The n-Category Café

Now I am quite fond of the Holographic Principle, but your formulation of it is just maths

But isn’t that like saying

“I am quite fond of Maxwell’s theory of electromagnetism, but saying that it is a theory of line bundles with connection is just math.”

?

How does physics work, since Newton? We see a phenomenon in nature and are looking for the math modelling it. The math by itself is always “just math”, but that’s why physics is successful.

A physics formulation would discuss […] the meaning of inertia

Right. But before I can do so, I have to understand what is actually going on. Before I can really understand the role of inertia, or anything else for that matter, in the context of the holographic principle, I need to know exactly what that principle actually is.

Right at the moment, I don’t see that this is the case. From the standpoint of mathematical physics (which is not at all the standpoint of most practicing physicists) the holographic principle is a mystery at the moment. Try to explain to a mathematician what the statement of the principle is, to quickly see that there is really no precise statement yet.

For instance, one reads sentences about how the principle is supposed to relate “observables” of one theory to correlators of another. But the observables here are that of a gravitational theory. It’s hard to find precise statement about what is actually meant by “observables” here. I don’t think people really know.

Just look at Witten’s latest paper, whose comments triggered my above entry. He quite openly admits at several places how at least one side of the supposed duality is by itself quite a mystery, conceptually. Without much ado, he seems to accept the point of view that “observable” on the side of the gravitational theory has to mean “S-matrix”.

It seems clear that in order to make progress beyond this proverbial handwaving (nonwithstanding the dramatic success this handwaving may have, in particular when it’s Witten’s hands that are being waved) we need to

- first understand what QFT actually is

- and then identitfy that property of the space of all QFTs which we want to identity as the “holographic principle”.

There are essentially two approaches for giving QFT a rigorous underpinning:

A) what is called “AQFT”: this is built on the Heisenberg picture of quantum mechanics, the basic building blocks are certain cosheaves of “local observable” algebras.

B) functorial QFT (as conceived originally by Atiyah and Segal): this is built on the Schrödinger picture of quantum mechanics, the basic building blocks are representations of cobordism categories.

The problem with looking for axiomatics is that one is likely to fall short of the goal. Maybe both these approaches are insufficient for capturing whatever is meant by the holographic principle. But maybe not. We can at least try.

Karl-Henning Rehren once tried in the context of A). While it seems he found some kind of relation between higher and lower dimensional QFTs which depends on AdS structures, my impression is that his axiomatics does not really capture the idea of the holographic principle.

So here I am trying to understand what the axiomatics of the holographic principle would be in the context of B).

And I am saying: there is i) a natural candidate mechanism there which relates $d$-dimensional to $(d-1)$-dimensional QFTs (namely pseudonatural transformation), and, ii), there is a bit of evidence that this actually does describe the holographic principle in the situation which is best understood, namely in $d=2$/$d=3$.