Lee Smolin's LQG may reproduce the standard model

[Kea]

To use Marcus' expression...aarrgghh! You've worked it all out. It's going to take me a long while to work through this. But I can't wait to see the Chandra pictures!

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...

...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.

Arrrgggh!

 

[CarlB]

Did you catch part where primitive idempotents came into the twistor string theory picture?

I saw it, can't claim to understand it. That issue with bilinearity reminds me of what Koide said about his formula. See the bottom of page 2 "Suggestion (A)" on this paper:

Challenge to the Mystery of the Charged Lepton Mass Formula
http://www.arxiv.org/abs/hep-ph/0506247

I should put a more philosophical explanation for why primitive idempotents should be studied with respect to the elementary particles:

The way that the standard model is put together is by making guesses at symmetries that apply to the elementary particles. If it weren't for relativity, the number of possible symmetries would be huge, but relativity cuts most of them down. But there is still quite a lot of freedom left.

One could look at the primitive idempotent structure of Clifford algebras as just another way of defining a symmetry but it's a little deeper than that. The primitive idempotents arise not from the symmetry of the elementary particles, but instead from the symmetry of a more primitive object, the equation of motion.

If you know the equations of motion, you can derive the symmetries but the reverse is not always true. In any case, the equations of motion are closer to the real world and if we are assuming that the real world is simple, then we should apply that rule to the equations of motion rather than to the symmetries.

As a particular example of this line of reasoning, the equation of motion of Newton's gravitation is exquisitely simple. The stuff that is conserved is not quite so. So since the equations of motion are simpler than the conservation laws, we may have more sucess finding the theory of everything by looking for simple equations of motion rather than looking for simple symmetries.

The basic idea here is that the equation of motion should provide very arbitrary motion, but that as humans, we call a "particle" a movement in the field that is conserved. We start with the usual Dirac equation which we will call the "spinor Dirac equation":

$$(\gamma^0\partial_t + \gamma^1\partial_x + \gamma^2\partial_y + \gamma^3\partial_z)\; \psi(x,y,z,t) = 0.$$

Previously we noted that moving from a spinor representation to a density matrix representation has the advantage of eliminating the unphysical U(1) gauge freedom.

In translating this change (spinor to density matrix) into the Dirac equation we have two choices. The usual method is to stick to pure density matrices, which results in a sort of double sided equation. The resulting equation doesn't increase the number of degrees of freedom beyond the usual density matrix form and so is boring as far as explaining symmetries between different particles.

The more obvious generalization of the Dirac equation to matrix form is seen less often. The idea is to simply replace the spinors with matrices. The resulting equation is the "matrix Dirac equation", which we will distinguish by capitalizing the wave function:

$$(\gamma^0\partial_t + \gamma^1\partial_x + \gamma^2\partial_y + \gamma^3\partial_z)\; \Psi(x,y,z,t) = 0.$$

The above two equations are identical except for the number of degrees of freedom held in the wave function. Instead of 4 complex degrees of freedom, the matrix has 16.

Suppose that we have a solution to matrix Dirac equation. There is an obvious way of extracting four spinor solutions from this one matrix solution, and that is to take each of the four columns of the matrix as a spinor.

A way of doing this splitting is to use the diagonal primitive idempotents, that is, the four matrices that are all zero except for a single 1 on the diagonal. For example, to extract the third spinor, we can use the diagonal projection operator with the 1 in the third position on the diagonal:

$$\Psi = \left(\begin{array}{cccc} \psi_{00}&\psi_{01}&\psi_{02}&\psi_{03}\\ \psi_{10}&\psi_{11}&\psi_{12}&\psi_{13}\\ \psi_{20}&\psi_{21}&\psi_{22}&\psi_{23}\\ \psi_{30}&\psi_{31}&\psi_{32}&\psi_{33}\end{array}\right)$$

$$\psi_2 = \left(\begin{array}{c} \psi_{02}\\\psi_{12}\\\psi_{22}\\\psi_{32}\end{array}\right) = \Psi\;\rho_2 = \Psi\; \left(\begin{array}{cccc}0&0&0&0\\0&0&0&0\\0&0&1&0\\0&0&0&0\end{array}\right)$$

I've used density matrix "rho" notation because I want to point out how the spinors themselves are associated with the projection operators that pull them out of the matrix.

Turning the Dirac equation into a matrix equation gives us a very natural way of defining a multi-particle equation. From this we can derive geometric relationships between the particles. But we only have four spin-1/2 particles and this isn't enough for even one generation of fermions.

We can increase the number of particles held in the matrix by going to a more complicated Clifford algebra. This was done by Trayling, and later Baylis and Trayling:

A Geometric Approach to the Standard Model
http://www.arxiv.org/abs/hep-th/9912231
A geometric basis for the standard-model gauge group
http://www.arxiv.org/abs/hep-th/0103137
also see citations:
http://www.arxiv.org/cits/hep-th/0103137

But as you can see from reading the above, there is not a good explanation for why the particles are different. This question, why are the particles so different is the buggaboo at the heart of analyzing the symmetries of the Dirac equation as an explanation for the standard model. In addition, the above method provides no explanation for the number of generations.

Another problem with this method is that it requies a fairly large number of hidden dimensions (i.e. increases in the number of canonical basis vectors in the mathematical language, or increases in the number of spatial vectors $$\gamma^1, \gamma^2, \gamma^3 ...$$ in the physics language) to get the spinor structure complicated enough to give the needed number of degrees of freedom. The reason for the large number of hidden dimension has to do with the structure of the primitive idempotents a topic I will turn to next.

In order to understand the structure of primitive idempotents (and therefore spinors) in a Clifford algebra, you have to read just a small amount of not very complicated mathematics. My source for this is:

Clifford Algebras and Spinors
London Mathematical Society Lecture Note Series 286
Pertti Lounesto, Cambridge University Press, 1997[?]
pp 226-228
https://www.amazon.com/gp/product/0521005515/?tag=pfamazon01-20

This theory is the basis for all those charts you've all seen that give what matrix groups the various real Clifford algebras are equivalent to. I try to keep close to the physics so rather than having my efforts depend on particular representations (or worse, use division algebras other than the complex numbers as elements of the matrices). You may be able to find a description of the structure of the primitive idempotents, by looking on the web for "Radon Hurwitz Clifford". Also, the primitive idempotents define the ideals of the Clifford algebra so you can look for that too. But the above book is sufficiently useful and easy to read that I think everyone should own a well thumbed copy.

To go further I need to define a "mutually annihilating set" of primitive idempotents. These are a set of primitive idempotents that (a) add up to unity, and (b) give zero when any two different ones are multiplied. The diagonal primitive idempotents are clearly a mutually annihilating set. But it turns out that given any primitive idempotent, one can always find a set of mutually annihilating primitive idempotents which it is a member of.

The Radon Hurwitz numbers tell us why Trayling and Baylis had to add so many hidden dimensions in order to model just the 1st generation of fermions. It (approximately) turns out that if you add two hidden dimensions, you end up doubling the number of primitive idempotents in a "mutually annihilating" set. The reason for the (more or less) doubling requiring two extra dimensions is because each time we add a hidden dimension we double the number of degrees of freedom in $$\Psi$$, but in order to double a spinor we have to double the matrix, and doubling the size of a matrix increases its number of degrees of freedom by four. Thus adding two dimensions makes the matrices big enough to double the size of the spinors.

The way I prefer to organize the project from the other side. Instead of taking the elementary particles and packing them into matrices as if they were commuters on the Tokyo subway, I think it is more elegant to look at what the natural symmetries between the different idempotents of the Dirac equation and to assume that these are preons that make up the elementary particles.

There is another reason for approaching the problem this way. If you spend some time playing around extending the Dirac algebra into a more general Clifford algebra by adding hidden dimensions, you will find that the canonical basis elements (i.e. "Dirac bilinears") that you get are very naturally classified into four different types according to their symmetry with respect to the 3 spatial coordinates: [x, y, z, 1].

The mapping is done by first assigning the canonical basis vectors to the above four categories, and then assigning products according to the very simple and obvious rule:

$$\begin{array}{ccccc} \times&1&x&y&z\\ 1&1&x&y&z\\ x&x&1&z&y\\ y&y&z&1&x\\ z&z&y&x&1\end{array}$$

One can easily verify that the above multiplication table is consistent with Clifford algebra multilication so that it gives a consistent classification of the canonical basis elements (Dirac bilinears) and therefore a consistent classification of the degrees of freedom of Psi.

Given any particular Clifford algebra, a set of mutually annihilating primitive idempotents is defined by a set of what the mathematicians call "commuting roots of unity". I've been calling them "normal operators" but that's lousy notation. In either case, they are canonical basis elements that square to unity and commute. When considered as operators acting on the set of primitive idempotents (i.e. density matrices) that they generate by (either side) multilication, they produce eigenvalues of +/- 1. It is these operators that I assume to be the natural operators for the primitive idempotents. As a concrete example of a set of commuting roots of unity in the Dirac algebra, each of the following sets is such a set (I use (-+++) signature if you don't then multiply everything by i):

$$\begin{array}{rccl} (&i\gamma^0,& i\gamma^1\gamma^2&)\\ (&\gamma^1,& \gamma^0\gamma^3&)\\ (&i\gamma^0\gamma^1\gamma^2\gamma^3,& \gamma^0\gamma^1&)\\ (&0.6\gamma^2+0.8\gamma^3,&\gamma^0\gamma^3&)\end{array}$$

There is a big problem here. As you can see from the above (which really only scratches the surface of the variation in available in commuting roots of unity) any given Clifford algebra has an infinite number of distinct sets of commuting roots of unity. Consequently, we don't have any reason to choose one over another.

In assuming that the elementary particles are made up of preons defined by primitive idempotents, we also have the problem that the set of commuting roots of unity that define one set of primitive idempotents may not be compatible (in the quantum mechanical sense of commuting operators) from the commuting roots of unity that define another preon in the same particle.

Given any two primitive idempotents, there may or may not be a set of commuting roots of unity that define them. It is easy to determine if the two primitive idempotents are compatible. If they multiply to zero (or are identical) then they are compatible, otherwise not.

Gosh, I think this is more than enough about primitive idempotents and Clifford algebras.

Carl

 

[Kea]

Chandra

kneemo

I'm guessing you have library access of some sort. Quick! We need to calculate

$$M_{\textrm{BH}} / M_{\textrm{pl}} \equiv \mu$$

so that we can get an exact generalised uncertainty principle [1]

$$\Delta x \geq \frac{\hbar}{\Delta p} + \mu^{2} L_{\textrm{pl}}^{2} \frac{\Delta p}{\hbar}$$

Now I remember playing around with hyperbolic volumes and knot invariants a few years ago, but I don't have my notes with me! But let me think...quandles?

[1] http://132.236.180.11/abs/astro-ph/0406514 Chen

No, hang on a minute...

Shouldn't $\mu = 1$ ?

 

[CarlB]

generalised uncertainty principle

Interesting stuff. Being hung up on density matrices, I found this article:

Generalized Uncertainty Relations,Fundamental Length and Density Matrix
A.E.Shalyt-Margolin, A.Ya.Tregubovich, 2003
http://arxiv.org/PS_cache/gr-qc/pdf/0207/0207068.pdf

Carl

 

[marcus]

Bananan quoted this from the New Scientist earlier:

http://www.newscientist.com/channel/fundamentals/mg19125645.800

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

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." ...

Here is another sample exerpt from the article that bananan mentioned at the start of the thread:

...an attempt to merge general relativity and quantum mechanics into a single consistent theory.

... Loop quantum gravity then defines space-time as a network of abstract links that connect these volumes of space, rather like nodes linked on an airline route map.

From the start, physicists noticed that these links could wrap around one another to form braid-like structures. Curious as these braids were, however, no one understood their meaning. "We knew about braiding in 1987," says Smolin, "but we didn't know if it corresponded to anything physical."

Enter Sundance Bilson-Thompson, a theoretical particle physicist at the University of Adelaide in South Australia. He knew little about quantum gravity when, in 2004, he began studying an old problem from particle physics. Bilson-Thompson was trying to understand the true nature of what physicists think of as the elementary particles - those with no known sub-components. He was perplexed by the plethora of these particles in the standard model, and began wondering just how elementary they really were. As a first step towards answering this question, he dusted off some models developed in the 1970s that postulated the existence of more fundamental entities called preons.

Just as the nuclei of different elements are built from protons and neutrons, these preon models suggest that electrons, quarks, neutrinos and the like are built from smaller, hypothetical particles that carry electric charge and interact with each other. The models eventually ran into trouble, however, because they predicted that preons would have vastly more energy than the particles they were supposed to be part of. This fatal flaw saw the models abandoned, although not entirely forgotten.

Bilson-Thompson took a different tack. Instead of thinking of preons as particles that join together like Lego bricks, he concentrated on how they interact. After all, what we call a particle's properties are really nothing more than shorthand for the way it interacts with everything around it. Perhaps, he thought, he could work out how preons interact, and from that work out what they are.

To do this, Bilson-Thompson abandoned the idea that preons are point-like particles and theorized that they in fact possesses length and width, like ribbons that could somehow interact by wrapping around each other. He supposed that these ribbons could cross over and under each other to form a braid when three preons come together to make a particle. Individual ribbons can also twist clockwise or anticlockwise along their length. Each twist, he imagined, would endow the preon with a charge equivalent to one-third of the charge on an electron, and the sign of the charge depends on the direction of the twist.

The simplest braid possible in Bilson-Thompson's model looks like a deformed pretzel and corresponds to an electron neutrino (see Graphic). Flip it over in a mirror and you have its antimatter counterpart, the electron anti-neutrino. Add three clockwise twists and you have something that behaves just like an electron; three anticlockwise twists and you have a positron. Bilson-Thompson's model also produces photons and the W and Z bosons, the particles that carry the electromagnetic and weak forces. In fact, these braided ribbons seem to map out the entire zoo of particles in the standard model.

Bilson-Thompson published his work online last year (www.arxiv.org/abs/hep-ph/0503213)[/URL]. Despite its achievements, however, he still didn't know what the preons were. Or what his braids were really made from. "I toyed with the idea of them being micro-wormholes, which wrapped round each other. Or some other extreme distortions in the structure of space-time," he recalls.

It was at this point that Smolin stumbled across Bilson-Thompson's paper. "When we saw this, we got very excited because we had been looking for anything that might explain braiding," says Smolin. Were the two types of braids one and the same? Are particles nothing more than tangled plaits in space-time?
...
...
...

[/QUOTE]

 

[pelastration]

The simplest braid possible in Bilson-Thompson's model looks like a deformed pretzel and corresponds to an electron neutrino (see Graphic).

That is still too complicated. You can do it more easy. Starting from only one giant string/brane of spacetime.

 

[Kea]

Hi Marcus and pelastration (and CarlB and kneemo)

pelastration, you have some nice ideas. I haven't been able to sleep at all for the last few days, trying to work this out, and I haven't gotten far yet! Surely there are a few thousand people out there who could set up an extremal black hole dark matter galactic collision simulation...before Monday.

Another quick summary: M-theory is the understanding of ribbon graph moduli in a twistor String picture (and eventually as a quantum topos). Recall that the Hughston/Hurd approach to mass generation in the 1980s was to take two twistor particles (using sheaf $H^{1}$) and combine them using a Kunneth formula to produce a massive state. This didn't work well, but it indicated that higher non-Abelian cohomology was the key to understanding mass generation.

 

[Kea]

Consequently, I have eight snuarks while they have 3 helons...

So the 16 primitive idempotents are like 4 special ribbons (up and down $H^{0}$, a left twist and a right twist) with $\mathbb{Z}_{4}$ labels. The 4 labels are very reminiscent of the allowable spin labels in the Freedman et al. approach to universal quantum computation, where one looks at the Jones polynomial at a 5th root of unity.

Moreover, an important element in their analysis is the connection between qubits and trouser pants diagrams. Here we note that the trouser pants have a zero dimensional moduli (there is only one of them), and this is the $\mathbb{P}^{1}$ model that appears above.

 

[CarlB]

So the 16 primitive idempotents are like 4 special ribbons

Looks like I've lost the ability to enter LaTex. Test "m_e": $$m_e$$

The first big difference between me and them is that I treat the "mass" interaction (i.e. $$m_e \bar{e_L} e_R + h.c.$$ ) as if it were like any other Lagrangian interaction term, that is, with the $$e_L$$ and $$e_R$$ as particles by themselves that turn into each other with a vertex strength of $$m_e.$$ (I am very simple minded.) The helons, by contrast, are particles with mass.

Now it seems to me that since I've broken the particles down farther, you have to sew them back up to get where Sundance is. I have trouble getting my mind around the meaning of the braids that they use, but I can at least imagine that when you have a pair of preons, one left handed and the other right handed, with a "mass interaction" that converts one to the other, you could draw that as a loop.

But what actually happens with my snuarks is considerably more complicated than that. As soon as one supposes that left handed particles are spontaneously transforming themselves into right handed particles, when you break a fermion into preons you also must take into account the possibility that your various preons won't preserve their identities when they go through a cycle.

That is, if you begin with the three snuarks $$R_L, G_L, B_L$$ which together form a left handed fermion, and R, G, B define the color of the three snuarks, then you might suppose that under the mass interaction, these becomes $$R_R, G_R, B_R$$. But it is also possible that they could become $$G_R, B_R, R_R$$.

According to the rules of QM, you have to sum over all the possible ways of doing this. The old assumption about how potential energy (I haven't shown you any of the arithmetic for this because I found the rules by running computer searches for bound states. Now if my computer programs are right then the fermions fall out automatically but I can hardly expect anyone to read my Java.) keeps the fermion colorless, that is, it prevents two snuarks from both becoming red, for example. That means that there are only three possible things that can happen, which is less complicated than the Sundance braid.

But each of those snuarks has structure, that is, they are each made up of two binons. This means that a mass interaction actually has 6 binons turning into 6 binons, and with that, I think you have enough to define braids.

Where I have difficulty connecting this with Sundance & Lee's theory is that they seem to associate each elementary fermion with only a single braid shape. What I would want is to associate each elementary fermion with a set of possible transformations, and to sum over those sets.

In other words, my method is to make up a set of Feynman diagrams that one must sum to get the overall propagator for the fermion. To do this, you have to assign amplitudes to all those possible interactions, and then sum over them. This seems difficult, but it really isn't so bad.

First, since the binons all amount to Clifford algebra density matrices for which you can trivially generate spinors, you can use the usual $$(1 + \cos(\theta))/2$$ rule for defining the probabilities of making the transformation. In doing this, it turns out that it is best to stick with density matrix theory because otherwise the arbitrary phases will drive you nuts.

Second, in summing the Feynman diagrams, one can take advantage of a cool trick which takes two parts. Since the Feynman diagrams for each transformation is totally trivial in that it consists of an incoming propagator, a vertex, and an out going propagator, and since the propagator is just the projection operator itself (I'm dealing with point particles so I ignore space and time), and since the amplitude is just the trace of the product of the projection operators (i.e. the $$(1+\cos(\theta))/2$$ rule for density matrices), then the complex propagator is simply the product of the density matrices.

The second part of the cool trick is that since particle numbers are conserved in the mass interaction (remember, the binon assumption is that the energies involved are on the order of the Plank mass and so particle number is conserved), one can define the sum of the Feynman diagrams as a matrix (with entries defined as the above propagators, that is, as sequences of projection operators), and the requirement that the sum of the Feynman diagrams be consistent for an elementary particle is simply that the matrix satisfy the usual idempotency relation. Another way of putting this is that the currents must all cancel and the composite particle is therefore stable.

It is in this requirement that the composite particle be stable that one finds that there are always three solutions that can be associated with the three generations.

For the lepton masses, the above summations are done in the point particle assumption, that is, with only finite degrees of freedom as a point particle can have. This gets back to the method of providing mass to an electron from a massless electron propagator by summation. Let "1/p" be the massless electron propagator, "m" be the mass, then one simply sums over the Feynman diagrams which include the possibility of an electron having a trivial interaction with itself of amplitude m:

$$1/p + 1/p m 1/p + 1/p m 1/p m 1/p + ...$$
$$1/p(1 + (m/p) + (m/p)^2 + ...)$$
$$p/(1-m/p) = 1/(p-m)$$

The above is due to Feynman himself, who mentions it as a footnote in a popular book. I would guess that few of the audience understood it, and he might have included it as a subtle joke.

One can take the above and rewrite it with two propagators, p_L and p_R, for the right and left handed states, and then do the resummation. When one does this, one ends up with four results rather than just one. The four results are L->L, L->R, R->L, and R->R. If you assemble these four results into matrix form, you will have derived the Dirac propagator for the massive electron.

All I'm doing is taking the above one step lower, to the preon level, using a Clifford algebra to add a hidden dimension, and making the calculations with the usual product rule for density matrices suitable for spinors.

So, is it possible to write Sundance and Lee's theory in Feynman diagram form? If there is a connection, I think that that is where it will be found.

Carl

By the way, I found this interesting article on the zitterbewegung Dirac problem. The zitterbewegung model of the electron has the electron always traveling at speed c, but back and forth. It's equivalent to the above Feynman diagram approach (that is, the massless particles travel at speed c):
http://www.arxiv.org/abs/physics/0504008

 

[CarlB]

That last post lost its LaTex. Let me try again.

In my version of fermions, the left and right handed fermions convert into one another. Each of these handed fermions are composed of three snuarks, which are an equivalent to ribbons. Each snuark has two binons, and these correspond to the two edges on a ribbon.

In converting from left to right and back to left, the binons get tangled in a manner similar to braids. The tangling can occur in several different ways. As in standard QM, you have to take into account all the ways that the snuarks might get tangled up.

This is a difference with helon braid theory. They associate each elementary particle with a particular braid. I think it is more consistent with the rest of QM if instead the braids indicate transformations which must be summed up in order to include all possible ways of things happening.

Now in order to show how this is done I have to include a little mathematics. Let $$\rho_x, \rho_y, \rho_z$$ be three primitive idempotents (i.e. binons). These are elements of a Clifford algebra. Let M and N be arbitrary elements of the Clifford algebra. Then the following are very general theorems.

I will illustrate them with the following assignments in the Pauli algebra, but again, these are very general theorems about primitive idempotents of Clifford algebras.

$$\rho_z = \left(\begin{array}{cc}1&0\\0&0\end{array}\right)$$
$$\rho_x = \frac{1}{2}\left(\begin{array}{cc}1&1\\1&1\end{array}\right)$$
$$\rho_y = \frac{1}{2}\left(\begin{array}{cc}1&-i\\i&1\end{array}\right)$$

$$M = \left(\begin{array}{cc}a&b\\c&d\end{array}\right)$$
$$N = \left(\begin{array}{cc}e&f\\g&h\end{array}\right)$$

Theorem (1) Products of the form $$\rho_z M \rho_z$$ form a copy of the complex numbers under multiplication and addition. Example:

$$\rho_z M \rho_z + \rho_z N \rho_z = \rho_z (M + N) \rho_z$$
becomes
$$a\left(\begin{array}{cc}1&0\\0&0\end{array}\right) +e\left(\begin{array}{cc}1&0\\0&0\end{array}\right)$$
$$=(a+e)\left(\begin{array}{cc}1&0\\0&0\end{array}\right)$$

This is the basis for how one gets from a pure density matrix representation to a spinor representation, which I think I mentioned before:

Theorem (2) Define the linear superposition (+) sum of two arbitrary primitive idempotents as follows. First, choose a "vacuum" primitive idempotent. We will use $$\rho_z$$ as the vacuum. Then define the linear superposition as follows:

$$\rho_x (+) \rho_y = (\rho_x + \rho_y) \rho_z (\rho_x + \rho_y)$$

Then the above is a "complex" multiple of a primitive idempotent where "complex" means in the context of theorem (1).

For the Pauli algebra example discussed above, we have:

$$\rho_x (+) \rho_y = \frac{1}{2}\left(\begin{array}{cc}2&1-i\\1+i&2\end{array}\right)$$
The array on the right is 1.5 times a primitive idempotent as can be verified by seeing that it squares to 1.5 times itself.

The first theorem shows that an arbitrary product of primitive idempotents that begins and ends with the same primitive idempotent can be thought of as a complex multiple of that same primitive idempotent. The next theorem generalizes this to products of primitive idempotents that begin and end with different primitive idempotents.

Theorem (3) The set of all products of the form $$\rho_x M \rho_z$$ form a copy of the complex numbers. That is, products of these sorts commute, and when you put them into a particular representation of the Clifford algebra, such as the Pauli matrices, you find that you can parameterize them with the complex numbers.

As an example of this fact with the pauli matrices chosen above we have:
$$\rho_x M \rho_z = (a+c) \left(\begin{array}{cc}1&0\\1&0\end{array}\right)$$

By using "x" and "z" in the above theorem I do not mean to imply any particular angular relationship between the primitive idempotents. Maybe a better way of putting all this is simply to state that these are examples of one parameter subgroups of the Clifford algebra. But I found this stuff on my own and so the notation is lousy.

The above gives us methods for reducing products of primitive idempotents. These theorems say that any such product can be reduced to a sort of generic example where only the first and last primitive idempotents are mentioned, multiplied by something that acts just like a complex number. The next step is to apply these to physics.

In Feynman diagrams, one computes a bunch of complex constants from various diagrams and adds them up. In the above, we see the density matrix equivalent of this. The complex numbers of Feynman diagrams become one parameter subgroups defined by products of the beginning and ending primitive idempotents of the Clifford algebra. Thus the complex numbers have been completely geometrized, but in a manner that depends on the particular primitive idempotents.

Carl

 

[CarlB]

Continuing in showing how to get from snuarks to braids...

The problem with using Feynman diagrams for this is that they are usually used for perturbative calculations where there are huge numbers of degrees of freedom, and here we want to analyze how handed particles turn into each other, which is a very deeply bound problem that happens at a single point and so has finite degrees of freedom.

Imagine a 2-slit experiment were there were only two possible paths to the destination. So instead of a 2-slit experiment, it would really be a 2-path experiment. And imagine that we had the figures for how a neutron and how a proton reacted to the experiment. And imagine that we are at such extremely low energies that no particles can be created or destroyed.

The question is once we have got the technology working for analyzing how the neutron and proton interact with this 2 path experiment, how can we use our calculations to change to the nucleus of a helium-4 atom. That is, how can we replace the separate proton and neutron interactions with an interaction that has two protons and two neutrons that happen to be deeply bound?

Furthermore, let us simply the problem by supposing that the binding is so deep that the particles all have identical position eigenstates. If you measure one of the nucleons you've measured them all, at least as far as position goes.

Because of the Pauli exclusion principle we have to suppose that the two protons are not in the same situation initially or finally, and the same for the neutrons. This means that we have two ways each of them could end up swapped.

In addition, we have two paths that the particles could take. Does this mean that there are 4 independent paths for each particle? No, because of energy constraints we have to reject paths that let some of the four particles pass through one path while the rest take the other path.

In the snuark case, we begin with 6 binon positions in the original particle (say a left handed electron). These 6 "left handed" binons become 6 right handed binons. Now we can't say which becomes which, but because of energy limitations, we do know that each of the 6 right handed slots is filled by exactly one of the 6 left handed binons.

In this first transition, there is no twisting or braiding. But when those 6 right handed binons come back to a left handed state, there are various ways that they can do it. If they all come back to their initial condition that's the trivial case of no braiding. Various other cases amount to various other swaps.

But the thing to note here is that if we are only considering one left to right and back pair of transitions, then the amount of twisting that can occur in the binons is very limited.

There is a question here, and that is should we suppose that the two binons making up a snuark can split up? I've been assuming that they can't, but maybe this is wrong. My original assumption was that the snuarks were binons. It was only after I realized that I could only get the quantum numbers to work if I made them into pairs that I expanded it. If they can't break up, then you get a braid structure. But if they can, then you end up having your transitions correspond to permutations of 6 numbers.

Let me do the braid case. Then there are 3 possible permutations, namely the identity (1), and (123) and (132). These three permutations fit into 3x3 matrices:

$$\left(\begin{array}{ccc}(1)&(123)&(132)\\(132)&(1)&(123)\\(123)&(132)&(1)\end{array}\right)$$

In the above matrix, we associate the (i,j) entry with a left to right and back transition that takes the ith snuark and turns it into a jth snuark. First of all, what is the amplitude for this?

If you have spinors, one computes amplitudes by <i|j>. With density matrices, this becomes |i><i| |j><j|. Now suppose we want to compute an amplitude for a sequence of transitions where the particle begins as an i, becomes a k, and then becomes a j. In density matrix form, the amplitude for this more complicated transition (which corresponds to starting left, becoming right, back to left, again to right, and finally back again so that you make two full loops) is just the product:
$$|i><i| |k><k| |j><j| = \rho_i \rho_k \rho_j$$

From here we now see where the utility of the matrix notation is. Suppose that a particle begins in the i state, and goes through two loops, ending up in the j state. What is the total amplitude for the process? As is normal with quantum mechanics, we sum over the amplitudes of the various ways of getting there. The particle begins as a left in the ith state. Then it goes through the loop and returns as one of R, G, or B. Then it goes through the loop and returns as j. So the total amplitude is A:

$$A = <i|R><R|j> + <i|G><G|j> + <i|B><B|j>$$

Now to put this back into our density matrix form, we multiply on the left by |i> and on the right by <j| to get:

$$A (|i><j|) = |i><i| |(R><R|+ |G><G| + |B><B|)|j><j|$$

The LHS of the above is written in spinor notation, while the RHS is written in density matrix notation, which is what we want. Let us write this explicitly and write it as the dot product of two vectors (taking into account the assumption that $$\rho_R^2 = \rho_R$$, etc.):

$$= \rho_i(\rho_R + \rho_G + \rho_B) \rho_j$$
$$= \left(\begin{array}{ccc}\rho_i\rho_R&\rho_i\rho_G&\rho_i\rho_B\end{array}\right) \; \left(\begin{array}{c}\rho_R\rho_j\\\rho_G\rho_j\\\rho_B\rho_j\end{array}\right)$$

This last is just the equation for matrix multiplication if we write our matrices as with each entry being a complex multiple of the corresponding Clifford algebra numbers of this matrix:

$$\left(\begin{array}{ccc}\rho_R&\rho_R\rho_G&\rho_R\rho_B\\ \rho_G\rho_R&\rho_G&\rho_G\rho_B\\ \rho_B\rho_R&\rho_B\rho_G&\rho_B\end{array}\right)$$

In other words, if we make our arrays into arrays of Clifford algebraic elements, then the square of the matrix will correspond to going through two loops of right to left and back instead of just one.

Note that while there is summation of Feynman diagrams going on here, there are no complex numbers in this, it is purely done inside the Clifford algebra. The previous section showed that products of the form |R><R| |G><G| form one-parameter subgroups that act just like the complex numbers.

With the above machinery in place, we can now say how the braid diagrams become particles. One takes an arbitrary 3x3 matrix of complex numbers, converts them into Clifford algebra constants according to the above prescription, and then require that the particle be unchanged after making a transition from left to right and back.

In short, you solve the density matrix equation $$\rho^2 = \rho$$. But in doing this, you have to remember that the entries of the "complex" matrix, are not really complex numbers, but instead are comlex multiples of Clifford algebra numbers. So the products are not exactly what you're used to getting when you multiply complex numbers. And when you solve this equation, you will find that there are three solutions, one for each generation.

It turns out that these products of Clifford algebra idempotents are not totally trivial. There is a beautiful theorem that tells you what you get when you multiply these sorts of things. I don't know who to attribute it to because I found it myself (by computer simulation) but it's obvious enough that someone out there has undoubtedly proved it.

Carl

 

[CarlB]

There is a beautiful theorem that tells you what you get when you multiply these sorts of things.

I'm going to show this theorem in the Pauli algebra. Now part of the reason for doing this is that the Pauli algebra is very simple. But it turns out that the generalization is trivial.

Let $$\rho_R, \rho_G, \rho_B$$ be three primitive idempotents (i.e. pure density matrices or projection operators) of the Pauli algebra. Then we've already shown that the product takes the following form:

$$\rho_R \;\rho_G \;\rho_B = \sqrt{P} \exp( i S /2)\;\rho_R\;\rho_B$$

where P and S are real numbers. Then

$$P = (1 + \cos(RG))(+\cos(GB))/4$$

that is, the usual spinor amplitude, and S is equal to the area of the oriented spherical triangle defined by the spin vectors of R, G, and B. As a corollary, products of the form $$\rho_R \;\rho_G \;\rho_R$$ are always real multiples of $$\rho_R$$.

As an example of this, let R=z, G=y and B=x in the usual Pauli matrices. Then the spherical triangle is an octant and its area is therefore 4pi/8 so S = pi/2 and the complex phase of the product taken is S/2 = pi/4. This is easily checked:

$$\left(\begin{array}{cc}1&0\\0&0\end{array}\right) \frac{1}{2}\left(\begin{array}{cc}1&-i\\i&1\end{array}\right) \frac{1}{2}\left(\begin{array}{cc}1&1\\1&1\end{array}\right)$$
$$=\frac{1-i}{4}\left(\begin{array}{cc}1&1\0&0\end{array}\right)$$
$$=\sqrt{\frac{1}{2}}\;\exp(-i\pi/4)\;\frac{1}{2}\left(\begin{array}{cc}1&1\0&0\end{array}\right)$$
$$=\sqrt{\frac{1}{2}}\;\exp(-i\pi/4)\;\rho_z\rho_x$$
and the orientation turns out to be negative.

Uh oh, looks like I'd better get some work done here.

Carl

 

[Dcase]

Since the expression
exp(((-i) * pi) / 4) = 0.707106781 - 0.707106781 i
merely corresponds to a specific quadrant of the Euler identity circle, I do not understand why you are dismayed about negative orientation?

The expression
exp((i * (-pi)) / 4) = 0.707106781 - 0.707106781 i
results in the same quadrant.

The second expression may be more likely due to a counter-clockwise rotation while the first expression may be more likely due to a clockwise rotation.

 

[CarlB]

Since the expression
exp(((-i) * pi) / 4) = 0.707106781 - 0.707106781 i
merely corresponds to a specific quadrant of the Euler identity circle, I do not understand why you are dismayed about negative orientation?

You are quite right that it's just an orientation issue. And, I wasn't dismayed, just had to get back to work. Talking about physics is "play". "Work" is writing up the air quality permit applications for a fuel ethanol plant. It's supposed to be done today.

Carl

 

[CarlB]

Okay, the ethanol plant air quality permit has been sent in, the state engineer has not yet found any big problems with it, and I'm bored and in the mood to write some more about physics.

An interpretation of primitive idempotents known as "Schwinger's Measurement Algebra" that is well described by Julian Schwinger in his book "Quantum Kinematics and Dynamics" is as Stern-Gerlach filters. Such a filter allows only one particular type of particle to pass. A filter is an object that has geometric properties (for example, a direction in which the magnetic field is inhomogeneous). In associating the particle with the filter that picks it out, we have a natural geometric designation of the particle.

In assuming a preon model based on "deeply bound" primitive idempotents, we have to decide on some way of binding them together; that is, we have to define a potential energy function. We are using the multiplication in the Clifford algebra to model what happens when a particle of one type is measured in some other way (as in consecutive Stern-Gerlach filters). In that model, summation corresponds to making a Stern-Gerlach experiment that allows the passage of more general particles.

For example, one could arrange for one Stern-Gerlach filter that only passed electrons with spin +1/2 in the z direction and another Stern-Gerlach filter that only passed neutrinos with spin +1/2 in the x direction. Both of these are represented by primitive idempotents. The "sum" of these two primitive idempotents would no longer be primitive, but in this case it would still be an idempotent.

Note that in general, the sum of two primitive idempotents is not necessarily an idempotent. If the two primitive idempotents annihilate each other, that is, if they multiply to zero, that is, if no particle can traverse the two filters consecutively, then their sum is still an idempotent (or projection operator).

The simplest example of two such primitive idempotents adding together to produce an idempotent would be two spin-1/2 Pauli algebra projection operators oriented in opposite directions, for example, spin +/- 1/2 in the x direction. When we sum up these two primitive idempotents we get unity:

$$\begin{array}{rcl} \rho_{+x} &=& (1 + \sigma_x)/2\\ \rho_{-x} &=& (1 - \sigma_x)/2\\ \rho_{+x} + \rho_{-x} &=& 1 \end{array}$$

In Schwinger's measurement algebra, 1 is a free beam (while 0 is a complete beam stop). Thus the Stern-Gerlach experiment that corresponds to the sum of the two experiments is a much simpler experiment to set up. Instead of having inhomogeneous magnetic fields in the +x and -x directions, we have no need of magnetic fields at all.

As a more general example (that requires more general Clifford algebra), let Q be the operator that measures the charge of a particle and let us consider a set of particles that includes both positively and negatively charged particles of some sort such as electrons and positrons. Then the projection operators that pick out the + and - charged particles are:

$$\begin{array}{rcl} \rho_+Q &=& (1+Q)/2\\ \rho_-Q &=& (1-Q)/2 \end{array}$$

Again, the sum of these two projection operators is unity. When we put a positively charged particle near a negatively charged one, we expect them to bind together. It is therefore natural to consider "1" as a bound state, while the primitive idempotents are unbound states.

The natural value for the potential energy of unbound states is the Planck energy, while the observed elementary particles have energies much much smaller. Therefore we will assume that the potential energy of a collection of primitive idempotents is going to be zero if the sum of their Clifford algebra numbers adds to unity (I used to think "zero", but I now think this is better), and to be of the order of the Planck mass otherwise.

We will therefore write the potential energy as a function of the sum of the primitive idempotents making up the particle, and the function will be zero if the sum is unity, and otherwise greater. We need the potential energy function to be zero when the sum "A" of the primitive idempotents is unity, and otherwise positive:

$$\begin{array}{rcl} V(1) &=& 0\\ V(A \neq 1) & > & 0 \end{array}$$

The natural way to accomplish this is to write A as a sum of complex multiples of Dirac bilinears (or the generalization of Dirac bilinears to the Clifford algebra), and then to define the potential energy as the sum of the squares of the coefficients, ignoring the scalar coefficient. For example:

$$\begin{array}{rcl} V( (1+\sigma_x)/2) &=& V(1/2 + \sigma_x/2)\\ &=& V(\sigma_x/2)\\ &=& (1/2)^2\\ &=& 1/4. \end{array}$$

A more general example: consider a set of spin-1/2 charged particles that come in +1 and -1 charges. This is the sort of thing you could get with your Clifford algebra chosen to be the Dirac algebra. The potential energy of a primitive idempotent that is charge +1 and spin +1/2 in the y direction is:

$$\begin{array}{rcl} V( (1+\sigma_y)(1+Q)/4) &=& V(1/4 + \sigma_y/4 + Q/4 + \sigma_y Q/4)\\ &=& V(\sigma_y/4 + Q/4 + \sigma_y Q/4) \\ &=& (1/4)^2 + (1/4)^2 + (1/4)^2\\ &=& 3/16. \end{array}$$

As an extemporaneous (i.e. possibly bad) side note giving the potential energy of primitive idempotents in general, one notes that in the above (and in general for primitive idempotents of Clifford algebras), $$Q$$ and $$\sigma_y$$ each square to unity, and they commute. In addition, if the above is primitive, then you can't find another operator that squares to unity and commutes with these, other than obvious stuff like $$\sigma_y Q$$ or 1 (and that give quantum numbers that are already determined by $$Q, \sigma_y$$). In other words, $$Q, \sigma_y$$ are a "complete set of commuting roots of unity". When looking for complete sets of commuting roots of unity, one finds that each Clifford algebra gives you sets of a particular size, say N. The Dirac algebra gives sets of size 2 while the Pauli algebra gives sets of size 1. Typically, but not always, one has to add two dimensions to increase N by one. Counting out the terms in the above calculations, one finds that the potential energy of a primitive idempotent in a Clifford algebra which has N elements in its complete sets of commuting roots of unity, is equal to $$V_N = (2^N - 1)2^{-2N}.$$

In my next post, I will hopefully show how these simple rules reproduce the structure of the quarks and leptons. I say "hopefully" because writing the above changed some of the way I look at this theory. I like the interpretation V(1) = 0 more than the way I was doing it before, and intuitively I'm guessing that the structure will work out the same way. It seems to me that this way I will get a cleaner interpretation of the Pauli exclusion principle. Meanwhile, tomorrow I meet with the engineers who will design the concrete load structures for our plant.

Carl

 

[Dcase]

The concept of V(1)=0 for idempotents is intriging.

The MathWorld page on Idempotent appears to suggest that opeartors such as minus [-] or absolute value [| |] could also seve to satisfy x^2=x
suggesting that -x may be an idempotent or some type of relative?
http://mathworld.wolfram.com/Idempotent.html

 

[CarlB]

The structure of idempotents of Clifford algebras is related to the elements that square to unity. Given the set of real functions with multiplication given by product of composition, an element that squares to unity (i.e. f(x)=x) is negation, that is, the function f(x) = - x. From that function I guess the generated idempotents would be f(x) = (x-x)/2 = 0 and f(x) = (x+x)/2 = x. In other words, the Clifford algebraic method of defining idempotents from roots of unity doesn't work so well here. But all this is kind of off topic. I think that there is way too much math that is done for the pleasure of finding unimportant relationships between things of not much importance. I guess I'll continue to the next step in the derivation of the structure of the elementary fermions.

With the above definition of potential energy, it is apparent that a simple solution to it is to just have a complete set of annihilating primitive idempotents. Such a complete set adds to unity, so would have a potential energy of zero. But this sort of solution is not so simple. It is made up of a collection of individual primitive idempotents, and these primitive idempotents cannot stay together becaue they will move in different directions. Such a solution would minimize energy, but only for a moment, it would fly apart. The problem is that the particle has to reverse its direction, and using the spinor probability function $$P = (1+\cos(\theta))/2$$, the probability of this is zero.

The simplest wave equation in a Clifford algebra is the "generalized massless Dirac" equation. The Dirac equation includes mass, and this makes it somewhat more complicated than what will be described here. Eliminating mass, as is appropriate for a theory that treats mass as an interaction between particles gives the simple equation:

$$\nabla \Psi = 0$$

The left hand side is the usual Dirac operator, generalized to a Clifford algebra. Rather than writing it in Dirac's gamma notation, I will write it in notation that brings out the geometric quality of the equation:

$$(\hat{x}\partial_x + \hat{y}\partial_y + \hat{z}\partial_z + \hat{s}\partial_s + \hat{t}\partial_t) \Psi = 0$$

In the above, s is a coordinate corresponding to a hidden dimension. This single hidden dimension (used like Kaluza-Klein in some ways) is needed to get the elementary particles as will be seen later. For now, treat it as just like the other spatial dimensions. Since we're working on a mass theory for point particles (i.e. a finite dimensional system) we don't need to treat s as special in any way. The various hatted objects are the canonical basis vectors for the Clifford algebra. In the Dirac notation, these are $$\gamma^\mu$$. They square to the signature, in the order above this is [++++-], and they anticommute with each other.

If $$\Psi$$ has no space-time dependence on s, then you can eliminate that element and what you have would look suspiciously like the usual Dirac equation. But it would not be so, for two reasons. First, even though Psi has no dependence on the coordinate s, its wave function still carries the extra degrees of freedom associated with the 2x larger Clifford algebra that s implies. Second, Psi is taken to be a member of the Clifford algebra, rather than a spinor. This is similar to the "square spinor" discussed by Lounesto in his classic text on spinors, but without the restriction that Psi be the square of a spinor.

Upgrading Psi to be a Clifford algebra element instead of a spinor means that there are a lot more degrees of freedom available in Psi. In terms of relating the above "massless generalized Dirac" equation to the usual Dirac equation, this means that in a single equation we are simultaneously writing the Dirac equations for a bunch of different spin-1/2 particles. For an example of the literature where this is done (but with spinors instead of the density matrix formalism that I follow) see Trayling and Baylis, and citations: http://arxiv.org/abs/hep-th/0103137
Since Trayling and Baylis model the fermions directly, rather than use a preon model, they require a much larger number of hidden dimensions, and of course they do not get the three generations.

The classic approach to putting the Dirac equation into geometric form is the famouse "Dirac-Hestenes" equation. This formula looks very similar to the above, except that Psi is restricted to an even subalgebra. You can learn more about this by searching for "Dirac-Hestenes" on arXiv. My favorite article on this is Bayls' critque of another paper on the subject: http://www.arxiv.org/abs/quant-ph/0202060 .

Psi is composed of multiple copies of the Dirac equation. If we are given a solution for the massless generalized Dirac equation and we want to split it into a set of Dirac equations, (note that there are an infinite number of ways of doing this), we can use a complete set of primitive idempotents.

In adding one hidden dimension, the number of primitive idempotents in a complete set has doubled, so instead of the four that one would have for the Dirac algebra, we will have eight. I will write these as $$\rho_{---}, \rho_{--+}, ... \rho_{+++}.$$ To make sure that the notation is familiar to the reader, let me write down the natural primitive idempotents (i.e. the diagonalized PIs) for some particular representation of the Dirac algebra.

A quick google search found gamma matrix definitions at http://www.answers.com/topic/gamma-matrices so I look for two diagonal matrices, $$\mu_1, \mu_2$$ that (a) square to unity, and (b) commute, and (c) cannot be written in terms of one another. These are the "commuting roots of unity" that are so important for Clifford algebra primitive idempotents. Since (b) and (c) are trivial for diagonal 4x4 matrices, the solution is:

$$\mu_1 = i \hat{t} = \gamma^0 = \left(\begin{array}{cccc} 1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{array}\right)$$

$$\mu_2 = i \hat{x} \hat{y} = i\gamma^1\gamma^2 = \left(\begin{array}{cccc} 1&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&-1\end{array}\right)$$

In the above, I've also written in the geometric description of the roots of unity in hat (Clifford algebra) form which I prefer to use. We can now write the four primitive idempotents in terms of the above matrices as:

$$\rho_{--} = (1-\mu_2)(1-\mu_1)/4$$
$$\rho_{-+} = (1-\mu_2)(1+\mu_1)/4$$
$$\rho_{+-} = (1+\mu_2)(1-\mu_1)/4$$
$$\rho_{++} = (1+\mu_2)(1+\mu_1)/4$$

or
$$\rho_{\pm\pm} = (1\pm\mu_2)(1\pm\mu_1)/4$$

Adding the hidden dimension s doubles the number of primitive idempotents (i.e. adds one more commuting root of unity) so we can write them as:

$$\rho_{\pm\pm\pm} = (1\pm\mu_3)(1\pm\mu_2)(1\pm\mu_1)/8$$

where $$\mu_3$$ is a root of unity that commutes with the other two and isn't in the group they generate. (In the above example, we could have $$\mu_3 = i\hat{z}\hat{s}$$ or $$\mu_3 = \hat{z}\hat{s}\hat{t}$$ but not $$\mu_3 = \hat{x}\hat{y}\hat{t}$$ as this is in the group generated by the other two. )

With all that preparation out of the way, one splits the generalized Dirac equation into Dirac equations by right multiplying by the primitive idempotents. But there are an infinite number of ways of choosing those primitive idempotents.

Plane Wave Solutions and Feynman's Checkerboard

The propagators of our theory have to preserve particle identity. In the usual model, with a propagator devoted to each particle type, this is not a problem, but with a single Dirac equation that handles multiple particls, it is important that they not leak into each other. Accordingly, we now solve for the primitive idempotents that correspond to plane wave solutions to the generalized Dirac equation.

Let Psi be a function of z and t that satisfies the generalized massless Dirac equation as follows:

$$\Psi(x,y,z,s,t) = \Psi_0\; \sin(z-ct).$$

where $$\Psi_0$$ is a Clifford algebra constant and c is a real constant (to be interpreted as the speed of the wave). Applying this to the massless generalized Dirac equation gives:

$$(\hat{z} -c\hat{t})\;\Psi_0 = 0.$$

Multiplying the above equation on the right by $$\hat{t}$$, (an operation which is reversible and so does not change the equation's solutions) and rearranging terms by using anticommutation gives:

$$(-\widehat{zt} +c)\;\Psi_0 = 0.$$

which has a solution only for c=1:

$$\Psi_0 = (1 + \widehat{zt})/2.$$

where the overall sign and the 2 have been chosen to put the answer into idempotent form. To translate this back into the language of the representation of the Dirac algebra linked above, we have:

$$\Psi_0 = \frac{1}{2}\left(\begin{array}{cccc} 1&0&-1&0\\ 0&1&0&1\\ -1&0&1&0\\ 0&1&0&1\end{array}\right)$$

Note that the above matrix is similar to the chirality matrices, $$(1\pm\gamma^5)/2.$$ If all the off diagonal signs had been +, it would be the projection operator for right handed states, and if all the off diagonal signs had been -, it would be the projection operator for left handed states.

Breaking the above matrix into spinors (i.e. column vectors and ignoring repeats with different arbitrary complex phases), we see that there are two, (1,0,-1,0) and (0,1,0,1), transposed. In this representation of the Dirac algebra, these spinors are the right handed particle and the left handed antiparticle traveling in the +z direction. (I have about a 50% chance of reversing my particle and antiparticle in that last sentence, but it doesn't matter.) The fact that the above projection operator picks out different handed states depending on particles or antiparticles is why I called this quantum number "anti handedness" in post #32 of this thread. In some of my writings, I use "L" and "R" to refer to anti handedness and this can be confusing to people used to dealing with handedness.

When one converts the Dirac equation into an equation appropriate to density matrices, one finds that the density matrix has to be hit on both sides by the operator. I.e. $$i\partial_t \rho = H \rho -\rho H.$$ For this reason, any density matrix plane wave +z traveling wave solution of the density matrix equivalent to the generalized Dirac equation will have to have the above as an idempotent factor.

We therefore choose $$\widehat{zt}$$ as one of our three commuting root of unity. In order to match my notes, in which all this was derived sort of backwards, I put:

$$\mu_3 = \widehat{zt}$$

For choosing the other two commuting roots of unity, first note that the elements of the Clifford algebra that commute with the above consist of any products of $$\hat{x},\hat{y},\hat{s},\widehat{zt}.$$

The above root of unity defines two (non primitive) idempotents, $$(1\pm \widehat{zt})/2.$$ The + sign gives stuff that is traveling in the +z direction, while the - sign gives stuff traveling in the -z direction. This is reminiscent of the Feynman checkerboard (or chessboard) model of the one-dimensional Dirac propagator: "Feynman, R. P. and A. R. Hibbs. Quantum Mechanics and Path Integrals. New York: McGraw-Hill, 1965". To learn more about this, do a google search for Feynman+checkerboard or Feynman+chess+board or see Tony Smith:
http://www.valdostamuseum.org/hamsmith/Sets2Quarks7.html

Since the Feynman checkerboard model of a (one dimensional!) elementary particle consists of something that alternately goes in two different directions, and since we are looking here at splitting the elementary fermions into right handed and left handed parts (which must go in opposite directions in order to make a particle with spin defined in that direction), our choice of this as a commuting root of unity is very natural. Naively, the spread of the wave function in z is caused by the points at which the particle switches from being left handed to right handed being non deterministic.

But Feynman checkerboard is only a one-dimensional model of a quantum particle. Getting away from the plane wave solutions, actual electrons are 3-dimensional particles (that is, their wave functions spread in all three dimensions). This suggests that in modeling the electron we are going to have to also take into account idempotents for the other two dimensions, that is, $$(1\pm\hat{xt})/2$$ and $$(1\pm\hat{yt})/2.$$ The name that we will use to call the stuff that shares one of these idempotents is "snuark". From this the reader may have realized that snuarks do not obey the Heisenberg uncertainty principle. Instead, only the bound combination of them can do so, and to get uncertainty in all directions requires three snuarks for the left handed particle and three more for the right handed particle. A single pair of snuarks would satisfy the Heisenburg uncertainty principle in one dimension only, and would be classical in the other two. But since the rules we give here allow snuarks to convert from one type to another, such a monstrosity would not be stable.

The necessity of having three snuarks per handed fermion thus comes from the following reasons: (a) to allow the Heisenburg uncertainty principle to apply in all three directions rather than just one, (b) to allow for quarks to be intermediate to the leptons in quantum numbers, that is, to make the quarks and leptons from bound states of three preons, (c) to allow a minimization of the potential energy (to be discussed in next post I guess), and (d) to use up enough degrees of freedom in the Clifford algebra to explain why only two electrons (spin up and down) can fit into a given position (the Pauli exclusion principle for spin-1/2, to be discussed later). In some of this reasoning, there are hints of a sort of classical behavior among the preons.

Carl

 

[Kea]

Hi CarlB

I'm thinking of reviving this thread again soon. What do you think?

 

[CarlB]

I felt kind of bad about hijacking the thread. Do you mean to rejack it back to the original topic? By the way, I put up a new copy of my book, but it's not worth looking at. The major change is that I added the chapter divisions, with poetry.

 

[Sauron]

I have a question about the original topic.

It basically states that some configurations of pure gravity LQG could behave as particles.

On the other hand we have the work of baratin and Freidel (whose 4-d paper has been realized these days) which is discused in another thread and which states that any "ordinary" point particle feyman diagram can be expresed as an amplitud in a pure LQG spin-foam model.

These two results seem to me very similars in the conclusions, but aparently very diferent in the way they arrive to them. That looks at least I see it so, an invitation to try to search a relation betwen the two aproachs. Anyone is doint it? Or I am missing something and what I wonder makes no sense?

 

[marcus]

...
These two results seem to me very similars in the conclusions, but aparently very diferent in the way they arrive to them. That looks at least I see it so, an invitation to try to search a relation betwen the two aproachs. Anyone is doint it? Or I am missing something and what I wonder makes no sense?

the two approaches differ quite a bit, as you remarked, and they are both unfinished or incomplete. I think it would be difficult to match them up at this point.

Smolin et al's paper IIRC only involves spin NETWORKS, and there remains the problem of specifying how the networks evolve.

The paper by Baratin and Freidel investigates a spinFOAM model, and proves a result about the zero-gravity limit.

In principle the B&F model should have two parameters, h-bar, and Newton G, which one can take to zero and check the behavior.
I believe that they have done half the work: they have shown that if you make G -> 0 you get something recognizable as closed Feynman diagrams. Maybe this result can be strengthened but it is already a positive indication.

But they also now must show that if you do not make G -> 0, but instead make h-bar -> 0, you get something like classical gravity.

You know that Rovelli et al recently put out some work about spinFOAM where they derived gravitons. This is suggestive, but it is no guarantee that B & F, with their spinfoam model, can get classical gravity in the limit of small h-bar.

Speaking not as an expert, and hoping others will correct me if I am mistaken, I would say that B & F paper is very different from Smolin et al in the sense that B & F is closer to having dynamics.

It seems to me that the virtue of the Smolin et al paper is that it gives more or less the right types of particles in a conceptually simple QG scheme. The Smolin picture, I think, is of a spinnetwork evolving by LOCAL MOVES which reconnect nearby vertices in different ways or which insert and remove vertices. To have a structured dynamics, they still must assign amplitudes to these moves, time would presumably be some index of the rate that these local moves are occurring. It is an elegant and evocative picture of the universe which has seeds of ideas for a theory more fundamental than quantum mechanics (see Smolin's most recent preprint) and better adapted to handling cosmology. But I think it is not at a comparable stage of definition so that one could connect it with Freidel et al's work.

 

[bananan]

Hi CarlB

I'm thinking of reviving this thread again soon. What do you think?

Bilson updated his article Oct 26, 2006

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

A topological model of composite preons

 

[bananan]

Hi CarlB

I'm thinking of reviving this thread again soon. What do you think?

As Yershov correctly noted in a private email to me,
I personally wrote (or substantially added) this for wikipedia preon
as " 65.26.44.75 " and "216.16.237.110 "

Preon research is motivated by the desire to explain already existing facts (postdiction), which include:

To reduce the large number of particles, many that differ only in charge, to a smaller number of more fundamental particles. For example, the electron and positron are identical except for charge, and preon research is motivated by explaining that electrons and positrons are composed of similar preons with the relevant difference accounting for charge. The hope is to reproduce the reductionist strategy that has worked for the periodic table of elements.
The second and third generation fermions are supposedly fundamental, yet they have have higher masses than those of the first generation, and the quarks are unstable and decay into their first generation counterparts. Historically, the instability and radiactivity of some chemical elements were explained in terms of isotopes. By analogy this suggests a more fundamental structure for at least some fermions. [[1]]
To unify particle physics with gravity, for example, Bilson-Thompson model with loop quantum gravity.
To give prediction for parameters that are otherwise unexplained by the Standard Model, such as particle masses and charges and color, and reduce the number of experimental input parameters required by the standard model.
To provide reasons for the very large differences in energy-masses observed in supposedly fundamental particles, from the electron neutrino to the top quark.
To explain the number of generations of fermions.
To provide alternative explanations for the electro-weak symmetry breaking without invoking a Higgs field, which in turn possibly needs a supersymmetry to correct the theoretical problems involved with the Higgs field. Supersymmetry itself has theoretical problems.
To explain the features of particle physics without the need for higher dimensions, supersymmetry, higgs field, or string theory.
To account for neutrino oscillation and mass.
The desire to make new nontrivial predictions, for example, to provide possible cold dark matter candidates, or to predict that the LHC will not observe a Higgs boson or superpartners.
The desire to reproduce only observed particles, and to prevent prediction within its framework for non-observed particles (which is a theoretical problem with supersymmetry).
The experimental falsification of certain grand unified theories of particle physics as the result of not observing proton decay may suggest that the grand unification scenario, which string theory is predicated on, and supersymmetry, may be false, and different solutions and thinking will be required for the progress of particle physics.
Were string theory successful in its original objectives, preon theory research would not be necessary. String theory was supposed to account for the above issues in terms of string dynamics. The different particles of the standard model were accounted for as different frequencies (tension) of a Planck-scale string, particle dynamics were explained in terms of the worldsheet diagrams, (the string theory equivalent of Feynman diagrams) and the three generations of fermions were explained in terms of strings "wrapping around" specific configuration of higher-dimensional moduli. The continuing failure of string theory to achieve the above objectives as a theory of particle physics Relevant literature include: Peter Woit Not Even Wrong, or Lee Smolin's The Trouble with Physics, or Daniel Friedan's "String theory is a complete scientific failure". Andrew Oh-Willeke states "as string theory develops more doubters, I think [preon theory] will be an obvious direction for non-string theory investigators and theorists."

The vast bulk of recent theoretical research into the particle zoo has been string theory. It was thought string theory has completely supplanted preon research, and that one dimensional supersymmetric strings can reproduce all the particles of the standard model, and their superpartners, the MSSM, their properties, color, charge, parity, chirality, and energy-masses, obviating any need for preon research. To date, string theory has been unable to reproduce the standard model.

A search through Spires and Arxiv, show that approximately over 30, 000 papers in string theory or supersymmetry since 1982, with several hundred new papers being published every month. In comparison, in 2006, since 2003, there have been about a dozen papers in preon theory listed as such in arxiv.

String theories continuing failure to reproduce the particle spectrum of the standard model has given some life for preon theories, and there have been recent papers on preon theory. As of 2006, Yershov, Fredriksson, and Bilson-Thompson have published papers in Preon theory within the past 5 years: a 2003 paper by Fredriksson [4], and a 2005 paper by Bilson-Thompson [5].

When the term "preon" was coined, it was primarily to explain the two families of spin 1/2 fermions: leptons and quarks. More recent preon models also account for spin-1 bosons, and are still called "preons". The term "preon" is the term of choice for Bilson-Thompson, Yershov, and Fredrickson, although they expand the meaning of the term, in addition to accounting for spin 1/2 fermions of leptons and quarks, which the term was used in its early history, the latter theories also accounts for spin-1 bosons.

Yershov's model is patterned after the idea naked singularities in general relativity, and closely resembles geon from John Archibald Wheeler research program into Geometrodynamics. Electron structure in Yershov's theory was further elaborated on in 2006 [6]. 2003 papers by Yershov [7] [8] are notable for being some of the only papers in the field to use the Preon model as a basis for providing specific numerical values from first principles for the masses of the particles described in the Standard Model. Yershov's model does not predict the mass of the Higgs Boson, and does not need the Higgs boson, and predicts it will not be found. Yershov's model deals with the mass paradox by prosposing a huge binding energy for his preons, which acts as a source of mass-energy, as through mass defect. To get around the mass paradox, Yershov's model proposes a new force that is 10^5 stronger than the strong nuclear force, that binds his preons together.

Fredriksson preon theory does not need the Higgs boson, and explains the electro-weak breaking as the rearrangement of preons, rather a Higgs-mediated field. In fact, Fredriksson preon model predicts that the Higgs boson does not exist. In the above cited paper, Fredricksson acknowledges the mass paradox represents a problem in his accounting for neutrino mass, however, he proposes a specific arrangement of preons in his model, which he calls the X-quark, which his theory suggests could be a stable good cold, dark matter candidate.


[edit] Loop quantum gravity and Bilson-Thompson Preon theory
In a 2006 paper [9] Sundance Bilson-Thompson, Fotini Markopolou, and Lee Smolin suggested that in any of a class of quantum gravity theories similar to loop quantum gravity (LQG) in which spacetime comes in discrete chunks, excitations of spacetime itself may play the role of preons, and give rise to the standard model of particle physics as an emergent property of the quantum gravity theory.

Sundance preon model was inspired by the Harari Rishon Model but posits ribbon-like structures that braid in groups of three, rather than point-particles. Proposing extended ribbon-like braided structures helps explains why ordering matters whereas the older point-particle preon model, the Harari Rishon Model, is unable to do so. It has been shown that the properties of Sundance ribbon-like structure can be derived from coherent states of spin foam, which may also give rise to gravity. His ribbon like structures have been described as "pieces of spacetime ribbon-tape", in that the Bilson-Thompson ribbons are made of the same structure that makes up spacetime itself. [10] While Sundance papers do offer braiding and an explanation on how to get fermions and spin-1 bosons, he does not show a braiding that would account for the Higgs boson [11].


Specifically, Bilson-Thompson et al proposed that loop quantum gravity could reproduce the standard model. The first generation of fermions (leptons and quarks) with correct charge and parity properties have been modeled using preons constituted of braids of spacetime as the building blocks[1]. Bilson-Thompson's original paper suggested that the higher-generation fermions could be represented by more complicated braidings, although explicit constructions of these structures were not given. The electric charge, colour, and parity properties of such fermions would arise in the same way as for the first generation. Utilization of quantum computing concepts made it possible to demonstrate that the particles are able to survive quantum fluctuations.[2]

In a 2006 paper [12], L. Freidel, J. Kowalski--Glikman, A. Starodubtsev suggests that elementary particles are Wilson lines of gravitational field, which implies that the properties of elementary particles, such as mass, energy, and spin, can be described by LQG's Wilson loops, and particle dynamics can be modeled on breaks in these Wilson loops, adding theoretical support to Bilson-Thompson's preon proposals.

Bilson-Thompson's ribbon preon scheme is intended to provide a picture diagram to represent coherent phases of spin foam dynamics whose description is quantum mechanical, not classical. For example, Bilson-Thompson's picture diagram of a preon with a twist, representing a U(1) charge equal to 1/9 of an electron charge, is to map to an eigenstate of spin foam. The spin foam formalism allows for the derivation of certain other particles of the standard model, the spin-1 bosons, such as photons and gluons, [[13]] and gravitons [[14]], [[15]] from loop quantum gravity's fundamental principles, and independent of Bilson-Thompson's braiding scheme for fermions. However, as of 2006, there is not a derivation of Bilson-Thompsons from spin foam formalism, including a derivation of 1/9 e- U(1) charge, and dynamics, as described by braiding. Bilson-Thompsons' braiding scheme does not offer a braiding that would account for a Higgs, but does not rule out the possibility of a Higgs boson. Bilson-Thompson himself observes that since the preons that have mass have charge as part of its internal "structure", it is possible it is this internal structure of charge that interacts with an electric field to give rise to inertial mass, or perhaps interacts with the Higgs field to give rise to inertial mass. et al. (The massless photon is untwisted in Bilson's preon scheme). As of 2006, it remains to be seen whether the derivation of the photon from the spin foam formalism in [[16]] can be matched with Bilson-Thompson's braiding of three untwisted ribbons [17], or perhaps, there are multiple ways to derive photons from the spin foam formalism.

When the term "preon" was first coined, it was used to describe pointlike subparticles that describe spin-1/2 fermions that include leptons and quarks. Such sub-quark pointlike particles would suffer from the mass paradox described below. It is observed that Bilson's ribbon structures are not actually "classical" preons, as defined in the introduction to this article as "pointlike structures or objects" of fermions, but Bilson-Thompson chooses to call his extended ribbon like structures of space-time "preons" in his research papers in the second sense of definition of preon as being more fundamental "subparticles" than elementary particles, and to maintain continuity in terminology with the larger physics community. His braiding also accounts for spin-1 bosons. In many respects, Bilson-Thompson's topological "preon" model resemble the geon more strongly than classical preon, and follows more closely a program inspired by Einstein and John Archibald Wheeler, in which particles are reduced to geometry through Wheeler's program Geometrodynamics (which has its roots in Lord Kelvin's knotting theory of atoms, and possibly to Spinoza's belief that reality is geometrical in structure) and its model of geons and continued through loop quantum gravity. The Wheeler's original Geometrodynamics suffers from the fact that it does not take quantum theory into account, whereas loop quantum gravity does.


[edit] Theoretical objections to preon theories: The mass paradox, chirality, and T'Hooft anomaly matching constraints
Heisenberg's uncertainty principle states that xp >= h bar/2 and thus anything confined to a box smaller than x would have a momentum of uncertainty proportionately greater. Some candidate preon models propose particles smaller than the elementary particles they make up, therefore, the momentum of uncertainty p should be greater than the particles themselves.

One preon model started as an internal paper at the Collider Detector at Fermilab (CDF) around 1994. The paper was written after the occurrence of an unexpected and inexplicable excess of jets with energies above 200 GeV were detected in the 1992—1993 running period.

Scattering experiments have shown that quarks and leptons are "pointlike" down to distance scales of less than 10−18 m (or 1/1000 of a proton diameter). The momentum uncertainty of a preon (of whatever mass) confined to a box of this size is about 200 GeV, 50,000 times larger than the rest mass of an up-quark and 400,000 times larger than the rest mass of an electron.

Thus, the preon model represents a mass paradox: How could quarks or electrons be made of smaller particles that would have many orders of magnitude greater mass-energies arising from their enormous momenta? Yershov's model, referenced above, proposes that when both particles and anti-particles of the proposed Y-particles in the theory are present, such as in the model's proposed neutrino composition, the mass of the constituent parts "cancels out", but can appear again when the structure of the Y-particles is changed. Yershov's model also proposes that particle mass arises as a mass defect binding energy among his preons, which helps account for the mass paradox.

Sundance preon model may avoid this by denying that preons are pointlike particles confined in a box less than 10−18 m, and instead positing that preons are extended 2-dimensional ribbon-like structures, not necessarily smaller than the elementary particles they compose, not necessarily confined in a small box as point particles preon models propose, and not necessarily "particle-like", but more like glitches and topological folds of spacetime that exist in three-fold bound states that interact as though they were point particles when braided in groups of three as a bound state with other particle properties such as mass and pointlike interatcion arising as an emergent property so that their momentum uncertainty would be on the same order as the elementary particles themselves.

String theory posits one-dimensional strings on the order of the Planck scale as giving rise to all the particles of the Standard Model, which would appear to also have the mass paradox problem. String theorist Lubos Motl has offered explanations as to how string theory gets around the mass paradox [18].

Any candidate preon theory must address particle chirality and T'Hooft anomaly matching constraints, and ideally be more parsimonious in theoretical structure than the Standard Model itself. Often, preon models propose additional unobserved forces or dynamics to account for their proposed preons compose the particle zoo, which may make the theory even more complicated than the Standard Model, or have implications in conflict with observation. One specific example: should the LHC observe a Higgs boson, or superpartners, or both, the observation would be in conflict with the predictions of many preon models, which predict the Higgs boson does not exist, or are unable to derive a combination of preons which would give rise to a Higgs Boson.


[edit] String theory and preon theory
String theory proposes that a one dimensional string on the order of a Planck scale has a tension, and differences in tension give rise directly to all the particles of the standard model and their super partners, in interaction with the proper compactified 6 or 7 dimensional Yau-Calabi mainfold and SUSY breaking. To date, string theory has been no more successful than preon theory in achieving this goal. John Baez and Lubos Motl have discussed the possibility that [19] that should preon theory prove successful, it may be possible to formulate a version of string theory that gives rise to a successful model of preons.

There have been recent research papers that have proposed preon models that are made of superstrings in Arxiv [[20]], [[21]] or supersymmetry [[22]]

 

[bananan]

I felt kind of bad about hijacking the thread. Do you mean to rejack it back to the original topic? By the way, I put up a new copy of my book, but it's not worth looking at. The major change is that I added the chapter divisions, with poetry.

You shouldnt' feel bad, you can talk about whatever you wish :)

 

[bananan]

Is there a spin network/spin foam state that maps to a "twist" in Bilson's ribbon which creates a 1/9 electron U(1) charge, and an explanation why they "braid" (presumably state-sum) in groups of 3, as opposed to 2 or 4?

the two approaches differ quite a bit, as you remarked, and they are both unfinished or incomplete. I think it would be difficult to match them up at this point.

Smolin et al's paper IIRC only involves spin NETWORKS, and there remains the problem of specifying how the networks evolve.

The paper by Baratin and Freidel investigates a spinFOAM model, and proves a result about the zero-gravity limit.

In principle the B&F model should have two parameters, h-bar, and Newton G, which one can take to zero and check the behavior.
I believe that they have done half the work: they have shown that if you make G -> 0 you get something recognizable as closed Feynman diagrams. Maybe this result can be strengthened but it is already a positive indication.

But they also now must show that if you do not make G -> 0, but instead make h-bar -> 0, you get something like classical gravity.

You know that Rovelli et al recently put out some work about spinFOAM where they derived gravitons. This is suggestive, but it is no guarantee that B & F, with their spinfoam model, can get classical gravity in the limit of small h-bar.

Speaking not as an expert, and hoping others will correct me if I am mistaken, I would say that B & F paper is very different from Smolin et al in the sense that B & F is closer to having dynamics.

It seems to me that the virtue of the Smolin et al paper is that it gives more or less the right types of particles in a conceptually simple QG scheme. The Smolin picture, I think, is of a spinnetwork evolving by LOCAL MOVES which reconnect nearby vertices in different ways or which insert and remove vertices. To have a structured dynamics, they still must assign amplitudes to these moves, time would presumably be some index of the rate that these local moves are occurring. It is an elegant and evocative picture of the universe which has seeds of ideas for a theory more fundamental than quantum mechanics (see Smolin's most recent preprint) and better adapted to handling cosmology. But I think it is not at a comparable stage of definition so that one could connect it with Freidel et al's work.

 

[marcus]

Bilson updated his article Oct 26, 2006

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

A topological model of composite preons

Thanks for flagging that!

The paper has been substantially rewritten at least in the introduction and the conclusion sections. Also a new section was added called
"Unresolved Issues"

It seemed sufficiently different to print out the new version.

Maybe we should have a thread about the new version, just to call attention to it.

 

[R.X.]

Any candidate preon theory must address particle chirality and T'Hooft anomaly matching constraints...

Only too true.. that's why the Rishon model of Harari took a sudden nose dive, namely when it was realized by his student Nati Seiberg that the anomalies do not match. It simply disappered since then.

This is what I thought til today. However, in the paper 0503213 mentioned above it made a surprise reapperance:

"The rishon model explained the number
of leptons and quarks, the precise ratios of their elec-
tric charges, and the origin and nature of colour charge.
The helon model does all this, but in additionin the framework of Loop Quantum Gravity [11]...
"

Christ... but well, it kind of makes sense. Since LQG people do not seem to care about anomalies, why bother and not re-introduce an inconsistent preon theory?

 

[bananan]

Only too true.. that's why the Rishon model of Harari took a sudden nose dive, namely when it was realized by his student Nati Seiberg that the anomalies do not match. It simply disappered since then.

This is what I thought til today. However, in the paper 0503213 mentioned above it made a surprise reapperance:

"The rishon model explained the number
of leptons and quarks, the precise ratios of their elec-
tric charges, and the origin and nature of colour charge.
The helon model does all this, but in additionin the framework of Loop Quantum Gravity [11]...
"

Christ... but well, it kind of makes sense. Since LQG people do not seem to care about anomalies, why bother and not re-introduce an inconsistent preon theory?

I recentally added this to wiki.

Bilson-Thompson has recently updated his paper dated October 27, 2006, [[18]] and acknowledges that his model, while not preon in the strict sense of the term, nevertheless is preon-inspired model, and is open to the possibility other more fundamental theories, such as M-Theory, may account for his topological diagrams, as well as the Higgs boson and gravity. The theoretical objections that apply to classic preon models do not necessarily apply to his preon inspired model, as it is not the particles themselves, but the relations between his preons (braiding) that give rise to the properties of particles. In this newer version of his paper, he has added a new section, section IV, called "unresolved issues" and acknowledges that open issues include mass, spin, cabbibo mixing, and grounding in a more fundamental theory. He states that grounding preons in M-theory is a possibility, as well as loop quantum gravity.

 

[bananan]

Only too true.. that's why the Rishon model of Harari took a sudden nose dive, namely when it was realized by his student Nati Seiberg that the anomalies do not match. It simply disappered since then.

This is what I thought til today. However, in the paper 0503213 mentioned above it made a surprise reapperance:

"The rishon model explained the number
of leptons and quarks, the precise ratios of their elec-
tric charges, and the origin and nature of colour charge.
The helon model does all this, but in additionin the framework of Loop Quantum Gravity [11]...
"

Christ... but well, it kind of makes sense. Since LQG people do not seem to care about anomalies, why bother and not re-introduce an inconsistent preon theory?

What sort of anomalies do you have in mind, in LQG?

 

[R.X.]

What sort of anomalies do you have in mind, in LQG?

Anomaly matching refers to chiral anomalies.

But yours is a tricky question, because it has been suggested (from what I gather from Thiemann's and other's papers), that anomalies simply do not exist in LQG, due to the kind of quantization procedure applied there.

One can only wonder how a standard quantum field theory that suffers from chiral anomalies and thus is inconsistent (gauge invariance broken, longitudinal modes do not decouple, path integral and thus correlation functions ill defined) should suddenly become consistent when treated with those methoids ...well, frankly they do not make much sense, see comments in Distlers' and Helling's blogs.

I think the least what one would require from any extension of ordinary QFT that it should reproduce the known features of QFT, and not plainly contradict them.

 

[selfAdjoint]

think the least what one would require from any extension of ordinary QFT that it should reproduce the known features of QFT, and not plainly contradict them.

Well, reproduced or better the RESULTS but there's no requirement to reproduce the details of the calculations. Particle physicists have over the decades taught themselves to believe six impossible things before breakfast, like Popov ghosts for example.

And this may be off base, but isn't the problem with QCD that it theoretically should have a chiral anomaly but phenomenologically doesn't, so they have to speculate on fine tuning schemes (like axions) to carefully cancel out the anomaly by two counterterms?

 

[R.X.]

Well, reproduced or better the RESULTS but there's no requirement to reproduce the details of the calculations. Particle physicists have over the decades taught themselves to believe six impossible things before breakfast, like Popov ghosts for example.

What's wrong with this neat computational trick? It is simply a clever way to deal with gauge fixing. No one ever has claimed that FP ghosts would be more than ficticious degrees of freedom and be at the same level es eg electrons. The rules of QFT work so well so that one can call it the most accurate theory of all of natural sciences. And giving up proven fundamental principles, like requiring the path integral be well defined, without very very strong reasons does not hold a lot of promise.

And this may be off base, but isn't the problem with QCD that it theoretically should have a chiral anomaly but phenomenologically doesn't, so they have to speculate on fine tuning schemes (like axions) to carefully cancel out the anomaly by two counterterms?

I don't know what you mean. Perhaps there is a confusion between global and local anomalies? Perturbative QCD at high energies is a very well established and experimentally proven theory, it is pointless to argue against its validity.

Frankly, in order to understand these things and hopefully proceed with one's own research in the future, there is no way other than really learning this stuff from the ground, and this takes many years of hard work. I understand the temptation to avoid this by simply declaring the results of many thousand hard working people as misguided, failed and not even wrong, in order to cook up "alternative" theories that are in contradiction with those results.

Well, science just works in a different way, and those who don't recognize this won't get anywhere.

 

[Careful]

The rules of QFT work so well so that one can call it the most accurate theory of all of natural sciences. And giving up proven fundamental principles, like requiring the path integral be well defined, without very very strong reasons does not hold a lot of promise.

I always love this argument, especially when it is known that all QED results such as Lamb shift, g factor, Casimir effect ... can be derived by treating the EM field classically while obtaining similar precision. Nobody really argues that the results of QED and QCD don't come out well perturbatively, but another thing is to appreciate this fact for the right reasons ! The latter might very well shake up some fundamental assumptions seriously (actually I am sure of this). But I agree with you, on the other hand, that in doing so you must always keep QED in mind and progress comes in small steps. I moreover concur that ``background independence'' is not going to solve the problems at hand. On the other hand, there are very good reasons from the theoretical side to protest against QED : sometimes it is good to go against something in order to learn to appreciate it for the right reasons, progress often comes from that direction.

Careful

 

[Kea]

And giving up proven fundamental principles, like requiring the path integral be well defined, without very very strong reasons does not hold a lot of promise.

Indeed.

 

[Chronos]

I'm not willing to surrender SA's point without resistance. Careful is very bright but a bit reckless, IMO. Going against things is all about science, going through the motions is another matter.