Uniqueness of deSitter spacetime and the Standard Model (hints from two papers)

[marcus]

Loll will deliver three one-hour talks at Oporto in mid July Here's the abstract
Renate Loll, Quantum Gravity from Causal Dynamical Triangulations
Abstract:
I discuss motivation, implementation and results of the nonperturbative approach to quantum gravity based on Causal Dynamical Triangulations, including the recent reconstruction of de Sitter space from quantum fluctuations.

A published version of some of what she will be talking about is here:

http://arxiv.org/abs/0712.2485
Planckian Birth of the Quantum de Sitter Universe
J. Ambjorn, A. Gorlich, J. Jurkiewicz, R. Loll
published in Physical Review Letters, 7 March 2008
10 pages, 3 figures
(Submitted on 17 Dec 2007)

"We show that the quantum universe emerging from a nonperturbative, Lorentzian sum-over-geometries can be described with high accuracy by a four-dimensional de Sitter spacetime. By a scaling analysis involving Newton's constant, we establish that the linear size of the quantum universes under study is in between 17 and 28 Planck lengths. Somewhat surprisingly, the measured quantum fluctuations around the de Sitter universe in this regime are to good approximation still describable semiclassically. The numerical evidence presented comes from a regularization of quantum gravity in terms of causal dynamical triangulations."
==========================

The sum-over-geometries is analogous to a Feynman path integral. The universe takes a path through a regularized collection of possible spatial geometries. Each path is a possible spacetime. In effect integrating to get the "average" path gives de Sitter spacetime. That is an oversimplification but the key idea is that it picks out de Sitter as a special distinguished background geometry---that was not put in by hand at the beginning.

what one puts in at the beginning are local dynamics of microscopic geometry. there is no guarantee that any recognizable global spacetime will result, or even that you will get something four dimensional. But something recognizable does indeed come out of the path integral. So there is a suggestion of uniqueness here.
============================

Chamseddine will also be giving three one-hour talks at Oporto: on research hinting at the uniqueness of the standard particle model. I am not announcing this. (I already announced the Oporto Meeting lineup.) I am initiating a discussion, or at least proposing one. Hopefully people will find these two matters of interest. Here is the Oporto minicourse abstract:

Ali Chamseddine, Classification of discrete noncommutative geometries and the uniqueness of the standard model
Abstract:
Assuming that space-time is a product of a continuous four-dimensional manifold times a discrete space F, we classify the irreducible geometries F consistent with imposing reality and chiral conditions on spinors. Remarkably we find that the noncommutative geometry of the standard model results almost uniquely, with all the necessary details. In particular we prove that the number of fermions per generation is 16, the square of an even integer. The spectral action of this geometry is constructed, and the model is analyzed.
=====================

This corresponds to a published paper also:
http://arxiv.org/abs/0706.3688
Why the Standard Model
Ali H. Chamseddine, Alain Connes
13 pages
(Submitted on 25 Jun 2007)

"The Standard Model is based on the gauge invariance principle with gauge group U(1)xSU(2)xSU(3) and suitable representations for fermions and bosons, which are begging for a conceptual understanding. We propose a purely gravitational explanation: space-time has a fine structure given as a product of a four dimensional continuum by a finite noncommutative geometry F. ...Under an additional hypothesis of quaternion linearity, the geometry which reproduces the Standard Model is singled out (and one gets k=4)with the correct quantum numbers for all fields. The spectral action applied to the product MxF delivers the full Standard Model,with neutrino mixing, coupled to gravity, and makes predictions(the number of generations is still an input). "

Both these lines of research, highlighted by the Oporto Meeting, are clearly driving towards establishing a kind of uniqueness. In terms of such and such a theory (CDT, NCG) the world can only be one way, the way we see it. That is the general idea or thrust---naturally there are qualifications and stuff to be worked out, but that's where both are heading.

I want to do some explication and I'm going to start with a few quotes from Loll's paper.

 

[marcus]

Over the years, here at PF, I've done a bunch of explication of the basics of Loll's approach. If you want to get the basic physics of it explained, you might check out her website. She has links to her papers and the information is pretty well organized.
http://www.phys.uu.nl/~loll/Web/title/title.html
I will skip to what is special about this paper---maybe review basics lightly later on if there is interest.

To provide context here are portions of the Introduction and Conclusions.
==quote==

1 Introduction
To show that the physical spacetime surrounding us can be derived from some
fundamental, quantum-dynamical principle is one of the holy grails of theoretical
physics. The fact that this goal has been eluding us for the better part of the
last half century could be taken as an indication that we have not as yet gone
far enough in postulating new, exotic ingredients and inventing radically new
construction principles governing physics at the relevant, ultra-high Planckian
energy scale. – In this letter, we add to previous evidence that such a conclusion
may be premature.
Our results are obtained in the framework of Lorentzian simplicial quantum
gravity, based on the concept of causal dynamical triangulations (CDT). While
referring to [1, 2, 3] for details, briefly, it defines a nonperturbative way of doing
the sum over four-geometries, assembled from triangular building blocks such that
only causal spacetime histories are included. To perform the actual summation,
one rotates them to spacetimes of Euclidean signature. The building blocks are
four-simplices characterized by a cut-off a, the side length of the simplices. The
continuum limit of the regularized path integral corresponds to the limit a → 0,
possibly accompanied by a readjustment of bare coupling constants, and such that
the physics stays invariant. The challenge of a quantum field theory of gravity is
to find a theory which behaves in this way, and suitable observables to test it.
How can we judge whether CDT can be taken seriously as a regularized quantum
field theory of gravity? Our knowledge of the physical world suggests that a
viable theory should generate a ‘background geometry’ with positive cosmological
constant, superposed with small quantum fluctuations. The challenge is to obtain
this from a background-independent formulation where no background spacetime
is put in by hand. We have earlier provided indirect evidence for such a scenario
[4, 5]. Here, we present new computer simulations which confirm this picture
much more directly, by establishing the de Sitter nature of the background spacetime,
quantifying the fluctuations around it, and setting a scale for the universes
we are dealing with...

4 Discussion
The CDT model of quantum gravity is extremely simple, namely, the path integral
over the class of causal geometries with a global time foliation. In order to perform
this summation explicitly, we introduce a grid of piecewise linear geometries, much
in the same way as when defining the path integral in quantum mechanics. Next,
we rotate each of these geometries to Euclidean signature and use as bare action
the Einstein-Hilbert action 2 in Regge form. Nothing else is put in.
The resulting superposition exhibits scaling behaviour as function of the fourvolume,
and we observe the appearance of a well-defined average geometry, that
of de Sitter space. We are definitely in a quantum regime, since the fluctuations
around de Sitter space are sizeable, as can be seen in Fig. 1. Both the average
geometry and the quantum fluctuations are well described by the mini-superspace
action (4). Unlike in standard cosmological treatments, this description is the out-
come of a nonperturbative evaluation of the full path integral, with everything
but the scale factor (equivalently, V3(t)) summed over...

...Renormalization group methods have produced predictions for the scaling violations of
G in the context of asymptotic safety [7], which in principle we should be able to
test. In this context it would be ideal to have an observable with an associated
correlation length that could be kept constant when expressed in terms of V₄^1/4.

A further step will be to include matter in the model and verify directly that G
can indeed be interpreted as Newton’s constant, perhaps along the lines pursued
earlier in Euclidean quantum gravity [8]. All of these issues are currently under
investigation.Footnote 2 Of course, the full, effective action, including measure contributions, will contain all higher-derivative terms.


==endquote==

Reference [7] is to the Asymptotic Safety approach of Reuter, Percacci, and others. This is a different QG approach which corroborates some of the findings of CDT, regarding reduced spacetime dimensionality at very small scale. The V term is the fourth root of the fourvolume, proportional to the number of simplices in the Monte Carlo run.

 

[marcus]

that brief passage in the Discussion section really does give a nice concise description of the CDT approach, which is in fact rather spare and elementary-----a straightforward application of path integral ideas to evolving geometry:
==quote==
The CDT model of quantum gravity is extremely simple, namely, the path integral
over the class of causal geometries with a global time foliation. In order to perform
this summation explicitly, we introduce a grid of piecewise linear geometries, much
in the same way as when defining the path integral in quantum mechanics. Next,
we rotate each of these geometries to Euclidean signature and use as bare action
the Einstein-Hilbert action in Regge form. Nothing else is put in.
==endquote==

What she means by "rotate...to Euclidean signature" is Wick rotation. In effect switching time between real numbers and complex numbers. It means incidentally that deSitter spacetime rotates to become a Euclidean sphere.

What she means by the "grid" is the subset of all geometries which are constructed using a large number of simplicial building blocks. Lego masonry, if you ever played with Lego blocks. It is a brilliant simplification IMO and gets rid of coordinates entirely. Regge showed how to implement the Einstein equation or Einstein Hilbert action without coordinates, essentially by COUNTING various numbers of blocks.

 

[arivero]

Not sure if it is the same. Perhaps you could ask Chamseddine for the blue-pre-prints of the course. What is clear is that with the new model, the previous classification (mainly from german and french teams) was to need a review.

The unexpected thing of the models two years ago is that they are six-dimensional, instead of seven as one could expect from KaluzaKlein.

Ali Chamseddine, Classification of discrete noncommutative geometries and the uniqueness of the standard model
Abstract:
Assuming that space-time is a product of a continuous four-dimensional manifold times a discrete space F, we classify the irreducible geometries F consistent with imposing reality and chiral conditions on spinors. Remarkably we find that the noncommutative geometry of the standard model results almost uniquely, with all the necessary details. In particular we prove that the number of fermions per generation is 16, the square of an even integer. The spectral action of this geometry is constructed, and the model is analyzed.
=====================

This corresponds to a published paper also:
http://arxiv.org/abs/0706.3688
Why the Standard Model
Ali H. Chamseddine, Alain Connes
13 pages
(Submitted on 25 Jun 2007)

Under an additional hypothesis of quaternion linearity, the geometry which reproduces the Standard Model is singled out