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MARKUS KLUTE: Hello and
welcome back to 8.701.

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So in this class we're going to
talk about the range of forces

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and specifically how
the range of forces

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depends on the mass of
the particle involved

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in transmitting the force.

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We have seen this table
before, the different forces,

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the strong force, the
electromagnetic force,

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and the weak force,
and the boson

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which carries the force, gluons,
photons, and the W and the Z

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bosons.

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But now we want to
actually just look

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into the aspect of the range
and how masses interplay here.

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So you all know the
electromagnetic potential

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due to a point charge given
by the Maxwell equation.

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We have seen this
equation before

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and the time-independent
potential here.

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You also know that the
massless photon gives rise

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to an infinite range of
electromagnetic [INAUDIBLE]..

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But the question now
is, how would this

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change, how would
this be modified

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if the photon were massive?

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But to do this we have to
generalize just a little bit.

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First of all, we have to look
at the time-dependent equation

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or the wave equation.

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And this is wave equation.

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You have also seen this before.

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Wave equation, and by
adding a mass term.

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So we are using here
an equation which

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has to be fulfilled
by our particle

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simulation between energy,
momentum, and mass.

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We have discussed this in the
context of special relativity

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before.

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What we are trying
to do now is build

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a Schrodinger-like
equation by using

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the quantum mechanical operators
for energy and momentum.

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So we just add
this here and find

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a new equation, which is called
the Klein-Gordon equation.

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So this equation has to be in
all particle waves or particles

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have to fulfill the equation.

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So the question is,
what kind of solutions

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does this kind of equation have?

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How does this look like?

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And if now start from, again,
a time-independent equation,

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you find solutions
which look like this.

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And you see again
very similar, then,

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to before the charge
over some constant

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as a function of radius, but you
also see this exponential term.

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And what we see here
is this potential,

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this form of potential is
called Yukawa potential.

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And what's nicely
shown in this plot here

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is, again, the potential
or the function

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of radius in units
of centimeters,

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the dependency of
the mass [INAUDIBLE]

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the range of the
force and the mass.

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So you see here as an example
for the mass equal to here.

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Mass is going to 1 GeV and
mass is going to 10 GeV.

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And you can easily see
the range of the force

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is reduced by the fact that
the particle actually had mass.

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Now, the gauge boson, the boson
of the weak interaction, the W

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and the Z boson
are quite massive.

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They have masses in
the order of 80, 90,

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in the order of 100 GeV.

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So you can see that
the masses actually

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leads to a reduction of
the range of this force.

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And yet, we find--

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here, just don't look.

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Don't look at the
gravitational part here.

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It's not part of
the standard model.

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If you look at
electromagnetic interaction,

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where the range is
infinite, we find

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that the range of
the weak interaction

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is 10 to the minus 18 meters.

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So this is greatly
reduced, because

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of the masses of the particles.

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If you just look at
the charge itself,

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we would find that
the weak interaction

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and the electromagnetic
interaction

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are actually quite comparable.

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We will also talk about
the strong interaction.

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We see here there's
a strong interaction

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that the coupling, the
alpha, is in the order of 1.

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And when we talked
about Feynman diagrams,

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we talked about
perturbation theory.

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If you do a perturbation theory
for a interaction of order

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of 1, couplings of order 1,
your vertices on the order of 1,

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you will see that perturbation
theory might break down.

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So, OK.

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So what we have seen in this
lecture is how a mass, or how

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masses, reduce the
range of a force.

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We have simply built
Klein-Gordon equation here,

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looked at a solution, and found
that there is this diminishing

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of the range of the force.

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Later, we will look at one
additional complication,

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and another equation
which has to do

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with [INAUDIBLE]
particles, which

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is the so-called
Dirac equation, which

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has to be fulfilled or
hold up by the fermions.

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But that's beyond the
scope of this class.

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We'll look at this
later in more detail.