A unified quantum theory of mechanics and thermodynamics. Part IIb. Stable equilibrium states

Part IIb presents some of the most important theorems for stable equilibrium states that can be deduced from the four postulates of the unified theory presented in Part I. It is shown for the first time that the canonical and grand canonical distributions are the only distributions that are stable. Moreover, it is shown that reversible adiabatic processes exist which cannot be described by the dynamical equation of quantum mechanics. A number of conditions are discussed that must be satisfied by the general equation of motion which is yet to be discovered.Part I of this paper appeared inFound. Phys. 6(1), 15 (1976). Part IIa appeared inFound. Phys. 6(2), 127 (1976). The numbering of the sections, equations, and references in this part of the paper continues from those in Part IIa.     

Quantum World with Quantum Time

Since the beginning of quantum mechanics there have been a lot of attempts to quantize time. In this paper we refer to the little known concept of quantum time proposed by E.Kapuscik [Hadronic J. 8 (1985) 75]. We analyze some properties of systems with quantum time. Moreover we comment and discuss the idea of quantum time.     

Formalism and Interpretation in Quantum Theory

Quantum Mechanics can be viewed as a linear dynamical theory having a familiar mathematical framework but a mysterious probabilistic interpretation, or as a probabilistic theory having a familiar interpretation but a mysterious formal framework. These points of view are usually taken to be somewhat in tension with one another. The first has generated a vast literature aiming at a “realistic” and “collapse-free” interpretation of quantum mechanics that will account for its statistical predictions. The second has generated an at least equally large literature aiming to derive, or at any rate motivate, the formal structure of quantum theory in probabilistically intelligible terms. In this paper I explore, in a preliminary way, the possibility that these two programmes have something to offer one another. In particular, I show that a version of the measurement problem occurs in essentially any non-classical probabilistic theory, and ask to what extent various interpretations of quantum mechanics continue to make sense in such a general setting. I make a start on answering this question in the case of a rudimentary version of the Everett interpretation.     

Cryptanalysis of fair quantum blind signatures

We investigate the fair quantum blind signature scheme proposed by Wang and Wen [Wang T Y and Wen Q Y 2010 Chin.Phys.B 19 060307],which uses the fundamental properties of quantum mechanics and the availability of a trusted arbitrator.However,in this paper,we find that the protocol cannot satisfy the property of non-forgeability even under the condition that the trusted arbitrator is totally credible.Moreover,a simple feasible suggestion for improving the protocol is proposed.     

Erwin Schrödinger and the rise of wave mechanics. I. Schrödinger's scientific work before the creation of wave mechanics

This article is in three parts. Part I gives an account of Erwin Schrödinger's growing up and studies in Vienna, his scientific work—first in Vienna from 1911 to 1920, then in Zurich from 1920 to 1925—on the dielectric properties of matter, atmospheric electricity and radioactivity, general relativity, color theory and physiological optics, and on kinetic theory and statistical mechanics. Part II deals with the creation of the theory of wave mechanics by Schrödinger in Zurich during the early months of 1926; he laid the foundations of this theory in his first two communications toAnnalen der Physik. Part III deals with the early applications of wave mechanics to atomic problems—including the demonstration of equivalence of wave mechanics with the quantum mechanics of Born, Heisenberg, and Jordan, and that of Dirac—by Schrödinger himself and others. The new theory was immediately accepted by the scientific community.This article (in three parts) is an expanded version of the Schrödinger Centenary Lecture delivered by me at CERN (Organisation Européenne pour la Recherche Nucléaire), 1211 Geneva 23, Switzerland, on July 30, 1987.     

Quantum phenomena and the zeropoint radiation field. II

A previous paper was devoted to the discussion of a new version of stochastic electrodynamics (SED) and to the study of the conditions under which quantum mechanics can be derived from it, in the radiationless approximation. In this paper further effects on matter due to the zeropoint field are studied, such as atomic stability, radiative transitions, the Lamb shift, etc., and are shown to be correctly described by the proposed version of SED. Also, a detailed energy-balance condition and a fluctuation-dissipation relation are established; it is shown in particular that equilibrium is attained only with a field spectrum ³.The proposed approach is shown to suggest an understanding of quantum mechanics as a kind of limitcycle theory. Finally, a brief discussion is included about the nonchaotic behavior of the (bounded) SED system in the quantum regime, as measured by Lyapunov exponents.On Leave of absence at Mathematics Department, University College London, srGower Street, London WC1, United Kingdom.     

Toward a micro realistic version of quantum mechanics. Part I

This paper investigates the possibility of developing a fully micro realistic version of elementary quantum mechanics. I argue that it is highly desirable to develop such a version of quantum mechanics, and that the failure of all current versions and interpretations of quantum mechanics to constitute micro realistic theories is at the root of many of the interpretative problems associated with quantum mechanics, in particular the problem of measurement. I put forward a propensity micro realistic version of quantum mechanics, and suggest how it might be possible to discriminate, on experimental grounds, between this theory and other versions of quantum mechanics.     

A Hilbert Space Realization of Nonlinear Quantum Mechanics as Classical Extension of Its Linear Counterpart

This paper studies the state-effect-probability structure associated with thequantum mechanics of nonlinear (homogeneous, in general nonadditive) operatorson a Hilbert space. Its aim is twofold: to provide a concrete representation ofthe features of nonlinear quantum mechanics on a Hilbert space, and to showthat the properties of the nonlinear version of quantum mechanics here describedhave the structure of a classical logic.     

Numerical analysis of trajectories of a quantum particle in two-slit experiment

It is well known that the concept of a trajectory of a quantum particle is itself nonsense in the so-called Copenhagen interpretation. However, if the interpretation proposed by Ishikawa [International Journal of Theoretical Physics,30(4), 401 (1991)] can be accepted in quantum mechanics, the trajectory of a quantum particle is significant (though it includes errors). In this paper we numerically analyze discrete trajectories of a quantum particle in a two-slit experiment under this new interpretation.     

A unified quantum theory of mechanics and thermodynamics. Part IIa. Available energy

Part II of this three-part paper presents some of the most important theorems that can be deduced from the four postulates of the unified theory discussed in Part I. In Part IIa, it is shown that the maximum energy that can be extracted adiabatically from any system in any state is solely a function of the density operator associated with the state. Moreover, it is shown that for any state of a system, nonequilibrium, equilibrium or stable equilibrium, a unique propertyS exists which is proportional to the total energy of the system minus the maximum energy that can be extracted adiabatically from the system in combination with a reservoir. For statistically independent systems, propertyS is extensive, it is invariant during all reversible processes, and it increases during all irreversible processes.Part I of this paper appeared inFound. Phys. 6(1) (1976). The numbering of the sections, equations, and references in this part of the paper continues from those in Part I.     

On Quantum Phase Transition. I. Spinless Electrons Strongly Correlated with Ions

We study a large class F of models of the quantum statistical mechanics dealing with two types of particles. First the spinless electrons are quantum particles obeying to the Fermi statistics, they can hop. Secondly the ions which cannot move, are classical particles. The Falicov–Kimball (FK) model⁽¹⁾ is a well known model belonging to F, for which the existence of an antiferomagnetic phase transition was proven in the seminal paper of Kennedy and Lieb.⁽²⁾ This result was extended by Lebowitz and Macris.⁽³⁾ A new approach to this problem based on quantum selection of the ground states was proposed in ref. 4. In this paper we extend this approach to show that, under the strong insulating condition, any hamiltonian of the class F admits, at every temperature, an effective hamiltonian, which governs the behaviour of the ions interacting through forces mediated by the electrons. The effective hamiltonians are long range many body Ising hamiltonians, which can be computed by a cluster expansion expressed in term of the quantum fluctuations. Our main result is that we can apply the powerfull results of the classical statistical mechanics to our quantum models. In particular we can use the classical Pirogov–Sinai theory to establish a hierarchy of phase diagrams, we can also study of the behaviour of the quantum inter- faces,⁽²⁹⁾ and so on...     

Quantum mechanics on Riemannian manifold in Schwinger's quantization approach II

The extended Schwinger quantization procedure is used for constructing quantum mechanics on a manifold with a group structure. The considered manifold M is a homogeneous Riemannian space with the given action of an isometry transformation group. Using the identification of M with the quotient space G/H, where H is the isotropy group of an arbitrary fixed point of M, we show that quantum mechanics on G/H possesses a gauge structure, described by a gauge potential that is the connection 1-form of the principal fiber bundle G(G/H, H). The coordinate representation of quantum mechanics and the procedure for selecting the physical sector of the states are developed. Received: 27 June 2000 / Revised version: 10 May 2001 / Published online: 19 July 2001     

New transformation of Wigner operator in phase space quantum mechanics for the two-mode entangled case

As a natural and important extension of Fan's paper (Fan Hong-Yi 2010 Chin. Phys. B 19 040305) by employing the formula of operators' Weyl ordering expansion and the bipartite entangled state representation this paper finds a new two-fold complex integration transformation about the Wigner operator Δ ( μ,v ) (in its entangled form) in phase space quantum mechanics, and its inverse transformation. In this way, some operator ordering problems regarding to ( a₁⁺-a₂) and (a₁+a₂⁺) can be solved and the contents of phase space quantum mechanics can be enriched, where a_i,a_i⁺ are bosonic creation and annihilation operators, respectively.     

Quantum mechanics on Riemannian manifold in Schwinger's quantization approach III

Using the extended Schwinger quantization approach, quantum mechanics on a Riemannian manifold M with the given action of an intransitive group of isometries is developed. It was shown that quantum mechanics can be determined unequivocally only on submanifolds of M where G acts simply transitively (orbits of G action). The remaining part of the degrees of freedom can be described unequivocally after introducing some additional assumptions. Being logically unmotivated, these assumptions are similar to the canonical quantization postulates. Besides this ambiguity which is of a geometrical nature there is an undetermined gauge field of the order of (or higher), vanishing in the classical limit . Received: 19 February 2001 / Revised version: 10 May 2001 / Published online: 6 July 2001     

The conceptual analysis (CA) method in theories of microchannels: Application to quantum theory. Part II. Idealizations. “Perfect measurements”

The application of the conceptual analysis (CA) method outlined in Part I is illustrated on the example of quantum mechanics. In Part II, we deduce the complete-lattice structure in quantum mechanics from postulates specifying the idealizations that are accepted in the theory. The idealized abstract concepts are introduced by means of a topological extension of the basic structure (obtained in Part I) in accord with the “approximation principle”; the relevant topologies are not arbitrarily chosen; they are fixed by the choice of the idealizations. There is a typical topological asymmetry in the mathematical scheme. Convexity or linear structures do not play any role in the mathematical methods of this approach. The essential concept in Part II is the idealization of “perfect measurement” suggested by our conceptual analysis in Part I. The Hilbert-space representation will be deduced in Part III. In our papers, we keep to the tenet: The mathematical scheme of a physical theory must be rigorously formulated. However, for physics, mathematics is only a nice and useful tool; it is not purpose.     

Popper's Experiment Revisited

Recently, there has been increasing theoretical and experimental interest in Popper's gedanken experiment. We calculate in this paper, using the path integral approach, the diffraction patterns predicted by quantum mechanics for this arrangement. The calculations confirm the narrowing of the width of the pattern in absence of the slit obtained experimentally by Kim and Shih (Y. Kim and Y. Shih, Found. Phys. 29, 1849 (1999)).     

Quantum mechanics as a matrix symplectic geometry

The main purpose of this work is to describe the quantum analog of the usual classical symplectic geometry and then to formulate quantum mechanics as a noncommutative symplectic geometry. First, we describe a discrete Weyl-Schwinger realization of the Heisenberg group and we develop a discrete version of the Weyl-Wigner-Moyal formalism. We also study the continuous limit and the case of higher degrees of freedom. In analogy with the classical case, we present the noncommutative (quantum) symplectic geometry associated with the matrix algebraM _N (C) generated by the Schwinger matrices.     

The S-matrix in classical mechanics

In the Hilbert-space version of classical mechanics, scattering theory forN-particle systems is developed in close analogy to the quantum case. Asymptotic completeness is proved for forces of finite range. Infinite-range forces lead to the problem of stability of bound states and can be dealt with only in some simple cases.It is a pleasure to thank Prof.L. Motchane for his kind hospitality at the I.H.E.S., where most of this work was done, and where the author profited from discussions withD. Ruelle andO. E. Lanford.     

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.     

Modeling Modal Talk in Quantum Mechanics

In this paper, modal and counterfactual logical connectives are defined in an extended framework of branching space-time (Belnap, N. D. (1992). Branching space-time. Synthese 92, 385–434). It is shown that a variety of definitions of the counterfactual can be given. The validity of certain modal statements occurring in quantum mechanics depends on the choice of definition. These considerations can be applied to an analysis of Stapp’s premises LOC1 and LOC2 from his purported proof of non-locality (Stapp, H. P. (1997). Nonlocal character of quantum theory. American Journal of Physics 65, 300–304). It is shown that while the validity of LOC1 depends on the choice of the definition of the counterfactual, LOC2 is absolutely invalid.