Fortschritte der Physik / Progress of Physics: Band 14, Heft 8/9 1966 [Reprint 2021 ed.] 9783112500361, 9783112500354

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE DER PHYSIKALISCHEN GESELLSCHAFT IN DER DEUTSCHEN DEMOKRATISCHEN REPUBLIK VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

B A N D 14 • H E F T 8/9 • 1966

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W. GÛTTINGER : Generalized Functions and Dispersion Relations in Physios

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Fortschritte der Physik 14, 483—602 (1966)

Generalized Functions and Dispersion Relations in Physics*) W . GÜTTINGEB

Institute of Theoretical Physics,

University of Munich

München

Preface The aim of this article ist to present in a coherent and self-contained fashion the recent developments of the theory of generalized functions and their applications to dispersion relations, system theory and elementary particle physics. Details of the presented material, the major part of which is published here for the first time, may be inferred from the table of contents. I have tried to make the various chapters reflect the actual situation in this field of research. However, there is no attempt at completeness either in coverage or references. The paper is based in part on lectures given by the author at the University of Munich. I t is a pleasure to acknowledge many helpful discussions with Drs. C. Braga, R. Blomer, F. Boop, A. P. Contogouris, H. Cornille, V. Glaser, M. B. Halpern, E. Pfaffelhuber, A. Rieckers and G. Wiegand. Particular thanks are due to the Deutsche Forschungsgemeinschaft for a research grant and to the Theoretical Division at CERN for the kind hospitality. Finally I wish to express my appreciation to the editorial staff of the publisher for the efficient handling of a difficult manuscript.

Contents Introduction

485

Chapter I. Definition of Generalized Functions

486

1.1. 1.2. 1.3. 1.4.

Test Functions Test Function Spaces Generalized Functions Local Properties of Generalized Functions

Chapter II. Operations with Generalized Functions 2.1. 2.2. 2.3. 2.4.

Addition and Multiplication by a Number and by a Function . Differentiation of Generalized Functions Series of Generalized Functions Generalized Functions Depending Analytically on a Parameter

) Supported in part by the Deutsche Forschungsgemeinschaft. 36

Zeitschrift „Fortschritte der Physik", Heft 8/9

486 487 490 493 495 495 496 498 500

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W . GÜTTINGER

2.5. Transformation of Variables 1. Regular Transformations 2. Singular Transformations 2.6. Convolution of Generalized Functions 2.7. Division of a Generalized Function by a Function 2.8. Products of Generalized Functions

501 501 502 505 506 509

Chapter I I I . Fourier Transforms of Generalized Functions 512 3.1. Fourier Transforms of Test Functions 512 3.2. Fourier Transforms of Generalized Functions 513 3.3. Fourier Transforms and Analytic Functionals 517 1. The Fourier Transform of a Taylor Series 517 2. Infinite Series of Derivatives of Delta-Functions and Fourier Transforms of Rapidly Increasing Functions 520 3.4. Laplace-Transforms of Generalized Functions 522 • Chapter IV. Singular Generalized Functions 4.1. Generalized Functions with Algebraic Singularities. A Single Variable 4.2. Integral Representation of Singular Generalized Functions 4.3. Generalized Functions with Algebraic Singularities in R„ 4.4. Fourier Transforms of Generalized Functions with Algebraic Singularities in B n . 4.5. Products of Causal Generalized Functions. Convolution Algebras in Momentum Space Chapter V. Positive and Positive-Definite Generalized Functions 5.1. Positive Generalized Functions 5.2. Positive-Definite Generalized Functions Chapter VI. Dispersion Relations and Boundary Values of Analytic Functions 6.1. Boundary Values of Analytic Functions 6.2. Analytic Functionals and Boundary Values of Analytic Functions 6.3. Hilbert-Stieltjes Transforms of Generalized Functions 6.4. Spectral Representations and Dispersion Relations 6.5. Do Dispersion Relations Require Substractions?

534 535 535 536

. . . .

Chapter VII. Differential Equations of Physics 7.1. Green's Functions 7.2. Plane-Wave Expansions and Potential Theory in B n 7.3. Initial Value Problems. Feynman Kernels 7.4. Wave Equations in R„. Generalized Bochner Formula. Cerenkov Radiation. 7.5. Singular Differential Equations and Instabilities Chapter V I I I . Passive System S-Matrix Theory 8.1. Passive Systems in Terms of an S-Matrix 8.2. Dispersion Theory of Passive (Unitary) Systems 8.3. Discontinuous Functionals and Instable Systems

522 522 528 530 532

539 539 543 547 549 552

554 554 557 559 . . 561 566 567 567 570 573

Chapter I X . Lorentz Invariant Generalized Functions 574 9.1. Definition and Properties of Lorentz Invariant Generalized Functions . . . . . 575 9.2. Fourier Transforms of Lorentz Invariant Generalized Functions and Related Problems 578 Chapter X . Rigged Hilbert Spaces and All That 10.1. Formal Series and Quantum Mechanics 10.2. Rigged Hilbert Spaces and Group Representations. SL(2, c) 10.3. Spectral Decompositions and Analytic Functionals

580 581 583 585

Generalized Functions and Dispersion Relations

485

Chapter XI. Problems of Quantum Field Theory and S-Matrix Theory 585 11.1. Generalized Lehmann Representation and Dispersion Relations without Subtractions 585 11.2. High Energy Bound on Scattering Amplitudes 588 11.3. Renormalizable Theories. JZ"1 as a Differential Operator 588 11.4. Unrenormalizable Theories 590 11.5. Peratization. Singular Potentials and Singular NjD Equations 591 Appendix 1. Summary of Fundamental Formula

594

Appendix 2. Fourier Transforms and Spectral Representations

597

References

601

Introduction Perfectly rigorous mathematical schemes do not always satisfy the physicist. For, owing to the precise delimitation between what is allowed and what is not, such schemes are often too narrow to give free play to formal entities and techniques the success of which was uncontestable. I t is therefore not surprising that new, strange but efficient objects and methods have been conceived by mathematically undisciplined spirits, mostly by impatient theoretical physicists. Defended as a kind of shorthand or heuristic means for obtaining tentative solutions, such entities and techniques have become popular by Dirac's delta-function, Hadamard's finite parts of divergent integrals and Heaviside's symbolic calculus. These conceptions are at the origin of what is now being called the theory of generalized functions. Physical necessities are always within the mathematical possibilities. During the last decades, however, the gap between mathematics and physics has steadily widened and it has taken a long time till L. Schwartz generalized classical analysis by his theory of distributions to include singular functions and operations with them. Right now, also distribution theory becomes too narrow to satisfy all the needs of the physicists. Thus the story starts anew: Unpleasant subtraction constants in dispersion relations, essential singularities in physical systems and broken symmetries, non-compact groups and all that are urging theoretical physicists to look for further generalizations of familiar mathematical objects, and some progress in this direction has already been made by Gel'fand and others. The ultimate objective of those conceptions ought to be the determination of a new and more appropriate language in which to express the mathematical content of any particular physical theory. The present article is designed to serve this purpose. I t develops the theory and techniques of generalized functions in an intimate connection with current problems of theoretical physics, in particular, with dispersion relations, passive systems and elementary particle physics. Physics also serves as a guide to the concept of generalied functions: When one asks for the properties of a physical system (a 'black box') one operates on it with a testing body and watches how the system responds to this stimulus. This response is a number / (cp) depending on both the object / and the testing body 0. Since the functions

) = (/,