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Yes, the subject of my lecture theory is conformal quantum field theory.

And we have already heard in the previous lecture...

We have already heard in the previous lecture about topological field theory

and there are in fact some relations between my subject and Christoph's subject

but these relations are pretty loose and my emphasis will be on the relativistic quantum side

of this big area of fields whereas topological field theory is located in a somewhat different context.

Nevertheless, some of the models are the same.

Let me start from the total scratch and tell you what is quantum field theory all about.

And I will just start with quantum field theory and only in the course of the presentation

it will be clear what makes the difference, what makes the quantum field theory conformal.

Just a special case of general quantum field theory.

And I want to emphasize that quantum field theory is a unification of several very important

general and fundamental concepts of physics mainly concentrated around the issue of energy.

For instance, when we start about quantum mechanics or quantum theory in general

there is the prime object is the Hamiltonian which is an operator on a Hilbert space.

It's the same letter but a different font.

And the role of the Hamiltonian is that it describes the dynamics.

So it describes how the system evolves in time.

And one way of writing is to write the commutator.

So we have any quantity phi.

This is a dynamical quantity.

And the commutator tells us how this quantity changes in time.

This is the most fundamental equation of quantum mechanics and also actually of classical mechanics

if you replace the commutator bracket by a Poisson bracket.

And the Hamiltonian age is not just any operator on the Hilbert space but it has a physical meaning.

Namely, it describes the energy.

Physical meaning, it is the operator which represents the energy.

In other words, the eigenvalues of these operators would give the possible values of the energy of the system.

So here we have the Hamiltonian in quantum theory.

The second area of physics which comes into play is field theory.

And field theory says quantities of interest are not just global quantities but local quantities, fields that depends on space and time.

So for instance, phi depending on t and x, such quantities are called fields.

And in particular, the Hamiltonian, the total energy, is something like an integral of a density.

Only a spatial integral at any fixed time.

It doesn't matter which time I take because by this very formula here we see that the time dependence of the Hamiltonian is given by the commutator with itself.

So it does not depend on time.

So I may put any time here. It doesn't matter which one.

But the density here is a field.

And for instance, you may think of the case of electrodynamics where the fundamental fields are the electric field and the magnetic field E and B.

And then maybe you remember from your physics course that this is the formula for the energy density of the electromagnetic field.

So that's a field that depends on space and time. It means that the energy density is redistributing all the time over space and time.

But there is a conservation of total energy which says that you cannot just create energy from nothing.

You can only put energy somewhere at the cost of taking it somewhere else.

And that's the continuity equation.

And then it says that the time derivative of the energy density is the divergence of an energy current.

So this is an energy of flow, energy current or energy flow, matter of taste.

This is the energy density.

And this energy current, which tells us how energy flows about, is given in Maxwell's theory by the formula.

It's known as the pointing field or pointing vector field.