Feynman amplitudes, periods, and motives : International Research Conference on Periods and Motives : a modern perspective on renormalization : July 2-6, 2012, Institute de Ciencias Matemáticas, Madrid, Spain / Luis Álvarez-Cónsul, Jose Ignacio Burgos-Gil, Kurusch Ebrahimi-Fard, editors.

Format:

Book

Conference/Event

Author/Creator:

International Research Workshop on Periods and Motives - A Modern Perspective on Renormalization, Corporate Author.

Contributor:

Álvarez-Cónsul, Luis, 1970- editor.

Burgos Gil, José I. (José Ignacio), 1962- editor.

Ebrahimi-Fard, Kurusch, 1973- editor.

Conference Name:

International Research Workshop on Periods and Motives - A Modern Perspective on Renormalization (2012 : Madrid, Spain), issuing body.

International Research Workshop on Periods and Motives - A Modern Perspective on Renormalization

Series:

Contemporary mathematics (American Mathematical Society). 0271-4132 648

Contemporary mathematics, 648 0271-4132

Language:

English

Subjects (All):

Mathematical physics--Congresses.

Mathematical physics.

Perturbation (Quantum dynamics)--Congresses.

Perturbation (Quantum dynamics).

Perturbation (Mathematics)--Congresses.

Perturbation (Mathematics).

Quantum field theory--Congresses.

Quantum field theory.

Physical Description:

1 online resource (302 pages) : illustrations.

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2015]

Language Note:

English

Summary:

This volume contains the proceedings of the International Research Workshop on Periods and Motives-A Modern Perspective on Renormalization, held from July 2-6, 2012, at the Instituto de Ciencias Matemáticas, Madrid, Spain. Feynman amplitudes are integrals attached to Feynman diagrams by means of Feynman rules. They form a central part of perturbative quantum field theory, where they appear as coefficients of power series expansions of probability amplitudes for physical processes. The efficient computation of Feynman amplitudes is pivotal for theoretical predictions in particle physics. Periods are numbers computed as integrals of algebraic differential forms over topological cycles on algebraic varieties. The term originated from the period of a periodic elliptic function, which can be computed as an elliptic integral. Motives emerged from Grothendieck's "universal cohomology theory", where they describe an intermediate step between algebraic varieties and their linear invariants (cohomology). The theory of motives provides a conceptual framework for the study of periods. In recent work, a beautiful relation between Feynman amplitudes, motives and periods has emerged. The articles provide an exciting panoramic view on recent developments in this fascinating and fruitful interaction between pure mathematics and modern theoretical physics.

Contents:

Cover

Title page

Contents

Preface

A note on twistor integrals

1. Introduction

2. Linear Algebra

3. The Twistor Integral

4. Proof of theorem 1.1

References

Multiple polylogarithms and linearly reducible Feynman graphs

2. Multiple Polylogarithms and Feynman Integrals

3. Linear Reducibility

4. Towards a Classification by Critical Minors

5. Conclusions

Comparison of motivic and simplicial operations in mod- -motivic and étale cohomology

2. Cohomology of the classifying space for a finite group

3. The total power operations: I

4. The total power operations: II

5. Comparison with the operadic definition of simplicial cohomology operations: properties of simplicial operations

6. Comparison between the motivic and simplicial operations

7. Cohomological operations that commute with proper push-forwards and Examples

On the Broadhurst-Kreimer generating series for multiple zeta values

2. Period polynomials and the special depth filtration

3. Distributivity conjecture and Broadhurst-Kreimer dimensions

4. Shuffle subspaces of ℱ

5. Proof of Theorem 3.4.

6. Proofs of Lemmas 5.2 and 5.3.

7. Multiple zeta values and their duals

Dyson-Schwinger equations in the theory of computation

2. Primitive recursive functions and the Hopf algebra of flow charts

3. Flow charts, templates, and algorithms

4. Dyson-Schwinger equations in the Hopf algebra of flow charts

5. Operadic viewpoint

6. Renormalization of the halting problem

Acknowledgment

Scattering amplitudes, Feynman integrals and multiple polylogarithms

2. Scattering amplitudes and Feynman integrals

3. Feynman integrals and multiple polylogarithms.

4. Functional equations for multiple polylogarithms

5. The Hopf algebra of multiple polylogarithms and Feynman integrals

6. Conclusion

Equations D3 and spectral elliptic curves

2. Determinantal differential equations

3. The Beukers-Zagier equation as a D2 equation

4. Modular D2 equations

5. Differential equations of type D3

6. Nondegenerate modular D3 equations

7. All solutions of the multiplicativity equations for D3

8. From D2's to D3's

Quantum fields, periods and algebraic geometry

2. Graphs and algebras

3. Feynman Rules

4. Examples

5. Feynman rules from a Lie viewpoint

Acknowledgments

Renormalization, Hopf algebras and Mellin transforms

Motivation: The renormalization problem

1. Notations and preliminaries

2. Finiteness of renormalization by kinematic subtraction

3. Regularization and Mellin transforms

4. Hopf algebra morphisms and the renormalization group

5. Locality, finiteness and minimal subtraction

6. Dyson-Schwinger equations and correlation functions

7. Extensions towards \qft

8. Summary

Appendix A. The Hopf algebra of rooted trees

Appendix B. The Hopf algebra of polynomials

Appendix C. The Dynkin operator D=S*Y

Multiple zeta value cycles in low weight

2. Combinatorial situation

3. Algebraic cycles

4. Parametric and combinatorial representation for the cycles: trees with colored edges

5. Bar construction settings

6. Integrals and multiple zeta values

Periods and Hodge structures in perturbative quantum field theory

1. Periods

2. Hodge structures

3. Picard-Fuchs equations

Some combinatorial interpretations in perturbative quantum field theory

1. Introduction.

2. Dyson-Schwinger equations and the chord diagram expansion

3. Denominator reduction and special changes of variables

Back Cover.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-2727-3