Lee Smolin's LQG may reproduce the standard model

[bananan]

http://www.newscientist.com/channel/fundamentals/mg19125645.800
sounds like string theory, all particles from vibrations. how successful has this theory been in comparison to string theory, on particle physics?

"physical particles may seem very different from the space-time they inhabit, but what if the two are one and the same thing? New Scientist investigates

LEE SMOLIN is no magician. Yet he and his colleagues have pulled off one of the greatest tricks imaginable. Starting from nothing more than Einstein's general theory of relativity, they have conjured up the universe. Everything from the fabric of space to the matter that makes up wands and rabbits emerges as if out of an empty hat.

It is an impressive feat. Not only does it tell us about the origins of space and matter, it might help us understand where the laws of the universe come from. Not surprisingly, Smolin, who is a theoretical physicist at the Perimeter Institute in Waterloo, Ontario, is very excited. "I've been jumping up and down about these ideas," he says.

This promising approach to understanding the cosmos is based on a collection of theories called loop quantum gravity, an attempt to merge general relativity and quantum mechanics into a single consistent theory." ...

 

[marcus]

He is evidently talking about work that first appeared this year. It is very much in the start-up stages.

Lee Smolin was invited to a special conference in July celebrating the 60th birthday of Gerard 't Hooft, and the talk he gave was about this new idea.
http://www1.phys.uu.nl/gerard60/program/
See title of Smolin's talk on 15 July
"Emergence of chiral matter from quantum gravity"

Many of the people at the conference were Nobel laureates or other big name theoreticians. I wonder how Smolin's idea of matter emerging from network states of the gravitational field went over!

It is a very novel approach, based on work that appeared in 2005 by a young Australian physicist named Sundance Bilson-Thompson.

It has been discussed in some PF threads, but very little has been written about it so far.

http://arxiv.org/abs/hep-ph/0503213

http://arxiv.org/abs/hep-th/0603022

the quantum state of the geometry of space is represented by a web or NETWORK
durable matter particles exist as twists and tangles ( braids) in this network.
semipermanent topological snarls in the geometry of space, in other words, represent matter.

In this approach the evolution of geometry (in interaction with matter) is conjectured to be explainable in terms of
elementary local "moves" by which the network modifies itself
(by breaking a link between a pair of nodes and reconnecting them some other way, or inserting a new node etc )
it modifies itself by quantum (loosely speaking probabilistic) local reconnections according to these "moves"
matter is special knots in the web that these moves can't usually untangle
only when two knots come together can something happen and them might cancel each other or turn into something else.
A quantum state of geometry is represented as a network of pure spatial relationship----what is between what, what is adjacent to what---a network or graph of nodes connected by links. It has not been proven that this SUCCEEDS as a quantum spacetime dynamics.
==============

that said, notice that what we're discussing is a small PRELIMINARY investigation of a possible quantum theory of space and matter----whether or not it can be brought to full conclusion, people are likely to learn something from investigating it.
AFAIK only two faculty and some 3 or 4 postdoc/gradstudent researchers have looked into this.
so far, half a dozen "man-years" or less
the payoff in interesting provocative results has been high IMHO given such a small investment

another thing one might remark is that a lot of what comes out in New Scientist is not especially meaningful.

If someone wanted to find out about this, I would probably not recommend the New Scientist. I'd suggest instead reading Sundance original paper and watching the video of his November 2005 talk at Perimeter Institute

If you want the video, go here
http://streamer.perimeterinstitute.ca:81/mediasite/viewer/FrontEnd/Front.aspx?&shouldResize=False
select "seminar series" from the menu at the left
click on "presenter" to get a list of seminar speakers alphabetical by first name
scroll down to "Sundance" click on the name and press "search"
You will get the November 16, 2005 talk---it is about an hour long.

or look on page 11 out of 24 in the whole listing of seminar talks
it will usually be about 13 from the end, so when there are 30 pages of talks it will be on page 17.

If you want a general overview by Lee Smolin, a recent paper is
http://arxiv.org/abs/hep-th/0605052

that would have references to other papers, including the two I mentioned

 

[alexsok]

Thx for that. So is there a full version of this article elsewhere on the web?

 

[bananan]

Thanks Marcus,

incidentally, do you have an opinion on the Heim- Dröscher Theory, and its possible connections with LQG/spinfoam, perhaps its forumla for mass of elementary particles can be re-derived from LQG, to give a boost of LQG over its enemy, lubos

He is evidently talking about work that first appeared this year. It is very much in the start-up stages.

Lee Smolin was invited to a special conference in July celebrating the 60th birthday of Gerard 't Hooft, and the talk he gave was about this new idea.
http://www1.phys.uu.nl/gerard60/program/
See title of Smolin's talk on 15 July
"Emergence of chiral matter from quantum gravity"

Many of the people at the conference were Nobel laureates or other big name theoreticians. I wonder how Smolin's idea of matter emerging from network states of the gravitational field went over!

It is a very novel approach, based on work that appeared in 2005 by a young Australian physicist named Sundance Bilson-Thompson.

It has been discussed in some PF threads, but very little has been written about it so far.

http://arxiv.org/abs/hep-ph/0503213

http://arxiv.org/abs/hep-th/0603022

the quantum state of the geometry of space is represented by a web or NETWORK
durable matter particles exist as twists and tangles ( braids) in this network.
semipermanent topological snarls in the geometry of space, in other words, represent matter.

In this approach the evolution of geometry (in interaction with matter) is conjectured to be explainable in terms of
elementary local "moves" by which the network modifies itself
(by breaking a link between a pair of nodes and reconnecting them some other way, or inserting a new node etc )
it modifies itself by quantum (loosely speaking probabilistic) local reconnections according to these "moves"
matter is special knots in the web that these moves can't usually untangle
only when two knots come together can something happen and them might cancel each other or turn into something else.
A quantum state of geometry is represented as a network of pure spatial relationship----what is between what, what is adjacent to what---a network or graph of nodes connected by links. It has not been proven that this SUCCEEDS as a quantum spacetime dynamics.
==============

that said, notice that what we're discussing is a small PRELIMINARY investigation of a possible quantum theory of space and matter----whether or not it can be brought to full conclusion, people are likely to learn something from investigating it.
AFAIK only two faculty and some 3 or 4 postdoc/gradstudent researchers have looked into this.
so far, half a dozen "man-years" or less
the payoff in interesting provocative results has been high IMHO given such a small investment

another thing one might remark is that a lot of what comes out in New Scientist is not especially meaningful.

If someone wanted to find out about this, I would probably not recommend the New Scientist. I'd suggest instead reading Sundance original paper and watching the video of his November 2005 talk at Perimeter Institute

If you want the video, go here
http://streamer.perimeterinstitute.ca:81/mediasite/viewer/FrontEnd/Front.aspx?&shouldResize=False
select "seminar series" from the menu at the left
click on "presenter" to get a list of seminar speakers alphabetical by first name
scroll down to "Sundance" click on the name and press "search"
You will get the November 16, 2005 talk---it is about an hour long.

or look on page 11 out of 24 in the whole listing of seminar talks
it will usually be about 13 from the end, so when there are 30 pages of talks it will be on page 17.

If you want a general overview by Lee Smolin, a recent paper is
http://arxiv.org/abs/hep-th/0605052

that would have references to other papers, including the two I mentioned

 

[CarlB]

Yet he and his colleagues have pulled off one of the greatest tricks imaginable.

I saw his lecture on this a few months ago at the May APS meeting in Dallas, Texas. It was not well received largely because (a) it makes no new predictions, and (b) they don't have any clue on how to get neutrinos to work, and (c) getting 3 generations seemed iffy.

The primary complaint that arose in the question and answer section was that the theory suggested that there should be an infinite number of generations and had great difficulty explaining why there appear to be exactly three generations, particularly, exactly three neutrinos.

I guess I should admit that I have a pony in this race.

Carl

 

[bananan]

I saw his lecture on this a few months ago at the May APS meeting in Dallas, Texas. It was not well received largely because (a) it makes no new predictions, and (b) they don't have any clue on how to get neutrinos to work, and (c) getting 3 generations seemed iffy.

The primary complaint that arose in the question and answer section was that the theory suggested that there should be an infinite number of generations and had great difficulty explaining why there appear to be exactly three generations, particularly, exactly three neutrinos.

I guess I should admit that I have a pony in this race.

Carl

has string theory been more successful in this regard? (points abc) what's the pony you have in this race?

 

[Dcase]

From 'Generic predictions of quantum yjroties of garvity; referenced 8-11 "If you want a general overview by Lee Smolin, a recent paper is" by marcus
http://arxiv.org/abs/hep-th/0605052

For item 6 - the emergence of matter from quantum geometry -
from my perspective - sounds as though ‘quantum geometry’ is a form of energy - and may relate to string vibrations?

Smolin may yet unify LGQ and causal spin networks with the various string theories?

I truly hope that he or someone is able to do this.

 

[Kea]

I guess I should admit that I have a pony in this race.

CarlB, please give our readers a summary of your argument.

 

[CarlB]

CarlB, please give our readers a summary of your argument.

The idea behind "particle physics" is that as a physical object is made more and more energetic, it becomes simpler because it is broken into smaller and smaller parts that can be treated alone. For example, on a cold morning your car might have extravagant patterns of frost on it. As your car warms up, these change to dewdrops, which are simpler because the water molecules aren't stuck together any more. Warming further, the dewdrops evaporate into the air, which is simpler yet. Heat that water up higher, it disassociates into atoms. Hotter, and it becomes nuclei and electrons. Hotter yet, and you get neutrons, protons and electrons. Hotter still, and you presumably get up and down quarks and electrons. As the material gets hotter, it splits into simpler subparts, conversely, as it cools, it condenses back again.

But at the level of the electrons and up and down quarks, the splitting/condensation sequence somehow gets complicated. As you add more energy to the system, instead of continuing to break into smaller parts, one finds that these particles are "point particles", and in adding energy one must suddenly deal with a whole plethora of unusual particles from the other two generations. The point here is that this is contrary to our expectations that things should get simpler as we heat them up. Instead, what is happening is that the extra energy allows new degrees of freedom to show up.

I assume that the three generations of elementary fermions must be made up of simpler particles, or "preons". Lee and Sundance's idea is similar in that they also assume preons, and their preons have some similarities to mine in that both ideas assume that the quarks and leptons are unified and are composites more or less formed from triples of subparticles.

Their idea is that the quarks and leptons are each made up of three subparticles which they call Helons, $$H_0, H_+, H_-,$$ where the suffix gives the electric charge in units of 1/3 e. These are braid groups, more or less. The three generations arise from differing complexity in the number of crossings of the braid group. They are unable to explain why there are exactly three generations or why they have the relative masses.

I model the subparticles based on a type of Clifford algebra called a "geometric algebra", a theory which was started by David Hestenes 25 years ago. This is different from braids in that it is a heck of a lot simpler to do calculations. If you don't want to learn a bunch of math, you can think of a great example of a Geometric Algebra as being the set of functions that map from space time to the set of 4x4 matrices that you could describe by taking sums of products of Dirac matrices.

My first splitting/condensation is that I break the quarks and leptons into their "chiral" halves. That is, I assume that the electron is a composite particle composed of two parts, a left handed electron and a right handed electron. These two parts convert back and forth into each other with a probability that is proportional to the mass (more or less, neutrinos are a bit odd). The energy required to break, for example, an electron is approximately infinite (i.e. Plank mass) because the chiral states are massless and therefore travel at speed c.

This division of quark or lepton into two "constituents" is similar in that it is assuming that the quarks and leptons are composite particles that are "condensed" from simpler particles, but this layer of condensation is different from the ones following in two ways. First, the binding energy is "infinite", and second, there is a phase change here in that mass appears. The unbound particles travel at c while the bound particles travels have mass and travel at slower speeds. I like this way of going to the next preon state because it is compatible with the quarks and leptons being point particles (at energies smaller than the Plank mass), and it is the natural division of the quarks and leptons into subparts (i.e. the weak interactions treat the chiral particles very differently). This method treats mass as just another interaction between particles.

My second splitting/condensation is to break the chiral particles into three subparticles that I am now calling "snuarks". You can get my snuarks by taking Lee and Sundance's helons, and (a) splitting them into left and right halves, and (b) splitting $$H_0$$ into a particle / antiparticle pair. Consequently, I have eight snuarks while they have 3 helons, but as far as preon models go, the two models are fairly similar. For a while I thought that I could get an algebraic model that would underlie their theory, after splitting the $$H_0$$ into two particles, and after combining the right and left handed snuarks. The basic problem with getting my version to line up with theirs is that I have complex numbers (or things that act like complex numbers anyway), and I can't see how to get that in their theory.

My final splitting is to take the snuarks, and break each of them up into two particles that are assumed to be the truly elementary particles which I call "binons". In terms of the Geometric algebra, these are "primitive idempotents", which is what the mathematicians use when they want to say "$$\rho^2 = \rho$$ and you can't make them any simpler". In terms of the Dirac matrices, the binons are the set of all possible density matrices, which is why I am constantly harping on density matrices on physics forums. In standard quantum mechanics, the spinor wave states are fundamental and the density matrices are derived from these. In my version of QM, these are reversed. My version is much more elegant, as you can see by reading the short discussion in https://www.physicsforums.com/showthread.php?t=124904 but this simplicitly and elegance only happens with spin-1/2 density matrices, which is why I use them. This gives a geometric foundation for the elementary particles which is a great feature of my theory.

I call them "binons" because their quantum numbers of Clifford algebra primitive idempotents are binary (look up the "spectral decomposition theorem" for Clifford algebras to see a proof). As with any symmetry class, there are a bunch of different ways to choose good quantum numbers. You need 8 different types of binons to get the quarks and leptons. There quantum numbers are therefore $$(\pm 1, \pm 1, \pm 1),$$ where the signs are chosen independently. Quantum numbers are additive. We can assume that the first of these quantum numbers depends on orientation, that is, it is the usual spin-1/2.

Binons are bound together by a potential energy. Since the binons are represented by Clifford algebra numbers, it is natural to use those numbers to define the potential energy. The definition is very simple. One adds together the Clifford algebra numbers, and then computes the "absolute value squared" of the sum. For the Dirac algebra, you could define the "absolute value squared" as the function which takes a matrix and gives the sum of the absolute squares of all its entries. That is a rather ugly definition (since it depends on the choice of representation etc.), but it turns out to be compatible (at least for the usual representations that physicists use) with the unique natural definition. The scale of the potential energy is the Plank mass. It is the potential energy that determines how binons have to be combined to make low energy particles and it is a great success of this theory that one can derive the structure of the quarks and leptons from first principles this way, with such a simple definition.

The snuarks are composed of two binons that are "compatible" in that their direction of travel is identical. In this, my theory is similar to the old "zitterbewegung" theory. The direction in which a chiral spin-1/2 particle travels is completely determined by its spin orientation, so the requirement that the snuark be compatible in their direction of travel is equivalent to requiring that their first quantum number be the same.

This leaves the other two quantum numbers arbitrary. I suppose that the next quantum number is "weak isospin". There are four cases: (+1+1, +1-1, -1+1, -1,-1). These four cases divide into a weak isospin doublet (+1+1, -1-1), and two weak isospin singlets (+1-1), (-1+1). The usual weak isospin quantum numbers are obtained by dividing these values, (2,-2,0,0) by 4. This simple derivation gives both the correct SU(2) symmetry and the correct pattern of representations, a great success of this idea.

The three snuarks that make up half of a quark or lepton are oriented in different directions. It turns out that this is necessary from the way that the potential energy is defined; otherwise there would be no bound states with energies less than Plank mass scale. So my version of preons has that when you successivley break an electron up into its components you first find 2 chiral states, then 6 snuarks, and finally 12 binons. That three snuarks are bound this way gives rise to an SU(3) symmetry, which is a great success of this theory.

As Clifford algebraic numbers, the snuarks can be multiplied by complex constants. That means that there are multiple ways of combining them together. Since this is a theory based on (pure) density matrices, one finds the number of ways that one can combine three snuarks together by solving the basic pure density matrix equation: $$\rho^2 = \rho.$$ It is a great success of this theory that when you do this, you obtain three solutions, which correspond to the three generations of elementary particles.

The derivation in the previous paragraph requires solving equations which have 3x3 matrices. One of the side effects of this is that one obtains a new way of expressing the masses of the leptons. Having done this, one finds that there are remarkable patterns in the lepton masses. In particular, it appears that there are discrete symmetries that define the hierarchy of the generations, and also the hierarchy from the neutrinos to the charged leptons. This is a great success of the theory and is discussed here: https://www.physicsforums.com/showthread.php?t=117787

Carl

 

[kneemo]

My final splitting is to take the snuarks, and break each of them up into two particles that are assumed to be the truly elementary particles which I call "binons". In terms of the Geometric algebra, these are "primitive idempotents", which is what the mathematicians use when they want to say "$$\rho^2 = \rho$$ and you can't make them any simpler". In terms of the Dirac matrices, the binons are the set of all possible density matrices, which is why I am constantly harping on density matrices on physics forums. In standard quantum mechanics, the spinor wave states are fundamental and the density matrices are derived from these.

Hi CarlB

Density matrices (primitive idempotents) certainly do provide a nice geometrical picture of quantum processes. This is because primitive idempotents give points in projective space. Of course, the question of which projective space, depends on the underlying division algebra you're working with as well as the dimension of your Hilbert space. In the finite dimensional complex case, in which the Hilbert space is $$\mathbb{C}^n$$, we recover points in the projective space $$\mathbb{CP}^{n-1}$$ (see http://math.ucr.edu/home/baez/octonions/node8.html" ). The most natural way they arise, is as you've noted, in the spectral decomposition of Hermitian operators:

$$\Phi=\lambda_1P_1+...+\lambda_nP_n$$.

Density matrix formalism plays an integral role in matrix models for string theory. In matrix models, fluctuations of D0-branes (preons) are given by scalar fields $$\Phi_m$$ (m=1,...,d) (Hermitian matrices), which are in the adjoint representation of the unbroken $$U(n)$$ gauge symmetry group.

One diagonalizes a scalar field $$\Phi$$ by a $$U(n)$$ gauge transformation yielding:

$$\Phi = U\left(\begin{array}{ccc}\lambda_1 & & 0 \\ & \ddots & \\ 0 & & \lambda_n \end{array}\right)U^{\dagger}$$.

The real eigenvalue $$\lambda_i$$ describes the classical position of D0-brane i. The unitary matrix $$U$$ with entries $$U_{ij}(i\neq j)$$ describe fluctuations about classical spacetime arising from the short open strings connecting D0-branes i and j.

Of course the diagonalization is equivalent to the spectral decomposition in terms of primitive idempotents. So recalling the decomposition:$$\Phi=\lambda_1P_1+...+\lambda_nP_n$$.

we can see a Hermitian operator expanded as a linear combination of primitive idempotents, which in turn can be associated with n fluctuating D0-branes (preons).

 

[CarlB]

Density matrices (primitive idempotents) certainly do provide a nice geometrical picture of quantum processes. This is because primitive idempotents give points in projective space. Of course, the question of which projective space, depends on the underlying division algebra you're working with as well as the dimension of your Hilbert space.

I've been trying to get away from the concepts of both Hilbert space and the apparently arbitrary choice of division algebra in favor of a purely algebraic approach.

The ideals of a Clifford algebra naturally form a copy of the complex numbers without any need to assume them, provided that the algebra has an odd number of canonical basis vectors. The Pauli algebra provides a good example of this. For example, the matrix

$$\rho_x = \frac{1}{2}\left(\begin{array}{cc}1&1\\1&1\end{array}\right)$$

is the density matrix for spin-1/2 in the +x direction. Any product of the form

$$\rho_x M \rho_x$$

will be a complex mutiple of $$\rho_x.$$ For example:

$$\rho_x\left(\begin{array}{cc}a+b&c-id\\c+id&a-b\end{array}\right)\rho_x$$

$$= (a+c)\; \rho_x$$

where a, b, c, and d are complex constants. Now the above was written in the usual representation of the Pauli algebra. If you translate it into purely geometric (density matrix) form it is somewhat more elegant:

$$\rho_x(a + b\rho_z + c\rho_x + d\rho_y)\rho_x = (a+c)\;\rho_x$$

In the above, a and c are "complex" numbers in that they are of the form

$$a = a_R + a_I\sigma_x\sigma_y\sigma_z$$

The underlying idea here is to eliminate unphysical gauge freedom. But the whole essence of Hilbert space implies a gauge freedom, so I'm avoiding Hilbert space. The density matrix formalism provides a way of doing this.

By the way, the method I'm using to extend the density matrix formalism to more complicated cases than spin-1/2 is Schwinger's measurement algebra. There is an interesting paper out by LP Horwitz on the use of the quaternions as the division algebra:
http://www.arxiv.org/abs/hep-th/9702080

The book by Schwinger, "Quantum Kinematics and Dynamics" has the arbitrary choice of complex numbers as the division algebra. I think that the way I'm doing it, relying on the density matrices themselves to define the division algebra, is far more natural and geometric.

There is only one problem with this, and that is that the number of canonical basis vectors for the usual spacetime is 4, and that does not contain a complex unit. (That is, there is no element of the Clifford algebra CL(3,1) that squares to -1 and commutes with everything in the algebra). For this reason, but more importantly because I need it to get the right complexity for the binons, I have to assume one hidden dimension.

My suspicion for the hidden dimensions in string theory is that the large number they require comes from the fact that they are not assuming preons. The preons add hidden degrees of freedom, it is natural that those degrees of freedom would show up as compacted dimensions in a theory that ignored them.

Density matrix formalism plays an integral role in matrix models for string theory. In matrix models, fluctuations of D0-branes (preons) are given by scalar fields $$\Phi_m$$ (m=1,...,d) (Hermitian matrices), which are in the adjoint representation of the unbroken $$U(n)$$ gauge symmetry group.

One diagonalizes a scalar field $$\Phi$$ by a $$U(n)$$ gauge transformation yielding:

$$\Phi = U\left(\begin{array}{ccc}\lambda_1 & & 0 \\ & \ddots & \\ 0 & & \lambda_n \end{array}\right)U^{\dagger}$$.

The real eigenvalue $$\lambda_i$$ describes the classical position of D0-brane i. The unitary matrix $$U$$ with entries $$U_{ij}(i\neq j)$$ describe fluctuations about classical spacetime arising from the short open strings connecting D0-branes i and j.

Of course the diagonalization is equivalent to the spectral decomposition in terms of primitive idempotents. So recalling the decomposition:


$$\Phi=\lambda_1P_1+...\lambda_nP_n$$.

we can see a Hermitian operator expanded as a linear combination of primitive idempotents, which in turn can be associated with n fluctuating D0-branes (preons).

This is very informative. The reason I started working on particle physics again after a break of 25 years is because I bought a copy of Polchinski's two volume textbook and wasn't satisfied with it. But I didn't get to a part that got into matrix models.

By restricting myself to density matrices, what I am doing is only working with the operators and avoiding the things that they operate on, (other than each other). Well, most of the time, but sometimes it is useful to work in state vector form, but I always keep an eye on how this relates to the density operator form.

Carl

 

[Kea]

...we can see a Hermitian operator expanded as a linear combination of primitive idempotents, which in turn can be associated with n fluctuating D0-branes (preons).

And matrix model D0-branes (preons) show up as objects in the ribbon graph description of moduli spaces (for Riemann surfaces with punctures). This is best understood in terms of labelled (metric) ribbon graphs, where the labels on edges take values in the positive real numbers. The use of twisted ribbons for nonorientable surfaces allows a consideration of the quaternionic ensemble.

The application of T-duality essentially means a restriction to the Hermitean case. See Mulase's papers, including the related paper
http://www.math.ucdavis.edu/~mulase/courses/2001modulich01.pdf which looks especially at the theory of elliptic curves.

In Smolin's picture, invariance under loop-addition moves would also indicate the presence of a duality. A ribbon vertex with three legs is typically dual to a triangle, which sort of introduces a loop.

 

[Kea]

...oh, and there is an association between the local pieces of $\mathcal{M}_{g,n} \times \mathbb{R}_{+}^{n}$ and the Stasheff associahedra of planar trees. This is in Mulase's papers. In the elliptic curve paper it is observed that the moduli $\mathcal{M}_{1,1}$ is like a cylinder with 2 corners. Corners are the reason one needs to do extended (2-categorical) TFTs, such as those studied by Pfeiffer and Lauda, but here the surface is a moduli space. The cylinder is what one gets by gluing the lines $Re(\tau) = \frac{1}{2}$ and $Re(\tau) = - \frac{1}{2}$ above the radius 1 semicircle, where $\tau$ is the usual parameter for the upper half plane. This picks out points with $y$ coordinates $\textrm{exp}(\pm 2 \pi i / 3)$.

Considering the special points $0,1, \infty$ on $\mathbb{P}^{1}$, then the inverse image under the $j$ invariant gives for $j^{-1}(0)$ one of $\textrm{exp}(\pm 2 \pi i / 3)$ (multiplicity 3 = trivalent vertex) and for $j^{-1}(\infty)$ the three puncture points $0,1, \infty$, each of which gets something Mulase calls a bigon (multiplicity 2) etc.

I think that this is all really about operads, which are introduced in
On operad structures of moduli spaces and string theory
http://www.citebase.org/fulltext?format=application/pdf&identifier=oai:arXiv.org:hep-th/9307114
Kimura, Stasheff, Voronov

This paper has a particularly nice theorem, namely, to quote: String vertices exist.

So, although the interpretation does change, it would appear that CarlB's analysis is on a fairly solid theoretical footing.

 

[kneemo]

Eigenmatrices

By restricting myself to density matrices, what I am doing is only working with the operators and avoiding the things that they operate on, (other than each other). Well, most of the time, but sometimes it is useful to work in state vector form, but I always keep an eye on how this relates to the density operator form.

Indeed, it is desirable to think in terms of operators only, so that one can study operators acting on operators, yielding more operators (the regular representation). If we wish our operators to be Hermitian, we want to find a product that enables the composition of a Hermitian operator with another Hermitian operator, to produce a Hermitian operator. Pascual Jordan, John von Neumann and Eugene Wigner did this in 1934 [1] and classfied all the "Jordan algebras" (observable algebras) over the finite dimensional division algebras. The maximal octonionic case (3x3 Hermitian matrices) was the most subtle however, as it cannot be recovered by assigning the Jordan product to an associative algebra. For this reason it is called the exceptional Jordan algebra (for a matrix model see Smolin's http://arxiv.org/abs/hep-th/0104050" ).

The Jordan algebras over the quaternions and octonions are very exciting to work with as the quaternions are noncommutative and the octonions are noncommutative and nonassociative. However, when it comes to Hermitian operators in these cases, we are not guaranteed that all eigenvalues are real. In fact, we can have quaternionic and octonionic eigenvalues [2]. This is likely a reason why quantum mechanics over such division algebras didn't go mainstream.

In [2], the authors proposed the Jordan eigenvalue problem:

$$A\circ V=\lambda V$$.

In essence, we are considering eigenvalues under the regular representation of the Jordan algebra. The corresponding matrices $$V$$ are called eigenmatrices. In [2], it turns out we can get real eigenvalues for Hermitian matrices over the quaternions and octonions, when we restrict the eigenmatrices to be primitive idempotents (hence points in projective space).

The complex case is much easier to study, as the eigenvalues for Hermitian matrices are real for the Hilbert space eigenvalue problem. This facilitates the process of bridging the Hilbert space formalism and the purely operator formalism. Here's a little theorem from my thesis:

Theorem.
Let $$A$$ be an $$n\times n$$ Hermitian matrix over $$\mathbb{C}$$ with a complete orthonormal set of eigenvectors $$v_1,v_2,...,v_n$$ and corresponding eigenvalues $$\lambda_1,\lambda_2,...,\lambda_n\in\mathbb{R}$$. Then the set of matrices $$W_{ij}=\frac{1}{2}(v_iv_j^*+v_jv_i^*)$$ for $$i\leq j$$ is a complete orthogonal set of Hermitian eigenmatrices of $$A$$ with corresponding eigenvalues $$\lambda_{ij}=\frac{1}{2}(\lambda_i+\lambda_j)$$.

Here's a cool lemma that follows.

Lemma.
For Hermitian eigenmatrices $$P_i=W_{ij}(i=j)$$, $$P_i$$ satisfy $$P_i^2=P_i$$ and $$tr(P_i)=1$$. That is, the $$P_i$$ are primitive idempotents.

Proof.
Expanding we have $$P_i=\frac{1}{2}(v_iv_i^*+v_iv_i^*)=v_iv_i^*$$. Squaring this and invoking orthonormality of eigenvectors yields $$P_i^2=v_iv_i^*v_iv_i^*=v_iv_i^*=P_i$$ and $$tr(P_i)=v_i^*v_i=1$$.

In summary, over the complex field we can see that solving the eigenvalue problem in the state vector formalism leads to the solution of the eigenvalue problem in the operator formalism. Moreover, we see that the set of density eigenmatrices are in one-to-one correspondence with the set of eigenvectors (as well as their corresponding eigenvalues). This begs the question, "what is the physical interpretation of the non-density Hermitian eigenmatrices (with their real eigenvalues)?" Certainly, for the non-density eigenmatrices, we lose the nice geometrical interpretation as points in projective space--which may complicate the cool ribbon graph connection that Kea nicely noted .
[1] Pascual Jordan, John von Neumann, Eugene Wigner, On an algebraic
generalization of the quantum mechanical formalism, Ann. Math. 35
(1934), 29-64.

[2] Tevian Dray, Corinne A. Manogue, The Exceptional Jordan Eigenvalue Problem, http://arxiv.org/abs/math-ph/9910004"

 

[Kea]

Cool stuff, kneemo! I knew I needed to get into those exceptional algebras for some reason!

Certainly, for the non-density eigenmatrices, we lose the nice geometrical interpretation as points in projective space...

Oh, but the usual complex moduli only need ordinary flat ribbons. We can then do twisted ribbons (quaternionic) for the interesting case which includes noncommutativity ... the mathematicians are still working all this out ... and, let's see, for octonionic? Good category theory question!

Hang on a minute ... we only need associahedra up to (roughly) the dimension of the moduli space, so the $\mathbb{P}^{1}$ case only needs simple polygons. Going up higher in dimension should give us a 'basic template' for the quaternionic and octonionic cases. After all, the associativity breaking etc. is exactly what associahedra and permutohedra are all about!

What if we took a triple of ribbons for the quaternionic case, using the idea of hyperkahler geometry? Does this make sense? And then add twists...hmm...

 

[kneemo]

What if we took a triple of ribbons for the quaternionic case, using the idea of hyperkahler geometry? Does this make sense? And then add twists...hmm...

I'm thinking the triple of ribbons arises in all the Jordan algebras of degree 3, that is, all the 3x3 Hermitian Jordan algebra cases. The quaternionic and octonionic cases are likely the ones with twists. I'll sketch a little something.

In the quaternionic 3x3 case, we get 3 vertices (D0-branes) connected by projective lines, producing a kind of triangle in the 8D space HP^2 (with isometry group Sp(3)). Each line is a copy of HP^1 (with isometry group Sp(2)). The resulting "loop" will thus have dimension 4. This is likely dual to the ribbon vertex you mentioned. To make the connection to hyperkähler manifolds, we only need a Riemannian manifold of dimension 4x2=8 or 4x3=12 with holonomy group contained in Sp(2) or Sp(3). And since the holonomy group is a subgroup of the isometry group, it's likely we'll find a hyperkähler manifold somewhere (maybe the 8D HP^2 has holonomy group Sp(2)?).

In the octonionic case, we get a triangle in the space OP^2 (which has isometry group F4, E6(-26) as a collineation group, E7(-25) as a conformal group, and E8(-24) as a quasiconformal group). The triangle is made of 8D OP^1 lines, making the "loop" 8-dimensional. (Perhaps there is a G2 manifold in the loop somewhere.)

HP^2 and OP^2 describe microscopic BPS black holes in the recent string literature (see http://arxiv.org/abs/hep-th/0512296" ). The connection is not explicit however, as stringy people haven't learned about eigenmatrices and their relation to ribbon graphs yet.

 

[Kea]

Cool, cool, cool!

The connection is not explicit however, as stringy people haven't learned about eigenmatrices and their relation to ribbon graphs yet.

Really? Oh, dear.

 

[selfAdjoint]

I am so happy the two of you, a couple of my big heroes, are sharing. May you be blessed with a breakthrough!

 

[Kea]

May you be blessed with a breakthrough!

Dear selfAdjoint

I'm so happy that kneemo is here ... maybe this is a breakthrough! Well, I'm just having fun. Look, if we take kneemo's idea and we think about complex moduli spaces modeled on $\mathbb{P}^{3}$ then, guess what? There are only three 3-dimensional (complex) moduli that matter, namely

$$\mathcal{M}_{1,3} \hspace{2cm} \mathcal{M}_{0,6} \hspace{2cm} \mathcal{M}_{2,0}$$

According to Mulase, these have orbifold Euler characteristics of, respectively (if I've calculated right - I'm not usually up this late)

[tex]- \frac{1}{6} \hspace{2cm} - 6 \hspace{2cm} - \frac{1}{120}[/itex]

so ... let's see ... that might not be the most interesting thing ... must sleep ...

 

[kneemo]

May you be blessed with a breakthrough!

Thanks for the words of support sA.

Dear selfAdjoint

I'm so happy that kneemo is here ... maybe this is a breakthrough! Well, I'm just having fun. Look, if we take kneemo's idea and we think about complex moduli spaces modeled on $\mathbb{P}^{3}$ then, guess what? There are only three 3-dimensional (complex) moduli that matter, namely

$$\mathfrak{M}_{1,3} \hspace{2cm} \mathfrak{M}_{0,6} \hspace{2cm} \mathfrak{M}_{2,0}$$

According to Mulase, these have orbifold Euler characteristics of, respectively (if I've calculated right - I'm not usually up this late)

[tex]- \frac{1}{6} \hspace{2cm} - 6 \hspace{2cm} - \frac{1}{120}[/itex]

Kea, I like the -6 Euler characteristic, and so do string theorists. Following the arguments in [1], I'll explain why.

Begin with the adjoint representation of $$E_8$$ and break it down to $$SU(3)\times E_6$$. Under this subgroup, we have the decomposition:

$$(8,1)\oplus(3,27)\oplus(\overline{3},\overline{27})\oplus(1,78)$$.

The first and last terms are the adjoint representations of $$SU(3)$$ and $$E_6$$, respectively. The two middle terms are tensor products of the three-dimensional representations of $$SU(3)$$ with 27-dimensional representations of $$E_6$$.

The difference of massless modes with distinct chirality in $$(3,27)$$ is given by:

$$n_{27}^L-n_{27}^R=\mathtr{index}(iD_3^{(6)})=\frac{1}{48}(2\pi)^3\int tr_3F\wedge F\wedge F=\frac{-\chi(K)}{2}$$

The Dirac operator and trace are taken in the vector bundle associated to the three-dimensional rep of $$SU(3)$$. The massive modes are recovered from combining modes of opposite chirality, where the absolute value of $$\frac{-\chi(K)}{2}$$ gives the number of particle generations. This is why string theorists are so interested in Calabi-Yau's with Euler characteristic +/-6.

You recovered Euler characteristic -6 (=$$\chi(\mathfrak{M}_{0,6})=(-1)^{6-1}(6-3)!$$) for $$\mathfrak{M}_{0,6}$$ (the moduli space of Riemann surfaces of genus 0 with 6 marked points), which plugs nicely into the index formula and gives three particle generations. [1] Candelas, Horowitz, Strominger, Witten, Vacuum Configurations for Superstrings, Nucl. Phys. B 258 (1985), pp. 46-47.

 

[Kea]

which plugs nicely into the index formula and gives three particle generations...

Well, that will do for a start! Kneemo, did you ever think about the Riemann hypothesis? :shy:

I like the 6-punctured sphere because it looks like a closed string version of a 6-valent ribbon vertex, and a piece of moduli for the latter is a cell decomposition of $\mathbb{R}^{3}$ which is naturally dual to the Stasheff K4 associahedron on 5-leaved trees.

 

[CarlB]

In summary, over the complex field we can see that solving the eigenvalue problem in the state vector formalism leads to the solution of the eigenvalue problem in the operator formalism. Moreover, we see that the set of density eigenmatrices are in one-to-one correspondence with the set of eigenvectors (as well as their corresponding eigenvalues).

When you break a Clifford algebra into primitive idempotents by the spectral decomposition, you end up with the primitive idempotents in this form:

$$\rho_{abc} = (1+aS)(1+bT)(1+bU)/8$$

where a,b and c are +/- 1, and S, T and U are operators that commute with one another and satisfy:

$$S^2 = T^2 = U^2 = 1$$

The 8 is to make rho square to itself. Then S, T, and U are particularly natural operators as the primitive idempotents take quantum numbers of +/- 1 with respect to them.

To go the reverse direction, given three values of {a,b,c}, one gets the eigenmatrix as the above $$\rho_{abc}$$. In other words, the eigenmatrix is trivial to solve for. It is these operators that I use for weak isospin, weak hypercharge and spin.

On the other hand, to get the eigenvectors from the eigenmatrices, one must first put the eigenmatrices into matrix form by choosing a representation of the Clifford algebra. Then any nonzero column of the eigenmatrix is an eigenvalue. But the point is that you can get the eigenmatrices without ever having solved for the eigenvectors, and without ever having chosen a representation. This is how you can avoid unphysical gauge freedoms in the density matrix form.

This begs the question, "what is the physical interpretation of the non-density Hermitian eigenmatrices (with their real eigenvalues)?" Certainly, for the non-density eigenmatrices, we lose the nice geometrical interpretation as points in projective space--which may complicate the cool ribbon graph connection that Kea nicely noted.

I guess you have to bow to tradition and call them statistical mixtures.

Carl

 

[CarlB]

I'm thinking the triple of ribbons arises in all the Jordan algebras of degree 3, that is, all the 3x3 Hermitian Jordan algebra cases. The quaternionic and octonionic cases are likely the ones with twists. I'll sketch a little something.

Starting from this and getting to three generations gets my approval. The 3x3 matrices that show up in my insane paper that calculates the neutrino masses to 7 decimal place accuracy, i.e. http://www.brannenworks.com/MASSES2.pdf have entries taken from the Pauli algebra, which if I recall, fits into your 3x3 Hermitian Jordan algebra case because the Pauli algebra is equivalent to the quaternions.

But quaternions are a little outside my favorite hunting ground, Clifford algebra, and I say this with tentativeness. My preference for Clifford algebra is in their use as Geometric Algebra in that they have immediate geometric interpretations.

Carl

 

[Kea]

Hi CarlB

My preference for Clifford algebra is in their use as Geometric Algebra in that they have immediate geometric interpretations...

...which of course is fantastic! Personally, I'm fond of category theory, which gives us lots of geometry. For instance, the K4 associahedron that I just mentioned is a nice polytope with 6 pentagonal faces (K4 can represent a cocycle condition for parity cubes in a tricategory) which might be paired up using a sort of handedness flip on trees...

 

[Kea]

CarlB

Could you update us on allowable (ranges) values for $\delta_{0}$ and $\delta_{1}$?

Cheers

 

[CarlB]

Could you update us on allowable (ranges) values for $\delta_{0}$ and $\delta_{1}$?

You know, I haven't worried too much about that. It takes so long for experimentalists to measure neutrino oscillations and tau masses that I think that looking at it once per year is enough. So I'll go looking for new data in May 2007.

What particularly interests me about the geometry of Geometric (Clifford) algebra is that the generalized Dirac equation if you squint your eyes a little, looks a bit like a linearized stress / strain equation. Of course, any wave equation of the usual sort can be put into stress strain form. But let me do the example of the "Dirac" equation for CL(2,0) with time as in independent variable (i.e. time not a part of the geometry).

First of all, the canonical basis vectors are $$\hat{x}, \hat{y}$$, and the canonical basis elements (i.e. the basis set for the whole algebra) is

$$1, \hat{x}, \hat{y}, \widehat{xy}$$

the stresses I associate with these four degrees of freedom are as follows:

$$1:$$ This is a scalar. It corresponds to a small region with a localized increase / decrease in density as the sign is +/-.

$$\hat{x}:$$ This is a vector. It corresponds to a small region which has been pulled off in the direction of +x or -x, according to the sign. Same with y.

$$\widehat{xy}:$$ This is a bivector or (with 2 dimensions) a psuedo-scalar. It corresponds to a small region that has been twisted clockwise or counter-clockwise, according to sign.

Now the "Dirac" equation is:

$$(\hat{x}\partial_x + \hat{y}\partial_y) \Psi(x,y,t) = -\partial_t \Psi$$

Now suppose that Psi is initially zero except around a small region where there is a scalar value. For example,
$$\Psi(x,y,0) = \exp(-(x^2 + y^2))$$

You expect that this small region of high density will spread out. You get:

$$\partial_t \Psi(x,y,0) = (\hat{x}\; x + \hat{y}\; y) \Psi(x,y,0)$$

If the notation bothers you, I can rewrite this in a sort of half Pauli half Dirac notation:

$$\partial_t \Psi(x,y,0) = (\gamma_x\; x + \gamma_y\; y) \Psi(x,y,0)$$


where I've left off a factor of two. Thus the localized increase in density at the origin causes a stress corresopnding to movement away from the origin. You can go through the remaining degrees of freedom and they will also "make sense" in terms of these sorts of stresses. For example, if the initial condition is a localized region where you start with $$\widehat{xy}$$, then the result is a twisting force trying to remove the initial condition.

In other words, what I'm trying to do is to think of things in terms of the geometry of real stresses and strains in spacetime, rather than in a geometry that has a mathematical interpretation only.

Carl

 

[Kea]

In other words, what I'm trying to do is to think of things in terms of the geometry of real stresses and strains in spacetime, rather than in a geometry that has a mathematical interpretation only.

Oh, it doesn't only have a mathematical interpretation...

 

[Kea]

M-theory

I hope kneemo comes back soon with everything worked out! I don't have any books or library access here...

And I saw when the Lamb opened one of the seals, and I heard, as it were the noise of thunder, one of the four beasts saying, Come and see. Apocalypse of John (Rev 6.)

Anyway, the point of this folks, is that we can do M-theory.

Imagine a world in which quantum gravity algorithms can factorise primes, or build an AI mind of greater than human power...

 

[kneemo]

In search of M-theory

I hope kneemo comes back soon with everything worked out! I don't have any books or library access here...

Ok, I'm back. I was working on a few things, and PF wasn't working for me till now.

Anyway, the point of this folks, is that we can do M-theory.

Yes, that elusive beast that sees nothing but branes. A good way to approach this beast is by seeking advice from one of its greatest hunters, Ed Witten. Back in Dec. 2003 Witten showed http://arxiv.org/abs/hep-th/0312171" that the perturbative expansion of $$\mathcal{N}=4$$ super Yang-Mills theory is equivalent to the $$D$$-instanton expansion of the topological $$B$$ model with target space $$\mathbb{CP}^{3|4}$$. (selfAdjoint referred me to this paper originally)

Witten considered the scattering amplitudes of gluons in four dimensions. As the gluons are scalar particles, the initial and final states of the particles are completely determined by their momenta. So Witten identified the momentum vector of a gluon with a "bi-spinor" $$P_{a\dot{a}}=p_{\mu}\sigma_{a\dot{a}}^{\mu}$$. He then stated that a lightlike bi-spinor can be written in the form $$P_{a\dot{a}}=\lambda_{a}\tilde{\lambda}_{\dot{a}}$$ so that it is a point in $$\mathbb{CP}^3$$. (In the language of this thread, Witten associated the momentum vector with a primitive idempotent matrix). In an $$n$$ gluon scattering process, the amplitudes thus become functions of the "projective twistors" $$P_i$$, that is, functions defined on the product of $$n$$ copies of the projective twistor space $$\mathbb{CP}^3$$, one for each particle.

Witten found that the minimal nonzero Yang-Mills tree amplitudes vanish unless some genus zero curve of degree one in $$\mathbb{CP}^3$$ contains all $$n$$ points $$P_i$$. In other words, the amplitude vanishes unless the gluon projective twistors are collinear on some $$\mathbb{CP}^1$$ (Riemann sphere) in $$\mathbb{CP}^{3}$$.

The general conjecture is that an $$l$$-loop amplitude with $$p$$ gluons of positive helicity and $$q$$ gluons of negative helicity is supported on a holomorphic curve in $$\mathbb{CP}^3$$, with the degree given by $$d=q-1+l$$ and genus bounded as $$g\leq l$$.

The interpretation given by string theorists is that the curve is a worldsheet of a string, where the amplitudes arise from the coupling of gluons to a string. Witten interpreted the string as a D1-string, while N. Berkovits called it a fundamental string.

Now going back to our earlier discussion about moduli spaces, we saw genus zero Riemann surfaces arise in the discussion of the moduli space $$\mathfrak{M}_{0,6}$$, the set of isomorphism classes of Riemann surfaces of genus 0 with 6 marked points. In light of twistor string theory, one may be able to view $$\mathfrak{M}_{0,6}$$ as the moduli space of 6 gluons on a holomorphic curve. Even if not, the condition of collinearity in twistor space is interesting enough.

Collinearity in Jordan algebras is determined by a cubic form that is related to the determinant. The cubic form is written as $$(X,Y,Z)=tr(X\circ(Y\ast Z))$$ for $$X,Y,Z$$ in the Jordan algebra. (The determinant is just the special case $$(X,X,X)=tr(X\circ(X\ast X))$$.) For the exceptional Jordan algebra $$\mathfrak{h}_3(\mathbb{O})$$, linear transformations that preserve the cubic form (and hence the determinant) comprise the exceptional Lie group $$E_{6(-26)}$$. For this reason, I'm interested if there is a generalization of twistor string theory with twister space $$\mathbb{OP}^2$$. If so, we can study amplitudes that are supported on holomorphic curves in $$\mathbb{OP}^2$$ (i.e., 8-sphere copies of $$\mathbb{OP}^1$$), and transform these holomorphic curves via $$E_{6(-26)}$$ collineations.

Recall that $$E_{6(-26)}$$ is the U-duality group of little black holes in $$N=2$$ homogenous supergravity. So twistor string theory might be the way to compute scattering amplitudes of little black holes with points identified. Also, the cubic form (which measures collinearity) shows up as the M-theory Chern-Simons term of an E8 gauge field http://arxiv.org/abs/hep-th/0312069" . This gives yet another connection to topological quantum field theory.

Hence, in searching for M-theory, it seems one must first identify its various guises, and learn to study them as a whole.[1] E. Witten, Perturbative Gauge Theory As A String Theory In Twistor, Space, http://arxiv.org/abs/hep-th/0312171"

[2] E. Diaconescu, D. Freed, G. Moore, The M-theory 3-form and E8 gauge theory, http://arxiv.org/abs/hep-th/0312069"

 

[Kea]

Hence, in searching for M-theory, it seems one must first identify its various guises, and learn to study them as a whole.

Thank goodness for PF. I wrote a whole lot of comments on CV, but they were deleted! And then Peter Woit told me off for not sticking to the topic. It must be the Chandra excitement, or something...

From a topos theory point of view, there should be a special quantum monad with a structure map that does just this sort of thing with $\mathbb{CP}^{1}$ and $\mathbb{CP}^{3}$. For the classical case of Set one of the special arrows would be

$$\Omega \rightarrow \Omega^{\Omega}$$

where, remember, that $\Omega$ is the 2 point set. Thinking of the adjunction between sets and vector spaces (pre-projectivisation) this would have the analogue $\mathbb{C}^{2} \rightarrow \mathbb{C}^{4}$. So the projective geometry is to be thought of as the canonical model for some good notion of quantum topos.

Also of interest, I think, is the fact that the 3-punctured torus is a Siefert fibre for the complement of the Borromean rings (knotty Venn diagram), just like the 1-punctured torus was a fibre for the complement of the figure 8 knot. That is, the whole complexity of 3D geometry is peeking out from under the seal...

 

[Kea]

Lawvere famously showed that one could think of the positive reals (plus $\infty$) as a symmetric monoidal category. If one enriches in this category instead of in the simple category 2 one gets (generalised) metric spaces...

It is important to realize that an M-theory topos would not just be a 2-topos or some obvious generalisation of the usual topos...it's the quantum topos.

 

[CarlB]

I went back and reread the thread and realized that I didn't describe the quantum numbers of binons very well.

Define the "natural operators" of a Clifford algebra as the canonical basis elements that square to +1. For the Dirac algebra, the natural operators would include spin, handedness, charge, and in general, any Dirac bilinear that squares to +1 rather than -1. Note that if you only consider the bilinears that can be written as products of gamma matrices, (that is, if you ignore things like $$(0.6\gamma_1 +0.8\gamma_2)$$ which some might call a Dirac bilinear and some might not) then half of them will be natural operators and the other half will square to -1.

A primitive idempotents of a Clifford algebra can be written as
$$\rho_{a,b, ...} = (1 + aO_a)/2(1 + bO_b)/2 ...$$
where a, b, ... are each +1 or -1, and O_a, O_b, ... are natural operators. Such a primitive idempotent has quantum number a, b, ...
with respect to the natural operators.

To get binons that will combine to make the quarks and leptons, you have to start with a Clifford algebra that has one more quantum number than the Dirac algebra. You can get this by adding one or two hidden dimensions. This gives you three natural quantum numbers needed to describe a binon.

As mentioned above, the organization is to make a handed elementary fermion out of 6 binons, with the 6 binons organized as three composites of two, and I am calling a composite of two a "snuark".

The first quantum number is related to weak isospin. Each snuark in a quark or lepton carries the same weak isospin number. Consequently, the weak isospin numbers for a snuark are: (+1+1, +1-1, -1+1, -1-1). Since a lepton has three identical snuarks, their quantum numbers are three times this or (+6, 0, 0 -6). To get the usual weak isospin numbers, multiply by 1/12.

The second quantum number is related to electric charge. As before, there are four cases for each snuark, (+1+1, +1-1, -1+1, -1-1) = (+2, 0, 0, -2). To get the usual electric charge, multiply by 1/6. Thus the snuarks carry electric charges of +1/3, 0, or -1/3. If all three snuarks are identical, the particle is a lepton. The quarks are obtained when two of the snuarks are of one type and the other is of another type.

The third quantum number is "anti handedness", the product of handedness and the quantum number that distinguishes particles from anti partilcles. Anti handedness is +1 for right handed particles and left handed anti particles, and -1 for left handed particles and right handed anti particles.

The rule that tells which different snuarks may combine to form a quark is that their weak isospin quantum numbers must all be the same, and their anti handedness quantum numbers must all be the same. Thus the electric charge quantum number is less robust than the other two.

The above rules will generate a set of particles whose quantum numbers match the set of quarks and leptons and their anti particles. But one should note that the rules for making quarks requires them to be a sort of combination of a neutrino and a positron (or an anti neutrino and an electron). In the above, I've ignored the extra generations.

I haven't written the above up but I do have a very obsolete paper that includes a chart of the elementary fermions of the 1st generation according to weak hypercharge and weak isospin. It shows how the quarks appear as intermediate states between electrons and quarks. See page 6:
http://brannenworks.com/a_fer.pdf

By the way, I've been enjoying the mathematical conversations but much of them are over my head. And this is so much a better place than Peter Woit's blog to do this.

Carl

 

[kneemo]

By the way, I've been enjoying the mathematical conversations but much of them are over my head. And this is so much a better place than Peter Woit's blog to do this.

Did you catch part where primitive idempotents came into the twistor string theory picture? Witten assigned the gluon's momentum to a lightlike bi-spinor (4x4 primitive idempotent), which geometrically is a point in the projective space $$\mathbb{CP}^3$$.

The alternative approach would identify the 8-dimensional "twistor space" $$\mathbb{C}^4$$ with its 8-dimensional quaternion analog $$\mathbb{H}^2$$. In the quaternionic representation the Dirac wave function is a 2-component column matrix (bi-quaternion). So lightlike bi-spinors in the quaternionic formulation look like:

$$P=\left(\begin{array}{c} q_1 \\ q_2 \end{array}\right)\left(\begin{array}{cc} \overline{q}_1 & \overline{q}_2 \end{array}\right)=\left(\begin{array}{cc} q_1\overline{q}_1 & q_1\overline{q}_2 \\ q_2\overline{q}_1 & q_2\overline{q}_2 \end{array}\right)$$

for $$||q_1||^2+||q_2||^2=1$$. The quaternionic lightlike bi-spinors are points of the projective twistor space $$\mathbb{HP}^1$$.

Were we to go on and describe quarks with the bi-quaternion representation, a colour triplet would take the form:

$$\mathbf{q}=\left(\begin{array}{c} q_{1}^r+iq_{2}^r \\ q_{1}^g+iq_{2}^g \\ q_{1}^b+iq_{2}^b \end{array}\right)$$

There's more we can do, but I'll stop here because I'm falling asleep.

 

[Kea]

There's more we can do...

No kidding. John B ... Help! We only have until Monday to predict what the Chandra people have seen ... and I work very slowly ...

Recall that in Louis Crane's geometrization of matter proposal for spin foam models, matter arose from the links associated to singularities. Now picture an ordinary manifold point: the surface drawn about the point (vertex) is like a sphere; in 3D it will be a 2-sphere, in 2D it will be a circle. But if the point is not a manifold point, then the surface might be of higher genus. One can also draw links about an edge, and putting vertices and edges together one can draw Feynman diagrams.

In 4D, the vertex link will be a 3-manifold, maybe a link complement and so with a surface boundary. Such a link complement might be a fibration with a punctured torus as a fibre. Maybe non-orientable surfaces should be in there as well (these come from twists in the ribbons mentioned above). Anyway, by thinking about this a bit one could see that the genus 1 (torus) case might have a lot to do with the SM spectrum (a la Connes, for instance).

The question was: what did the higher genus surfaces represent? The guess was that they should have something to do with so-called dark matter. But that was all rather ad hoc. Notice here that instead of 3-manifolds in 4D spin foams corresponding to vertices we now have the twistor version of a point, namely a projective line, and this is only allowed to represent a sphere or a torus because the moduli for higher genus have higher dimension.

Back to the level of twistor space: kneemo has been discussing the 6 punctures on the sphere (gluons). But remember that twistor space can also deal with the moduli $\mathcal{M}_{1,3}$ and $\mathcal{M}_{2,0}$, that is the 3-punctured torus and the genus 2 surface with no punctures.

Recall that cohomology on twistor space can be mapped back to ordinary spacetime via the twistor correspondence, so we expect the appearance of these three moduli to tell us something. By taking only one copy of $\mathbb{CP}^{3}$ we are dealing with single objects, as kneemo has pointed out.

What do the other moduli give us? Think of the Bilson-Thompson preons. The photon was flat ribbons, but the $W^{\pm}$ and $Z^{0}$ had twists, so we only expect the photon to show up in this example.

 

[kneemo]

Dark matter and the Higgs mechanism

No kidding. John B ... Help! We only have until Monday to predict what the Chandra people have seen ... and I work very slowly ...

Yes, I'm extremely curious what Sean and other Chandra people will announce Monday concerning dark matter. The ideas in this thread are related to the idea that dark matter is in the form of extremal black holes http://arxiv.org/abs/astro-ph/0303422" , and their "dark" nature is due to the fact that such black holes are in their ground state and no longer Hawking radiate. Such "extremal" (dark) matter interacts with other extremal (dark) matter via repulsive velocity-dependent forces which results in an acceleration, effectively accounting for dark energy. On the other hand, the extremal (dark) matter interacts with ordinary (non-extremal) matter via the attractive gravitational force. Accelerated expansion occurs when the repulsive dark matter-dark matter force overcomes the attractive gravitational force between dark matter and ordinary matter.

Recall that in Louis Crane's geometrization of matter proposal for spin foam models, matter arose from the links associated to singularities.

In the stringy context, Louis Crane's "links" seem equivalent to the picture of a fundamental string stretched between two branes (D0-branes) whose excitations have masses proportional to their length times their tension.

When the branes are coincident, the fundamental string between the branes is no longer stretched, and effectively has length zero. Since the string excitations have mass = length x tension, we see nothing but massless particles in this case. For N branes, with (complex) orientable fundamental strings, the gauge symmetry is U(N). If we use (real) non-orientable strings, the gauge symmetry is SO(N). Moving the branes away from each other in the transverse space, breaks the gauge symmetry. This is the Higgs mechanism. (see Clifford Johnson's website http://maths.dur.ac.uk/~dma0cvj/Rutherford/page2.html" )

Back to the level of twistor space: kneemo has been discussing the 6 punctures on the sphere (gluons). But remember that twistor space can also deal with the moduli $\mathcal{M}_{1,3}$ and $\mathcal{M}_{2,0}$, that is the 3-punctured torus and the genus 2 surface with no punctures.

What do the other moduli give us?

In Bigelow and Budny's paper http://citeseer.ist.psu.edu/cache/p...erszSzgenus2.pdf/the-mapping-class-group.pdf", the mapping class group of the genus 2 surface is shown to be $$\mathbb{Z}_2$$ extension of the mapping class group of the 6 punctured sphere. This results in Dehn twists in the genus 2 surface being mapped to half Dehn twists in the 6 punctured sphere.

[1] Ramzi R. Khuri, Dark Matter as Dark Energy, http://arxiv.org/abs/astro-ph/0303422"

[2] S. Bigelow, R. Budney, The Mapping Class Group of a Genus Two Surface is Linear, http://citeseer.ist.psu.edu/485445.html".