66 research outputs found
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
-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 expresses the interaction strength
as a linking number 2-form. There is room for both Coulomb constant and
Newton's gravitational constant , exponentially weaker then the reciprocal
of the fine structure constant . 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
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