Poincare Invariance from General QFT

**CarlB** · Sep11-06, 04:07 PM

Derivation of Poincare Invariance from general quantum field theory
C.D. Froggatt, H.B. Nielsen
Annalen der Physik, Volume 14, Issue 1-3 , Pages 115 - 147 (2005)
Special Issue commemorating Albert Einstein
Starting from a very general quantum field theory we seek to derive Poincare invariance in the limit of low energy excitations. We do not, of course, assume these symmetries at the outset, but rather only a very general second quantised model. Many of the degrees of freedom on which the fields depend turn out to correspond to a higher dimension. We are not yet perfectly successful. In particular, for the derivation of translational invariance, we need to assume that some background parameters, which a priori vary in space, can be interpreted as gravitational fields in a future extension of our model. Assuming translational invariance arises in this way, we essentially obtain quantum electrodynamics in just 3 + 1 dimensions from our model. The only remaining flaw in the model is that the photon and the various Weyl fermions turn out to have their own separate metric tensors.
http://www.arxiv.org/abs/hep-th/0501149
http://www3.interscience.wiley.com/c...84430/ABSTRACT

I found this elegant paper after seeing the following passage in Lee Smolin's new book, "The Trouble With Physics":

Originally Posted by Lee Smolin in The Trouble With Physics"

(p 315) One of the great seers is Holger Bech Nielsen of the Niels Bohr Institute. He was an inventor of string theory, and he has many other key discoveries to his credit. But for many years he has been isolated from the mainstream for advocating what he calls random dynamics. He believes that the most useful assumption we can make about the fundamental laws is that they are random. Everything we think of as intrinsically true, such as relativity and the principles of quantum mechanics, he thinks are just accidental facts that are emergent from a fundamental theory so beyond our imagining that we might as well assume that its laws are random. His models are the laws of thermodynamics, which used to be based on principles but now are understood as the most likely way that large numbers of atoms in random motion will behave. This may not be right, but Nielsen has come remarkably far in his antiunification program.

How can I describe the joy I experienced on reading this paper? An oasis after a long trek in the desert? Reunion with first and true love after long absence? Child nearly died but now safe? Peace and victory after long hard war? None of these examples are strong enough.

When I say that my calculation for the lepton masses is "insane", it is because it is based on an underlying QFT that is not Lorentz invariant and needs preons that travel at $LaTeX Code: c\\sqrt{3}$ . I knew that this was the big problem. Even the professor who taught me QFT a couple decades ago (and gave me the high grade in the class) wouldn't consider my idea because of this fact. I figured that getting it read (or even allowed into arXiv) would be impossible because of this deviation from the norm, but the above article was not only put onto arXiv, but published in a peer reviewed journal. Oh happy day. I will return to writing down the theory in all its non Lorentz invariant glory, to hell with hiding the heresy. Look for it to be complete in a month or two. It will begin with a geometric version of the density matrix formalism and end with the masses of the leptons calculated to 6 decimal places.

Carl

**Kea** · Sep11-06, 05:37 PM

Originally Posted by CarlB

Oh happy day. I will return to writing down the theory...

I'm very glad to hear that, Carl.

**CarlB** · Sep13-06, 07:51 PM

The laws of physics are all very closely connected in multiple ways. This makes it difficult to rewrite any part of the foundation without having to rewrite the whole thing. An example of the interaction is how electron spin is connected to relativity. One asks, “if relativity isn’t correct, then how did it predict the electron spin?”

Electron spin comes from the geometry of spinors and spinors arrive as part of the ideal structure of Clifford algebras. Where the Clifford algebra connects to relativity is in its signature. So, in the spirit of Froggatt and Nielsen, let me write down here a simplistic sort of way of getting the gamma matrices as an accidental symmetry.

Begin with a Euclidean Clifford algebra R(4,0), that is, the real Clifford algebra with signature (++++). In this time is a parameter, not a member of the geometry, and I've added one hidden spatial dimension (which is needed as will be seen in a moment). To get specific, I'll label the spatial coordinates x, y, z and s. Since time is treated differently from the spatial coordinates, this is a space-time with a preferred reference frame.

Classically, to define a wave equation we need a position and a momentum. Let both of them be in the above Clifford algebra. Then a natural way of writing the simplest possible equations of motion is to suppose two wave functions, one for position and the other for momentum:

$LaTeX Code: \\begin{array}{rcl} \\partial_t \\Psi_P &=& \\nabla \\Psi_Q, \\\\ \\partial_t \\Psi_Q &=& \\nabla \\Psi_P \\end{array}$

You can mess around with the signs in the above, it doesn't change the result any. I like the symmetric version. In the above,

$LaTeX Code: \\nabla = \\hat{x}\\partial_x + \\hat{y}\\partial_y + \\hat_z\\partial_z + \\hat_s\\partial_s$

Now the above was written with a purely positive signature (totally classical mechanics as opposed to relativistic mechanics) Clifford algebra. To unite our two waves, $LaTeX Code: \\Psi_Q, \\Psi_P$ into a single wave function, it would be convenient to add a degree of freedom to the Clifford algebra. We can do this by adding another canonical basis vector, but that would have messy commutation relations with the rest of the algebra. What we really want to do is to add a canonical basis vector that commutes with all the rest of the algebra. Then we can make this degree of freedom distinguish between canonical momentum and canonical position.

Doing this takes us outside of the definition of a Clifford algebra, but all it really amounts to mathematically is modifying the "presentation" of the algebra. Accordingly, let us add another canonical basis vector that commutes with R(4,0) and let us call it by the not quite arbitrarily chosen value $LaTeX Code: (\\hat{x}\\hat{y}\\hat{z}\\hat{s}\\hat{t})$ . That is, we are not adding a $LaTeX Code: \\hat{t}$ basis vector, but instead are adding the thing with the long name that is to always remain as a unit inside parentheses.

From the point of view of characterizing the elementary particles as primitive idempotents (which mathematically define the spinors), it should be clear that the addition of the non Clifford algebraic canonical vector basis element is to be treated the same as any Clifford algebra canonical basis vector element. For example, one splits the combined wave function into canonical momentum and canonical position by multiplication by idempotents made from the new element:

$LaTeX Code: \\begin{array}{rcl} \\Psi_Q &=& \\frac{1}{2}(1+(\\hat{x}\\hat{y}\\hat{z}\\hat{s}\\hat{t} ))\\;\\;\\Psi \\\\ \\Psi_P &=& \\frac{1}{2}(1-(\\hat{x}\\hat{y}\\hat{z}\\hat{s}\\hat{t}))\\;\\;\\Psi. \\end{array}$

The presentation of the algebra is now as follows. First the usual Clifford algebra part of the presentation:
$LaTeX Code: \\begin{array}{rcl} \\hat{x}^2 &=& \\hat{y}^2 = \\hat{z}^2 = \\hat{s}^2 =1\\\\ \\hat{x}\\hat{y} &=& -\\hat{y}\\hat{x}\\\\ \\hat{x}\\hat{z} &=& -\\hat{z}\\hat{x}\\\\ \\hat{x}\\hat{s} &=& -\\hat{s}\\hat{x}\\\\ \\hat{y}\\hat{z} &=& -\\hat{z}\\hat{y}\\\\ \\hat{y}\\hat{s} &=& -\\hat{s}\\hat{y}\\\\ \\hat{z}\\hat{s} &=& -\\hat{s}\\hat{z} \\end{array}$

And we add the new, non-Clifford algebraic presentation to tell what to do with the new basis vector:

$LaTeX Code: \\begin{array}{rcl} (\\hat{x}\\hat{y}\\hat{z}\\hat{s}\\hat{t})^2 &=& 1\\\\ (\\hat{x}\\hat{y}\\hat{z}\\hat{s}\\hat{t})\\hat{x} &=& \\hat{x}(\\hat{x}\\hat{y}\\hat{z}\\hat{s}\\hat{t})\\\\ (\\hat{x}\\hat{y}\\hat{z}\\hat{s}\\hat{t})\\hat{y} &=& \\hat{y}(\\hat{x}\\hat{y}\\hat{z}\\hat{s}\\hat{t})\\\\ (\\hat{x}\\hat{y}\\hat{z}\\hat{s}\\hat{t})\\hat{z} &=& \\hat{z}(\\hat{x}\\hat{y}\\hat{z}\\hat{s}\\hat{t})\\\\ (\\hat{x}\\hat{y}\\hat{z}\\hat{s}\\hat{t})\\hat{s} &=& \\hat{s}(\\hat{x}\\hat{y}\\hat{z}\\hat{s}\\hat{t}) \\end{array}$

But as the audience has undoubtedly realized, these new presentation rules can be obtained from the usual Clifford algebra presentation rules when a time dimensions, $LaTeX Code: \\hat{t},$ is added to the other four spatial dimensions, namely:

$LaTeX Code: \\begin{array}{rcl} \\hat{t}^2 &=& -1\\\\ \\hat{t}\\hat{x} &=& -\\hat{x}\\hat{t}\\\\ \\hat{t}\\hat{y} &=& -\\hat{y}\\hat{t}\\\\ \\hat{t}\\hat{z} &=& -\\hat{z}\\hat{t}\\\\ \\hat{t}\\hat{s} &=& -\\hat{s}\\hat{t} \\end{array}$

That is, to convert the above rules into the rules for $LaTeX Code: (\\hat{x}\\hat{y}\\hat{z}\\hat{s}\\hat{t}),$ simply multiply on the left by $LaTeX Code: \\hat{x}\\hat{y}\\hat{z}\\hat{s}$ , then note that this constant anticommutes with $LaTeX Code: \\hat{x}, \\hat{y}, \\hat{z}, \\hat{s}$ and commutes with $LaTeX Code: \\hat{t}.$ This allows you to rearrange terms, insert parentheses, and obtain the presentation rules for the preferred reference frame version of Clifford algebra.

Now how does this relate to the usual gamma matrices? In the above we have assumed real Clifford algebra. The gamma matrices are complex and so have twice the degrees of freedom of a real Clifford algebra. While the above has 5 canonical basis vectors, the gamma matrices (when written in complex form) have only 4. The equivalency, with the gamma matrices written in the (-+++) signature, is as follows:

$LaTeX Code: \\begin{array}{rcl} \\gamma_0 &\\equiv& \\hat{t}\\\\ \\gamma_1 &\\equiv& \\hat{x}\\\\ \\gamma_2 &\\equiv& \\hat{y}\\\\ \\gamma_3 &\\equiv& \\hat{z}\\\\ i\\gamma_0\\gamma_1\\gamma_2\\gamma_3 &\\equiv& \\hat{s} \\end{array}$

Acknowledgements: I should mention that the above was largely pointed out to me in conversations with Jose Almeida and Larry Horwitz.

Kea, your encouragement is very important to me. If you live out in the wilderness for too long, it is easy to forget that enthusiasm is contagious.

Carl

**Dcase** · Sep13-06, 08:19 PM

In some respect this thread [Poincare Invariance] resembles a 1995 ArXiv paper

‘p-Adic TGD: Mathematical Ideas’
Authors: M.Pitkänen (Helsinki,Finland)

http://arxiv.org/abs/hep-th/9506097

2 - Matti Pitkänen, Department of Physics, Theoretical Physics Division,
University of Helsinki, Finland has updated this idea to ‘Topological Geometrodynamics ]TGD] on the web in 7 on-line books at

http://www.helsinki.fi/~matpitka/index.html

**Careful** · Sep14-06, 03:09 AM

Originally Posted by Dcase

In some respect this thread [Poincare Invariance] resembles a 1995 ArXiv paper

‘p-Adic TGD: Mathematical Ideas’
Authors: M.Pitkänen (Helsinki,Finland)

http://arxiv.org/abs/hep-th/9506097

2 - Matti Pitkänen, Department of Physics, Theoretical Physics Division,
University of Helsinki, Finland has updated this idea to ‘Topological Geometrodynamics ]TGD] on the web in 7 on-line books at

http://www.helsinki.fi/~matpitka/index.html

Looked at the webpage for some 10 minutes. Why do such authors never write some very concise papers (of 30 pages or so) for the layman (even professional physicists are layman regarding his theory) where they start explaining from zero the very basic premises of their theory as well as the physical motivations behind them. Nobody is going to read this gigantic book if it is not made clear from the start what the physics is about.

Careful

**Kea** · Sep16-06, 05:58 AM

Originally Posted by CarlB

If you live out in the wilderness for too long, it is easy to forget that enthusiasm is contagious.

Ah, don't I know it.

Pitkanen's theory should tie in somehow. He is thinking along quantum topos lines. But, yes, I can't say I've made any attempt to wade through it all, because there is simply too much.

**CarlB** · Sep16-06, 08:59 AM

The algebraic topology is too far from the physics for me. It seems, well, bloodless and I don't have any intuition for it. In fact, sometimes mathematics seems to be a search for the least intuitive ways of understanding things.

Meanwhile, some progress on my derivation of the neutrino masses. Here's chapter 1:

http://www.brannenworks.com/dmaa.pdf

Any comments welcome. The next chapter title is to be "Primitive Idempotents of the Dirac Algebra" and will transition the reader to thinking in terms of Clifford algebra instead of matrix representations. The chapter I'm really looking forward to writing is the one that gives a geometric interpretation to Feynman diagrams.

By the way, Hestenes told me that he didn't work in QFT because he thought it was wrong.

Carl

**Kea** · Sep16-06, 05:16 PM

Originally Posted by CarlB

In fact, sometimes mathematics seems to be a search for the least intuitive ways of understanding things.

Yes, I constantly find that myself, although I see a lot of physics in basic Algebraic Topology. Hopefully sometime soon I'll be writing some articles myself. I'll take a look at your chapter 1. Thanks for updating us as you go along!

Later: it looks good so far. At this rate you'll have a whole book of teachable material. There are a couple of simple typos, and I thought that the introduction could be expanded a little with more explanation of where you are heading.

**Dcase** · Sep18-06, 05:48 PM

This comment is very speculative, a crude attempt to unify Mathematical Game Theory with Differential Geometry by analogy to comments on bra- -ket or +z -z direction or negative positive spin in section 1.4 Bras and Kets pages 8-11 chapter 1 of the CarlB reference:
http://www.brannenworks.com/dmaa.pdf

A - Reasons for this comment:
The same mind conceived of two diverse mathematical representations suggesting the possibility of a relationship between these representations of mathematical objects:
1 - John Nash is probably best known for
a - Nash Equilibrium [Nobel Economics 1994] in Mathematical Game Theory
http://nobelprize.org/nobel_prizes/e...on-speech.html
b - Nash “embedding theorems (or imbedding theorems) ... state that every Riemannian manifold can be isometrically embedded in a Euclidean space” in Differential Geometry
http://en.wikipedia.org/wiki/Nash_embedding_theorem

2 - Some game theorists think that saddle points [which may be found in many curves, including helicoids / helices] are equivalent to Nash Equilibria.
a - Game Theory in the History of Mathematics of 20-th Century from the Department of Mathematics, University of Rhode Island discusses the relationship to John von Neumann's Minimax Theorem and to the saddle point as ‘... an equilibrium decision point ..." for "... in which the maximum of the row minimax equals the minimum of the column maxima ..."
http://www.math.uri.edu/~kulenm/mth3...ametheory.html
b - Gerhard Gierz [Mathematics 121] University of California Department of Mathematics Riverside, CA 92521 in “Saddle Points and Nash Equilibria”
http://ggierz.ucr.edu/Math121/Winter...tsNashEqui.pdf

B - Why this may be a type of game theory or bifurcation category theory:
1 - +z or -z directions may represent an attractor-dissipator or braid-unbraid relationship [negative or positive spin may do something similar, bra- or -ket seems less likely to do so0].
2 - Energy games with mathematical or geometrical representations may be consistent with D Hestenes ideas in ‘The Zitterbewegung Interpretation of Quantum Mechanics’ abstract:
“The zitterbewegung is a local circulatory motion of the electron presumed to be the basis of the electron spin and magnetic moment. A reformulation of the Dirac theory shows that the zitterbewegung need not be attributed to interference between positive and negative energy states as originally proposed by Schroedinger. Rather, it provides a physical interpretation for the complex phase factor in the Dirac wave function generally. Moreover, it extends to a coherent physical interpretation of the entire Dirac theory, and it implies a zitterbewegung interpretation for the Schroedinger theory as well“.
3 - Perhaps if Einstein's most famous equation were modified as a chaotic game,
v^2 * E-attracting = E-dissipating
lim v: 0 -> c
where E-attracting is m and E-dissipating is E,
it might represent the transformation of a star into a supernova as welll as have application to nuclear physics with gauge as the only difference?

C - Rather than continue this remotely related subject matter in your thread, I will post more on this remote but seemingly not impossible idea on the STROOP thread.[/url]

**CarlB** · Sep18-06, 07:49 PM

Originally Posted by Kea

Later: it looks good so far. At this rate you'll have a whole book of teachable material. There are a couple of simple typos, and I thought that the introduction could be expanded a little with more explanation of where you are heading.

Thanks for looking. Yes, I want this material to be so clear and obvious that any graduate student or good senior can understand it without a lot of effort. By the way, I think the difference between a 30 page paper and a 150 page paper is "teachable", so we are probably in agreement at how much stuff needs to be in a paper that derives the standard model.

I'd have put more of an explanation of where it was heading but I didn't want to scare anybody off. It's a rewrite of all the foundations of physics but putting that in the introduction just gives people an excuse not to read it. Basically, I still don't have the GUTs to write down the truth.

I'm redoing chapter 1 by adding in everything I can think of that can be derived in the context of the Pauli algebra. It's now about 23 pages, with the index. The sections I've added have to do with the probability interpretation, the different sorts of complex numbers that appear in this, transition probabilities for state changes, geometric definition of Hermitian operators (i.e. under the Clifford algebraic "blade" splitting, the Pauli algebra Hermitian operators are those which are made of scalars and vectors, while the anti Hermitian operators are those which are made of psuedoscalars and psuedovectors), and how you write Feynman diagrams in terms of density operators which is mostly about the geometric definition of "amplitudes". I still have to write up the beautiful theory of the products of primitive idempotents. That is, in Clifford algebras the product of the vectors is what is normally defined, i.e. $LaTeX Code: \\sigma_x\\sigma_y$ , but what you need in a particle theory is the product of primitive idempotents, which is what gives you transition amplitudes that go into Feynman diagrams, i.e.
$LaTeX Code: 0.5(1+\\sigma_x) * 0.5(1+\\sigma_y)*0.5(1+\\sigma_z) * 0.5(1+\\sigma_x)$

Transition probabilities are defined as stuff like
$LaTeX Code: 0.5(1+\\sigma_x)* 0.5(1+\\sigma_y)$ which is where the $LaTeX Code: P = 0.5(1+\\cos(\\theta))$ comes from. When you want to use density matrices in Feynman path integral expansions, you have to take into account longer products of primitive idempotens (pure density matrices), and it turns out that longer products will generate complex phases.

Then Chapter 2 will cover the mathematical facts that require the Dirac algebra to describe. That would be the general structure of primitive idempotents of Clifford algebras.

One of the things that is difficult for me in writing these things is getting a decent balance between putting too much explanation down and not putting down enough. When I've actually seen people try to read stuff I've written they seem to find too much stuff skipped over so I am trying to make things as clear and complete as possible.

What I'd like to see from you is some notes on an the algebraic topology someday. Oh, good news from the Joint Pacific Particle Physics Conference, they've accepted my abstract so I guess I will be giving a talk. The subject will be in neutrino physics. In retrospect, I wish I had offered a contribution in cosmic rays as those are the talks I will be attending, and I am certain that they have already observed these preons in certain experiments (i.e. Centauros, and the AGASA energy calibration problem). I also am an engineer by trade and I love talking with experimentalists who are very practical people.

Oh, I should mention that the reason it's called a "Operator Guide to the Standard Model" is so that it is consistent with the picture that will go on the front cover. This will involve yours truly sitting on a piece of standard equipment that is used to dig foundations, technically, a "ground breaking" machine:
http://brannenworks.com/IMG_0340.JPG

Carl

**Kea** · Sep18-06, 08:21 PM

Originally Posted by CarlB

What I'd like to see from you is some notes on an the algebraic topology someday.

Yes, I would like to write some simple notes on calculating topological invariants for physics. Alas, at present I am still fighting battles on many fronts.

Oh, good news from the Joint Pacific Particle Physics Conference, they've accepted my abstract so I guess I will be giving a talk.

Great. Exposure is what is needed here.

In retrospect, I wish I had offered a contribution in cosmic rays as those are the talks I will be attending, and I am certain that they have already observed these preons in certain experiments (i.e. Centauros, and the AGASA energy calibration problem).

Well, good luck and all that, but I can't say that I'm ready to jump to such conclusions yet, especially since our M-theory approach to the particle spectrum will undoubtedly end up looking quite different to what one might guess at first. To us, it isn't the preons that are fundamental... it's the ribbons, because they form the logical symbols that express the experimental question.

**CarlB** · Sep18-06, 08:43 PM

Originally Posted by Dcase

B - Why this may be a type of game theory or bifurcation category theory:
1 - +z or -z directions may represent an attractor-dissipator or braid-unbraid relationship [negative or positive spin may do something similar, bra- or -ket seems less likely to do so0].
2 - Energy games with mathematical or geometrical representations may be consistent with D Hestenes ideas in ‘The Zitterbewegung Interpretation of Quantum Mechanics’ abstract:

This is interesting, could you fix the link to the STROOP discussion?

Bifurcation theory. I'll let it spin around in the back of my head for a couple weeks, maybe something will come to me. It does seem like there is an attractor going on in there.

And I like the idea of the elementary particles as attractors. But where I think this comes from is in the conversion of wave to particle, that is, the wave function collapse. My sense of this is that the idempotent type representation of the particles is a "bistable" description. The two stable states are 0 and 1.

Ah, now I recall where an attractor comes up sort of naturally in quantum mechanics. When we, as callow youth, were inculcated in the arcana of quantum mechanics, one of the facts that were drilled into our heads is that $LaTeX Code: \\Delta x \\Delta p > \\hbar.$

Being somewhat more stupid than most, I decided to build a wave function that would fail to satisfy this inequality for its initial condition. As everyone else knows, this is impossible, the problem is in the wave itself. But then I found out about Bohmian mechanics. If you rewrite the wave as:
$LaTeX Code: \\psi = R(x,t)e^{iS(x,t)}$
where R and S are real valued functions of position and time, then you CAN set up initial conditions that violate the HUP. And it's well known among the Bohmian mechanics that when you do this, the HUP becomes a stable point. Furthermore, if you put a very small amount of dissipation, then you can arrange for a stiuation that violates the HUP to evolve by the modified Schroedinger equation, into one that meets it to high precision. And that's an attractor.

The difference between what Hestenes is doing and I am doing, philosophically, is that I hold that the density operators are fundamental and so I geometrize them. Hestenes geometrized the spinors which I think was a great idea. In fact I thought it was such a great idea that that is why I learned Clifford algebra. But I think that the density operators make a much more natural target for geometrization as you can avoid the very arbitrary assumptions Hestenes has to make about even subalgebras and all that. In addition, the Hestens geometrization is as old as string theory and, like string theory, has produced no new results (but much more efficiently in terms of money and time wasted on what the petroleum engineers call a "dry hole"); so I think it's time to explore something new.

Carl

**CarlB** · Sep18-06, 08:51 PM

Originally Posted by Kea

To us, it isn't the preons that are fundamental... it's the ribbons, because they form the logical symbols that express the experimental question.

So what does it take to motivate you to write that introductory material? PhD theses are some of the best teachable material I've ever seen. Is it ethical / legal / a good idea/ to put unfinished PhD thesis material on the web?

And I saw some more work on braid theory and was shocked at realizing again some of the relations to what I'm doing. It was in that superconducting paper you and ZapperZ were talking about. The shock came at seeing a situation where multiple particles were assumed to move from one point to another. That reminded me of what I am doing, modifying the single particle propagators so that they carry multiple particles. In my case they are so deeply bound together that you can ignore the (uh, "pre-Higgs"?) bosons that bind them together. Another way of saying that is to say that the pre-Higgs have such high mass (Planck mass) that the creation and annihilation operators for it have to occur at the same point in spacetime.

Carl

**Kea** · Sep18-06, 11:12 PM

Originally Posted by CarlB

PhD theses are some of the best teachable material I've ever seen.

Yes, but alas the nightmare thesis is neither (a) finished nor (b) in any way pedagogical.

Another way of saying that is to say that the pre-Higgs have such high mass (Planck mass) that the creation and annihilation operators for it have to occur at the same point in spacetime.

Or perhaps that a Planck mass black hole is like an instanton. The braid maths, as you observe, also puts multiparticles on a fundamental footing.

**CarlB** · Sep19-06, 01:00 AM

Things that make you go hmmmmm.

Carl

**Chronos** · Sep19-06, 01:51 AM

It seems plausible to me that the most 'fundamental' building blocks of our perceived, macroscopic universe need not require 3+1 dimension descriptions, perhaps not even integer numbers of dimensions. Perhaps it will turn out that a landscape of backgrounds is the most mathematical precise definition of background independence. It may well also be incomprensibly difficult to solve the 'correct' mathematical model with greater precision than existing models permit.