LectureNotes dirac.pdf

Lecture notes to the 1-st year master course

Particle Physics 1

Nikhef - Autumn 2013

Marcel Merk ([email protected])

Wouter Hulsbergen ([email protected])

Contents

Preliminaries i

1 Particles and Forces 11.1 The Yukawa Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Strange Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 The Eightfold Way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 The Quark Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Units in particle physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Wave Equations and Anti-Particles 192.1 Particle-wave duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 The Schrodinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Intermezzo: four-vector notation . . . . . . . . . . . . . . . . . . . . . . . 252.4 The Klein-Gordon equation . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Interpretation of negative energy solutions . . . . . . . . . . . . . . . . . 28

2.5.1 Diracs interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 292.5.2 Pauli-Weisskopf interpretation . . . . . . . . . . . . . . . . . . . . 302.5.3 Feynman-Stuckelberg interpretation . . . . . . . . . . . . . . . . . 30

3 The Electromagnetic Field 353.1 The Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3 The photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Electrodynamics in quantum mechanics . . . . . . . . . . . . . . . . . . . 413.5 The Aharanov-Bohm Effect . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Perturbation Theory and Fermis Golden Rule 474.1 Decay and scattering observables . . . . . . . . . . . . . . . . . . . . . . 474.2 Non-relativistic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Relativistic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3.1 Normalisation of the Wave Function . . . . . . . . . . . . . . . . 554.3.2 The Flux Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3.3 The Phase Space Factor . . . . . . . . . . . . . . . . . . . . . . . 57

i

ii CONTENTS

4.3.4 Golden rules for cross-section and decay . . . . . . . . . . . . . . 59

5 Electromagnetic Scattering of Spinless Particles 635.1 Electromagnetic current . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Coulomb scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.3 Spinless pi K Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 675.4 Feynman calculus: propagators and vertex factors . . . . . . . . . . . . . 705.5 Form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.6 Particles and Anti-Particles . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 The Dirac Equation 776.1 Spin, spinors and the gyromagnetic ratio . . . . . . . . . . . . . . . . . . 776.2 Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.3 Covariant form of the Dirac equation . . . . . . . . . . . . . . . . . . . . 816.4 Dirac algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.5 Adjoint spinors and current density . . . . . . . . . . . . . . . . . . . . . 836.6 Bilinear covariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.7 Charge current and anti-particles . . . . . . . . . . . . . . . . . . . . . . 85

7 Solutions of the Dirac Equation 897.1 Plane waves solutions with p = 0 . . . . . . . . . . . . . . . . . . . . . . 897.2 Plane wave solutions for p 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . 907.3 Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.4 Antiparticle spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937.5 Normalization of the wave function . . . . . . . . . . . . . . . . . . . . . 947.6 The completeness relation . . . . . . . . . . . . . . . . . . . . . . . . . . 957.7 The charge conjugation operation . . . . . . . . . . . . . . . . . . . . . . 96

8 Spin-1/2 Electrodynamics 1018.1 Feynman rules for fermion scattering . . . . . . . . . . . . . . . . . . . . 1018.2 Electron-muon scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048.3 Crossing: the process ee+ + . . . . . . . . . . . . . . . . . . . . . 1098.4 Summary of QED Feynman rules . . . . . . . . . . . . . . . . . . . . . . 111

9 The Weak Interaction 1179.1 The 4-point interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199.2 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219.3 Covariance of the wave equations under parity . . . . . . . . . . . . . . . 1239.4 The V A interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1259.5 The propagator of the weak interaction . . . . . . . . . . . . . . . . . . . 1259.6 Muon decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1269.7 Quark mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

9.7.1 Cabibbo - GIM mechanism . . . . . . . . . . . . . . . . . . . . . . 1319.7.2 The Cabibbo - Kobayashi - Maskawa (CKM) matrix . . . . . . . 132

CONTENTS iii

10 Local Gauge Invariance 13710.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13710.2 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13810.3 Global phase invariance and Noethers theorem . . . . . . . . . . . . . . 14010.4 Local phase invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14110.5 Application to the Lagrangian for a Dirac field . . . . . . . . . . . . . . . 14210.6 Yang-Mills theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14310.7 Historical interlude 1: isospin, QCD and weak isospin . . . . . . . . . . . 14710.8 Historical interlude 2: the origin of the name gauge theory . . . . . . . 148

11 Electroweak Theory 15311.1 SU(2) symmetry for left-handed douplets . . . . . . . . . . . . . . . . . . 15411.2 The Charged Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15611.3 The Neutral Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15811.4 Couplings for Z ff . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16111.5 The mass of the W and Z bosons . . . . . . . . . . . . . . . . . . . . . . 162

12 The Process ee+ + 16512.1 Helicity conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16512.2 The cross section of ee+ + . . . . . . . . . . . . . . . . . . . . . 166

12.2.1 Photon contribution . . . . . . . . . . . . . . . . . . . . . . . . . 16712.2.2 Z0 contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16912.2.3 Correcting for the finite width of the Z0 . . . . . . . . . . . . . . 17112.2.4 Total unpolarized cross-section . . . . . . . . . . . . . . . . . . . 17112.2.5 Near the resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 173

12.3 The Z0 decay widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17412.4 Forward-backward asymmetry . . . . . . . . . . . . . . . . . . . . . . . . 17512.5 The number of light neutrinos . . . . . . . . . . . . . . . . . . . . . . . . 175

A Summary of electroweak theory 179

B Some properties of Dirac matrices i and 183

Preliminaries

These are the lecture notes for the Particle Physics 1 (PP1) master course of that istaught at Nikhef in the autumn semester of 2013. The notes mainly follows the materialas discussed in the books of Halzen and Martin and Griffiths. They have been writtenby Marcel Merk in the period 2000-2011. Some parts have been edited by WouterHulsbergen for the master courses of 2012 and 2013.

The notes can be used by the students but should not be distributed. The originalmaterial is found in the books used to prepare the lectures (see below).

The contents of particle physics 1 is the following:

Lecture 1: Concepts and History

Lecture 2 - 5: Electrodynamics of spinless particles

Lecture 6 - 8: Electrodynamics of spin 1/2 particles

Lecture 9: The Weak interaction

Lecture 10 - 12: Electroweak scattering: The Standard Model

Each lecture of 2 45 minutes is followed by a 1 hour problem solving session.

The particle physics 2 course contains the following topics:

The Higgs Mechanism

Quantum Chromodynamics

This course will be given by Michiel Botje en Ivo van Vulpen.

In addition the master offers in the next semester topical courses (not obligatory) onthe particle physics subjects: CP Violation, Neutrino Physics and Physics Beyond theStandard Model

i

ii PRELIMINARIES

Examination

The examination consists of two parts: Homework (weight=1/3) and an Exam (weight=2/3).

Literature

The following literature is used in the preparation of this course (the comments reflectmy personal opinion):

Halzen & Martin: Quarks & Leptons: an Introductory Course in Modern ParticlePhysics :Although it is somewhat out of date (1984), I consider it to be the best book in the field

for a master course. It is somewhat of a theoretical nature. It builds on the earlier work

of Aitchison (see below). Most of the course follows this book.

Griffiths: Introduction to Elementary Particle Physics, second, revised ed.The text is somewhat easier to read than H & M and is more up-to-date (2008) (e.g.

neutrino oscillations) but on the other hand has a somewhat less robust treatment in

deriving the equations.

Perkins: Introduction to High Energy Physics, (1987) 3-rd ed., (2000) 4-th ed.The first three editions were a standard text for all experimental particle physics. It is

dated, but gives an excellent description of, in particular, the experiments. The fourth

edition is updated with more modern results, while some older material is omitted.

Aitchison: Relativistic Quantum Mechanics(1972) A classical, very good, but old book, often referred to by H & M.

Aitchison & Hey: Gauge Theories in Particle Physics(1982) 2nd edition: An updated version of the book of Aitchison; a bit more theoretical.

(2003) 3rd edition (2 volumes): major rewrite in two volumes; very good but even more

theoretical. It includes an introduction to quantum field theory.

Burcham & Jobes: Nuclear & Particle Physics(1995) An extensive text on nuclear physics and particle physics. It contains more

(modern) material than H & M. Formulas are explained rather than derived and more

text is spent to explain concepts.

Das & Ferbel: Introduction to Nuclear and Particle Physics(2006) A book that is half on experimental techniques and half on theory. It is more

suitable for a bachelor level course and does not contain a treatment of scattering theory

for particles with spin.

Martin and Shaw: Particle Physics , 2-nd ed.(1997) A textbook that is somewhere inbetween Perkins and Das & Ferbel. In my

opinion it has the level inbetween bachelor and master.

iii

Particle Data Group: Review of Particle PhysicsThis book appears every two years in two versions: the book and the booklet. Both of

them list all aspects of the known particles and forces. The book also contains concise,

but excellent short reviews of theories, experiments, accellerators, analysis techniques,

statistics etc. There is also a version on the web: http://pdg.lbl.gov

The Internet:In particular Wikipedia contains a lot of information. However, one should notethat Wikipedia does not contain original articles and they are certainly not re-viewed! This means that they cannot be used for formal citations.

In addition, have a look at google books, where (parts of) books are online avail-able.

iv PRELIMINARIES

About Nikhef

Nikhef is the Dutch institute for subatomic physics. (The name was originally anacronym for Nationaal Instituut voor Kern en Hoge Energie Fysica.) The nameNikhef is used to indicate simultaneously two overlapping organisations:

Nikhef is a national research lab (institute) funded by the foundation FOM; thedutch foundation for fundamental research of matter.

Nikhef is also a collaboration between the Nikhef institute and the particle physicsdepartements of the UvA (Adam), the VU (Adam), the UU (Utrecht) and theRU (Nijmegen) contribute. In this collaboration all dutch activities in particlephysics are coordinated.

In addition there is a collaboration between Nikhef and the Rijksuniversiteit Groningen(the former FOM nuclear physics institute KVI) and there are contacts with the Uni-versities of Twente, Leiden and Eindhoven. For more information see the Nikhef webpage: http://www.nikhef.nl.

The research at Nikhef includes both accelerator based particle physics and astro-particlephysics. A strategic plan, describing the research programmes at Nikhef can be foundon the web, from: www.nikhef.nl/fileadmin/Doc/Docs & pdf/StrategicPlan.pdf .

The accelerator physics research of Nikhef is currently focusing on the LHC experiments:Alice (Quark gluon plasma), Atlas (Higgs) and LHCb (CP violation). Each ofthese experiments search answers for open issues in particle physics (the state of matterat high temperature, the origin of mass, the mechanism behind missing antimatter) andhope to discover new phenomena (eg supersymmetry, extra dimensions).

The LHC has started taking data in 2009 and the first LHC physics run has officiallyended in winter 2013. In a joint meeting in summer 2012 the ATLAS and CMS experi-ments presented their biggest discovery so far, a new particle, called the Higgs particle,with a mass of approximately 125 GeV. The existence of this particle was a predictionof the Standard Model. Further research concentrates on signs of physics beyond theStandard Model, sometimes called New Physics.

In preparation of these LHC experiments Nikhef has also been active at other labs:STAR (Brookhaven), D0 (Fermilab) and Babar (SLAC). Previous experiments thatended their activities are: L3 and Delphi at LEP, and Zeus, Hermes and HERA-B atDesy.

A more recent development is the research field of astroparticle physics. It includesAntares & KM3NeT (cosmic neutrino sources), Pierre Auger (high energy cosmicrays), Virgo & ET (gravitational waves) and Xenon (dark matter).

Nikhef houses a theory departement with research on quantum field theory and gravity,string theory, QCD (perturbative and lattice) and B-physics.

vDriven by the massive computing challenge of the LHC, Nikhef also has a scientificcomputing departement: the Physics Data Processing group. They are active in thedevelopment of a worldwide computing network to analyze the large datastreams fromthe (LHC-) experiments (The Grid).

Nikhef program leaders/contact persons:

Name office phone email

Nikhef director Frank Linde H232 5001 [email protected] Eric Laenen H323 5127 [email protected] Paul de Jong H253 2087 [email protected] Marcel Merk N243 5107 [email protected] Thomas Peitzmann N325 5050 [email protected] Aart Heijboer H342 5116 [email protected] Auger Charles Timmermans - - [email protected] and ET Jo van den Brand N247 2015 [email protected] Patrick Decowski H349 2145 [email protected] R&D Frank Linde H232 5001 [email protected] Computing Jeff Templon H158 2092 [email protected]

vi PRELIMINARIES

A very brief history of particle physics

The book of Griffiths starts with a nice historical overview of particle physics in theprevious century. This is a summary of key events:

Atomic Models

1897 Thomson: Discovery of Electron. The atom contains electrons as plums ina pudding.

1911 Rutherford: The atom mainly consists of empty space with a hard and heavy,positively charged nucleus.

1913 Bohr: First quantum model of the atom in which electrons circled in stableorbits, quatized as: L = ~ n

1932 Chadwick: Discovery of the neutron. The atomic nucleus contains bothprotons and neutrons. The role of the neutrons is associated with the bindingforce between the positively charged protons.

The Photon

1900 Planck: Description blackbody spectrum with quantized radiation. No inter-pretation.

1905 Einstein: Realization that electromagnetic radiation itself is fundamentallyquantized, explaining the photoelectric effect. His theory received scepticism.

1916 Millikan: Measurement of the photo electric effect agrees with Einsteinstheory.

1923 Compton: Scattering of photons on particles confirmed corpuscular characterof light: the Compton wavelength.

Mesons

1934 Yukawa: Nuclear binding potential described with the exchange of a quan-tized field: the pi-meson or pion.

1937 Anderson & Neddermeyer: Search for the pion in cosmic rays but he finds aweakly interacting particle: the muon. (Rabi: Who ordered that?)

1947 Powell: Finds both the pion and the muon in an analysis of cosmic radiationwith photo emulsions.

Anti matter

1927 Dirac interprets negative energy solutions of Klein Gordon equation as energylevels of holes in an infinite electron sea: positron.

1931 Anderson observes the positron.

vii

1940-1950 Feynman and Stuckelberg interpret negative energy solutions as the positiveenergy of the anti-particle: QED.

Neutrinos

1930 Pauli and Fermi propose neutrinos to be produced in -decay (m = 0).

1958 Cowan and Reines observe inverse beta decay.

1962 Lederman and Schwarz showed that e 6= . Conservation of lepton number.Strangeness

1947 Rochester and Butler observe V 0 events: K0 meson.

1950 Anderson observes V 0 events: baryon.

The Eightfold Way

1961 Gell-Mann makes particle multiplets and predicts the .

1964 particle found.

The Quark Model

1964 Gell-Mann and Zweig postulate the existance of quarks

1968 Discovery of quarks in electron-proton collisions (SLAC).

1974 Discovery charm quark (J/) in SLAC & Brookhaven.

1977 Discovery bottom quarks ( ) in Fermilab.

1979 Discovery of the gluon in 3-jet events (Desy).

1995 Discovery of top quark (Fermilab).

Broken Symmetry

1956 Lee and Yang postulate parity violation in weak interaction.

1957 Wu et. al. observe parity violation in beta decay.

1964 Christenson, Cronin, Fitch & Turlay observe CP violation in neutral K mesondecays.

The Standard Model

1978 Glashow, Weinberg, Salam formulate Standard Model for electroweak inter-actions

1983 W-boson has been found at CERN.

1984 Z-boson has been found at CERN.

1989-2000 LEP collider has verified Standard Model to high precision.

viii PRELIMINARIES

Lecture 1

Particles and Forces

After Chadwick had discovered the neutron in 1932, the elementary constituents ofmatter were the proton and the neutron inside the atomic nucleus, and the electroncircling around it. The force responsible for interactions between charged particleswas the electromagnetic force. Moving charges emitted electromagnetic waves, whichhappened to be quantized in energy and were called photons. With these constituentsthe atomic elements could be described, as well as their chemistry. The answer to thequestion: What is the world made of? was indeed rather simple.

However, there were already some signs that there were more fundamental particles thanjust protons, neutrons, electrons and photons:

Dirac had postulated in 1927 the existence of anti-matter as a consequence of hisrelativistic version of the Schrodinger equation in quantum mechanics. (We willcome back to the Dirac theory later on.) The anti-matter partner of the electron,the positron, was actually discovered in 1932 by Anderson (see Fig. 1.1).

Pauli had postulated the existence of an invisible particle that was produced innuclear beta decay: the neutrino. In a nuclear beta decay process:

NA NB + e

the energy of the emitted electron is determined by the mass difference of the nucleiNA and NB. It was observed that the kinetic energy of the electrons, however,showed a broad mass spectrum (see Fig. 1.2), of which the maximum was equalto the expected kinetic energy. It was as if an additional invisible particle of lowmass is produced in the same process: the (anti-) neutrino.

1.1 The Yukawa Potential

Though the constituents of atoms were fairly well established, there was somethingpuzzling about atoms: What was keeping the nucleus together? It clearly had to be

1

2 LECTURE 1. PARTICLES AND FORCES

Figure 1.1: The discovery of the positron as reported by Anderson in 1932. Knowing thedirection of the B field Anderson deduced that the trace was originating from an anti electron.Question: how?

a new force, something beyond electromagnetism. Rutherfords scattering experimentshad given an estimate of the size of the nucleus, of about 1 fm. With protons packed thisclose, the new force had to be very strong to overcome the repulsive coulomb interactionof the protons. (Being imaginative, physicists simply called it the strong nuclear force.)Yet, to explain scattering experiments, the range of the force had to be small, boundjust to the nucleus itself.

In an attempt to solve this problem Japanese physicist Yukawa published in 1935 a fun-damentally new view of interactions. His idea was that forces, like the electromagneticforce and the nuclear force, could be described by the exchange of virtual particles, asillustrated in Fig. 1.3. These particles (or rather, their field) would follow a relativisticwave-equation, just like the electromagnetic field.

In this picture, the massless photon was the carrier of the electromagnetic field. As wewill see in exercise 1.6 the relativistic wave equation for a massless particle leads to anelectrostatic potential of the form

V (r) = e2 1r

(1.1)

where e is the fundamental unit of charge, also called the electromagnetic couplingconstant. Because of its 1/r dependence, the force is said to be of infinite range.

By contrast, in Yukawas proposal the strong force were to be carried by a massiveparticle, later called the pion. A massive force carrier leads to a potential of the form

U(r) = g2 er/R

r(1.2)

1.1. THE YUKAWA POTENTIAL 3

0.8

0.6

0.4

0.2

1.0

02 6 10 14 18

Energy (keV)

0.00004

018.45 18.50 18.55 18.60

0.00012

0.00008

Mass = 0

Mass = 30 eV

Rel

ativ

e D

ecay

Pro

babi

lity

Figure 1. The Beta Decay Spectrum for Molecular Tritium The plot on the left shows the probability that the emerging electron has a particular energy. If the electron were neutral, the spectrum would peak at higher energy andwould be centered roughly on that peak. But because the electron is negativelycharged, the positively charged nucleus exerts a drag on it, pulling the peak to alower energy and generating a lopsided spectrum. A close-up of the endpoint (plot on the right) shows the subtle difference between the expected spectra for a massless neutrino and for a neutrino with a mass of 30 electron volts.

Figure 1.2: The beta spectrum as observed in tritium decay to helium. The endpoint of thespectrum can be used to set a limit of the neutrino mass. Question: how?

which is called the One Pion Exchange Potential. Since it falls of exponentially, it hasa finite range. The range R is inversely proportional to the mass of the force carrier andfor a massless carrier the expression reduces to that for the electrostatic potential.

In the exercise you will derive the relation between mass and range properly, but itcan also be obtained with a heuristic argument, following the Heisenberg uncertaintyprinciple. (As you can read in Griffiths, whenever a physicists refers to the uncertaintyprinciple to explain something, take all results with a grain of salt.) In some interpreta-tion, the principle states that we can borrow the energy E = mc2 to create a virtualparticle from the vacuum, as long as we give it back within a time t ~/E. Withthe particle traveling at the speed of light, this leads to a range R = ct = c~/mc2.

From the size of the nucleus, Yukawa estimated the mass of the force carrier to be ap-proximately 100 MeV/c2. He called the particle a meson, since its mass was somewherein between the mass of the electron and the nucleon.

In 1937 Anderson and Neddermeyer, as well as Street and Stevenson, found that cosmicrays indeed consist of such a middle weight particle. However, in the years after, itbecame clear that this particle could not really be Yukawas meson, since it did notinteract strongly, which was very strange for a carrier of the strong force. In fact thisparticle turned out to be the muon, the heavier brother of the electron.

Only in 1947 Powell (as well as Perkins) found Yukawas pion in cosmic rays. Theytook their photographic emulsions to mountain tops to study the contents of cosmic


Figure 1.3: Illustration of the interaction between protons and neutrons by charged pionexchange. (From Aichison and Hey.)

rays (see Fig. 1.4). (In a cosmic ray event a cosmic proton scatters with high energyon an atmospheric nucleon and produces many secondary particles.) Pions produced inthe atmosphere decay long before they reach sea level, which is why they had not beenobserved before.

As a carrier of the strong force Yukawas meson did not stand the test of time. We nowknow that the pion is a composite particle and that the true carrier for the strong force isthe massless gluon. The range of the strong force is small, not because the force carrieris massive, but because gluons carry a strong interaction charge themselves. However,even if Yukawas original meson model did not survive, his interpretation of forces asthe exchange of virtual particles is still central to the description of particle interactionsin quantum field theory.

1.2 Strange Particles

After the pion had been identified as Yukawas strong force carrier and the anti-electronwas observed to confirm Diracs theory, things seemed reasonably well under control.The muon was a bit of a mystery. It lead to a famous quote of Isidore Rabi at theconference: Who ordered that?

But in December 1947 things became a lot more complicated when Rochester and Butlerobserved so-called V 0 events in cloud chamber photographs. It turned out that cosmicparticles hitting a lead larget plate could produce many different types of new particles.Those new particles were classified as:

baryons: particles whose decay products ultimately include a proton;

mesons: particles whose decay products ultimately include only leptons or photons.

1.2. STRANGE PARTICLES 5

Figure 1.4: A pion entering from the left decays into a muon and an invisible neutrino.


These events were strange because of an apparent mismatch between their productioncross-section and their decay time. The observed yield indicated that the cross-sectionfor these events was large, 1027cm2, typical for a strong nuclear interaction process.On the other hand, the long lifetime of these particles was typical for a weak interactiondecay. So, if these were strongly interacting particles, why did not they decay faster?

The explanation to this puzzle was given by Abraham Pais in 1952 and is called asso-ciated production. Pais suggested that these particles carried a new additive quantumnumber, called strangeness, that is conserved in strong interactions but not in weakinteractions. The production of particles with non-zero strangeness via the strong inter-action can only occur in pairs: a particle with strangeness S = +1 is always producedtogether with a particle with strangeness is S = 1 such that total strangeness is con-served. Such particles with non-zero strangeness would be stable if not for the weakinteraction: each of the created particles decays through the weak interaction, leadingto a large lifetime. An example of an associated production event is seen in Fig. 1.5.

In the years 1950 - 1960 many elementary particles were discovered and one started tospeak of the particle zoo. A quote: The finder of a new particle used to be awardedthe Nobel prize, but such a discovery now ought to be punished by a $10.000 fine.

1.3 The Eightfold Way

In the early 60s Murray Gell-Mann (at the same time also Yuval Neeman) observedpatterns of symmetry in the discovered mesons and baryons. He plotted the spin 1/2baryons in a so-called octet (the eightfold way after the eightfold way to Nirvana inBuddhism). The octet of the lightest baryons and mesons is displayed in Fig. 1.6 andFig. 1.7. In these graphs the strangeness quantum number is plotted vertically.

Also heavier hadrons could be given a place in multiplets. The baryons with spin=3/2were seen to form a decuplet, see Fig. 1.8. The particle at the bottom (at S=-3) had notbeen observed. Not only was it found later on, but also its predicted mass was found tobe correct! The discovery of the particle is shown in Fig. 1.9.

1.4 The Quark Model

The observed structure of hadrons in multiplets hinted at an underlying structure. Gell-Mann and Zweig postulated indeed that hadrons consist of more fundamental partons:the quarks. Initially three quarks and their anti-particle were assumed to exist (see Fig.1.10). A baryon consists of 3 quarks: (q, q, q), while a meson consists of a quark and anantiquark: (q, q). Mesons can be their own anti-particle, baryons cannot.

How does this explain that baryons and mesons appear in the form of octets, decuplets,nonets etc.? For example a baryon, consisting of 3 quarks with 3 flavours (u, d, s),

1.4. THE QUARK MODEL 7

Figure 1.5: A bubble chamber picture of associated production.

n

p

+

+0

0

Q=1 Q=0

S=0

S=1

S=2

Q=+1Figure 1.6: Octet of lightest baryons with spin=1/2.


S=1

+

Q=1 Q=0 Q=1

K

0

K0

0

+S=1

S=0

Figure 1.7: Octet with lightest mesons of spin=0

+

0

0 + ++

0

Q=1Q=0

Q=+1Q=+2

S=0

S=1

S=2

S=3

mass

~1230 MeV

~1380 MeV

~1530 MeV

~1680 MeV

Figure 1.8: Decuplet of baryons with spin=3/2. The was not yet observed when thismodel was introduced. Its mass was predicted.


Figure 1.9: Discovery of the omega particle.

S=0

S=1 Q=+2/3Q=1/3

s

d u S=+1 s

d

Q=+1/3Q=2/3

S=0u

Figure 1.10: The fundamental quarks: u,d,s.


could in principle lead to 3x3x3=27 combinations. The answer lies in the fact that thewave function of fermions is subject to a symmetry under exchange of fermions. Sincebaryons are fermions (half-integer spin) the total wave function must be anti-symmetricwith respect to the interchange of two quarks. The multiplets are grouped by thesymmetry of the quark wave functions, as illustrated in Fig. 1.11. The assumptionthat u, d and s quarks are interchangeable leads to a symmetry called SU(3) and themultiplets represent different representations of that group, identified by their total spin.(In the exercises you will work out an example yourself.) In group theory language thedecomposition is written as

3 3 3 = 10 8 8 1for the baryons and as

3 3 = 8 1for the mesons. For more information on the static quark model read 2.10 and 2.11in H&M, 5.5 and 5.6 in Griffiths, or chapter 5 in the book of Perkins.

Figure 1.11: Quark wave functions of the baryon multiplets. (From Griffiths.)

The baryon wave function can be schematically written as

(baryon) = (space) (spin) (flavour)


On closer inspection (not detailed here) this leads to a problem for the states with threeequal quarks in the decouplet, like the +++: their wave function would not be anti-symmetric. A solution is to introduce yet another SU(3) symmetry for quarks: eachquark actually comes in three different configurations, labeled by the quantum numbercolour. The colour quantum number takes values red (r), green (g) or blue (b) for quarksand anti-red (r), anti-green (g) or anti-blue (b) for anti-quarks. Of course, in the newSU(3) symmetry we would also expect a division in a multiplets. However, for a reasonthat is still not entirely understood all naturally occurring hadrons are in the coloursinglet state only. For example, the colour wave function for baryons is given by

(colour) = (rgb rbg + gbr grb+ brg bgr)/

6 (1.3)

This is anti-symmetric in the exchange of two quarks, which solves the problem forthe wave-functions of qqq baryons. Likewise, mesons occur only in a colour singletsuperposition of rr, bb and gg.

Of course, when the idea of colour was introduced, people saw it just as a somewhatartificial way to solve a problem in a model that was anyway not based on very solidfooting. However, besides the theoretical arguments, there is also direct experimentalfor this hidden degree of freedom of quarks. Consider the ratio of the cross-section ofe+e to hadrons and muons:

R (e+e hadrons)

(e+e +) = NCi

Q2i (1.4)

Since the initial state is a lepton pair, for energies well below the Z boson this cross-section is fully dominated by the electromagnetic interaction. Knowing the electriccharge of quarks and leptons, we know the coupling strength and we can predict thisratio given the number of quarks. Only quarks with a mass less than half the centre-of-momentum energy (

s) can be created. Therefore, we expect R to depend on

s and

jump with exactly the charge Qi of the quark when the energy passes the threshold toproduce that quark. On the other hand, if there are actually identical quarks with thesame mass, then we expect the jump to be higher. From the result, shown in Fig. 1.12,one can extract that NC = 3.

We now identify the SU(3)-colour symmetry with the gauge symmetry of the stronginteraction. Colour is the charge of the strong interaction, just like electric charge isthe charge of the electromagnetic interaction. The carrier of the strong interaction isthe massless gluon. Unlike the carrier of the electromagnetic interaction, which is theelectrically neutral photon, the gluon carries itself colour charge. It is believed that as aresult of the gluon self-coupling quarks are confined in colourless objects. This propertyof the strong interaction is called confinement. It implies that one can never observe afree quark or gluon!


10-1

1

10

10 2

10 3

1 10 10 2

R

J/ (2S)

Z

s [GeV]

Figure 1.12: The ratio R as a function ofs.

1.5 The Standard Model

In the Standard Model (SM) of particle physics all matter particles are spin-12

fermionsand all force carriers are spin-1 bosons. The fermions are the quarks and leptons,organized in three families:

charge Quarks

23

u (up) c (charm) t (top)1.54 MeV 1.151.35 GeV (174.3 5.1) GeV

13

d (down) s (strange) b (bottom)48 MeV 80130 MeV 4.14.4 GeV

charge Leptons

0 e (e neutrino) ( neutrino) ( neutrino)< 3 eV < 0.19 MeV < 18.2 MeV

1 e (electron) (muon) (tau)0.511 MeV 106 MeV 1.78 GeV

The force carriers are the photon, the Z and W and the gluons:

Force Boson Relative strengthStrong g (8 gluons) s O(1)

Electromagnetic (photon) O(102)Weak Z0,W (weak bosons) W O(106)

In the SM forces originate from a mechanism called local gauge invariance, which willbe discussed later on in the course. The strong force (or colour force) is mediated by

1.5. THE STANDARD MODEL 13

gluons, the weak force by intermediate vector bosons, and the electromagnetic forceby photons. Only the charged weak interaction can change the flavour of quarks andleptons: it allows for transitions between an up-type quark and a down-type quark, andbetween charged leptons and neutrinos.

Some of the fundamental diagrams are represented below:

a:

e+

e

+

b: W

e

e

c:g

q

q

q

q

Figure 1.13: Feynman diagrams of fundamental lowest order perturbation theory processesin a: electromagnetic, b: weak and c: strong interaction.

As also indicated above, there is an important difference between the electromagneticforce on one hand, and the weak and strong force on the other hand. The photon doesnot carry charge and, therefore, does not interact with itself. The gluons, however, carrycolour and do interact amongst each other. Also, the weak vector bosons carry weakisospin and undergo a self-coupling.

The strength of an interaction is determined by the coupling constant as well as themass of the vector boson. Contrary to its name the couplings are not constant, butvary as a function of energy. At a momentum transfer of 1015 GeV the couplings ofelectromagnetic, weak and strong interaction all obtain approximately the same value.In the quest of unification it is often assumed that the three forces unify to a single force(i.e. a single gauge symmetry group) at this energy.

Due to the self-coupling of the force carriers the running of the coupling constants of theweak and strong interaction are opposite to that of electromagnetism. Electromagnetismbecomes weaker at low momentum (i.e. at large distance), the weak and the strong forcebecome stronger at low momentum or large distance. The strong interaction couplingeven diverges at momenta less than a few 100 MeV (the perturbative QCD descriptionbreaks down), leading to confinement.

Finally, the Standard Model includes a scalar boson field, the Higgs field, which providesmass to the vector bosons and fermions in the Brout-Englert-Higgs mechanism. A newparticle consistent with the Higgs particle was discovered in summer 2012 by the ATLASand CMS collaborations.

Despite the success of the standard model in describing all physics at low energy scale,there are still many open questions, such as

Why are the masses of the particles what they are? Why are there 3 generations of fermions? Are quarks and leptons truly fundamental?


Figure 1.14: Running of the coupling constants and possible unification point. On the left:Standard Model. On the right: Supersymmetric Standard Model.

Is there really only one Higgs particle? Why is the charge of the electron exactly opposite to that of the proton? Or

phrased differently: why is the total charge of leptons and quarks in one generationexactly zero?

Is a neutrino its own anti-particle? Can all forces be described by a single gauge symmetry (unification)? Why is there no anti matter in the universe? What is the source of dark matter? What is the source of dark energy?

Particle physicists try to address these questions both with scattering experiments inthe laboratory and by studying high energy phenomena in the cosmos.

1.6 Units in particle physics

In particle physics we often make use of natural units to simplify expressions. In thissystem of units the action is expressed in units of Plancks constant

~ 1.055 1034Js (1.5)

and velocity is expressed in units of the speed of light in vacuum

c = 2.998 108m/s. (1.6)

such that all factors ~ and c can be omitted. As a consequence (see textbooks), there isonly one basic unit for length (L), time (T), mass (M), energy and momentum. In highenergy physics this basic unit is often chosen to be the energy in MeV or GeV, where

1.6. UNITS IN PARTICLE PHYSICS 15

1 eV is the kinetic energy an electron obtains when it is accelerated over a voltage of1V. Momentum and mass then get units of energy, while length and time get units ofinverse energy.

To confront the result of calculation with experiments the factors ~ and c usually need tobe reintroduced. There are two ways to do this. First one can take the final expressionsin natural units and then use the table 1.1 to convert the quantities for space, time,mass, energy and momentum back to their original counterparts. (For the positroncharge, see below.)

quantity symbol in natural units equivalent symbol in ordinary unitsspace x x/~ctime t t/~mass m mc2

momentum p pcenergy E E

positron charge e e~c/0

Table 1.1: Conversion of basic quantities between natural and ordinary units.

Alternatively, one can express all results in GeV, then use the following table withconversion factors to translate it into SI units:

quantity conversion factor natural unit normal unitmass 1 kg = 5.61 1026 GeV GeV GeV/c2length 1 m = 5.07 1015 GeV1 GeV1 ~c/GeVtime 1 s = 1.52 1024 GeV1 GeV1 ~/GeVTable 1.2: Conversion factors from natural units to ordinary units.

Where it concerns electromagnetic interactions, there is also freedom in choosing theunit of electric charge. The electrostatic force between two electrons in vacuum is givenby Coulombs law

F =e2

4pi0

1

r2(1.7)

where 0 is the vacuum permittivity. The dimension of the factor e2/0 is fixed it is

[L3M/T2] but this still leaves a choice of what to put in the charges and what in thevacuum permittivity.

In the SI system the unit of charge is the Coulomb. (It is currently defined via theAmpere, which in turn is defined as the current leading to a particular force betweentwo current-carrying wires. In the near future, this definition will probably be replacedby the charge corresponding to a fixed number of particles with the positron charge.)The positron charge expressed in Coulombs is about

e 1.6023 1019C (1.8)


while the vacuum permittivity is

0 8.854 1012C2s2kg1m3. (1.9)

As we shall see in Lecture 3 the Maxwell equations look much more neat if, in additionto c = 1, we choose 0 = 1. This is called the Heaviside-Lorentz system. Obviously, thischoice affects the numerical value of e. However, note that the factor

=e2

4pi0~c(1.10)

is dimensionless. This parameter is called the fine structure constant. Its value isapproximately 1/137 and independent of the system of units. It is because of the factthat 1 that perturbation theory works so well in quantum electrodynamics.Finally, it is customary to express cross sections in barn, which is equal to 1024cm2.

Glossary

hadron (greek: strong) particle that feels the strong interactionlepton (greek: light, weak) particle that feels only EM and weak interactionbaryon (greek: heavy) particle consisting of three quarksmeson (greek: middle) particle consisting of a quark and an anti-quarkpentaquark a hypothetical particle consisting of 4 quarks and an anti-quarkfermion half-integer spin particleboson integer spin particlegauge-boson force carrier as predicted from local gauge invariance

1.6. UNITS IN PARTICLE PHYSICS 17

Exercises

Exercise 1.1 (Conversion factors)Derive the conversion factors for mass, length and time in table 1.2.

Exercise 1.2 (Kinematics of Z production)The Z-boson has a mass of 91.1 GeV. It can be produced by annihilation of an electronand a positron. The mass of an electron, as well as that of a positron, is 0.511 MeV.

(a) Draw the (dominant) Feynman diagram for this process.

(b) Assume that an electron and a positron are accelerated in opposite directions andcollide head-on to produce a Z-boson in the lab frame. Calculate the beam energyrequired for the electron and the positron to produce a Z-boson.

(c) Assume that a beam of positron particles is shot on a target containing electrons.Calculate the beam energy required for the positron beam to produce Z-bosons.

(d) This experiment was carried out in the 1990s. Which method (b or c) do youthink was used? Why?

Exercise 1.3 (The Yukawa potential)

(a) The wave equation for an electromagnetic wave in vacuum is given by:

2 V = 0 ; 2 2

t22

which in the static case can be written in the form of Laplace equation:

2 V = 0Now consider a point charge in vacuum. Exploiting spherical symmetry, show thatthis equation leads to a potential V (r) 1/r.Hint: look up the expression for the Laplace operator in spherical coordinates.

(b) The wave equation for a massive field is the Klein Gordon equation:

2 U +m2 U = 0

which, again in the static case can be written in the form:

2 U m2 U = 0Show, again assuming spherical symmetry, that Yukawas potential is a solutionof the equation for a massive force carrier. What is the relation between the massm of the force carrier and the range R of the force?

(c) Estimate the mass of the pi-meson assuming that the range of the nucleon force is1.5 1015 m = 1.5 fm.


Exercise 1.4 (Quark content of hadrons)

(a.) Assign the quark contents to the particles in the baryon decuplet (Fig. 1.8) andthe meson octet (Fig. 1.7).

(b.) Using the concept of strangeness conservation, explain why the threshold energy(for pi incident on stationary protons) for

pi + p K0 + anything

is less than forpi + p K0 + anything

assuming that both processes proceed through the strong interaction. (From Aichi-son and Hey, chapter 1.)

Exercise 1.5 (The Quark Model)

(a) Quarks are fermions with spin 1/2. Show that the spin of a meson (2 quarks) canbe either a triplet of spin 1 or a singlet of spin 0.Hint: Remember the Clebsch Gordon coefficients in adding quantum numbers.In group theory this is often represented as the product of two doublets leads tothe sum of a triplet and a singlet: 2 2 = 3 1 or, in terms of quantum numbers:1/2 1/2 = 1 0.

(b) Show that for baryon spin states we can write: 1/2 1/2 1/2 = 3/2 1/2 1/2or equivalently 2 2 2 = 4 2 2

(c) Let us restrict ourselves to two quark flavours: u and d. We introduce a newquantum number, called isospin in complete analogy with spin, and we refer tothe u quark as the isospin +1/2 component and the d quark to the isospin -1/2component (or u= isospin up and d=isospin down). What are the possibleisospin values for the resulting baryon?

(d) The ++ particle is in the lowest angular momentum state (L = 0) and hasspin J3 = 3/2 and isospin I3 = 3/2. The overall wavefunction (Lspace-part,Sspin-part, Iisospin-part) must be anti-symmetric under exchange of any ofthe quarks. The symmetry of the space, spin and isospin part has a consequencefor the required symmetry of the Colour part of the wave function. Write downthe colour part of the wave-function taking into account that the particle is colourneutral.

(e) In the case that we include the s quark the flavour part of the wave functionbecomes: 3 3 3 = 10 8 8 1.In the case that we include all 6 quarks it becomes: 6 6 6. However, this isnot a good symmetry. Why not?

Lecture 2

Wave Equations and Anti-Particles

As briefly discussed in Lecture 1, most of our knowledge of the physics of elementaryparticles comes from scattering experiments, from decays and from the spectroscopyof bound states. In this course we will only consider the electroweak theory, leavingquantum chromodynamics, the theory of the strong interaction, to the Particle Physics2 course.

The theory of bound states in electrodynamics is essentially the hydrogen atom, which(apart from a few subtle phenomena) is well described by non-relativistic quantummechanics (QM). We do not discuss the hydrogen atom: instead we concentrate onprocesses at high energies, which requires an extension to relativistic velocities. Thetheoretical framework that allows this is called quantum field theory (QFT). The quan-tum field theory for electrodynamics is called Quantum Electrodynamics (QED).

To understand when classical mechanics breaks down it is useful to look at a few typicaldistance scales of electromagnetic interactions in the quantum world. The first distancescale is the Bohr radius, the distance at which an electron circles around an infinitelyheavy object (a proton) of opposite charge. Using just classical mechanics and imposingquantization of angular momentum by requiring that rp = ~, you will find (try!) thatthis distance is given by

rBohr =~

mec. (2.1)

(A proper treatment in QM tells you that the expectation value for the radius is notexactly the Bohr radius, but it comes close.) Hence, the velocity of the electron is

vBohr = p/m = c (2.2)

which indeed makes the electron in the hydrogen atom notably non-relativistic.

The second distance scale is the Compton wavelength of the electron. Suppose thatyou study electrons by shooting photons at zero velocity electrons. The smaller thewavelength of the photon, the more precise you look. However, at some point theenergy of the photons becomes large enough that you can create a new electron. (In

19

20 LECTURE 2. WAVE EQUATIONS AND ANTI-PARTICLES

our real theory, you can only create pairs, but that factor 2 is not important now.) Theenergy at which this happens is when ~ = mec2, or at a wavelength

e =2pi~mec

(2.3)

Usually, we divide both sides by 2pi and call this the reduced Compton wavelength,just like ~ is usually called the reduced Plancks constant. In electromagnetic collisionsat this energy, quantum mechanics no longer suffices: as soon as collisions involve thecreation of new particles, one needs QFT.

Finally, consider the collisions of two electrons at even higher energy. If the electronsget close enough, the Coulomb energy is sufficient to create a new electron. (Again,ignore the factor two required for pair production.) Expressing the Coulomb potentialas V (r) = ~c/r, and setting this equal to mec2, one obtains for the distance

re =~mc

(2.4)

Note that, taking into account the definition of , this expression does not explicitlydepend on ~: you do not need quantization to compute this distance, which is why itis usually called the classical radius of the electron. At energies this high lowest or-der perturbation theory may not be sufficient to compute a cross-section. The effectof screening (see Lecture 1) becomes important, amplitudes described by Feynman di-agrams with loops contribute and QED needs renormalization to provide meaningfulanswers. We will not discuss renormalization in this course.

In fact, we will hardly discuss quantum field theory at all! Do not be disappointed,there are two pragmatic reasons this. First, a proper treatment requires a proper coursewith some non-trivial math, which would leave insufficient time for other things thatwe do need to address. Second, if you accept a little handwaving here and there, thenwe do not actually need QFT: starting from quantum mechanics and special relativitywe can derive the Born level that is, leading order cross-sections, following aroute that allows us to introduce new concepts in a somewhat historical, and hopefullyenlightening, order.

However, before continuing and setting aside the field theory completely until chapter10, it is worthwhile to briefly discuss some gross features of QFT, in particular those thatdistinguish it from ordinary quantum mechanics. In QM particles are represented bywaves, or wave packets. Quantization happens through the fundamental postulate ofquantum mechanics that says that the operators for space coordinates and momentumcoordinates do not commute,

[x, p] = i~ (2.5)

The dynamics of the waves is described by the Schrodinger equation. Scattering cross-sections are derived by solving, in perturbation theory, a Schrodinger equation with aHamiltonian operator that includes terms for kinetic and potential energy. Usually weexpand the solution around the solution for a free particle and write the solution as a

21

sum of plane waves. This is exactly what you have learned in your QM course and wewill come back to this in Lecture 3.

In QFT particles are represented as excitations (or quanta) of a field q(x), a functionof spacetime coordinates x. There are only a finite number of fields, one for each type ofparticle, and one for each force carier. This solves one imminent problem, namely whyall electrons are exactly identical. In its simplest form QED has only two fields: one fora spin-1

2electron and one for the photon. The dynamics of these field are encoded in

a Lagrangian density L. Equations of motions are obtained with the principle of leastaction. Those for the free fields (in a Lagrangian without interaction terms) leads towave equations, reminiscent of the Schrodinger equation, but now Lorentz covariant.Again, solutions are written as superpositions of plane waves. The fields are quantizedby interpreting the fields as operators and imposing a quantization rule similar to thatin ordinary quantum mechanics, namely

[q, p] = i~ (2.6)

where the momentum p = L/q is the so-called adjoint coordinate to q. (You mayremember that you used similar notation to arrive at Hamiltons principle in your classi-cal or quantum mechanics course.) The Fourier components of the quantized fields canbe identified as operators that create or destruct field excitations, exactly what we needfor a theory in which the number of particles is not conserved. The relation to classicalQM can be made by identifying the result of a creation operator acting on the vacuumas the QM wave in the Schrodinger equation.

That was a mouth full and you can to forget most of it. One last thing, though: one veryimportant aspect of quantum field theory is the role of symmetries in the Lagrangian.In fact, as we shall see in Lecture 10 through 12, the concept of phase invariance allowsto define the standard model Lagrangian by specifying only the matter field and thesymmetries: once the symmetries are defined, the dynamics (the force carriers) comefor free.

That said, we leave the formal theory of quantum fields alone for now. In this lecture,Lecture 2, we formulate a relativistic wave equation for a spin-0 particle. In Lecture 3,we show how the Maxwell equations take a very simple form when expressed in termsof a new spin-1 field, which we identify as the photon. In Lecture 4 we discuss classicalQM perturbation theory and Fermis Golden rule, which allows us to formalize thecomputation of a cross-section. In lecture 5 we apply the developed tools to computethe scattering of spin-0 particles. In Lectures 6, 7 and 8 we turn to spin-1

2field, which

are considerably more realistic given that all SM matter fields are indeed fermions.Finally, in Lectures 9 through 12, we introduce the weak interaction, gauge theory andelectroweak unification.


2.1 Particle-wave duality

Ever since Maxwell we know that electromagnetic fields propagating in a vacuum aredescribed by a wave equation [

1

c22

t22

](x, t) = 0 (2.7)

The solution to this equation is given by plane waves of the form

(x, t) = eikxit (2.8)

where the wave-vector k and the angular frequency are related by the dispersionrelation

= c|k| (2.9)(Of course, since the equation above is real, we can restrict ourselves to real solutions.In fact, the photon field is real. However, it is often more convenient to work withcomplex waves.) Maxwell identified propagating electromagnetic fields with light, andthereby firmly established what everybody already knew: light behaves as a wave.

However, to explain the photo-electric effect Einstein hypothesized in 1904 that light isalso a particle with zero mass. For a given frequency, lights comes in packets (quanta)with a fixed energy. The energy of a quantum is related to the frequency by

E = h = ~ (2.10)

while its momentum is related to the wave-number

p = ~k (2.11)

In terms of energy and momentum the dispersion relation takes the familiar form E = pc.The idea of light as a particle was received with much skepticism and only generallyaccepted after Compton showed in 1923 that photons scattering of electrons behave asone would expect from colliding particles.

So, by 1923 light was a wave and a particle: it satisfied a wave equation, yet it only cameabout in packets of discrete energy. That lead De Broglie in 1924 to make another boldpreposition: if light is both a wave and a particle, then why wouldnt matter particlesbe waves as well? It took another few years before physicists established the wave-likecharacter of electrons in diffraction experiments, but well before that people took DeBroglie hypothesis seriously and started looking for a suitable wave-equation for massiveparticles.

The crucial element is to establish the dispersion relation for the wave. Schrodingerstarted with the relativistic equation for the total energy

E2 = m2c4 + p2c2 (2.12)

2.2. THE SCHRODINGER EQUATION 23

but abandoned the idea, for reasons we will see later. He then continued with theequation for the kinetic energy in the non-relativistic limit

E =p2

2m(2.13)

which, as we shall see now, led to his famous equation.

2.2 The Schrodinger equation

One heuristic way to quantize a classical theory is to take the classical equations ofmotion and substitute energy and momentum by their operators in the coordinate rep-resentation,

E E = i~ t

and p p = i~ . (2.14)Inserting these operators in Eq. (2.13), leads to the Schrodinger equation for a freeparticle,

i~

t = ~

2

2m2 . (2.15)

In quantum mechanics we interprete the square of the wave function as a probabilitydensity. The probability to find a particle at time t in a box of finite size V is given bythe volume integral

P (particle in volume V , t) =

V

(x, t)d3x (2.16)

where the density is

(x, t) = |(x, t)|2 (2.17)Since total probability is conserved, the density must satisfy a so-called continuity equa-tion

t+ j = 0 (2.18)

where j is the density current or flux. When considering charged particles you canthink of as the charge per volume and j as the charge times velocity per volume. Thecontinuity equation can then be stated in words as The change of charge in a givenvolume equals the current through the surrounding surface.

What is the current corresponding to a quantum mechanical wave ? It is straightfor-ward to obtain this current from the continuity equation by writing /t = /t+/t and inserting the Schrodinger equation. However, because this is useful lateron, we follow a slightly different approach. First, rewrite the Schrodinger equation as

t =

i~2m2.


Now multiply both sides on the left by and add the expression to its complex conju-gate

t=

(i~2m

)2

t=

(i~2m

)2

+

t( )

= [i~2m

( )]

j

where in the last step we have used that ( ) = 2 2.In the result we can recognize the continuity equation if we interpret the density andcurrent as indicated.

Plane waves of the form

= N ei(pxEt)/~ (2.19)

with E = p2/2m are solutions to the free Schrodinger equation. (In fact, startingfrom the idea of particle-wave duality, Schrodinger took the plane wave form above andderived his equation as the equation that described its time evolution.) We will leavethe definition of the normalization constant N for the next lecture: as the plane waveis not localized in space (it has precise momentum, and infinitely imprecise position!),it can only be normalized on a finite volume.

To get rid of the inconvenient factor ~ in the exponent, we usually express energy andmomentum in terms of the wave vector and angular frequency defined above. Of course,in natural units there is no difference. However, since it is sometimes useful to verifyexpressions with a dimensional analysis, we shall keep the factors ~ for now.

For the density of the plane wave we obtain

= |N |2 (2.20)

j i~2m

( ) = |N |2

mp (2.21)

Note that, as expected, the density current is equal to the density times the non-relativistic velocity v = p/m.

Any solution to the free Schrodinger can be written as a superposition of plane waves.Ignoring boundary conditions (which usually limit the energy to quantized values), thedecomposition is written as the convolution integral

(x, t) =(~

2pi)3

(p)ei(pxEt)/~d3p (2.22)

2.3. INTERMEZZO: FOUR-VECTOR NOTATION 25

with E = p2/2m. Note that for t = 0 this is just the usual Fourier transform. Forthe exercises, remember that in one dimension the Fourier transform and its inverse aregiven by (Plancherels theorem),

f(x) =12pi

+

F (k)eikxdk F (k) = 12pi

+

f(x)eikxdx (2.23)

If we replace p with p in the plane wave definition

out = N ei(pxEt)/~ (2.24)

we still have a solution to the Schrodinger equation, since the latter is quadratic in coor-dinate derivatives. Note that these solutions are already included in the decompositionin Eq. (2.22).

By convention when describing scattering in terms of plane waves we identify those with+p x in the exponent as incoming waves and those with p x as outgoing waves. Inone dimension, incoming waves travel in the positive x direction and outgoing waves inthe negative x direction.

Note that waves with E E are not solutions of the Schrodinger equation, but onlyto its complex conjugate. That is different for solutions to the Klein-Gordon equation,which we will describe next.

2.3 Intermezzo: four-vector notation

We define the coordinate four-vector x as

x = (x0, x1, x2, x3) (2.25)

where the first component x0 = ct is the time coordinate and the latter three componentsare the spatial coordinates (x1, x2, x3) = x. Under a Lorentz transformation along thex1 axis with velocity = v/c, x transforms as

x0= (x0 x1)

x1= (x1 x0)

x2= x2

x3= x3

(2.26)

where = 1/

1 2.A general contravariant four-vector is defined to be any set of four quantities A =(A0, A1, A2, A3) = (A0,A) which transforms under Lorentz transformations exactly as


the corresponding components of the coordinate four-vector x. Note that it is thetransformation property that defines what a contravariant vector is.

Lorentz transformations leave the quantity

|A|2 = A02 |A|2 (2.27)invariant. This expression may be regarded as the scalar product of A with a relatedcovariant vector A = (A

0,A), such that

A A |A|2 =

AA. (2.28)

From now on we omit the summation sign and implicitly sum over any index thatappears twice. Defining the metric tensor

g = g =

1 0 0 00 1 0 00 0 1 00 0 0 1

(2.29)we have A = gA

and A = gA . A scalar product of two four-vectors A and B

can then be written asA B = AB = gAB . (2.30)

One can show that such a scalar product is indeed also a Lorentz invariant.

You will show in exercise 2.1 that if the contravariant and covariant four-vectors for thecoordinates are defined as above, then the four-vectors of their derivatives are given by

=

(1

c

t,

)and =

(1

c

t,). (2.31)

Note that the position of the minus sign is opposite to that of the coordinate four-vectoritself.

2.4 The Klein-Gordon equation

To find the wave equation for massive particles Schrodinger and others had originallystarted from the relativistic relation between energy and momentum, Eq. (2.12). Usingagain the operator substitution in Eq. (2.14) one obtains a wave equation

1c22

t2 = 2+ m

2c4

~2 (2.32)

This equation is called the Klein-Gordon equation. Having seen it with the factors ~and c included once, we will from now on omit them.

2.4. THE KLEIN-GORDON EQUATION 27

The Klein-Gordon equation can then be efficiently written in four-vector notation as(2+m2

)(x) = 0 (2.33)

where

2 1c22

t22 (2.34)

is the so-called dAlembert operator.

Planes waves of the form

(x) = N ei(pxEt) = eipx

(2.35)

with p = (E,p) are solutions of the KG equation provided that they satisfy the disper-sion relation E2 = p2 +m2. Note that nothing restricts solution to have positive energy:We will discuss the interpretation of negative energy solutions later in this lecture.

Any solution to the KG equation can be written as a superposition of plane waves, likefor the Schrodinger equation. However, in contrast to the classical case, the complexconjugate of the plane wave above

(x) = N ei(px+Et) = eipx

(2.36)

is also a solution to the KG equation and need to be accounted for in the decomposition.Note that it is not independent though, since (p, E) = (p,E). Consequently, wecan write the generic decomposition restricting ourselves to positive energy solutions, ifwe write

(x) =

d3p

[A(p) eipx

+ B(p) eipx]

(2.37)

with E = +p2 +m2. By popular convention, motovated later, we identify the first

exponent as an incoming particle wave, or an outgoing anti-particle wave, and vice-versafor the second exponent.

In analogy to the procedure applied above for the non-relativistic free particle, we nowderive a continuity equation. We multiply the Klein Gorden equation for from theleft by i, then add to the complex conjugate equation:

i(

2

t2

)= i (2+m2 )

i

(

2

t2

)= i

(2 +m2)+

ti

(

t

t

)

= [i ( )] j


where we can recognize again the continuity equation. In four-vector notation theconserved current becomes

j = (, j) = i [ () ()] (2.38)

while the continuity equation is simply

j = 0 (2.39)

You may wonder why we introduced the factor i in the current: this is in order to makethe density real.

Substituting the plane wave solution gives

= 2 |N |2 Ej = 2 |N |2 p (2.40)

or in four-vector notationj = 2 |N |2 p. (2.41)

Note that, like for the the classical Schrodinger equation, the ratio of the current to thedensity is still a velocity since v = p/E. However, in contrast to the non-relativisticcase, the density of the Klein-Gordon wave is proportional to the energy. This is a directconsequence of the Klein-Gordon equation being second order in the time derivative.

We write the conserved current as a four-vector assuming that it transforms underLorentz transformation the way four-vector are supposed to do. It is not so hard to showthis by looking at how a volume and velocity change under Lorentz transformations (seee.g. the discussion in Feynmans Lectures, Vol. 2, sec. 13.7.) The short argumentis that since is a Lorentz-scalar, and a Lorentz vector, their product must be aLorentz vector.

You may remember that conservation rules in physics are related to symmetries. Thatmakes you wonder which symmetry leads to the conserved currents for the Schrodingerand Klein-Gordon equations. In Lecture 11 we discuss Noethers theorem and showthat it is the phase invariance of the Lagrangian. The phase of the wave functions isnot a physical observable. For QM wave functions the conserved current means thatprobability is conserved. For the QED Lagrangian it implies that charge is conserved.

2.5 Interpretation of negative energy solutions

The constraint E2 = p2 +m2 leaves the sign of the energy ambiguous. This leads to aninterpretation problem: what is the meaning of the states with E = p2 +m2 whichhave a negative density? We cannot just leave those states away since we need to workwith a complete set of states.

2.5. INTERPRETATION OF NEGATIVE ENERGY SOLUTIONS 29

m

E

+m

E

+m

m

Figure 2.1: Diracs interpretation of negative energy solutions: holes

2.5.1 Diracs interpretation

In 1927 Dirac offered an interpretation of the negative energy states. To circumventthe problem of a negative density he developed a wave equation that was linear in timeand space. The Dirac equation turned out to describe particles with spin 1/2. (Atthis point in the course we consider spinless particles. The wave function is a scalarquantity as there is no individual spin up or spin down component. We shall discussthe Dirac equation later in this course.) Unfortunately, this did not solve the problemof negative energy states.

In a feat that is illustrative for his ingenuity Dirac turned to Paulis exclusion principle.The exclusion principle states that identical fermions cannot occupy the same quantumstate. Diracs picture of the vacuum and of a particle are schematically represented inFig. 2.1.

The plot shows all the available energy levels of an electron. Its lowest absolute energylevel is given by |E| = m. Dirac imagined the vacuum to contain an infinite numberof states with negative energy which are all occupied. Since an electron is a spin-1/2particle each state can only contain one spin up electron and one spin-down electron.All the negative energy levels are filled. Such a vacuum (sea) is not detectable sincethe electrons in it cannot interact, i.e. go to another state.

If energy is added to the system, an electron can be kicked out of the sea. It now getsa positive energy with E > m. This means this electron becomes visible as it can nowinteract. At the same time a hole in the sea has appeared. This hole can be interpretedas a positive charge at that position: an anti-electron! Diracs original hope was that hecould describe the proton in such a way, but it is essential that the anti-particle massis identical to that of the electron. Thus, Dirac predicted the positron, a particle thatcan be created by pair production. The positron was discovered in 1931 by Anderson.


There is one problem with the Dirac interpretation: it only works for fermions!

2.5.2 Pauli-Weisskopf interpretation

Pauli and Weiskopf proposed in 1934 that the density should be regarded as a chargedensity. For an electron the charge density is written as

j = ie( ). (2.42)To describe electromagnetic interactions of charged particles we do not need to consideranything but the movement of charge. This motivates the interpretation as a chargecurrent. Clearly, in this interpretation solutions with a negative density pose no longera concern. However, it does not yet solve the issue of negative energies.

2.5.3 Feynman-Stuckelberg interpretation

Stuckelberg and later Feynman took this approach one step further. Consider the currentfor a plane wave describing an electron with momentum p and energy E. Since theelectron has charge e, this current is

j(e) = 2e |N |2 p = 2e |N |2 (E,p) . (2.43)Now consider the current for a positron with momentum p. Its current is

j(+e) = +2e |N |2 p = 2e |N |2 (E,p) . (2.44)Consequently, the current for the positron is identical to the current for the electronbut with negative energy and traveling in the opposite direction. Or, in terms of theplane waves, to go from the positron current to the electron current, we just need tochange the sign in the exponent of eixp

. By our earlier convention, this is equivalent

to saying that the ingoing plane wave of a positron is identical to the outgoing wave ofan electron.

Now consider what happens to the electron wave function if we change the direction oftime: We will have ct ct and p p. You immediately notice that this has exactlythe same effect on the plane wave exponent as the transformation (E, p) (E,p).In other words, we can interprete the negative energy current of the electron as anelectron moving backward in time. This current is identical to that of a positron movingforward in time.

This interpretation, illustrated in Fig. 2.2, is very convenient when computing scatteringamplitudes: in our calculations with Feynman diagrams we can now express everythingin terms of particle waves, replacing every anti-particle with momentum p by a particlewith momentum p, as if it were traveling backward in time. For example, the processof an absorption of a positron with energy E is the same as the emission of an electron


e e

E>0 E


the figure. These two contributions are different and both of them must be included ina relativistically covariant computation.

The time-ordering on the right can be viewed in two ways:

The electron scatters at time t2 runs back in time and scatters at t1. First at time t1 spontaneously an ee+ pair is created from the vacuum. Later-

on, at time t2, the produced positron annihilates with the incoming electron, whilethe produced electron emerges from the scattering process.

The second interpretation would allow the process to be computed in terms of particlesand anti-particles that travel forward in time. However, the first interpretation is justmore economic. We realize that the vacuum has become a complex environment sinceparticle pairs can spontaneously emerge from it and dissolve into it!

2

x

e

time

e

x

x

e e

e

x

xtt12

tt1

Figure 2.5: First and second order scattering.

Exercises

Exercise 2.1 (From A&H, chapter 3. see also Griffiths, exercise 7.1)Write down the inverse of the Lorentz transformation along the x1 axis defined inEq. (2.26) ( i.e. express (x0, x1) in (x0

, x1)). Then use the chain rule of partial differen-

tiation to show that, under the Lorentz transformation, the operators (/x0,/x1)transform in the same way as (x0, x1).


Exercise 2.2 (KG conserved current)Verify that the conserved current Eq. (2.38) satisfies the continuity equation: computej

explicitly and make use of the KG equation to show that the result is zero.Hint: note that that for any fourvectors A and B we have AB

= AB.

Exercise 2.3 (The Dirac -Function)

Consider a function defined by the following prescription

(x) = lim0

{1/ for |x| < /20 otherwise

0

surface = 1

infinite

The integral of this function is normalized

(x) dx = 1 (2.45)

and for any (reasonable) function f(x) we have

f(x) (x) dx = f(0). (2.46)

These last two properties define the Dirac -function. The prescription above gives anapproximation of the -function. We shall encounter more of those prescriptions whichall have in common that they are the limit of a sequence of functions whose propertiesconverge to those given here.

(a) Starting from the defining properties of the -function, prove that

(kx) =1

|k|(x) . (2.47)

(b) Prove that

(g (x)) =ni=1

1

|g (xi)| (x xi) , (2.48)

where the sum i runs over the 0-points of g(x), i.e.:g(xi) = 0.Hint: make a Taylor expansion of g around the 0-points.

Exercise 2.4 (Some exercises with the -function)

(a) Calculate 3

0ln(1 + x) (pi x) dx

(b) Calculate 3

0(2x2 + 7x+ 4) (x 1) dx

(c) Calculate 3

0ln(x3) (x/e 1) dx

(d) Simplify (

(5x 1) x 1)


(e) Simplify (sinx) and draw the function

(Note: just writing a number is not enough!)

Exercise 2.5 (Wave packets)In the coordinate representation a plane wave with momentum p = ~k is infinitelydislocalised in space. This does not quite correspond to our picture of a particle, whichis why we usually visualize particles as wave packets, superpositions of plane waves thathave a finite spread both in momentum and coordinate space. As we shall see in thefollowing, the irony is that such wave packets are dispersive in QM: their size increasesas a function of time. So, even wave packets can hardly be thought of as representingparticles.

(a) Consider a one-dimensional Gaussian wave packet that at time t = 0 is given by

(x, 0) = Aeax2+ik0x

with a real and positive. Compute the normalization constant A such that +|(x, 0)|2dx = 1.

Hint:

ey2

dy =pi

(b) Take the Fourier transform to derive the wave function in momentum space att = 0,

(k) =

(1

2api

)1/4e(kk0)

2/4a

Hint: You can write

exp(ax2 + bx) = exp [a(x b/2a)2 + b2/4a]

and then move the integration boundaries by b/2a. (Dont mind that b is com-plex.)

(c) Use this result and Eq. (2.22) to show that the solution to the Schrodinger equation(with E(p) = p2/2m or (k) = ~k2/2m) is given by

(x, t) =

(2a

pi

)1/4(1 + it)1/2 exp

(ax2 + ik0x itk20/4a1 + it

)with 2~a/m.

(d) Compute |(x, t)|2. Qualitatively, what happens to 2 as time goes on?(e) Now compute the same for a solution to the massless Klein-Gordon equation ( =

ck). Note that the wave packet maintains its size as a function of time.

Lecture 3

The Electromagnetic Field

3.1 The Maxwell Equations

In classical electrodynamics the movement of a point particle with charge q in an electricfield E and magnetic field B follows from the equation of motion

dp

dt= q (E + v B). (3.1)

The Maxwell equations tell us how electric and magnetic field are induced by staticcharges and currents. In vacuum they can be written as:

Gauss law E = 0

(3.2)

No magnetic charges B = 0 (3.3)Faradays law of induction E + B

t= 0 (3.4)

Modified Amperes law c2B Et

=j

0(3.5)

where 0 is the vacuum permittivity. From the first and the fourth equation we canderive the continuity equation for electric charges, j =

t. Historically, it was

the continuity equation that lead Maxwell to add the time dependent term to Ampe`reslaw.

The constant c in the Maxwell equations is, of course, the velocity of light. WhenMaxwell formulated his laws, he did not anticipate this. He did realize that c is thevelocity of a propagating electromagnetic wave. The value of c2 can be computed frommeasurements of 0 (e.g. with the force between static charges) and measurements ofc20 (e.g. from measurements of the force between static currents). From the fact thatthe result was close to the known speed of light Maxwell concluded that electromagneticwaves and light were closely related. He had, in fact, made one of the great unifications

35

36 LECTURE 3. THE ELECTROMAGNETIC FIELD

of physics! For a very readable account, including an explanation of how electromagneticwaves travel, see the Feynman lectures, Vol.2, section 18. From now on we choose unitsof charge such that we can set 0 = 1 and velocities such that c = 1. (That is, we useso-called Heaviside-Lorentz rationalised units. See section 1.6.)

For what follows it is convenient to write the Maxwell equations in a covariant way (i.e.in a manifestly Lorentz invariant way). As shown below we can formulate them in termsof a single 4 component vector field, which we denote by A = (V/c,A). As suggestedby our notation, the components of this field transform as a Lorentz vector.

Remember that the following identities are valid for any vector field A and scalar fieldV :

divergence of rotation is 0: (A) = 0 (3.6)rotation of gradient is 0: (V ) = 0 (3.7)

From electrostatics you may remember that, because the rotation of E is zero (whichis the same as saying that E is a conservative vector field), all physics can be derivedby considering a scalar potential field V . The electric field becomes the gradient of thepotential, E = V . The potential V is not unique: we can add an arbitrary constantand the physics will not change. Likewise, you may have seen in your electrodynamicscourse that, because the divergence of the B field is zero, we can always find a vectorfield A such that B is the rotation of A.

So, lets choose a vector field A such that

B = A (3.8)

and a scalar field V such that

E = AtV (3.9)

Then, by virtue of the vector identities above, the Maxwell equations 3.3 and 3.4 areautomatically satisfied.

What remains is to write the other two equations, those that involve the charge densityand the charge current density, in components of A and A0 = V/c as well. You willshow in exercise 3.1 that these can be written very efficiently as

A A = j . (3.10)

We have argued in the previous lecture that a conserved current j transforms as aLorentz vector. (It is easy to work this out for yourself. See also Feynman, Vol.2,section 13.6.) The derivative also transforms as a Lorentz vector. Therefore, itfollows that if the equation above is Lorentz covariant, then A must transform as aLorentz vector as well. Showing that the electromagnetic field indeed transform thisway is outside the scope of these lecture, but you may know that the transformation

3.2. GAUGE TRANSFORMATIONS 37

properties of the fields were an important clue when Einstein formulated his theory ofspecial relativity.

The expressions can be made even more compact by introducing the tensor

F A A. (3.11)

such that

F = j . (3.12)

Just as the potential V in electrostatics was not unique, neither is the field A. Imposingadditional constrains on A is called choosing a gauge. In the next section we shalldiscuss this freedom in more detail. Written out in terms of the components E and Bthe (4 4) matrix for the electromagnetic field tensor F is given by

F =

0 Ex Ey EzEx 0 Bz ByEy Bz 0 BxEz By Bx 0

. (3.13)Note that F is uniquely specified in terms of E and B. In other words, it does notdepend on the choice of the gauge.

3.2 Gauge transformations

As stated above the choice of the field A is not unique. Transformations of the field A

that leave the electric and magnetic fields invariant are called gauge transformations. Inexercise 3.2 you will show that for any scalar field (t,x), the transformations

V = V +

tA = A.

(3.14)

or in terms of four-vectors

A A = A + (3.15)do not change E and B.

If the laws of electrodynamics only involve the electric and magnetic fields, then, whenexpressed in terms of the field A, the laws must be gauge invariant: physical observablesshould not depend on . Sometimes we choose a particular gauge in order to make theexpressions in calculations simpler. In other cases, we require gauge invariance to imposeconstraints on a solution, as with the photon below.


A gauge choice that is often made is called the Lorentz gauge1. In exercise 3.3 you willshow that it is always possible to choose the gauge field such that A satisfies thecondition

Lorentz condition: A = 0. (3.16)

Note that this implies that A becomes a conserved current. In the Lorentz gauge theMaxwell equations simplify further:

Maxwell in Lorentz gauge: A = j (3.17)

However, as you will see in the exercise, A still has some freedom since the Lorentzcondition requires us to fix only (

) and not itself. In other words a gaugetransformation of the form

A A = A + with 2 = = 0 (3.18)

is still allowed within the Lorentz gauge A = 0. Consequently, we can in addition

impose the Coulomb condition:

Coulomb condition: A0 = 0 (3.19)

(Note that, given the Lorentz condition, also A = 0 with this choice of gauge.) Notethat this choice of gauge is not Lorentz invariant. This is allowed since the choice ofthe gauge is irrelevant for the physics observables, but it is sometimes considered notelegant.

3.3 The photon

Let us now turn to electromagnetic waves and consider Maxwells equations in vacuumin the Lorentz gauge,

vacuum: j = 0 = 2A = 0. (3.20)

We recognize in each component the Klein-Gordon equation of a particle with massm = 0. (See Eq. (2.32) in the previous lecture.) This particle is the photon. Itrepresents an electromagnetic wave, a bundle of electric and magnetic field that travelsfreely through space, no longer connected to the source. Using results below you canshow that the E and B fields of such a wave are perpendicular to the wave front andperpendicular to each other. Furthermore, the magnitudes are related by the speed oflight, |E| = c|B|.

1It is actually called the Lorenz condition, named after Ludvig Lorenz (without the letter t). Itis a Lorentz invariant condition, and is frequently called the Lorentz condition because of confusionwith Hendrik Lorentz, after whom Lorentz covariance is named. Since almost every reference has thiswrong, we will use Lorentz as well.

3.3. THE PHOTON 39

We have seen before that the following complex plane waves are solutions of the Klein-Gorden equation,

(x) eipx and (x) eipx (3.21)For a given momentum vector p any solution in the complex plane is a linear combinationof these two plane waves. However, you may have noticed that, in contrast to theSchrodinger equation, the Klein-Gorden equation is actually real. Since the E and Bfields are real, we restrict ourselves to solutions with a real field A.

We could write down the solution to 2A = 0 considering only the real axis, but it iscustomary (and usually more efficient) to form the real solutions by combining the twocomplex solutions,

A(x) = a(p)eipx + a(p)eipx

= 2


At this point it is customary to uniquely factorize a(p) as follows

a(p) N(p) (p) (3.29)such that the vector has unit length and N(p) is real. The normalization N(p) dependsonly on the magnitude of the momentum and is essentially just the energy density ofthe wave. The vector depends only on the direction of p and is called the polarizationvector. Choosing the z axis along the direction of the momentum vector and imposingthe gauge conditions, the latter can be parameterized as

= (c1ei1 , c2e

i2 , 0) . (3.30)

where ci and i are all real and c21 + c

22 = 1. Note that we can remove one phase by

moving the origin. (Just look at how a shift of the origin affects the factors eipx.)Therefore, only two parameters of the polarization vector are physically meaningful:these are the two polarization degrees of freedom of the photon.

Any polarization vector can be written as a (complex) linear combination of the twotransverse polarization vectors

1 = (1, 0, 0) 2 = (0, 1, 0) . (3.31)

If the phases of the two components are identical, the light is said to be linearly polarized.If the two components have equal size (c1 = c2 =

2) but a phase difference of pi/2,

the light wave is circularly polarized. The corresponding circular polarization vectorsare

+ =1 i2

2 =

+1 i22

(3.32)

You will show in exercise 3.4 that the circular polarization vectors + and transformunder a rotation with angle around the z-axis (the momentum direction) as

+ + = ei+ = ei

(3.33)

We now show that this means that these polarization states correspond to the twohelicity eigenstates of the photon.

You may remember from your QM course that the z component of the angular momen-tum operator, Jz is the generator of rotations around the z-axis. That means that fora wavefunction (x) the effect of an infinitesimal rotation around the z axis is given by

(x) U()(x) (1 iJ3)(x). (3.34)A arbitrarily large rotation may be built up from infinitesimal rotations by dividingit in infinitesimal steps

U() = limn

(U(/n))n

= limninfty

(1 i

(

n

)Jz)

)n= eiJz

(3.35)

3.4. ELECTRODYNAMICS IN QUANTUM MECHANICS 41

Consequently, if is an eigen vector of Jz

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