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I was making a little excursion into the realm of Euclidean conformal field theory,

which formally just consists in an analytic continuation in the coordinate variable.

So instead of talking about space and time, one talks about a complex variable z and its complex conjugate z bar,

and fields generally will be functions of both z and z bar.

What used to be a chiral field, a field that depends only on x plus, like t plus x,

will become a field that depends only on z and not on z bar.

So that explains the terminology holomorphic field.

So when we have a chiral field, phi of x plus,

then under this passage to Euclidean conformal field theory,

it becomes a field as a function of z only, and this is called holomorphic.

For obvious reasons.

And it is quite common to specify a model.

In the relativistic setting, we have to specify a model by prescribing commutators,

which is only partial information about the algebra, of course.

But in some cases, specifying the commutators is already sufficient,

because once you construct a representation on a Hilbert space,

then you also have products in this construction, not only the commutator.

So the specification of the commutator tells us something about the product.

Sure. In the Euclidean, one could instead of specifying the commutators,

also specify the product, as long as this is consistent.

And I mentioned to you that the proper formulation of a product of two fields

is in terms of the operator product expansion.

The product of two fields is never just a field again,

but you have to expand in the difference variable, x minus y,

or z minus w in the Euclidean language.

And the coefficient of this expansion will be other fields,

and you may specify these fields occurring in the expansion.

We are its OPE, its operator product expansion.

And here maybe I should say the OPE of its holomorphic fields.

And that would look in general something like this.

I have some holomorphic field, this variable z.

I have another holomorphic field, and a variable w.

And this is something like a sum over all, a priori all integers z,

n and z with powers of the difference times new fields taking at the point w.

And as I said, this kind of expansion holds only in a certain formal sense,

but that should not worry us too much here.

And just let me say that if you know this expand, then first of all,

this n cannot run over all integers because one has the dimension of this field here.

This field has dimension hn, which equals hA plus hB plus this n.

Just basically you count the powers of the scaling parameter,

which you get when you scale everything.

So you have some homogeneous transformation and there's a scale transformation.

And now we know that dimensions must not be negative.

And if dimensions must not be negative, then this n must not be too negative.

It can be at most minus hA minus hB.

And therefore, this is here a sum bounded below.

And so you have a finite singular part of this expansion,

the negative exponents of z minus w.

And from this, one can also recover the commutators.