Lecture 4: Antiparticles & Virtual Particles

• Klein-Gordon Equation

• Antiparticles & Their Asymmetry in Nature

• Yukawa Potential & The Pion

• The Bound State of the Deuteron

• Virtual Particles

• Feynman Diagrams

Chapter 1

Useful Sections in Martin & Shaw:

Aei(kx-t) = Ae (px-Et)ℏi

Free particle

iℏ∂t∂Note that: -iℏ p

∂x∂&

So define: E iℏ∂t∂ p -iℏ ∇

E = p2/2m ∇2 ℏ 2m∂t

∂ i

Schrodinger Equation (non-relativistic)

E2 p2c2 + m2c4

To make relativistic, try the same trick with

c2 ∂t∂ ∇2 m2c2

ℏ2

Klein-Gordon Equation

1 first proposed

by de Broglie in 1924

Aexp(- Et)ℏiFor every plane-wave solution of the form

(positive E)

(negative E)

There is another solution of the form Aexp( Et)ℏi

Try again, but attempt to force a linear form:

Where n and are determined by requiring that solutions

of this equation also satisfy the Klein-Gordon equation

iℏ∂t∂ -iℏ ∂x

n

∂ mc2 3

n=1c

n

Dirac Equation

and need to be 4x4 matrices and

still have positiveand negative energystates but now also have spin!

How do you prevent transitions into ''negative energy" states?

0

E

Dirac ''Hole" Theory

''sea" of negative energy states

Nowdays we don’tthink of it this way!

Instead we can say thatenergy always remains positive, but solutionsexist with time reversed(Feynman-Stukelberg)

..

Antimatter

Anderson 1933

The Earth

The Moon

The Planets

Ouside the Solar System

Another Part of the Galaxy

Other Galaxies

Larger Scales

Spontaneous combustion is relatively rare

Neil Armstrong survived

Space probes, solar wind...

Comets...

Cosmic Rays...

Mergers, cosmic rays...

Diffuse ray background

Where’s the Antimatter ???

∇2m2c4

ℏ2

For a static solution, Klein-Gordon reduces to

in this case the solution is

V(r) = g2 e-r/R

4 r

Yukawa Potential

note that if m=0, we would have the equivalent of anelectromagnetic potential:

∇whose solution is

V(r) er e2 1

4 r

where R ℏ/mc So this gives us a new ''charge" g and an effective range Rhmmm... sounds like the

''neutron-proton" problem

n pnp

whatever keeps them together must be very strong and short-ranged

EM ''carrier" of electromagnetic field = photon (massless boson)

Strong nuclear force ''carrier" of field must be some massive boson

R 10-15m 1fmℏc = 197 MeV fm mc2 = 100 MeV ''meson"

Yukawa (1934)

-meson (muon) Anderson & Neddermeyer (1936)

m = 105.6 MeV ! ...but a fermion, doesn’t interact strongly (looks like a heavy electron)

''Who ordered that ?!" (I. I. Rabi)

e = 0.511 MeV ''lepton" p = 938 MeV ''baryon"

-meson (pion), m=140 MeV Powell et al. (1947)

Cecil Powell Marietta BlauDon Perkins

ED = p2/ 2 + V(r)

reduced mass

assume mp ≃ m

n M, so

= (MM)/(M+M) = M/2 also take p ≃ ℏ/r(de Broglie wavelength)

''Bohr Condition"

ED = exp(mcr/ℏ)

ℏ2 g2

Mr2 4r

let: x mcr/ℏ

E = e-x m2c2 g2mc

Mx2 4ℏx

n pThe Bound State of the Deuteron(from Bowler)

ED = e-x m

2c2 g2mc Mx2 4ℏx

for a bound state to exist, ED < 0

( ) e-x g2

4ℏc m

M 1 x

m

M 1 x2

> ( )2

ex g2

4ℏc 1 x

> m

M( ) this is a minimum when x=1

g2

4ℏc > 140 MeV938 MeV( )(2.718)

s

g2

4ℏc > 0.4 compare with e2 14ℏc 137

= Mc ( ) ( ) e-x g2

4ℏc m

M m

M 1 x

1 x2 [ ]2 2

What does ''carrier of the field" mean ??

Note: the time it would take for the carrier of the strong force to propagate over the distance R is t R/c

Heisenberg uncertainty E t ℏ>

so R ~ ℏc/E

if we associate E with the rest mass energy of the pion, then

R ~ ℏ/mc

which is what enters into the Yukawa potential !

This implies we are ''borrowing" energy over a ''Heisenberg time"

''virtual particle"

+

EM(infinite range)

p n

Strong Nuclear Force (finite range)

''Field Lines"

e+ e+

e- e-

e+

e-

e+

e-

p1

p2

p3

p4

p1

p2

p3

p4

q q

Leading order diagrams for Bhabha Scattering e+ + e e+ + e

x

t

Feynman Diagrams

e+ e+

e- e-

e+

e-

e+

e-

p1

p2

p3

p4

p1

p2

p3

p4

q q

Leading order diagrams for Bhabha Scattering: e+ + e e+ + e

x

t

1) Energy & momentum are conserved at each vertex

2) Charge is conserved

3) Straight lines with arrows pointing towards increasing time represent fermions. Those pointing backwards in time represent anti-fermions

4) Broken, wavy or curly lines represent bosons

5) External lines (one end free) represent real particles

6) Internal lines generally represent virtual particles

Some Rules for the Construction & Interpretation of Feynman Diagrams

7) Time ordering of internal lines is unobservable and, quantum mechanically, all possibilities must be summed together. However, by convention, only one unordered diagram is actually drawn

8) Incoming/outgoing particles typically have their 4-momenta labelled as p

n and internal lines as q

n

9) Associate each vertex with the square root of the appropriate coupling constant,

x , so when the amplitude is squared to yield a

cross-section, there will be a factor of x

n , where n is the number of vertices (also known as the ''order" of the diagram)

Some Rules for the Construction & Interpretation of Feynman Diagrams

e+ e+

e- e-

e+

e-

e+

e-

p1

p2

p3

p4

p1

p2

p3

p4

q q

Leading order diagrams for Bhabha Scattering: e+ + e e+ + e

x

t

10) Associate an appropriate propagator of the general form 1/(q2 + M2) with each internal line, where M is the mass of mediating boson

11) Source vertices of indistinguishable particles may be re-associated to form new diagrams (often implied) which are added to the sum

Thus, the leading orderdiagrams for pair annihilation

( e- + e+ + ) are: and

Some Rules for the Construction & Interpretation of Feynman Diagrams

e+ e+

e- e-

e+

e-

e+

e-

p1

p2

p3

p4

p1

p2

p3

p4

q q

Leading order diagrams for Bhabha Scattering: e+ + e e+ + e

x

t

The ''play catch" idea seems to work intuitively when it comes to understanding how like charges repel.

The ''play catch" idea seems to work intuitively when it comes to understanding how like charges repel.

But what about attractive forces between dissimilar charges??Are you somehow exchanging ''negative momentum" ???!

The best I can offer: Note from Feynman diagrams (and later CPT) that a particle travelling forward in time is equivalent to an anti-particle, going in the opposite direction, travelling backwards in time.

e+

e-

Feynman-Stuckelberg interpretation is thatthe photon scatters the electron back in time!

..

More Bhabha Scattering...

So this basically a perturbative expansion in powers of the coupling constant. You can see how this will work well for QED since ~ 1/137, but things are going to get dicey with the strong interaction, where

s ~ 1 !!

Richard Feynman

(Baron) Ernest Stuckelberg..

von Breidenbach zu Breidenstein und Melsbach