Identity Rule for Classical and Quantum Theories

It is shown that the identity rule — arule of inference which has the form of modus ponens butwith the operation of identity substituted for theoperation of implication — turns any ortholatticeinto either an orthomodular lattice (a model of a quantumtheory) or a distributive lattice (a model of aclassical theory). It is also shown that — asopposed to the implication algebras — one cannotconstruct an identity algebra although the identity rule contains theoperation of identity as the only operation.     

Orthomodular Lattices and a Quantum Algebra

We show that one can formulate an algebra with lattice ordering so as to contain one quantum and five classical operations as opposed to the standard formulation of the Hilbert space subspace algebra. The standard orthomodular lattice is embeddable into the algebra. To obtain this result we devised algorithms and computer programs for obtaining expressions of all quantum and classical operations within an orthomodular lattice in terms of each other, many of which are presented in the paper. For quantum disjunction and conjunction we prove their associativity in an orthomodular lattice for any triple in which one of the elements commutes with the other two and their distributivity for any triple in which a particular element commutes with the other two. We also prove that the distributivity of symmetric identity holds in Hilbert space, although whether or not it holds in all orthomodular lattices remains an open problem, as it does not fail in any of over 50 million Greechie diagrams we tested.     

Free Massive Fermions¶Inside the Quantum Discrete sine-Gordon Model

We extend the notion of space shifts introduced in [FV3] for certain quantum light cone lattice equations of sine-Gordon type at root of unity (e.g. [FV1,FV2,BKP,BBR]). As a result, we obtain a compatibility equation for the roots of central elements within the algebra of observables (also called current algebra). The equation, which is obtained by exponentiating these roots, is exactly the evolution equation for the?“classical background” as described in [BBR]. As an application for the introduced constructions, we derive a one to one correspondence between a special case of the quantum light cone lattice equations of sine-Gordon type and free massive fermions on a lattice, as a special case of the lattice Thirring model constructed in [DV]. Received: 2 December 1996 / Accepted: 19 January 1999     

Stochastic Gravity

We give a summary of the status of currentresearch in stochastic semiclassical gravity and suggestdirections for further investigations. This theorygeneralizes the semiclassical Einstein equation to an Einstein-Langevin equation with a stochasticsource term arising from the fluctuations of theenergy-momentum tensor of quantum fields. We mentionrecent efforts in applying this theory to the study of black hole fluctuation and backreactionproblems, linear response of hot flat space, andstructure formation in inflationary cosmology. Toexplore the physical meaning and implications of thisstochastic regime in relation to both classical andquantum gravity, we find it useful to take the view thatsemiclassical gravity is mesoscopic physics and thatgeneral relativity is the hydrodynamic limit of certain spacetime quantum substructures. We view theclassical spacetime depicted by general relativity as acollective state and the metric or connection functionsas collective variables. Three basic issues —stochasticity, collectivity, correlations — andthree processes — dissipation, fluctuations,decoherence — underscore the transformation fromquantum microstructure and interaction to the emergenceof classical macrostructure and dynamics. We discuss ways toprobe into the high-energy activity from below and maketwo suggestions: via effective field theory and thecorrelation hierarchy. We discuss how stochastic behavior at low energy in an effective theoryand how correlation noise associated with coarse-grainedhigher correlation functions in an interacting quantumfield could carry nontrivial information about the high-energy sector. Finally, we describeprocesses deemed important at the Planck scale,including tunneling and pair creation, wave scatteringin random geometry, growth of fluctuations and forms, Planck-scale resonance states, and spacetimefoams.     

A new (in)finite-dimensional algebra for quantum integrable models

A new (in)finite-dimensional algebra which is a fundamental dynamical symmetry of a large class of (continuum or lattice) quantum integrable models is introduced and studied in details. Finite-dimensional representations are constructed and mutually commuting quantities—which ensure the integrability of the system—are written in terms of the fundamental generators of the new algebra. Relation with the deformed Dolan–Grady integrable structure recently discovered by one of the authors and Terwilliger's tridiagonal algebras is described. Remarkably, this (in)finite-dimensional algebra is a “q-deformed” analogue of the original Onsager's algebra arising in the planar Ising model. Consequently, it provides a new and alternative algebraic framework for studying massive, as well as conformal, quantum integrable models.     

Digital Signatures with Quantum Candies

Quantum candies (qandies) represent a type of pedagogical simple model that describes many concepts from quantum information processing (QIP) intuitively without the need to understand or make use of superpositions and without the need of using complex algebra. One of the topics in quantum cryptography that has gained research attention in recent years is quantum digital signatures (QDS), which involve protocols to securely sign classical bits using quantum methods. In this paper, we show how the “qandy model” can be used to describe three QDS protocols in order to provide an important and potentially practical example of the power of “superpositionless” quantum information processing for individuals without background knowledge in the field.     

Two dimensional lattice gauge theory based on a quantum group

In this article we analyse a two dimensional lattice gauge theory based on a quantum group. The algebra generated by gauge fields is the lattice algebra introduced recently by A.Yu. Alekseev, H. Grosse and V. Schomerus in [1]. We define and study Wilson loops. This theory is quasi-topological as in the classical case, which allows us to compute the correlation functions of this theory on an arbitrary surface.Laboratoire Propre du CNRS UPR 14     

Operational foundation of quantum logic

The logic of quantum mechanical propositions—called quantum logic—is constructed on the basis of the operational foundation of logic. Some obvious modifications of the operational method, which come from the incommensurability of the quantum mechanical propositions, lead to the effective quantum logic. It is shown in this paper that in the framework of a calculization of this effective quantum logic the negation of a proposition is uniquely defined (Theorem I), and that a weak form of the quasimodular law can be derived (Theorem II). Taking account of the definiteness of truth values for quantum mechanical propositions, the calculus of full quantum logic can be derived (Theorem III). This calculus represents an orthocomplemented quasimodular lattice which has as a model the lattice of subspaces of Hilbert space.     

Supersymmetry algebras that include topological charges

We show that in supersymmetric theories with solitons, the usual supersymmetry algebra is not valid; the algebra is modified to include the topological quantum numbers as central charges. Using the corrected algebra, we are able to show that in certain four dimensional gauge theories, there are no quantum corrections to the classical mass spectrum. These are theories for which Bogomolny has derived a classical bound; the argument involves showing that Bogomolny's bound is valid quantum mechanically and that it is saturated.     

q-Deformation of algebraic structures of quantum and classical mechanics

There exists a coassociative and cocommutative coproduct in the linear space spanned by the two algebraic products of a classical Hamilton algebra (the algebraic structure underlying classical mechanics [1]). The transition from classical to quantum Hamilton algebra (the algebraic structure underlying quantum mechanics) is anħ-deformation which preserves not only the Lie property of the classical Hamilton algebra but also the coassociativity and cocommutativity of the above coproduct. By explicit construction we obtain the algebraic structures of theq-deformed Hamilton algebras which preserve the said properties of the coproduct. Some algorithms of these structures are obtained and their implications discussed. The problem of consistency of time evolution with theq-deformed kinematical structure is discussed. A characteristic distinction between the parametersħ andq is brought out to stress the fact thatq cannot be regarded as a fundamental constant.     

Equilibrium states for a classical lattice gas

Various definitions of thermodynamic equilibrium states for a classical lattice gas are given and are proved to be equivalent. In all cases, a set of equations is given, the solutions of which are by definition equilibrium states. Examples are the condition of Lanford and Ruelle, and the KMS boundary condition. In connection with this, it is shown that the time translation for classical interactions exists as an automorphism of the quantum algebra of observables, under conditions which are weaker than those found for quantum interactions.     

Quantum Logic for Quantum Computers

The following results obtained within a project of finding the algebra of statesin a general-purpose quantum computer are reported: (1) All operations of anorthomodular lattice, including the identity, are fivefold-defined; (2) there arenonorthomodular models for both quantum and classical logics; (3) there is afour-variable orthoarguesian lattice condition which contains all known orthoarguesianlattice conditions including six- and five-variable ones. Repercussions to quantumcomputers operating as quantum simulators are discussed.     

Exact quantum three-point function of Liouville highest weight states

In their earlier works on the quantum Liouville theory, Gervais and Neveu derived the exact spectrum of highest weight states of the conformal algebra. In the present paper, we determine the interaction between three of these states exactly, in the weak coupling regime of the quantum Liouville dynamics. It is first studied, at the classical level, by computing the time delays from an appropriate classical solution. The result is then extended to the quantum case, by following a path taken some time ago by Faddeev, Kulish, and Korepin, for the sine-Gordon theory: the coupling constant is replaced by the renormalized one, and the classical action that takes the form of a three-dimensional line integral is replaced by a discrete sum running over the exact quantum spectrum of the three asymptotic states that forms a three-dimensional lattice. At the quantum level, the classical S-matrix, that is the exponential of the action, becomes a product to be computed along a line on this lattice. It must only depend upon the end points and this completely determines the three-point function at the quantum level. Its structure is reminiscent of the other exact S-matrices that have been discovered earlier.     

Algebraically Self-Consistent Quasiclassical Approximation on Phase Space

The Wigner–Weyl mapping of quantum operators to classical phase space functions preserves the algebra, when operator multiplication is mapped to the binary * operation. However, this isomorphism is destroyed under the quasiclassical substitution of * with conventional multiplication; consequently, an approximate mapping is required if algebraic relations are to be preserved. Such a mapping is uniquely determined by the fundamental relations of quantum mechanics, as is shown in this paper. The resultant quasiclassical approximation leads to an algebraic derivation of Thomas–Fermi theory, and a new quantization rule which—unlike semiclassical quantization—is non-invariant under action transformations of the Hamiltonian, in the same qualitative manner as the true eigenvalues. The quasiclassical eigenvalues are shown to be significantly more accurate than the corresponding semiclassical values, for a variety of 1D and 2D systems. In addition, certain standard refinements of semiclassical theory are shown to be easily incorporated into the quasiclassical formalism.     

Quantum behavior of deterministic systems with information loss: Path integral approach

’t Hooft’s derivation of quantum from classical physics is analyzed by means of the classical path integral of Gozzi et al. It is shown how the key element of this procedure—the loss of information constraint—can be implemented by means of Faddeev–Jackiw’s treatment of constrained systems. It is argued that the emergent quantum systems are identical with systems obtained in Blasone et al. [Phys. Rev. A 71 (2005) 052507] through Dirac–Bergmann’s analysis. We illustrate our approach with two simple examples—free particle and linear harmonic oscillator. Potential Liouville anomalies are shown to be absent.     

Quantization as Selection Problem

Quantum systems exhibit a smaller number of energetic states than classical systems (A. Einstein, 1907, Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme, Ann. Phys. 22, 180ff). We take up the selection criterion for this in two parts. (1) The selection problem between classical and nonclassical mechanical systems is formulated in terms of possible and impossible configurations (among others, this overcomes the difficulties occurring when discussing the behavior of quantum particles in terms of paths). (2) The (nonclassical) selection of the quantum states is formulated, using recurrence relations and the energy law. The reformulation of “quantization as eigenvalue problem” in terms of “quantization as selection problem” allows one to derive Schrödinger’s stationary equation from classical mechanics through a straightforward and unique procedure; the nonstationary and multibody equations are subsequently acquired within the same frame. In contrast to the (classical) eigenvalue problem, the (nonclassical) selection problem can be formulated and solved without any reference to additional a priori assumptions on the nature of the quantum system, such as the wave-corpuscle dualism or an underlying wave equation or the existence of Planck’s finite action parameter. The existence of such an additional parameter—as the only additional one—is inherent in the procedure. Within our axiomatic-deductive approach, we modify classical mechanics only where it itself indicates an inherent limitation.     

S-Dominating Effect Algebras

A special type of effect algebra called anS-dominating effect algebra is introduced. It is shownthat an S-dominating effect algebra P has a naturallydefined Brouwer-complementation that gives P thestructure of a Brouwer–Zadeh poset. This enables usto prove that the sharp elements of P form anorthomodular lattice. We then show that a standardHilbert space effect algebra is S-dominating. Weconclude that S-dominating effect algebras may be usefulabstract models for sets of quantum effects in physicalsystems.     

Equilibrium states of a semi-quantum lattice system

As an intermediate situation between quantum and classical lattice systems we investigate a semi-quantum lattice system (such as an alloy of several kinds of atoms). The relevant observable algebra is a special type of AF algebras. We study equilibrium states of the system and obtain some equivalent equilibrium conditions.     

Polynomial poisson algebras for two-dimensional classical superintegrable systems and polynomial associative algebras for quantum superintegrable systems

The integrals of motion of the classical two-dimensional superintegrable systems close in a restrained polynomial Poisson algebra, whose general form is discussed. Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system. The polynomial Poisson algebra is deformed to a polynomial associative algebra, the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. It is conjectured that the finite-dimensional representations of the polynomial algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the solution of algebraic equations, which are universal for a large number of two-dimensional superintegrable systems. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

A Topos Perspective on the Kochen-Specker Theorem II. Conceptual Aspects and Classical Analogues

In a previous paper, we proposed assigning asthe value of a physical quantity in quantum theory acertain kind of set (a sieve) of quantities that arefunctions of the given quantity. The motivation was in part physical — such a valuationilluminates the Kochen–Specker theorem — andin part mathematical — the valuation arisesnaturally in the topos theory of presheaves. This paperdiscusses the conceptual aspects of this proposal. We also undertake two othertasks. First, we explain how the proposed valuationscould arise much more generally than just in quantumphysics; in particular, they arise as naturally in classical physics. Second, we give anothermotivation for such valuations (that applies equally toclassical and quantum physics). This arises fromapplying to propositions about the values of physical quantities some general axioms governingpartial truth for any kind of proposition.