Particles and the Universe

From the Ionian School to the Higgs Boson and Beyond

Stephan Narison

• Publisher: WSPC
• Year: 2015
• eBook ISBN:
  9789814644716
• Topic: Physical Sciences
• Subtopic: Science History

This book aims to present the history and developments of particle physics from the introduction of the notion of particles by the Ionian school until the discovery of the Higgs boson at LHC in 2012. Neutrino experiments and particle accelerators where different particles have been discovered are reviewed. In particular, details about the CERN accelerators are presented. This book also discusses the future developments of the field and the work to popularize high energy physics. A short presentation of some features of astrophysics and its connection to particle physics is also included. At the end of the book, some useful tools in the research of particle physics are given for the advanced readers.

Part II

Modern Theories of Forces

Chapter 4

Quantum Electrodynamics (QED)

4.1 QED is a U (1) Gauge Theory

• Prototype gauge theory

Quantum electrodynamics (QED)¹ is the theory of the electron and photon quantum fields. It is the prototype of the quantum field and gauge theories and is a quantum and relativistic version of classical electromagnetism.

• Abelian U (1) group and gauge invariance

QED is described by the Abelian U (1) rotation group on a circle.

— It remains invariant by multiplying the electron field ψ(x, t) by a phase exp [iθ(x)] where θ(x) is the rotation angle as in Fig. 3.2 but depends on the position x of the particle.

— It is also invariant by changing the electromagnetic field A_µ(x) (called gauge field) by a new field: A_µ(x) = A_µ(x) − ∂_µθ(x) where ∂_µ

∂/∂x_µ is the derivative (small variation) of θ(x) in the space–time.

— These transformations are called gauge transformations.

Figure 4.1: (a) P. de Fermat, (b) R.P. Feynman, (c) F. Villars.

4.2 Path Integral and Feynman Diagrams

Principle of Least Action

In 1948, Feynman (Fig. 4.1) pointed out that if light is a wave and hence a field, principle of Fermat (1603/1608–1665) (Fig. 4.1) (the light path is the minimum optical path) can be deduced from the more general principle of Huygens (Fig. 1.5), which stipulates that the amplitude of a field is the sum of the contributions of all the field waves that depart from a source S and arrive at the observer

after reflection on the mirror. Waves will then browse multiple paths, the sum allows light to choose one that minimizes its time of travel. In the case of very low wavelengths (geometrical optics), this minimum path is that of the so-called distance of Fermat. By applying this summation to the motion of the particles considered as quanta of the field, Feynman gets, in the classical limit (the reduced Planck constant ħ = 0), the principle of least action corresponding to Fermat’s principle in geometrical optics. He thus managed to unify the wave and the particle.

Amplitudes (S-Matrix) and Path Integral

In the approach of least action, to calculate the transition amplitude

(or S-matrix) to go from initial point A at the time t_i to a final point B at the time t_f, must be considered the sum of all paths that meet the initial and final conditions. Each path is characterized by its weight that is the exponential of the classical action

with respect to the variation of the Lagrangian L (difference between kinetic E_c and potential energy V) in 3-dimensional space multiplied by (i/ħ). In terms of an equation, it means that:

= (i/ħ) ∫ dt L (see more in Appendix D). The complete path or functional integral corresponds to the summation over an infinite number of complex weights (infinite number of variables of integration). Then, it permits to calculate the transition amplitude

.

Perturbation Theory

However, the summation for getting the transition amplitude

is, in principle, infinite and, then, it is impossible in practice to extract the transition amplitude. In order to circumvent this problem, one proceeds by approximation series in terms of a small parameter expansion. In the case of QED, this small parameter is the square of the electric charge e (in natural units: ħ = c = 1):

where α = 1/137 is the fine structure constant. It enters via the tree level (lowest order) interaction Lagrangian

between the electron ψ and photon A^µ fields (see more in Appendix D), where the index zero indicates that the interaction is at the lowest order (tree level) of the perturbation theory. Here we have introduced the normal ordered product notation : : which means that the annihilation operators are to the right of the creation operators (see Appendix B). Therefore, in natural units, the amplitude

can be written as a sum of amplitudes evaluated at each order of the perturbation theory:

when expressed in terms of the interaction Lagrangian

is called chronological ordering or time ordered product which ...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Dedication
• Foreword
• Outline
• Contents
• I General Introduction
• II Modern Theories of Forces
• III Higgs Boson Discovery and Beyond
• IV Experiments to Go Back in Time
• V A Brief Cosmos Tour
• VI Fundamental Researches and Society
• VII Epilogue
• VIII Appendix: Useful Notes for Advanced Readers
• IX Annex
• X Bibliography and Index
• XI About the Author
• Stephan Narison