on this phase space. To illustrate the procedure, we examine two examples of weights. The first one corresponds to 2-D coherent state families, while the second one corresponds to the affine inversion in the punctured plane. The later yields the usual canonical quantization and a quasi-probability distribution (2-D affine Wigner function) which is real, marginal in both position and momentum., Comment: In the Addendum " Comment on 2-D Covariant Affine Integral Quantization(s)" a few errors have been identified in the article [Adv. Oper. Theory (2020) 5:901-935] and they are corrected. Furthermore, some notations have been modified in order to avoid any confusion

In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analyse observed sequences of q-triplets, or q-doublets if one of them is the unity, in terms of cycles of successive M\"obius transforms of the line preserving unity ( q=1 corresponds to the BG theory). Such transforms have the form q --> (aq + 1-a)/[(1+a)q -a], where a is a real number; the particular cases a=-1 and a=0 yield respectively q --> (2-q) and q --> 1/q, currently known as additive and multiplicative dualities. This approach seemingly enables the organisation of various complex phenomena into different classes, named N-complete or incomplete. The classification that we propose here hopefully constitutes a useful guideline in the search, for non-BG systems whenever well described through q-indices, of new possibly observable physical properties., Comment: Keyword: non-additive entropy; q-statistics; M\"obius transform; complex systems

We present a covariant quantization of the "massive" spin-${\frac{3}{2}}$ Rarita-Schwinger field in de Sitter (dS) spacetime. The dS group representation theory and its Wigner interpretation combined with the Wightman-G$\mbox{\"{a}}$rding axiomatic and analyticity requirements in the complexified pseudo-Riemanian manifold constitute the basis of the quantization scheme, while the whole procedure is carried out in terms of coordinate-independent dS plane waves. We make explicit the correspondence between unitary irreducible representations (UIRs) of the dS group and the field theory in dS spacetime: by "massive" is meant a field that carries a particular principal series representation of the dS group. We drive the plane-wave representation of the dS massive Rarita-Schwinger field in a manifestly dS-invariant manner. We show that it exactly reduces to its Minkowskian counterpart when the curvature tends to zero as far as the analyticity domain conveniently chosen. We then present the Wightman two-point function fulfilling the minimal requirements of local anticommutativity, covariance, and normal analyticity. The Hilbert space structure and the unsmeared field operator are also defined. The analyticity properties of the waves and the two-point function that we discuss in this paper allow for a detailed study of the Hilbert space of the theory, and give rise to the thermal physical interpretation., Comment: 17 pages, no figure, version accepted for publication in Physical Review D

In a recent Letter, we have pointed out that the linearized Einstein gravity in de Sitter (dS) spacetime besides the spacetime symmetries generated by the Killing vectors and the evident gauge symmetry also possesses a hitherto `hidden' local (gauge-like) symmetry which becomes anomalous on the quantum level. This gauge-like anomaly makes the theory inconsistent and must be canceled at all costs. In this companion paper, we first review our argument and discuss it in more detail. We argue that the cancelation of this anomaly makes it impossible to preserve dS symmetry in linearized quantum gravity through the usual canonical quantization in a consistent manner. Then, demanding that all the classical symmetries to survive in the quantized theory, we set up a coordinate-independent formalism \`{a} \emph{la} Gupta-Bleuler which allows for preserving the (manifest) dS covariance in the presence of the gauge and the gauge-like invariance of the theory. On this basis, considering a new representation of the canonical commutation relations, we present a graviton quantum field on dS space, transforming correctly under isometries, gauge transformations, and gauge-like transformations, which acts on a state space containing a vacuum invariant under all of them. Despite the appearance of negative norm states in this quantization scheme, the energy operator is positive in all physical states, and vanishes in the vacuum., Comment: 19 pages, no figure, version accepted for publication in Physical Review D

We implement a SU(1,1) covariant integral quantization of functions or distributions on the unit disk. The latter can be viewed as the phase space for the motion of a test "massive" particle on 1+1 Anti de Sitter space-time, and the relevant unitary irreducible representations of SU(1,1) corresponding to the quantum version of such motions are found in the discrete series and its lower limits. Our quantization method depends on a weight function on the phase space, and it includes Perelomov coherent states quantization. Semi-classical portraits or lower symbols of main physically relevant operators are determined., Comment: 26 pages

In this survey, various generalisations of Glauber-Sudarshan coherent states are described in a unified way, with their statistical properties and their possible role in non-standard quantisations of the classical electromagnetic field. Some statistical photon-counting aspects of Perelomov SU(2) and SU(1,1) coherent states are emphasized., Comment: 37 pages, submitted as a contribution to "Integrability, Supersymmetry and Coherent States", a volume in honour of V. Hussin

We propose a deformed version of the generalized Heisenberg algebra by using techniques borrowed from the theory of pseudo-bosons. In particular, this analysis is relevant when non self-adjoint Hamiltonians are needed to describe a given physical system. We also discuss relations with nonlinear pseudo-bosons. Several examples are discussed., Comment: Accepted in Journal of Physics A