Sharp Elements in Effect Algebras

We show that if for an arbitrary pair of orthogonal sharp elements of an effect algebra E its join exists and is sharp, then the set E_S of all sharp elements of E is a subeffect algebra of E that is an orthomodular poset. Such effect algebras need not be sharply dominating but S-dominating. Further, we show that in every nonproper effect algebra E, E_S is a subeffect algebra that is an orthomodular poset. Moreover, a general theorem for E_S is proved.     

Not each sequential effect algebra is sharply dominating

Let E be an effect algebra and Es be the set of all sharp elements of E. E is said to be sharply dominating if for each a∈E there exists a smallest element such that . In 2002, Professors Gudder and Greechie proved that each σ-sequential effect algebra is sharply dominating. In 2005, Professor Gudder presented 25 open problems in [S. Gudder, Int. J. Theory Phys. 44 (2005) 2219], the 3rd problem asked: Is each sequential effect algebra sharply dominating? Now, we construct an example to answer the problem negatively.     

Measures on the Splitting Subspaces of an Inner Product Space

Let S be an inner product space and let E(S) (resp. F(S)) be the orthocomplemented poset of all splitting (resp. orthogonally closed) subspaces of S. In this article we study the possible states/charges that E(S) can admit. We first prove that when S is an incomplete inner product space such that dim S/S < , then E(S) admits at least one state with a finite range. This is very much in contrast to states on F(S). We then go on showing that two-valued states can exist on E(S) not only in the case when E(S) consists of the complete/cocomplete subspaces of S. Finally we show that the well known result which states that every regular state on L(H) is necessarily -additive cannot be directly generalized for charges and we conclude by giving a sufficient condition for a regular charge on L(H) to be -additive.     

Pastings of MV-Effect Algebras

We give a method of constructing a lattice effect algebra E with given family of blocks. The given MV-effect algebras are pasted along some common sub-MV-effect algebras in a such manner that there exists an ((o)-continuous) state on the pasting E.     

A New Algebraic Criterion for Completeness of Inner Product Spaces

We show that an inner product space S is complete whenever the system E(S) of all splitting subspaces of S, i.e., of all subspaces M of S such that M + M = S holds, satisfies the -Riesz interpolation property. This generalizes the result of H. Gross and H. Keller who required E(S) to be a complete lattice, of G. Cattaneo and G. Marino who required E(S) to be a -complete lattice, and that of the author who required E(S) to be a -orthocomplete OMP.     

Invariant states and conditional expectations of the anticommutation relations

The groupG of unitary elements of a maximal abelian von Neumann algebra on a separable, complex Hilbert spaceH acts as a group of automorphisms on the CAR algebraA(H) overH. It is shown that the set ofG-invariant states is a simplex, isomorphic to the set of regular probability measures on aw*-compact setS ofG-invariant generalized free states. The GNS Hilbert space induced by an arbitraryG-invariant state onA(H) supports a *-representation ofC(S); the canonical map ofA(H) intoC(S) can then be locally implemented by a normal,G-invariant conditional expectation.     

Nonequilibrium phase transition in stochastic lattice gases: Simulation of a three-dimensional system

We report results of computer simulations of a three-dimensional lattice gas of interacting particles subject to a uniform external fieldE. The dynamics of the system is given by hoppings of particles to nearby empty sites with rates biased for jumps in the direction ofE. As for the two-dimensional system we find that here too there exists a critical temperature,T _c (E) such that forT < T _c (E) the systems orders in a very anisotropic phase with striplike typical configurations parallel to the field.T _c (E) increases withE but substantially less strongly than in two dimensions. There is a break in the slope of the saturation current atT _c (E). Our data are consistent with the critical exponent being mean field.     

Symmetries of physical theories

A physical theory is, by definition, a complete orthomodular atomic lattice having the covering property. GivenL a quantum logic andS _L the set of all its states, a theorem is proved which asserts that, if certain reasonable assumptions concerningS _L are satisfied, then for any bijective convex mappingU: S _LS_L, satisfying also certain physically meaningful conditions, there exists a unique automorphismV: L L such thatU(p)=p oV ^–1 for allp S _L.     

Finite-dimensional representations of the euclidean group in the plane

In this article two theorems are given which permit, together with the concept of a representation vector diagram, to classify all (linear) finite-dimensional representations of the algebra and group E ₂ which are induced by a master representation on the place of the universal enveloping algebra of the algebra E ₂. Apart from a classification of the finite-dimensional representations, the two theorems make it possible to obtain the matrix elements of these representations for both, algebra and group, in explicit form. The material contained in this letter forms part of an analysis of indecomposable (finite- and infinite-dimensional) representations of the algebra and group E ₂ which is contained in Reference [1]. No proofs will be given in this letter. We refer instead to [1].     

Invariant states in statistical mechanics

Mathematically we consider aC*-algebra , acted upon by the groupT of space-translations, which has an asymptotic abelian property. We analyse invariant states over . Physically this programme can be considered as a kinematical study of equilibrium states in statistical mechanics. Each invariant state can be uniquely decomposed into elementary invariant states (E-states). These elementary states have, amongst other characteristics, the physical property that space-averages of local observables are constants in the corresponding representations. In anE-state the discrete spectrum S _D of space-translations is additive which gives rise to the classificationE _I,E _II, andE _III corresponding to the three possibilities that S _D contains one point, a lattice of points, or a set with accumulation points. AnE _II-state can be uniquely decomposed into states (L-states) having a symmetry with respect to a closed subgroupT _L of (S _D and T _L are reciprocal lattices).L-states have properties with respect toT _L analogous to the properties ofE _I-states with respect toT. The decomposition intoL-states is the inverse process of homogenizing a lattice state by smearing it over a lattice distance. The mathematical methods which we employ have more general applications.     

The q-boson operator algebra and q-Hermite polynomials

The q-boson algebra is defined as an associative algebra with generators and relations. Some examples are given, and then the q-boson algebra is extended such that the roots of the diagonal generators are also defined. It is shown that a family of transformations exist mapping one set of standard generators of the q-boson algebra to another set of standard generators. Using such a transformation, one obtains expressions for q-bosons for which the kth q-boson state is expressed in terms of a q-Hermite polynomial p _k (x; q) which reduces to the ordinary Hermite polynomial of degree k when q=1.     

Symmetries of Einstein-Yang-Mills fields and dimensional reduction

LetE be a manifold on which a compact Lie groupS acts simply (all orbits of the same type);E can be written locally asM×S/I,M being the manifold of orbits (space-time) andI a typical isotropy group for theS action. We study the geometrical structure given by anS-invariant metric and anS-invariant Yang Mills field onE with gauge groupR. We show that there is a one to one correspondence between such structures and quadruplets of fields defined solely onM; _v is a metric onM,h are scalar fields characterizing the geometry of the orbits (internal spaces), ⁱ are other scalar fields (Higgs fields) characterizing theS invariance of the Lie(R)-valued Yang Mills field and is a Yang Mills field for the gauge groupN(I)|I×Z((I)),N(I) being the normalizer ofI inS, is a homomorphism ofI intoR associated to theS action, andZ((I)) is the centralizer of(I) inR. We express the Einstein-Yang-Mills Lagrangian ofE in terms of the component fields onM. Examples and model building recipes are given.     

Extensions of Ordering Sets of States from Effect Algebras onto Their MacNeille Completions

In (Rie?anová and Zajac in Rep. Math. Phys. 70(2):283–290, 2012) it was shown that an effect algebra E with an ordering set $\mathcal{M}$ of states can by embedded into a Hilbert space effect algebra $\mathcal{E}(l_{2}(\mathcal{M}))$ . We consider the problem when its effect algebraic MacNeille completion $\hat{E}$ can be also embedded into the same Hilbert space effect algebra $\mathcal {E}(l_{2}(\mathcal{M}))$ . That is when the ordering set $\mathcal{M}$ of states on E can be extended to an ordering set of states on $\hat{E}$ . We give an answer for all Archimedean MV-effect algebras and Archimedean atomic lattice effect algebras.     

Spin physics with internal targets and the blast detector

The aim of this paper is to give a set of central elements of the algebras U_q(so_m) and U _q(iso _m ) when q is a root of unity. They are surprisingly arise from a single polynomial Casimir element of the algebra U_q(so₃). It is conjectured that the Casimir elements of these algebras under any values of q (not only for q a root of unity) and the central elements for q a root of unity derived in this paper generate the centers of U_q(so_m) and U _q(iso _m ) when q is a root of unity.     

A Finitely Additive State Criterion for the Completeness of Inner Product Spaces

We show that an inner product space S (real, complex or quaternion) is complete if, and only if, the system of all orthogonally closed subspaces in S, denoted by F(S), admits at least one finitely additive state which is not vanishing on the set of all finite dimensional subspaces of S. Although it gives only a partial solution to the problem formulated by Pták on the existence of a finitely additive state on F(S) for incomplete S, this gives an important insight into the structure of the set of states on F(S). This criterion has no analogue whatsoever in E(S), the system of splitting subspaces of S.     

Range of Charges on Orthogonally Closed Subspaces of an Inner Product Space

It is still an open question whether the complete lattice F(S) of all orthogonally closed subspaces of an incomplete inner product space S admits a nonzero charge. A negative answer would result in a new way of completeness characterization of inner product spaces. Many partial results have been established regarding what has now turned to be a highly nontrivial problem. Recently, in Dvureenskij and Ptak (Letters in Mathematical Physics, 62, 63–70, 2002) the range of a finitely additive state s on F(S), dim S = , was shown to include the whole interval [0, 1]. This was then generalized in Dvureenskij (International Journal of Theoretical Physics, 2003) for general inner product spaces satisfying the Gleason property. Motivated by these results, we give a thorough investigation of the possible ranges of charges on F(S), dim S q3. We show that if the nonzero charge m is bounded, then for infinite dimensional inner product spaces, Range(m) is always convex. We also show that this need not be the case with unbounded charges. Finally, in the last section, we investigate the range of charges on F(S), dim S = , satisfying the sign-preserving and Jauch-Piron properties. We show that for such measures the range is again always convex.     

Order Topology and Frink Ideal Topology of Effect Algebras

In this paper, the following results are proved: (1) If E is a complete atomic lattice effect algebra, then E is (o)-continuous iff E is order-topological iff E is totally order-disconnected iff E is algebraic. (2) If E is a complete atomic distributive lattice effect algebra, then its Frink ideal topology τ _id is Hausdorff topology and τ _id is finer than its order topology τ _o , and τ _id =τ _o iff 1 is finite iff every element of E is finite iff τ _id and τ _o are both discrete topologies. (3) If E is a complete (o)-continuous lattice effect algebra and the operation ⊕ is order topology τ _o continuous, then its order topology τ _o is Hausdorff topology. (4) If E is a (o)-continuous complete atomic lattice effect algebra, then ⊕ is order topology continuous.     

Electrodynamics Under a Possible Alternative to the Lorentz Transformation

A generalization of the classical electrodynamics for systems in absolute motion in presented using a possible alternative to the Lorentz transformation. The main hypothesis assumed in this work are: a) The inertial transformations relate two inertial frames: the privileged frame S and the moving frame S with velocity v with respect to S. b) The transformation of the fields from S to the moving frame S is given by H = a(H – v × D) and E = a(E + v × B), where a is a matrix whose elements depend of the absolute velocity of the system. c) The constitutive relations in the moving frame S are given by D = E, B = H and J = E. It is found that Maxwell's equations, which are transformed to the moving frame, take a new form depending on the absolute velocity of the system. Moreover, differing from classical electrodynamics, it is proven that the electrodynamics proposed explains satisfactorily the Wilson effect.     

Models of particle states

Two different types of particle state models are discussed. In the first type, particles are considered to be dynamically bound systems of a small set of physical constituents. In the second type, particle states are constructed from tensor products of symmetry constituents, i.e., states that are the basis elements of finite irreducible representations of an internal algebra. These states need not represent physical particles. We present three models of the first type. For the second type, we discuss in detail the main thrust of this paper, a new version of the quark-lepton model based on the algebrasu(4)_flaourXsu(6)_flavour. The quark color-triplet and a lepton color-singlet are united by a single irreducible representation of su(4)_colour. Thesu(6)_colour algebra is an extension of the originalsu(3)_flavor. All observed ground-state hadron multiplets are in full accord with the predictions of this model. The numbers of hadron ground states it predicts are 36 spin-0 mesons, 36 spin-1 mesons, 70 spin-1/2 baryons, and 56 spin-3/2 baryons.Professor Barut passed away suddenly on December 5, 1994.     

Open Problems for Sequential Effect Algebras

A sequential effect algebra (SEA) is an effect algebra on which a sequential product with certain natural properties is defined. In such structures, we can study combinations of simple measurements that are series as well as parallel. This article presents some open problems for SEAs together with background material, comments and partial results. Two examples of open problems are the following: is A° B = A ^1/2 BA ^1/2 the only sequential product on a Hilbert space SEA? It is known that the sharp elements of a SEA form an orthomodular poset. Is every orthomodular poset isomorphic to the set of sharp elements for some SEA?