Ebook Quantum field theory II: Quantum electrodynamics - Part 1

Part 1 of ebook Quantum field theory II: Quantum electrodynamics provide readers with content about: mathematical principles of modern natural philosophy; the basic strategy of extracting finite information from infinities – ariadne’s thread in renormalization theory; the power of combinatorics; the strategy of equivalence classes in mathematics; basic ideas in classical mechanics;...

Quantum Electrodynamics

Quantum Field Theory II: Quantum Electrodynamics

A Bridge between Mathematicians

and Physicists

for Mathematics in the Sciences

Library of Congress Control Number: 2006929535

Mathematics Subject Classification (2000): 35-XX, 47-XX, 49-XX, 51-XX, 55-XX, 81-XX, 82-XX c

 2009 Springer-Verlag Berlin Heidelberg

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IN GRATITUDE

And God said, Let there be light; and there was light.

Genesis 1,3

Light is not only the basis of our biological existence, but also an essentialsource of our knowledge about the physical laws of nature, ranging fromthe seventeenth century geometrical optics up to the twentieth centurytheory of general relativity and quantum electrodynamics

Folklore

Don’t give us numbers: give us insight!

A contemporary natural scientist to a mathematician

The present book is the second volume of a comprehensive introduction tothe mathematical and physical aspects of modern quantum field theory whichcomprehends the following six volumes:

Volume I: Basics in Mathematics and Physics

Volume II: Quantum Electrodynamics

Volume III: Gauge Theory

Volume IV: Quantum Mathematics

Volume V: The Physics of the Standard Model

Volume VI: Quantum Gravitation and String Theory

It is our goal to build a bridge between mathematicians and physicists based

on the challenging question about the fundamental forces in

• macrocosmos (the universe) and

• microcosmos (the world of elementary particles).

The six volumes address a broad audience of readers, including both graduate and graduate students, as well as experienced scientists who want

under-to become familiar with quantum field theory, which is a fascinating under-topic inmodern mathematics and physics

For students of mathematics, it is shown that detailed knowledge of thephysical background helps to motivate the mathematical subjects and to

discover interesting interrelationships between quite different mathematicaltopics For students of physics, fairly advanced mathematics are presented,which is beyond the usual curriculum in physics The strategies and thestructure of the six volumes are thoroughly discussed in the Prologue toVolume I In particular, we will try to help the reader to understand thebasic ideas behind the technicalities In this connection, the famous ancientstory of Ariadne’s thread is discussed in the Preface to Volume I In terms

of this story, we want to put the beginning of Ariadne’s thread in quantumfield theory into the hands of the reader

The present volume is devoted to the physics and mathematics of light.

It contains the following material:

Part I: Introduction

• Chapter 1: Mathematical Principles of Modern Natural Philosophy

• Chapter 2: The Basic Strategy of Extracting Finite Information from

Infinities – Ariadne’s Thread in Renormalization Theory

• Chapter 3: The Power of Combinatorics

• Chapter 4: The Strategy of Equivalence Classes in Mathematics

Part II: Basic Ideas in Classical Mechanics

• Chapter 5: Geometrical Optics

• Chapter 6: The Principle of Critical Action and the Harmonic

Oscilla-tor as a Paradigm

Part III: Basic Ideas in Quantum Mechanics

• Chapter 7: Quantization of the Harmonic Oscillator – Ariadne’s Thread

in Quantization

• Chapter 8: Quantum Particles on the Real Line – Ariadne’s Thread in

Scattering Theory

• Chapter 9: A Glance at General Scattering Theory.

Part IV: Quantum Electrodynamics (QED)

• Chapter 10: Creation and Annihilation Operators

• Chapter 11: The Basic Equations in Quantum Electrodynamics

• Chapter 12: The Free Quantum Fields of Electrons, Positrons, and

Photons

• Chapter 13: The Interacting Quantum Field, and the Magic Dyson Series for the S-Matrix

• Chapter 14: The Beauty of Feynman Diagrams in QED

• Chapter 15: Applications to Physical Effects

Part V: Renormalization

• Chapter 16: The Continuum Limit

• Chapter 17: Radiative Corrections of Lowest Order

• Chapter 18: A Glance at Renormalization to all Orders of Perturbation

Theory

• Chapter 19: Perspectives

We try to find the right balance between the mathematical theory and itsapplications to interesting physical effects observed in experiments In par-ticular, we do not consider purely mathematical models in this volume.

It is our philosophy that the reader should learn quantum field theory

by studying a realistic model, as given by quantum electrodynamics.

Let us discuss the main structure of the present volume In Chapters 1through 4, we consider topics from classical mathematics which are closelyrelated to modern quantum field theory This should help the reader to un-derstand the basic ideas behind quantum field theory to be considered inthis volume and the volumes to follow In Chapter 1 on the mathematicalprinciples of modern natural philosophy, we discuss

• the infinitesimal strategy due to Newton and Leibniz,

• the optimality principle for processes in nature (the principle of critical action)

and the calculus of variations due to Euler and Lagrange, which leads to thefundamental differential equations in classical field theory,

• the propagation of physical effects and the method of the Green’s function,

• harmonic analysis and the Fourier method for computing the Green’s functions,

• Laurent Schwartz’s theory of generalized functions (distributions) which is

re-lated to the idea that the measurement of physical quantities by devices is based

on averaging,

• global symmetry and conservation laws,

• local symmetry and the basic ideas of modern gauge field theory, and

• the Planck quantum of action and the idea of quantizing classical field theories.

Gauge field theory is behind both

• the Standard Model in elementary particle physics and

• Einstein’s theory of gravitation (i.e., the theory of general relativity).

In quantum field theory, a crucial role is played by renormalization In terms

of physics, this is based on the following two steps:

• the regularization of divergent integrals, and

• the computation of effective physical parameters measured in experiments (e.g.,

the effective mass and the effective electric charge of the electron)

Renormalization is a highly technical subject For example, the full proof

on the renormalizability of the electroweak sector of the Standard Model inparticle physics needs 100 pages This can be found in:

E Kraus, Renormalization of the electroweak standard model to all orders,

Annals of Physics 262 (1998), 155–259.

Whoever wants to understand quantum field theory has to understand theprocedure of renormalization Therefore, the different aspects of renormal-ization theory will be studied in all of the six volumes of this series of mono-graphs This ranges from

• resonance phenomena for the anharmonic oscillator (classical bifurcation theory),

• the Poincar´e–Lindstedt series (including small divisors) in celestial mechanics,

• and the Kolmogorov–Arnold–Moser (KAM) theory for perturbed quasi-periodic

oscillations (e.g., in celestial mechanics) based on sophisticated iterative niques (the hard implicit function theorem)

tech-to the following fairly advanced subjects:

• the Feynman functional integral (the Faddeev–Popov approach),

• the Wiener functional integral (the Glimm–Jaffe approach),

• the theory of higher-dimensional Abelian integrals (algebraic Feynman integrals),

• Hopf algebras and Rota–Baxter algebras in combinatorics (the modern

vari-ant of the Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) approach due toKreimer),

• the Riemann–Hilbert problem and the Birkhoff decomposition (the Connes–

Kreimer approach),

• Hopf superalgebras (the Brouder–Fauser–Frabetti–Oeckl (BFFO) approach),

• characterization of physical states by cohomology and algebraic renormalization

(the Becchi–Rouet–Stora–Tyutin (BRST) approach),

• the Riesz–Gelfand theory of distribution-valued meromorphic functions

(con-struction of the Green’s functions),

• wave front sets and H¨ormander’s multiplication of distributions (the Stueckelberg–

Bogoliubov–Epstein–Glaser–Scharf approach),

• the Master Ward identity as a highly non-trivial renormalization condition and

the generalized Dyson–Schwinger equation (the D¨utsch–Fredenhagen approach),

• q-deformed quantum field theory (the Wess–Majid–Wachter–Schmidt approach

based on the q-deformed Poincar´ e group, quantum groups, and the q-analysis on specific classes of q-deformed quantum spaces),

• deformation of bundles and quantization (the Weyl–Flato–Sternheimer–Fedosov–

Kontsevich approach),

• microlocal analysis and renormalization on curved space-times (the Radzikowski–

Brunetti–Fredenhagen–K¨ohler approach),

• renormalized operator products on curved space-times (the Wilson–Hollands–

Wald approach to quantum field theory),

• natural transformations of functors in category theory and covariant quantum

field theory on curved space-time manifolds (the Brunetti–Fredenhagen–Verchapproach),

as well as

• one-parameter Lie groups and the renormalization group,

• attractors of dynamical systems in the space of physical theories (the Wilson–

Polchinski–Kopper–Rivasseau approach to renormalization based on the malization group),

renor-• the Master Ward Identity and the Stueckelberg–Petermann renormalization

group (the D¨utsch–Fredenhagen approach),

• motives in number theory and algebraic geometry, the Tannakian category, and

the cosmic Galois group as a universal (motivic) renormalization group (theConnes–Marcolli approach),

• noncommutative geometry and renormalization (the Grosse–Wulkenhaar

ap-proach)

The recent work of Alain Connes, Dirk Kreimer, and Matilde Marcolli showsconvincingly that renormalization is rooted in highly nontrivial mathemati-cal structures We also want to emphasize that the theory of many-particlesystems (considered in statistical physics and quantum field theory) is deeplyrooted in the theory of operator algebras This concerns

• von Neumann algebras (the von Neumann approach),

• C ∗-algebras (the Gelfand–Naimark–Segal approach),

• local nets of operator algebras (the Haag–Kastler approach) and,

• noncommutative geometry (the Connes approach).

As a warmup, we show in Chapter 2 that the regularization of divergentexpressions represents a main subject in the history of mathematics startingwith Euler in the eighteenth century In this connection, we will consider

• the regularization of divergent series, and

• the regularization of divergent integrals.

In particular, in Sect 2.1.3, we will discuss the classical Mittag–Leffler

theo-rem on meromorphic functions f If the function f has merely a finite number

of poles, then the method of partial fraction decomposition works well

How-ever, as a rule, this method fails if the function f has an infinite number of

poles In this case, Mittag–Leffler showed in the late nineteenth century thatone has to subtract special terms, which are called subtractions by physicists

The subtractions force the convergence of the infinite series.

This is the prototype of the method of iteratively adding subtractions in theBogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) approach to renormaliza-tion theory The corresponding iterative algorithm (called the Bogoliubov

R-operation) has to be constructed carefully (because of nasty overlapping

divergences) This was done by Nikolai Bogoliubov in the 1950s An ingeniousexplicit solution formula for this iterative method was found by Wolfhart Zim-

mermann in 1969 This is the famous Zimmermann forest formula In the

late 1990s, it was discovered by Dirk Kreimer that the sophisticated natorics of the Zimmermann forest formula can be understood best in terms

combi-of a Hopf algebra generated by Feynman diagrams By this discovery, themodern formulation of the BPHZ approach is based on both Hopf algebrasand Rota–Baxter algebras

As a warmup, in Chapter 3, we give an introduction to the modern binatorial theory, which was founded by Gian-Carlo Rota (MIT, Cambridge,Massachusetts) in the 1960s This includes both Hopf algebras and Rota–Baxter algebras

com-Surprisingly enough, it turns out that the Zimmermann forest mula is closely related to methods developed by Lagrange in the eigh- teenth century when studying the solution of the Kepler equation for the motion of planets in celestial mechanics.

for-In modern terminology, the Lagrange inversion formula for power series pansions is based on the so-called Fa`a di Bruno Hopf algebra.1 This will bestudied in Sect 3.4.3

ex-1

The Italian priest and mathematician Francesco Fa`a di Bruno (1825–1888) wasbeatified in 1988

In physics, symmetries are basic For describing symmetries in terms ofmathematics, there are two approaches based on

• groups and

• Hopf algebras.

In 1941, Heinz Hopf wanted to compute the cohomology of topological groups.Hopf discovered that the cohomology ring of topological groups is equippedwith an additional structure which is called a Hopf algebra structure today.This additional algebraic structure is based on the notion of the coproduct.Roughly speaking, the concept of Hopf algebra is dual to the concept ofgroup Hopf algebras are intimately related to quantum groups We will show

• the coproduct is related to the Leibniz product rule of differentiation, and

• the coinverse (also called the antipode) is related to the integration-by-parts

formula (see Sect 3.3.1)

The integration-by-parts formula is a special case of the general Stokes gral theorem, which lies at the heart of the duality between homology andcohomology in topology This duality plays a key role for the mathematicaldescription of processes in nature In particular, cohomology is deeply rooted

inte-in Maxwell’s theory of electrodynamics (see Sect 4.4.7)

Incredible cancellations When doing computations in renormalization

theory, as a big surprise, physicists and mathematicians encounter incrediblecancellations of a large amount of terms This dramatically simplifies thefinal result In terms of mathematics, a sophisticated combinatorics is behindthese cancellations The prototype for this is given by the Fa`a di Bruno Hopfalgebra mentioned above

The language of modern mathematics We do not assume that the

reader of this series of monographs is familiar with the language used in ern mathematics In this connection, we want to help the reader For example,many notions in advanced mathematics and hence in modern mathematicalphysics are based on mathematical operations applied to equivalence classes.For example, this concerns

mod-• the construction of quantum states as equivalence classes of elements of Hilbert

spaces (and the relation to projective geometry),

• the Gelfand–Naimark–Segal (GNS) construction for representing the elements of

an abstract C ∗-algebra as observables on a Hilbert space (the algebraic approach

to quantum theory),

• the Wightman reconstruction theorem for axiomatically defined quantum fields

(via the GNS-construction),

• moduli spaces of Riemann surfaces (modulo conformal equivalence) and physical

states in string theory

This leads to quotient spaces in algebra, analysis, geometry, and topology,which will be encountered again and again in this series of monographs (e.g.,

homology groups, cohomology groups, homotopy groups, and K-theory in

topology) Chapter 4 serves as an introduction to quotient structures in ematics and physics The idea of the quotient ring (modulo a fixed integer)

math-can be traced back at least to the Disquisitiones arithmeticae written by the

young Gauss (1777–1855) in 1801.2 In order to give the reader a feel for theusefulness of working with equivalence classes, we will consider the followingexamples:

• Gaussian quotient rings (modulo a prime number) and coding theory (as warmup

for quantum information),

• quotient fields and Heaviside’s symbolic method in electrical engineering (the

Mikusi´nski operational calculus),

• physical fields, observers, bundles, and cocycles,

• deformation, mapping classes, and topological charges,

• loop spaces and higher homotopy groups,

• the projective and the injective limit of mathematical structures (e.g., topological

The language of theoretical physics Chapters 5 through 9 are devoted

to the basic ideas of

• classical geometric optics,

• classical mechanics, and

• quantum mechanics.

Here, we want to help mathematicians who are not familiar with theoreticalphysics In Chapter 5, we study Carath´eodory’s royal road to geometricaloptics based on the fundamental duality between

• light rays and

• wave fronts

which can be traced back to the work of Huygens in the seventeenth tury In string theory, K¨ahler manifolds play a crucial role In Chapter 5, wewill show how Poincar´e’s non-Euclidean geometry on the upper half-plane isrelated to both geometrical optics and K¨ahler geometry

cen-2

The enormous influence of Gauss’ first masterpiece on the development of ematics is described in the monograph by C Goldstein, N Schappacher, and J.Schwermer: The Shaping of Arithmetic after Gauss’ Disquisitiones Arithmeticae,Springer, Berlin 2007

math-Since all the models of quantum fields are based on the study of an infinitenumber of (slightly perturbed) harmonic oscillators in the setting of pertur-bation theory, we use the harmonic oscillator as a paradigm for both classicalmechanics and quantum mechanics In Chapter 6 on classical mechanics, wewill study the following topics:

• Newtonian mechanics,

• Lagrangian mechanics (the Euler–Lagrange equation, the Jacobi accessory

eigen-value problem and Morse theory),

• Hamiltonian mechanics (the canonical dynamical system and the Hamilton–

Jacobi partial differential equation), and

• Poissonian mechanics.

In particular, this concerns

• the Legendre transformation and contact geometry,

• the Hamiltonian flow and symplectic geometry,

• the tangent bundle of the position space (the position-velocity space also called

the state space),

• the cotangent bundle of the position space (the position-momentum space also

called the phase space),

• the Legendre transformation as a transformation from the tangent bundle to the

cotangent bundle; the latter is equipped with a natural symplectic structure

In terms of mathematics, the fundamental relation between symmetry andconservation laws in physics is related to

• the Noether theorem, and

• Poisson brackets and Lie’s momentum map.

Quantum mechanics The comprehensive Chapter 7 lies at the heart of

this series of monographs This chapter should help the reader to understandthe different aspects of the passage from classical physics to quantum physics,

by using the different procedures of quantization We will use the paradigm

of the harmonic oscillator in order to explain the basic ideas of the followingapproaches:

• Heisenberg’s quantum mechanic (via creation and annihilation operators),

• Schr¨odinger’s quantum mechanics (via the Schr¨odinger partial differential

equa-tion),

• Feynman’s quantum mechanics (via the path integral),

• von Neumann’s functional-analytic approach (via the spectral theory for

self-adjoint operators in Hilbert spaces),

• von Neumann’s density operator in statistical physics (via trace class operators),

• Weyl’s symbolic calculus for pseudo-differential operators (deformation

quanti-zation),

• the Poincar´e–Wirtinger calculus and Bargmann’s holomorphic quantization,

• the Stone-von Neumann uniqueness theorem (for the fundamental commutation

relations in quantum mechanics) and the Weyl functor3 based on symplecticgeometry,

• supersymmetric quantization.

3 At this place, the general theory of mathematical structures (also called categorytheory) enters the theory of quantization (also called quantum mathematics)

Concerning the Feynman path integral as a fundamental tool in quantumphysics, we will study the following:

• Brownian motion and the infinite-dimensional rigorous Wiener integral based on

measure theory,

• the rigorous Feynman–Kac formula for diffusion processes,

• rigorous finite-dimensional Gaussian integrals, the computation of correlations

and moments, the Wick theorem, and Feynman diagrams,

• rigorous definition of infinite-dimensional Gaussian integrals via zeta function

regularization,

• the Wentzel–Kramers–Brioullin (WKB) method of stationary phase for the

com-putation of Gaussian integrals, and the approximate comcom-putation of Feynmanpath integrals

The Feynman path integral can be obtained from the Wiener integral byusing formal analytic continuation from real time to imaginary time Thiscorresponds to the fact that the Schr¨odinger equation describes diffusionprocesses in imaginary time Furthermore, in Chapter 7, we discuss the basic

ideas of the algebraic approach to quantum mechanics by using C ∗-algebras

and von Neumann algebras In this connection, we consider:

• applications to statistical mechanics (Boltzmann statistics, Bose–Einstein

statis-tics, and Fermi–Dirac statistics),

• thermodynamic equilibrium states (Kubo–Martin–Schwinger (KMS) states) and

the Tomita–Takesaki theory for von Neumann algebras,

• the Murray–von Neumann classification of factors in the theory of von Neumann

algebras,

• projection operators and the main theorem of quantum logic (Gleason’s extension

theorem for C ∗-algebras)

The modern theory of operator algebras culminates in Alain Connes’s commutative geometry, which represents the appropriate mathematical struc-ture for a deeper understanding of the Standard Model in elementary particlephysics This will be investigated in Volume IV on quantum mathematics.For the interested reader, we refer to the following fundamental monograph:

non-A Connes and M Marcolli, Noncommutative Geometry, Quantum Fields,and Motives, American Mathematical Society 2008

Internet: http://www.math.fsu.edu/∼ marcolli/bookjune4.pdf

Chapters 8 and 9 serve as an introduction to scattering theory, which plays

a crucial role in elementary particle physics As a paradigm for general tering theory, we consider the scattering of a quantum particle on the realline We consider

scat-• the energy levels of bound states,

• the energy levels of scattering states, and distributions as generalized

eigenfunc-tions of the Schr¨odinger equation,

• the transition matrix,

• the unitary S-matrix and transition probabilities for scattering processes,

• the relation between the singularities of the S-matrix in the complex energetic

plane and the energy levels of stable bound states,

• unstable particles (resonances) and the second sheet of the energetic Riemann

surface (the Breit–Wigner formula for the energy levels and the mean lifetime ofresonances),

• stationary scattering theory, the Green’s function of the Helmholtz equation, and

the Lippmann–Schwinger integral equation,

• instationary scattering theory, wave operators, the absolutely continuous

spec-trum of the Hamiltonian, and the S-matrix in functional analysis.

Here, we do not assume that the reader is familiar with

• von Neumann’s functional-analytic spectral theory for self-adjoint operators in

Hilbert spaces,

• the Gelfand–Kostyuchenko theory of generalized eigenfunctions for self-adjoint

operators,

• the Møller–Kato theory of wave operators in scattering theory, and

• the Weyl–Kodaira theory for singular differential operators.

For the convenience of the reader, the necessary material will be summarized

at the proper place when it is needed in Volumes II and III

Quantum electrodynamics In the present volume, it is our main goal to

illustrate the beauty of quantum electrodynamics by proceeding cally

pragmati-We do not start with an abstract approach, but with the tion of important physical effects which are observed in experiments, including radiative corrections in lowest order of renormalization the- ory.

computa-This should help the reader in getting a feel for the essential questions Moresophisticated approaches are postponed to later volumes of this series ofmonographs In the introductory Chapter 10, we study creation and anni-hilation operators for electrons, positrons, and photons In Chapter 11, weformulate the classical field equations of quantum electrodynamics on the in-teraction between electrons and photons, by coupling the Maxwell equations

of the electromagnetic field to the Dirac equation of the electron wave tion This equation depends on the gauge fixing of the four-potential for theelectromagnetic field However, it turns out that physical effects measured inexperiments are independent of the choice of the gauge fixing The point isthat:

func-The classical field equations of quantum electrodynamics have to be quantized.

In this connection, we have to distinguish between

• the single free quantum fields for electrons, positrons, and photons, and

• the total interacting quantum field for electrons, positrons, and photons.

In Chapter 12, we construct free quantum fields by using the method ofFourier quantization based on creation and annihilation operators

In Chapter 13, we study the interacting quantum field of electrons,positrons, and photons by using

• the magic Dyson formula for the S-matrix (scattering matrix), and

• the main Wick theorem for the S-matrix, which implies the Feynman diagrams.

This is Dyson’s classical approach to understanding the Feynman diagrams.4

Originally, Feynman invented his exciting diagram technique on the basis ofingenious physical intuition In Dyson’s mathematical setting, the Feynmandiagrams are nothing other than graphical representations of well-defined an-alytic expressions, which are effectively produced by the main Wick theorem

Feynman’s use of propagators and Dyson’s magic formula for the S-matrix

are closely related to Lagrange’s variation-of-parameter formula in celestialmechanics Many mathematicians complain about the following situation:

In the physics textbooks, one reads the Feynman rules for Feynman diagrams, but it is not clear where the Feynman rules come from.

In the present textbook, we will thoroughly study the mathematical andphysical origin of both the Feynman diagrams and the Feynman rules Wewill also consider applications to interesting physical effects

In Chapter 15, we investigate the following physical effects in lowest order

of perturbation theory:

• the cross section for Compton scattering between photons and electrons

(im-provement of the Thomson formula in classical electrodynamics),

• the cross section for the scattering of electrons in an external electromagnetic

field,

• the intensity of spectral lines for bound states in an external electromagnetic

field, and

• the Cherenkov radiation.

For the computation of terms corresponding to higher order of tion theory, renormalization theory is needed In Chapter 17, we discuss thephysics behind the following radiative corrections in lowest possible order ofrenormalization theory:

perturba-• the screening of the Coulomb potential by vacuum polarization (the Uehling

potential),

• the anomalous magnetic moment of the electron (the Schwinger formula),

• the anomalous magnetic moment of the muon, and

• the Lamb shift in the spectrum of the hydrogen atom.

Unfortunately, the explicit computations (in the framework of dimensionalregularization in renormalization theory) are lengthy We will postpone thesedetailed computations to Volume III

In Chapter 18, we discuss the main result telling us that quantum trodynamics can be renormalized to all orders of perturbation theory Thefinal result consists of getting finite expressions in each order of perturbationtheory (e.g., cross sections for scattering processes), which depend on the twofundamental free parameters

elec-4 Dyson’s discovery of this approach is described by himself in his book, F Dyson,

Disturbing the Universe, Harper & Row, New York, 1979 (see page 27 of Volume

I for this fascinating story)

• meff (effective mass of the electron) and

• −eeff (effective electric charge of the electron)

Observe the crucial fact that:

The free parameters meff and eeff cannot be determined theoretically

by quantum electrodynamics.

They have to be determined by physical experiments In the SI system, oneobtains the following values:

meff = 0.511 MeV/c2, eeff = 1.602 · 10 −19 As.

A reader who wants to become familiar with quantum electrodynamics asquickly as possible should start reading with Chapter 10

The incredible effectiveness of perturbation theory in physics

Sur-prisingly enough, low-order radiative corrections are sufficient for getting afantastic coincidence between theory and experimental data For example,the anomalous magnetic moment of the electron measured in experiments ispredicted very precisely by fourth-order radiative corrections (up to 9 digits).However, the necessary amount of computations is enormous One has to eval-uate high-dimensional integrals which correspond to 891 Feynman diagrams;this needs years of supercomputer time

A warning to the reader In summer 1976, Arthur Wightman (Princeton

University) organized a famous conference in Erice (Sicily/Italy) on malization theory In the introduction to the Proceedings of this conference,

renor-he writes:5

Renormalization theory is a notoriously complicated and technical ject I want to tell stories with a moral for the earnest student: Renor-malization theory has a history of egregious errors by distinguished savants(see page 967) It has a justified reputation for perversity; a method thatworks up to 13th order in the perturbation theory fails in the 14th order.Arguments that sound plausible often dissolve into mush when examinedclosely The worst that can happen often happens The prudent studentwould do well to distinguish sharply between what has been proved andwhat has been plausible, and in general he should watch out!

sub-In 1999 Gerardus ’t Hooft and Martinus Veltman were awarded the Nobelprize in physics for their contributions to the renormalization of the theory

of electroweak interaction and for the computation of radiative corrections inthe Standard Model of particle physics

Perspectives More advanced approaches to renormalization theory will be

systematically studied in the following volumes of this series of monographs

In particular, this concerns the new approaches to perturbative quantum fieldtheory due to Connes and Kreimer (Hopf algebras), and Brunetti, D¨utsch,

5

A Wightman, Orientation In: Renormalization Theory, pp 1–20 Edited by G.Velo and A Wightman, Reidel, Dordrecht, 1976 (reprinted with permission)

and Fredenhagen (microlocal analysis and the Master Ward Identity) Inorder to give the reader an overview on the large variety of different ap-proaches to renormalization theory, we summarize important references inSection 19.11, and in Chapter 19 we sketch some basic ideas.

The propagation of light in the universe, namely,

• the deflection of light at the sun, and

• the red shift of spectral lines as a consequence of the expansion of the universe

(the Hubble effect)

will be investigated in Volume III in terms of Einstein’s theory of generalrelativity

The basic idea of our approach to quantum electrodynamics As a

rule, mathematicians have trouble with reading some textbooks on quantumfield theory written by physicists The point is that:

In mathematics, one never does computations with quantities which

do not exist.6

In order to respect this basic principle in mathematics, we will use the lattice approach That is, roughly speaking, we will proceed in the following two

steps

Step 1: The discretized physical system We put the physical system in a cubic

box of finite side length L and volume V = L3 The boundary conditions are

given by periodicity

• We observe the physical system in a finite time interval [− T

2, T2].

• We choose a maximal energy Emax.

• In the 3-dimensional momentum space, we introduce a finite lattice of spacing

Δp and maximal momentum Pmax In this setting, the Fourier integral transform

is replaced by a discrete Fourier transform via finite Fourier series expansions

• We define Dyson’s S-matrix for this situation.

• The main Wick theorem allows us to compute the S-matrix elements (i.e., the

transition amplitudes) in an elegant manner, by eliminating the creation andannihilation operators, and replacing them by propagators (i.e., correlation func-tions for free fields)

• The point is that the propagators are discrete algebraic Feynman integrals, which

are indeed well-defined finite sums

• The transition amplitudes can be graphically represented by Feynman diagrams.

• The Feynman rules allow us to translate the Feynman diagrams into well-defined

finite sums

• From the transition amplitudes, we obtain the transition probabilities which

yield the cross sections for scattering processes Note that cross sections can bemeasured in particle accelerators

Step 2: The delicate continuum limit Explicitly, we have to study the ing limits:

follow-• L → ∞ (the volume L3 of the cubic box becomes infinite),

6 For example, this concerns infinite renormalization constants or ill-definedinfinite-dimensional functional/path integrals

• T → ∞ (the time interval becomes infinite),

• Pmax → ∞ (i.e., Emax → ∞) (high-energy limit),

• Δp → 0 (low-energy limit).

In order to force the convergence of the discrete algebraic Feynman integrals

to well-defined expressions, we modify the classical Lagrangian density bysetting

meff := m e + δm, eeff := e + δe.

That is, we replace the so-called bare electron mass m eand the so-called bareelectron charge−e in the Lagrangian density by

m e = meff− δm, −e = −eeff + δe,

respectively This way, the classical Lagrangian density

L(ψ, ∂ψ, A, ∂A; m e , e)

is modified by the function

Lmodified (ψ, ∂ψ, A, ∂A; meff, eeff ; δm, δe).

The terms multiplied by δm, δe are called counterterms of the classical

La-grangian density L Note that in this lattice approach, δe and δm are real

parameters which depend on the shape of the lattice, that is, they depend on

the maximal energy Emax Now consider the high-energy limit

Emax → +∞.

Roughly speaking, we have to show that δm(Emax) and δe(Emax) can be

chosen in such a way that the finite continuum limit exists for the S-matrix elements (i.e., the transition elements) This is the procedure of renormaliza- tion.

Observe the following peculiarity By the Stone–von Neumann uniqueness

theorem, a finite number of creation and annihilation operators is uniquely

determined by the commutation relations (up to unitary equivalence) This is

not true anymore for an infinite number of creation and annihilation

opera-tors, as was shown by Lars G˚arding and Arthur Wightman in 1954 However,our approach avoids the latter ambiguity, since we only work with a finitenumber of creation and annihilation operators before passing to the contin-uum limit (of the vacuum expectation values) We also would like to empha-size that our approach differs only slightly from the usual approach used byphysicists In particular, we use a notation for discrete Fourier integrals suchthat the formal passage to the language used by physicists is possible at eachstage of our procedure

For the physical quantities which can be measured in experiments, our final formulas coincide with the formulas used by physicists.

Moreover, in each step of our procedure it is easy to pass formally to theexpressions used by physicists, since the Feynman diagrams are the same.

This way, we hope to help mathematicians in getting a better standing for the ingenious and beautiful approach invented by physi- cists.

under-From the physical point of view, the modification of the classical Lagrangiandensity reflects the fact that:

Quantum effects have to be added to the classical theory.

Intuitively, this means that the quantum fluctuations of the ground state

of the quantum field of electrons, positrons, and photons influence physicaleffects observed in experiments For example, this concerns the anomalousmagnetic moment of the electron and the spectrum of the hydrogen atom(Lamb shift)

Convention If we do not expressively state the opposite, we will use the SI

system of physical units (international system of units) which can be found

in the Appendix to Volume I In particular, note that in Chapters 10–19

on quantum electrodynamics, we will use the energetic system with c = 1

(velocity of light in a vacuum), = h/2π = 1 (Planck’s quantum of action),

k = 1 (Boltzmann constant), ε0 = μ0= 1 (see page 790)

The Poincar´ e Seminar The best way of getting information about

re-cent developments in modern physics is to look at the books which reportthe lectures given at the Poincar´e Seminar in Paris Starting in 2002, this

seminar has been organized by l’Institut Henri Poincar´ e in Paris (see page

1050) Bertrand Duplantier and Vincent Rivasseau write in the Foreword to

Quantum Spaces, Birkh¨auser, Basel, 2007:

This book is the seventh in a series of lectures of the S´ eminaire Poincar´ e,

which is directed towards a large audience of physicists and cians

mathemati-The goal of this seminar is to provide up-to-date information about eral topics of great interest in physics Both the theoretical and experimen-tal aspects are covered, with some historical background Inspired by the

gen-S´ eminaire Bourbaki in mathematics in its organization, hence nicknamed

“S´eminaire Bourbaphy,” the Poincar´e Seminar is held twice a year at the

Institut Henri Poincar´ e in Paris, with contributions prepared in advance.

Particular care is devoted to the pedagogical nature of the presentation so

as to fulfill the goal of being readable by a large audience of scientists

Two recent survey volumes The following two volumes try to reflect

the state of the art by summarizing the most important approaches used inmodern quantum field theory:

• B Fauser, J Tolksdorf, and E Zeidler (Eds.), Quantum Gravity: Mathematical

Models and Experimental Bounds, Birkh¨auser, Basel, 2006

• B Fauser, J Tolksdorf, and E Zeidler (Eds.), Quantum Field Theory –

Com-petitive Methods, Birkh¨auser, Basel, 2008

Acknowledgements I am very grateful to my collaborators Bertfried

Fauser and J¨urgen Tolksdorf for organizing two workshops in 2005 and 2007

on recent developments in quantum field theory, and for both doing an cellent editorial job and telling me new research strategies and new results

ex-in the algebraic and geometric approach to quantum field theory In ular, I would like to thank Bertfried Fauser for drawing my attention to therelation between quantum field theory and the following topics: Gian-CarloRota’s work on combinatorics, the ubiquitous Hopf algebras, Rota–Baxteralgebras, quantum Clifford algebras, the Rota–Stein cliffordization process,Littlewood’s work on the representation theory of the symmetric group, mo-tivic number theory, topos theory, and category theory in algebraic geometry.For illuminating discussions about promising new approaches to (perturba-tive) quantum field theory in both flat and curved space-time, I would like

partic-to thank Klaus Fredenhagen and the late Julius Wess, as well as DetlevBuchholz, Christian Brouder, Romeo Brunetti, Michael D¨utsch, KuruschEbrahimi-Fard, Joel Feldman, Felix Finster, Christian Fleischhack, Alessan-dra Frabetti, Stefan Hollands, Harald Grosse, Jerzy Kijowski, Dirk Kreimer,Elisabeth Kraus, Alexander Lange, Matilde Marcolli, Mario Paschke, KlausRehren, Gerd Rudolph, Manfred Salmhofer, Alexander Schmidt, Klaus Si-bold, Rainer Verch, Hartmut Wachter, and Raimar Wulkenhaar

I would like to thank Thomas Hahn from the Max Planck Institute forPhysics, Werner Heisenberg, in Munich for informing me on the state of theart in automated multi-loop computations in perturbation theory Such so-phisticated computer programs are used for preparing the experiments at theLHC (Large Hadron Collider) of CERN (European Organization for NuclearResearch at Geneva, Switzerland) (see Sect 18.4)

On the occasion of Professor Friedrich Hirzebruch’s 80th birthday on tober 17 in 2007, I would like to dedicate this volume to him in gratitude Hisscientific work deeply influenced the development of mathematics in the sec-ond half of the twentieth century Nowadays physicists frequently use Hirze-bruch’s results in order to study the topological structure of physical fields

Oc-In 1982, Friedrich Hirzebruch founded the Max Planck Oc-Institute for ematics in Bonn (Germany) Mathematicians from all over the world enjoydoing research in the relaxed and highly stimulating atmosphere of this insti-tute In 1996, the Max Planck Institute for Mathematics in the Sciences wasfounded in Leipzig Friedrich Hirzebruch was the chairman of the Founders’Committee The staff and the visitors of our institute are very grateful toProfessor Hirzebruch for his efforts made as chairman

Math-For helping me to save a lot of time, I am very grateful to my secretaryRegine L¨ubke (invaluable support), Katarzyna Baier (answering patientlyalmost infinitely many bibliographical questions), the library team (steadilysupport), Kerstin F¨olting (graphics and tables), Micaela Krieger-Hauwede(answering patiently my LATEX questions), Katrin Scholz (internet searching),and Thomas Heid (computer expert) I also would like to thank the staff of

the Springer publishing house in Heidelberg, Ruth Allewelt, Joachim Heinze,and Martin Peters, for the harmonious collaboration Many years ago, myCzech colleague from Prague, the late Svatopluk Fuˇcik, wrote the followingsentence in the preface to his book on nonlinear partial differential equations:

“Finally, I would like to thank myself for typing the manuscript.” I resist thetemptation of repeating this sentence with respect to my LATEX file But Iwould like to thank Donald Knuth for enjoying the mathematical communitywith the beautiful gift of the TEX tool

In the Proverbs 31, 10 of the Bible, one reads:

Who can find a virtuous woman? For her price is far above rubies Theheart of her husband does safely trust in her, so that he shall have no need

of spoil She will do him good and not evil all the days of her life

I am very grateful to my beloved wife, Christine, who has been taking care

of me for 40 years

Part I Introduction

Prologue 1

1. Mathematical Principles of Modern Natural Philosophy 111.1 Basic Principles 121.2 The Infinitesimal Strategy and Differential Equations 141.3 The Optimality Principle 141.4 The Basic Notion of Action in Physics and the Idea of

Quantization 151.5 The Method of the Green’s Function 171.6 Harmonic Analysis and the Fourier Method 211.7 The Method of Averaging and the Theory of Distributions 261.8 The Symbolic Method 281.9 Gauge Theory – Local Symmetry and the Description of

Interactions by Gauge Fields 341.10 The Challenge of Dark Matter 46

2 The Basic Strategy of Extracting Finite Information from Infinities – Ariadne’s Thread in Renormalization Theory 472.1 Renormalization Theory in a Nutshell 482.1.1 Effective Frequency and Running Coupling Constant

of an Anharmonic Oscillator 482.1.2 The Zeta Function and Riemann’s Idea of AnalyticContinuation 542.1.3 Meromorphic Functions and Mittag-Leffler’s Idea

of Subtractions 562.1.4 The Square of the Dirac Delta Function 582.2 Regularization of Divergent Integrals in Baby

Renormalization Theory 60

Power-Counting 602.2.2 The Choice of the Normalization Momentum 632.2.3 The Method of Differentiating Parameter Integrals 632.2.4 The Method of Taylor Subtraction 64

2.2.5 Overlapping Divergences 652.2.6 The Role of Counterterms 672.2.7 Euler’s Gamma Function 672.2.8 Integration Tricks 692.2.9 Dimensional Regularization via Analytic Continua-tion 732.2.10 Pauli–Villars Regularization 762.2.11 Analytic Regularization 772.2.12 Application to Algebraic Feynman Integrals in

Minkowski Space 802.2.13 Distribution-Valued Meromorphic Functions 812.2.14 Application to Newton’s Equation of Motion 872.2.15 Hints for Further Reading 922.3 Further Regularization Methods in Mathematics 932.3.1 Euler’s Philosophy 932.3.2 Adiabatic Regularization of Divergent Series 942.3.3 Adiabatic Regularization of Oscillating Integrals 952.3.4 Regularization by Averaging 962.3.5 Borel Regularization 982.3.6 Hadamard’s Finite Part of Divergent Integrals 1002.3.7 Infinite-Dimensional Gaussian Integrals and the ZetaFunction Regularization 1012.4 Trouble in Mathematics 1022.4.1 Interchanging Limits 1022.4.2 The Ambiguity of Regularization Methods 1042.4.3 Pseudo-Convergence 1042.4.4 Ill-Posed Problems 1052.5 Mathemagics 109

3. The Power of Combinatorics 1153.1 Algebras 1153.2 The Algebra of Multilinear Functionals 1173.3 Fusion, Splitting, and Hopf Algebras 1223.3.1 The Bialgebra of Linear Differential Operators 1233.3.2 The Definition of Hopf Algebras 1283.4 Power Series Expansion and Hopf Algebras 1313.4.1 The Importance of Cancellations 1313.4.2 The Kepler Equation and the Lagrange

Inversion Formula 1323.4.3 The Composition Formula for Power Series 1343.4.4 The Fa`a di Bruno Hopf Algebra for the Formal

Diffeomorphism Group of the Complex Plane 1363.4.5 The Generalized Zimmermann Forest Formula 1383.4.6 The Logarithmic Function and Schur Polynomials 1403.4.7 Correlation Functions in Quantum Field Theory 141

3.4.8 Random Variables, Moments, and Cumulants 1433.5 Symmetry and Hopf Algebras 1463.5.1 The Strategy of Coordinatization in Mathematicsand Physics 1463.5.2 The Coordinate Hopf Algebra of a Finite Group 1483.5.3 The Coordinate Hopf Algebra of an Operator Group 1503.5.4 The Tannaka–Krein Duality for Compact Lie Groups 1523.6 Regularization and Rota–Baxter Algebras 1543.6.1 Regularization of the Laurent Series 1573.6.2 Projection Operators 1583.6.3 The q-Integral 1583.6.4 The Volterra–Spitzer Exponential Formula 1603.6.5 The Importance of the Exponential Function in

Mathematics and Physics 1613.7 Partially Ordered Sets and Combinatorics 1623.7.1 Incidence Algebras and the Zeta Function 1623.7.2 The M¨obius Function as an Inverse Function 1633.7.3 The Inclusion–Exclusion Principle in Combinatorics 1643.7.4 Applications to Number Theory 1663.8 Hints for Further Reading 167

4. The Strategy of Equivalence Classes in Mathematics 1754.1 Equivalence Classes in Algebra 1784.1.1 The Gaussian Quotient Ring and the Quadratic

Reciprocity Law in Number Theory 1784.1.2 Application of the Fermat–Euler Theorem in CodingTheory 1824.1.3 Quotient Rings, Quotient Groups, and Quotient Fields 1844.1.4 Linear Quotient Spaces 1884.1.5 Ideals and Quotient Algebras 1904.2 Superfunctions and the Heaviside Calculus in Electrical

Engineering 1914.3 Equivalence Classes in Geometry 1944.3.1 The Basic Idea of Geometry Epitomized by Klein’sErlangen Program 1944.3.2 Symmetry Spaces, Orbit Spaces, and HomogeneousSpaces 1944.3.3 The Space of Quantum States 1994.3.4 Real Projective Spaces 2004.3.5 Complex Projective Spaces 2034.3.6 The Shape of the Universe 2044.4 Equivalence Classes in Topology 2054.4.1 Topological Quotient Spaces 2054.4.2 Physical Fields, Observers, Bundles, and Cocycles 2084.4.3 Generalized Physical Fields and Sheaves 216

4.4.4 Deformations, Mapping Classes, and Topological

Charges 2194.4.5 Poincar´e’s Fundamental Group 2234.4.6 Loop Spaces and Higher Homotopy Groups 2254.4.7 Homology, Cohomology, and Electrodynamics 2274.4.8 Bott’s Periodicity Theorem 2274.4.9 K-Theory 2284.4.10 Application to Fredholm Operators 2334.4.11 Hints for Further Reading 2354.5 The Strategy of Partial Ordering 2374.5.1 Feynman Diagrams 2384.5.2 The Abstract Entropy Principle in Thermodynamics 2394.5.3 Convergence of Generalized Sequences 2404.5.4 Inductive and Projective Topologies 2414.5.5 Inductive and Projective Limits 2434.5.6 Classes, Sets, and Non-Sets 2454.5.7 The Fixed-Point Theorem of Bourbaki–Kneser 2474.5.8 Zorn’s Lemma 2484.6 Leibniz’s Infinitesimals and Non-Standard Analysis 2484.6.1 Filters and Ultrafilters 2504.6.2 The Full-Rigged Real Line 251

Part II Basic Ideas in Classical Mechanics

5. Geometrical Optics 2635.1 Ariadne’s Thread in Geometrical Optics 2645.2 Fermat’s Principle of Least Time 2685.3 Huygens’ Principle on Wave Fronts 2705.4 Carath´eodory’s Royal Road to Geometrical Optics 2715.5 The Duality between Light Rays and Wave Fronts 2745.5.1 From Wave Fronts to Light Rays 2755.5.2 From Light Rays to Wave Fronts 2765.6 The Jacobi Approach to Focal Points 2765.7 Lie’s Contact Geometry 2795.7.1 Basic Ideas 2795.7.2 Contact Manifolds and Contact Transformations 2835.7.3 Applications to Geometrical Optics 2845.7.4 Equilibrium Thermodynamics and Legendre

Submanifolds 2855.8 Light Rays and Non-Euclidean Geometry 2895.8.1 Linear Symplectic Spaces 2915.8.2 The K¨ahler Form of a Complex Hilbert Space 2955.8.3 The Refraction Index and Geodesics 2975.8.4 The Trick of Gauge Fixing 299

5.8.5 Geodesic Flow 2995.8.6 Hamilton’s Duality Trick and Cogeodesic Flow 3005.8.7 The Principle of Minimal Geodesic Energy 3015.9 Spherical Geometry 3025.9.1 The Global Gauss–Bonnet Theorem 3035.9.2 De Rham Cohomology and the Chern Class of

the Sphere 3055.9.3 The Beltrami Model 3085.10 The Poincar´e Model of Hyperbolic Geometry 3145.10.1 K¨ahler Geometry and the Gaussian Curvature 3185.10.2 K¨ahler–Einstein Geometry 3235.10.3 Symplectic Geometry 3235.10.4 Riemannian Geometry 3245.11 Ariadne’s Thread in Gauge Theory 3335.11.1 Parallel Transport of Physical Information – the Key

to Modern Physics 3345.11.2 The Phase Equation and Fiber Bundles 3375.11.3 Gauge Transformations and Gauge-Invariant

Differential Forms 3385.11.4 Perspectives 3415.12 Classification of Two-Dimensional Compact Manifolds 3435.13 The Poincar´e Conjecture and the Ricci Flow 3465.14 A Glance at Modern Optimization Theory 3485.15 Hints for Further Reading 348

6 The Principle of Critical Action and the Harmonic

Oscillator – Ariadne’s Thread in Classical Mechanics 3596.1 Prototypes of Extremal Problems 3606.2 The Motion of a Particle 3646.3 Newtonian Mechanics 3666.4 A Glance at the History of the Calculus of Variations 3706.5 Lagrangian Mechanics 3726.5.1 The Harmonic Oscillator 3736.5.2 The Euler–Lagrange Equation 3756.5.3 Jacobi’s Accessory Eigenvalue Problem 3766.5.4 The Morse Index 3776.5.5 The Anharmonic Oscillator 3786.5.6 The Ginzburg–Landau Potential and the

Higgs Potential 3806.5.7 Damped Oscillations, Stability, and Energy

Dissipation 3826.5.8 Resonance and Small Divisors 3826.6 Symmetry and Conservation Laws 3836.6.1 The Symmetries of the Harmonic Oscillator 3846.6.2 The Noether Theorem 384

6.7 The Pendulum and Dynamical Systems 3906.7.1 The Equation of Motion 3906.7.2 Elliptic Integrals and Elliptic Functions 3916.7.3 The Phase Space of the Pendulum and Bundles 3966.8 Hamiltonian Mechanics 4026.8.1 The Canonical Equation 4046.8.2 The Hamiltonian Flow 4046.8.3 The Hamilton–Jacobi Partial Differential Equation 4056.9 Poissonian Mechanics 4066.9.1 Poisson Brackets and the Equation of Motion 4076.9.2 Conservation Laws 4076.10 Symplectic Geometry 4076.10.1 The Canonical Equations 4086.10.2 Symplectic Transformations 4096.11 The Spherical Pendulum 4116.11.1 The Gaussian Principle of Critical Constraint 4116.11.2 The Lagrangian Approach 4126.11.3 The Hamiltonian Approach 4146.11.4 Geodesics of Shortest Length 415

6.12 The Lie Group SU (E3) of Rotations 4166.12.1 Conservation of Angular Momentum 4166.12.2 Lie’s Momentum Map 4196.13 Carath´eodory’s Royal Road to the Calculus of Variations 4196.13.1 The Fundamental Equation 4196.13.2 Lagrangian Submanifolds in Symplectic Geometry 4216.13.3 The Initial-Value Problem for the Hamilton–JacobiEquation 4236.13.4 Solution of Carath´eodory’s Fundamental Equation 4236.14 Hints for Further Reading 424

Part III Basic Ideas in Quantum Mechanics

7 Quantization of the Harmonic Oscillator – Ariadne’s Thread

in Quantization 427

7.1 Complete Orthonormal Systems 4307.2 Bosonic Creation and Annihilation Operators 4327.3 Heisenberg’s Quantum Mechanics 4407.3.1 Heisenberg’s Equation of Motion 4437.3.2 Heisenberg’s Uncertainty Inequality for the HarmonicOscillator 4467.3.3 Quantization of Energy 4477.3.4 The Transition Probabilities 4497.3.5 The Wightman Functions 4517.3.6 The Correlation Functions 456

7.4 Schr¨odinger’s Quantum Mechanics 4597.4.1 The Schr¨odinger Equation 4597.4.2 States, Observables, and Measurements 4627.4.3 The Free Motion of a Quantum Particle 4647.4.4 The Harmonic Oscillator 4677.4.5 The Passage to the Heisenberg Picture 4737.4.6 Heisenberg’s Uncertainty Principle 4757.4.7 Unstable Quantum States and the Energy-Time

Uncertainty Relation 4767.4.8 Schr¨odinger’s Coherent States 4787.5 Feynman’s Quantum Mechanics 4797.5.1 Main Ideas 4807.5.2 The Diffusion Kernel and the Euclidean Strategy inQuantum Physics 4877.5.3 Probability Amplitudes and the Formal PropagatorTheory 4887.6 Von Neumann’s Rigorous Approach 4957.6.1 The Prototype of the Operator Calculus 4967.6.2 The General Operator Calculus 4997.6.3 Rigorous Propagator Theory 5057.6.4 The Free Quantum Particle as a Paradigm of

Functional Analysis 5097.6.5 The Free Hamiltonian 5247.6.6 The Rescaled Fourier Transform 5327.6.7 The Quantized Harmonic Oscillator and the MaslovIndex 5347.6.8 Ideal Gases and von Neumann’s Density Operator 5407.7 The Feynman Path Integral 5477.7.1 The Basic Strategy 5477.7.2 The Basic Definition 5497.7.3 Application to the Free Quantum Particle 5507.7.4 Application to the Harmonic Oscillator 5527.7.5 The Propagator Hypothesis 5557.7.6 Motivation of Feynman’s Path Integral 5557.8 Finite-Dimensional Gaussian Integrals 5597.8.1 Basic Formulas 560

Diagrams 5647.8.3 Full Moments and Perturbation Theory 5677.9 Rigorous Infinite-Dimensional Gaussian Integrals 5707.9.1 The Infinite-Dimensional Dispersion Operator 5717.9.2 Zeta Function Regularization and Infinite-DimensionalDeterminants 5727.9.3 Application to the Free Quantum Particle 5747.9.4 Application to the Quantized Harmonic Oscillator 576

7.9.5 The Spectral Hypothesis 5797.10 The Semi-Classical WKB Method 5807.11 Brownian Motion 5847.11.1 The Macroscopic Diffusion Law 5847.11.2 Einstein’s Key Formulas for the Brownian Motion 5857.11.3 The Random Walk of Particles 5857.11.4 The Rigorous Wiener Path Integral 5867.11.5 The Feynman–Kac Formula 5887.12 Weyl Quantization 5907.12.1 The Formal Moyal Star Product 5917.12.2 Deformation Quantization of the Harmonic Oscillator 5927.12.3 Weyl Ordering 5967.12.4 Operator Kernels 5997.12.5 The Formal Weyl Calculus 6027.12.6 The Rigorous Weyl Calculus 6067.13 Two Magic Formulas 6087.13.1 The Formal Feynman Path Integral for the Propaga-tor Kernel 6117.13.2 The Relation between the Scattering Kernel and thePropagator Kernel 6147.14 The Poincar´e–Wirtinger Calculus 6167.15 Bargmann’s Holomorphic Quantization 6177.16 The Stone–Von Neumann Uniqueness Theorem 6217.16.1 The Prototype of the Weyl Relation 6217.16.2 The Main Theorem 626

7.16.3 C ∗-Algebras 627

7.16.4 Operator Ideals 6297.16.5 Symplectic Geometry and the Weyl Quantization

Functor 6307.17 A Glance at the Algebraic Approach to Quantum Physics 6337.17.1 States and Observables 6337.17.2 Gleason’s Extension Theorem – the Main Theorem

of Quantum Logic 6377.17.3 The Finite Standard Model in Statistical Physics as

a Paradigm 6387.17.4 Information, Entropy, and the Measure of Disorder 6407.17.5 Semiclassical Statistical Physics 6457.17.6 The Classical Ideal Gas 6487.17.7 Bose–Einstein Statistics 6497.17.8 Fermi–Dirac Statistics 6507.17.9 Thermodynamic Equilibrium and KMS-States 6517.17.10 Quasi-Stationary Thermodynamic Processes and

Irreversibility 6527.17.11 The Photon Mill on Earth 6547.18 Von Neumann Algebras 654

7.18.1 Von Neumann’s Bicommutant Theorem 6557.18.2 The Murray–von Neumann Classification of Factors 6587.18.3 The Tomita–Takesaki Theory and KMS-States 6597.19 Connes’ Noncommutative Geometry 6607.20 Jordan Algebras 6627.21 The Supersymmetric Harmonic Oscillator 6637.22 Hints for Further Reading 667

8 Quantum Particles on the Real Line – Ariadne’s Thread

in Scattering Theory 6998.1 Classical Dynamics Versus Quantum Dynamics 6998.2 The Stationary Schr¨odinger Equation 7038.3 One-Dimensional Quantum Motion in a Square-Well

Potential 7048.3.1 Free Motion 7048.3.2 Scattering States and the S-Matrix 7058.3.3 Bound States 7108.3.4 Bound-State Energies and the Singularities of the

S-Matrix 7128.3.5 The Energetic Riemann Surface, Resonances, and theBreit–Wigner Formula 7138.3.6 The Jost Functions 7188.3.7 The Fourier–Stieltjes Transformation 7198.3.8 Generalized Eigenfunctions of the Hamiltonian 7208.3.9 Quantum Dynamics and the Scattering Operator 7228.3.10 The Feynman Propagator 7268.4 Tunnelling of Quantum Particles and Radioactive Decay 7278.5 The Method of the Green’s Function in a Nutshell 7298.5.1 The Inhomogeneous Helmholtz Equation 7308.5.2 The Retarded Green’s Function, and the Existenceand Uniqueness Theorem 7318.5.3 The Advanced Green’s Function 7368.5.4 Perturbation of the Retarded and Advanced Green’sFunction 7378.5.5 Feynman’s Regularized Fourier Method 7398.6 The Lippmann–Schwinger Integral Equation 7438.6.1 The Born Approximation 7438.6.2 The Existence and Uniqueness Theorem via Banach’sFixed Point Theorem 7448.6.3 Hypoellipticity 745

9. A Glance at General Scattering Theory 7479.1 The Formal Basic Idea 7499.2 The Rigorous Time-Dependent Approach 7519.3 The Rigorous Time-Independent Approach 753

9.4 Applications to Quantum Mechanics 7549.5 A Glance at Quantum Field Theory 7579.6 Hints for Further Reading 758

Part IV Quantum Electrodynamics (QED)

10 Creation and Annihilation Operators 77110.1 The Bosonic Fock Space 77110.1.1 The Particle Number Operator 77410.1.2 The Ground State 77410.2 The Fermionic Fock Space and the Pauli Principle 77910.3 General Construction 78410.4 The Main Strategy of Quantum Electrodynamics 788

11 The Basic Equations in Quantum Electrodynamics 79311.1 The Classical Lagrangian 79311.2 The Gauge Condition 796

12 The Free Quantum Fields of Electrons, Positrons,

and Photons 799

12.1 Classical Free Fields 79912.1.1 The Lattice Strategy in Quantum Electrodynamics 79912.1.2 The High-Energy Limit and the Low-Energy Limit 80212.1.3 The Free Electromagnetic Field 80312.1.4 The Free Electron Field 80612.2 Quantization 81112.2.1 The Free Photon Quantum Field 81212.2.2 The Free Electron Quantum Field and Antiparticles 81412.2.3 The Spin of Photons 81912.3 The Ground State Energy and the Normal Product 82212.4 The Importance of Mathematical Models 82412.4.1 The Trouble with Virtual Photons 82512.4.2 Indefinite Inner Product Spaces 82612.4.3 Representation of the Creation and Annihilation

Operators in QED 82612.4.4 Gupta–Bleuler Quantization 831

13 The Interacting Quantum Field, and the Magic

Dyson Series for the S-Matrix 83513.1 Dyson’s Key Formula 83513.2 The Basic Strategy of Reduction Formulas 84113.3 The Wick Theorem 84613.4 Feynman Propagators 856

13.4.1 Discrete Feynman Propagators for Photons and

Electrons 85613.4.2 Regularized Discrete Propagators 86213.4.3 The Continuum Limit of Feynman Propagators 86413.4.4 Classical Wave Propagation versus Feynman

Propagator 870

14 The Beauty of Feynman Diagrams in QED 87514.1 Compton Effect and Feynman Rules in Position Space 87614.2 Symmetry Properties 88114.3 Summary of the Feynman Rules in Momentum Space 88214.4 Typical Examples 88514.5 The Formal Language of Physicists 89014.6 Transition Probabilities and Cross Sections of Scattering

Processes 89114.7 The Crucial Limits 89414.8 Appendix: Table of Feynman Rules 896

15 Applications to Physical Effects 89915.1 Compton Effect 89915.1.1 Duality between Light Waves and Light Particles

in the History of Physics 90215.1.2 The Trace Method for Computing Cross Sections 90315.1.3 Relativistic Invariance 91215.2 Asymptotically Free Electrons in an External

Electromagnetic Field 91415.2.1 The Key Formula for the Cross Section 91415.2.2 Application to Yukawa Scattering 91515.2.3 Application to Coulomb Scattering 915

15.2.4 Motivation of the Key Formula via S-Matrix 91615.2.5 Perspectives 92115.3 Bound Electrons in an External Electromagnetic

Field 92215.3.1 The Spontaneous Emission of Photons by the Atom 92215.3.2 Motivation of the Key Formula 92315.3.3 Intensity of Spectral Lines 92515.4 Cherenkov Radiation 926

Part V Renormalization

16 The Continuum Limit 94516.1 The Fundamental Limits 94516.2 The Formal Limits Fail 94616.3 Basic Ideas of Renormalization 947

16.3.1 The Effective Mass and the Effective Charge of theElectron 94716.3.2 The Counterterms of the Modified Lagrangian 94716.3.3 The Compensation Principle 94816.3.4 Fundamental Invariance Principles 94916.3.5 Dimensional Regularization of Discrete Algebraic

Feynman Integrals 94916.3.6 Multiplicative Renormalization 95016.4 The Theory of Approximation Schemes in Mathematics 951

17 Radiative Corrections of Lowest Order 95317.1 Primitive Divergent Feynman Graphs 95317.2 Vacuum Polarization 95417.3 Radiative Corrections of the Propagators 95517.3.1 The Photon Propagator 95617.3.2 The Electron Propagator 95617.3.3 The Vertex Correction and the Ward Identity 95717.4 The Counterterms of the Lagrangian and the CompensationPrinciple 95717.5 Application to Physical Problems 95817.5.1 Radiative Correction of the Coulomb Potential 95817.5.2 The Anomalous Magnetic Moment of the

Electron 95917.5.3 The Anomalous Magnetic Moment of the Muon 96117.5.4 The Lamb Shift 96217.5.5 Photon-Photon Scattering 964

18 A Glance at Renormalization to all Orders of

Perturbation Theory 96718.1 One-Particle Irreducible Feynman Graphs and

Divergences 97018.2 Overlapping Divergences and Manoukian’s Equivalence

Principle 97218.3 The Renormalizability of Quantum Electrodynamics 97518.4 Automated Multi-Loop Computations in Perturbation

19.2 The Causal Epstein–Glaser S-Matrix Approach 98719.3 Kreimer’s Hopf Algebra Revolution 99019.3.1 The History of the Hopf Algebra Approach 991

19.3.2 Renormalization and the Iterative Birkhoff

Factorization for Complex Lie Groups 99319.3.3 The Renormalization of Quantum

Electrodynamics 99619.4 The Scope of the Riemann–Hilbert Problem 99719.4.1 The Gaussian Hypergeometric Differential

Equation 99819.4.2 The Confluent Hypergeometric Function and the

Spectrum of the Hydrogen Atom 100419.4.3 Hilbert’s 21th Problem 100419.4.4 The Transport of Information in Nature 100719.4.5 Stable Transport of Energy and Solitons 100719.4.6 Ariadne’s Thread in Soliton Theory 100919.4.7 Resonances 101419.4.8 The Role of Integrable Systems in Nature 101419.5 The BFFO Hopf Superalgebra Approach 101619.6 The BRST Approach and Algebraic Renormalization 101919.7 Analytic Renormalization and Distribution-Valued

Analytic Functions 102219.8 Computational Strategies 102319.8.1 The Renormalization Group 102319.8.2 Operator Product Expansions 102419.8.3 Binary Planar Graphs and the Renormalization

of Quantum Electrodynamics 102619.9 The Master Ward Identity 102719.10 Trouble in Quantum Electrodynamics 102719.10.1 The Landau Inconsistency Problem in Quantum

Electrodynamics 102719.10.2 The Lack of Asymptotic Freedom in Quantum

Electrodynamics 102919.11 Hints for Further Reading 1029

Epilogue 1045 References 1049 List of Symbols 1061 Index 1069

One thing I have learned in a long life: that all our science, measuredagainst reality, is primitive and childlike – and yet it is the most precisething we have.

Albert Einstein (1879–1955)

The development of quantum mechanics in the years 1925 and 1926 hadproduced rules for the description of systems of microscopic particles,which involved promoting the fundamental dynamical variables of a corre-sponding classical system into operators with specified commutators Bythis means, a system, described initially in classical particle language, ac-quires characteristics associated with the complementary classical wavepicture It was also known that electromagnetic radiation contained in anenclosure, when considered as a classical dynamical system, was equiva-lent energetically to a denumerably infinite number of harmonic oscillators.With the application of the quantization process to these fictitious oscil-lators, the classical radiation field assumed characteristics describable inthe complementary classical particle language The ensuing theory of lightquantum emission and absorption by atomic systems7 marked the begin-ning of quantum electrodynamics

When it was attempted to quantize the complete electromagnetic field,8

difficulties were encountered that stem from the gauge ambiguity of thepotentials that appear in the Lagrangian formulation of the Maxwell equa-tions

From the origin of quantum electrodynamics, in the classical theory ofpoint charges, came a legacy of difficulties.9 The coupling of an electronwith the electromagnetic field implied an infinite displacement, and, in-deed, an infinite shift of all spectral lines emitted by an atomic system;10

7 P Dirac, The quantum theory of the emission and absorption of radiation, Proc

Royal Soc Ser A 14 (1927), 244–265.

8 W Heisenberg and W Pauli, On the quantum electrodynamics of wave fields

(in German), Z Phys 56 (1929), 1–61; 59 (1930), 108–190.

9 H Lorentz, Theory of Electrons, Dover, New York, 1915

10

R Oppenheimer, Note on the interaction of field and matter, Phys Rev 35

(1930), 461–477

in the reaction of the electromagnetic field stimulated by the presence

of the electron, arbitrary short wave lengths play a disproportionate and

divergent role The phenomenon of electron-positron pair creation, which

finds a natural place in the relativistic electron field theory, contributes tothis situation in virtue of the fluctuating densities of charge and current

that occur even in the vacuum state11 as the matter-field counterpart ofthe fluctuations in electric and magnetic field strengths.12

In computing the energy of a single electron relative to that of the vacuumstate, it is of significance that the presence of the electron tends to suppressthe charge-current fluctuations induced by the fluctuating electromagneticfield The resulting electron energy, while still divergent in its dependenceupon the contributions of arbitrarily short wave lengths exhibits only a

logarithmic infinity;13the combination of quantum and relativistic effectshas destroyed all correspondence with the classical theory and its stronglystructured-dependent electromagnetic mass

The existence of current fluctuations in the vacuum has other tions, since the introduction of an electromagnetic field induces currents

implica-that tend to modify the initial field; the “vacuum” acts as a polarizable

medium.14

New nonlinear electromagnetic phenomena appear, such as the scattering

of one light beam by another, or by an electrostatic field .

It is not likely that future developments will change drastically the tical results of the electron theory, which gives contemporary quantumelectrodynamics a certain enduring value Yet the real significance of thework of the past decade lies in the recognition of the ultimate problems

prac-facing electrodynamics, the problems of conceptual consistency and of

phys-ical completeness No final solution can be anticipated until physphys-ical

sci-ence has met the heroic challenge to comprehend the structure of thesub-microscopic world that nuclear exploration has revealed.15

Julian Schwinger, 1958

This quotation is taken from a beautiful collection of 34 papers which played

a fundamental role in the development of quantum electrodynamics Thisvolume was edited by Julian Schwinger from Harvard University who himselfmade fundamental contributions to this fascinating field of contemporaryphysics

In the present volume, we will use Dyson’s extremely elegant approach to

quantum electrodynamics based on the Dyson series for the S-matrix

(scat-11

The ground state of a quantum field is also called the vacuum state, by abuse oflanguage Note that the vacuum state is full of physics In particular, quantumfluctuations of the vacuum state cause the essential physical effects observed inphysical experiments

tering matrix) In the beginning of his Selected Papers, Freeman Dyson (born

1923) describes the history of quantum electrodynamics:16

My first stroke of luck was to find Nicholas Kemmer in Cambridge land) in 1946 He was the teacher I needed He rapidly became a friend

(Eng-as well (Eng-as a teacher.17Our friendship is still alive and well after 45 years.Kemmer gave two courses of lectures in Cambridge, one on nuclear physicsand one on quantum field theory In 1946, the only existing text-book onquantum field theory was the book “Quantentheorie der Wellenfelder”, byGregor Wentzel (1898–1978) written in Z¨urich and published in 1943 inVienna in the middle of the war Kemmer had been a student of Wentzeland possessed a copy of Wentzel’s book It was at that time a treasurewithout price I believe there were then only two copies in England It waslater reprinted in America and translated into English.18But in 1946, fewpeople in America knew of its existence and fewer considered it important.Kemmer not only possessed a copy, he also lent it to me and explainedwhy it was important

In 1947, I arrived at Cornell as a student and found myself, thanks toKemmer, the only person in the whole university who knew about quantumfield theory The great Hans Bethe (1906–2005) and the brilliant RichardFeynman (1918–1988) taught me a tremendous lot about many areas ofphysics, but when we were dealing with quantum field theory, I was theteacher and they were the students19

Julian Schwinger (1918–1994) had known about quantum field theory longbefore But he shared the American view that it was a mathematical ex-travagance, better avoided unless it should turn out to be essential In 1948,

he understood that it could be useful He used it for calculations of theenergy level shifts20revealed by the experiments of Lamb and Retherford,

16

F Dyson, Selected Papers of Freeman Dyson with Commentary, American ematical Society, Providence, Rhode Island, and International Press, Cambridge,Massachusetts (reprinted with permission)

18 G Wentzel, Quantum Theory of Wave Fields, Interscience, New York, 1949

19 Hans Bethe was born in Strasbourg (Alsace) in 1906 He studied at the versity of Frankfurt/Main (Germany), and he obtained his Ph.D at the Uni-versity of Munich in 1928 In 1934, he emigrated to the United States, and hewas appointed to a professorship at the Cornell University (Ithaca, New York).From 1943 until 1946, he worked in Los Alamos (New Mexico) (the Manhattanproject for constructing the atomic bomb) In 1946, Bethe returned to Cornelland brought with him a group of brilliant young experimental and theoreti-cal physicists Among them was Richard Feynman In 1967, Hans Bethe wasawarded the Nobel prize in physics for his contributions to the theory of nuclearreactions, especially his discoveries concerning the energy production in stars.See H Bethe, R Bacher, and M Livingstone, Basic Bethe: Seminal Articles onNuclear Physics 1936–1937, American Institute of Physics, 1986

Uni-20

The first calculations of the Lamb shift were accomplished by Bethe in 1947; thiswas a highlight in quantum electrodynamics

Foley and Kusch at Columbia.21 But he used it grudgingly In his cations, he preferred not to speak explicitly about quantum field theory.

publi-Instead, he spoke about Green’s Functions It turned out that the Green’s

Functions that Schwinger talked about and the quantum field theory thatKemmer talked about were fundamentally the same thing

At Cornell, I was learning Richard Feynman’s quite different way of lating atomic processes Feynman had never been interested in quantumfield theory He had his own private way of doing calculations His waywas based on things that he called “Propagators,” which were probabil-ity amplitudes for particles to propagate themselves from one space-timepoint to another He calculated the probabilities of physical processes byadding up the propagators He had rules for calculating the propagators.Each propagator was represented graphically by a collection of diagrams.Each diagram gave a pictorial view of particles moving along straight linesand colliding with one another at points where the straight lines met.When I learned this technique of drawing diagrams and calculating prop-agators from Feynman, I found it completely baffling, because it alwaysgave the right answer, but did not seem based on any solid mathemati-cal foundation Feynman called his way of calculating physical processes

calcu-“the space-time approach,” because his diagrams represented events asoccurring at particular places and at particular times The propagatorsdescribed sequences of events in space and time It later turned out thatFeynman’s propagators were merely another kind of Green’s Functions.Feynman had been talking the language of Green’s Functions all his lifewithout knowing it

Green’s Functions also appeared in the work of Sin-Itiro Tomonaga (1906–1979), who had developed independently a new elegant version of relativis-tic quantum field theory His work was done in the complete isolation ofwar-time Japan, and was published in Japanese in 1943 The rest of theworld became aware of it only in the spring of 1948, when an Englishtranslation of it arrived at Princeton sent by Hideki Yukawa (1907–1981)

to Robert Oppenheimer (1904–1967) Tomonaga was a physicist in theEuropean tradition, having worked as a student with Heisenberg (1901–1976) at Leipzig before the war For him, in contrast to Schwinger andFeynman, quantum field theory was a familiar and natural language inwhich to think

After the war, Tomonaga’s students had been applying his ideas to late the properties of atoms and electrons with high accuracy, and werereaching the same results as Schwinger and Feynman When Tomonaga’spapers began to arrive in America, I was delighted to see that he wasspeaking the language of quantum field theory that I had learned fromKemmer It did not take us long to put the various ingredients of the pud-ding together When the pudding was cooked, all three versions of the newtheory of atoms and electrons turned out to be different ways of express-ing the same basic idea The basic idea was to calculate Green’s Functionsfor all atomic processes that could be directly observed Green’s Functionsappeared as the essential link between the methods of Schwinger and Feyn-man, and Tomonaga’s relativistic quantum field theory provided the firmmathematical foundation for all three versions of quantum electrodynam-ics

calcu-21Columbia University, New York

Dyson wrote two fundamental papers on the foundations of quantum trodynamics, which are now classics:

elec-F Dyson, The radiation theories of Tomonaga, Schwinger, and Feynman,

are completely described by the S-matrix (scattering matrix) of

Heisen-berg.22 It is shown that the elements of this matrix can be calculated by

a consistent use of perturbation theory to any order in the fine-structureconstant Detailed rules are given for carrying out such calculations, and

it is shown that divergences arising from higher order radiative corrections

can be removed from the S-matrix by a consistent use of the ideas of mass

and charge renormalization.

Not considered in this paper are the problems of extending the treatment

to bound-state phenomena, and of proving the convergence of the theory

as the order of perturbation itself tends to infinity.23

In 1950, John Ward published a short note where he used a highly formalargument in order to get a specific identity:24

It has been recently proved by Dyson that all divergences in the S-matrix

may be removed by a renormalization of mass and charge Dyson defines

certain fundamental divergent operators Γ μ , S F  , D  F and gives a procedure

for their finite parts Γ μ1 , S F 1  , D F 1  by a process of successive tion It is then shown that

approxima-Γ μ = Z1−1 Γ μ1 (e1), S F  = Z2S F 1  (e1), D F  = Z3D  F 1 (e1),

e1 = Z1−1 Z2Z31/2 e,

where Z1, Z2, and Z3 are certain infinite constants and e1 is the (finite)

renormalized electronic charge Dyson conjectured that Z1= Z2 and it is

proposed here to give a formal proof of this relation.

22

W Heisenberg, The observable quantities in particle physics I–III, Z f Phys

120 (1943), 513–538, 673–702; 123 (1944), 93–112 (in German).

23

Bound states were considered in:

H Bethe and E Salpeter, A relativistic equation for bound-state problems Phys

Rev 84 (1951), 1232–1242.

H Bethe and E Salpeter, Quantum Mechanics of One- and Two-Electron Atoms,Springer, Berlin, 1957

24

J Ward, An identity in quantum electrodynamics, Phys Rev 78 (1950), p 182

(letter to the editor)