The Existence of Finitely Additive States on Orthogonally Closed Subspaces of Incomplete Inner Product Spaces

We give a positive answer to an old problem of whether there exists an incomplete inner product space S such that its system of orthogonally closed subspaces—denoted by F(S)—admits finitely additive states. Indeed, we show that every infinite-dimensional separable Hilbert space H contains an incomplete dense hyperplane S H such that F(S) admits finitely additive states. We also show that the system of orthogonally closed subspaces of any inner product space with countably infinite linear dimension always admits finitely additive states.     

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.     

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.     

On the (Non)Existence of States on Orthogonally Closed Subspaces in an Inner Product Space

Suppose that S is an incomplete inner product space. In (Dvurečenskij, 1992, Gleason's Theorem and Its Applications, Ister Science Press, Bratislava, Kluwer Academic Publishers, Dordrecht), A. Dvurečenskij shows that there are no finitely additive states on orthogonally closed subspaces, F(S), of S that are regular with respect to finitely dimensional spaces. In this note we show that the most important special case of the former result—the case of the evaluations given by vectors in the “Gleason manner”—allows for a relatively simple proof. This result further reinforces the conjecture that there are no finitely additive states on F(S) at all.     

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.     

States on Subspaces of Inner Product Spaces with the Gleason Property

We show that the range of every finitely additive state on the system of all orthogonally closed subspaces of an infinite-dimensional inner product space E satisfying the Gleason property is equal to the real interval [0, 1]. Every pre-Hilbert space satisfies the Gleason property, and in Keller spaces it fails to hold.     

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.     

Finitely Generated Free Orthomodular Lattices. III

The finitely generated free algebras F _V(Lk)(n) (k 2, n 3) in the varieties V(L _k )of orthomodular lattices generated by the ortholattices L _k which are horizontalsums of one block 2³ and k – 1 blocks 2² are described as abstract algebras. Thisis a continuation of earlier work and indicates the complexity one must expectwhen describing the finitely generated free algebras in finitely generated varietiesof orthomodular lattices generated by ortholattices containing Boolean blockslarger than 2².     

New inequalities from local realism

The probabilistic formulation of local realism is shown to imply the existence of physically meaningful limits for arbitrary linear combinations of joint probabilities. The set of the so generated inequalities (setA) is wider than the previously known set of inequalities for linear combinations of correlation functions (setB). One particular inequality of the setA is shown to be violated by the probabilities of the Garg-Mermin model. The same model satisfies instead all the inequalities of the setB. As a consequence, the Garg-Mermin model is nonlocal and the setA provides physical restrictions not contained in the setB. 1. In the adopted formalism it is implicitly assumed that physical properties of the type are not created in the act of measurement. IfB(b) is measured on the systems, the setT is split into two parts,T(b _±), corresponding to the resultsB(b) = ±1, respectively. AlsoS is split intoS(b _±) from the existing correlation between and systems. If it is possible to predict that a measurement ofA(a) on the's of, say,S(b ₊) will give the results ±1 with respective probabilitiesP _±, then, on the basis of the probabilistic criterion of reality, we can attribute a physical property ₊ toS(b ₊) such that p(a ₊, ₊) is the probability ofA(a) = +1 inS(b ₊), p(a _–, ₊) is the probability ofA(a) = –1 inS(b ₊).It is natural to assume that ₊ belongs toS(b ₊) also ifA(a) isnot measured. In so doing, we exclude that future measurements create, with a retroaction in time, the physical properties of the statistical ensembles on which these measurements are performed.     

One-dimensional Ising chain with competing interactions: Exact results and connection with other statistical models

We study the ground state properties of a one-dimensional Ising chain with a nearest-neighbor ferromagnetic interactionJ ₁, and akth neighboranti-ferromagnetic interactionJ _k . WhenJ _k/J₁=–1/k, there exists a highly degenerate ground state with a residual entropy per spin. For the finite chain with free boundary conditions, we calculate the degeneracy of this state exactly, and find that it is proportional to the (N+k–1)th term in a generalized Fibonacci sequence defined by,F _N ^(k) =F _N–1 ^(k) +F _N–k ^(k) . In addition, we show that this one-dimensional model is closely related to the following problems: (a) a fully frustrated two-dimensional Ising system with a periodic arrangement of nearest-neighbor ferro- and antiferromagnetic bonds, (b) close-packing of dimers on a ladder, a 2× strip of the square lattice, and (c) directed self-avoiding walks on finite lattice strips.Work partially supported by grants from AFOSR and ARO.     

On Strong Superadditivity of the Entanglement of Formation

We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation E_F and the conjugate function E^* of the entanglement function E()=S(Tr_A). We then consider the conjectured strong superadditivity of the entanglement of formation E_F()E_F(_I)+E_F(_II), where _I and _II are the reductions of to the different Hilbert space copies, and prove that it is equivalent with subadditivity of E^*. Furthermore, we show that strong superadditivity would follow from multiplicativity of the maximal channel output purity for quantum filtering operations, when purity is measured by Schatten p-norms for p tending to 1.     

First experiment to test whether space-time is flat. II. Effects of orbit

The gyroscope in an orbiting satellite will be acted on by additional gravitational fields due to the rotation of the earth and due to the orbital velocity of the satellite. According to special relativistic gravitational theory, we deduce _L ^(S) —the gyroscope's precession rate due to the orbital velocity—and _S ^(S) —the gyroscope's precession rate due to the earth's rotation in the polar orbit case. The results are _L ^(S) = (2/3) _L ^(G) , _S ^(S) = (3/2) cos (1 - sin² cos²)^1/2 _S ^(G) , where and are the gyroscope's polar angles, and _L ^(G) and _S ^(G) are the geodetic and frame-dragging precession rates predicted by general relativity, respectively.     

Finitely additive states and completeness of inner product spaces

For any unit vector in an inner product space S, we define a mapping on the system of all -closed subspaces of S, F(S), whose restriction on the system of all splitting subspaces of S, E(S), is always a finitely additive state. We show that S is complete iff at least one such mapping is a finitely additive state on F(S). Moreover, we give a completeness criterion via the existence of a regular finitely additive state on appropriate systems of subspaces. Finally, the result will be generalized to general inner product spaces.     

Self-Gravitating String-Like Configurations of Nonlinear Spinor Fields

We consider the problem of the existence of soliton-like self-gravitating cylindrically symmetric configurations of a classical spinor field with the nonlinearity F(S) ( , F is an arbitrary function). Soliton-like configurations should have, by definition, a regular axis of symmetry and a flat or string-like geometry far from the axis (i.e., an asymptotically Minkowskian metric with a possible angular defect). It is shown that these conditions can be fulfilled if F(S) is finite as S and decreases faster than S ² as S 0. The set of field equations is entirely integrated, and some explicit examples are considered. A regularizing role of gravity is discussed.     

Statistical theory of force-induced unzipping of DNA

The unzipping transition under the influence of external force of a dsDNA molecule has been studied using the Peyrard-Bishop Hamiltonian. The critical force F_c(T) for unzipping calculated in the constant force ensemble is found to depend on the potential parameter k which measures the stiffness associated with a single strand of DNA and on D, the well depth of the on-site potential representing the strength of hydrogen bonds in a base pair. The dependence on temperature of F_c(T) is found to be (T_D - T)^1/2 (T_D being the thermal denaturation temperature) with F_c(T_D) = 0 and F_c(0) = . We used the constant extension ensemble to calculate the average force F(y) required to stretch a base pair a y distance apart. The value of F(y) needed to stretch a base pair located far away from the ends of a dsDNA molecule is found twice the value of the force needed to stretch a base pair located at one of the ends to the same distance for y 1.0 . The force F(y) in both cases is found to have a very large value for y 0.2 compared to the critical force found from the constant force ensemble to which F(y) approaches for large values of y. It is shown that the value of F(y) at the peak depends on the value of k which measures the energy barrier associated with the reduction in DNA strand rigidity as one passes from dsDNA to ssDNA and on the value of the depth of the on-site potential. The effect of defects on the position and height of the peak in the F(y) curve is investigated by replacing some of the base pairs including the one being stretched by defect base pairs. The formation and behaviour of a loop of Y shape when one of the ends base pair is stretched and a bubble of ssDNA with the shape of an eye when a base pair far from ends is stretched are investigated.     

Entropy andP-particle observables. II. The two-particle entropyS 2

This paper defines, and then evaluates perturbatively, an information-theoretic notion of entropyS ₂ for a system ofN interacting particles which assesses an observer's limited knowledge of the state of the system, assuming that he or she can measure with arbitrary precision all one-particle observables and correlations involving pairs of particles, but is completely ignorant of the form of any higher-order correlations involving three or more particles. By construction, thisS ₂(t) involves only the reduced two-particle distribution functions, or density matrices,f ₂(i,j) at timet, and, though the implementation of a subdynamics,dS ₂ (t)/dt can be realized in terms of thef ₂(i, j)'s at retarded timest–. A similar line of reasoning demonstrates that the most probable three-particlef ₃(i,j, k) consistent with a knowledge of thef ₂'s is precisely thatf ₃ suggested by the Kirkwood, or cluster, decomposition.     

Quantum deformation of Lorentz group

A one parameter quantum deformationS L(2,) ofSL(2,) is introduced and investigated. An analog of the Iwasawa decomposition is proved. The compact part of this decomposition coincides withS U(2), whereas the solvable part is identified as a Pontryagin dual ofS U(2). It shows thatS L(2,) is the result of the dual version of Drinfeld's double group construction applied toS U(2). The same construction applied to any compact quantum groupG _c is discussed in detail. In particular the explicit formulae for the Haar measures on the Pontryagin dualG _d ofG _c and on the double groupG are given. We show that there exists remarkable 1-1 correspondence between representations ofG and bicovariant bimodules (tensor bundles) overG _c . The theory of smooth representations ofS L(2,) is the same as that ofSL(2,) (Clebsh-Gordon coefficients are however modified). The corresponding tame bicovariant bimodules onS U(2) are classified. An application to 4D ₊ differential calculus is presented. The nonsmooth case is also discussed.     

Smearings of States Defined on Sharp Elements Onto Effect Algebras

In some sense, a lattice effect algebra E is a smeared orthomodular lattice S(E), which then becomes the set of all sharp elements of the effect algebra E. We show that if E is complete, atomic, and (o)-continuous, then a state on E exists iff there exists a state on S(E). Further, it is shown that such an effect algebra E is an algebraic lattice compactly generated by finite elements of E. Moreover, every element of E has a unique basic decomposition into a sum of a sharp element and a -orthogonal set of unsharp multiples of atoms.     

Photoproduction of $\eta$-mesons on protons in the resonance region: The background problem and the third S11-resonance

We have constructed an isobar model for the -photoproduction on the proton in the energy region up to the photon lab energy K ₀ = 3 GeV. The database involved into the fitting procedure includes precise results for the cross-section and for the T-asymmetry of the process near threshold obtained at MAMI and ELSA as well as recent results for the -asymmetry and for the angular distribution measured at higher energies in Grenoble and also more recent measurements performed at JLab for the photon energies up to 2 GeV. The model includes twelve nucleon resonances: S ₁₁(1535), S ₁₁(1650), S ₁₁(1825), P ₁₁(1440), P ₁₃(1720), D ₁₃(1520), D ₁₅(1675), F ₁₅(1680), F ₁₇(1990), G ₁₇(2190), G ₁₉(2250), H ₁₉(2220), and the background consisting of the nucleon pole term and the vector meson exchange in the t-channel. To explain the observed energy dependence of the integrated cross-section, two s-wave resonances, S ₁₁(1650) and S ₁₁(1825), have to be taken into account along with the dominating S ₁₁(1535). The integrated cross-section as well as the angular distribution and -asymmetry predicted by the model are in good agreement with the data. Above the photon energy K ₀ = 2 GeV, the calculated cross-section exhibits an appreciable dependence on the - and -meson contribution, whose coupling with nucleons is not well defined. Several versions of extending the model to higher energies are considered.Received: 20 January 2004, Revised: 13 April 2004, Published online: 12 October 2004PACS: 13.60.Le Meson production - 25.20.Lj Photoproduction reactions     

Additivity of vector gleason measures

We study the degree of additivity of orthogonal Hilbert-space-valued measures on the latticeL(H) of all projections acting on a Hilbert spaceH. We present criteria for such measures to be completely additive and we establish the connection between the additivity of orthogonal measures and the size of almost disjoint families on dimH. [For example, we show that everyH-valued orthogonal measure is weakly-additive iff (dimH) > dim H]. As a corollary we see that finitely additive orthogonal measures distinguish dimensions of Hilbert spaces (this can be viewed as a generalization of a theorem by Kruszynski). As a further corollary, we obtain that, for cardinals, with >,3, there is no Jordan homomorphism from a typeI -factor into a typeI -factor. Finally, we show that every latticeL(H) with (dimH) = dimH admits a nonzero free orthogonal measure with values inH. Our results contribute to the noncommutative probability theory and also may find applications in the theory of the representation ofC ^*-algebras.