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MARKUS KLUTE: All right.

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So welcome back to 8.701.

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So we have all ingredients
now to prepare Feynman rules

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for QED.

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So that's the toolkit
we need in order

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to make calculations to
calculate scattering processes

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and decays.

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And we've already seen Feynman
rules for our toy theory.

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Again, now the situation
is a little bit more

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complicated, because
we can consider

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the spin of
particles in addition

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to their energy and momentum.

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The rules' sequence of things
are very much the same.

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There is, however, a few
caveats to keep in mind,

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and I'll point those out.

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OK.

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So the very first thing is to
be very clear in our notation.

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So this is an arbitrary or
generic QED Feynman diagram.

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We have only pointed
out the incoming

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and the outgoing lines.

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There is internal lines
which I didn't mention here.

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Important to note the
momentum and the directions.

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The directions are arbitrary.

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We just have to be
clear on them and then

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treat them consistently.

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All right?

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So this is not different
in our previous discussion.

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Then, here comes the difference.

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Our external lines
either electron,

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positrons, or photons.

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All right?

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You can-- fermions and photons,
charged fermions and photons.

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So we discussed how
the solutions look

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like, our spinors u and v.
And for outgoing electrons,

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for outgoing particles, we have
this adjunct vector here, which

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is given by u dagger gamma 0.

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And similarly for the
incoming antiparticle--

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v dagger gamma 0.

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For the photon, we have
the polarization vectors

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for incoming and
outgoing photons.

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All right.

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Then we have a vertex factor.

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Here, now, g e is a constant
and a dimensionless property.

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But we do have to
have a gamma mu here

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as part of our vertex factor.

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For the propagator,
our internal lines,

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we have a difference between
electrons, positrons,

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and photons.

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And that comes from the fact
that electrons and positrons

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are massive particles.

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So we have vertex
vectors which now

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have this 1 over
q square behavior,

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or 1 over q square
minus m square behavior.

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So here, you can
already see that there's

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going to be a complication later
when we evaluate or integrate

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over momentum--

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simply the same
discussion I had before.

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And we already know how to
solve this problem of infinities

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by renormalizing-- by having a
cut-off and renormalizing it.

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Excellent.

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So the next step, then,
is very much the same.

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There's no change.

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We have to make sure that
there's energy and momentum

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conservation, and
we enforce this

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by introducing delta functions.

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We have to integrate over each
and every internal momenta,

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and each internal line gets one
of those integration factors.

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And then after we integrate, we
are left with a delta function,

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and we have to cancel
that delta function.

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All right.

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In our toy experiment, the
order of things didn't matter.

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Everything we had in there
was scalar numbers, right?

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Here we do have a little bit
more complicated problem.

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So there's an
importance in the order

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of which we execute things.

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So what we want to do
is form fermion lines.

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We just follow a fermion as we
go from the left to the right.

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And then we find things
which are always of the form

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an adjoint spinor, a 4-times-4
matrix, and a spinor.

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And the result of that
is going to be a number.

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All right?

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Great.

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There is one additional
complication,

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is accounting for
duplications and making sure

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that the sign is [INAUDIBLE].

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I'm just mentioning this here.

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This will become more clear
as you work through examples.

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So there is an
antisymmetrization going on,

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where we have to
introduce a minus

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sign between different
diagrams that differ only

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by the interchange or the
exchange of two incoming or two

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outgoing electrons or positrons
and/or the incoming electron

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with an outgoing positron.

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So if you have a diagram
which is exactly the same,

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but the two incoming
electrons are interchanged,

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you have to add
those two diagrams.

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You have to add all
matrix elements together

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for recalculating amplitude.

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But you have to introduce
a minus sign when

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you change those two particles.

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So with that, we can now
just basically calculate

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whatever QED process we want.

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All the tools are already here.

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And what we want to do now
next, in the next video,

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and also in the
recitation and homework,

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is to go through a
few examples to get

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a little practice with this.

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There's a number of tricks
which will come in handy,

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and I'll explain those
in a separate video.

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They are just
mathematical tricks

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which allow us to
quickly evaluate

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the multiplication of spinors
and matrix elements and so on.

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All right.

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That's it for this video.

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Again, there is going to
be another two or three

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videos which deal with actually
evaluating or calculating

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matrix elements.