Zeno's arrow and classical phase space logics

We analyze the Zeno's familiar paradox of the arrow using recently developed non-Boolean derived logics for classical systems. We show that the paradox depends upon a premise that is identically false in such logics, so that the language of experimental propositions is immune to the paradox.     

Characterizations of classical and quantum logics

In classical logic (Boolean algebras) probability systems involving correlations are fully characterized by the system of generalized Bell inequalities. On the other hand, probability systems with pairwise correlations on orthomodular lattices (OML) representing quantum logics are so general that the only inequalities that hold universally are the trivial inequalities 0p _i1, 0p _ijmin {p _i,p _j}. In this paper it is shown that every correlation sequence p=(p₁,...,p _n,...,P _ij,...) satisfying the above inequalities can be represented by a probability measure on an orthomodular latticeL admitting a full set of {0,1}-valued probability measures with the additional property that isL ortho-Arguesian.     

Monodromy in non-integrable systems on certain compact classical phase spaces

On the example of bending vibrational polyads of the acetylene molecule (C₂H₂) in the approximation of the $1:1:1:1$ resonant oscillator with axial symmetry, whose geometry is similar to the n-shell approximation of the perturbed hydrogen atom, we show how remaining invariant tori of the underlying classical non-integrable system form a nontrivial continuous family with monodromy. We read this monodromy off the quantum energy spectrum which was observed experimentally by spectroscopists, and we uncover its origins through the particular topology, geometry, and symmetry. We explain how monodromy characterizes the chaotic region surrounded by the tori. We detail the explicit correspondence between the bending polyads of C₂H₂ and the n-shells of the hydrogen atom, and uncover the dynamical SO(3) symmetry of the bending polyads and the corresponding spherically localized vibrational states.     

Quantum logics derived from asymmetric Mielnik forms

It is shown that a logic will possess a rich set of states if and only if it can be derived from a Mielnik form, not necessarily symmetric.     

Quantum logics and completeness criteria of inner product spaces

We present the survey of measure-theoretic completeness criteria for inner product spaces using methods and notions important for quantum logics. Moreover, some new criteria and open problems are given.     

Transition amplitude spaces and quantum logics with vector-valued states

Relations between transition amplitude spaces and quantum logics are studied. It is shown that transition amplitude spaces correspond to quantum logics with rich enough sets of vector-valued states.     

Uncertainty relations for general phase spaces

We describe a setup for obtaining uncertainty relations for arbitrary pairs of observables related by a Fourier transform. The physical examples discussed here are the standard position and momentum, number and angle, finite qudit systems, and strings of qubits for quantum information applications. The uncertainty relations allow for an arbitrary choice of metric for the outcome distance, and the choice of an exponent distinguishing, e.g., absolute and root mean square deviations. The emphasis of this article is on developing a unified treatment, in which one observable takes on values in an arbitrary locally compact Abelian group and the other in the dual group. In all cases, the phase space symmetry implies the equality of measurement and preparation uncertainty bounds. There is also a straightforward method for determining the optimal bounds.     

Twisted classical phase space

We consider relativistic phase space constructed by the twist procedure from the translation sector of the standard, nondeformed Poincaré algebra. Using the concept of cross product algebra we derive two kinds of phase space with noncommuting configuration space. The generalized uncertainty relations are formulated.     

Twistor flag spaces as phase spaces of conformal particles

This paper contains a kinematic interpretation of certain open orbits of conformal group on twistor flag spaces. In a particular case, the complexified Minkowski space IM _inf4c ^supc becomes the phase space of massive scalar conformal particle.This research has been supported by M. Skodowska-Curie Found.,-Grant No. OIP 74-01416.     

On the university phase spaces for homogeneous principal bundles

It is shown that the universal phase space for certain homogeneous principal bundles can be interpreted as a co-adjoint orbit in a semi-direct product.     

Stochastic phase spaces and master Liouville spaces in statistical mechanics

The concept of probability space is generalized to that of stochastic probability space. This enables the introduction of representations of quantum mechanics on stochastic phase spaces. The resulting formulation of quantum statistical mechanics in terms of -distribution functions bears a remarkable resemblance to its classical counterpart. Furthermore, both classical and quantum statistical mechanics can be formulated in one and the same master Liouville space overL ²(). A joint derivation of a classical and quantum Boltzman equation provides an illustration of the practical uses of these formalisms.Supported in part by an NRC grant.     

Faithful actions of locally compact quantum groups on classical spaces

We construct examples of locally compact quantum groups coming from bicrossed product construction, including non-Kac ones, which can faithfully and ergodically act on connected classical (noncompact) smooth manifolds. However, none of these actions can be isometric in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009), leading to the conjecture that the result obtained by Goswami and Joardar (Rigidity of action of compact quantum groups on compact, connected manifolds, 2013. arXiv:1309.1294) about nonexistence of genuine quantum isometry of classical compact connected Riemannian manifolds may hold in the noncompact case as well.     

Mathematical foundations of classical embeddings of quantum mechanical state spaces

In this paper we review the basic mathematical properties that allow the embedding of quantum state spaces into spaces of classical probability measures. In particular, the precise topological structures used for these immersions are described.     

Stochastic phase spaces,fuzzy sets,and statistical metric spaces

This paper is devoted to the study of the notion of the phase-space representation of quantum theory in both the nonrelativisitic and the relativisitic cases. Then, as a derived concept, the stochastic phase space is introduced and its connections with fuzzy set theory and probabilistic topological (in particular, metric) spaces are discussed.Supported by NSERC Grant No. A5206.On leave of absence from the Theoretical Physics Institute, University of Gdask, 80-952 Gdask, Poland.     

Low temperature phase diagrams for quantum perturbations of classical spin systems

We consider a quantum spin system with Hamiltonian $$H = H^{(0)} + \lambda V,$$ whereH ⁽⁰⁾ is diagonal in a basis ∣s〉=? _x ∣s _x 〉 which may be labeled by the configurationss={s_x} of a suitable classical spin system on ? ^d , $$H^{(0)} |s\rangle = H^{(0)} (s)|s\rangle .$$ We assume thatH ⁽⁰⁾(s) is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitation, whileV is a finite range or exponentially decaying quantum perturbation. Mapping thed dimensional quantum system onto aclassical contour system on ad+1 dimensional lattice, we use standard Pirogov-Sinai theory to show that the low temperature phase diagram of the quantum spin system is a small perturbation of the zero temperature phase diagram of the classical HamiltonianH ⁽⁰⁾, provided λ is sufficiently small. Our method can be applied to bosonic systems without substantial change. The extension to fermionic systems will be discussed in a subsequent paper.     

An order-by-order construction of low-temperature phase diagrams for classical lattice systems

An inductive algorithm is presented for the construction of phase diagrams by means of the low-temperature expansion technique. First the phase diagram is studied in the set of formal series. In each step, properties of this phase diagram are related to extremal elements of some family of convex sets. Approximations of the phase diagram in orderN are obtained by truncating all formal series at theNth term.This paper was presented at the Trebon, Czechoslovakia, Symposium September 1–6, 1986.     

Boolean machinery for quantum logics

The arithmetical tools based on Boolean matrices are described. They are applied to finite ortholattices to decompose them into products and sums, and to check atomisticity and orthomodularity.