for s=1

1
for s1
for EM-waves total energy density
? of photons
the Klein-Gordon equation
wave number vector
2
(or any Dirac, i.e. spin ½ particle muons, taus,
quarks)
for FREE ELECTRONS
electrons
positrons
1 or 1,2
u
v
a spinor satisfying
u
v
Note for each with
-s
-E,-p
for u3, u4
i.e. we write
Notice now we express all terms of the
physical (positive) energy of positrons!
3
The most GENERAL solutions will be LINEAR
COMBINATIONS
?
k
?
g
h
s
linear expansion coefficients
where g?g(k,s), h?h(k,s),
Insisting ?(r,t), ?(r,t) ?(r,t),
?(r,t)0 (in recognition of the Pauli
exclusion principal), or, equivalently ?(r,t),
?(r,t) d3(r r?) respecting the condition on
Fourier conjugate fields
forces the g, h to obey the same basic
commutation relation (in the conjugate
momentum space)
the g?g(k,s), h?h(k,s) coefficients cannot
simple be numbers!
4
?
k
?
g
h
s
?


?
g
h
s
5
We were able to solve Diracs (free particle)
Equation by looking for solutions of the form
This form automatically satisfied the
Klein-Gordon equation. But the appearance of the
Dirac spinors means the factoring effort isolated
what very special class of particles?
?(x) ue-ix?p?/h
u(p?)
cpz E-mc2 c(pxipy) E-mc2
c(px-ipy) E-mc2 -cpz E-mc2
1 0
0 1
c(px-ipy) Emc2 -cpz Emc2
cpz Emc2 c(pxipy) Emc2
1 0
1 0
u(p?) a spinor describing either spin up or
down components
6
The fundamental mediators of forces the VECTOR
BOSONS
What about vector (spin 1) particles?
Again try to look for solutions of the form
?(x) ?m(p)e-i x?p?/h
Polarization vector (again characterizing
SPIN somehow)
but by just returning to the Dirac-factored form
of the Klein-Gordon equation, will we learn
anything new?
What about MASSLESS vector particles? (the
photon!)
the Klein-Gordon equation
2? 0
becomes
or
Where the dAlembertian operator 2
7
is a differential equation you have already
solved in Mechanics and E M
2? 0
Classical Electrodynamics J.D.Jackson (Wiley)
derives the relativistic (4-vector) expressions
for Maxwells equations
can both be guaranteed by introducing the scalar
V and vector A potentials
which form a 4-vector (VA)
along with the charge and current densities
(c?J)
Then the single relation
completely summarizes
8
( )
for example, the arbitrary assignment of zero
gravitational potential energy
Potentials can be changed by a constant
leaving everything invariant.
or even
In solving problems this gives us the flexibility
to adjust potentials for our convenience
0
0
The Lorentz Gauge
The Coulomb Gauge
In the Lorentz Gauge
0
a vector particle with 4 components (VA)
and a FREE PHOTON satisfies
The VECTOR POTENTIAL from EM is the wave
function in quantum mechanics for the free photon!
9
so continuing (with our assumed form of a
solution)
? ?(p)
like the Dirac u, a polarization
vector characterizing spin
Substituting into our specialized Klein-Gordon
equation
(for massless vector particles)
E2p2c2
just as it should for a massless particle!
10
? ?(p)
Like we saw with the Dirac u before, ? has
components!
4
but not all of them are independent!
How many?
??A??0
The Lorentz gauge constrains
p m e m 0
p0e 0-e .p 0
which you should recognize as the familiar
condition on em waves
? ? A 0
? ? p 0
while the Coulomb Gauge
only for free photons
Obviously only 2 of these 3-dim vectors can
be linearly independent such that
? ? p 0
Why cant we have a basis of 3 distinct
polarization directions? Were trying to describe
spin 1 particles! (mspin -1, 0, 1)
11
spin 1 particles mspin -1 , 0 , 1
anti- aligned
aligned
The m0 imposes a harsher constraint (adding yet
another zero to all the constraints on the
previous page!)
The masslessness of our vector particle implies
v c
???
In the photons own frame longitudinal distances
collapse.
How can you distinguish mspin ?1 ?
Furthermore with no frame traveling faster than
c, can never change a ?s spin by changing
frames.
What 2 independent polarizations are then
possible?
12
The most general solution
e s where s 1, 2 or s ?1
moving forward
moving backward
Notice here no separate ANTI-PARTICLE (just one
kind of particle with 2 spin states)
Massless force carriers have no anti-particles.
13
Finding a Klein-Gordon Lagrangian
or
The Klein-Gordon Equation
L
Provided we can identify the appropriate this
should be derivable by The Euler-Lagrange Equation
L
L
14
I claim the expression
L
serves this purpose
15
L
L
L
L
L
L
L
L
16
You can show (and will for homework!) show the
Dirac Equation can be derived from
LDIRAC(r,t)
We might expect a realistic Lagrangian that
involves systems of particles
LK-G LDIRAC
L(r,t)
but each term describes free non-interacting parti
cles
describes ee- objects
describes photons
L
LINT
But what does terms look like? How do we
introduce the interactions the experience?
17
Well follow (Jackson) EMs lead
A charge interacts with a field through
current-field interactions
the fermion (electron)
the boson (photon) field
from the Dirac expression for J?
particle state
antiparticle (hermitian conjugate) state
Recall the state functions Have coefficients
that must satisfy anticommutation relations. They
must involve operators!
What does such a PRODUCT of states mean?