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MARKUS KLUTE: Welcome
back to 8.701.

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So we switch gears now and talk
about quantum electrodynamics,

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QED.

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And we start the
discussion by going back

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to free wave equations.

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Now could argue that we are
interested in collisions

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and we're interested
in decays of particles.

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So why do we discuss
free wave equations?

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But the theory we
discussed last week,

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which we used in order to get
a hold on Feynman diagrams

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and calculations,
was very simplified.

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And one of the aspects not
considered in this theory

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was the fact that
particles carry spin.

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So we had a theory which not
only was applicable to scalars.

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Now by walking through
wave equations,

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we can see how we can
incorporate or make

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use of the fact that particles
actually do carry spin.

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So let's do this one by one.

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So we start off with our
relativistic energy-momentum

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relation--

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e squared is equal to p
squared plus m squared.

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We express energy
and momentum via

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quantum mechanical operators.

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And so immediately
by putting this in,

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we find this
equation here, which

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is the so-called
Klein-Gordon wave equation.

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So if you look at
this equation, we

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see that the second
derivative here in time,

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there's no derivative in space.

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So there's an asymmetry
between space and time.

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And that is a not really useful
feature of our wave equation

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as we want them to be, Lorentz
invariant, for example.

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So what we want is a first-order
equation in both derivatives.

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So we'll just start writing
this down in general terms,

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and then make sure that
this equation holds

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to the relativistic
information we just

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saw on the previous slide.

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We'll just write this down here.

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We have a first derivative time
and a first derivative space.

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And we'll just say there's a
constant between those two,

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relating those two.

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So the sigmas are just
unknown constants.

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So if you now try to find
by squaring this, trying

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to find the
Klein-Gordon equation

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and relates the
coefficients, you

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find this relationship here.

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So the sigma squared are
all the same and equal to 1.

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But you also see
that the sigmas,

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they're anti-commutate-- sorry--

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which is not
possible for numbers.

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So sigmas need to be matrices.

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You also see that this
is only holding true here

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for m equal to 0.

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So this equation here is
true for a massless particle.

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All right, so if we then
try to find solutions

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for those relations, we find
that they can be fulfilled

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by the 2 by 2 Pauli matrices.

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We might have seen
this already hopefully

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in the discussion
of atomic physics.

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And there, those Pauli matrices
associate spin to electrons.

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So this is exactly what
we have in mind here also.

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Now using this definition, we
can rewrite the Weyl equations

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to energy times the
field is equal to minus

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sigma times the momentum
times the field.

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And to find a second equation,
we'll just design flips.

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The chi here and
the phi spinors,

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they're two-dimensional
vectors and the sigma

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are our Pauli matrices.

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Good.

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So we have the relation
of [INAUDIBLE]..

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So we can go a step further.

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Now we want to introduce
mass term as well.

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Those hold for
massless particles,

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so we're going introduce mass.

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So we can rewrite this equation
and introduce its mass term

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here, again, with
the coefficient.

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And we find now this
alpha here being--

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sorry.

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So this phi here is a
core component spinor.

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And it stands for the particle,
its antiparticle, and the two

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spin states.

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So that's combining that
two equations we had here.

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So you'll see one
is for particles

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and one is for antiparticles,
for the two spin states.

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So we combine this
in one equation

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and we added this mass term.

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So if you try to find
the solutions here,

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you find that
alpha is a matrix--

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4 by 4 matrix which has the
sigma, supporting matrices

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on the off-diagonal elements.

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And beta is a
diagonal matrix-- a 4

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by 4 matrix with identity
on the upper two components,

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and minus 1 in the
lower two components.

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So now with this, this is
already the Dirac equation.

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We can rewrite the Dirac
equation in the covariant form

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where you have just
defined a new matrix here,

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so-called gamma matrix
which you build out

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of this matrix beta and alpha.

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Which are defined on
the previous slide.

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Good, so good.

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So we have this new
matrix, this new equation

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here, which is the
Dirac equation.

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And it holds for particles
with two spin states, examples

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with spin half states.

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And it holds for particles
which have masses.

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So that's great.

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So this is now on starting
point for the discussion.

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The next lecture we'll look
at solutions of this equation.