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Papers by Torsten Asselmeyer-Maluga
The topology of the unitary groups and the Hamilton representation of unitary lattices
Verhandlungen der Deutschen Physikalischen Gesellschaft, 2005
Between Quantum Mechanics and Cosmology
Gauge Theory and Moduli Space
WORLD SCIENTIFIC eBooks, 2007
A Guide to the Classification of Manifolds
WORLD SCIENTIFIC eBooks, 2007
Algebraic Tools for Topology
Smooth Manifolds, Geometry
Bundles, Geometry, Gauge Theory
Bundles, Geometry, Gauge Theory
Seiberg-Witten Theory: The Modern Approach
Lecture Notes in Computer Science, 1996
We report a rigorous derivation of an analytical expression for the wake fields of accelerated el... more We report a rigorous derivation of an analytical expression for the wake fields of accelerated electron beams that takes into account the discontinuity of the accelerating cavities of linear accelerators. The approach employs Fourier's transforms, Green's function techniques, and Laurent series in order to solve the full set of Maxwell's equations in the Lorentz gauge. This represents a significant step toward a realistic description of the evolution of wake fields in linear accelerators. The expression is applied to the ''ELSA'' photo injector to obtain relevant numerical results.
arXiv (Cornell University), Aug 10, 1995
We investigate an simple evolutionary game of sequences and demonstrate on this example the struc... more We investigate an simple evolutionary game of sequences and demonstrate on this example the structure of fitness landscapes in discrete problems. We show the smoothing action of the genotype-phenotype mapping which still makes it feasible for evolution to work. Further we propose the density of sequence states as a classifying measure of fitness landscapes.
Foundations of Science, Jun 27, 2022
Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probab... more Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin-Löf. We extend this result and demonstrate that QM is algorithmic ω-random and generic, precisely as described by the 'miniaturisation' of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo-Fraenkel Solovay random on infinite-dimensional Hilbert spaces. Moreover, it is more likely that there exists a standard transitive ZFC model M , where QM is expressed in reality, than in the universe V of sets. Then every generic quantum measurement adds to M the infinite sequence, i.e. random real r ∈ 2 ω , and the model undergoes random forcing extensions M [r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit, therefore showing the structural resemblance of both in this limit. We discuss several questions regarding measurability and possible practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC; however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself, the issues considered here remain hidden to a great extent.
arXiv (Cornell University), Jan 24, 2016
Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic R 4 ,... more Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic R 4 , and quantum field theory were found. Some of these relations are rooted in a relation to superstring theory and quantum gravity. Therefore one would expect that exotic smoothness is directly related to the quantization of general relativity. In this article we will support this conjecture and develop a new approach to quantum gravity called smooth quantum gravity by using smooth 4-manifolds with an exotic smoothness structure. In particular we discuss the appearance of a wildly embedded 3-manifold which we identify with a quantum state. Furthermore, we analyze this quantum state by using foliation theory and relate it to an element in an operator algebra. Then we describe a set of geometric, non-commutative operators, the skein algebra, which can be used to determine the geometry of a 3-manifold. This operator algebra can be understood as a deformation quantization of the classical Poisson algebra of observables given by holonomies. The structure of this operator algebra induces an action by using the quantized calculus of Connes. The scaling behavior of this action is analyzed to obtain the classical theory of General Relativity (GRT) for large scales. This approach has some obvious properties: there are non-linear gravitons, a connection to lattice gauge field theory and a dimensional reduction from 4D to 2D. Some cosmological consequences like the appearance of an inflationary phase are also discussed. At the end we will get the simple picture that the change from the standard R 4 to the exotic R 4 is a quantization of geometry.
Symmetry, Sep 9, 2022
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Symmetry, Jan 5, 2020
Various experimentally verified values of physical parameters indicate that the universe evolves ... more Various experimentally verified values of physical parameters indicate that the universe evolves close to the topological phase of exotic smoothness structures on R 4 and K3 surface. The structures determine the α parameter of the Starobinski model, the number of e-folds, the spectral tilt, the scalar-to-tensor ratio and the GUT and electroweak energy scales, as topologically supported quantities. Neglecting exotic R 4 and K3 leaves these free parameters undetermined. We present general physical and mathematical reasons for such preference of exotic smoothness. It appears that the spacetime should be formed on open domains of smooth K3#CP 2 at extra-large scales possibly exceeding our direct observational capacities. Such potent explanatory power of the formalism is not that surprising since there exist natural physical conditions, which we state explicitly, that allow for the unique determination of a spacetime within the exotic K3.
International Journal of Geometric Methods in Modern Physics, May 1, 2012
In this paper, we present the idea that the formalism of string theory is connected with the dime... more 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.
Classical and Quantum Gravity, Mar 1, 1997
In this paper the relation between the choice of a differential structure and a smooth connection... more In this paper the relation between the choice of a differential structure and a smooth connection in the tangential bundle is discussed. For the case of an exotic S 7 one obtains corrections to the curvature after the change of the differential structure, which can not be neglected by a gauge transformation. In the more interesting case of four dimensions we obtain a correction of the connection constructed by intersections of embedded surfaces. This correction produce a source term in the equation of the general relativity theory which can be interpreted as the energy-momentum tensor of a embedded surface.
Exotic Smoothness and Physics - Differential Topology and Spacetime Models
The recent revolution in differential topology related to the discovery of non-standard ("ex... more The recent revolution in differential topology related to the discovery of non-standard ("exotic") smoothness structures on topologically trivial manifolds such as R4 suggests many exciting opportunities for applications of potentially deep importance for the spacetime models of theoretical physics, especially general relativity. This rich panoply of new differentiable structures lies in the previously unexplored region between topology and geometry. Just as physical geometry was thought to be trivial before Einstein, physicists have continued to work under the tacit - but now shown to be incorrect - assumption that differentiability is uniquely determined by topology for simple four-manifolds. Since diffeomorphisms are the mathematical models for physical coordinate transformations, Einstein's relativity principle requires that these models be physically inequivalent. This book provides an introductory survey of some of the relevant mathematics and presents preliminary results and suggestions for further applications to spacetime models
Entropy, Mar 11, 2022
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
The topology of the unitary groups and the Hamilton representation of unitary lattices
Verhandlungen der Deutschen Physikalischen Gesellschaft, 2005
Between Quantum Mechanics and Cosmology
Gauge Theory and Moduli Space
WORLD SCIENTIFIC eBooks, 2007
A Guide to the Classification of Manifolds
WORLD SCIENTIFIC eBooks, 2007
Algebraic Tools for Topology
Smooth Manifolds, Geometry
Bundles, Geometry, Gauge Theory
Bundles, Geometry, Gauge Theory
Seiberg-Witten Theory: The Modern Approach
Lecture Notes in Computer Science, 1996
We report a rigorous derivation of an analytical expression for the wake fields of accelerated el... more We report a rigorous derivation of an analytical expression for the wake fields of accelerated electron beams that takes into account the discontinuity of the accelerating cavities of linear accelerators. The approach employs Fourier's transforms, Green's function techniques, and Laurent series in order to solve the full set of Maxwell's equations in the Lorentz gauge. This represents a significant step toward a realistic description of the evolution of wake fields in linear accelerators. The expression is applied to the ''ELSA'' photo injector to obtain relevant numerical results.
arXiv (Cornell University), Aug 10, 1995
We investigate an simple evolutionary game of sequences and demonstrate on this example the struc... more We investigate an simple evolutionary game of sequences and demonstrate on this example the structure of fitness landscapes in discrete problems. We show the smoothing action of the genotype-phenotype mapping which still makes it feasible for evolution to work. Further we propose the density of sequence states as a classifying measure of fitness landscapes.
Foundations of Science, Jun 27, 2022
Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probab... more Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin-Löf. We extend this result and demonstrate that QM is algorithmic ω-random and generic, precisely as described by the 'miniaturisation' of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo-Fraenkel Solovay random on infinite-dimensional Hilbert spaces. Moreover, it is more likely that there exists a standard transitive ZFC model M , where QM is expressed in reality, than in the universe V of sets. Then every generic quantum measurement adds to M the infinite sequence, i.e. random real r ∈ 2 ω , and the model undergoes random forcing extensions M [r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit, therefore showing the structural resemblance of both in this limit. We discuss several questions regarding measurability and possible practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC; however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself, the issues considered here remain hidden to a great extent.
arXiv (Cornell University), Jan 24, 2016
Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic R 4 ,... more Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic R 4 , and quantum field theory were found. Some of these relations are rooted in a relation to superstring theory and quantum gravity. Therefore one would expect that exotic smoothness is directly related to the quantization of general relativity. In this article we will support this conjecture and develop a new approach to quantum gravity called smooth quantum gravity by using smooth 4-manifolds with an exotic smoothness structure. In particular we discuss the appearance of a wildly embedded 3-manifold which we identify with a quantum state. Furthermore, we analyze this quantum state by using foliation theory and relate it to an element in an operator algebra. Then we describe a set of geometric, non-commutative operators, the skein algebra, which can be used to determine the geometry of a 3-manifold. This operator algebra can be understood as a deformation quantization of the classical Poisson algebra of observables given by holonomies. The structure of this operator algebra induces an action by using the quantized calculus of Connes. The scaling behavior of this action is analyzed to obtain the classical theory of General Relativity (GRT) for large scales. This approach has some obvious properties: there are non-linear gravitons, a connection to lattice gauge field theory and a dimensional reduction from 4D to 2D. Some cosmological consequences like the appearance of an inflationary phase are also discussed. At the end we will get the simple picture that the change from the standard R 4 to the exotic R 4 is a quantization of geometry.
Symmetry, Sep 9, 2022
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Symmetry, Jan 5, 2020
Various experimentally verified values of physical parameters indicate that the universe evolves ... more Various experimentally verified values of physical parameters indicate that the universe evolves close to the topological phase of exotic smoothness structures on R 4 and K3 surface. The structures determine the α parameter of the Starobinski model, the number of e-folds, the spectral tilt, the scalar-to-tensor ratio and the GUT and electroweak energy scales, as topologically supported quantities. Neglecting exotic R 4 and K3 leaves these free parameters undetermined. We present general physical and mathematical reasons for such preference of exotic smoothness. It appears that the spacetime should be formed on open domains of smooth K3#CP 2 at extra-large scales possibly exceeding our direct observational capacities. Such potent explanatory power of the formalism is not that surprising since there exist natural physical conditions, which we state explicitly, that allow for the unique determination of a spacetime within the exotic K3.
International Journal of Geometric Methods in Modern Physics, May 1, 2012
In this paper, we present the idea that the formalism of string theory is connected with the dime... more 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.
Classical and Quantum Gravity, Mar 1, 1997
In this paper the relation between the choice of a differential structure and a smooth connection... more In this paper the relation between the choice of a differential structure and a smooth connection in the tangential bundle is discussed. For the case of an exotic S 7 one obtains corrections to the curvature after the change of the differential structure, which can not be neglected by a gauge transformation. In the more interesting case of four dimensions we obtain a correction of the connection constructed by intersections of embedded surfaces. This correction produce a source term in the equation of the general relativity theory which can be interpreted as the energy-momentum tensor of a embedded surface.
Exotic Smoothness and Physics - Differential Topology and Spacetime Models
The recent revolution in differential topology related to the discovery of non-standard ("ex... more The recent revolution in differential topology related to the discovery of non-standard ("exotic") smoothness structures on topologically trivial manifolds such as R4 suggests many exciting opportunities for applications of potentially deep importance for the spacetime models of theoretical physics, especially general relativity. This rich panoply of new differentiable structures lies in the previously unexplored region between topology and geometry. Just as physical geometry was thought to be trivial before Einstein, physicists have continued to work under the tacit - but now shown to be incorrect - assumption that differentiability is uniquely determined by topology for simple four-manifolds. Since diffeomorphisms are the mathematical models for physical coordinate transformations, Einstein's relativity principle requires that these models be physically inequivalent. This book provides an introductory survey of some of the relevant mathematics and presents preliminary results and suggestions for further applications to spacetime models
Entropy, Mar 11, 2022
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY