And it turns out that the Schrodinger equation has many,
many interesting properties.
And one of them is called supersymmetry or SUSY as physicists call it.
Not only Schrodinger equation has supersymmetry, but
this one obtained from the classical stochastic dynamics does have it.
An the supersymmetry of the Schrodinger equation is based on the observation that
are Hamiltonian H can be factorized as a product of two operators,
A and A+, as shown in equations 55, and 56 here.
These operators are sometimes called the supercharge generators.
And the function, U prime is called the superpotential.
Now if we have these generators we can swap the order and
get a new Hamiltonian H+ shown in equation 57.
And the most interesting thing about this pair of Hamiltonians H and
H+ is that they have the generate spectra for all states excluding
the lowest energy state with zero energy, if such state exists.
And these can be seen using the simple chain of
transformations shown in equation 58.
If Si n is an eigenstate of H, who is an eigenvalue En,
then we can form a new state, A times this state, and this state
will be an eigenstate of the SUSY partner Hamiltonian H+ with the same energy.
And this means that all eigenstates of H with non zero energy
should be the degenerate with eigenstates of H+.
And now SUSY can be unbroken or spontaneously broken.
If it's unbroken, a ground state with zero energy exists.
On the other hand, if the energy of the ground state is larger than zero,
supersymmetry is broken.
And it turns out that mechanisms of breaking supersymmetry in
quantum mechanics and quantum field theory are the same mechanisms
that lead to tunneling and escape from metastable potential.
So how the escape looks like.
If we go back to the language of classical statistical physics,
then the process is described as an event when due to thermal fluctuations,
a particle gets enough energy to jump over the barrier.
And the probability of such event will be obtained as a product
of two factors, the Arrhenius factor B and pre-factor A.
The Arrhenius factor B is shown here in equation 60.
And its exponential in parameter Eb that gives the height of the barrier.
So if a barrier is very high then the actual escape probability can be
very tiny.
And vice versa, if a barrier is not too high, or
the energy of a particle is such that it's near the top of the barrier
then the escape probability might become quite noticeable.
And the remaining pre-factor A can also be computed for
one dimensional diffusion it turns out that this factor is proportional to
the frequency of oscillations, omega near the bottom of the potential well.
That's shown in equation 61 here.
It turns out that the same expression can also
be obtained from an equivalent quantum mechanical formulation.
And in this case it turns out that tunneling can also be described by
the laws of classical mechanics.
But applied in imaginary time.
In imaginary time, the kinetic energy becomes negative, and
the action becomes imaginary.
As you can see, if you look again at our equation,
for, which I did here as equation 62, for your convenience.
So for this case, the expression in the square root in there
integral is negative and therefore the action itself is imaginary.
But because the weight of the action is I times S,
this produces exponentially suppressed tunneling in quantum mechanics.
And finally, a few more words about the tunneling effect.
This effect is non perturbative as we said, so
it cannot be obtained as an expansion in small values of parameters kappa and
g around a model with a trivial vacuum x=0.
It turns out if we still start developing such
perturbative schemes they become divergence series and
the origin of these divergence of perturbative series and
tunneling turns out to be the same.
A mechanism for this is similar to divergence of Quantum Electro-Dynamics
that was discovered by Freeman Dyson in 1950s.
You can read more about such problems since statistical physics and
quantum mechanics in your weekly reading quiz for this week.
And for now we just want to summarize.
So we saw that reinforcement learning and inverse reinforcement learning can be used
not only to compute specific numbers in finals, but also to construct new models.
And we presented one such simple model for market dynamics that is inspired or
kind of derived from reinforcement learning in our previous course.
Now, in this week we took another look at the same model and
found the need to extend it by introducing the cubic non-linearity.
And this falls from the analysis of behavior of the model and
is needed for stability.
This cubic non-linearity can also probably be derived directly from
the enforcement learning approach.
But here we take a simpler route and just added this to our phenomenological grounds
using arguments based on asymptotic analysis on electricity and symmetries.
And all these are common are useful tools in physics.
In particular symmetries play a major role in determining characters or
phase transitions between different phases of matter.
Here we use such sort of analysis in a similar way to using
prior Bayesian statistics.
This is something that is not directly in the data but
should hold anyway based on some more general arguments.
In the course project for this course you will analyze the model that you
did in your previous course, but this time with keeping a non-zero value of g.
And hence with keeping a cubic non linearity.
And this will be it for this lesson.
And see you the next week.
