Nontrivial Space Research Articles

Fourier duality as a quantization principle

The Weyl-Wigner prescription for quantization on Euclidean phase spaces makes essential use of Fourier duality. The extension of this property to more general phase spaces requires the use of Kac algebras, which provide the necessary background for the implementation of Fourier duality on general locally compact groups. Kac algebras -- and the duality they incorporate -- are consequently examined as candidates for a general quantization framework extending the usual formalism. Using as a test case the simplest non-trivial phase space, the half-plane, it is shown how the structures present in the complete-plane case must be modified. Traces, for example, must be replaced by their noncommutative generalizations - weights - and the correspondence embodied in the Weyl-Wigner formalism is no more complete. Provided the underlying algebraic structure is suitably adapted to each case, Fourier duality is shown to be indeed a very powerful guide to the quantization of general physical systems.

Comments on Neveu–Schwarz five-branes

We study the theory of NS five-branes in string theory with a smooth non-trivial transverse space. We show that in the limit that the bulk physics decouples, these theories become equivalent to theories with a flat and non-compact transverse space. We present a matrix model description of the type IIA theory on $\IR^9\times S^1$ with NS five-branes located at points on the circle. Consequently, we obtain a description of the dual configuration of Kaluza-Klein monopoles in the type IIB theory.

The picture of a specific maximal monotone set

Let E be a nontrivial Banach space. The concept of ‘picture’ has been used to provide a new proof of the surjectivity of S+J, for E reflexive and S: E→2E* maximal monotone. It is known that if E is reflexive, then the picture of a maximal monotone subset of E×E* is a singleton. We calculate an example showing that in the nonreflexive case, the picture of a maximal monotone subset can be quite substantial.

Reflexive objects in topological categories

This paper presents several basic results about the non-existence of reflexive objects in cartesian closed topological categories of Hausdorff spaces. In particular, we prove that there are no non-trivial countably compact reflexive objects in the category of Hausdorff k-spaces and, more generally, that any non-trivial reflexive Tychonoff space in this category contains a closed discrete subspace corresponding to a numeral system in the sense of Wadsworth. In addition, we establish that a reflexive Tychonoff space in a cartesian-closed topological category cannot contain a non-trivial continuous image of the unit interval. Therefore, if there exists a non-trivial reflexive Tychonoff space, it does not have a nice geometric structure.

Closed superstrings in magnetic flux tube background

We consider a 2-parameter class of solvable closed superstring models which `interpolate' between Kaluza-Klein and dilatonic Melvin magnetic flux tube backgrounds. The spectrum of string states has similarities with Landau spectrum for a charged particle in a uniform magnetic field. The presence of spin-dependent `gyromagnetic' interaction implies breaking of supersymmetry and possible existence (for certain values of magnetic parameters) of tachyonic instabilities. We study in detail the simplest example of the Kaluza-Klein Melvin model describing a superstring moving in flat but non-trivial 10-d space containing a 3-d factor which is a `twisted' product of a 2-plane and an internal circle. We also discuss the compact version of this model constructed by `twisting' the product of the two groups in SU(2) x U(1) WZNW theory without changing the local geometry (and thus the central charge). We explain how the supersymmetry is broken by continuous `magnetic' twist parameters and comment on possible implications for internal space compactification models. (Contribution to the Proceedings of the 1995 Erice School "String Gravity and Physics at the Planck Scale")

Reduced SL(2,R) WZNW quantum mechanics

The SL(2,R) WZNW→Liouville reduction leads to a nontrivial phase space on the classical level both in 0+1 and 1+1 dimensions. To study the consequences in the quantum theory, the quantum mechanics of the 0+1-dimensional, point particle version of the constrained WZNW model is investigated. The spectrum and the eigenfunctions of the obtained (rather nontrivial) theory are given, and the physical connection between the pieces of the reduced configuration space is discussed in all the possible cases of the constraint parameters.

Spectra of selfadjoint operators in constructive analysis

The approximate point spectrum σ a ( T) of a selfadjoint operator T on a nontrivial separable Hilbert space is examined constructively with the help of the functional calculus for T. In particular, it is proved that σ a ( T) is compact if and only if ∥ f( T)∥ can be computed for each f in C[− b, b], where b > 0 is a bound for T. The paper culminates in a full constructive analysis of the spectrum of a compact selfadjoint operator with infinite-dimensional range. Brouwerian examples show that the results are the best possible in the constructive setting.

Non-negative differentially constrained entropy-like regularization

Classical maximum entropy is quite popular, not only because it is based on appropriate physical and phenomenological concepts, but also because it yields, among other things, a natural way in which to invoke, or even force, a positivity constraint. However, though it is known to converge with reasonable rates of convergence and to exhibit good stability properties, it is, from a practical point of view, flawed, because of its inherent shrinkage properties. This has been well documented along with the fact that it can, when the circumstances are appropriate, exhibit super-resolution behaviour. Thus, there is a clear need to examine whether the maximum entropy functional can be modified in a way which, on the one hand, removes the shrinkage difficulty and, on the other hand, guarantees good asymptotics for the oversmoothed solution as well as reasonable rates of convergence and appropriate stability. This is the goal of the present paper. In particular, it is shown how such results are obtained by simply replacing the function u in the classical entropy regularizor by a differential operator , which may have a non-trivial null space. Such regularizors will be called non-negative differentially constrained entropy-like regularizors. The utility of such regularizors is examined not only in terms of the standard stability, convergence and rates of convergence, but also in terms of the important practical measures of asymptotics and shrinkage. In particular, it is shown that the unwanted shrinkage does not, in general, hold for such entropy-like regularizors, the choice of which can be more specifically adapted to the problem context of the regularization than its classical counterpart. Some practical aspects of these entropy-like regularizors are briefly discussed in the introduction. In particular, because such regularizors automatically guarantee the positivity of , the regularized approximations thereby generated will retain the monotonicity or convexity of the underlying solution u if is taken to be or , respectively.

Magnetic flux tube models in superstring theory

Superstring models describing curved 4-dimensional magnetic flux tube backgrounds are exactly solvable in terms of free fields. We first consider the simplest model of this type (corresponding to a ‘Kaluza-Klein’ a = √3 Melvin background). Its 2d action has a flat but topologically non-trivial 10-dimensional target space (there is a mixing of the angular coordinate of the 2-plane with an internal compact coordinate). We demonstrate that this theory has broken supersymmetry but is perturbatively stable if the radius R of the internal coordinate is larger than R 0 = √2 α′. In the Green-Schwarz formulation the supersymmetry breaking is a consequence of the presence of a flat but non-trivial connection in the fermionic terms in the action. For R < R 0 and the magnetic field strength parameter q > R/2 α′, instabilities appear corresponding to tachyonic winding states. The torus partition function Z( q, R) is finite for R > R 0 and vanishes for qR = 2 n ( n integer). At the special points qR = 2 n (2 n + 1) the model is equivalent to the free superstring theory compactified on a circle with periodic (antiperiodic) boundary conditions for space-time fermions. Analogous results are obtained for a more general class of static magnetic flux tube geometries including the a = 1 Melvin model.

MULTISTRING VERTICES AND HYPERBOLIC KAC-MOODY ALGEBRAS

Multistring vertices and the overlap identities which they satisfy are exploited to understand properties of hyperbolic Kac-Moody algebras, in particular E10. Since any such algebra can be embedded in the larger Lie algebra of physical states of an associated completely compactified subcritical bosonic string, one can in principle determine the root spaces by analyzing which (positive norm) physical states decouple from the N-string vertex. Consequently, the Lie algebra of physical states decomposes into a direct sum of the hyperbolic algebra and the space of decoupled states. Both these spaces contain transversal and longitudinal states. Longitudinal decoupling holds more generally, and may also be valid for uncompactified strings, with possible consequences for Liouville theory; the identification of the decoupled states simply amounts to finding the zeroes of certain “decoupling polynomials.” This is not the case for transversal decoupling, which crucially depends on special properties of the root lattice, as we explicitly demonstrate for a nontrivial root space of E10· Because the N-vertices of the compactified string contain the complete information about decoupling, all the properties of the hyperbolic algebra are encoded into them. In view of the integer grading of hyperbolic algebras such as E10 by the level, these algebras can be interpreted as interacting strings moving on the respective group manifolds associated with the underlying finite-dimensional Lie algebras.

Solution of a non-trivial space charge problem by the hodographic method

A numerical technique for solving the discontinuous injection space-charge problem is described. This technique is based upon the hodographic method studied by Budd and Wheeler. In order to avoid the difficulties introduced by the vanishing charge density the domain is split into two subdomains with an unknown common boundary. Each of these subproblems is then solved by the hodographic method, and an extra boundary condition is used to update the position of the unknown boundary.

Phase space structure and the path integral for gauge theories on a cylinder

The physical phase space of gauge field theories on a cylindrical spacetime with an arbitrary compact simple gauge group is shown to be the quotient R 2r W A , r a rank of the gauge group, W A the affine Weyl group. The path integral (PI) formula resulting from Dirac's operator method contains a symmetrization with respect to W A rather than the integration domain reduction. It gives a natural solution to Gribov's problem. Some features of fermion quantum dynamics caused by the nontrivial phase space geometry are briefly discussed.

Flow alignment of nematics under oscillatory shear

Using a simple averaging method we show that under plane oscillatory shear a net torque acts on the director of a nematic liquid crystal leading to unexpected flow alignment. Under quite general conditions the usual flow-alignment orientations are unstable, and the ori- entation parallel to the shear (perpendicular to the velocity) becomes most stable within the shear plane. The effect drops out exactly under (unrealistic) linear shear unless the director has a nontrivial space dependence. An explicit application to plane Poiseuifle flow is presented. Pre- liminary experimental data are shown which demonstrate a transition under planar oscillatory shear from a homeotropically aligned nematic to a spatially homogeneously deformed state.

Additive mappings preserving operators of rank one

Let B( X) be the algebra of all bounded linear operators on a nontrivial real or complex Banach space, and let F( X) be the subalgebra of all finite-rank operators. A characterization of additive mappings on F( X) which preserve operators of rank one or projections of rank one is given. In the real case such mappings are automatically linear.

Infrared singularities and breaking of the Poincaré group: The massless dipole field

A four‐dimensional quantum field theory model is studied that exhibits infrared singularities expected to occur in realistic models of confinement. The Fock–Hilbert–Krein structure associated with the model in a local and covariant formulation is constructed and its unconventional features, like the existence of translationally invariant field operators and states (other than the vacuum), the implementation of the symmetries, etc., are discussed. Also, a canonical quantization of the model which improves the existing ones is derived. Finally, it is shown that the infrared singularities are responsible of the breaking of the Poincaré group in every nontrivial physical space; two explicit examples of possible physical spaces are constructed and it is shown that they have the same gauge invariant content.

Loop space representation of quantum general relativity

We define a new representation for quantum general relativity, in which exact solutions of the quantum constraints may be obtained. The representation is constructed by means of a noncanonical graded Poisson algebra of classical observables, defined in terms of Ashtekar's new variables. The observables in this algebra are nonlocal and involve parallel transport around loops in a three-manifold Σ. The theory is quantized by constructing a linear representation of a deformation of this algebra. This representation is given in terms of an algebra of linear operators defined on a state space which consists of functionals of sets of loops in Σ. The construction is general and can be applied also to Yang-Mills theories. The diffeomorphism constraint is defined in terms of a natural representation of the diffeomorphism group. The hamiltonian constraint, which contains the dynamics of quantum gravity, is constructed as a limit of a sequence of observables which incorporates a regularization prescription. We give the general solution of the diffeomorphism constraint in closed form. It is spanned by a countable basis which is in one-to-one correspondence with the diffeomorphism equivalence classes of multiple loops, which are a generalization of the link classes studied in knot theory. Then we explicitly construct, in closed form, a large space of solution of the entire set of constraints, including the hamiltonian constraint. These turn out to be classified by the ordinary knot and link classes of Σ. The space of solutions that we find is a sector of the physical states space of nonperturbative quantum general relativity. The failure of perturbation theory is thus shown to be not relevant to the problem of the existence of a nontrivial physical state space in quantum gravity. The relationship between this new loop representation and the self-dual representation of Ashtekar is illuminated by means of a functional transform between states in the two representations. Questions of the completeness of the solution space, the meaning of the physical operators and the physical inner product, are discussed, but not, so far, resolved.

Relativistic bursts.

In this Letter we obtain an exact solution for a relativistic Langmuir wave with nontrivial space and time dependence. We find explosive behavior in the electron density and hence also in the electric field gradient. Relativistic effects are thus found a possible candidate to explain this interesting type of behavior, recently found in numerical simulations (though it is too early to give a quantitative treatment).

Quasar-galaxy associations with discordant redshifts as a topological effect. II - A closed hyperbolic model

view Abstract Citations (32) References (18) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Quasar-Galaxy Associations with Discordant Redshifts as a Topological Effect. II. A Closed Hyperbolic Model Fagundes, Helio V. Abstract This is the second of two papers developing a conjecture that might solve the quasar redshift controversy. The quasar-galaxy or quasar-quasar associations with discordant redshifts reported by Arp and others would be explained as the effect of multiple image production in a cosmological model with nontrivial space topology. The model here explored has the same local metric as Friedmann's open model, but the latter's infinite space is replaced by a closed space, constructed as usual from a polyhedron with faces pairwise identified. The results exhibit, semiquantitatively, some of the characteristics of the mentioned associations, and are suggestive of other effects such as an inhomogeneous distribution of QSO images on the sky. But no actual fitting of data is attempted, for these are too rich and varied, and probably will require a more complex topology for their explanation than the one used here. So the present model will be seen as a prototype for future developments in this direction. A possible connection with the ideas of small universe of Gott and Ellis is hinted at. Publication: The Astrophysical Journal Pub Date: March 1989 DOI: 10.1086/167223 Bibcode: 1989ApJ...338..618F Keywords: Computational Astrophysics; Galaxies; Quasars; Red Shift; Topology; Distribution (Property); Icosahedrons; Image Analysis; Astrophysics; COSMOLOGY; GALAXIES: CLUSTERING; GALAXIES: REDSHIFTS; QUASARS full text sources ADS | Related Materials (1) Erratum: 1990ApJ...349..678F

Current algebra representation for the 3+1 dimensional Dirac-Yang-Mills theory

The structure of the current algebra representation in the state space of fermions in an external Yang-Mills field in 3+1 space-time dimensions is analyzed; the topology of the vector space is determined by a countable family of semi-definite inner products. We show that there is no hermitian non-trivial Hilbert space representation such that the energy is bounded from below. The structure of the Hilbert space for the quantized coupled Dirac-Yang-Mills system is discussed and the existence of the vacuum vector and the cancellation of commutator anomalies is described in terms of complex line bundles over infinite-dimensional Grassmannians.

Commutativity Preserving Maps of Factors

By a von Neumann algebra M we mean a weakly closed, self-adjoint algebra of operators on a Hilbert space which contains I, the identity operator. A factor is a von Neumann algebra whose centre consists of scalar multiples of I.In all that follows ϕ:M → N will be a one to one, *-linear map from the von Neumann factor M onto the von Neumann algebra N such that both ϕ and ϕ−1 preserve commutativity. Our main result states that if M is not of type I2 then where is an isomorphism or an antiisomorphism, c is a non-zero scalar, and λ is a *-linear map from M into ZN, the centre of N.Our interest in this problem was aroused by several recent results. In [1], Choi, Jafarian, and Radjavi proved that if S is the real linear space of n × n matrices over any algebraically closed field, n ≧ 3, and ψ a linear operator on S which preserves commuting pairs of matrices, then either ψ(S) is commutative or there exists a unitary matrix U such thatfor all A in S. They proved an analogous result for the collection of all bounded self-adjoint operators on an infinite dimensional Hilbert space when ψ is one to one. Subsequently, Omladic [7] proved that if ψ:L(X) → L(X) is a bijective linear operator preserving commuting pairs of operators where X is a non-trivial Banach space, thenwhere U is a bounded invertible operator on X and A′ is the adjoint of A.