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Noncommutative Geometry, Quantum Fields, and Motives, (2008)

by A Connes, M Marcolli
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From polygons and symbols to polylogarithmic functions

by Claude Duhr, Herbert Gangl, John R. Rhodes - JHEP 1210 (2012) 075, arXiv:1110.0458 [math-ph
"... Abstract: We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an ..."
Abstract - Cited by 43 (8 self) - Add to MetaCart
Abstract: We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an associated rooted decorated polygon. We also outline a systematic approach to constructing a function corresponding to a given symbol, and illus-trate it in the particular case of harmonic polylogarithms up to weight four. Furthermore, part of the ambiguity of this process is highlighted by exhibiting a family of non-trivial elements in the kernel of the symbol map for arbitrary weight.
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...espondence between Broadhurst and Deligne [84] resulting e.g. in ref. [85], work of Belkale–Brosnan [86] or more recently by Brown [87] and others. One should also mention work of Connes and Marcolli =-=[88]-=- in this direction. – 5 – thereof [26, 27]. As harmonic polylogarithms are just a special case of the multiple polylogarithms introduced at the beginning of this section, all HPL’s through weight thre...

A Topos for Algebraic Quantum Theory

by Chris Heunen, Nicolaas P. Landsman, Bas Spitters - 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 ..."
Abstract - Cited by 31 (5 self) - Add to MetaCart
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 topos-theoretical 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 self-adjoint 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 topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.

Renormalization and effective field theory

by Kevin Costello , 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 ..."
Abstract - Cited by 22 (1 self) - Add to MetaCart
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

Scalar curvature for the noncommutative two torus

by Farzad Fathizadeh - J. Noncommut. Geom
"... Abstract. The scalar curvature for the noncommutative four torus T4Θ, where its flat geometry is conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3-sphere. This method is more convenient since it does not require the ..."
Abstract - Cited by 16 (5 self) - Add to MetaCart
Abstract. The scalar curvature for the noncommutative four torus T4Θ, where its flat geometry is conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3-sphere. This method is more convenient since it does not require the rearrangement lemma and it is advantageous as it explains the simplicity of the final functions of one and two variables, which describe the curvature with the help of a modular automorphism. In particular, it readily allows to write the function of two variables as the sum of a finite difference and a finite product of the one variable function. The curvature formula is simplified for dilatons of the form sp, where s is a real parameter and p ∈ C∞(T4Θ) is an arbitrary projection, and it is observed that, in contrast to the two dimensional case studied by A. Connes and H. Moscovici, unbounded functions of the parameter s appear in the final formula. An explicit formula for the gradient of the analog
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...mately related to twisted spectral triples and we refer to [8, 10, 9, 27, 26, 21] for detailed discussions. Also it is closely related to the spectral action computations in the presence of a dilaton =-=[5, 6]-=-. For noncommutative four tori T4Θ, the scalar curvature is computed in [15] and it is shown that flat metrics are the critical points of the analog of the Einstein-Hilbert action. Also noncommutative...

Spectral triples on the super-Virasoro algebra

by Sebastiano Carpi, Robin Hillier, Yasuyuki Kawahigashi, Roberto Longo
"... We construct infinite dimensional spectral triples associated with representations of the super-Virasoro 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 ..."
Abstract - Cited by 16 (8 self) - Add to MetaCart
We construct infinite dimensional spectral triples associated with representations of the super-Virasoro 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 θ-summable spectral triples with non-zero Fredholm index. The irreducible unitary positive energy representations of the Neveu-Schwarz algebra give rise to nets of even θ-summable generalised spectral triples where there is no Dirac operator but only a superderivation.
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...ility). The operator Q is called the supercharge operator, its square the Hamiltonian. Remark 3.2. Restricting to the even subalgebra A+ of A, the above definition is essentially Connes [6] (see also =-=[7]-=-) definition of a (even) spectral triple (A+, H,Q). This is the fundamental object for index theorems and evaluating on K-theory elements. In this case the supercharge Q is traditionally called Dirac ...

Suijlekom, The noncommutative geometry of Yang-Mills fields

by Jord Boeijink, Walter, D. Van Suijlekom - J. Geom. Phys
"... ar ..."
Abstract - Cited by 14 (3 self) - Add to MetaCart
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.../ is the Dirac operator on the spinor bundle. For even dimensional spin-manifolds there exists a grading γ on L2(M,S). A spectral triple can have additional structure such as reality. Definition 2.3 (=-=[11]-=-, Definition 1.124). A real structure on a spectral triple (A,H, D) is an anti-unitary operator J : H → H, with the property that J2 = ε, JD = ε′DJ, and Jγ = ε′′γJ, (even case), where the numbers ε,ε′...

Feynman motives and deletion-contraction relations

by Paolo Aluffi, Matilde Marcolli
"... We prove a deletion-contraction 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 ..."
Abstract - Cited by 13 (8 self) - Add to MetaCart
We prove a deletion-contraction 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 deletion-contraction 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.

The Witt construction in characteristic one and quantization

by Alain Connes - In Noncommutative geometry and global analysis, volume 546 of Contemp. Math , 2011
"... Abstract. We develop the analogue of the Witt construction in characteristic one. We construct a functor from pairs (R, ρ) of a perfect semi-ring R of characteristic one and an element ρ > 1 of R to real Banach algebras. We find that the entropy function occurs uniquely as the analogue of the Te ..."
Abstract - Cited by 13 (4 self) - Add to MetaCart
Abstract. We develop the analogue of the Witt construction in characteristic one. We construct a functor from pairs (R, ρ) of a perfect semi-ring R of characteristic one and an element ρ > 1 of R to real Banach algebras. We find that the entropy function occurs uniquely as the analogue of the Teichmüller polynomials in characteristic one. We then apply the construction to the semi-field R max + which plays a central role in idempotent analysis and tropical geometry. Our construction gives the inverse process of the "dequantization" and provides a first hint towards an extension R un of the field of real numbers relevant both in number theory and quantum physics.
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... construction is intimately related to idempotent analysis 5 (cf. [13], [16]). In fact our analogue in characteristic one of the Witt construction provides the inverse process of the “dequantization” of idempotent analysis, a fact which justifies the word “quantization” appearing in the title of this paper. In conclusion we conjecture that the extension Run of R is the natural home for the “values” of the many ~-dependent physical quantities arising in quantum field theory. This fits with the previous understanding of renormalization from the Riemann-Hilbert correspondence (cf. [4], [5], [8], [9]). 3. Sum of Teichmuller representatives Our goal in this section is to recall a formula which goes back to Teichmuller [20], for the sum of Teichmuller representatives in the context of strict p-rings. We begin by recalling briefly the simplest instance of the polynomials with integral coefficients which express the addition in the Witt theory (We refer to [19] and [17] for a quick introduction to that theory). One defines the polynomials with integral coefficients Sn(x, y) ∈ Z[x, y] by the equality (1− tx)(1− ty) = ∞∏ n=1 (1− Sn(x, y)tn) (28) The first few are of the form S1 = x+ y S2 = −...

EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS

by Jeffrey C. Lagarias , 2013
"... This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Eul ..."
Abstract - Cited by 12 (1 self) - Add to MetaCart
This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler–Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler’s constant.
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...[70]. There is a statistical mechanics interpretation of the Riemann zeta function as a partition function given in Bost and Connes [40], with subsequent developments described in Connes and Marcolli =-=[64]-=-. For some other views on arithmetical dynamical systems related to zeta functions, see Lagarias [190], [192] and Lapidus [197, Chapter 5]. Here we note the coincidence that in formulations of the Rie...

Quantum statistical mechanics, L-series and anabelian geometry

by Gunther Cornelissen, Matilde Marcolli , 2010
"... 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 auto ..."
Abstract - Cited by 11 (7 self) - Add to MetaCart
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 Neukirch-Uchida 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-
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.... . . . . . . group scheme ofn-th roots of unity (n integer) Part A. QSM-ISOMORPHISM OF NUMBER FIELDS 1. Isomorphism of QSM systems We recall some definitions and refer to [7], [11], and Chapter 3 of =-=[12]-=- for more information and for some physics background. After that, we introduce isomorphism of QSM-systems, and prove they preserve KMS-states (cf. infra). 1.1. Definition. A quantum statistical mecha...

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