arXiv subjects

H. E. Winkelnkemper

Publications and source records attributed to H. E. Winkelnkemper.

3 recordsLinked to original sources

arXiv2010

AP Theory IV: Intrinsic Topological Quantum Langlands Theory

Without using any moduli, sheaves, stacks, nor any analytic, nor category-type arguments, we exhibit an analogue to Geometric Langlands Theory in an entirely model-independent, non-perturbative,purely smooth topological context in Artin Presentation Theory. A basic initial feature is that AP Theory, as a whole, is already, ab initio, a universal canonical 2D sigma-model, targeting smooth, compact, simply-connected 4-manifolds with a connected boundary, and its topological Planckian quantum starting point, as well as its cone-like, infinitely-generated at each stage, graded group of homology-preserving, but topology-changing transitions/interactions, exhibit the most general qualitative S-duality. We first point out the numerous mathematically rigorous, model-free, (i.e., intrinsic), topological AP analogues with the heuristic Kapustin-Witten version of Geometric Langlands theory, as well as the crucial differences between the two theories. The latter have to exist since AP Theory deals, a priori, essentially only with discrete group-theoretic presentation theory, not Lie group representation theory, not with category nor moduli theory, does not need classical SUSY, nor Feynman integrals, lattice models, etc., and furthermore is model-independent and characteristic of real dimension four. It will become clear that AP Langlands theory is a model-independent, cone-like, graded, compact completion and topological 'envelope' of 4D N=4 SUSY YM theory, their starting point, anchored, kept in place, so to speak, by a purely discrete group-theoretic analogue of Donaldson/Seiberg-Witten theory. Due to its discrete group-theoretic conceptual simplicity and universality, the AP Langlands program should also be considered to be a new type of intrinsic model-free symmetry, a universal Erlanger Program for Modern Physics.

math.GM↗

arXiv2010

AP Theory III: Cone-like Graded SUSY, Dynamic Dark Energy and the YM Millenium Problem

Artin Presentation Theory, (AP Theory), is a new, direct infusion, via pure braid theory, of discrete group theory, (i.e., symmetry in its purest form), into the theory of {\it smooth} 4-manifolds, (i.e.,$(3+1)$-Quantum Gravity in its purest topological form), thus exhibiting the most basic, rigorous, universal, model-free intrinsic {\it gauge-gravity} duality in a non-infinitesimal, cone-like graded, as holographic as possible, model-independent, non-perturbative, background-independent, parameter-free manner. {\it In AP Theory even smooth topology change becomes gauge-theoretic, setting the stage for a rigorous smooth topological $(3+1)$-QFT of Dynamic Dark Energy.} In this theory, the rigid $\infty$ of the dimension of classical Hilbert space is substituted by the dynamic $\infty$ of the $\infty$ generation at each stage of a cone-like graded subgroup of topology-changing transitions/interactions. As a corollary, the Cosmological Constant problem and the YM Millenium Mass Gap problem, two of the most perplexing main problems of modern physics, become rigorously, intimately mathematically related, by having the same qualitative {\it dynamical} roots. Ultimately our main point is meta-mathematical, as far as modern physics is concerned: due to the discrete group-theoretic conceptual simplicity of the theory, with its group-theoretic 'Planckian membrane/discreteness' starting point, {\it the fact that it is not just a mere mathematical model,} and all its properties above, any other {\it mathematically rigorous} approach has to built on AP Theory and be topologically absorbed and enveloped by it.

math.GM↗

arXiv2008

AP Theory II:Intrinsic 4D Quantum YM Theory with Mass Gap

We describe a sub-theory of Artin Presentation Theory (AP Theory), which has many genuine,discrete,group-theoretic,non-infinitesimal, qualitative analogues (including with the mass gap) of the main desiderata of the hypothetical quantitative infinitesimal '4D Quantum YM Theory' for the so-called Clay Millenium 'YM Existence and Mass Gap' problem. Our entirely mathematically rigorous theory is not a model, no new axioms or measures are introduced,does not rely on SUSY,is free of smooth 4D singularities,moduli spaces,path integrals, graph/lattice combinatorics and probabilistic,category,twistor or topos arguments and is intimately related to the theory of pure framed braids. Despite being based on a rigorous, radical,universal Holographic Principle,the theory still contains an analogue of Donaldson/Seiberg-Witten Theory, an infinitely generated, at each stage, graded group of topology-changing transitions and interactions and more. Our main contention is: the radical,universal AP-holography, with its strong topology changing interactions, which reach all the way to the 'vacuum' of discrete group theory, may destroy any infinitesimal, PDE based Field Theory,required for solving the actual Clay YM problem in its present quantitative form as a problem of so-called 'constructive' 3+1 QFT. More generally,due to the fact that AP Theory is not a model,e.g., does not introduce any new axioms,any rigorous axiomatic 3+1 QFT has to align itself with it in a mathematically rigorous fashion.

math.GM↗