State on splitting subspaces and completeness of inner product spaces

We show that an inner product spaceV is complete iff the system of all splitting subspaces, i.e., of all subspacesM for whichM + M =V, possesses at least one completely additive state.     

Completeness of inner product spaces and quantum logic of splitting subspaces

We show that an inner product space S is complete whenever its system E(S) of all splitting subspaces, i.e., of all subspaces E for which EE =S holds, is a quantum logic, that is, an orthocomplemented orthomodular -orthoposet. It is well known that the quantum logic is an important axiomatic model of quantum mechanics. This generalizes the result of G. Cattaneo and G. Marino (Lett. Math. Phys. 11, 15–20 (1986)) who required that E(S) be a lattice. Moreover, the conditions are weakened to show that S is complete whenever E(S) contains the join of any sequence of one-dimentional orthogonal subspaces.     

Completeness of inner product spaces with respect to splitting subspaces

The set E(S) of all splitting subspaces, i.e., of all subspaces M of an inner product space S for which MM =S holds, is an orthocomplemented orthomodular orthoposet and it has already been shown that the ordering property on E(S) of being a complete lattice characterizes the completeness of inner product spaces. In this work this last result is generalized proving that S is a Hilbert space under the weaker request that E(S) is a -lattice. As a marginal result, we also prove that an inner product space is complete if and only if the complete lattice (S) of all subspaces is orthomodular.     

Gleason's theorem and completeness of inner product spaces

We generalize the result of Hamhalter and Pták showing that an inner product space whose dimension is either a nonmeasurable cardinal or an arbitrary cardinal is complete iff its lattice of strongly closed subspaces possesses at least one state or one completely additive state, respectively. Moreover, we show that this lattice of any separable space possesses many-finite measures, and we give the Gleason analogue for them.     

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.     

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.     

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.     

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.     

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.     

Magnetic correlations in three-dimensional ising spin glasses

By a recursive method numerically exact free energies are calculated forL×L×M Ising lattices with random bonds andL=4, 4M10, applying free boundaries in the direction where the lattice is less small and otherwise periodic boundary conditions. Both for the±J model and the gaussian model the specific heat is in fair agreement with Monte Carlo results obtained for much larger lattices. However, the correlation function [S ₀ S _{R T} ² ]_av is found to decay exponentially with distanceR [for 1R9] at temperatures far below the apparent freezing temperatures of the Monte Carlo simulations, implying that there is no nonzero Edwards-Anderson order parameter in equilibrium. This behavior is qualitatively different from Mattis spin glasses (or Ising ferromagnets) where even smaller lattices show pronounced magnetic order at low temperatures. As the Monte Carlo results give evidence for a nonzero Edwards-Anderson order parameter (for not too long observation times), which is fairly independent of lattice size down to sizes of 4³, we suggest that Edwards-Anderson ordering is a nonequilibrium phenomenon visible only in studying dynamic properties.     

Conformal regularization of the Kepler Problem

The manifoldM of null rays through the origin of ^2,n+1 is diffeomorphic toS ¹×S ⁿ , and it is a homogeneous space of SO(2,n+1). This group therefore acts onT*M, which we show to be the generating manifold of the extended phase space of the regularized Kepler Problem. A local canonical chart inT*M is found such that the restriction to the subbundle of the null nonvanishing covectors is given byp ₀+H(q,p)=0, whereH(q,p) is the Hamiltonian of the Kepler Problem. By means of this construction, we get some results that clarify and complete the previous approaches to the problem.     

Converse of the Eilers-Horst theorem

We generalize the theorem of Eilers and Horst, showing that any finite as well as any-finite measure on a quantum logic of all closed subspaces of a Hilbert spaceH of dimension 2 is a Gleason one iff the dimension ofH is a nonmeasurable cardinal.     

States on families of subspaces of pre-Hilbert spaces

We present six families of closed subspaces of a pre-Hilbert space and we show that if arbitrary, one of them possesses at least one completely additive state, then the pre-Hilbert space is complete.     

On States on Orthogonally Closed Subspaces of an Inner Product Space

We show that every finitely additive state on the system F(S) of all orthogonally closed subspaces of an infinite-dimensional inner product space S attains all values from the real interval [0,1]. In particular, we show that there is no finitely additive countably-valued state on F(S) whenever dim S=. The main technique we use is an embedding of L(H _n ) into F(S).     

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.     

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.     

Transparent potentials at fixed energy in dimension two. Fixed-energy dispersion relations for the fast decaying potentials

For the two-dimensional Schrödinger equation $$[ - \Delta + v(x)]\psi = E\psi , x \in \mathbb{R}^2 , E = E_{fixed} > 0 (*)$$ at a fixed positive energy with a fast decaying at infinity potentialv(x) dispersion relations on the scattering data are given. Under “small norm” assumption using these dispersion relations we give (without a complete proof of sufficiency) a characterization of scattering data for the potentials from the Schwartz classS=C _∞ ^(∞) (?²). For the potentials with zero scattering amplitude at a fixed energyE _fixed (transparent potentials) we give a complete proof of this characterization. As a consequence we construct a family (parametrized by a function of one variable) of two-dimensional spherically-symmetric real potentials from the Schwartz classS transparent at a given energy. For the two-dimensional case (without assumption that the potential is small) we show that there are no nonzero real exponentially decreasing, at infinity, potentials transparent at a fixed energy. For any dimension greater or equal to 1 we prove that there are no nonzero real potentials with zero forward scattering amplitude at an energy interval. We show that KdV-type equations in dimension 2+1 related with the scattering problem (*) (the Novikov-Veselov equations) do not preserve, in general, these dispersion relations starting from the second one. As a corollary these equations do not preserve, in general, the decay rate faster than |x|^?3 for initial data from the Schwartz class.     

Scaling function for the energy density correlation at large momentum

The static energy-density correlation function S ²(0)S ²(x)–S ²² is calculated in the critical region for theS ⁴ Ginzburg-Landau-Wilson model at short distances and for nonzero field. Short distance expansion is used and its structure for more complex vertex functions is given. Goldstone mode singularities present at the magnetization curve are taken into account. The main application is given in the theory of polymer solutions. Here, S ²(0)S ²(x)–S ²² becomes the Fourier transform of the densitydensity correlation.     

X-Band Pulsed ENDOR Study ofFe-Substituted Sodalite— The Effect of the Zero-Field Splitting

X-band (∼9.3 GHz) pulsed ENDOR measurements were carried out on⁵⁷Fe-substituted sodalite (FeSOD) which contains only one type of Fe(III) (S=) located at a framework site. The ENDOR spectrum recorded atg= 2 shows three doublets corresponding to the sixM_Smanifolds. The assignment of these signals was confirmed by hyperfine-selective and triple ENDOR experiments. The components of each of the doublets had different intensities, reflecting the different populations of the EPR energy levels at the measurement temperature, 1.8 K. ENDOR spectra were recorded at magnetic fields within the EPR powder pattern, and the field dependence observed showed an anisotropic behavior, unexpected from the isotropic character of the⁵⁷Fe(III) hyperfine coupling. This dependence was attributed to the high-order effects of the zero-field splitting (ZFS) interaction on the ENDOR frequencies. Three different theoretical approaches were used to account for the dependence of the ENDOR spectrum on the ZFS interaction. The first involves the exact diagonalization of the total spin Hamiltonian, the second uses third-order perturbation approximations, and the third employs an effective nuclear Hamiltonian for each of theM_Smanifolds. The simulations showed that the ENDOR signals of theM_S= ±5/2 (ν_±5/2) manifold are the least sensitive to the magnitude of the ZFS parameterDand are therefore the most appropriate for the determination ofa_iso. It is shown that at X band anda_isovalues of about 30 MHz, the perturbation approach is valid up toDvalues of 500 MHz if all three doublets are concerned. However, if only the ν_±5/2doublet is considered, then this approach is valid forD< 1000 MHz. The third approach was found inappropriate fora_isovalues of ∼30 MHz. Using the method of exact diagonalization together with orientation selectivity, the trends observed in the experimental spectra could be reproduced. The ENDOR spectra of the⁵⁷Fe-substituted zeolites ZSM5, L, and mazzite showed broad and ill-defined peaks since the ZFS of Fe(III) in these zeolites is significantly larger than that of FeSOD. Because this broadening is a high-order effect, it can be significantly reduced at higher spectrometer frequencies.