The Nambu mechanics as a class of singular generalized dynamical formalisms

In a recent work Nambu has proposed ac-number dynamical formalism which can allow an odd numbern of canonical variables. Naturally associated to this new mechanics there exists ann-linear bracket whose study opens interesting possibilities. The purpose of this work is to show that besides this bracket another one which is bilinear and in fact a Lie bracket can also be associated with the Nambu mechanics. For anyn, however, this bracket is singular. In a sense previously used by the present author, this result exhibits the Nambu mechanics as an interesting class of singular generalized dynamical formalisms irrespective of the number of phase space variables. Reasons are given suggesting that such singular formalisms would be, within our context, the only ones capable of describing classical analogues of generalized quantum systems.     

A Wigner-function formulation of finite-state quantum mechanics

For a non-relativistic system with only continous degrees of freedom (no spin, for example), the original Wigner function can be used as an alternative to the density matrix to represent an arbitrary quantum state. Indeed, the quantum mechanics of such systems can be formulated entirely in terms of the Wigner function and other functions on phase space, with no mention of state vectors or operators. In the present paper this Wigner-function formulation is extended to systems having only a finite number of orthogonal states. The “phase space” for such a system is taken to be not continuous but discrete. In the simplest cases it can be pictured as an N×N array of points, where N is the number of orthogonal states. The Wigner function is a real function on this phase space, defined so that its properties are closely analogous to those of the original Wigner function. In this formulation, observables, like states, are represented by real functions on the discrete phase space. The complex numbers still play an important role: they appear in an essential way in the rule for forming composite systems.     

Tunneling time in space fractional quantum mechanics

We calculate the time taken by a wave packet to travel through a classically forbidden region of space in space fractional quantum mechanics. We obtain the close form expression of tunneling time from a rectangular barrier by stationary phase method. We show that tunneling time depends upon the width b of the barrier for $b\to \infty$ and therefore Hartman effect doesn't exist in space fractional quantum mechanics. Interestingly we found that the tunneling time monotonically reduces with increasing b. The tunneling time is smaller in space fractional quantum mechanics as compared to the case of standard quantum mechanics. We recover the Hartman effect of standard quantum mechanics as a special case of space fractional quantum mechanics.     

Geometry of time-dependent PT-symmetric quantum mechanics

A new type of quantum theory known as time-dependent PT-symmetric quantum mechanics has received much attention recently. It has a conceptually intriguing feature of equipping the Hilbert space of a PT-symmetric system with a time-varying inner product. In this work, we explore the geometry of time-dependent PT-symmetric quantum mechanics. We find that a geometric phase can emerge naturally from the cyclic evolution of a PT-symmetric system,and further formulate a series of related differential-geometry concepts, including connection, curvature, parallel transport,metric tensor, and quantum geometric tensor. These findings constitute a useful, perhaps indispensible, tool to investigate geometric properties of PT-symmetric systems with time-varying system's parameters. To exemplify the application of our findings, we show that the unconventional geometric phase [Phys. Rev. Lett. 91 187902(2003)], which is the sum of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase unveiled in this work.     

Geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics

A new type of quantum theory known as time-dependent $\mathcal{PT}$-symmetric quantum mechanics has received much attention recently. It has a conceptually intriguing feature of equipping the Hilbert space of a $\mathcal{PT}$-symmetric system with a time-varying inner product. In this work, we explore the geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics. We find that a geometric phase can emerge naturally from the cyclic evolution of a $\mathcal{PT}$-symmetric system, and further formulate a series of related differential-geometry concepts, including connection, curvature, parallel transport, metric tensor, and quantum geometric tensor. These findings constitute a useful, perhaps indispensible, tool to investigate geometric properties of $\mathcal{PT}$-symmetric systems with time-varying system's parameters. To exemplify the application of our findings, we show that the unconventional geometric phase [Phys. Rev. Lett. 91 187902 (2003)], which is the sum of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase unveiled in this work.     

Amplitude phase-space model for quantum mechanics

We show that there is a close relationship between quantum mechanics and ordinary probability theory. The main difference is that in quantum mechanics the probability is computed in terms of an amplitude function, while in probability theory a probability distribution is used. Applying this idea, we then construct an amplitude model for quantum mechanics on phase space. In this model, states are represented by amplitude functions and observables are represented by functions on phase space. If we now postulate a conjugation condition, the model provides the same predictions as conventional quantum mechanics. In particular, we obtain the usual quantum marginal probabilities, conditional probabilities and expectations. The commutation relations and uncertainty principle also follow. Moreover Schrödinger's equation is shown to be an averaged version of Hamilton's equation in classical mechanics.     

Stochasticity,decoherence and an arrow of time from the discretization of time?

Certain intriguing consequences of the discreteness of time on the time evolution of dynamical systems are discussed. In the discrete-time classical mechanics proposed here, there is an arrow of time that follows from the fact that the replacement of the time derivative by the backward difference operator alone can preserve the non-negativity of the phase space density. It is seen that, even for free particles, all the degrees of freedom are correlated in principle. The forward evolution of functions of phase space variables by a finite number of time steps, in this discrete-time mechanics, depends on the entire continuous-time history in the interval [0, ∞]. In this sense, discrete time evolution is nonlocal in time from a continuous-time point of view. A corresponding quantum mechanical treatment is possible via the density matrix approach. The interference between nondegenerate quantum mechanical states decays exponentially. This decoherence is present, in principle, for all systems; however, it is of practical importance only in macroscopic systems, or in processes involving large energy changes.     

Identifying Quantum Structures in the Ellsberg Paradox

Empirical evidence has confirmed that quantum effects occur frequently also outside the microscopic domain, while quantum structures satisfactorily model various situations in several areas of science, including biological, cognitive and social processes. In this paper, we elaborate a quantum mechanical model which faithfully describes the Ellsberg paradox in economics, showing that the mathematical formalism of quantum mechanics is capable to represent the ambiguity present in this kind of situations, because of the presence of contextuality. Then, we analyze the data collected in a concrete experiment we performed on the Ellsberg paradox and work out a complete representation of them in complex Hilbert space. We prove that the presence of quantum structure is genuine, that is, interference and superposition in a complex Hilbert space are really necessary to describe the conceptual situation presented by Ellsberg. Moreover, our approach sheds light on ‘ambiguity laden’ decision processes in economics and decision theory, and allows to deal with different Ellsberg-type generalizations, e.g., the Machina paradox.     

Polarization of light and topological phases

The phase shifts experienced by a polarized light wave when it propagates through media with arbitrary birefringence, dichroism and depolarizing properties, while on the one hand provide the basis for a variety of optical devices and experiments, on the other provide a powerful means of understanding unitary evolution, nonunitary evolution and decoherence of two-state quantum systems by virtue of a mathematical isomorphism of the two systems. These also help understand aspects of evolution of classical systems under the group of rotations in three-dimensional space, namely the SO(3) group, by virtue of its homomorphism with the group SU(2) governing unitary evolution of polarized light waves. In this review we present a survey and analysis of recent work on topological phases with polarization of light which has revealed several counterintuitive features of such phase shifts such as 2nπ anholonomies, nonlinear and discontinuous behaviour originating hi singularities, peculiar spectral dependence, etc. We point out several areas where these results may find practical application, for example endless phase correction in interferometric sensors, fast switching spatial light modulators, phase shifters with unusual chromatic properties, phasing of antenna arrays, etc. Several useful theoretical insights relevant to polarization optics, quantum mechanics, classical mechanics and other areas of physics, obtained from the work on polarization states are described and some directions for future work are indicated.     

Evaluation of entanglement of CV-pair

Practical realizations of qubits and systems of qubits are implement by particles with states from infinite-dimensional space, such as space of wave functions for coordinate representation of quantum mechanics. Properties of the entangled states of continuous variables differ from such for discrete variables. That occurs due to existence of unlimited operators of observables and their uncertainties. In this work we used and extended idea of entanglement estimation for the case of continuous variables with the help of covariance matrix norm. Well-known concurrence was calculated with the help of covariance of observables. The goal of this paper is to check if it is possible to apply this idea to continuous variables. We consider coherent electron pair as the object of our investigation. We also consider coordinate and momentum of coherent electron pair as basis of observables. In this work, a new characteristic of entanglement is proposed. We evaluated that using correlation function instead of covariance. For the small distances between particles, it coincides with covariance matrix norm but for the big distances, it remains restricted.     

Contact geometry in Lagrangian mechanics

We present a picture of Lagrangian mechanics, free of some unnatural features (such as complete divergences). As a byproduct, a completely natural U (1)-bundle over the phase space appears. The correspondence between classical and quantum mechanics is very clear, e.g. no topological ambiguities remain. Contact geometry is the basic tool.     

tt *-Geometry on the Big Phase Space

The big phase space, the geometric setting for the study of quantum cohomology with gravitational descendants, is a complex manifold and consists of an infinite number of copies of the small phase space. The aim of this paper is to define a Hermitian geometry on the big phase space. Using the approach of Dijkgraaf and Witten (Nucl Phys B 342:486–522, 1990), we lift various geometric structures of the small phase space to the big phase space. The main results of our paper state that various notions from tt ^*-geometry are preserved under such liftings.     

The Scattering Matrix and Associated Formulas in Hamiltonian Mechanics

We prove two new identities in scattering theory in Hamiltonian mechanics and discuss the analogy between these identities and their counterparts in quantum scattering theory. These identities involve the Poincaré scattering map, which is analogous to the scattering matrix. The first of our identities states that the Calabi invariant of the Poincaré scattering map can be expressed as the regularised phase space volume. This is analogous to the Birman-Krein formula. The second identity relates the Poincaré scattering map to the total time delay and is analogous to the Eisenbud-Wigner formula.     

What is Quantum Mechanics? A Minimal Formulation

This paper presents a minimal formulation of nonrelativistic quantum mechanics, by which is meant a formulation which describes the theory in a succinct, self-contained, clear, unambiguous and of course correct manner. The bulk of the presentation is the so-called “microscopic theory”, applicable to any closed system S of arbitrary size N, using concepts referring to S alone, without resort to external apparatus or external agents. An example of a similar minimal microscopic theory is the standard formulation of classical mechanics, which serves as the template for a minimal quantum theory. The only substantive assumption required is the replacement of the classical Euclidean phase space by Hilbert space in the quantum case, with the attendant all-important phenomenon of quantum incompatibility. Two fundamental theorems of Hilbert space, the Kochen–Specker–Bell theorem and Gleason’s theorem, then lead inevitably to the well-known Born probability rule. For both classical and quantum mechanics, questions of physical implementation and experimental verification of the predictions of the theories are the domain of the macroscopic theory, which is argued to be a special case or application of the more general microscopic theory.     

The foundation of quantum theory and noncommutative spectral theory. Part I

The present paper is the first part of a work which follows up on H. Kummer: A constructive approach to the foundations of quantum mechanics,Found. Phys. 17, 1–63 (1987). In that paper we deduced the JB-algebra structure of the space of observables (=detector space) of quantum mechanics within an axiomatic theory which uses the concept of a filter as primitive under the restrictive assumption that the detector space is finite-dimensional. This additional hypothesis will be dropped in the present paper.It turns out that the relevant mathematics for our approach to a quantum mechanical system with infinite-dimensional detector space is the noncommutative spectral theory of Alfsen and Shultz.We start off with the same situation as in the previous paper (cf. Sects. 1 and 2 of the present paper). By postulating four axioms (Axioms S, DP, R, and SP of Sec. 3), we arrive in a natural way at the mathematical setting of Alfsen and Shultz, which consists of a dual pair of real ordered linear spaces Y, M: A base norm space, called the strong source space (which, however, in slight contrast to the setting of Alfsen and Shultz, is not 1-additive) and an order unit space, called the weak detector space, which is the norm and order dual space of Y. The last section of part I contains the guiding example suggested by orthodox quantum mechanics. We observe that our axioms are satisfied in this example. In the second part of this work (which will appear in the next issue of this journal) we shall postulate three further axioms and derive the JB-algebra structure of quantum mechanics.     

Quantum tomography as a new approach to simulating quantum processes

A new method is proposed for ab initio calculations of nonstationary quantum processes on the basis of a probability representation of quantum mechanics with the help of a positive definite function (quantum tomogram). The essence of the method is that an ensemble of trajectories associated with the characteristics of the evolution equation for the quantum tomogram is considered in the space where the quantum tomogram is defined. The method is applied for detailed analysis of transient tunneling of a wave packet. The results are in good agreement with the exact numerical solution to the Schrödinger equation for this system. The probability density distributions are obtained in the coordinate and momentum spaces at consecutive instances. For transient tunneling of a wave packet, the probability of penetration behind the barrier and the time of tunneling are calculated as functions of the initial energy.     

Prequantum Classical Statistical Field Theory: Complex Representation, Hamilton-Schrödinger Equation, and Interpretation of Stationary States

We develop a prequantum classical statistical model in that the role of hidden variables is played by classical (vector) fields. We call this model Prequantum Classical Statistical Field Theory (PCSFT). The correspondence between classical and quantum quantities is asymptotic, so we call our approach asymptotic dequantization. We construct the complex representation of PCSFT. In particular, the conventional Schrödinger equation is obtained as the complex representation of the system of Hamilton equations on the infinite-dimensional phase space. In this note we pay the main attention to interpretation of so called pure quantum states (wave functions) in PCSFT, especially stationary states. We show, see Theorem 2, that pure states of QM can be considered as labels for Gaussian measures concentrated on one dimensional complex subspaces of phase space that are invariant with respect to the Schrödinger dynamics. “A quantum system in a stationary state ψ” in PCSFT is nothing else than a Gaussian ensemble of classical fields (fluctuations of the vacuum field of a very small magnitude) which is not changed in the process of Schrödinger's evolution. We interpret in this way the problem of stability of hydrogen atom. One of unexpected consequences of PCSFT is the infinite dimension of physical space on the prequantum scale.     

Quantum Blobs

Quantum blobs are the smallest phase space units of phase space compatible with the uncertainty principle of quantum mechanics and having the symplectic group as group of symmetries. Quantum blobs are in a bijective correspondence with the squeezed coherent states from standard quantum mechanics, of which they are a phase space picture. This allows us to propose a substitute for phase space in quantum mechanics. We study the relationship between quantum blobs with a certain class of level sets defined by Fermi for the purpose of representing geometrically quantum states.