Lecture 4:Antiparticles & Virtual Particles · 2006. 1. 13. · Lecture 4:Antiparticles & Virtual...
Transcript of Lecture 4:Antiparticles & Virtual Particles · 2006. 1. 13. · Lecture 4:Antiparticles & Virtual...
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)ℏi
Free particle⇒
iℏ Ψ = Ε∂t∂Note that: -iℏ Ψ = p
∂x∂&
So define: E ≡ iℏ∂t∂ p ≡ -iℏ ∇
E=p2/2m ⇒ ∇2Ψ ℏ2m∂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 = Ae (px-Et)ℏi
For every plane-wave solution of the form
Ψ∗ = Ae (-px+Et)ℏi
There is another solution of the form
(momentum p and positive E)
(momentum −p and negative E)
Try again, but attempt to force a linear form:
Where αnand β are determined by requiring that solutions
of this equation also satisfy the Klein-Gordon equation
iℏ Ψ = ∂t∂
-iℏ Σ Ψ ∂x
n
∂ + βmc2 Ψ
3
n=1
c αn
Dirac Equation
⇒ α and β need to be 4x4 matrices and
Ψ1Ψ2Ψ3Ψ4
Ψ =still have positive
and negative energy
states but now also
have spin!
How do you prevent transitions into ''negative energy" states?
0
E
m
−m
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-Stukleberg)..
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=0whose solution is
V(r) = eΦ(r)= −e2 1
4πεο r
where R ≡≡≡≡ ℏℏℏℏ/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)
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 BrauDon Perkins
ED= p2/ 2µ + V(r)
reduced mass
assume mp ≃ m
n ≡ M, so
µ = (MM)/(M+M) = M/2also take p ≃ ℏ/r(de Broglie wavelength)
''Bohr Condition"
ED= − exp(−mπcr/ℏ)ℏ2 g2
Mr2 4πr
let: x ≡ mπcr/ℏ
E = − e-xmπ
2c2 g2mπc
Mx2 4πℏx
n pThe Bound State of the Deuteron
ED= − e-x
mπ2c2 g2mπc
Mx2 4πℏx
for a bound state to exist, ED< 0
( ) e-x g2
4πℏcmπM
1
x
mπM
1
x2> ( )2
ex g2
4πℏc1
x >
mπM( )
this is a
⇒ minimumwhen x=1
g2
4πℏc >140 MeV
938 MeV( )(2.718)
αs≡
g2
4πℏc > 0.4 compare with α ≡ = e2 1
4πℏc 137
= Mc ( ) − ( ) e-xg2
4πℏcmπM
mπM
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/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
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
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
e+ e+
e- e-
e+
e-
e+
e-
p1
p2
p3
p4
p1
p2
p3
p4
Leading order diagrams for Bhabha Scattering: e+ + e− → e+ + e−
x
t
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
pnand internal lines as q
n
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 αxn , 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
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+→ γ + γ ) are: and
Some Rules for the Construction & Interpretation of Feynman Diagrams
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 ''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 α
s~ 1 !!
Richard Feynman
(Baron) Ernest Stuckelberg..
von Breidenbach zu Breidenstein und Melsbach