Transcript Lecture 4: Antiparticles & Virtual Particles

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
Useful Sections in Martin & Shaw:
Chapter 1
Free particle 
Aei(kx-t)
iℏ ∂ 
∂t
Note that:
So define:
E = p2/2m

i
= Ae ℏ (px-Et)
&
∂
-iℏ p
∂x
∂
E iℏ
∂t
∂
ℏ 2
i 2m
∇

∂t
Schrodinger Equation
(non-relativistic)
p  -iℏ ∇
To make relativistic, try the same trick with
E2  p2c2 + m2c4

m2c2
∂   ∇2
ℏ2
c2 ∂t
1
first proposed
by de Broglie
in 1924
Klein-Gordon Equation
For every plane-wave solution of the form
Aexp(- ℏ Et)
i
(positive E)
There is another solution of the form
Aexp( ℏEt)
i
(negative E)
Try again, but attempt to force a linear form:
3
∂
∂
iℏ  -iℏ  c n  mc2 
∂xn
∂t
n=1
Dirac Equation
Where n and  are determined by requiring that solutions
of this equation also satisfy the Klein-Gordon equation
  and  need to be 4x4 matrices and





still have positive
and negative energy
states but now also
have spin!
How do you prevent transitions into ''negative energy" states?
E
Dirac
''Hole" Theory
Nowdays we don’t
think of it this way!
0
''sea" of negative
energy states
Instead we can say that
energy always remains
positive, but solutions
exist with time
.. reversed
(Feynman-Stukelberg)
Antimatter
Anderson
1933
Where’s the Antimatter ???
The Earth 
Spontaneous combustion is relatively rare
The Moon 
Neil Armstrong survived
The Planets 
Space probes, solar wind...
Ouside the Solar System 
Another Part of the Galaxy 
Other Galaxies 
Larger Scales 
Comets...
Cosmic Rays...
Mergers, cosmic rays...
Diffuse ray background
For a static solution,
Klein-Gordon reduces to
note that if m=0, we would
have the equivalent of an
electromagnetic potential:
2 4
mc
2
∇ 
∇
ℏ2
whose solution is
in this case the solution is
g2 e-r/R
  V(r) = 
4 r
Yukawa Potential
where R  ℏ/mc
hmmm... sounds like the
''neutron-proton" problem
n p
p n
e2
1
V(r) er
4 r
So this gives us a
new ''charge" g and
an effective range R
 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
Yukawa (1934)
-meson (muon)
e = 0.511 MeV ''lepton"
''meson"
p = 938 MeV ''baryon"
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)
-meson (pion), m=140 MeV
Cecil Powell
Don Perkins
Powell et al. (1947)
Marietta Blau
(from Bowler)
The Bound State of the Deuteron
n p
ED = p2/ 2 + V(r)
reduced mass
assume mp ≃ mn M, so
 = (MM)/(M+M) = M/2
''Bohr Condition"
also take p ≃ ℏ/r
(de Broglie wavelength)
ℏ2
g2
ED = Mr2  4r exp(mcr/ℏ)
let: x mcr/ℏ
m2c2
g2mc -x
E = Mx2  4ℏx e
m2c2
g2mc -x
ED = Mx2  4ℏx e
2
= Mc
g2 m 1

(
)
2
M
x
4ℏc
x
m 2 1
[(M )
e-x]
for a bound state to exist, ED < 0
m 2 1
g2 m 1 -x
(
)
e
(
)
>
M
M
4ℏc
x
x2
this is a
g2
m 1
x
e
 minimum
>
(
)
M
4ℏc
x
when x=1
g2
140 MeV (2.718)
>
4ℏc (938 MeV )
g2
s 
4ℏc
>
0.4
e2
1
compare with  
4ℏc 137
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
>ℏ
E t 
Heisenberg uncertainty 
so
R ~ ℏc/E
if we associate E with the rest mass energy of the pion, then
R ~ ℏ/mc
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"
Feynman Diagrams
e+
p1
p3
e+
e+
p1
q
p2
e-
x
t
p2
p4
e+
q
p3
p4
ee-
Leading order diagrams for Bhabha Scattering
e+ + e  e+ + e
e-
e+
p1
p3
e+
e+
p1
q
e+
q
p2
x
t
e-
p2
p4
p3
p4
e-
e-
e-
Leading order diagrams for Bhabha Scattering:
e+ + e   e+ + e 
Some Rules for the Construction & Interpretation of Feynman Diagrams
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
e+
p1
p3
e+
e+
p1
q
p2
x
t
e-
p2
p4
e+
q
p3
p4
e-
e-
e-
Leading order diagrams for Bhabha Scattering:
e+ + e   e+ + e 
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, x , so when the amplitude is squared to yield a
cross-section, there will be a factor of xn , where n is the number of
vertices (also known as the ''order" of the diagram)
e+
p1
p3
e+
e+
p1
q
e+
q
p2
x
t
e-
p2
p4
p3
p4
e-
e-
e-
Leading order diagrams for Bhabha Scattering:
e+ + e   e+ + e 
Some Rules for the Construction & Interpretation of Feynman Diagrams
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
form new diagrams (often implied) which are added to the sum
Thus, the leading order
diagrams for pair annihilation
( e- + e+   + ) are:
and
to
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
e+
opposite direction, travelling backwards in time.

..
Feynman-Stuckelberg interpretation is that
the photon scatters the electron back in time!
e-
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