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MARKUS KLUTE: Welcome
back to 8.701.

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So we'll continue the
discussion on QED.

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In the last video, we
looked at wave equations

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and we discussed
Dirac equations.

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Now we want to look at solutions
to the Dirac equations.

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All right, so remember
that the overall goal now

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is to find a description of
spin-half particles, which

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we can then use in our
matrix element calculation

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in order to get to
cross sections or decay

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rates of particles.

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If you just ad hoc or
natural choice for a solution

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would be a wave
equation, which is

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a product of a spinor, which
depends on energy and momentum,

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and an exponent.

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So we have a free plane wave as
a solution to our free particle

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base waveform.

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We have to show or
we have to make sure

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that this wave equation
satisfies the Dirac equation

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as shown here.

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Since the spinor depends only
on energy and momentum here,

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it's rather simple to
write down the derivatives,

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because they only
depend on the exponent.

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So we can do this here.

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And we find those solutions
here for the four components.

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We can rewrite this by putting
those derivatives here back

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into the Dirac equation.

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What we find then here's this
simplified form for the spinor.

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Note that this does not
depend on derivatives anymore.

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So this is a rather simple form.

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And then we can study--

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what happens now if they
have a particle at rest.

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So it further simplifies
the Dirac equation.

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It further simplifies
here to items E

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times gamma 0, u-- that's
our spinor-- equal m times u.

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Since gamma 0 is a
diagonal, or is diagonal,

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we can immediately find the
eigenstates to this equation.

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So we find four
different eigenstates,

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and they are orthogonal.

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And you find that they
look very similar.

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So n here is just the
normalization factor,

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which is the same for all four.

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And we find those four
different values here.

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You can find two with a negative
sign in the exponent and two

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with a positive sign
in the exponent.

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Now this is for
particles at rest, fine.

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We can interpret those solutions
as positive and negative energy

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states of a spin-half particles
or a particle with two spin

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states.

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But now we want to
see what happens

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if you have a particle
which is not addressed.

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So the way to approach
this is, so once, we

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can just apply
Lorentz transformation

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and see how the
solutions transform.

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But it is even easier
to just look directly

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at the Dirac equations
for the spinor.

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So we start again from
our equation here.

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We just write this down.

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And then we rewrite the
equation using the gamma

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matrices until we find those
factors p times gamma 1--

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px times gamma 1, py
gamma 2, p3 gamma 3.

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Can we write this using the
Pauli matrices in this form?

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OK, so what this gives us is
this coupled form of equations.

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So here we revised our spinor
as a two vector, uA and uB.

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And you find the coupled
form between those two

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if we look at those
set of equations.

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Great.

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But this is rather
cumbersome and complicated.

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However, now we can try
to find the solutions

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or try to find eigenstates
to the equation.

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We know that the solutions
are of this form here.

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That's how we started.

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If you then try to find
a specific [? state, ?]

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you can start from the
simplest alternate solution,

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which is uA equal 1, 0.

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And then just put this in here.

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And you find for u1
those solutions here.

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And then you turn this around.

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For uB, you find 1 and 0 and
you find the other solution.

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So similarly as for
the solutions at rest,

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we find here for our
spinors that there

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is four different spinors,
which are independent.

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You can interpret
them now as, again,

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the positive and
negative energy states.

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So if you then,
for example, say,

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OK, let's make sure that
this is all consistent,

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we want to see that
when the momentum is 0,

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you come back to the
previous solution.

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If you look at the
momentum, if they're 0,

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those components are all here--

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become 0, you find
the very same solution

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as we had on the previous page.

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You can also ask
yourself what happens now

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if you don't use this idea of
positive and negative energy

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solutions.

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If you want to do
that and you define

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that as all are either
positive for energy solutions,

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and not two positive
and two negative,

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you find that you
can divide them

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as linear combinations
of the others,

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so they're not
independent solutions.

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So in order to have four
independent solution

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of the Dirac equations,
two have to be positive

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and two have to be
negative in energy.

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All right, so I
recommend just trying

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the exercise of playing
around with the Pauli matrices

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and the gamma matrices.

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If you have not
seen this before,

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it's not easy to
follow the algebra.

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But once you get
a hang of it, it's

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actually not that complicated.

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In the next lecture, we look
at the solutions-- specifically

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the solutions for
antiparticles a little bit more

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and discuss interpretations
of those solutions.