Dynamically Triangulating Gravity hep-th/0105267

[marcus]

this thread focuses on the May 2001 paper of Ambjorn Jurkiewicz Loll called
Dynamically Triangulating Lorentzian Quantum Gravity
http://arxiv.org/hep-th/0105267

It may be that this will turn out to have been the historical landmark paper
because this year we have had two confirming followup papers from AJL
showing that they got their approach (mapped out in 2001) to work in 4D.

It starts with no space and time, but with hundreds of thousands of things like (4D) tetrahedrons. And a combinatorial spacetime grows from them.
according to a propagator. and more conventional macroscopic space and time features emerge at large scale from this more fundamental microscopic picture.

so it has the kind of features you'd like to see in a theory of Quantum Gravity----a general relativistic quantum theory of space and time---and it provides an interesting venue where matter could be added to the picture and studied in this new quantum spacetime context.

In this 2001 paper, AJL are laying out their program. they give a lot more detail here than they do later in the papers where they report that the program is working. what they are proposing in this (I think landmark) early paper is the Monte Carlo computer runs of emerging 4D spacetime that they then went and did and reported later.

Because its more detailed and nittygritty, it seems more accessible. this is the paper that the ones this year refer back to. So I guess it is basic. Let's see if we can get somewhere with it.

Remember that when this paper appeared, String approaches had been attempted for over 20 years, and Loop approaches for over 10 years.

 

[marcus]

snatches of the abstract

"Fruitful ideas on how to quantize gravity are few and far between. In this paper, we give a complete description of a recently introduced non-perturbative gravitational path integral whose continuum limit has already been investigated extensively in d<4, with promising results.

It is based on a simplicial regularization of Lorentzian space-times and, most importantly, possesses a well-defined, non-perturbative Wick rotation.
...
... We define a transfer matrix for the system and show that it leads to a well-defined self-adjoint Hamiltonian.

In view of numerical simulations, we also suggest sets of Lorentzian Monte Carlo moves.

We demonstrate that certain pathological phases found previously in Euclidean models of dynamical triangulations cannot be realized in the Lorentzian case."
========

the Monte Carlo moves are "shuffles" of the spacetime that help to randomize it----there are 5 or 10 basic shuffles and if you do enough of them in random order it gets to be ergodic (you thoroughly explore the possibilities of all the different spacetime geometries)

do enough changes and you get to any weird geometry whatever, and also most often to the most typical, so it gives a practical way arriving at a probablility distribution on random evolutions of geometry.
======

the pathological spacetime evolutions were what they were always encountering for almost 10 years prior to their 2001 Eureka moment. they would set up computer simulations of geometry of the universe and the geometry would either (1) crumple into a supercompact wad or (2) get all branchy and leafy like some fractal fern
Ambjorn apparently is a very patient person. he endured this for some 10 years. Maybe it is a Danish virtue.
then around 2001 the three of them got together and decided to try it with the extra provision that the spacetime had to be "Lorentzian" in the sense of having some CAUSAL ORIENTATION. the analog of having future and past lightcones so you can say who is in the causal past of whom.

It basically means that the path integral, or the spacetime, is sliceable into ordered layers

but it is still going to consist of little 4D tetrahedrons

=======
a 4D "tetrahedron" is technically called a "4-simplex" and it has 5 vertices and it has 5 faces---each face being an ordinary 3D tetrahedron.
========

A "wick rotation" just means going back and forth between complex time and real time-------you erase the imaginary number i (sqr root of -1), and time becomes realnumber valued----you write the number i back in, and it is imaginarynumber valued.

It is named after Mr. Wick.

It doesn't mean, like it sounds, that you have to rotate your wick.
=============

the transfer matrix mentioned above is very important.
the propagator has the "semigroup property" which means
that it is a very proper propagator
=============

some of this is explained reasonably clearly in the main part of
the paper, and it turns out to be not nearly as bad as you thought
it might be

 

[marcus]

some parts of the introduction

sometimes a paper from a few years back can even sound prophetic...

---quotes from Introduction---

Despite an ever-increasing arsenal of sophisticated mathematical machinery, the nonperturbative quantization of gravity remains an elusive goal to theorists.

Although seemingly negligible in most physical situations, quantum-gravitational phenomena may well provide a key to a more profound understanding of nature.

Unfortunately, the entanglement of technical problems with more fundamental issues concerning the structure of a theory of quantum gravity often makes it difficult to pinpoint why any particular quantization program has not been successful. The failure of perturbative methods to define a fundamental theory that includes gravity has led to alternative, non-perturbative approaches which seek to describe the quantum dynamics of gravity on the mother of all spaces, the “space of geometries”.

By a “geometry” we mean a space-time with Lorentzian metric properties. Classically, such space-times usually come as smooth manifolds M equipped with a metric tensor field $$g_{\mu \nu}(x)$$. Since any two such metrics are physically equivalent if they can be mapped onto each other by a diffeomorphism of M, the physical degrees of freedom are precisely the “geometries”, namely, the equivalence classes of metrics with respect to the action of Diff(M)...
---end quote---

Loop gravity is an example of one of the non-perturbative approaches where the configuration space (upon which quantum states are defined) is the "space of geometries"---more precisely the space of connections on the underlying manifold.

Stringy perturbative approaches use a fixed background geometry and superimpose small changes or perturbations on top of the fixed background.

They are talking about the failure of perturbative approaches and the increased effort applied to non-perturbative ones using a space of all geometries.

but they intend to go one better than either.

---more quote---
In what follows, we will describe a path-integral approach to quantum gravity that works much more directly with the “geometries” themselves, without using coordinates [Footnote 2]. It is related in spirit to other constructions based on simplicial “Regge” geometries [3], in particular, the method of dynamical triangulations ...

Footnote 2: There is no residual gauge invariance since the state sum is taken over inequivalent discretized geometries. In this sense, the formalism is manifestly diffeomorphism-invariant. ...
---end quote---

 

[marcus]

I had better link to some bibliography.
this PF thread has links to other papers by the authors
https://www.physicsforums.com/showthread.php?t=53974