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We study the spectral problem in deformed supersymmetric quantum mechanics with polynomial superpotential by using the exact WKB method and the TBA equations. We apply the ODE/IM correspondence to the Schrödinger equation with an effective potential deformed by integrating out the fermions, which admits a continuous deformation parameter. We find that the TBA equations are described by the ℤ4-extended ones. For cubic superpotential corresponding to the symmetric double-well potential, the TBA system splits into the two D3-type TBA equations. We investigate in detail this example based on the TBA equations and their analytic continuation as well as the massless limit. We find that the energy spectrum obtained from the exact quantization condition is in good agreement with the diagonalization approach of the Hamiltonian. [ABSTRACT FROM AUTHOR]

Following the lines of the recent papers (Daneshmand and Tavassoly, Int. J. Theor. Phys. 56, 1218 (2017)), we study quantum mechanical treatments of an interaction between a two-level atom with a single-mode field in the two-photon Jaynes-Cummings model, where the Hamiltonian of the field is considered to be the quantized Caldirola-Kanai (CK) Hamiltonian. As a result, we would expect that the quantum dynamics of the two-photon JCM in terms of the CK Hamiltonian is qualitatively different from that of the usual one-photon case. We analytically calculate the explicit form of the atom-field entangled state and numerically evaluate the dynamics of its physical properties. The degree of entanglement, atomic population as well as sub-Poissonian statistics and quadrature squeezing of the field are analyzed. We adjust the latter evolved parameters by appropriately tuning the damping parameter within the CK Hamiltonian and detuning factor. Finally, we report a field detuning asymmetry in the collective statistical behavior. [ABSTRACT FROM AUTHOR]

This work develops a new method to calculate non-perturbative corrections in one-dimensional Quantum Mechanics, based on trans-series solutions to the refined holomorphic anomaly equations of topological string theory. The method can be applied to traditional spectral problems governed by the Schrödinger equation, where it both reproduces and extends the results of well-established approaches, such as the exact WKB method. It can be also applied to spectral problems based on the quantization of mirror curves, where it leads to new results on the trans-series structure of the spectrum. Various examples are discussed, including the modified Mathieu equation, the double-well potential and the quantum mirror curves of local P2 and local F0. In all these examples, it is verified in detail that the trans-series obtained with this new method correctly predict the large-order behavior of the corresponding perturbative sectors. [ABSTRACT FROM AUTHOR]

Observables have a dual nature in both classical and quantum kinematics: they are at the same time quantities, allowing to separate states by means of their numerical values, and generators of transformations, establishing relations between different states. In this work, we show how this twofold role of observables constitutes a key feature in the conceptual analysis of classical and quantum kinematics, shedding a new light on the distinguishing feature of the quantum at the kinematical level. We first take a look at the algebraic description of both classical and quantum observables in terms of Jordan-Lie algebras and show how the two algebraic structures are the precise mathematical manifestation of the twofold role of observables. Then, we turn to the geometric reformulation of quantum kinematics in terms of Kähler manifolds. A key achievement of this reformulation is to show that the twofold role of observables is the constitutive ingredient defining what an observable is. Moreover, it points to the fact that, from the restricted point of view of the transformational role of observables, classical and quantum kinematics behave in exactly the same way. Finally, we present Landsman’s general framework of Poisson spaces with transition probability, which highlights with unmatched clarity that the crucial difference between the two kinematics lies in the way the two roles of observables are related to each other. [ABSTRACT FROM AUTHOR]

We consider the relationship between the unitary quantization scheme and the para-Fermi statistics of order 2. We propose an appropriate generalization of Green’s ansatz, which has made it possible to transform bilinear and trilinear commutation relations for the creation and annihilation operators for two different para-Fermi fields φa and φb into identities. We also propose a method for incorporating para-Grassmann numbers ξk into the general unitary quantization scheme. For the parastatistics of order 2, a new fact has been revealed: the trilinear relations containing both para-Grassmann variables ξk and field operators ak and bm are transformed under a certain reversible mapping into unitary equivalent relations in which commutators are replaced by anticommutators, and vice versa. It is shown that this leads to the existence of two alternative definitions of the coherent state for para-Fermi oscillators. The Klein transformation for Green’s components of operators ak and bm is constructed in explicit form, which enabled us to reduce the initial commutation rules for the components to the normal commutation relations for ordinary Fermi fields. We have analyzed a nontrivial relationship between the trilinear commutation relations of the unitary quantization scheme and the so-called Lie supertriple system. The possibility of incorporating the Duffin-Kemmer-Petiau theory into the unitary quantization scheme is discussed briefly. [ABSTRACT FROM AUTHOR]

These notes are an expanded version of a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularities and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterparts are also presented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can one recover the classical system? [ABSTRACT FROM AUTHOR]

This study attempts to spell out more explicitly than has been done previously the connection between two types of formal correspondence that arise in the study of quantum-classical relations: one the one hand, deformation quantization and the associated continuity between quantum and classical algebras of observables in the limit ħ→0, and, on the other, a certain generalization of Ehrenfest’s Theorem and the result that expectation values of position and momentum evolve approximately classically for narrow wave packet states. While deformation quantization establishes a direct continuity between the abstract algebras of quantum and classical observables, the latter result makes in-eliminable reference to the quantum and classical state spaces on which these structures act—specifically, via restriction to narrow wave packet states. Here, we describe a certain geometrical re-formulation and extension of the result that expectation values evolve approximately classically for narrow wave packet states, which relies essentially on the postulates of deformation quantization, but describes a relationship between the actions of quantum and classical algebras and groups over their respective state spaces that is non-trivially distinct from deformation quantization. The goals of the discussion are partly pedagogical in that it aims to provide a clear, explicit synthesis of known results; however, the particular synthesis offered aspires to some novelty in its emphasis on a certain general type of mathematical and physical relationship between the state spaces of different models that represent the same physical system, and in the explicitness with which it details the above-mentioned connection between quantum and classical models. [ABSTRACT FROM AUTHOR]

Despite its age, quantum theory still suffers from serious conceptual difficulties. To create clarity, mathematical physicists have been attempting to formulate quantum theory geometrically and to find a rigorous method of quantization, but this has not resolved the problem. In this article we argue that a quantum theory recursing to quantization algorithms is necessarily incomplete. To provide an alternative approach, we show that the Schrödinger equation is a consequence of three partial differential equations governing the time evolution of a given probability density. These equations, discovered by Madelung, naturally ground the Schrödinger theory in Newtonian mechanics and Kolmogorovian probability theory. A variety of far-reaching consequences for the projection postulate, the correspondence principle, the measurement problem, the uncertainty principle, and the modeling of particle creation and annihilation are immediate. We also give a speculative interpretation of the equations following Bohm, Vigier and Tsekov, by claiming that quantum mechanical behavior is possibly caused by gravitational background noise. [ABSTRACT FROM AUTHOR]

The interaction between particles in the mean-field approximation of the many-body theory is often taken into account with the use of the semiclassical description of the particle motion. However, quantization of a part of the degrees of freedom becomes essential in certain cases. In this work, two such cases where nonlinear wave equations appear have been considered: electrons in a quantum well and excitons in a trap. In the case of indirect excitons in an annular trap, the one-dimensional Gross-Pitaevskii equation permits an analytical solution and it turns out that there can be no bound state in a one-dimensional symmetric potential well. This makes the problem qualitatively different from a similar one-body problem. In the case of electrons in a quantum well, the nonlinear integro-differential equation does not have an exact solution and the allowed energy levels have been found by the direct variational method. [ABSTRACT FROM AUTHOR]

In this paper we study the third quantized super-group field cosmology, a model in multiverse scenario, in Batalin-Vilkovisky (BV) formulation. Further, we propose the superfield/super-antifield dependent BRST symmetry transformations. Within this formulation we establish connection between the two different solutions of the quantum master equation within the BV formulation. [ABSTRACT FROM AUTHOR]

We study the semiclassical mechanics of systems with second-class constraints. We assume that the quantum mechanics of such a system is constructed by the Batalin-Fradkin-Tyutin method, where some additional coordinates and momenta are introduced and second-class constraints are converted into firstclass constraints. We also assume that the algebraic quantization method is used to quantize the extended system. To construct the semiclassical approximation, we use the Maslov complex germ theory. We study the semiclassical states of a system with second-class constraints, their scalar product, and the action of semiclassical observables in the first order of the semiclassical expansion. We consider the transformation of semiclassical states in the course of evolution. [ABSTRACT FROM AUTHOR]

In this article we present a possible way to make usual quantum mechanics fully compatible with physical realism, defined as the statement that the goal of physics is to study entities of the natural world, existing independently from any particular observer's perception, and obeying universal and intelligible rules. Rather than elaborating on the quantum formalism itself, we propose a new quantum ontology, where physical properties are attributed jointly to the system, and to the context in which it is embedded. In combination with a quantization principle, this non-classical definition of physical reality sheds new light on counter-intuitive features of quantum mechanics such as the origin of probabilities, non-locality, and the quantum-classical boundary. [ABSTRACT FROM AUTHOR]

Rather than consider space-time as an a priori arena in which events take place, it is a construction of our mind making possible a particular kind of ordering of events. As quantum entanglement is a property of states independent of classical distances, the notion of space and time has to be revised to represent the holistic interconnectedness of quanta. We also speculate about various forms of reprogramming, or reconfiguring, the propagation of information for multipartite statistics and in quantum field theory. [ABSTRACT FROM AUTHOR]

Historically, quantization has meant turning the dynamical variables of classical mechanics that are represented by numbers into their corresponding operators. Thus the relationships between classical variables determine the relationships between the corresponding quantum mechanical operators. Here, we take a radically different approach to this conventional quantization procedure. Our approach does not rely on any relations based on classical Hamiltonian or Lagrangian mechanics nor on any canonical quantization relations, nor even on any preconceptions of particle trajectories in space and time. Instead we examine the symmetry properties of certain Hermitian operators with respect to phase changes. This introduces harmonic operators that can be identified with a variety of cyclic systems, from clocks to quantum fields. These operators are shown to have the characteristics of creation and annihilation operators that constitute the primitive fields of quantum field theory. Such an approach not only allows us to recover the Hamiltonian equations of classical mechanics and the Schrödinger wave equation from the fundamental quantization relations, but also, by freeing the quantum formalism from any physical connotation, makes it more directly applicable to non-physical, so-called quantum-like systems. Over the past decade or so, there has been a rapid growth of interest in such applications. These include, the use of the Schrödinger equation in finance, second quantization and the number operator in social interactions, population dynamics and financial trading, and quantum probability models in cognitive processes and decision-making. In this paper we try to look beyond physical analogies to provide a foundational underpinning of such applications. [ABSTRACT FROM AUTHOR]

We describe the behavior of a quantum particle in a random medium using the Green's function in the functional integral representation and propose methods for evaluating it. We calculate the localization length in the case of motion in a disordered medium and in a medium described by a random Gaussiantype potential. We consider the motion of the quantum particle in the quantized-field vacuum and show the difference between the relativistic and nonrelativistic approaches. [ABSTRACT FROM AUTHOR]

We propose a group-theoretical interpretation of the fact that the transition from classical to quantum mechanics entails a reduction in the number of observables needed to define a physical state (e.g. from $$q$$ and $$p$$ to $$q$$ or $$p$$ in the simplest case). We argue that, in analogy to gauge theories, such a reduction results from the action of a symmetry group. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry and group representation theory, notably Souriau's moment map, the Mardsen-Weinstein symplectic reduction, the symplectic 'category' introduced by Weinstein, and the conjecture (proposed by Guillemin and Sternberg) according to which 'quantization commutes with reduction'. By using the generalization of this conjecture to the non-zero coadjoint orbits of an abelian Hamiltonian action, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric underpinning, namely the symplectic reduction formalism. This stance points towards a gauge-theoretical interpretation of Heisenberg indeterminacy principle. We revisit (the extreme cases of) this principle in the light of the difference between the set-theoretic points of a phase space and its category-theoretic symplectic points. [ABSTRACT FROM AUTHOR]

The aim of the famous Born and Jordan 1925 paper was to put Heisenberg's matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg's theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger's wave mechanics cannot be equivalent to Heisenberg's more physically motivated matrix mechanics unless its observables are quantized using this rule, and not the more symmetric prescription proposed by Weyl in 1926, which has become the standard procedure in quantum mechanics. This observation confirms the superiority of Born-Jordan quantization, as already suggested by Kauffmann. We also show how to explicitly determine the Born-Jordan quantization of arbitrary classical variables, and discuss the conceptual advantages in using this quantization scheme. We finally suggest that it might be possible to determine the correct quantization scheme by using the results of weak measurement experiments. [ABSTRACT FROM AUTHOR]

The standard quantization formalism in spacetimes with event horizons implies a non-unitary evolution of quantum states, as initial pure states may evolve into thermal states. This phenomenon is behind the famous black hole information loss paradox which provoked long-standing debates on the compatibility of quantum mechanics and gravity. In this paper we demonstrate that within an alternative time-symmetric quantization formalism thermal radiation is absent and states evolve unitarily in spacetimes with event horizons. We also discuss the theoretical consistency of the proposed formalism. We explicitly demonstrate that the theory preserves the microcausality condition and suggest a 'reinterpretation postulate' to resolve other apparent pathologies associated with negative energy states. Accordingly as there is a consistent alternative, we argue that choosing to use time-asymmetric quantization is a necessary condition for the black hole information loss paradox. [ABSTRACT FROM AUTHOR]

We show that the mathematical formalisms of general relativity and of quantum mechanics can be reconciled based on an algebraic approach. In this case, gravity does not need to be quantized. [ABSTRACT FROM AUTHOR]

We provide a proof for the necessity of quantizing fundamental interactions demonstrating that a quantum version is needed for any non trivial conservative interaction whose strength depends on the relative distance between two objects. Our proof is based on a consistency argument that in the presence of a classical field two interacting objects in a separable state could not develop entanglement. This requirement can be cast in the form of a holonomic constraint that cannot be satisfied by generic interparticle potentials. Extending this picture of local holonomic constraints, we design a protocol that allows to measure the terms of a multipole expansion of the interaction of two composite bodies. The results presented in this work can pave the way for a study of fundamental interactions based on the analysis of entanglement properties. [ABSTRACT FROM AUTHOR]