Hamiltonian

It takes a massive, specific spike in energy to kick that ball completely out of the valley and over the mountain into a new phase.

So two materials are in the same phase if their Hamiltonians can smoothly connect without eliminating that protective energy valley.

It ensures stability.

Exactly.

And when it becomes a mixed state, that deep valley, the Hamiltonian, literally ceases to exist mathematically.

The map is just gone.

So what does this all mean? If pure states are like a perfectly sealed terrarium, but mixed states are like opening the window to a hurricane, and we've just lost our mathematical map because the Hamiltonians are gone, how do scientists even begin to categorize this mess? You're basically blindfolded.

That is the exact wall the field has been slamming into.

The sum from I equals one to N of L of X sub I comma W with the population loss defined via the standard N right arrow infinity limit.

One thing about the loss that sometimes goes underappreciated is that it looks a lot like what we might call a parameterized Hamiltonian.

To see this, note that we can read the empirical loss as a total energy, written as a sum of energies of individual particles in the system.

Each data point is a particle, and L of X sub I comma W is its energy when the model has parameters W.

And the last part, there's this segment right at the end of Sabina's little rant when she talks about a different area, something called loop quantum gravity.

So she says, at the end, seeming to compare this lack of a well-defined Lagrangian in GU, partially due to this undefined Shiab operator, to some other difficulties in this other theory, loop quantum gravity, where there are issues in finding a well-defined Hamiltonian, which is something a bit like a Lagrangian.

So she asks,

or how about loop quantum gravity? Do they have a well-defined Hamiltonian? Where is the classical limit?

So here I partly agree.

I agree with Sabina that there are serious problems with loop quantum gravity that have not yet been overcome, which is why personally I don't work on it.

And few people work on it nowadays compared to other areas like string theory.

And the two criticisms that she leveled are essentially correct, that loop quantum gravity, it does have what's called a well-defined kinematical Hilbert space, but they have not yet realized the Hamiltonian constraint in an anomaly-free way.

Well, uh, so the way that we learn quantum field theory in graduate school is, uh, that generaliz- the way that you generalize quantum mechanics to this spatial field.

So in quantum mechanics, as you know, you have classical dynamical system described by Lagrangian or Hamiltonian, and then you apply the, uh, procedure called quantization, and then, uh, formulate time evolution, wave functions, and et cetera.

And the way we learn quantum mechanics in graduate school, and it's the way that, uh, all the textbook of quantum field theory is written, is to apply that to the dynamics of field, like, uh, electromagnetic fields or, uh, fi- fields of quarks and leptons in the case of a standard model.

And, uh, so, so you start with Lagrangian, you apply the methods of quantization, and then the, the theory is a quantum field theory.

Essentially, just as F equals ma applies to Newtonian systems, the Schrödinger equation applies to quantum systems by describing the system's three-dimensional wave function, represented by the Greek letter psi.

In this equation, this term is called the Hamiltonian operator, which is a set of mathematical operations that describes all the interactions that are affecting the state of the system, which can be interpreted as the total energy of a particle.

But while the Schrödinger equation can calculate the wave function of a system, it does not specifically reveal what the wave function is.

Max Born proposed that we interpret the wave function as a probability amplitude, where the square of the magnitude of the wave function describes the probability of an electron existing in a particular location.

Okay.

So the oracle, the Hamiltonian, the, the black box, you have to really think hard about what you want the quantum computer to do for you.

Way harder thinking that you, than you ever had to do.

Now, I, we, I wish you had the benefit of being here earlier today.

And just before this talk, Bruno, you and I were talking about AI and its role in this.

Yeah.

AI is the perfect companion to help to stand up the Hamiltonian, the, the black box, the quantum circuit necessary.

I absolutely would never have been able to create a, um, a quantum circuit capable of solving for Grover's algorithm just on my own.

you bring it to linear that all of that is classical process.

Once you are ready to put that into quantum, then from there on, you basically have to make it a Hamiltonian system.

So Hamiltonian can be solved in the NISQ error through like variational circuit and in quantum native error through like HHL type of algorithms.

But that's how things work.

Now, when we started doing that, because the approach is how we are linearizing the problem is still your classical method of finite element, finite volume type of methods.

So, valence shell electron pair repulsion, the VSEPR, V-S-E-P-R, is a well-known theory.

And according to that theory, where like spin electrons avoid each other and unlike spin electrons form pairs, we have to understand that wave mechanics have to then invoke what's known as Coulomb and exchange integrals of the kinetic and electrostatic terms that ultimately arise in what's called the Hamiltonian.

So the purpose of all of this is to bridge that dialectic.

In particular, to be able to experimentally, of course, not just theoretically, isolate the influences of electron anti-symmetric spatiotemporal moments that we know must be occurring within biological membrane molecular structures.

Analytical mechanics.

Advanced formulations like Lagrangian mechanics and Hamiltonian mechanics use energy and calculus of variations instead of direct forces to solve complex motion.

Most users agree that rigorous college courses distinguish themselves by moving past basic forces into these advanced mathematical frameworks.

I want to get into Hamiltonian mechanics.

That's one of the ones written down on my list.

We want to know about quantum mechanics.

Hamilton's Principle.

Conserved Quantities, Rigid Bodies and Tops.