1

Particle Physics 1

U23525, Particle Physics, Year 3

University of Portsmouth, 2013 - 2014

Prof. Glenn Patrick

𝝁

𝑱/𝝍

𝒁𝟎

𝑾±

𝜸 𝑯𝟎 𝝉 𝝂𝒆

𝒆

𝝓 𝝎

𝑩

𝑲+ 𝒈

𝒒 𝝌

𝝂𝝁

𝝂𝝉

2

Apologies

Course should have started last week, but I was at CERN.

We have plenty of time to cover the material over 2 teaching blocks.

3

Course Outline Major particle physics option (U23525), but some overlap

with particle component of 2nd year quantum course.

1 The Standard Model of Particle Physics.

2 Strong Interactions and Quantum Chromodynamics (QCD).

3 Electromagnetic interactions including fundamental electron-

positron process.

4 Weak and Electroweak Interactions.

5 Antimatter and Quark Flavour Physics.

6 Neutrino Physics.

7 Frameworks for Beyond the Standard Model (BSM) physics.

8 High Energy Particle Accelerators and Beams.

9 Particle Detectors.

10 Particle Astrophysics.

4

Preliminaries - Assessment

40 hours of lectures across two teaching blocks

plus 8 hours of tutorial classes.

The main aim is to improve your understanding of

fundamental physics.

However, we cannot forget the small matter of your degree….

1 Final written examination (2 hours) – 80%

2 Coursework questions and problems – 20%

Main thing is that you enjoy the course.

We will try and focus on understanding the underlying concepts.

Extra material/maths shown mainly to aid understanding.

Guidance will be given over essential knowledge needed for exam.

5

Telling the Difference

It is important to also attempt any non-assessed questions and do

some background reading as this will be the best way of checking

your understanding of material and prepare for assessments.

Otherwise, how do you tell the difference?

6

Particle Physics, B.R. Martin and G. Shaw,

3rd edition, Wiley, ISBN: 978-0-470-03293-0

Main book we are following – at least at the start.

Modern Particle Physics, Mark Thomson, New!

Cambridge University Press, ISBN-13: 978-1107034266

New book – only just appeared (Sep 2013) – looks good.

Introduction to High Energy Physics, D.H. Perkins,

4th edition, Cambridge, ISBN: 9780521621960

Good, classic and readable text – a bit dated.

Introduction to Elementary Particles, David Griffiths,

Wiley VCH, 2nd revised edition, ISBN-13:978-3527406012

Good clear text on theoretical aspects.

Quarks & Leptons, Francis Halzen & Alan D. Martin,

John Wiley, ISBN: 0471887412

Advanced for this course, but some sections very good.

The Experimental Foundations of Particle Physics,

Robert Cahn & Gerson Goldhaber, 2nd edition,

ISBN: 9780521521475

Preliminaries - Books

7

Particle Astrophysics, D.H. Perkins, 2nd edition,

Oxford, ISBN: 978-0-19-850951-6

The Physics of Particle Accelerators, Klaus Wille,

Oxford, ISBN: 978-0-19-850549-5

An Introduction to Particle Accelerators, Edmund

Wilson, Oxford, ISBN: 978-0-19-850829-8

Particle Detectors, Claus Grupen & Boris Shwarz, 2nd

edition, Cambridge, ISBN:9780521187954

The Physics of Particle Detectors, Dan Green, ISBN:

9780521675680

Supplementary Books - Later When we get deeper into the course, there are some

supplementary books for more specialised topics.

I will remind you about them later.

The lecture slides and material should, however,

be sufficient for your study.

8

Preliminaries - Course Material WEB PAGE for material

http://hepwww.rl.ac.uk/gpatrick/portsmouth/courses.htm

MOODLE

When set up.

9

Timetable – Teaching Block 1

Week Date Start Finish Building Room Size

9 26.09.2013 09:00 11:00 Buckingham (BK) 3.05 35

10 03.10.2013 09:00 11:00 Buckingham (BK) 3.05 35

11 10.10.2013 09:00 11:00 Portland (PO) 0.36 35

12 17.10.2013 09:00 11:00 Portland (PO) 0.36 35

13 24.10.2013 09:00 11:00 Portland (PO) 0.36 35

14 31.10.2013 09:00 11:00 Portland (PO) 0.36 35

15 07.11.2013 09:00 11:00 Buckingham (BK) 3.05 35

16 14.11.2013 09:00 11:00 Buckingham (BK) 3.05 35

17 21.11.2013 09:00 11:00 Portland (PO) 0.36 35

18 28.11.2013 09:00 11:00 Portland (PO) 0.36 35

19 05.12.2013 09:00 11:00 Burnaby (BB) 2.24 35

20 12.12.2013 09:00 11.00 Anglesea (AA) 2.06 35

10

Timetable – Teaching Block 2 Week Date Start Finish Building Room Size

24 09.01.2014 09:00 11:00 Park (PK) 2.01 35

25 16.01.2014 09:00 11:00 Park (PK) 2.01 35

26 23.01.2014 09:00 11:00 St. Andrew’s Court (SA) 0.04 35

27 30.01.2014 09:00 11:00 Lion Gate 2.05 35

28 06.02.2014 09:00 11:00 Lion Gate 2.05 35

29 13.02.2014 09:00 11:00 Lion Gate 2.05 35

30 20.02.2014 09:00 11:00 Buckingham (BK) 3.04 35

31 27.02.2014 09:00 11:00 Buckingham (BK) 3.04 35

32 06.03.2014 09:00 11:00 Park (PK) 3.23 35

33 13.03.2014 09:00 11:00 Buckingham (BK) 3.04 35

34 19.03.2014 09:00 11:00 Buckingham (BK) 3.04 35 Wed!

35 27.03.2014 09:00 11:00 Buckingham (BK) 3.04 35

Will try and start lectures at 09:05, but bear in kind that I have to travel 70 miles

to Portsmouth.

I also teach the quantum/nuclear course on Thursdays (weeks 12-20).

11

Today’s Plan Particle Physics 1

Course Outline

Preliminaries - Assessment

Preliminaries - Books

Preliminaries - Course Material

Particle Physics, Cosmology & Particle Astrophysics

Natural Units

Rationalised Heaviside-Lorentz EM Units

Special Relativity and Lorentz Invariance

Mandelstam Variables

Spin and Spin Statistics Theorem – Fermions and Bosons

Addition of Angular Momenta – Clebsch Gordon Coefficients

Crossing Symmetry and s, t & u Channels

Non-Relativistic Quantum Mechanics (Schrödinger Equation)

Relativistic Quantum Mechanics (Klein-Gordon Equation)

Feynman-Stückelberg Interpretation of Negative Energy States

12

What you should remember from Year 2

H0

Quantum

Force Carriers

[Bosons]

(Spin 1)

Higgs

Particles

[Boson(s)]

(Spin 0)

Standard Model

of Particle Physics

First fundamental

scalar particle.

Nobel Prize next

week?

13

The Big Picture?

14

Fabric of the Universe

Particle Physics is the study of the fundamental constituents

of matter and the forces that hold them together.

15

Boltzmann

Mean Energy

where,

k = 8.6 x 10-5 eV K-1

Particle Physics and Energy Particle accelerators probe further and further back

in time towards the Big Bang.

kTE

Particle

Physics

LHC energy=14 TeV

T ~ 1.6 x 1017 K

but focussed in a tiny

space-time element.

16

Particle Physics and Time

~1 sec after Big

Bang, T ~1010 K

Neutrinos

Interact weakly &

decoupled

2

220

exp

104

T

sKt

LHC ~10-14 s after Big Bang

Transparent to photons.

380,000 years

T ~3,000 K Opaque to photons.

17

Accelerator experiments can be complex

Beam

dumps

RF Collimation

Collimation

1720 Power converters > 9000 magnetic elements 7568 Quench detection systems 1088 Beam position monitors ~4000 Beam loss monitors

150 tonnes Helium, ~90 tonnes at 1.9 K 140 MJ stored beam energy in 2012 450 MJ magnetic energy per sector at 4 TeV

LHC: big, cold, high energy

Injection B2

Injection B1

courtesy

Mike Lamont

18

LHC is vital for 21st Century Physics and

will rewrite some textbooks,

but there is much more to the subject…

19

Particle Astrophysics

Particle Astrophysics (or Astroparticle Physics) is a branch of particle

physics that studies elementary particles of astronomical origin and

their relation to astrophysics and cosmology (Wikipedia)

In practice, there are two types of particle astrophysics research project.

1) Direct detectors of particles from space or 'particle telescopes':

Gravity wave telescopes

Neutrino detectors and telescopes

Cosmic ray telescopes

Gamma ray telescopes

Dark matter detectors and telescopes

Other exotic particle searches (e.g. axions, magnetic monopoles)

2) Indirect detection of the effects of particles on astronomical objects:

Binary pulsars and pulsar timing measurements for gravity waves.

CMB measurements for dark matter, dark energy, neutrinos, gravity waves.

Timing of gamma ray and TeV emission for Lorentz invariance tests.

Non-accelerator methods for measuring/constraining the properties of

fundamental particles.

20

“Small” specialised experiments still important

HESS II Observatory (Namibia)

Studies very high energy cosmic gamma rays

21

Theory and Experiment

It doesn’t matter how beautiful your theory is,

it doesn’t matter how smart you are.

If it doesn’t agree with experiment, it’s wrong.

Richard P. Feynman

A theory is something nobody believes except

the person who made it,

An experiment is something everybody believes

except the person who made it.

Albert Einstein

22

Natural Units

18 sec m 10 x 998.2 c

sec10 x 055.12

34- Jh

The fundamental constant of quantum mechanics is Planck’s

constant, h, and the fundamental constant of special relativity is

the velocity of light in vacuum, c:

Unit of action (ML2/T)

Unit of velocity (L/T)

In particle physics, we commonly

measure energy (ML2/T2) in units of

GeV (109 electron volts).

Mass of proton ~ 1 GeV

These so-called natural units are therefore based on the

fundamental constants of quantum mechanics and special

relativity, i.e. ℏ, c and GeV.

We can simplify matters even further by setting ℏ = 𝒄 = 𝟏

and then all quantities are expressed as powers of GeV.

23

Natural Units

FUNDAMENTAL UNITS

Quantity S.I. unit [kg, m, s] Natural

[ħ, c, GeV]

Natural

[ħ = c = 1]

Energy kg m-2 s-2 GeV GeV

Momentum kg m s-1 GeV/c GeV

Mass kg GeV/c2 GeV

Time s ħ/GeV GeV-1

Length m ħc/GeV GeV-1

Area m2 (ħc/GeV)2 GeV-2

24

Natural Units - Converting

Example: The root-mean-square charge radius of the proton is

calculating using natural units to be: 𝒓𝟐

𝟏𝟐 = 𝟒. 𝟏 𝑮𝒆𝑽−𝟏

To convert to S.I. units, we just have to reinsert the missing factors

of ħ and c:

𝐿𝑒𝑛𝑔𝑡ℎ =ℏ𝑐

𝐺𝑒𝑉

𝒓𝟐𝟏

𝟐 = 4.1 ×1.055 × 10−34 × 2.998 × 108

1.602 × 10−10= 𝟎. 𝟖 × 𝟏𝟎−𝟏𝟓𝒎

Conversion factors: s MeV x106.582 -22

m MeV x10973.1 -13c

JeVGeV 109 10602.110 1

25

Rationalised Heaviside-Lorentz EM Units

Fine structure constant, α :

035999074.137

1

44

2

0

2

HLSI

e

c

e

Usually, we will always be dealing with particles in vacuo.

Therefore we use the rationalised Heaviside-Lorentz system

of EM units where permittivity ε0 = 1 and permeability μ0 = 1.

We will be dealing with the interactions between charges. These can be the

familiar electric charge of electromagnetic interactions, or the strong charge of

the strong interaction or the weak charge of the weak interaction.

Maxwell’s Equations then become:

Clearly, the numerical

values of e are different

in each system…

𝑒 𝑆𝐼

→ 𝑒√4𝜋 𝐻𝐿

E

t

BE

0 B

JBt

E

𝑭 =𝒆𝟐

𝟒𝝅𝜺𝟎𝒓𝟐→

𝒆𝟐

𝟒𝝅𝒓𝟐

Coulomb’s Law then becomes:

where 𝜀0 has been absorbed into the

definition of the electron charge.

2

0

0

1

c

26

Special Relativity

A proton with energy of “only” 1 GeV already has a velocity of ~0.9c.

In particle physics, the consequences and effects of

Einstein’s theory of Special Relativity

have to be taken into account.

27

Special Relativity Fundamental laws have the same form in all Lorentz frames (i.e. reference

frames which have a uniform relative velocity).

• In particle physics we inevitably deal with relativistic particles.

• Need all calculations to be Lorentz invariant.

• Lorentz invariant quantities formed from scalar product of four-vectors.

z

y

ctx

xct

z

y

x

tc

z

y

x

ct

x

ctx

)(

)(

21

1

c

vwhere,

e.g. Lorentz boost along x-axis

xx

3

0

Alternatively,

in matrix form: where

1000

0100

00

00

xx Using the Einstein summation convention we can write

(repeated indices summed over)

28

Aside: Einstein Summation Convention

A notational convention that implies summation over a set of indexed terms in a

formula, thus achieving notational brevity.

i. Omit summation signs.

ii. If a suffix appears twice with one of the pair a superscript and the other a

subscript, a summation is implied.

e.g. 𝐴𝑖𝐵𝑖 = 𝐴1𝐵1 + 𝐴2𝐵2 + 𝐴3𝐵3

iii. If the index is a Greek letter, the summation extends over all 4

components (from 0 to 3), i.e. 4-vectors.

iv. If the index is a Latin letter, the summation only extends over the 3 spatial

components (1 to 3), i.e. 3 vectors.

v. If a suffix appears only once, it can take any value.

e.g. 𝐴𝑖 = 𝐵𝑖 holds for 𝑖 = 1,2,3

vi. A suffix CANNOT appear more than twice.

3

3

2

2

1

1

0

0

3

0

xxxxxxxxxx

e.g.

An index,

NOT a power.

29

Special Relativity

contravariant

covariant

where gμν is the metric tensor…

1000

0100

0010

0001

gg

The interval ds2 is invariant: iiii xdxdtdcdxdxdtcds 22222

),,,( 3210 xxxxx

),,,( 3210 xxxxxgx

where μ = 0,1,2,3

Defining space-time 4 vectors:

dxdxgds 2

We can now instead write: tensorsymmetric a is :Note gg

Quantities such as ds2 are scalars and, just as in 3D space, we can talk of a

scalar product of two 4-vectors, which are Lorentz invariant if they transform

the same way as the space-time 4 vector :

33221100 BABABABABAgBAgBABABA

Any expression that can be written in terms of 4-vector scalar products is

guaranteed to be Lorentz invariant.

30

Special Relativity In relativistic kinematics, we form a four vector, where energy plays the role of

the “time” component.

),,,(),( zyx pppEpEp

),,,(),( zyx pppEpEpgp

contravariant (four momentum)

covariant

2

0

22 mpEpp

Lorentz invariant since mo is the rest mass

or invariant mass.

timelike spacelike

rpEtpx

Phase of a plane wave.

We can form 4-vector scalar products, which are Lorentz invariant:

31

Mandelstam Variables

Consider the kinematics of the general 2-body

scattering process of the form 1 + 2 → 3 + 4

p1

p2

p3

p4

2

21 )( pps 2

31 )( ppt

2

41 )( ppu

i.e. square of total CMS energy (∴ 𝑠 = total energy in CMS).

i.e. square of four momentum transfer between particles 1 & 3

i.e. square of four momentum transfer between particles 1 & 4

However, it is convenient to define three Lorentz invariant quantities s, t and u

– the Mandelstam Variables:

We have 4 four-vectors of the form:

),()4,3,2,1( iii pEip

10 scalar products can be formed: jipp ji where

These are constrained by: 𝑝1 + 𝑝2 + 𝑝3 + 𝑝4 = 0 and )4..1 22 (imppp i

i

ii

Only two independent variables are sufficient to describe a 2-body process

(with no polarisation).

4

1

2

i

imutsThey are connected by the sum:

Mass shell condition Conservation of momentum/energy

32

Crossing Symmetry

Principle of Crossing Symmetry states that the 3 channels:

channel)-(u 2341

channel)-(t 4231

channel)-(s 4321

are described by a single transition

matrix and can be related by the same

analytic function of s, t and u for the

scattering amplitude in all 3 channels.

The channels are denoted by the variable which is positive (i.e. timelike) for

the channel in question.

1

2

3

4

1 𝟑

4 𝟐

1

𝟐

3

𝟒

s-channel t-channel u-channel

33

e+

e-

e-

𝜸

e+

s, t and u channels

p1

p2

p3

p4

s-channel

p1 p3

p2 p4

t-channel

p4

p3

p1

p2

u-channel

Classify diagrams according to the four momentum of the exchanged particle

(or “propagator” – see Lecture 2).

(“Annihilation” Diagram) (“Scattering” Diagram)

e+ e+

e-

𝜸

e-

e+ e+

e- e-

𝜸

Example: 𝒆+𝒆− → 𝒆+𝒆−

34

Quantum Mechanics An electron has a size of <10-18 m

We inevitably have to incorporate Quantum Mechanics when

studying particle physics.

Developed by some of the giants of physics.

As mere mortals, we need to have some

understanding and apply as a tool……

35

Spin - Reminder! • Spin is the intrinsic angular momentum of a

particle.

• It is Quantised.

• Can only measure magnitude(S) and one

component(usually Sz). i.e. S2 and Sz commute.

Total Spin: 𝑆2 = 𝑺. 𝑺 = 𝑠(𝑠 + 1)ℏ2

where, 𝑠 = 0,1

2, 1,

3

2, 2,

5

2, …

𝑆𝑧 = 𝑚𝑠ℏ where, 𝑚𝑠 = −𝑠, −𝑠 + 1, … , 𝑠 − 1, 𝑠

and

We have:

36

Spin Statistics Theorem

The Spin Statistics Theorem Systems of identical particles with integer spin (0ℏ, 1ℏ, 2ℏ, 3ℏ

…..), known as bosons, have wave functions which are

symmetric under interchange of any pair of particle labels.

The wave function is said to obey Bose-Einstein statistics.

Systems of identical particles with half-odd-integer-spin (1/2ℏ,

3/2ℏ, 5/2ℏ….), known as fermions, have wave functions which

are antisymmetric under interchange of any pair of particle

labels. The wave function is said to obey Fermi-Dirac statistics.

Under exchange of identical bosons 𝜓 → +𝜓; 𝜓 is symmetric

Under exchange of identical fermions 𝜓 → −𝜓; 𝜓 is antisymmetric

Pauli Exclusion Principle (1925)

Two fermions cannot exist in the same quantum state

37

Fermions and Bosons

Fermi Dirac Statistics Bose Einstein Statistics

Bose Einstein

condensate

Fundamental

difference in what

quantum states

fermions and bosons

can occupy.

38

Addition of Angular Momenta

llmUse “Ket” notation for orbital ssmand spin state

1,3 21,

21

Orbital l = 3, ml = -1 and Spin s = ½, ms = ½ e.g.

• We may be interested in the total angular momentum: 𝑱 = 𝑳 + 𝑺

• May be interested in combining the spins of quarks in a meson: 𝑺 = 𝑺𝟏 + 𝑺𝟐.

BUT HOW DO WE ADD TWO ANGULAR MOMENTA? J= J1+J2

In QM, we can only work with one component and magnitude.

The z components still add, so we have: 𝑚 = 𝑚1 + 𝑚2

But the magnitudes do NOT add (depends on orientation of J1 and J2.)

Turns out we get every j from 𝑗1 + 𝑗2 down to 𝑗1 − 𝑗2 in integer steps.

i.e. 𝑗 = 𝑗1 − 𝑗2 , 𝑗1 − 𝑗2 + 1, … , 𝑗1 + 𝑗2 − 1, 𝑗1 + 𝑗2

If we want the explicit decomposition of into states of angular

momentum : 2211 mjmj

JM

)(

2211

21

21

21

21

jj

jjj

jjj

mmm JMCmjmj

where 21

21

jjj

mmmC are known as Clebsch-Gordon coefficients.

39

Clebsch-Gordon Coefficients

http://pdg.lbl.gov/2013/reviews/

rpp2012-rev-clebsch-gordan-coefs.pdf

Deriving Clebsch-Gordon

coefficients is tedious.

Luckily, they have been tabulated

and we can just look them up!

40

Example: Meson A quark and an anti-quark are bound together in a state of zero orbital angular

momentum to form a meson. What are the possible values of the meson’s spin?

Quarks carry spin ½, so we can get 1

2+

1

2= 1 or

1

2−

1

2= 0. The spin-0

combination gives us the “pseudo-scalar” mesons. The spin-1 combination

gives the “vector” mesons.

Clebsch-Gordon coefficients for

j1 = ½ and j2=½. (square root

sign over each number implied)

112

12

12

12

1

002

1102

12

12

12

12

1

002

1102

12

12

12

12

1

112

12

12

12

1

21

21

21

2111

21

21

21

2111

2

12

12

12

12

12

12

12

12

110

2

12

12

12

12

12

12

12

12

100

The spin 0 state

The three spin 1 states are:

J

M

J

M

m1 m2

J

M m1 m2

m1 m2

m1 m2

J

M j1 x j2

41

Aside: Quick Reminders

iBAC 1 ii

iBAC * ConjugateComplex

22*2))(( BAiBAiBACCC

Complex Numbers

Vector Calculus

zk

yj

xidel

Operator

kVjViV zyx z)y,V(x,Vector

z

V

y

V

x

VV zyx

Divergence

kz

Sj

y

Six

SS

S Grad

ky

V

x

Vj

x

V

z

Vi

z

V

y

V xyzxyz

V Curl

2

2

2

2

2

22Laplacian

zyx

42

Non-Relativistic Quantum Mechanics

For a free particle of mass m, we can substitute the differential operators

m

pE

2

2

into the classical energy-momentum relation

ti

tiE

iip and

),(2

1),( 2 trmt

tri

We then obtain:

The non-relativistic

Schrödinger equation

acting on a complex

wave function Ψ(r, t).

This equation violates Lorentz covariance and is not suitable for a particle

moving relativistically, which is the usual situation in particle physics.

Also, with moving particles, we often need (e.g. in collisions) to calculate

the probability density, ρ, and the probability current density, j.

0

jdt

(CC) We can make use of the continuity condition:

43

Non-Relativistic Quantum Mechanics

),(2

1),( 2 trmt

tri

: SE x - SE x **

),(2

1),( *2*

trmt

tri

(SE) (SE)*

Complex conjugate of SE

)()(2

1 **22*

tti

m

0)(2

1)( ***

mtiRearranging

By comparison with the continuity condition (CC), this gives:

**

2

1

mij

2*

For a plane wave: )(),( EtrpiNetr

2N 2

2N

m

pj

Number of particles

per unit volume.

Number of particles

per unit area per unit

time.

Construct

44

Relativistic Quantum Mechanics

For a relativistic particle: 2

0

22 mpE , where m0 is the rest mass

),(),(),( 22

2

2

trmtrt

tr

The Klein-Gordon equation

or alternatively the

relativistic Schrödinger

equation.

and we can now write:

As before, take the

complex conjugate:

(KE)

),(),(),( *2*2

2

*2

trmtrt

tr

KE x - KE x ** Search for candidate continuity equation by forming:

(KE)*

*2*222*

2

*2

2

2* - -

mm

tt

***

*

ttt

0

jdt

Comparing with (CC) we get:

)( **

**

ij

tti

45

Relativistic Quantum Mechanics

For every plane wave solution of the form:

)(),( EtrpiNetr with 22 mpE

there is also a solution

)(** ),(),(~ EtrpieNtrtr with

22 mpE

PROBLEMS

• The negative energy solutions are a consequence of the quadratic mass-

energy relationship and cannot be avoided in a relativistic theory.

• This looks disastrous as it implies transitions could take place to lower

(more negative) energies.

• The E < 0 solutions are also associated with a negative probability

density (or in the jargon “not positive definite”).

What can all of this mean?....

With 2

2 NE and pNj 2Negative

energy!

Negative

E and p

46

Feynman-Stückelberg Interpretation Classically, the concept of negative energies for free particles is meaningless.

However, a prescription for handling these states was proposed by

Stückelberg and Feynman:

In quantum mechanics, we can think of particles with –E and –p

propagating backwards in time and space.

This means negative-energy particle solutions going backward in time/space

describe positive-energy antiparticle solutions going forward in

time/space.

For example, by replacing Et by (-E)(-t) and p.r by (-p).(-r) we get…

)).()))(()()(( EtrpitErpi

eNeNe

)).()))(()()(( EtrpitErpi

eNeNe

Electron (E < 0) backward in time:

Positron (E > 0) forward in time:

0

E

e

0

E

etime

Pictorially:

47

Many Particle Cases Double scattering of an electron

Two diagrams for the same observation

with two different time orderings.

In second pic, at time t2 the electron

scatters backward in time with E < 0.

Can be interpreted as a positron with E

> 0 going forward in time.

First, at t1 an e-e+ pair is created, then at a later time, t2, the e+ annihilates with the

incident e-. Between t1 and t2, the electron trajectory describes 3 particles: the initial and

final electrons and a positron! Photon-Particle Scattering

(a) Particle (E1) comes in and at t1/x1

emits photon ( with Eγ < E1). Travels

forward in time & at t2/x2 absorbs initial

state photon giving photon-particle final

state.

(b) Particle emits photon (with Eγ > E1) &

is forced to travel backward in time. At

earlier time it absorbs initial state photon

rendering its energy positive again.

48

CONTACT

Professor Glenn Patrick

email: glenn.patrick@port.ac.uk

→ email: glenn.patrick@stfc.ac.uk

End

49

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