Physics Notes: Supersymmetry, Grand Unification, and String Theory

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Lecture 6: May 10, 2010 Back to PHY31

Topics: Vacuum, Supersymmetry Infrastructure

Questions

Q: Explain the Casmir effect

The Casmir effect is a predicted effect of having two conducting/reflecting surfaces in close proximity.

Without the surfaces you would just have a uniform vacuum energy, which is a QFT result of all the possible Feynman graphs with 0 particles in and 0 particles out. The plates interfere with some of the Feynman graphs in a way that modifies the energy density as a function of the separation. When energy is a function of displacement, you have a force.

Q: If we can think of as a coordinate transform, then if I have a large population of bosons, why can’t I transform all the bosons into fermions? This would obviously be a problem.

removes one boson and inserts one boson. Repeated application of simply results in 0 because .

What Does Supersymmetry Do

Supersymmetry has been around a long time without direct confirmation. However, there are 3 things that Supersymmetry does for us and which are likely to be measureable in the next decade.

1) Controls big numbers like the Higgs mass - it controls the divergent mass renormalization of the Higgs.

2) It predicts a set of stable particles that are good candidates for dark matter. We have good evidence that dark matter exists from the mass and spin of galaxys. They spin to fast for the observed mass and should fly apart. We also know that dark matter interacts very weakly with itself and with normal matter. We know this because the models say that the dark matter is in halos around the normal matter. If dark matter interacted with dark or normal matter, then kinetic energy would be lost in collisions and the orbit of the dark matter would be shrunk.

3) It produces a much higher precision unification crossing. Without Supersymmetry the running coupling constants for electric charge, weak charge and color charge cross within a couples of orders of magnitude up just short of the Planck Energy. With Supersymmetry included in the calculations the crossing of all three coupling constants is within 1%. Gravity crosses a bit later. In unified theories, there is just one coupling constant plus a broken symmetry that splits the single coupling out into the low energy coupling constants we know.

Symmetry breaking affects low-energy physics more than high-energy physics.

The reason that Supersymmetry produces running value curves for the coupling is that it introduces new diagrams containing loops of the superpartners. For example a photon can temporarily split into a super-partner/anti-super-partner pair, recombine into a photon and then interact with an electron.

Cosmology – Evolution of Energy Density for Different Types of Energy

Take a collection of ordinary or dark matter particles. The energy density is simply the mass density . As the universe expands, the particles simply spread out [on average – gravitationally or otherwise coupled collections of particles like galaxies don’t spread out]. So the ordinary matter energy density decreases inversely with the volume of the universe.

Now take a volume of space and measure the radiation energy content. We can assume that for every photon that leaves, that another enters. As the universe expands, the number of photons in our volume remains constant, but the wavelengths of the photons are stretched by the cube root of the volume increase, so radiation energy density decreases even faster than matter density.

What about vacuum energy density? It remains constant.

At the “beginning”, vacuum energy would have been a tiny fraction of the total energy in the universe. Now we estimate that it is 70% of the total.

Question: What happened to energy conservation?

In GR, the answer is simple, total energy is always 0. The forms of energy we have talked about are offset by the kinetic energy of expansion, which makes a negative contribution.

In a simple scalar form

This is the Hamiltonian for GR.

is the size of the universe, so the first term is the velocity of expansion.

You choose between the +, - or 0 to choose an open, closed, or static universe.

includes the density of ordinary and dark matter, vacuum energy, and radiation.

The Lagrangian of Fermion Fields with Supersymmetry

In Supersymmetry you have added dimensions of space. The Grassmann variables act like little vectors. Our field is now called a superfield. We can expand it as a polynomial of every possible combination of Grassmann variables that don’t yield a 0.

If we have Grassmann variables, then we have terms in our superfield.

Once we have a field, we make a Lagrangian:

Then we need to integrate to find the action

We have a small problem. We don’t know how to integrate with respect to Grassmann variables.

Integration with Grassmann Variables

The justification for the way we integrate with Grassmann variables is simply that it is useful. Integration is defined for definite integrals, which is what we need to compute the action anyway. [When we integrate with respect to a Grassmann variable there is only a single point in the space, there are no continuous ranges of Grassmann values.]

The first demand we make on integration is linearity.

We also make an analogy to ordinary integration where most physical functions we deal with go to 0 at infinity. In this case we have:

so we will assume that:

Example:

And:

Now integrate the general form of a function of a single Grassmann variable:

This seems trivial, but is just the right thing to do the bookkeeping for fermion fields.

Some other relevant examples with multiple variables:

You integrate from the inside out.

Because we have to swap variables first

Next class we will use this to build a more complex Supersymmetry example.