Quantum Gravity

[Johnny5]

Have a read.

 

Note: Im only 15' date=' so be kind.[/quote']

 

Very well written paper Ed, congratulations on it. I just read this part of the abstract here:

 

 

Gravity and The Quantum Vacuum Inertia Hypothesis, where they suggest that virtual photons, there as a result of Heisenberg’s, uncertainty principle, are turned into real particles by an accelerating object. The pressure caused by the particles hitting the object (and the resonance) causes inertia.

 

Do you undertand the meaning of this?

[ed84c]

Thanks and Indeed I do.

 

Its for an AS Paper, Science for Public Understanding, that GCSE pupils can do.

[Johnny5]

What is a virtual photon?

[ed84c]

Im aware that this may be constrived as diverting the Thread somewhat, So in this post and in yours, any further questions can be edited in, rather than posting again, is that OK?

 

Well a virtual photon is where as Heisenberg stated, the less time looked at a system the more unpredictable it is, and photons pop in and out of existance, borrowing and returning energy. This is also incidently how a black hole evaporates.

 

If your wondering HOW i know this, i read a book a couple of years back, called 'In Search of Shrodinger's Cat'. Seriously good read.

[Martin]

Im aware that this may be constrived as diverting the Thread somewhat,....

 

Ed and J5, in this case I am very happy with the focus so I do not feel as if this (Quantum Gravity) thread is being diverted. It seems to me that what you are talking about is very welcome. but suit yourselves about editing and how you want to maintain focus.

 

Ed, I have always heard of the radiation that arises purely from acceleratation as "Unruh radiation" after the Canadian (Uni british columbia) physicist Bill Unruh who published about it in 1975 or so in Physical Review series D.

He defined the socalled Unruh temperature.

 

Indeed Rindler's name is also associated with this because rindler defined the Rindler horizon (analogous to the BH event horizon) in connection with accelerating observer.

 

your essay does not mention Bill Unruh but does mention Paul Davis and Rindler. Maybe you should google "Unruh radiation" and add a footnote or a reference, just for completeness. You say:

 

"In the mid 1970s Paul Davis at the University of Newcastle made the discovery that an observer accelerated through the quantum vacuum is actually bathed in these photons that have become real due to the acceleration. Hiasch, being an astrophysicist, as he claimed, knew that, these photons can cause a pressure upon an object. The virtual photon flux was named the Rindler Flux."

 

have to go

 

I dont say anything with yr essay is wrong, only that it is not complete in references

[ed84c]

sorry, given it in now

[5614]

virtual photons, there as a result of Heisenberg’s, uncertainty principle, are turned into real particles by an accelerating object[/b']. The pressure caused by the particles hitting the object (and the resonance) causes inertia.

I understand most of this, but how can a virtual photon be turned into a real photon (or particle?)

 

I mean, I've just read about Unruh effect or Unruh radiation:

 

http://en.wikipedia.org/wiki/Unruh_radiation]In[/url'] modern terms, the concept of "vacuum" is not the same as "empty space", as all of space is filled with the quantized fields that make up a universe and the vacuum is simply the lowest possible energy state of these fields, a very different concept than "empty"....... According to special relativity, two observers moving relative to each other must use different time coordinates. If those observers are accelerating, there may be no shared coordinate system and, due to this, the observers will see different quantum states and thus different vacua.

 

OK, I accept that, but how does that make virtual photons turn into real photons/particles???

[Martin]

Here is an exerpt from Bojowald article "Big Bang and Black Hole" (2003)

 

http://arxiv.org/astro-ph/0309478

 

 

 

"1 Introduction

When the big bang is approached, the volume becomes smaller and smaller and one enters a regime of large energy densities. Classically, those conditions will become so severe that a singularity is reached; the theory simply breaks down. For a long time, the expectation has been that somewhere along the way quantum gravity takes over and introduces new effects, e.g. a discrete structure, which prevent the singularity to develop. This presumably happens at scales the size of the Planck length lP, i.e. when the universe has about a volume l^3 P. Since at the classical singularity space itself becomes singular and gravitational interactions are huge, such a quantum theory of gravity must be background independent and non-perturbative. A theory satisfying these conditions is in fact available in the form of loop quantum gravity/quantum geometry (see[1, 2] for reviews). One of its early successes was the derivation of discrete spectra of geometric operators like area and volume[3, 4, 5]. Thus, the spatial geometry is discrete in a precise sense. Furthermore, matter Hamiltonians exist as well-defined operators in the theory which implies that ultraviolet divergences are cured in the fundamental formulation[6, 7].

 

Both properties must be expected to have important consequences for cosmology. The discreteness leads to a new basic formulation valid at small volume, and since gravity couples to the matter Hamiltonian, its source term is modified at small scales when the good ultraviolet behavior is taken into account. It is possible to introduce both effects into a cosmological model in a systematic way, which allows us to test the cosmological consequences of quantum gravity (reviewed in[8, 9])."

[Martin]

Here is the abstract of this paper:

Quantum Gravity and the Big Bang

Martin Bojowald

6 pages, invited talk at the conference "Where Cosmology and Fundamental Physics Meet" at IUFM, Marseille, June 23 - 26, 2003

Report-no: CGPG-03/9-2, AEI-2003-077

 

"Quantum gravity has matured over the last decade to a theory which can tell in a precise and explicit way how cosmological singularities of general relativity are removed. A branch of the universe "before" the classical big bang is obtained which is connected to ours by quantum evolution through a region around the singularity where the classical space-time dissolves. We discuss the basic mechanism as well as applications ranging to new phenomenological scenarios of the early universe expansion, such as an inflationary period."

[Martin]

Here is a link to a translation into English of an article from the German popular science magazine "Bild der Wissenschaft"

http://arxiv.org/abs/physics/0407071

 

 

Here are exerpts from the translation:

---quote---

The Inverted Big-Bang

By Rüdiger Vaas

 

Our universe appears to have been created not out of nothing but from a strange space-time dust of quantum geometry

Summary:

Quantum geometry makes it possible to avoid the ominous beginning of our universe with its physically unrealistic – infinite – curvature, extreme temperature, and energy density. It could be the long sought after explanation of the big-bang.

 

It perhaps even opens a window into a time before the big-bang – space itself may have come from an earlier collapsing universe that turned inside out or inverted and began to expand again.

---end quote---

 

---------some quotes from the beginning of main text----

With the help of one equation, Martin Bojowald tries to look into a time that no one has ever seen – into a time before time, into the time before the big-bang. If this equation is correct, the big-bang was not the beginning of everything but merely a transition – the end of a previous universe collapsing into itself and at the same time turning inside out into a new universe expanding out.

 

The young physicist at Max-Planck-Institute for gravitational physics in Potsdam cannot yet say what happened exactly before the big-bang. But his results are already so promising, that he has received high recognition and collaboration from renowned physicists worldwide.

 

... Classical physics in the shape of Albert Einstein’s theory of general relativity first formulated this riddle physically. But it could not solve the problem – rather it pronounced, as it were, its own bankruptcy. Soon after the Russian physicist Alexander Friedmann formulated, in 1922 and 1924, two equations for the evolution of space in a highly simplified form within the framework of relativity theory, it shocked the science with its prediction of the big-bang singularity...

 

General relativity theory predicts a boundary at the big-bang, in that the laws of physics break down as does the applicability of the theory: space and time shrink to nothing while the curvature of space, energy density, pressure and temperature on the other hand grow tremendously...

 

"General relativity predicts a first moment of time,” comments Lee Smolin, physics professor at the Canadian University of Waterloo and the associated Perimeter Institute. "But this conclusion disregards quantum physics. For relativity theory is not a quantum theory” and thus, in the last few years the hopes of cosmologists to crack the mystery of the big-bang singularity have grown stronger....

 

---end quote---

[Martin]

these Lineweaver links are potentially useful in QG discussion as well as general astronomy, so let's have a copy here:

http://www.sciam.com/article.cfm?chanID=sa006&colID=1&articleID=0009F0CA-C523-1213-852383414B7F0147

 

Popular written feature article "Misconceptions about BigBang"

There are "sidebars" on the article each with one or more visual diagrams illustrating a wrong answer and a right answer to some frequently asked question.

 

 

http://www.sciam.com/media/inline/0009F0CA-C523-1213-852383414B7F0147_p39.gif

What kind of explosion was the big bang?

 

http://www.sciam.com/media/inline/0009F0CA-C523-1213-852383414B7F0147_p40.gif

Can galaxies recede faster than light?

 

http://www.sciam.com/media/inline/0009F0CA-C523-1213-852383414B7F0147_p42.gif

Can we see galaxies receding faster than light?

 

http://www.sciam.com/media/inline/0009F0CA-C523-1213-852383414B7F0147_p43.gif

Why is there a cosmic redshift?

 

http://www.sciam.com/media/inline/0009F0CA-C523-1213-852383414B7F0147_p44.gif

How large is the observable universe?

 

http://www.sciam.com/media/inline/0009F0CA-C523-1213-852383414B7F0147_p45.gif

Do objects inside the universe expand, too?

[Martin]

Quantum Gravity research is redefined by every major conference.

Emphasis shifts and some lines of investigation get important new results and push ahead.

The status of non-string QG approaches will be somewhat clarified in July 05 when the schedule of talks for the "Loops 05" conference is published.

 

right now you can tell quite a bit from the list of SPEAKERS that have been invited to give plenary talks:

http://loops05.aei.mpg.de/index_files/Programme.html

 

and the list of TOPICS to be focused on at Loops 05

http://loops05.aei.mpg.de/index_files/Home.html

 

As you can see at the website, the list of topics which the organizers of Loops 05 give is:

 

Background Independent Algebraic QFT

Causal Sets

Dynamical Triangulations

Loop Quantum Gravity

Non-perturbative Path Integrals

String Theory"

[Martin]

the most exciting new approach that will be featured at the Loops 05 conference is CDT (causal dynamical triangulations) and the main speaker about CDT will be Renate Loll

 

You can see her name on the list of the 15 organizers of the conference:

 

"International Organising Committee

 

Abhay Ashtekar (USA)

John Baez (USA)

John Barrett (UK)

Alejandro Corichi (MEX)

Fay Dowker (UK)

Laurent Freidel (FR and CA)

Chris Isham (UK)

Jurek Lewandowski (POL)

Renate Loll (NL)

Hugo Morales Tecotl (MEX)

Alejandro Perez (FR)

Jorge Pullin (USA)

Carlo Rovelli (FR)

Lee Smolin (CA)

Rafael Sorkin (USA)"

 

The CDT approach first appeared in 1998 and was at first "test-driven" by applying it only to lower spacetime dimensions: 2D and 3D.

 

the first results for 4D spacetime appeared in 2004.

 

there are a half dozen or more authors of CDT research but the main ones are "AJL" standing for Ambjorn, Jurkiewicz, and Loll. In fact "AJL" is almost synonymous with CDT. I will get a list of links to the main AJL papers.

 

Here is a short CDT reading list:

 

The first paper is long and detailed. It is better to begin with #2 on this list called "Emergence of a 4D world..."

 

1.

http://arxiv.org/hep-th/0105267

Dynamically Triangulating Lorentzian Quantum Gravity

J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (AEI, Golm)

41 pages, 14 figures

Nucl.Phys. B610 (2001) 347-382

"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 less than 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 present a detailed analysis of the geometric and mathematical properties of the discretized model in d=3,4..."

 

2.

http://arxiv.org/abs/hep-th/0404156

Emergence of a 4D World from Causal Quantum Gravity

J. Ambjorn (1 and 3), J. Jurkiewicz (2), R. Loll (3) ((1) Niels Bohr Institute, Copenhagen, (2) Jagellonian University, Krakow, (3) Spinoza Institute, Utrecht)

11 pages, 3 figures; final version to appear in Phys. Rev. Lett

Phys.Rev.Lett. 93 (2004) 131301

"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over geometries in nonperturbative quantum gravity, with a positive cosmological constant. We present evidence that a macroscopic four-dimensional world emerges from this theory dynamically."

 

3.

http://arxiv.org/abs/hep-th/0411152

Semiclassical Universe from First Principles

J. Ambjorn, J. Jurkiewicz, R. Loll

15 pages, 4 figures

Phys.Lett. B607 (2005) 205-213

"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over space-time geometries in nonperturbative quantum gravity. We show that the macroscopic four-dimensional world which emerges in the Euclidean sector of this theory is a bounce which satisfies a semiclassical equation. After integrating out all degrees of freedom except for a global scale factor, we obtain the ground state wave function of the universe as a function of this scale factor."

 

4.

http://arxiv.org/abs/hep-th/0505113

Spectral Dimension of the Universe

J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)

10 pages, 1 figure

SPIN-05/05, ITP-UU-05/07

 

"We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be "self-renormalizing" at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction."

[Martin]

4.

http://arxiv.org/abs/hep-th/0505113

Spectral Dimension of the Universe

J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)

10 pages, 1 figure

SPIN-05/05, ITP-UU-05/07

 

"We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be "self-renormalizing" at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction."

 

It has been an historical obstacle, which has plagued and discouraged people for several decades, that gravity is not renormalizable in 4D, although in D < = 2 it is renormalizable.

 

One way to approach that is to notice that in natural units the coupling constant G turns out to be expressed in terms L^{D - 2}, where D is the spacetime dimension. so if D > 2 then if you go to a smaller length scale the units you measure G in get smaller and the numerical value increases. this is part of a handwaving argument used by John Baez to explain why one needs D = 2 or less for gravity to be analyzable with finite perturbation series. Here is the whole Baez thing:

http://math.ucr.edu/home/baez/twf_ascii/week139

 

Strange as it seems, this historical obstacle may now have been overcome. If you look at

http://arxiv.org/abs/hep-th/0505113

you will see that even though the macroscopic dimension of spacetime is 4D just as we would expect, the dimension goes down at very small (planckian) scales in their computer simulations actually to BELOW 2.

 

microscopically their computer generated spacetimes have a kind of fractal character at very tiny scale, but look normal at large scale

 

this may be adequate to defeat the "ultraviolet" divergences of various field theories when they are moved into the CDT spacetime of Ambjorn Jurkiewicz Loll.

 

this is one of several reasons why the October Loop 05 conference should be interesting and why the invited talk by Renate Loll should be especially.

[Martin]

In an earlier post (#46 in this thread) I linked to some snapshots of Renate Loll at last year's Marseille Loop Quantum Gravity conference. In case anyone wonders what she looks like.

 

...

 

Here's a photo of Renate Loll with Thomas Thiemann:

http://perimeterinstitute.ca/images/marseille/marseille028.JPG

 

Here she is out for a walk with Julian Barbour and Don Marolf:

http://perimeterinstitute.ca/images/marseille/marseille103.JPG

[Martin]

Another paper by the CDT group at Utrecht has come out

http://arxiv.org/hep-th/0505154

Reconstructing the Universe

J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)

52 pages, 20 postscript figures

SPIN-05/14, ITP-UU-05/18

 

"We provide detailed evidence for the claim that nonperturbative quantum gravity, defined through state sums of causal triangulated geometries, possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the volume of the universe behaves semiclassically. This is a first step in reconstructing the universe from a dynamical principle at the Planck scale, and at the same time provides a nontrivial consistency check of the method of causal dynamical triangulations. A closer look at the quantum geometry reveals a number of highly nonclassical aspects, including a dynamical reduction of spacetime to two dimensions on short scales and a fractal structure of slices of constant time."

 

 

I am beginning to grasp how revolutionary this is.

 

William Occam (the guy who said to keep it simple) would favor this approach to quantizing gravity. By far the simplest.

 

Does not require hidden extra dimensions, or various other string and brane baloney. Is not built in a fairyland of abstract algebra but right here in spacetime with ordinary (real and complex) numbers and ordinary 3 and 4D constructions.

 

It is not EASY (though it gets easier as you get accustomed) but compared with other approaches it sure is simple! you don't have to assume a lot of extra paraphernalia.

 

And they have been running computer models of 4D universe for a couple years, and discovering stuff about it (computer-)experimentally, like a dimension change as you get down to very small scale. Getting some quite remarkable results-----some earlier articles linked in previous posts.

[Martin]

I've been watching the different approaches to Quantum Gravity for a few years, and know a bit about them, and I guess I feel fairly sure now that this new approach is going to have a big impact, may become the paradigm (at least for a few years) of how we think of QG and what a QG theory should look like.

 

It is a "path integral" approach. Like Feynman modeled the path of a particle in humble t-x-y-z space as a weighted average of all possible paths

 

for them, SPACETIME is a path, showing the evolution of the geometry of the universe from beginning to end

 

and they are able to write a weighted average of all possible spacetimes,

and program it into the computer, and calculate and get results

 

sounds fantastic, they do it by a clever "regularization" where they build approximate spacetimes with simple building blocks called SIMPLEXES

 

Simplexes are the key, and a technique for shuffling simplexes to get random geometries they invented that makes possible a "Monte Carlo" computation to evaluate the integral and add up the weighted average.

 

0 simplex = point

1 simplex = line

2 simplex = triangle

3 simplex = tetrahedron (a trianglebased pyramid, has 4 points)

4 simplex = ? (you think of a name, it has 5 points)

 

think by analogy what a 4-simplex is, it is the simplest possible 4D object.

 

the authors build their spacetimes out of these 4-simplex objects.

 

that is how they dont get swamped with too many geometries, they can explore essentially all shapes of spacetime, but because it is built out of these simplex building blocks they dont choke the computer with too many possibilities at once.

 

then they let the size of the simplexes go to zero and the number of simplexes grow, and in that way they approximate the CONTINUUM spacetime.

 

the ultimate goal is the continuum, you want to understand the continuum, not some model built of blocks. but you build the block model, and study it and learn stuff, and then you let the size of the blocks shrink towards zero.

[Martin]

this is a basic paper

http://arxiv.org/hep-th/0505154

Reconstructing the Universe

J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)

 

and so is this introduction that Renate Loll wrote for her grad students

http://arxiv.org/hep-th/0212340

 

it is a fundamental new approach to modeling the spacetime continuum

(with some similarities to the tried and true) and to what could be called

"quantum spacetime dynamics" path integral

 

the Ambjorn Jurkiewicz Loll papers open up a new area of research that is underpopulated and making rapid advances.

 

How to describe their style of "simplex gravity"? It is not clear where to begin or how to introduce the subject. I will try diving in at the level of one of their computer simulations.

 

Imagine half a million 4D simplexes stored in the computer

they are of two types---call it he-type and she-type

he-type is oriented like a pyramid sitting on its base---a 3D tetrahedron---with its point in the air. or the same thing upside down

 

they call the he-type simplexes "(4,1) type and (1,4) type" because they have their 4-pointed tetrahedron base at one level of spacetime and their point at next level up, or next level down.

 

(this approach uses a spacetime that is LAYERED by a timelike or causal ordering, events can only be caused events in earlier layers, this layering or "foliation" is a feature of usual cosmology spacetimes)

 

the simplexes are arranged in layers where in one layer the bases of all the (4,1) simplexes form a spacelike slice at level 't' and the bases of the (1,4) simplexes form a spacelike slice at level 't+1'

and sandwiched in between these two spacelike slices is a filling of he and she type simplexes.

 

the she-type ones are needed because otherwise there would be CHINKS between.

 

a she-type 4D simplex is very like a he-type except it has a different orientation. It doesnt sit on a spacelike tetrahedron base, it sits on a triangle.

And it doesnt have a point straight up in the air, it has a horizontal RIDGE, with two endpoints.

So a she-type is like a he-type but TILTED OVER slightly, and the authors call the she-types (3,2) and the upside down version of the same thing: (2,3).

 

(3,2) means it sits on a spacelike triangle (3 point) at layer 't' and it has a spacelike line segment (2 point) up in layer 't+1'

 

If you tried to make an all male sandwich with these he-type simplexes with their bases down in slice 't' making a triangulation of that slice and their bases up in slice 't+1' triangulating that slice, then you would find that THEIR BASES WOULD TRIANGULATE THE SPATIAL SLICES JUST FINE but the FILLING WOULD HAVE GAPS.

So you need about equal number of she-type to fill in these gaps to make a coherent whole

 

By "triangulate" is not meant to divide something up into triangles but more generally to map it out in simplexes, or to assemble simplexes together to build it

 

We are thinking of a quarter million IDENTICAL he-type simplexes in the computer, and about the same number, a quarter million IDENTICAL she-type simplexes. and the cosmic dynamical triangulation program is able to GLUE them into EXTREMELY CURVY layered spacetimes.

 

I have to go, will be back later

[Martin]

here is the most recent big CDT paper

http://arxiv.org/hep-th/0505154

Reconstructing the Universe

J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)

 

this is an introduction that Renate Loll wrote for her grad students a couple of years ago.

http://arxiv.org/hep-th/0212340

 

it is hard to know how to explain CDT to a layperson audience

amazingly though, the approach IS BASICALLY REAL SIMPLE and has been having remarkable success in past couple of years.

 

other approaches have a lot more paraphernalia and have mostly gotten bogged down (e.g. string/M is a mess and going no place etc.)

 

the CDT "Simplex" quantum spacetime could conceivably even be right!

It is still new and testable prediction need to be worked out. But it is certainly worthwhile learning about, for anyone with an interest in Quantum Gravity.

 

So I will quote some passages from CDT papers and try to paraphrase, in case anyone is interest. And it is good practice for me in any case.

[Martin]

I will quote some from this recent CDT paper:

http://arxiv.org/hep-th/0505154

Reconstructing the Universe

J. Ambjorn, J. Jurkiewicz, R. Loll

 

this is from the summary section at the end:

 

 

<<...This paper describes the currently known geometric properties of the quantum universe generated by the method of causal dynamical triangulations, as well as the general phase structure of the underlying statistical model of four-dimensional random geometries. The main results are as follows. An extended quantum universe exists in one of the three observed phases of the model, which occurs for sufficiently large values of the bare Newton’s constant G and of the asymmetry [tex]\Delta[/tex], which quantifies the finite relative length scale between the time and spatial directions. In the two other observed phases, the universe disintegrates into a rapid succession of spatial slices of vanishing and nonvanishing spatial volume (small G), or collapses in the time direction to a universe that only exists for an infinitesimal moment in time (large G, vanishing or small [tex]\Delta[/tex]). In either of these two cases, no macroscopically extended spacetime geometry is obtained. By measuring the (Euclidean) geometry of the dynamically generated quantum spacetime in the remaining phase, in which the universe appears to be extended in space and time, we collected strong evidence that it behaves as a four-dimensional quantity on large scales....

 

......

 

...In summary, what emerges from our formulation of nonperturbative quantum gravity as a continuum limit of causal dynamical triangulations is a compelling and rather concrete geometric picture of quantum spacetime. Quantum spacetime possesses a number of large-scale properties expected of a four-dimensional classical universe, but at the same time exhibits a nonclassical and nonsmooth behaviour microscopically, due to large quantum fluctuations of the geometry at small scales. These fluctuations “conspire” to create a quantum geometry that is effectively two-dimensional at short distances....>>

[Martin]

... they build approximate spacetimes with simple building blocks called SIMPLEXES

 

Simplexes are the key' date=' and a technique for shuffling simplexes to get random geometries they invented that makes possible a "Monte Carlo" computation to evaluate the integral and add up the weighted average.

 

0 simplex = point

1 simplex = line segment (has 2 endpoints)

2 simplex = triangle (a "pyramid" based on a line segment, 3 vertex points)

3 simplex = tetrahedron (a trianglebased pyramid, has 4 points)

4 simplex = ? (you think of a name, it has 5 points)

 

think by analogy what a 4-simplex is, it is the simplest possible 4D object.

 

the authors build their spacetimes out of these 4-simplex objects.

 

...[/quote']

 

each higher dimension simplex is made by taking the previous simplex as a base, and putting a new point above it and forming something like a cone or pyramid-----with the previous simplex as base.

 

I could use some feedback about what to call things. the usual math name for these CDT buildingblocks is "4-simplex" but we could use a made-up name instead.

for instance we could call the 4D building blocks "PENTAMIDS"

because they are 5-pointed pyramids.

[Martin]

 

I could use some feedback about what to call things. the usual math name for these CDT buildingblocks is "4-simplex" but we could use a made-up name instead.

for instance we could call the 4D building blocks "PENTAMIDS"

because they are 5-pointed pyramids.

 

the CDT authors build their model of quantum spacetime out of "PENTAMIDS"

 

and they build it up in layers or storeys' date=' like a building

where the floors/ceilings are spacelike sheets and the numbering of the sheets or layers plays the role of time.

 

an interesting wrinkle is that there are TWO TYPES of pentamid. there is the LEVEL kind that sits straightandlevel on its tetrahedron base, or else is the same thing upsidedown with its tetrahedron base on the ceiling and its point on the floor.

 

and then there is the TILT kind which is tipped slightly so that it just sits on one of its triangles, and at the top it has a horizontal line segment like the ridgepole of a roof. Or you can also have the same thing upside down, so it rests on its horizontal line segment ridge, and has its triangle up in the air..

 

you need both kinds to make a solid sandwich of pentamids filling in between two spacelike floor/ceiling sheets.

 

All the Level-type pentamids are identical, and all the Tilt-kind are also identical

 

In a typical computer run, the CDT people might have a halfmillion total pentamids, and there would about a quartermillion identical LEVEL kind and about a quartermillion identical TILT kind.

 

[b']the geometry is all in the gluing[/b] of the blocks

 

and the randomization of geometries that makes it a quantum spacetime is accomplished by a technique they discovered for SHUFFLING the assemblage of pentamid blocks. taking a part of the structure apart and re-arranging the blocks

 

when i say the geometry is all in the gluing of the blocks, I mean this assemblage of blocks is not IN some larger surrounding spacetime, rather it IS spacetime, and the issue is whether it is curved or flat (gravity is experienced because of a curvature of spacetime)

 

and the intuitive thing is that if you COUNT the pentamid blocks that all come together around some line and if their angles add up to 360 degrees then that is what you expect in a spacetime with no curvature.

BUT IF THEIR ANGLES ADD UP TO SOMETHING YOU DONT EXPECT because of how they are glued together, then the spacetime around there is CURVED.

 

and since the pentamid blocks are all IDENTICAL they have the same angles (all the level kind have the same, and all the tilt kind have the same) and so you can figure out the overall amounts of curvature by COUNTING how many various kind block you have.

 

so there are no frigging coordinate functions to futz around with and take drivatives etc etc. you have the frigging pentamids in the computer, and they fit together, and the computer counts them and that's it

 

then you do the scramble where they get shuffled about a million times (they call a pass where the computer does a million shuffles one SWEEP) so that the geometry is thoroughly randomized, and then you count the pentamids again, and so on.

 

Meanwhile they are measuring the DIMENSIONALITY and running diffusion processes in these sample spacetimes and comparing distances and volumes and stuff, so they can learn about experimentally about the geometry of this model quantum spacetime.

 

you can read about some of the results in that reading list I posted especially articles #2, #3, #4

[Martin]

in another thread Rajama asked about CDT and I posted this short reading list, it is an update of one posted earlier in this thread too

 

Martin

 

can I ask where I would find an overview of the 'new approach' you mentioned' date=' or maybe what search terms would provide relevent hits?[/quote']

 

there is no "Brian Greene-type" popularizer for it

there are only lecture notes and research papers

but here is a reading list, and the first half of #6 is pretty readable

(it is lecture notes to introduce grad students to it in case they

might want to get into it and do a thesis, so it explains more):

 

1.

http://arxiv.org/hep-th/0105267

Dynamically Triangulating Lorentzian Quantum Gravity

J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (AEI, Golm)

41 pages, 14 figures

Nucl.Phys. B610 (2001) 347-382

"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 less than 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 present a detailed analysis of the geometric and mathematical properties of the discretized model in d=3,4..."

 

2.

http://arxiv.org/abs/hep-th/0404156

Emergence of a 4D World from Causal Quantum Gravity

J. Ambjorn (1 and 3), J. Jurkiewicz (2), R. Loll (3) ((1) Niels Bohr Institute, Copenhagen, (2) Jagellonian University, Krakow, (3) Spinoza Institute, Utrecht)

11 pages, 3 figures; final version to appear in Phys. Rev. Lett

Phys.Rev.Lett. 93 (2004) 131301

"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over geometries in nonperturbative quantum gravity, with a positive cosmological constant. We present evidence that a macroscopic four-dimensional world emerges from this theory dynamically."

 

3.

http://arxiv.org/abs/hep-th/0411152

Semiclassical Universe from First Principles

J. Ambjorn, J. Jurkiewicz, R. Loll

15 pages, 4 figures

Phys.Lett. B607 (2005) 205-213

"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over space-time geometries in nonperturbative quantum gravity. We show that the macroscopic four-dimensional world which emerges in the Euclidean sector of this theory is a bounce which satisfies a semiclassical equation. After integrating out all degrees of freedom except for a global scale factor, we obtain the ground state wave function of the universe as a function of this scale factor."

 

4.

http://arxiv.org/abs/hep-th/0505113

Spectral Dimension of the Universe

J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)

10 pages, 1 figure

SPIN-05/05, ITP-UU-05/07

 

"We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be "self-renormalizing" at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction."

 

5.

http://arxiv.org/hep-th/0505154

Reconstructing the Universe

J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)

52 pages, 20 figures

Report-no: SPIN-05/14, ITP-UU-05/18

 

"We provide detailed evidence for the claim that nonperturbative quantum gravity, defined through state sums of causal triangulated geometries, possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the volume of the universe behaves semiclassically. This is a first step in reconstructing the universe from a dynamical principle at the Planck scale, and at the same time provides a nontrivial consistency check of the method of causal dynamical triangulations. A closer look at the quantum geometry reveals a number of highly nonclassical aspects, including a dynamical reduction of spacetime to two dimensions on short scales and a fractal structure of slices of constant time."

 

 

6.

http://arxiv.org/hep-th/0212340

A discrete history of the Lorentzian path integral

R. Loll (U. Utrecht)

38 pages, 16 figures

SPIN-2002/40

Lect.Notes Phys. 631 (2003) 137-171

"In these lecture notes, I describe the motivation behind a recent formulation of a non-perturbative gravitational path integral for Lorentzian (instead of the usual Euclidean) space-times, and give a pedagogical introduction to its main features. At the regularized, discrete level this approach solves the problems of (i) having a well-defined Wick rotation, (ii) possessing a coordinate-invariant cutoff, and (iii) leading to_convergent_ sums over geometries. Although little is known as yet about the existence and nature of an underlying continuum theory of quantum gravity in four dimensions, there are already a number of beautiful results in d=2 and d=3 where continuum limits have been found. They include an explicit example of the inequivalence of the Euclidean and Lorentzian path integrals, a non-perturbative mechanism for the cancellation of the conformal factor, and the discovery that causality can act as an effective regulator of quantum geometry."

 

Loll wrote this as an introduction to CDT for Utrecht graduate students who might want to get into her line of research. It is a good beginning. It is already 2 years out of date so it does not have the latest headline results but that is OK

[Martin]

To update the above list, when I posted it, the most recent Renate Loll paper was

 

http://arxiv.org/hep-th/0505154

Reconstructing the Universe

J. Ambjorn, J. Jurkiewicz, R. Loll

 

Another paper Loll paper appeared since then:

http://arxiv.org/gr-qc/0506035

Counting a black hole in Lorentzian product triangulations

B. Dittrich (AEI, Golm), R. Loll (U. Utrecht)

42 pages, 11 figures

 

"We take a step toward a nonperturbative gravitational path integral for black-hole geometries by deriving an expression for the expansion rate of null geodesic congruences in the approach of causal dynamical triangulations. We propose to use the integrated expansion rate in building a quantum horizon finder in the sum over spacetime geometries. It takes the form of a counting formula for various types of discrete building blocks which differ in how they focus and defocus light rays. In the course of the derivation, we introduce the concept of a Lorentzian dynamical triangulation of product type, whose applicability goes beyond that of describing black-hole configurations."

[Martin]

I decided against any kind of made-up name for the CDT building blocks. Loll calls them by the standard math term "4-simplex".

I've been looking for some popularized accounts of CDT, found some by John Baez, Dave Bacon, Adrian Cho (two professional physicists and a science journalist).

 

CDT is developing fast and probably the most important quantum gravity development at the moment. we really need popularized account of it. I will get some quotes and links

 

physicist Dave Bacon's blog

http://dabacon.org/pontiff/?p=706#comments

<<...One question which plagues theoretical physicists’ poor little minds is the question of why we see a macroscopic world of 3+1 dimensions. Mostly this is because physicists believe that at small enough length or time scales (large enough energies) the geometry of spacetime itself can exist in nontrivial states of connectivity.... “Spacetime foam” is what we call this strange state of affairs. How do we get from this spacetime foam up to where our experiments live and we seem to see a four dimensional universe?

 

Concerning this problem, I just today read the paper “Emergence of a 4D World from Causal Quantum Gravity,” by J. Ambjorn, J. Jurkiewicz, and R. Loll which was published in Physical Review Letters, (Volume 93, page 131301, 2004.) This paper attempts the following. Construct spacetime by glueing together a bunch of little four dimensional simplical spacetimes. Like I said earlier, if we glue a bunch of these four dimensional simplical spacetimes together, we get something which is not necessarily four dimensional. Now when we do this glueing we should insist on maintain causality (i.e. no closed time like curves and such.) So we can construct these crazy spacetimes, but what do they mean. Well now we associate with each of these spacetimes an amplitude. So there is some notion of an action S for the given simpical spacetime we have created and we assign to this an amplitude, Exp[iS]. Now what one would love to do is to sample over all of these crazy spacetimes and hence calculate the propogators for different such spacetimes. But this is hard. This is hard because of the fact that we have to sample over this crazy oscillating Exp[iS]. But sometimes it is not so hard. Sometimes it is possible to perform a “Wick” rotation and change Exp[iS] into Exp[-S]. This means the problem of calculating the total amplitude looks like adding up a bunch of different spacetimes with weights Exp[-S]: this looks just like classical statistical mechanics! What the authors of the above paper do is they insist that it is possible to perform such a rotation. They then perform Monte Carlo simulations of the resulting statistical mechanical system. And what do they find? They argue that what they find is that the resulting spacetime is indeed dominated by a spacetime of dimension “3+1!”

 

So starting out from something which had only a totally local sense of dimension (the original building blocks are “3+1?) you glue them together in pseudo-arbitrary (preserve causality, able to Wick rotate) ways (this is what is called “background independence”) and yet, you find, at the end of the day, that you have effectively a global “3+1? spacetime! Amazing, no?>>

 

If that is confusing, maybe Baez will be clearer. This is an exerpt from John Baez TWF #206:

 

<<...In other words, can we do quantum physics without choosing some fixed spacetime geometry from the start, a "background" on which small perturbations move like tiny quantum ripples on a calm pre-established lake? A background geometry is convenient: it lets us keep track of times and distances. It's like having a fixed stage on which the actors - gravitons, strings, branes, or whatever - cavort and dance. But, the main lesson of general relativity is that spacetime is not a fixed stage: it's a lively, dynamical entity! There's no good way to separate the ripples from the lake. This distinction is no more than a convenient approximation - and a dangerous one at that.

 

So, we should learn to make do without a background when studying quantum gravity. But it's tough! There are knotty conceptual issues like the "problem of time": how do we describe time evolution without using a fixed background to measure the passage of time? There are also practical problems: in most attempts to describe spacetime from the ground up in a quantum way, all hell breaks loose!

 

We can easily get spacetimes that crumple up into a tiny blob... or spacetimes that form endlessly branching fractal "polymers" of Hausdorff dimension 2... but it seems hard to get reasonably smooth spacetimes of dimension 4. It's even hard to get spacetimes of dimension 10 or 11... or anything remotely interesting!

 

It almost seems as if we need a solid background as a bed frame to keep the mattress of spacetime from rolling up, getting all lumpy, or otherwise misbehaving. Unfortunately, even with a background there are serious problems: we can use perturbation theory to write the answers to physics questions as power series, but these series diverge and nobody knows how to resum them.

 

String theorists are pragmatic in a certain sense: they don't mind using a background, and they don't mind doing what physicists always do: approximating a divergent series by the sum of the first couple of terms. But this attitude doesn't solve everything, because right now in string theory there is an enormous "landscape" of different backgrounds, with no firm principle for choosing one. Some estimates guess there are over 10100. Leonard Susskind guesses there are 10500, and argues that we'll need the anthropic principle to choose the one describing our world:

 

2) Leonard Susskind, The Landscape, article and interview on John Brockman's "EDGE" website, http://www.edge.org/3rd_culture/susskind03/susskind_index.html

 

This position is highly controversial, but my point here shouldn't be: developing a background-free theory of quantum gravity is tough, but working with a background has its own difficulties. And let's face it: we haven't spent nearly as much time thinking about background-free or nonperturbative physics as we've spent on background-dependent or perturbative physics. So, it's quite possible that our failures with the former are just a matter of inexperience.

 

Given all this, I'm delighted to see some real progress on getting 4d spacetime to emerge from nonperturbative quantum gravity:

 

3) Jan Ambjorn, Jerzy Jurkiewicz and Renate Loll, Emergence of a 4d world from causal quantum gravity, available as hep-th/0404156.

 

This trio of researchers have revitalized an approach called "dynamical triangulations" where we calculate path integrals in quantum gravity by summing over different ways of building spacetime out of little 4-simplices. They showed that if we restrict this sum to spacetimes with a well-behaved concept of causality, we get good results. This is a bit startling, because after decades of work, most researchers had despaired of getting general relativity to emerge at large distances starting from the dynamical triangulations approach. But, these people hadn't noticed a certain flaw in the approach... a flaw which Loll and collaborators noticed and fixed!

 

If you don't know what a path integral is, don't worry: it's pretty simple. Basically, in quantum physics we can calculate the expected value of any physical quantity by doing an average over all possible histories of the system in question, with each history weighted by a complex number called its "amplitude". For a particle, a history is just a path in space; to average over all histories is to integrate over all paths - hence the term "path integral". But in quantum gravity, a history is nothing other than a SPACETIME.

 

Mathematically, a "spacetime" is something like a 4-dimensional manifold equipped with a Lorentzian metric. But it's hard to integrate over all of these - there are just too darn many. So, sometimes people instead treat spacetime as made of little discrete building blocks, turning the path integral into a sum. You can either take this seriously or treat it as a kind of approximation. Luckily, the calculations work the same either way!

 

If you're looking to build spacetime out of some sort of discrete building block, a handy candidate is the "4-simplex": the 4-dimensional analogue of a tetrahedron. This shape is rigid once you fix the lengths of its 10 edges, which correspond to the 10 components of the metric tensor in general relativity.

 

There are lots of approaches to the path integrals in quantum gravity that start by chopping spacetime into 4-simplices. The weird special thing about dynamical triangulations is that here we usually assume every 4-simplex in spacetime has the same shape. The different spacetimes arise solely from different ways of sticking the 4-simplices together.

 

Why such a drastic simplifying assumption? To make calculations quick and easy! The goal is get models where you can simulate quantum geometry on your laptop - or at least a supercomputer. The hope is that simplifying assumptions about physics at the Planck scale will wash out and not make much difference on large length scales.

 

Computations using the so-called "renormalization group flow" suggest that this hope is true if the path integral is dominated by spacetimes that look, when viewed from afar, almost like 4d manifolds with smooth metrics. Given this, it seems we're bound to get general relativity at large distance scales - perhaps with a nonzero cosmological constant, and perhaps including various forms of matter.

 

Unfortunately, in all previous dynamical triangulation models, the path integral was not dominated by spacetimes that look like nice 4d manifolds from afar! Depending on the details, one either got a "crumpled phase" dominated by spacetimes where almost all the 4-simplices touch each other, or a "branched polymer phase" dominated by spacetimes where the 4-simplices form treelike structures. There's a transition between these two phases, but unfortunately it seems to be a 1st-order phase transition - not the sort we can get anything useful out of. For a nice review of these calculations, see:

 

4) Renate Loll, Discrete approaches to quantum gravity in four dimensions, available as gr-qc/9805049 or as a website at Living Reviews in Relativity, http://www.livingreviews.org/Articles/Volume1/1998-13loll/

 

Luckily, all these calculations shared a common flaw!

 

Computer calculations of path integrals become a lot easier if instead of assigning a complex "amplitude" to each history, we assign it a positive real number: a "relative probability". The basic reason is that unlike positive real numbers, complex numbers can cancel out when you sum them!

 

When we have relative probabilities, it's the highly probable histories that contribute most to the expected value of any physical quantity. We can use something called the "Metropolis algorithm" to spot these highly probable histories and spend most of our time focusing on them.

 

This doesn't work when we have complex amplitudes, since even a history with a big amplitude can be canceled out by a nearby history with the opposite big amplitude! Indeed, this happens all the time. So, instead of histories with big amplitudes, it's the bunches of histories that happen not to completely cancel out that really matter. Nobody knows an efficient general-purpose algorithm to deal with this!

 

For this reason, physicists often use a trick called "Wick rotation" that converts amplitudes to relative probabilities. To do this trick, we just replace time by imaginary time! In other words, wherever we see the variable "t" for time in any formula, we replace it by "it". Magically, this often does the job: our amplitudes turn into relative probabilities! We then go ahead and calculate stuff. Then we take this stuff and go back and replace "it" everywhere by "t" to get our final answers.

 

While the deep inner meaning of this trick is mysterious, it can be justified in a wide variety of contexts using the "Osterwalder-Schrader theorem". Here's a pretty general version of this theorem, suitable for quantum gravity:

 

5) Abhay Ashtekar, Donald Marolf, Jose Mourao and Thomas Thiemann, Constructing Hamiltonian quantum theories from path integrals in a diffeomorphism invariant context, Class. Quant. Grav. 17 (2000) 4919-4940. Also available as quant-ph/9904094.

 

People use Wick rotation in all work on dynamical triangulations. Unfortunately, this is not a context where you can justify this trick by appealing to the Osterwalder-Schrader theorem. The problem is that there's no good notion of a time coordinate "t" on your typical spacetime built by sticking together a bunch of 4-simplices!

 

The new work by Ambjorn, Jurkiewiecz and Loll deals with this by restricting to spacetimes that do have a time coordinate. More precisely, they fix a 3-dimensional manifold and consider all possible triangulations of this manifold by regular tetrahedra. These are the allowed "slices" of spacetime - they represent different possible geometries of space at a given time. They then consider spacetimes having slices of this form joined together by 4-simplices in a few simple ways.

 

The slicing gives a preferred time parameter "t". On the one hand this goes against our desire in general relativity to avoid a preferred time coordinate - but on the other hand, it allows Wick rotation. So, they can use the Metropolis algorithm to compute things to their hearts' content and then replace "it" by "t" at the end.

 

When they do this, they get convincing good evidence that the spacetimes which dominate the path integral look approximately like nice smooth 4-dimensional manifolds at large distances! Take a look at their graphs and pictures - a picture is worth a thousand words...>>

 

here is the link to Baez column "This Week's Finds in Mathematical Physics #206" from which I'm quoting

http://math.ucr.edu/home/baez/week206.html