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2009 | Buch

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Quantum Field Theory II: Quantum Electrodynamics

A Bridge between Mathematicians and Physicists

verfasst von: Eberhard Zeidler

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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 essential source of our knowledge about the physical laws of nature, ranging from the seventeenth century geometrical optics up to the twentieth century theory 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 to themathematicalandphysicalaspectsofmodernquantum?eldtheorywhich comprehends 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 und- graduate and graduate students, as well as experienced scientists who want to become familiar with quantum ?eld theory, which is a fascinating topic in modern mathematics and physics.

Inhaltsverzeichnis

Frontmatter

Introduction

Prologue

The development of quantum mechanics in the years 1925 and 1926 had produced rules for the description of systems of microscopic particles, which involved promoting the fundamental dynamical variables of a corresponding classical system into operators with specified commutators. By this means, a system, described initially in classical particle language, acquires characteristics associated with the complementary classical wave picture. It was also known that electromagnetic radiation contained in an enclosure, when considered as a classical dynamical system, was equivalent energetically to a denumerably infinite number of harmonic oscillators. With the application of the quantization process to these fictitious oscillators, the classical radiation field assumed characteristics describable in the complementary classical particle language. The ensuing theory of light quantum emission and absorption by atomic systems marked the beginning of quantum electrodynamics…

1. Mathematical Principles of Modern Natural Philosophy

There exist the following fundamental principles for the mathematical description of physical phenomena in nature.

2. The Basic Strategy of Extracting Finite Information from Infinities – Ariadne’s Thread in Renormalization Theory

3. The Power of Combinatorics

In this series of monographs, we will show that:

There are highly complex mathematical structures behind the idea of the renormalization of quantum field theories.

The combinatorial structure of Feynman diagrams lies at the heart of renormalization methods. In the standard Boguliubov–Parasiuk–Hepp–Zimmermann (BPHZ) approach, the regularization of algebraic Feynman integrals is carried out by an iterative method which was invented by Bogoliubov in the 1950s. It was shown by Zimmermann in 1969 that Bogoliubov’s iterative method can be solved in a closed form called the Zimmermann forest formula. Finally, it was discovered by Kreimer in 1998 that Zimmermann’s forest formula can be formulated by using the coinverse of an appropriate Hopf algebra for Feynman graphs. This will be thoroughly studied later on. In this chapter, we only want to discuss some basic ideas about Hopf algebras and Rota–Baxter algebras.

4. The Strategy of Equivalence Classes in Mathematics

One of the main strategies in the sciences consists in using classifications. This means that we put single objects into classes. Instead of studying individual objects, we investigate the properties of classes. This is a simple, but extremely powerful general strategy. For example, the first systematic classification of plants and animals was developed by the Swedish biologist Carl von Linné (1707–1778). Baron de la Brède et de Montesquieu (1689–1755) said:

Intelligence consists of this; that we recognize the similarity of different things and the difference between similar ones.

Basic Ideas in Classical Mechanics

5. Geometrical Optics

In 1636, the year in which Harvard College was founded, Réne Descartes (1596–1650) was putting his last hand to his Discourse sur la méthode de bien conduire sa raison which contained among others his geometry and also his dioptrics. In 1637, this book came into the hands of Pierre de Fermat (1601–1665). In 1657 Fermat received from the physician of King Louis XIV and of Mazarin, Cureau de la Chambre, in his time a very reputed man who was also a physicist of note, a treatise about optics. In the letter in which he acknowledged the receipt of this book, he stated for the first time his idea that the law of refraction might be deduced from the minimum principle of shortest time, just like the Greek engineer Heron of Alexandria (100 A.D.) had done for the reflection of light… In a letter dated January 1, 1662, he announces to Cureau de la Chambre that he found to his amazement that his principle was yielding a new demonstration of Descartes’ refraction law …

6. The Principle of Critical Action and the Harmonic Oscillator – Ariadne’s Thread in Classical Mechanics

The aim of this and the following chapter is to explain the basic physical and mathematical ideas of classical mechanics and quantum mechanics by considering the so-called harmonic oscillator. In all fields of physics, one encounters oscillating systems. Let us mention the following examples:

•

electromagnetic waves and light (photons);

•

laser beams (coherent states);

•

oscillating molecules in a gas or a liquid;

•

sound waves (phonons);

•

oscillations of a crystal lattice (phonons);

•

oscillations of a string (e.g., a violin string);

•

waves in a plasma (plasmons);

•

matter waves of elementary particles (e.g., electrons);

•

gravitational waves (gravitons).

Basic Ideas in Quantum Mechanics

7. Quantization of the Harmonic Oscillator – Ariadne’s Thread in Quantization

In this chapter, we will study the following quantization methods:

•

Heisenberg quantization (matrix mechanics; creation and annihilation operators),

•

Schrödinger quantization (wave mechanics; the Schrödinger partial differential equation),

•

Feynman quantization (integral representation of the wave function by means of the propagator kernel, the formal Feynman path integral, the rigorous infinite-dimensional Gaussian integral, and the rigorous Wiener path integral),

•

Weyl quantization (deformation of Poisson structures),

•

Weyl quantization functor from symplectic linear spaces to C^*-algebras,

•

Bargmann quantization (holomorphic quantization),

•

supersymmetric quantization (fermions and bosons).

8. Quantum Particles on the Real Line – Ariadne’s Thread in Scattering Theory

The S-Matrix is the most important quantity in elementary particle physics. The fundamental role played by the S-matrix was emphasized by Heisenberg in 1943. He was motivated by the following philosophy:

Use only such quantities in quantum physics which are closely related to physical experiments.

9. A Glance at General Scattering Theory

The main tool for studying experimentally the properties of elementary particles are scattering processes performed in particle accelerators. Physicists measure cross sections and the mathematical theory has to develop methods for computing cross sections.

Quantum Electrodynamics (QED)

10. Creation and Annihilation Operators

We want to study a mathematical formalism which describes creation and annihilation operators for many-particle systems. We have to distinguish between

•

bosons (particles with integer spin like photons, gluons, vector bosons, and gravitons) and

•

fermions (particles with half-integer spin like electrons, neutrinos, and quarks).

11. The Basic Equations in Quantum Electrodynamics

The Einstein convention. Let us choose a fixed inertial system with x = (x⁰, x¹,x²,x³). Here, x=x¹i+x²j+x³k is the position vector of a Cartesian coordinate system with the right-handed orthonormal basis i,j,k. We also set x⁰:=t where t denotes time. For the indices μ,ν=0,1,2,3, we set η₀₀:= 1, η₁₁= η₂₂= η₃₃:=-1, η_μν:= 0 if μ≠ν along with η^{μ
ν}:=η_{μ
ν}. As usual, we use the η-symbol for lifting and lowering of indices.

12. The Free Quantum Field of Electrons, Positrons, and Photons

For vanishing coupling constant, κ_QED=0 (free fields), we want to consider solutions of the classical field equations in the form of a finite Fourier series.

13. The Interacting Quantum Field, and the Magic Dyson Series for the S-Matrix

Let us first summarize the two key formulas (13.1) and (13.6). The motivation will be given below.

14. The Beauty of Feynman Diagrams in QED

In elementary particle physics, physicists use the highly intuitive language of Feynman diagrams as a universal tool. It is crucial to know that the geometric Feynman diagrams come from well-defined analytic expressions generated by applying the Wick theorem to the Dyson series.

15. Applications to Physical Effects

We want to apply the method of Feynman diagrams to the following problems:

•

the scattering of electrons and photons (Compton effect);

•

scattering of particles in an external electromagnetic field;

•

the spontaneous emission of photons by atoms, and

•

the Cherenkov effect.

Renormalization

16. The Continuum Limit

In Sect. 14.7 we applied this golden rule to the cross section of the Compton scattering between photons and electrons. This way, we obtained the famous Klein–Nishina formula in lowest order of perturbation theory. This formula can be established by physical experiments. In this connection, we argued as follows: We started with a lattice. Then we studied the three fundamental limits: P_max→+∞ (high-energy limit), mathcalV→+∞,Δp→0 (low-energy limit), and T→+∞ (long-time limit), by using the language of distributions. In addition, we included the two limits ε→+0 (electron propagator regularization) and m_ph→+0 (photon propagator regularization – the virtual photon mass goes to zero). In this chapter, we want to discuss how this approach can be applied to general problems in quantum electrodynamics.

17. Radiative Corrections of Lowest Order

In Chap. 15 we have studied some applications to physical processes by using the lowest order of perturbation theory. The next step is to compute corrections in higher order of perturbation theory. Such so-called radiative corrections suffer under the appearance of divergent algebraic Feynman integrals. Therefore, we have to use the method of renormalization in order to get finite expressions which depend on the free parameters m_eff and e_eff. These parameters must be determined by physical experiments. Typical radiative corrections in lowest possible order (so-called one-loop corrections) are:

•

the radiative correction of the Coulomb potential (Uehling 1935);

•

the Lamb shift of the spectrum of the hydrogen atom (Bethe 1947);

•

the anomalous magnetic moment of the electron (Schwinger 1947);

•

scattering of photons by photons (Euler 1936, Karpus and Neumann 1950).

18. A Glance at Renormalization to all Orders of Perturbation Theory

In this chapter, we will sketch the basic ideas of general renormalization theory. A detailed study will be postponed to Vol. IV on quantum mathematics. Renormalization theory cannot be understood without knowing its long and strange history. At this point we merely restrict ourselves to a few quotations and comments. In the next chapter, we will sketch some basic ideas together with detailed hints for further reading.

19. Perspectives

In this chapter we will take the cable railway. We postpone the mountain climbing to Vol. IV on quantum mathematics. There exist the following two basic approaches to quantum field theory:

(S)

the S-matrix approach (scattering matrix), and

(G)

the Green’s function approach (correlation functions).

The main ideas are discussed in Chaps. 14 and 15 of Vol. I. A detailed study will be carried out in Vol. IV. Roughly speaking, the two approaches (S) and (G) are equivalent to each other.

Backmatter

Titel: Quantum Field Theory II: Quantum Electrodynamics
verfasst von: Eberhard Zeidler
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-85377-0
Print ISBN: 978-3-540-85376-3
DOI: https://doi.org/10.1007/978-3-540-85377-0