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Torsten Asselmeyer-Maluga

Publications and source records attributed to Torsten Asselmeyer-Maluga.

30 records · Page 2Linked to original sources
arXiv2011

Constructing a quantum field theory from spacetime

The paper shows deep connections between exotic smoothings of a small R^4 (the spacetime), the leaf space of codimension-1 foliations (related to noncommutative algebras) and quantization. At first we relate a small exotic R^4 to codimension-1 foliations of the 3-sphere unique up to foliated cobordisms and characterized by the real-valued Godbillon-Vey invariant. Special care is taken for the integer case which is related to flat PSL(2,R)-$bundles. Then we discuss the leaf space of the foliation using noncommutative geometry. This leaf space contains the hyperfinite III_1 factor of Araki and Woods important for quantum field theory (QFT) and the I_{\infty} factor. Using Tomitas modular theory, one obtains a relation to a factor II_{\infty} algebra given by the horocycle foliation of the unit tangent bundle of a surface S of genus g>1. The relation to the exotic R^4 is used to construct the (classical) observable algebra as Poisson algebra of functions over the character variety of representations of the fundamental group π_{1}(S) into the SL(2,C). The Turaev-Drinfeld quantization (as deformation quantization) of this Poisson algebra is a (complex) skein algebra which is isomorphic to the hyperfinite factor II_{1} algebra determining the factor II_{\infty}=II_{1}\otimes I_{\infty} algebra of the horocycle foliation. Therefore our geometrically motivated hyperfinite III_1 factor algebra comes from the quantization of a Poisson algebra. Finally we discuss the states and operators to be knots and knot concordances, respectively.

hep-th↗
arXiv2011

Quantum D-branes and exotic smooth R^4

In this paper, we present the idea that the formalism of string theory is connected with the dimension 4 in a new way, not covered by phenomenological or model-building approaches. The main connection is given by structures induced by small exotic smooth R^4's having intrinsic meaning for physics in dimension 4. We extend the notion of stable quantum D-branes in a separable noncommutative C* algebras over convolution algebras corresponding to the codimension-1 foliations of S^3 which are mainly connected to small exotic R^4. The tools of topological K-homology and K-theory as well KK-theory describe stable quantum branes in the C* algebras when naturally extended to algebras. In case of convolution algebras, small exotic smooth R^4's embedded in exotic R^4 correspond to a generalized quantum branes on the algebras. These results extend the correspondence between exotic R^4 and classical D and NS branes from our previous work.

hep-th↗
arXiv2010

Exotic smooth R^4, geometry of string backgrounds and quantum D-branes

In this paper we make a first step toward determining 4-dimensional data from higher dimensional superstring theory and considering these as underlying structures for the theory. First, we explore connections of exotic smoothings of R^4 and certain configurations of NS and D-branes, both classical and (generalized) quantum using C* algebras. Effects of some small exotic R^4's, when localized on S^3, correspond to stringy geometries of B-fields on S^3. Exotic smoothness of R^4 acts as a non-vanishing B-field on S^3. The dynamics of D-branes in SU(2) WZW model at finite k indicates the exoticness of ambient R^4. Next, based on the relation of exotic smooth R^4 with integral levels of SU(2) WZW model we show the correspondence between exotic smoothness on 4-space, transversal to the world volume of NS5 branes, and the number of these NS5 branes. Relation with the calculations in holographically dual 6-dimensional little string theory is discussed. Generalized quantum D-branes in the noncommutative C* algebras corresponding to the codimension-1 foliations of S^3 are considered and these determine the KK invariants of exotic smooth R_h^4 for the case of non-integral $[h]\in H^3(S^3,R)$. Moreover, exotic smooth R^4's embedded in some exotic R^4 as open submanifolds, are shown to correspond to generalized quantum D-branes in the noncommutative C* algebra of the foliation. Finally, we show how exotic smoothness of R^4 is correlated with D6 brane charges in IIA string theory. In the last section we construct wild embeddings of spheres and relate them to D-brane charges as well to KK theory. We conjecture that a quantum D-brane is wild embedding. Then we construct an action for a quantum D-brane and show that the classical limit (the usual embedding) agrees with the Born-Infeld action.

hep-th↗
arXiv2010

Exotic Smoothness and Quantum Gravity

Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. Thus, the negative result of Duston (arXiv:0911.4068) was a surprise. A closer look into the semi-classical approach uncovered the implicit assumption of a close connection between geometry and smoothness structure. But both structures, geometry and smoothness, are independent of each other. In this paper we calculate the "smoothness structure" part of the path integral in quantum gravity assuming that the "sum over geometries" is already given. For that purpose we use the knot surgery of Fintushel and Stern applied to the class E(n) of elliptic surfaces. We mainly focus our attention to the K3 surfaces E(2). Then we assume that every exotic smoothness structure of the K3 surface can be generated by knot or link surgery a la Fintushel and Stern. The results are applied to the calculation of expectation values. Here we discuss the two observables, volume and Wilson loop, for the construction of an exotic 4-manifold using the knot $5_{2}$ and the Whitehead link $Wh$. By using Mostow rigidity, we obtain a topological contribution to the expectation value of the volume. Furthermore we obtain a justification of area quantization.

gr-qc↗
arXiv2010

Exotic smooth R^4, noncommutative algebras and quantization

The paper shows deep connections between exotic smoothings of small R^4, noncommutative algebras of foliations and quantization. At first, based on the close relation of foliations and noncommutative C*-algebras we show that cyclic cohomology invariants characterize some small exotic R^4. Certain exotic smooth R^4's define a generalized embedding into a space which is K-theoretic equivalent to a noncommutative Banach algebra. Furthermore, we show that a factor III von Neumann algebra is naturally related with nonstandard smoothing of a small R^4 and conjecture that this factor is the unique hyperfinite factor III_1. We also show how an exotic smoothing of a small R^4 is related to the Drinfeld-Turaev (deformation) quantization of the Poisson algebra (X(S,SL(2,C),{,}) of complex functions on the space of flat connections X(S,SL(2,C) over a surface S, and that the result of this quantization is the skein algebra (K_t(S),[,]) for the deformation parameter t=exp(h/4). This skein algebra is retrieved as a II_1 factor of horocycle flows which is Morita equivalent to the II_infty factor von Neumann algebra which in turn determines the unique factor III_1 as crossed product. Moreover, the structure of Casson handles determine the factor II_1 algebra too. Thus, the quantization of the Poisson algebra of closed circles in a leaf of the codimension 1 foliation of S^3 gives rise to the factor III_1 associated with exotic smoothness of R^4. Finally, the approach to quantization via exotic 4-smoothness is considered as a fundamental question in dimension 4 and compared with the topos approach to quantum theories.

hep-th↗
arXiv2009

Gerbes on orbifolds and exotic smooth R^4

By using the relation between foliations and exotic R^4, orbifold $K$-theory deformed by a gerbe can be interpreted as coming from the change in the smoothness of R^4. We give various interpretations of integral 3-rd cohomology classes on S^3 and discuss the difference between large and small exotic R^4. Then we show that $K$-theories deformed by gerbes of the Leray orbifold of S^3 are in one-to-one correspondence with some exotic smooth R^4's. The equivalence can be understood in the sense that stable isomorphisms classes of bundle gerbes on S^{3} whose codimension-1 foliations generates the foliations of the boundary of the Akbulut cork, correspond uniquely to these exotic R^{4}'s. Given the orbifold $SU(2)\times SU(2)\rightrightarrows SU(2)$ where SU(2) acts on itself by conjugation, the deformations of the equivariant $K$-theory on this orbifold by the elements of $H_{SU(2)}^{3}(SU(2),\mathbb{Z})$, correspond to the changes of suitable exotic smooth structures on R^4.

hep-th↗
arXiv2007

Dark energy and 3-manifold topology

We show that the differential-geometric description of matter by differential structures of spacetime leads to a unifying model of the three types of energy in the cosmos: matter, dark matter and dark energy. Using this model we are able to calculate the ratio of dark energy to the total energy of the cosmos.

gr-qc↗
arXiv2006

Differential Structures - the Geometrization of Quantum Mechanics

The usual quantization of a classical space-time field does not touch the non-geometrical character of quantum mechanics. We believe that the deep problems of unification of general relativity and quantum mechanics are rooted in this poor understanding of the geometrical character of quantum mechanics. In Einstein's theory gravitation is expressed by geometry of space-time, and the solutions of the field equation are invariant w.r.t. a certain equivalence class of reference frames. This class can be characterized by the differential structure of space-time. We will show that matter is the transition between reference frames that belong to different differential structures, that the set of transitions of the differential structure is given by a Temperley-Lieb algebra which is extensible to a $C^{*}$-algebra comprising the field operator algebra of quantum mechanics and that the state space of quantum mechanics is the linear space of the differential structures. Furthermore we are able to explain the appearance of the complex numbers in quantum theory. The strong relation to Loop Quantum Gravity is discussed in conclusion.

gr-qc↗
arXiv2005

There is no simulation of n-qubit operations by a single Hamiltonian with 2-spin interaction

Today's devices for quantum computing are still far from implementing useful and powerful quantum algorithms. Decoherence and the wish to resist the effects of errors in a system of quantum bits incurs a lot of overhead in the number of gates and qubits. From a theoretical perspective, controlled quantum simulation raises the hope to simulate the unitary quantum operationes generated by a Hamiltonian with 3-body interaction with a suitably designed element that is constructed of only 2-body interactions. That replacement would happen without any additional gates, and its possibility would be due to the ambiguity of the unit element of the Lie group connected with the algebra of traceless hermitian matrices. We show that this hope is void, and give a general proof for this for any order of interaction.

quant-ph↗
arXiv2004

The Fraunhofer Quantum Computing Portal - www.qc.fraunhofer.de - A web-based Simulator of Quantum Computing Processes

Fraunhofer FIRST develops a computing service and collaborative workspace providing a convenient tool for simulation and investigation of quantum algorithms. To broaden the twenty qubit limit of workstation-based simulations to the next qubit decade we provide a dedicated high memorized Linux cluster with fast Myrinet interconnection network together with a adapted parallel simulator engine. This simulation service supplemented by a collaborative workspace is usable everywhere via web interface and integrates both hardware and software as collaboration and investigation platform for the quantum community. The beta test version realizes all common one, two and three qubit gates, arbitrary one and two bit gates, orthogonal measurements as well as special gates like Oracle, Modulo function, Quantum Fourier Transformation and arbitrary Spin-Hamiltonians up to 31 qubits. For a restricted gate set it feasible to investigate circuits with up to sixty qubits. URL: http://www.qc.fraunhofer.de

quant-ph↗
arXiv2001

Cosmological anomalies and exotic smoothness structures

It seems to be generally accepted that apparently anomalous cosmological observations, such as accelerating expansion, etc., necessarily are inconsistent with standard general relativity and standard matter sources. Following the suggestions of Sładkowski, we point out that in addition to exotic theories and exotic matter there is another possibility. We refer to exotic differential structures on ${\mathbb R}^4$ which could be the source of the observed anomalies without changing the Einstein equations or introducing strange forms of matter.

gr-qc↗