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Manfred Requardt

Publications and source records attributed to Manfred Requardt.

47 records · Page 3Linked to original sources
arXiv2000

Graph-Laplacians and Dirac Operators on (Infinite) Graphs and the Calculation of the Connes-Distance-Functional

We develop a graph-Hilbert-space framework, inspired by non-commutative geometry, on (infinite) graphs and use it to study spectral properies of \tit{graph-Laplacians} and so-called \tit{graph-Dirac-operators}. Putting the various pieces together we define a {\it spectral triplet} sharing most (if not all, depending on the particular graph model) of the properties of what Connes calls a \tit{spectral triple}. With the help of this scheme we derive an explicit expression for the {\it Connes-distance function} on general graphs and prove both a variety of apriori estimates for it and calculate it for certain examples of graphs. As a possibly interesting aside, we show that the natural setting of approaching such problems may be the framework of \tit{(non-)linear programming} or \tit{optimization}. We compare our results (arrived at within our particular framework) with the results of other authors and show that the seeming differences depend on the use of different graph-geometries and/or Dirac operators.

math-ph↗
arXiv2000

Let's call it Nonlocal Quantum Physics

In the following we undertake to derive quantum theory as a stochastic low-energy and coarse-grained theory from a more primordial discrete and basically geometric theory living on the Planck scale and which (as we argue) possibly underlies also \tit{string theory}. We isolate the so-called \tit{ideal elements} which represent at the same time the cornerstones of the framework of ordinary quantum theory and show how and why they encode the \tit{non-local} aspects, being ubiquituous in the quantum realm, in a, on the surface, local way. We show that the quantum non-locality emerges in our approach as a natural consequence of the underlying \tit{two-storey} nature of space-time or the physical vacuum, that is, quantum theory turns out to be a residual effect of the geometric depth structure of space-time on the Planck scale. We indicate how the \tit{measurement problem} and the emergence of the \tit{macroscopic sub-regime} can be understood in this framework.

gr-qc↗
arXiv1999

(Quantum) Space-Time as a Statistical Geometry of Lumps in Random Networks

In the following we undertake to describe how macroscopic space-time (or rather, a microscopic protoform of it) is supposed to emerge as a superstructure of a web of lumps in a stochastic discrete network structure. As in preceding work (mentioned below), our analysis is based on the working philosophy that both physics and the corresponding mathematics have to be genuinely discrete on the primordial (Planck scale) level. This strategy is concretely implemented in the form of \tit{cellular networks} and \tit{random graphs}. One of our main themes is the development of the concept of \tit{physical (proto)points} or \tit{lumps} as densely entangled subcomplexes of the network and their respective web, establishing something like \tit{(proto)causality}. It may perhaps be said that certain parts of our programme are realisations of some early ideas of Menger and more recent ones sketched by Smolin a couple of years ago. We briefly indicate how this \tit{two-story-concept} of \tit{quantum} space-time can be used to encode the (at least in our view) existing non-local aspects of quantum theory without violating macroscopic space-time causality.

gr-qc↗
arXiv1999

Space-Time as an Orderparameter Manifold in Random Networks and the Emergence of Physical Points

In the following we are going to describe how macroscopic space-time is supposed to emerge as an orderparameter manifold or superstructure floating in a stochastic discrete network structure. As in preceeding work (mentioned below), our analysis is based on the working philosophy that both physics and the corresponding mathematics have to be genuinely discrete on the primordial (Planck scale) level. This strategy is concretely implemented in the form of cellular networks and random graphs. One of our main themes is the development of the concept of physical (proto)points as densely entangled subcomplexes of the network and their respective web, establishing something like (proto)causality. It max perhaps be said that certain parts of our programme are realisations of some old and qualitative ideas of Menger and more recent ones sketched by Smolin a couple of years ago. We briefly indicate how this two-story-concept of space-time can be used to encode the (at least in our view) existing non-local aspects of quantum theory without violating macroscopic space-time causality!

gr-qc↗
arXiv1999

Perturbation Theory of Schrödinger Operators in Infinitely Many Coupling Parameters

In this paper we study the behavior of Hamilton operators and their spectra which depend on infinitely many coupling parameters or, more generally, parameters taking values in some Banach space. One of the physical models which motivate this framework is a quantum particle moving in a more or less disordered medium. One may however also envisage other scenarios where operators are allowed to depend on interaction terms in a manner we are going to discuss below. The central idea is to vary the occurring infinitely many perturbing potentials independently. As a side aspect this then leads naturally to the analysis of a couple of interesting questions of a more or less purely mathematical flavor which belong to the field of infinite dimensional holomorphy or holomorphy in Banach spaces. In this general setting we study in particular the stability of selfadjointness of the operators under discussion and the analyticity of eigenvalues under the condition that the perturbing potentials belong to certain classes.

math-ph↗
arXiv1998

Cellular Networks as Models for Planck-Scale Physics

Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck scale, one of the many problems one has to face in this enterprise is to find the discrete protoforms of the building blocks of our ordinary continuum physics and mathematics. We base our own approach on what we call `cellular networks', consisting of cells (nodes) interacting with each other via bonds (figuring as elementary interactions) according to a certain `local law'. Geometrically our dynamical networks are living on graphs. Hence a substantial amount of the investigation is devoted to the developement of various versions of discrete (functional) analysis and geometry on such (almost random) webs. Another important topic we address is a suitable concept of intrinsic (fractal) dimension on erratic structures of this kind. In the course of the investigation we make comments concerning both different and related approaches to quantum gravity as, say, the spin network framework. It may perhaps be said that certain parts of our programme seem to be a realisation of ideas sketched by Smolin some time ago (see the introduction).

hep-th↗
arXiv1998

Pregeometric Concepts on Graphs and Cellular Networks as Possible Models of Space-Time at the Planck-Scale

Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face is to find the discrete protoforms of the building blocks of continuum physics and mathematics. In the following we embark on developing such concepts for irregular structures like (large) graphs or networks which are intended to emulate (some of) the generic properties of the presumed combinatorial substratum from which continuum physics is assumed to emerge as a coarse grained and secondary model theory. We briefly indicate how various concepts of discrete (functional) analysis and geometry can be naturally constructed within this framework, leaving a larger portion of the paper to the systematic developement of dimensional concepts and their properties, which may have a possible bearing on various branches of modern physics beyond quantum gravity.

hep-th↗
arXiv1997

Dimension Theory of Graphs and Networks

Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this enterprise is to find the discrete protoforms of the building blocks of continuum physics and mathematics. A core concept is the notion of dimension. In the following we develop such a notion for irregular structures like (large) graphs and networks and derive a number of its properties. Among other things we show its stability under a wide class of perturbations which is important if one has 'dimensional phase transitions' in mind. Furthermore we systematically construct graphs with almost arbitrary 'fractal dimension' which may be of some use in the context of 'dimensional renormalization' or statistical mechanics on irregular sets.

hep-th↗
arXiv1997

Discrete Mathematics and Physics on the Planck-Scale exemplified by means of a Class of 'Cellular Network Models' and their Dynamics

Starting from the hypothesis that both physics, in particular space-time and the physical vacuum, and the corresponding mathematics are discrete on the Planck scale we develop a certain framework in form of a class of ' cellular networks' consisting of cells (nodes) interacting with each other via bonds according to a certain 'local law' which governs their evolution. Both the internal states of the cells and the strength/orientation of the bonds are assumed to be dynamical variables. We introduce a couple of candidates of such local laws which, we think, are capable of catalyzing the unfolding of the network towards increasing complexity and pattern formation. In section 3 the basis is laid for a version of 'discrete analysis' on 'graphs' and 'networks' which, starting from different, perhaps more physically oriented principles, manages to make contact with the much more abstract machinery of Connes et al. and may complement the latter approach. In section 4 several more advanced geometric/topological concepts and tools are introduced which allow to study and classify such irregular structures as (random)graphs and networks. We show in particular that the systems under study carry in a natural way a 'groupoid structure'. In section 5 a, as far as we can see, promising concept of 'topological dimension' (or rather: ' fractal dimension') in form of a 'degree of connectivity' for graphs, networks and the like is developed. The possibility of dimensional phase transitions is discussed.

hep-th↗
arXiv1996

Emergence of Space-Time on the Planck Scale described as an Unfolding Phase Transition within the Scheme of Dynamical Cellular Networks and Random Graphs

As in an earlier paper we start from the hypothesis that physics on the Planck scale should be described by means of concepts taken from ``discrete mathematics''. This goal is realized by developing a scheme being based on the dynamical evolution of a particular class of ``cellular networks'' being capable of performing an ``unfolding phase transition'' from a (presumed) chaotic initial phase towards a new phase which acts as an ``attractor'' in total phase space and which carries a fine or super structure which is identified as the discrete substratum underlying ordinary continuous space-time (or rather, the physical vacuum). Among other things we analyze the internal structure of certain particular subclusters of nodes/bonds (maximal connected subsimplices, $mss$) which are the fundamental building blocks of this new phase and which are conjectured to correspond to the ``physical points'' of ordinary space-time. Their mutual entanglement generates a certain near- and far-order, viz. a causal structure within the network which is again set into relation with the topological/metrical and causal/geometrical structure of continuous space-time. The mathematical techniques to be employed consist mainly of a blend of a fair amount of ``stochastic mathematics'' with several relatively advanced topics of discrete mathematics like the ``theory of random graphs'' or ``combinatorial graph theory''. Our working philosophy is it to create a scenario in which it becomes possible to identify both gravity and quantum theory as the two dominant but derived(!) aspects of an underlying discrete and more primordial theory (dynamical cellular network) on a much coarser level of resolution, viz. continuous space-time.

hep-th↗