Perturbative Quantum Field Theory and Configuration Space Integrals

L-infinity morphisms are studied from the point of view of perturbative quantum field theory, as generalizations of Feynman expansions. The connection with the Hopf algebra approach to renormalization is exploited. Using the coalgebra structure (Forest Formula), the weights of the corresponding expansions are proved to be cycles of the DG-coalgebra of Feynman graphs. The properties of integrals over configuration spaces (Feynman integrals) are investigated. The aim is to develop a cohomological approach in order to construct the coefficients of formality morphisms using an algebraic machinery, as an alternative to the analytical approach using integrals over configuration spaces. The connection with a related TQFT is mentioned, supplementing the Feynman path integral interpretation of Kontsevich formula.Comment: AMS LaTeX, 26 page

On Categorification

We review several known categorification procedures, and introduce a functorial categorification of group extensions with applications to non-abelian group cohomology. Categorification of acyclic models and of topological spaces are briefly mentioned.Comment: AMS-LaTex, 10 page

The Search for a New Equivalence Principle

The new emerging quantum physics - quantum computing conceptual bridge, mandates a ``grand unification'' of space-time-matter and quantum information (all quantized), with deep implications for science in general. The major physics revolutions in our understanding of the universe are briefly reviewed and a ``missing'' equivalence principle is identified and its nature explained. An implementation as an external super-symmetry \C{E}=ic\C{P} is suggested, generalizing the Wick rotation ``trick''. Taking advantage of the interpretation of entropy as a measure of symmetry, it is naturally asimilated within the present Feynman Path Integral algebraic formalism.Comment: Essay, 13 pages, AMS LaTeX fil

A Natural Partial Order on The Prime Numbers

A natural partial order on the set of prime numbers was derived by the author from the internal symmetries of the primary finite fields, independently of Ford a.a., who investigated Pratt trees for primality tests. It leads to a correspondence with the Hopf algebra of rooted trees, and as an application, to an alternative approach to the Prime Number Theorem.Comment: 10 pages, conference: Number Theory at Illinois, June 5-7, 2014; submitted to IJN

Remarks on Quantum Physics and Noncommutative Geometry

The "quantum-event / prime ideal in a category/ noncommutative-point" alternative to "classical-event / commutative prime ideal/ point" is suggested. Ideals in additive categories, prime spectra and representation of quivers are considered as mathematical tools appropriate to model quantum mechanics. The space-time framework is to be reconstructed from the spectrum of the path category of a quiver. The interference experiment is considered as an example.Comment: 9 pages, AMS-LaTex, 1 eps figur

A combinatorial approach to coefficients in deformation quantization

Graph cocycles for star-products are investigated from the combinatorial point of view, using Connes-Kreimer renormalization techniques. The Hochschild complex, controlling the deformation theory of associative algebras, is the ``Kontsevich representation'' of a DGLA of graphs coming from a pre-Lie algebra structure defined by graph insertions. Properties of the dual of its UEA (an odd parity analog of Connes-Kreimer Hopf algebra), are investigated in order to find solutions of the deformation equation. The solution of the initial value deformation problem, at tree-level, is unique. For linear coefficients the resulting formulas are relevant to the Hausdorff series.Comment: 23 pages, AMS LaTeX, 8 eps figure

The Feynman Legacy

The article is an overview of the role of graph complexes in the Feynman path integral quantization. The underlying mathematical language is that of PROPs and operads, and their representations. The sum over histories approach, the Feynman Legacy, is the bridge between quantum physics and quantum computing, pointing towards a deeper understanding of the fundamental concepts of space, time and information.Comment: 22 pages, LaTeX2e; submitted to Adv. Theor. Math. Phy

Hochschild DGLAs and torsion algebras

The associator of a non-associative algebra is the curvature of the Hochschild quasi-complex. The relationship ``curvature-associator'' is investigated. Based on this generic example, we extend the geometric language of vector fields to a purely algebraic setting, similar to the context of Gerstenhaber algebras. We interprete the elements of a non-associative algebra with a Lie bracket as ``vector fields'' and the multiplication as a connection. We investigate conditions for the existance of an ``algebra of functions'' having as algebra of derivations the original non-associative algebra.Comment: AMS-LaTex, 15 page

Quantum Relativity

Quantum Relativity is supposed to be a new theory, which locally is a deformation of Special Relativity, and globally it is a background independent theory including the main ideas of General Relativity, with hindsight from Quantum Theory. The qubit viewed as a Hopf monopole bundle is considered as a unifying gauge "group". Breaking its chiral symmetry is conjectured to yield gravity as a deformation of electromagnetism. It is already a quantum theory in the context of Quantum Information Dynamics as a discrete, background independent theory, unifying classical and quantum physics. Based on the above, Quantum Gravity is sketched as an open project.Comment: 26 pages; Quantum Gravity Project added (Annex

Quantum Relativity: an essay

Is "Gravity" a deformation of "Electromagnetism"? Deformation theory suggests quantizing Special Relativity: formulate Quantum Information Dynamics $SL(2,C)_h$-gauge theory of dynamical lattices, with unifying gauge ``group'' the quantum bundle obtained from the Hopf monopole bundle underlying the quaternionic algebra and Dirac-Weyl spinors. The deformation parameter is the inverse of light speed 1/c, in duality with Planck's constant h. Then mass and electric charge form a complex coupling constant (m,q), for which the quantum determinant of the quantum group $SL(2,C)_h$ expresses the interaction strength as a linking number 2-form. There is room for both Coulomb constant $k_C$ and Newton's gravitational constant $G_N$, exponentially weaker then the reciprocal of the fine structure constant $\alpha$. Thus "Gravity" emerges already "quantum", in the discrete framework of QID, based on the quantized complex harmonic oscillator: the quantized qubit. All looks promising, but will the details backup this "grand design scheme"?Comment: 7 page