Finitary Čech-de Rham Cohomology

The present paper continues (Mallios & Raptis, International Journal of Theoretical Physics, 2001, 40, 1885) and studies the curved finitary spacetime sheaves of incidence algebras presented therein from a ech cohomological perspective. In particular, we entertain the possibility of constructing a nontrivial de Rham complex on these finite dimensional algebra sheaves along the lines of the first author's axiomatic approach to differential geometry via the theory of vector and algebra sheaves (Mallios, Geometry of Vector Sheaves: An Axiomtic Approach to Differential Geometry, Vols. 1–2, Kluwer, Dordrecht, 1998a; Mathematica Japonica (International Plaza), 1998b, 48, 93). The upshot of this study is that important classical differential geometric constructions and results usually thought of as being intimately associated with C-smooth manifolds carry through, virtually unaltered, to the finitary-algebraic regime with the help of some quite universal, because abstract, ideas taken mainly from sheaf-cohomology as developed in Mallios (1998a,b). At the end of the paper, and due to the fact that the incidence algebras involved have been interpreted as quantum causal sets (Raptis, International Journal of Theoretical Physics, 2000, 39, 1233; Mallios & Raptis, 2001), we discuss how these ideas may be used in certain aspects of current research on discrete Lorentzian quantum gravity.     

Finitary, Causal, and Quantal Vacuum Einstein Gravity

We continue recent work (Mallios and Raptis, International Journal of Theoretical Physics 40, 1885, 2001; in press) and formulate the gravitational vacuum Einstein equations over a locally finite space-time by using the basic axiomatics, techniques, ideas, and working philosophy of Abstract Differential Geometry. The main kinematical structure involved, originally introduced and explored in (Mallios and Raptis, International Journal of Theoretical Physics 40, 1885, 2001), is a curved principal finitary space-time sheaf of incidence algebras, which have been interpreted as quantum causal sets, together with a nontrivial locally finite spin-Loretzian connection on it which lays the structural foundation for the formulation of a covariant dynamics of quantum causality in terms of sheaf morphisms. Our scheme is innately algebraic and it supports a categorical version of the principle of general covariance that is manifestly independent of a background -smooth space-time manifold M. Thus, we entertain the possibility of developing a fully covariant path integral-type of quantum dynamical scenario for these connections that avoids ab initio various problems that such a dynamics encounters in other current quantization schemes for gravity—either canonical (Hamiltonian) or covariant (Lagrangian)—involving an external, base differential space-time manifold, namely, the choice of a diffeomorphism-invariant measure on the moduli space of gauge-equivalent (self-dual) gravitational spin-Lorentzian connections and the (Hilbert space) inner product that could in principle be constructed relative to that measure in the quantum theory—the so-called inner product problem, as well as the problem of time that also involves the Diff(M) structure group of the classical -smooth space-time continuum of general relativity. Hence, by using the inherently algebraico—sheaf—theoretic and calculus-free ideas of Abstract Differential Geometry, we are able to draw preliminary, albeit suggestive, connections between certain nonperturbative (canonical or covariant) approaches to quantum general relativity (e.g., Ashtekar's new variables and the loop formalism that has been developed along with them) and Sorkin et al.'s causal set program. As it were, we noncommutatively algebraize, differential geometrize and, as a result, dynamically vary causal sets. At the end, we anticipate various consequences that such a scenario for a locally finite, causal and quantal vacuum Einstein gravity might have for the obstinate (from the viewpoint of the smooth continuum) problem of -smooth space-time singularities.     

Fourier-Mukai Transform and Adiabatic Curvature of Spectral Bundles for Landau Hamiltonians on Riemann Surfaces

We study the family of Landau Hamiltonians on a Riemann surface S by means of a Nahm transform and an integral functor related to the Fourier-Mukai transform associated to its jacobian variety J(S). This approach allows us to explicitly determine the spectral bundles associated to the holomorphic Landau levels. As a first main result we prove that these spectral bundles are holomorphic stable bundles with respect to the canonical polarization of J(S) determined by the theta divisor .The spectral bundles are endowed with a natural connection and the adiabatic charge transport properties of the corresponding Landau levels are determined by the adiabatic curvature of , which coincides with the curvature of the determinant bundle det . By means of the theory of analytic torsion and determinant bundles developed by Bismut, Gillet and Soulé we compute the adiabatic curvature of the spectral bundles on an arbitrary Riemann surface. We prove that all the holomorphic Landau levels have the same charge transport coefficients but their adiabatic curvatures differ by a term which involves the relative analytic torsion of different powers of the canonical bundle of S twisted by a fixed line bundle.     

Finitary-Algebraic ‘Resolution’ of the Inner Schwarzschild Singularity

A ‘resolution’ of the interior singularity of the spherically symmetric Schwarzschild solution of the Einstein equations for the gravitational field of a point-particle is carried out entirely and solely by finitistic and algebraic means. To this end, the background differential spacetime manifold and, in extenso, Differential Calculus-free purely algebraic (:sheaf-theoretic) conceptual and technical machinery of Abstract Differential Geometry (ADG) is employed. As in previous works [Mallios, A. and Raptis, I. (2001). Finitary spacetime sheaves of quantum causal sets: Curving quantum causality. International Journal of Theoretical Physics, 40, 1885 [gr-qc/0102097]; Mallios, A. and Raptis, I. (2002). Finitary Čech-de Rham cohomology. International Journal of Theoretical Physics, 41, 1857 [gr-qc/0110033]; Mallios, A. and Raptis, I. (2003). Finitary, causal and quantal vacuum Einstein gravity. International Journal of Theoretical Physics 42, 1479 [gr-qc/0209048]], which this paper continues, the starting point for the present application of ADG is Sorkin's finitary (:locally finite) poset (:partially ordered set) substitutes of continuous manifolds in their Gel'fand-dual picture in terms of discrete differential incidence algebras and the finitary spacetime sheaves thereof. It is shown that the Einstein equations hold not only at the finitary poset level of ‘discrete events,’ but also at a suitable ‘classical spacetime continuum limit’ of the said finitary sheaves and the associated differential triads that they define ADG-theoretically. The upshot of this is two-fold: On the one hand, the field equations are seen to hold when only finitely many events or ‘degrees of freedom’ of the gravitational field are involved, so that no infinity or uncontrollable divergence of the latter arises at all in our inherently finitistic-algebraic scenario. On the other hand, the law of gravity—still modelled in ADG by a differential equation proper—does not break down in any (differential geometric) sense in the vicinity of the locus of the point-mass as it is traditionally maintained in the usual manifold-based analysis of spacetime singularities in General Relativity (GR). At the end, some brief remarks are made on the potential import of ADG-theoretic ideas in developing a genuinely background-independent Quantum Gravity (QG). A brief comparison between the ‘resolution’ proposed here and a recent resolution of the inner Schwarzschild singularity by Loop QG means concludes the paper. PACS numbers: 04.60.−m, 04.20.Gz, 04.20.−q     

Time Flow in a Noncommutative Regime

We develop an approach to dynamical and probabilistic properties of the model unifying general relativity and quantum mechanics, initiated in the paper (Heller et al. (2005) International Journal Theoretical Physics 44, 671). We construct the von Neumann algebra of random operators on a groupoid, which now is not related to a finite group G, but is the pair groupoid of the Lorentzian frame bundle E over spacetime M. We consider the time flow on depending on the state . The state defining the noncommutative dynamics is assumed to be normal and faithful. Then the pair is a noncommutative probabilistic space and can also be interpreted as an equilibrium thermal state, satisfying the Kubo-Martin-Schwinger condition. We argue that both the “time flow” and thermodynamics have their common roots in the noncommutative unification of dynamics and probability.     

On the corrections to the Casimir effect depending on the resolution of measurement

The Casimir force      

Beam-energy and system-size dependence of the CME

The energy dependence of the local and violation in Au + Au and Cu + Cu collisions in a large energy range is estimated within a simple phenomenological model. It is expected that at LHC the Chiral Magnetic effect (CME) will be about 20 times weaker than at RHIC. In the lower energy range this effect should vanish sharply at energy somewhere above the top SPS one. To elucidate CME background effects a transport model including magnetic field evolution is put forward.     

Soft topological objects in topological media

Topological invariants in terms of the Green’s function in momentum and real space determine properties of smooth textures within topological media. In space dimension d = 1 the topological invariant N ₃ in terms of the Green’s function (ω, k _x , x) determines the fermion number of the kink, while in space dimension d = 3 the topological invariant N ₅ in terms of the Green’s function (ω, k _x , k _y , k _z , z) determines quantization of Hall conductivity in the soliton plane within the topological insulators.     

Thermal Entanglement for Interacting Spin-1/2 Systems and its Quantum Criticality

In this work, we investigate the thermal entanglement for interacting spin systems , by varying the parameters of temperature T, direction and magnetic field B. PACS numbers: 03.67.Mn, 03.65.Ud, 05.30.Cd, 73.43.Nq     

Quantum Computational Semantics on Fock Space

In the Fock space semantics, meanings of sentences are identified with density operators of the (unsymmetrized) Fock space based on the Hilbert space ℂ². Generally, the meaning of a sentence is smeared over different sectors of . The standard quantum computational semantics is a limit case of the Fock space semantics, where the meaning of any sentence α only “lives” in one sector of , which is determined by the logical complexity of α. We prove that the global Fock space semantics and the standard quantum computational semantics characterize the same logic. PACS: 03.67.Lx.     

Some asymptotic formulae for one-electron two-center problem

Asymptotic formulae of some expectation values related to the relativistic corrections in inverse powers of the internuclear distance R for the lsσ _g electron state of hydrogen molecular ion H₂⁺ and the lsσ molecule-like state of antiprotonic helium atom He⁺ are obtained with the use of the first-order perturbation function. Using these asymptotic formulae, the relativistic correction of order mα⁶ for these states in reciprocal powers of the internuclear distance R is derived to accuracy of (R ⁻⁴). The article is published in the original.     

Vector Measures on the Logic of <Emphasis Type="Italic">J</Emphasis>-projections of a Krein Space

Let be a Hilbert space with an inner product . In Jajte, R., and Paszkiewicz, A. (1978, Vector measure on the closed subspaces of a Hilbert space, Studia Mathematica 63, 229–251), the -measure on the logic of all orthogonal projections on H was studied. We examine the -measure on the hyperbolic logic of all J-projections on a Krein space. PACS: 03.65.Ta, 03.65.Db, 03.65.Ca.     

Finitary Spacetime Sheaves of Quantum Causal Sets: Curving Quantum Causality

A locally finite, causal, and quantal substitute for a locally Minkowskian principal fiber bundle of modules of Cartan differential forms over a bounded region X of a curved C -smooth spacetime manifold M with structure group G that of orthochronous Lorentz transformations L ⁺ := SO(1,3), is presented. is usually regarded as the kinematical structure of classical Lorentzian gravity when the latter is viewed as a Yang-Mills type of gauge theory of a sl(2, {})-valued connection 1-form . The mathematical structure employed to model this replacement of is a principal finitary spacetime sheaf of quantum causal sets with structure group G _n, which is a finitary version of the continuous group G of local symmetries of General Relativity, and a finitary Lie algebra g _n-valued connection 1-form on it, which is a section of its subsheaf . is physically interpreted as the dynamical field of a locally finite quantum causality, whereas its associated curvature as some sort of finitary and causal Lorentzian quantum gravity.     

Topos-Theoretic Extension of a Modal Interpretation of Quantum Mechanics

This paper deals with topos-theoretic truth-value valuations of quantum propositions. Concretely, a mathematical framework of a specific type of modal approach is extended to the topos theory, and further, structures of the obtained truth-value valuations are investigated. What is taken up is the modal approach based on a determinate lattice , which is a sublattice of the lattice of all quantum propositions and is determined by a quantum state e and a preferred determinate observable R. Topos-theoretic extension is made in the functor category of which base category is determined by R. Each true atom, which determines truth values, true or false, of all propositions in , generates also a multi-valued valuation function of which domain and range are and a Heyting algebra given by the subobject classifier in , respectively. All true propositions in are assigned the top element of the Heyting algebra by the valuation function. False propositions including the null proposition are, however, assigned values larger than the bottom element. This defect can be removed by use of a subobject semi-classifier. Furthermore, in order to treat all possible determinate observables in a unified framework, another valuations are constructed in the functor category . Here, the base category includes all ’s as subcategories. Although has a structure apparently different from , a subobject semi-classifier of gives valuations completely equivalent to those in ’s.     

Temperature Effect of Hydrogen-Like Impurity on the Ground State Energy of Strong Coupling Polaron in a RbCl Quantum Pseudodot

We study the ground state energy and the mean number of LO phonons of the strong-coupling polaron in a RbCl quantum pseudodot (QPD) with hydrogen-like impurity at the center. The variations of the ground state energy and the mean number of LO phonons with the temperature and the strength of the Coulombic impurity potential are obtained by employing the variational method of Pekar type and the quantum statistical theory (VMPTQST). Our numerical results have displayed that the absolute value of the ground state energy increases (decreases) when the temperature increases at lower (higher) temperature regime, the mean number of the LO phonons increases with increasing temperature, the absolute value of ground state energy and the mean number of LO phonons are increasing functions of the strength of the Coulombic impurity potential.     

Orthocomplementation and Compound Systems

In their 1936 founding paper on quantum logic, Birkhoff and von Neumann postulated that the lattice describing the experimental propositions concerning a quantum system is orthocomplemented. We prove that this postulate fails for the lattice _sep describing a compound system consisting of so called separated quantum systems. By separated we mean two systems prepared in different “rooms” of the lab, and before any interaction takes place. In that case, the state of the compound system is necessarily a product state. As a consequence, Dirac’s superposition principle fails, and therefore _sep cannot satisfy all Piron’s axioms. In previous works, assuming that _sep is orthocomplemented, it was argued that _sep is not orthomodular and fails to have the covering property. Here we prove that _sep cannot admit an orthocomplementation. Moreover, we propose a natural model for _sep which has the covering property. PACS: 03.65.Ta, 03.65.Ca     

Geometric prequantization on the spin bundle based onN-symplectic geometry: The dirac equation

We present preliminary results for a prequantization procedure that leads in a natural way to the Dirac equation. The starting point is the recently introducedn-symplectic geometry on the bundle of linear framesLM of ann-dimensional manifoldM in which the ⁿ-valued soldering 1-form onLM plays the role of then-symplectic potential. On a 4-dimensional spacetime manifold we consider the tensorial ⁴⁴valued function onLM determined by the spacetime metric tensor g as the Hamiltonian for free observers and determine the associated ⁴-valued Hamiltonian vector field , Integration of theX ⁱ yields the dynamics of free observers on spacetime, namely parallel transport of linear frames along spacetime geodesies. In order to obtain a vector field on the spin bundleSM which is a lift of and which is induced by a vector field for an appropriate mapping , it is useful to define a prolongation of some bundleL ^o M of oriented frames ofM. IfGL ⁺(4, ) denotes the identity component ofGL(4, ), thenGL ⁺(4, ) is the structure group ofL ^o M and its double cover is the structure group of. We show that the lift of onL ^o M to induces a natural 4-symplectic potential on. If is the lift of g to, then we find the ⁴-valued Hamiltonian vector field on determined by and show that the vector fieldsX _g ⁱ on are tangent to the subbundleSM. Integration of the restriction of theX ⁱ toSM now yields parallel transport of spin frames and thus tetrads along spacetime geodesies of g. We consider a naive prequantization operator assignment acting on ⁴-spinors in the standard representation ofSL(2, ). The eigenvalue equation for the system of new Hilbert space operators yields the Dirac equation.     

Topos Perspective on the Kochen–Specker Theorem: IV. Interval Valuations

We extend the topos-theoretic treatment given in previous papers (Butterfield, J. and Isham, C. J. (1999). International Journal of Theoretical Physics 38, 827–859; Hamilton, J., Butterfield, J., and Isham, C. J. (2000). International Journal of Theoretical Physics 39, 1413–1436; Isham, C. J. and Butterfield, J. (1998). International Journal of Theoretical Physics 37, 2669–2733) of assigning values to quantities in quantum theory. In those papers, the main idea was to assign a sieve as a partial and contextual truth value to a proposition that the value of a quantity lies in a certain set . Here we relate such sieve-valued valuations to valuations that assign to quantities subsets, rather than single elements, of their spectra (we call these interval valuations). There are two main results. First, there is a natural correspondence between these two kinds of valuation, which uses the notion of a state's support for a quantity (Section 3). Second, if one starts with a more general notion of interval valuation, one sees that our interval valuations based on the notion of support (and correspondingly, our sieve-valued valuations) are a simple way to secure certain natural properties of valuations, such as monotonicity (Section 4).