Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 051, 70 pages      arXiv:0904.3644

Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review

Ion C. Baianu a, James F. Glazebrook b and Ronald Brown c
a) FSHN and NPRE Departments, University of Illinois at Urbana-Champaign, AFC-NMR & FT-NIR Microspectroscopy Facility, Urbana IL 61801 USA
b) Department of Mathematics and Computer Science, Eastern Illinois University, 600 Lincoln Avenue, Charleston IL 61920 USA
    Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana IL 61801 USA

c) School of Computer Science, University of Bangor, Dean Street, Bangor Gwynedd LL57 1UT UK

Received November 25, 2008, in final form April 09, 2009; Published online April 23, 2009

A novel algebraic topology approach to supersymmetry (SUSY) and symmetry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non-Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier-Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasi-triangular, quasi-Hopf algebras, bialgebroids, Grassmann-Hopf algebras and higher dimensional algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang-Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant general relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin) models with the extended quantum symmetry of entangled, 'string-net condensed' (ground) states.

Key words: extended quantum symmetries; groupoids and algebroids; quantum algebraic topology (QAT); algebraic topology of quantum systems; symmetry breaking, paracrystals, superfluids, spin networks and spin glasses; convolution algebras and quantum algebroids; nuclear Fréchet spaces and GNS representations of quantum state spaces (QSS); groupoid and functor representations in relation to extended quantum symmetries in QAT; quantization procedures; quantum algebras: von Neumann algebra factors; paragroups and Kac algebras; quantum groups and ring structures; Lie algebras; Lie algebroids; Grassmann-Hopf, weak C*-Hopf and graded Lie algebras; weak C*-Hopf algebroids; compact quantum groupoids; quantum groupoid C*-algebras; relativistic quantum gravity (RQG); supergravity and supersymmetry theories; fluctuating quantum spacetimes; intense gravitational fields; Hamiltonian algebroids in quantum gravity; Poisson-Lie manifolds and quantum gravity theories; quantum fundamental groupoids; tensor products of algebroids and categories; quantum double groupoids and algebroids; higher dimensional quantum symmetries; applications of generalized van Kampen theorem (GvKT) to quantum spacetime invariants.

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