The Genesis of Feynman Diagrams: 26 (Archimedes)

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Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number theoretically meaningful Hopf algebra. This can be modelled alternatively by employing Rota—Baxter operators. We stress the need for a characteristic-free development where possible, to have a sound starting point for generalizations of the algebraic structures.

The last section provides three key applications: symmetric function theory, quantum matrix mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural.

This representation uses a toy quantum field theory with infinitely many propagators and interaction vertices. The propagators involve Bernoulli polynomials and Clausen functions to arbitrary orders. The Feynman integrals of this model can be decomposed in terms of an algebra of elementary vertex integrals whose structure we investigate. We derive a large class of relations between multiple zeta values, of arbitrary lengths and weights, using only a certain set of graphical manipulations on Feynman diagrams.

Further uses and possible generalizations of the model are pointed out. In other words, Feynman diagrams evaluate to Multiple Zeta Values in all cases. This proves a recent conjecture of Connes-Kreimer, and others including Broadhurst and Kontsevich. This is a new construction which provides illumination to the relations between zeta values, associators, Feynman diagrams and moduli spaces. An immediate implication of our Main Theorem is that by applying Terasoma's result and using the construction of our Selberg integral-rooted trees functional, we prove that the Hermitian matrix integral as discussed in Mulase [18] evaluates to a Multiple Zeta Value in all 3 cases: Asymptotically, the limit as N goes to infinity, and in general.

Furthermore, this construction provides for a positive resolution to Goncharov's conjecture see [7] pg. The Selberg integral functional can be extended to map the special values to depth m multiple polylogarithms on X. One of the key steps in this process is to recognize the roles that zeta functions play in various arenas using transform methods. Other logical connections are provided by the the appearance of the Drinfeld associator, Hopf algebras, and techniques of conformal field theory and braid groups.

These recurring themes are subtly linked in a vast scheme of a logically woven tapestry. An immediate application of this framework is to provide an answer to a question of Kontsevich regarding the appearance of Drinfeld type integrals and in particular, multiple zeta values in: a Drinfeld's work on the KZ equation and the associator; b Etingof-Kazhdan's quantization of Poisson-Lie algebras; c Tamarkin's proof of formality theorems; d Kontsevich's quantization of Poisson manifolds. Combinatorial arguments relating Feynman diagrams to Selberg integrals, multiple zeta values, and finally Poisson manifolds provide an additional step in this framework.

Along the way, we provide additional insight into the various papers and theorems mentioned above. This paper represents an overall introduction to work currently in progress. More details to follow. See our paper math. Solomon, G. Duchamp, P.

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Blasiak, A. Horzela, K. See cs. The difficulty inherent in the complexities of a fully-fledged field theory such as PQFT means that the essential beauty of the relationships between these areas can be somewhat obscured. Our intention is to display some, although not all, of these structures in the context of a simple zero-dimensional field theory; i.

The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs, analogous to the Feynman diagrams of PQFT, which we show possess a Hopf algebra structure. Our approach is based on the partition function for a boson gas. In a subsequent note in these Proceedings we sketch the relationship between the Hopf algebra of our simple model and that of the PQFT algebra.

Kaufmann and A. These are the Hopf algebras of Goncharov for multiple zeta values, that of Connes—Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common framework.

Abdesselam, A. Chandra and G. The construction includes that of the associated squared field and our result shows this squared field has a dynamically generated anomalous dimension which rigorously confirms a prediction made more than forty years ago, in an essentially identical situation, by K. We also prove a mild form of universality for the model under consideration. Our main innovation is that our rigourous renormalization group formalism allows for space dependent couplings. We derive the relationship between mixed correlations and the dynamical systems features of our extended renormalization group transformation at a nontrivial fixed point.

The key to our control of the composite field is a partial linearization theorem which is an infinite-dimensional version of the Koenigs Theorem in holomorphic dynamics. This is akin to a nonperturbative construction of a nonlinear scaling field in the sense of F. Wegner infinitesimally near the critical surface.

Our presentation is essentially self-contained and geared towards a wider audience. While primarily concerning the areas of probability and mathematical physics we believe this article will be of interest to researchers in dynamical systems theory, harmonic analysis and number theory. It can also be profitably read by graduate students in theoretical physics with a craving for mathematical precision while struggling to learn the renormalization group. Lipatov, A. Sabio Vera, V. Velizhanin and G. The concept of reflexivity in these polyhedra is reviewed and translated into the theory of reflexive numbers.

A new approach based on recurrence relations and Quantum Field Theory methods is applied to the simply-laced and quasi-simply-laced subsets of the reflexive numbers.

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In the correspondence between the reflexive vectors and Berger graphs the role played by the generalized Coxeter labels is shown to be important. We investigate the positive roots of some of the Berger graphs to guess the algebraic structure hidden behind them. One part of the anomalous dimension was calculated in a usual way with the help of Asymptotic Bethe Ansatz. The rest part, related with the wrapping effects, was reconstructed from known constraints with the help of methods from number theory.

Gayral, B. Iochum and D. It turns out that this question is very sensitive to the number-theoretical aspect of the deformation parameters. The central condition we use is of a Diophantine type. More generally, the importance of number theory is made explicit on a few examples. Ashok, F. Cachazo, E. We identify the Seiberg-Witten curve at these points with polynomial equations used to construct what Grothendieck called "dessins d'enfants" or "children's drawings" on the Riemann sphere.

At each loop order, the form factor is expressed as a linear combination of only a handful scalar integrals, with small integer coefficients. Working in dimensional regularisation, the expansion coefficients of each integral exhibit homogeneous transcendentality in the Riemann zeta-function. We find that the logarithm of the form factor reproduces the correct values of the cusp and collinear anomalous dimensions.

Eilenberg's concept of completeness for semirings. We show that a complete ground semiring, a system of fields on manifolds and a system of action functionals on these fields determine a positive TFT. The main feature of such a theory is a semiring-valued topologically invariant state sum that satisfies a gluing formula. The abstract framework has been carefully designed to cover a wide range of phenomena. For instance, we derive Polya's counting theory in combinatorics from state sum identities in a suitable positive TFT. Several other concrete examples are discussed, among them Novikov signatures of fiber bundles over spacetimes and arithmetic functions in number theory.

In the future, we will employ the framework presented here in constructing a new differential topological invariant that detects exotic smooth structures on spheres. So far no such operator was found. In this paper we show that the functional integrals associated with a hypothetical class of physical systems described by self-adjoint operators associated with bosonic fields whose spectra is given by three different sequence of numbers cannot be constructed. In other words we show that the generating functional of connected Schwinger functions of the associated quantum field theories cannot be constructed.

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In another token, I advance a probabilistic interpretation of Weil's positivity criterion, as opposed to the usual geometrical analogies or goals. This is the notion of correlation functions of a quantum field. Well-known advances in the field of knot invariants and many other geometric arenas have shown that these correlation functions can be used to represent intersection products or linking numbers and many other geometrical things. On the other hand they have of course a probabilistic interpretation being complex amplitudes and rigorous mathematical developments of constructive quantum field theory have as their goal the construction of probability measures on spaces of distributions, as considered above.

For these reasons, and other more precise reflections on the subject, moving towards quantum fields is an urgent goal. In this context it is reassuring that the "-log x " formulation of the Explicit Formula enables a few additional comments. Indeed as is well-known "-log x " is the propagator of the free Boson field in 2 dimensions here we look at the complex place and x so that we have both holomorphic and anti-holomorphic sectors.

But let us rather consider the non-archimedean completion K v , of the number field K. I believe that techniques and philosophy more organic to Quantum Fields will be most relevant. We will see in this paper that the function Julia's 'number theoretical gas' which provides a surprising natural 'physical' framework for certain manipulations on arithmetic functions. Aref'eva, I. We treat the Riemann zeta-function as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of p-adic strings and by recent works on stringy cosmological models.

We show that the Lagrangian for the zeta-function field is equivalent to the sum of the Klein-Gordon Lagrangians with masses defined by the zeros of the Riemann zeta-function. Quantization of the mathematics of Fermat-Wiles and the Langlands program is indicated. The Beilinson conjectures on the values of L-functions of motives are interpreted as dealing with the cosmological constant problem.

Possible cosmological applications of the zeta-function field theory are discussed. Dragovich, "On p -adic and adelic generalization of quantum field theory" , Nuclear Physics B - Proceedings Supplements "A brief review of p -adic and adelic quantum mechanics is presented, and a new approach to p -adic and adelic quantum field theory is proposed.

Path integral method turns out to be generic for quantum dynamics on both archimedean and nonarchimedean spaces. Some basic classical field properties of these scalar fields are obtained. In particular, some trivial solutions of the equations of motion and their tachyon spectra are presented. Field theory with Riemann zeta function nonlocality is also interesting in its own right. The corresponding new objects we call zeta scalar strings.

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Some basic classical field properties of these fields are obtained and presented in this paper. In particular, some solutions of the equations of motion and their tachyon spectra are studied. Field theory with Riemann zeta function dynamics is interesting in its own right as well. As a starting point, we seek to construct the most general quantum theory on the rational numbers. Very general arguments land us in the adeles. Quantum amplitudes generically will not factorize into separate p -adic sectors. The vacuum amplitude, in particular, factorizes only in a linear regime.

We conclude with implications for the cosmological constant and adelic string sigma models. Brekke and P. Freund, " p -adic numbers in physics", Physics Reports , Vladimirov, I. Volovich , E. The literature covered includes: S. Albeverio, A.

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Khrennikov, and R. Cianci, "A representation of quantum field Hamiltonian in a p -adic Hilbert space", Theor. Physics no. Albeverio and A. Khrennikov, "A regularization of quantum field Hamiltonians with the aid of p -adic numbers", Acta Appl. Khrennikov, "Statistical interpretation of p -adic valued quantum field theory", Dokl.

Nauk , no. Chau and W. Smirnov, "Renormalization in p -adic quantum field theory", Mod. A6 V.

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Lerner and M. Missarov, "Scalar models in p -adic quantum field theory and hierarchical models", Theor. Nishino, Y. Okada and M. Ubriaco, "Effective field theory and a p -adic string", Phys. D40 Ubriaco, "Fermions on the field of p -adic numbers", Phys. D41 M. Elizalde, "Spectral zeta functions in non-commutative spacetime" , Nucl. This poses a challenge to the zeta-function regularization procedure. Cognola, E. Elizalde and S.

Zerbini, "Fluctuations of quantum fields via zeta function regularization" , Phys. D65 [abstract:] "Explicit expressions for the expectation values and the variances of some observables, which are bilinear quantities in the quantum fields on a D -dimensional manifold, are derived making use of zeta function regularization. Some illustrating examples are worked through. Elizalde, "Some uses of zeta-regularization in quantum gravity and cosmology" , Grav. This book is meant as a guide for the student of this subject.

Everything is explained in detail, in particular the mathematical difficulties and tricky points, and several applications are given to show how the procedure works in practice e. Casimir effect, gravity and string theory, high-temperature phase transition, topological symmetry breaking. The formulas some of which are new can be used for accurate numerical calculations. The book is to be considered as a basic introduction and a collection of exercises for those who want to apply this regularization procedure in practice.


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Elizalde, S. Odintsov, A. Romeo and S. Zerbini, Zeta Regularization Techniques With Applications World Scientific, "This book is the result of several years of work by the authors on different aspects of zeta functions and related topics. The aim is twofold.


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On one hand, a considerable number of useful formulas, essential for dealing with the different aspects of zeta-function regularization analytic continuation, asymptotic expansions , many of which appear here, in book format, for the first time are presented. On the other hand, the authors show explicitly how to make use of such formulas and techniques in practical applications to physical problems of very different nature.

Virtually all types of zeta functions are dealt with in the book. Moretti and D. Iellici, "Zeta-function regularization and one-loop renormalization of field fluctuations in curved space-times" , Phys. B "A method to regularize and renormalize the fluctuations of a quantum field in a curved background in the zeta-function approach is presented. The method produces finite quantities directly and finite scale-parametrized counterterms at most.

These finite counterterms are related to the presence of a particular pole of the effective-action zeta function as well as to the heat kernel coefficients. Wuethrich delves into the creative thinking of Richard Feynman and takes the reader on an exciting tour that leads from the early, groping attempts to understand quantum electrodynamics using inadequate graphical representations to the powerful diagrams that bear Feynman's name.

Guided by Wuethrich's reconstruction of this path, we develop a deeper understanding of why the Feynman diagrams are what they are: they mediate and represent a deep conceptual change that was brought about by modern QED. Wuethrich's approach to a better understanding of the core concepts of QED by focussing on the emergence of its key representational tool is a masterful combination of historical reconstruction and conceptual analysis. Tilman Sauer, Einstein Papers Project, California Institute of Technology, Pasadena, CA,, From the reviews: Wuethrich delves into the creative thinking of Richard Feynman and takes the reader on an exciting tour that leads from the early, groping attempts to understand quantum electrodynamics using inadequate graphical representations to the powerful diagrams that bear Feynman's name.

Alexander Blum, Philosophy in Review, Vol. Feynman in the post-war years Stefan Weinzierl, Mathematical Reviews, Issue j The aim of this book is to reconstruct the route that led Feynman, between approximately and , to devise his new methods of diagrams and to evaluate what was achieved.

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