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A Topos for Algebraic Quantum Theory
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2009
"... The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C ..."
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Cited by 9 (1 self)
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The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*algebra of observables A induces a topos T (A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*algebra A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum �(A) in T (A), which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topostheoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on �, and selfadjoint elements of A define continuous functions (more precisely, locale maps) from � to Scott’s interval domain. Noting that open subsets of �(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T (A). These results were inspired by the topostheoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
FEYNMAN MOTIVES AND DELETIONCONTRACTION RELATIONS
"... Abstract. We prove a deletioncontraction formula for motivic Feynman rules given by the classes of the affine graph hypersurface complement in the Grothendieck ring of varieties. We derive explicit recursions and generating series for these motivic Feynman rules under the operation of multiplying e ..."
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Cited by 8 (7 self)
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Abstract. We prove a deletioncontraction formula for motivic Feynman rules given by the classes of the affine graph hypersurface complement in the Grothendieck ring of varieties. We derive explicit recursions and generating series for these motivic Feynman rules under the operation of multiplying edges in a graph and we compare it with similar formulae for the Tutte polynomial of graphs, both being specializations of the same universal recursive relation. We obtain similar recursions for graphs that are chains of polygons and for graphs obtained by replacing an edge by a chain of triangles. We show that the deletioncontraction relation can be lifted to the level of the category of mixed motives in the form of a distinguished triangle, similarly to what happens in categorifications of graph invariants. 1.
Marcolli Feynman integrals and motives of configuration spaces
 Communications in Mathematical Physics, Vol.313 (2012) N.1, 35–70
"... ABSTRACT. We formulate the problem of renormalization of Feynman integrals and its relation to periods of motives in configuration space instead of momentum space. The algebrogeometric setting is provided by the wonderful compactifications ConfΓ (X) of arrangements of subvarieties associated to the ..."
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Cited by 6 (4 self)
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ABSTRACT. We formulate the problem of renormalization of Feynman integrals and its relation to periods of motives in configuration space instead of momentum space. The algebrogeometric setting is provided by the wonderful compactifications ConfΓ (X) of arrangements of subvarieties associated to the subgraphs of a Feynman graph Γ, with X a (quasi)projective variety. The motive and the class in the Grothendieck ring are computed explicitly for these wonderful compactifications, in terms of the motive of X and the combinatorics of the Feynman graph, using recent results of Li Li. The pullback to the wonderful compactification of the form defined by the unrenormalized Feynman amplitude has singularities along a hypersurface, whose real locus is contained in the exceptional divisors of the iterated blowup that gives the wonderful compactification. A regularization of the Feynman integrals can be obtained by modifying the cycle of integration, by replacing the divergent locus with a Leray coboundary. The ambiguities are then defined by Poincaré residues. While these residues give mixed Tate periods associated to the cohomology of the exceptional divisors and their intersections, the regularized integrals give rise to periods of the hypersurface complement in the wonderful compactification, which can be motivically more complicated. 1.
Quantum statistical mechanics, Lseries and anabelian geometry, in preparation
"... Abstract. It is known that two number fields with the same Dedekind zeta function are not necessarily isomorphic. The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical system, which is a C ∗algebra with a one parameter group ..."
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Cited by 5 (4 self)
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Abstract. It is known that two number fields with the same Dedekind zeta function are not necessarily isomorphic. The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical system, which is a C ∗algebra with a one parameter group of automorphisms, built from Artin reciprocity. In the first part of this paper, we prove that isomorphism of number fields is the same as isomorphism of these associated systems. Considering the systems as noncommutative analogues of topological spaces, this result can be seen as another version of Grothendieck’s “anabelian ” program, much like the NeukirchUchida theorem characterizes isomorphism of number fields by topological isomorphism of their associated absolute Galois groups. In the second part of the paper, we use these systems to prove the following. If there is a continuous bijection ψ: ˇ G ab K → ˇ G ab L between the character groups (viz., Pontrjagin duals) of the abelianized Galois groups of the two number fields that induces an equality of all corresponding L
Definite integrals by the method of brackets
 Part 1. Adv. Appl. Math
"... Abstract. A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for the evaluation of Feynman diagrams. We describe the operational rules and illustrate the method with several examples. The method of brackets reduce ..."
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Cited by 3 (2 self)
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Abstract. A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for the evaluation of Feynman diagrams. We describe the operational rules and illustrate the method with several examples. The method of brackets reduces the evaluation of a large class of definite integrals to the solution of a linear system of equations. 1.
Renormalization and effective field theory
, 2011
"... This is a preliminary version of the book Renormalization and Effective Field Theory published by the American Mathematical Society (AMS). This preliminary version is made available with the permission of the AMS and may not be changed, edited, or reposted at any other website without explicit writt ..."
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Cited by 3 (0 self)
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This is a preliminary version of the book Renormalization and Effective Field Theory published by the American Mathematical Society (AMS). This preliminary version is made available with the permission of the AMS and may not be changed, edited, or reposted at any other website without explicit written permission from the author and the AMS. Author's preliminary version made available with permission of the publisher, the American Mathematical SocietyContents
Spectral triples on the superVirasoro algebra
"... We construct infinite dimensional spectral triples associated with representations of the superVirasoro algebra. In particular the irreducible, unitary positive energy representation of the Ramond algebra with central charge c and minimal lowest weight h = c/24 is graded and gives rise to a net of ..."
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We construct infinite dimensional spectral triples associated with representations of the superVirasoro algebra. In particular the irreducible, unitary positive energy representation of the Ramond algebra with central charge c and minimal lowest weight h = c/24 is graded and gives rise to a net of even θsummable spectral triples with nonzero Fredholm index. The irreducible unitary positive energy representations of the NeveuSchwarz algebra give rise to nets of even θsummable generalised spectral triples where there is no Dirac operator but only a superderivation.
Definite integrals by the method of brackets. Part 1
 Adv. Appl. Math
, 2010
"... Abstract. A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for the evaluation of Feynman diagrams. We describe the operational rules and illustrate the method with several examples. The method of brackets reduce ..."
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Cited by 1 (1 self)
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Abstract. A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for the evaluation of Feynman diagrams. We describe the operational rules and illustrate the method with several examples. The method of brackets reduces the evaluation of a large class of definite integrals to the solution of a linear system of equations. 1.
Spectral action on SUq(2)
, 803
"... The spectral action on the equivariant real spectral triple over A ( SUq(2) ) is computed explicitly. Properties of the differential calculus arising from the Dirac operator are studied and the results are compared to the commutative case of the sphere S 3. ..."
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Cited by 1 (1 self)
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The spectral action on the equivariant real spectral triple over A ( SUq(2) ) is computed explicitly. Properties of the differential calculus arising from the Dirac operator are studied and the results are compared to the commutative case of the sphere S 3.