Poincare Invariance from General QFT

[CarlB]

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/cgi-bin/abstract/109884430/ABSTRACT

I found this elegant paper after seeing the following passage in Lee Smolin's new book, "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 $$c\sqrt{3}$$. 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]

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

I'm very glad to hear that, Carl.

 

[CarlB]

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:

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

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,

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

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, $$\Psi_Q, \Psi_P$$ 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 $$(\hat{x}\hat{y}\hat{z}\hat{s}\hat{t})$$. That is, we are not adding a $$\hat{t}$$ 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:

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

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

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

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

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, $$\hat{t},$$ is added to the other four spatial dimensions, namely:

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

That is, to convert the above rules into the rules for $$(\hat{x}\hat{y}\hat{z}\hat{s}\hat{t}),$$ simply multiply on the left by $$\hat{x}\hat{y}\hat{z}\hat{s}$$, then note that this constant anticommutes with $$\hat{x}, \hat{y}, \hat{z}, \hat{s}$$ and commutes with $$\hat{t}.$$ 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:

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

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]

Poincare Invariance - related to - TGD?

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]

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]

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]

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]

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]

Debsity Operators: attractors / dissipators?

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/economics/laureates/1994/presentation-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/mth381pr/GAMETH/gametheory.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/Winter06/LectureNotes/09SaddlePointsNashEqui.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]

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. $$\sigma_x\sigma_y$$, 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.
$$0.5(1+\sigma_x) * 0.5(1+\sigma_y)*0.5(1+\sigma_z) * 0.5(1+\sigma_x)$$

Transition probabilities are defined as stuff like
$$0.5(1+\sigma_x)* 0.5(1+\sigma_y)$$ which is where the $$P = 0.5(1+\cos(\theta))$$ 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]

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]

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 $$\Delta x \Delta p > \hbar.$$

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:
$$\psi = R(x,t)e^{iS(x,t)}$$
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]

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]

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]

Things that make you go hmmmmm.

Carl

 

[Chronos]

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.

 

[CarlB]

Loved your comment on mass at your blog, http://kea-monad.blogspot.com/ . Was it your intention to turn off comments? Or at least make them very difficult to put in?

I borrowed a fascinating book on mass at the library, now available on Amazon for $8.98:

Concepts of Mass in Classical and Modern Physics
https://www.amazon.com/dp/0486299988/?tag=pfamazon01-20

My perspective is that mass is the coupling constant between left and right handed things, a probability (actually a probability rate and there is your time unit dependence). The more likely the conversion is, the greater the mass. That is at a fundamental particle level. The idea doesn't scale well. As far as I know there is no left handed Carl.

The basic problem with these things is that they are soooo emergent. As Max Jammer says, measurements of mass are just ratios of other things.

For the bound states I've been playing with, mass is the probability of running through the left to right to left cycle. The idea is to normalize the preons so that they satisfy the idempotency relation $$\rho^2 = \rho.$$ The idea is that things which satisfy an idempotency relation preserve their identity over time, which is the working definition of a particle. That the idempotents do this comes from looking at the general non linear wave equation in a Clifford algebra which reminds me of the paper listed in post #1.

To model a composite of 3 of these things to make a left handed fermion you end up with a 3x3 matrix of primitive idempotents, call it $$\rho_L$$. There's another 3x3 matrix for the right handed fermion, $$\rho_R$$.

The Feynman diagram for the left becoming a right is given by simple matrix multiplication (keeping in mind that the elements of the matrices are Clifford algebra idempotents). To get the mass, you solve the equation: $$\rho_L\rho_R\rho_L = m\rho_L.$$ In other words, while the preons are idempotent, the bound states they create are not, but are proportional to an idempotent. The proportionality constant gives the mass, a sort of probability rate. To get the $$\sqrt{m}$$ of the Koide relation, replace the $$\rho_L, \rho_R$$ with spinors by $$\rho_L = |L\rangle\langle L|$$, same with right, and you get: $$\langle R|L\rangle = \sqrt{m}.$$

It's late and I may very well have left off a square root somewhere here, largely because it is easy for me to confuse amplitudes with probabilities, so read with skepticism.

Carl

 

[Kea]

Hi

I just got back from UMacq and a very interesting (at least for me) conversation with M. Batanin, which means I'm going to run away shortly...

Was it your intention to turn off comments?

No, that was the default. Problem now fixed.

 

[Severian]

Derivation of Poincare Invariance from general quantum field theory
C.D. Froggatt, H.B. Nielsen

I had lunch with Colin Froggatt today. We didn't discuss his work though... next time maybe.

 

[CarlB]

No, that was the default. Problem now fixed.

I think the reason that there's still no comments showing up is because you've set it so that they have to be "moderated", but then you're never actually moderating them.

And I've uploaded a 2nd version of 1st chapter of density matrix formalism book. Instead of calling it "density matrix", I've been calling it "density operator", but a better word would be just "operator", because the idea is to replace the usual formalism of state vector + operator, with a pure operator formalism.

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

The primary change is that I've added a section on "Amplitudes and Feynman Diagrams" since that is the main topic of the book and it can be addressed at the Pauli algebra level, and a section on "Products of Density Operators" as that gives a calculational trick that gets used over and over again throughout the book. There are minor changes. For example I added a little poetry, on the topic of foundation work, at the start of the chapter. The full poem is here:
http://whitewolf.newcastle.edu.au/words/authors/K/KiplingRudyard/verse/p1/proconsuls.html

Carl

 

[Kea]

I think the reason that there's still no comments showing up is because you've set it so that they have to be "moderated", but then you're never actually moderating them.

Oops! You're quite right. OK, now I've fixed it.

And I've uploaded a 2nd version of 1st chapter of density matrix formalism book.

Great. I've been busy having fun at AustMS06. Excellent Kipling, yes.

 

[CarlB]

Kea,

I've been looking around for a paper I saw (maybe around here) and then lost that had to do with preons eliminating the need for spontaneous symmetry breaking:

Higgs Pain? Take a Preon!
J.-J. Dugne, S. Fredriksson, J. Hansson, E. Predazzi, 1997
The Higgs mechanism is the favourite cure for the main problem with electroweak unification, namely how to reconcile a gauge theory with the need for massive gauge bosons. This problem does not exist in preon models for quark and lepton substructure with composite $$Z^0$$ and $$W$$s, which, consequently, also avoid all other theoretical complications and paradoxes with the Higgs mechanism. We present a new, minimal preon model, which explains the family structure, and predicts several new, heavy quarks, leptons and vector bosons. Our preons obey a phenomenological supersymmetry, but without so-called squarks and sleptons, since this SUSY is effective only on the composite scale.
http://arxiv.org/abs/hep-ph/9709227

Then I got to reading other papers by the authors and it has been quite entertaining:
A simple solution to color confinement
http://arxiv.org/abs/hep-ph/0011060

Yet another paper which gets relativity from non relativisitic assumptions (and which I am going to have to reference if not in the stuff I'm typing up now then in later stuff):
Spin zero particle propagator from a random walk in 3-D space
J. J. Dugne, V. A. Karmanov
Phys. Lett. A310 (2003) 131-138
Abstract: The propagator of a spin zero particle in coordinate space is derived supposing that the particle propagates rectilinearly always at the speed of light and changes its direction in some random points due to a scattering process.The average path between two scatterings is of the order of the Compton length.
Conclusion:
It turned out that we had to suppose that the scattering wave v.s. the distance r from the scattering center be decreased as 1/r, eq. (1). It looks like the diffusion of a spherical wave which seems natural, but, at first glance, it has no any hint to relativity. However, if, instead of 1/r, we would take another function of r, we would not reproduce the propagator, but also would loose the relativistic invariance.
http://arxiv.org/abs/hep-th/0303028

Preon Trinity - a new model of leptons and quarks
http://arxiv.org/abs/hep-ph/9909569

The above preon paper is pretty similar to mine but it shows what kind of gymnastics you have to go through when you assume that the preons have to be treated statistically the same as the particles of the standard model. That is, when you assume that the particles have to have a particular spin. Conceptually, when you break a particle up into left and right handed parts, the individual parts can no longer be described as having spin. They are all singlets and must be covered by spin-0. But to get it to work, you have to assume the Pauli exclusion principle (uh oh). Well, every time the latest fundamental particles get split into smaller particles, somebody's eggs become part of the new omelet.

Any theory of deeply bound preons faces the problem of modeling a nonlinear interaction that can't be done with perturbation theory. I've been going with my intuition on this, which basically follows the line of thinking; "operators are bilinear, spinors are linear, so if you want to go to the first thing more complicated than linear, look at a bilinear theory and examine density matrices".

In doing this, I am trying to use Feynman diagrams, which are designed pretty much for approximations of weak nonlinear theories. This is why there is a section in that paper devoted to the geometric way of defining Feynman amplitudes; when you add density matrices you are automatically nonlinear as far as what you've done to the states. Let me try and explain this better. My "solution" is to follow Julian Schwinger's measurement algebra and look at the Stern-Gerlach experiment and model the preons as primitive idempotents, i.e.:

|A><A|

models the particles that can pass through a S-G filter (i.e. S-G experiment with one output sent to a beam stop) that allows only particles of type A to pass. But the above is not to be thought of as an abstract particle. Instead it is a description of how the particle "A" modifies the fabric of spacetime. The S-G experiment picks out particles of type "A" by adding a field that somehow matches the particle. For example, to separate electrons from positrons you use an electric field, which, sure enough, can be generated by electrons and positrons.

Since the interpretation is that the above Clifford algebraic element stands not for the particle but for the particle's effect on spacetime, it is natural to model the emission of a gauge boson by hooking two S-G filters up in a row. The gauge boson (or combination of them) that converts a preon of type A to a preon of type B is modeled by the same sequence of projection operators that models the experiment consisting of the two consecutive S-G filters:

|B><B| |A><A|.

In the above, one is not treating the gauge boson as a separate particle. Instead, one is treating the emission of a gauge boson by a preon of type A as an entity. But the above is just as much a member of the Clifford algebra as |A><A| was, and we can consider adding them together. Consequently, in computing the overall potential energy, you can treat these sorts of things the same way you treat the preons themselves.

In order to get away with the above simplification, I use the fact that the fermions are point particles so there is no time dependence (and therefore Schwinger's measurement algebra applies), which also means that there is no messy Fourier transform issues. All the particles exchange gauge bosons at the same point in spacetime and consequently you don't have to deal with the propagators for the gauge bosons. And in going to density operator theory, I already lost linear superposition and that is the principle on which Fourier theory relies, which is why reading Hansson's papers brought me to type this up. Here's a paper that references his:

Nonperturbative calculational method in quantum field theory
V. Dzhunushaliev, D. Singleton, T. Nikulicheva, (2004)
An approximate procedure for performing nonperturbative calculations in quantum field theories is presented. The focus will be quantum non-Abelian gauge theories with the goal of understanding some of the open questions of these theories such as the confinement phenomenon and glueballs. One aspect of this nonperturbative method is the breaking down of the non-Abelian gauge group into smaller pieces. For example $$SU(2) \to U(1) + coset$$ or $$SU(3) \to SU(2) + coset$$. The procedure also uses some aspects of an old method by Heisenberg to calculate the n-point Green's function of a strongly interacting, non-linear theory. Using these ideas we will give approximate calculations of the 2 and 4-points Green's function of the theories considered.
http://arxiv.org/abs/hep-ph/0402205

Oops! You're quite right. OK, now I've fixed it.

I think you will find that your readers are a lot more talkative than you thought.

Carl

 

[Kea]

Higgs Pain? Take a Preon!
J.-J. Dugne, S. Fredriksson, J. Hansson, E. Predazzi, 1997

Yes, I remember seeing this at one point.

It turned out that we had to suppose that the scattering wave v.s. the distance r from the scattering center be decreased as 1/r. It looks like the diffusion of a spherical wave which seems natural, but, at first glance, it has no any hint to relativity. However, if, instead of 1/r, we would take another function of r, we would not reproduce the propagator, but also would loose the relativistic invariance.
http://arxiv.org/abs/hep-th/0303028

Thanks. This was quite cute! Actually I get the feeling there is a lot of stuff out there along these lines, and since I'm looking at this a bit differently I don't want to spend time on it, but it's great to know that a lot of different threads are being woven together slowly.

Any theory of deeply bound preons faces the problem of modeling a nonlinear interaction that can't be done with perturbation theory.

And I guess the term deeply bound preons here would include the more abstract interpretations for which particles in this sense are not the fundamental thing. You have tackled this from what I would call a purely physical viewpoint, without worrying too much about weird possibilities in quantum gravity. This is great because the ideas are so accessible. Personally, for a long time I have felt compelled to think more abstractly, hoping that eventually it would link up with other ventures. And now I see so many reasons for optimism!

In doing this, I am trying to use Feynman diagrams which are designed pretty much for approximations of weak nonlinear theories.

And I am similarly interested in Feynman diagrams, but as they appear in a much more mathematical setting, a la Getzler, Kapranov, Connes, Kreimer et al. This isn't because the fancy maths is essential as it stands. On the contrary, I think the simplest way to explain the computational techniques of the theory will be in such purely combinatorial terms, where everything comes down to simple counting.

My "solution" is to follow Julian Schwinger's measurement algebra and look at the Stern-Gerlach experiment and model the preons as primitive idempotents...|A><A|...but the above is not to be thought of as an abstract particle. Instead it is a description of how the particle "A" modifies the fabric of spacetime.

Yes, this is nice. I've never worked with the measurement algebra, but sometimes I picture particles roughly in this way. Even naively the picture of measurements 'generating' spacetime can take one a long way. When I started studying twistors a while back, it was because I realized that Penrose had been thinking carefully about how if the matter were absent, then spacetime should be also.

All the particles exchange gauge bosons at the same point in spacetime...

I didn't realize how important this was to your approach until you mentioned it a couple of weeks ago. The connection to operad maths here is the thing that really convinces me that your picture is compatible with the full M-theory.

Cheers
Kea

 

[CarlB]

And I guess the term deeply bound preons here would include the more abstract interpretations for which particles in this sense are not the fundamental thing.

Yes, the only reason we've made any progress on QM at all is that bound states are still quantum objects.

You have tackled this from what I would call a purely physical viewpoint, without worrying too much about weird possibilities in quantum gravity.

I've been thinking about how I can justify defining a potential energy that is zero on the scalars, and positive for anything else. Looking at the measurement algebra, this is too general. I can get the same effect if the potential gives zero for both 0 and 1, and then gets large for anything else. Like I mentioned before, I was seeing things in terms of Schwinger's measurement algebra where "1" is a free beam, and "0" is a complete beam stop. Both of these require no fields, more or less. But tonight I realized that another way of justifying it, sort of, is as a generalization of maximization of entropy:

$$S= -\tr(\rho\;\ln(\rho))$$

so a natural potential energy would be

$$V = |\rho\;\ln(\rho)|^2$$

where $$\rho$$ is a Clifford algebra element. The reason for exploring this is that when you compute a potential by adding together a bunch of Clifford algebraic primitive idempotents, the result is certainly not primitive, and may not be an idempotent. That implies that I'm looking at a potential defined on an ensemble instead of a potential defined on a pure state (which is certainly compatible with Schwinger's measurement algebra).

Zero or one makes V = 0, anything else is positive. Zero is boring. But "one", or more accurately 1/N is what makes S maximum. And as long as the natural unit of energy is the Planck energy, all that matters is how you get V to be zero == how do you maximize entropy. It makes me feel optimistic that there really is some statistical mechanics hiding around underneath the standard model.

I didn't realize how important this was to your approach until you mentioned it a couple of weeks ago. The connection to operad maths here is the thing that really convinces me that your picture is compatible with the full M-theory.

Can you elaborate on this?

Carl

 

[Kea]

But tonight I realized that another way of justifying it, sort of, is as a generalization of maximization of entropy:

$$S= -\tr(\rho\;\ln(\rho))$$

so a natural potential energy would be

$$V = |\rho\;\ln(\rho)|^2$$

where $$\rho$$ is a Clifford algebra element...That implies that I'm looking at a potential defined on an ensembled instead of a potential defined on a pure state.

Excellent. The notion of ensembles of systems (prior to any choices of particular system) has to be more basic than potential, because otherwise there is no way that spacetime can be emergent, which is a quality that we demand of it in QG.

Operads can describe ensembles extremely abstractly: no fields on background manifolds, no pre-assigned properties. The simplest picture (and it is literally a picture) is of trees. Trees have a single output (this can of course be generalised) and many branches, reaching upwards. A vertex is permitted to have only one root, but may have many branches. A special class of tree is one for which all vertices are trivalent. This vertex is the pair-of-pants diagram of String theory, or the trivalent ribbon vertex for ensemble matrix models. Any surface can be triangulated, or dually described by a trivalent graph. As one allows higher degree vertices one investigates other dimensions, but dimension here shouldn't be taken to mean that of an a priori spacetime. It is a categorical dimension, representing the increasing complexity of larger ensembles, for which new types of behaviour may emerge at different levels.

I am starting to talk about these things on my blog, where it is also possible to put large diagrams.

 

[Dcase]

Attempt to unify all science information into mathematical objects

Hi Kea:

and CarlB:
and all others:

I admire your knowledge of bio-mathematics.
I search for the unification with other types of mathematics.
I suspect this will result in a mathematical game of energy exchanges through attractors and dissipators of various gauges.

I prefer your tree branch analogy [of this thread] to your pair of pants analogy [of the Application of topology in nature - auditory energy mapping thread].
https://www.physicsforums.com/showthread.php?t=134962

Such an analogy, may allow physicists [particle, astro, geo, bio, et al], mathematicians, biologists, physicians and engineers of all types, to make translations among disciplines easier, if all science information is translated into mathematical objects.

I am greatly influenced by Nobel Laureate Christian De Duve, most recently.
‘Singularities: Landmarks on the Pathways of Life’ [Cambridge University Press]:
p161 - “ ... be they microbes, plants, fungi, or animals, including humans. It is clear that such similarities can be explained only ...”
p171 - “... Eukaryotes, which may be unicellular (protists) or multicellular (plants, fungi, and animals) ...”
p186 - “,,, eventually, with all plants, fungi and animals; without it we wouldn’t be around ...”
p214 - “... all fungi, and all animals are derived each from a single common ancestor ...”

I only have read the abstract of
“From sphere to torus: A topological view of the metazoan body plan”
Harald Jockusch (Corresponding author)
Email: h.jockusch@uni-bielefeld.de
Pages 57-65
Saturday, June 17, 2006
Journal Bulletin of Mathematical Biology
Publisher Springer New York
ISSN 0092-8240 (Print) 1522-9602 (Online)
Subject Mathematics and Statistics
Issue Volume 65, Number 1 / January, 2003
DOI 10.1006/bulm.2002.0319
Abstract From the viewpoint of mathematical topology, membrane systems in intact living cells can be described as closed and orientable surfaces, i.e., as surfaces with two sides and no boundary lines so that an ‘inside’ and an ‘outside’ can be distinguished. Usually, biomembranes represent topological spheres, often one embedded in another one. Toroidal membranes are occasionally observed, e.g., in specialized structures of plant cells like the prolamellar body. Here, we propose that rules analogous to those that govern the topology of biomembranes hold for the epithelial cell sheets that cover anatomically external as well as internal surfaces of multicellular animals. We suggest to study the emergence of morphological complexity during metazoan development using concepts from mathematical topology, and propose experimental analyses of those topological transitions that appear to be relevant in development and evolution.
Based on a lecture held by H J at the Max Planck-Institute for Physics of Complex Systems, Dresden, 26 April 2001.
http://www.springerlink.com/content/47q7pvkr03882252/

I tend to probably oversimplify [and perhaps confuse] the possible relationships in this effort to demonstrate that evolutionary biology has apparently used experimental mathematics [likely related to the GUT / TOE quest of QM and GR] that we as humans now attempt to understand.

These manifolds from different disciplines probably differ primarily in gauge with similar but not necessarily identical structure and function.

I find that my math ability is like using a foreign language - some understanding with visual aids and a dictionary or reference text - but clearly unable to express thoughts attempting to relate one discipline to another in the language of mathematics [I am trying to improve.

Analogies?:

A tree trunk from the upper most root to the lower most limb can have its tree rings almost perfectly represented by cylindrical coordinate tensor calculus.

Reference diagram - JD Fehribach, Vector & Tensor Calculus, Worcester Polytechnic Institute
http://www.math.wpi.edu/Course_Materials/MA2251C99/images/cylndrcl.gif

The only thing that appears to prevent perfect symmetry is a lack of background independence. Tree ring growth is influenced by magneto, gas, liquid and solid influences - sources of actual perturbations.

Consider the whole tree: roots, cylindrical trunk [with pores?], branches and leaves.

Reference diagram - from VA TECH
http://fwie.fw.vt.edu/rhgiles/trevey/img/TreeBasic2.gif

Omitting the leaves for a moment, we appear to have a smooth manifold, bounded but infinite [from the fractal perspective] in surface area with a finite volume.

Add leaves, the surface area may almost be doubled [more than one infinity as per George Cantor] and the volume possibly also doubled but still finite.

For simplicity, again omit the leaves.

Technically this tree manifold may be transformed into a fungus manifold.

Fungi appear to be intermediate to flora and fauna.
Reference diagram - from Sydney [text with links to some pictures]
http://bugs.bio.usyd.edu.au/Mycology/StructureFunction/hyphalStructure.shtml

The fungus can be transformed into porifera manifold.
The roots become a holdfast.
The branches [become reduced but tend to reappear in Cnidaria] with inflow / outflow pores as ostia and osculum [or branched oscula].

Cnidaria life cycle more tree-like

Reference diagram - Western Kentucky U
Porifera with Cnidaria life cycle - holdfast demonstrated but not named
http://asm.wku.edu/faculty/Lienesch/225/225lab2.html

Reference diagram - U Miami
Porifera [magnified illustrations] holdfast not demonstrated
http://www.bio.miami.edu/dana/pix/spongeanatomy.jpg

Other body plans omitted for some brevity.

Eventually transformation to sequentially segmented genus-1 toroid manifolds of vertebrate, chordates with respect to the gastrointestinal tract.
Holdfast becomes sphincter.
Genus-3 torus of nares and mouth with lungs more like porifera; oral cavity shared with GI torus.

Reference diagram - Central Middlesex Hospital, London, England
http://www.ibs-research-update.org.uk/ibs/digestion1ie4.html

Nerves and blood vessel are branch-like in providing nutrients and information to various organs and single cells. This symmetry is relatively reversed from flora.

Reference diagram - Netterimages.com
http://www.netterimages.com/images/vpv/000/000/004/4621-0550x0350.jpg

These life manifolds can interact with Earth [magneto, gas, liquid and solid] manifolds and star [sun] manifolds.

The finest roots and branches [capillaries] appear to serve as gauge interfaces - allowing trans-membrane flow; while coarser branches allow for a generally one way cis-membrane flow [regurgitation may occur, usually in illness].

Most of life might be considered as attractors [contained energy] with a predator v prey [dissipator or uncontained energy] relationship.

 

[Dcase]

Speculative response to question on potential energy

Hi CarlB:

RE:

I've been thinking about how I can justify defining a potential energy that is zero on the scalars, and positive for anything else. Looking at the measurement algebra, this is too general. I can get the same effect if the potential gives zero for both 0 and 1, and then gets large for anything else. Like I mentioned before, I was seeing things in terms of Schwinger's measurement algebra where "1" is a free beam, and "0" is a complete beam stop. Both of these require no fields, more or less. But tonight I realized that another way of justifying it, sort of, is as a generalization of maximization of entropy

Is total energy = potential energy + kinetic energy?
If so, consider:
potential energy = attractor
kinetic energy = dissipator

 

[Dcase]

0 or 1 could be a probability range

Hi CarlB:

RE:

... I can get the same effect if the potential gives zero for both 0 and 1, and then gets large for anything else...

Speculative analogy suggests that 0 or 1 may represent the probability limits.

Is the result large for numbers between the interval [0,1]?

 

[CarlB]

Speculative analogy suggests that 0 or 1 may represent the probability limits.

I hate to mix probabilities with multivectors. In my view, the multivectors (i.e. what are called "complex numbers" in the usual non geometric version of quantum mechanics) are stresses in spacetime and the particles are dislocations that propagate through spacetime. The probabililty interpretation arises from the dislocations being indeterminate. The idea is that a given chunk of spacetime dislocates with a probability that can be calculated from the stress, hence the probability interpretation. After dislocating, the stresses are relieved, hence the Pauli exclusion principle.

Another way of describing this is that you take the Bohmian interpretation, which combines wave and particle information, and split the information into wave information, which give the stress before the dislocation, and particle information, which gives the actual position and orientation of the resulting dislocation. I should admit that I went to school to study mining engineering, so putting things in terms of crystallography makes sense to me. It's been a hobby for most of a half century. As long as I'm on the subject, here's my crystal drawing program written in Java:
http://brannenworks.com/xtl

Is the result large for numbers between the interval [0,1]?

The whole idea is that the natural unit for the potential energy is the Planck energy, so it is effectively infinite everywhere where it is not zero. The observed particles are supposed to all have mass zero. The observed masses are due to a splitting of a degeneracy. This is why the observed particles can mix so easily.

An alternative approach is to suppose that the various generations are excitations off the 1st generation. This has the defect of being very difficult to explain why there is, according to very good neutrino data, no 4th generation.

Carl

 

[Dcase]

RECIPROCALS as probabilities and particle generations

Hi CarlB:

1 - I think that I understand your reluctance to “mix probabilities with multivectors”.

There may be an alternative perspective of dealing with this which may even satisfy the hypothesis of David Hestenes in ‘The Kinematic Origin of Complex Wave Functions’: RECIPROCALS!

The 1/R to R relationship suggests as many numbers between the interval [0,1] as between the interval [1,oo].

QM statistical probabilities may be equivalent to reciprocals.

Consider this representation of the real [complex similar?] number line:

[+] Positive --->...1.....1......n......
-----------------------+---------------1-------------------+------------------- +oo
|......n......1......1.....
ZERO (0)
|......1......1......n......
---------------------_---------------1--------------------_-------------------- -oo
[-] Negative --->...n.....1......1......

At some gauge / scale or time it may be possible for +oo to coincide with -oo.

Hyperphysics of GA STATE may even demonstrate the probability relationship with reciprocals in ‘The Correspondence Principle and the Quantum Oscillator’ and ‘Comparison of Classical and Quantum Probabilities for Harmonic Oscillator’.
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc6.html

Planck length may be the minimal distance [probably as a radius rather a diameter] of an EM torus, black hole or between any strings or manifolds.

If a black hole is an extreme concept similar to the eye of an hurricane, then the Planck mass may be around but not in the black hole as the winds are around but not in the eye of an hurricane. Yet for the black hole the Planck mass would be centered within the black hole as the hurricane winds are centered within the eye.

2 - Could the “the various generations” be phase transitions?
Could there be a plasma phase?

This is of course a speculative analogy.

 

[Dcase]

Physics, ME (fluid dynamics) and attractors

Philip Marcus [UC-Berkeley] is one of the few discussing dissipators as well as attractors.

Researcher predicts global climate change on Jupiter as giant planet's spots disappear [see merger video]
http://www.berkeley.edu/news/media/releases/2004/04/21_jupiter.shtml

or

'Chaos' James Gleick [1987 version]
[last page of color inserts, text preceding page; between p 114-115]

and

Jupiter's Great Red Spot - Temperature Structure of Vortex
http://www.gfdl.noaa.gov/products/vis/gallery/index.html

I wonder if Red Jr is related to Shoemaker-Levy-9 collision with Jupiter?

 

[CarlB]

I've discovered the secret of time travel.

Kea,

I've discovered the secret to time travel, at least one way. You sit down and start writing a book. Days just disappear.

But I've finished the 2nd chapter, even sent the thing through a spell checker:
http://www.brannenworks.com/dmaa.pdf

The planned topic of chapter 3 is "Measurement". Chapter 4 is supposed to be "Potential Energy", but seeing that far into the future is tough. Maybe it would be better to make chapter 4 cover the elementary fermions. Otherwise I would put them into the applications part of chapter 3.

At this time I'm ceasing working on book and preparing lecture for the meeting in Hawaii. They're giving me 25 minutes and I don't want to embarass self by insufficient preparation.

Liquafaction Corporation has kindly allowed me 3 weeks leave with pay to get ready for, and attend, the conference. Thank you, Mark Mollo. Which reminds me, I need to type up an acknowledgments section. And I need some sort of subsection break mark, to indicate when a multiparagarph argument has completed. And I need to type up some sort of exercises for the student at the end of each chapter. And I don't think I have enough marginal notes in the last half of chapter 2. And I haven't proof read the index (which is easily the most boring thing one can imagine doing with ones brief time on earth). And etc., etc., etc.

Carl

 

[Kea]

I've discovered the secret to time travel, at least one way. You sit down and start writing a book.

I must try it sometime. But please keep going!

At this time I'm ... preparing lecture for the meeting in Hawaii.

Great! Please report back on all the gossip. We just love conference blogging.

Good luck.

 

[Dcase]

I am naive!

The Reimann Sphere accomplishes what I was trying to say in my post of 10-11-2006 02:22 PM.

0 at the south pole

oo at the north pole

only reals along the longitude when a=0 for ai

complex along the longitude when a not= o for ai

equator = 1

45 degrees south = 1/2 in interval [0.1]

45 degres north = 2 in interval [1,oo]

Thus any complex number or vector may [?] be mapped to interval [0,1] gainng an insight into the probability of that number or vector whether in [0,1] or [1,oo]