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What I explained to you yesterday, or started explaining to you yesterday,

was how to use this beautiful idea of regis to describe curved geometry

in terms of triangulation data,

might be profitably used in formulating a non-perturbative path integral for gravity.

And I will be following now one specific way of doing this,

the approach of dynamical triangulations,

of which there is kind of a Euclidean and a Lorentzian version,

and the basic idea is that you have just a single type of building block.

So the edge lengths, so the continuous variables of the triangulations,

they're just fixed to a single value.

When you are in Euclidean space, you just choose all your building blocks to be equilateral.

So in two dimensions you would just choose equilateral Euclidean triangles,

in three dimensions equilateral tetrahedral and so on.

Now that we'll also be doing this in Lorentzian signature,

it's convenient and actually essential to this weak rotation I will perform later on

to keep two types of edge lengths, a time-like edge length,

which has negative length squared, and a space-like one.

And then the set of possible building blocks you have are objects with these two types of edges,

which have, and since the edge length determines the geometry on the inside,

there is just a handful of possibilities that actually give you a Minkowski metric,

flat metric on the inside with the right signature minus, plus, plus, plus in four dimensions.

For D larger or equal to three, I already told you that it's very difficult to evaluate

this path integral by analytical means, not even in an asymptotic sense,

say when the number of building blocks n becomes large or goes to infinity.

And in that case you have to resort to other means, namely numerical ones,

to approximate the path integral.

And I had briefly explained to you how this works via Monte Carlo simulations.

You do a kind of important sampling, and that's how you will compute or probe certain observables.

So an observable is just inserting kind of a geometric quantity into the path integral

and of course normalizing the whole thing.

And these objects are then studied in higher dimensions using Monte Carlo simulations.

To illustrate the power of having both this approach and having actually in higher dimensions

a quantitative way of producing numbers, of evaluating these, albeit of course with some error bars

because you're performing a simulation on a finite, on a computer that can only simulate ever finite systems,

I explained to you, I talked about a specific quantum observable you can look at,

and this observable was the Hausdorff dimension.

And the Hausdorff dimension was extracted by relating geometric properties that were easy to measure,

like the volume in this case, the volume of a geodesic ball of radius r,

and relating that to this geodesic radius itself and then kind of extracting the leading power in power law behavior.

And this seems very simple-minded, but let me tell you that's actually one of the few nice observables

we currently have to study quantum geometry in higher dimensions.

And the somewhat surprising, perhaps surprising thing was that when I evaluated this Hausdorff dimension,

in four dimensions, so the Euclidean dynamically triangulated path integral,

I got a value, I measured a value that was definitely not four.

So even with very large error bars, certainly nothing like four.

And well, you could say, well, so what's the big bother?

Well, the big bother is that, of course, at least on large scales,

whatever quantum geometry your path integral produces should reproduce certain basic features of the classical theory.

And now the dimension of the space-time manifold is even kind of, you could say, a pre-geometric feature, pre-metric feature.