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:

Ψ = Aei(kx-ωt) = Ae (px-Et) ℏ i

Free particle ⇒

iℏ Ψ = ΕΨ ∂t ∂ Note that: -iℏ Ψ = pΨ ∂x

∂ &

So define: E ≡ iℏ ∂t ∂ p ≡ -iℏ ∇

E = p2/2m ⇒ ∇2Ψ ℏ 2m ∂t

∂ Ψ = i

Schrodinger Equation (non-relativistic)

E2 = p2c2 + m2c4

To make relativistic, try the same trick with

c2 ∂t2 ∂ 2Ψ ∇2Ψ - Ψ =

m2c2

ℏ2

Klein-Gordon Equation

⇒ 1 first proposed

by de Broglie in 1924

Ψ = Aexp(- Et) ℏ i For every plane-wave solution of the form

(positive E)

(negative E) There is another solution of the form Ψʹ′ = Aexp( Et) ℏ

i

Try again, but attempt to force a linear form:

Where αn and β are determined by requiring that solutions of this equation also satisfy the Klein-Gordon equation

iℏ Ψ = ∂t ∂ -iℏ Σ Ψ ∂xn

∂ + βmc2 Ψ

3

n=1 c αn

Dirac Equation

⇒ α and β need to be 4x4 matrices and

Ψ1 Ψ2 Ψ3 Ψ4

Ψ = still have positive and negative energy states 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’t think of it this way!

Instead we can say that energy always remains positive, but solutions exist 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 γ-ray background

Where’s the Antimatter ???

∇2Ψ = Ψ m2c4

ℏ2

For a static solution, Klein-Gordon reduces to

in this case the solution is

Ψ ≡ V(r) = - g2 e-r/R

4π r Yukawa Potential

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

∇2Φ=0 whose solution is

V(r) = eΦ(r)= - e2 1 4πεο r

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

''neutron-proton" problem

n p n p 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 ≡ 1fm ℏc = 197 MeV fm ⇓ mc2 = 100 MeV ''meson"

Yukawa (1934) µ-meson (muon) Anderson & Neddermeyer (1936)

mµ = 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"

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

Cecil Powell Marietta Blau Don Perkins

ED = p2/ 2µ + V(r)

reduced mass

assume mp ≃ mn ≡ M, so µ = (MM)/(M+M) = M/2

also take p ≃ ℏ/r (de Broglie wavelength)

''Bohr Condition"

ED = - exp(-mπcr/ℏ) ℏ2 g2

Mr2 4πr

let: x ≡ mπcr/ℏ

E = - e-x mπ2c2 g2mπc

Mx2 4πℏx

n p The Bound State of the Deuteron (from Bowler)

ED = - e-x mπ2c2 g2mπc

Mx2 4πℏx

for a bound state to exist, ED < 0

( ) e-x g2

4πℏc mπ M

1 x

mπ M

1 x2 > ( )2

ex g2

4πℏc 1 x > mπ

M ( ) this is a ⇒ minimum when x=1

g2

4πℏc > 140 MeV 938 MeV ( ) (2.718)

αs ≡ g2

4πℏc > 0.4 compare with α ≡ = e2 1 4πℏc 137

= Mc ( ) - ( ) e-x g2

4πℏc mπ M

mπ 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 Δt ∼ R/c

Heisenberg uncertainty ⇒ ΔE Δt ℏ ∼ >

so R ~ ℏc/ΔE

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

R ~ ℏ/mπc 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 pn and internal lines as qn

9) Associate each vertex with the square root of the appropriate coupling constant, √αx , so when the amplitude is squared to yield a cross-section, there will be a factor of αx

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 order diagrams for pair annihilation ( e- + e+ → γ + γ ) 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 that the 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 α ~ 1/137, but things are going to get dicey with the strong interaction, where αs ~ 1 !!

Richard Feynman

(Baron) Ernest Stuckelberg ..

von Breidenbach zu Breidenstein und Melsbach