Mohamed:

Thanks for your thoughtful questions.  I hope the material below adequately addresses your questions.  I welcome feedback (regarding such adequacy, about the research, and so forth).

Regarding numbers of particles, I counted based on how people show particles in a display such as http://en.wikipedia.org/wiki/Elementary_particle shows:

* 6 quarks (or, 2 sets, each with 3 generations)

* 3 charged leptons

* 3 neutrinos

* 1 gluon (yes, people might consider this to be under-counting)

* 1 photon

* 1 Z boson

* 1 W boson (here, as elsewhere, people seem not to count distinct antiparticles)

* 1 Higgs boson.

(My apologies for any confusion.  In table 2.8.6 in the attachment [to the original question], I use a counting method that, for example, counts 8 gluons.  [Item 2.8.71]  The paragraph before the table explains how I count particles in that work.  Also, yours is the first claim I have seen regarding using a counting factor for color for fermions.)

Regarding "elementary and composite," I realized (while I was looking at what I saw in the math basis I am using), that I could interpret aspects of 'the math' as correlating with 10 Poincare-group generators for some, but not all, of the known elementary particles.  (See the particles [other than "composite"] I list in the findings section of the question.)  I then saw that by combining the list of such generators for quarks with the list of such generators for particles I call Y-family spin-1 particles (these correlate [mathematically] with gluons), I obtained a list of 10 generators that seem similar to the 10 generators for free-ranging particles, but that differ in some respects that people might consider a reversal of roles of energy-like (or time-like) and momentum-like (or, space-like).  Here, "reversal" is compared to the 10 generators correlating with each (or all) of the free-ranging elementary particles.  I then said that pairs or triplets of "1 quark + gluons" should correlate with 10 generators for free-ranging particles.  The pairs would correlate with pions and other composite particles containing exactly 2 quarks.  The triplets would correlate with protons and other composite particles containing exactly 3 quarks.

My apologies for any confusion I may have caused regarding applications of group theory.

* In the book (from which I extracted the attachment to the original question), I explore the extent to which my work recreates the Standard Model's SU(3)×SU(2)×U(1).  People might say that I succeed at recreating that symmetry (or, those symmetries).  People might say that these symmetries pertain to aspects related to interactions between particles.  I would not disagree.

* People associate the Poincare group with symmetries related to "imposing" or "superimposing" coordinate systems (or, observer-centric "frames or reference") on 4-dimensional space times (that people assume correlate with the Minkowski metric).  Discussions feature 4 generators for "translation symmetries" (1 with respect to time and 3 with respect to space), 3 generators for "boost symmetries," and 3 generators for either "rotation symmetries" or "reflection symmetries" (Apologies, I have seen people state one or the other of these 2 interpretations - and never both of them.).

Let me finish (at least for now) by noting the work correlates with the possibility that dark matter includes 5 "clones" of Standard Model particles and the possibility that dark-energy stuff includes 7 "clones of 'ordinary-matter plus dark-matter.'"  In the math, these concepts correlate with Poincare-group symmetries and (for fermions) with generations.  (All of this is covered in the attachment to the original question.)

Mohamed:

Some thoughts.

* I am having difficulties relating some of your remarks to my question.  Perhaps, you are trying to suggest, via your overall-third answer (above), that what I may have discovered has no value.  I can accept such a finding, if you can provide reasoning I can follow.  (I asked the question with hopes that include that I might learn something.)  Perhaps you will try to read my work [through section 2.8] and then comment?  Or, perhaps you know of some observed data or some fundamental theoretical concept that 'only-7 such generators' would violate?  Or, perhaps you can show that 'only-7 such generators' has no bearing on people's 'current and ultimate' understanding of nature?  Your thoughts?

* I tried to follow several of the links you suggest.  It appears that the links do not point to accessible webpages.

* Let's abandon discussing the number of gluons.  As I pointed out in my response to your overall-first response, I count 8 gluons in table 2.8.6 in the relevant description of my attempted research.  Again, I apologize for 'taking a shortcut (when formulating my question)' regarding counting, by reflecting a popular means to show a table of elementary particles.

Book Theory of Particles plus the Cosmos: Small Things and Vast E...

Mohamed:

Thanks for recommending that we try to increase mutual understanding and that I try to clarify some matters.

1. (Overview): Regarding traditional Standard Model symmetries, I think I found (or reproduced) in my work the traditional SU(3)×SU(2)×U(1) symmetries.

1. (Details):  In sections 2.1 through 2.6 (of the excerpt from "Theory of Particles plus the Cosmos"), I show what I call "core representations" for elementary particles.  These representations describe some interactions in which the particles partake.  Regarding SU(3)×SU(2)×U(1) and gluons, see the discussion after item (2.6.26) in section 2.6.  (In an earlier book, I pointed out SU(3)×SU(2)×U(1) regarding other elementary bosons.  In this book, I show the work regarding gluons and I return to the boarder-SU(3)×SU(2)×U(1) topic in section 5.6.)

2. (Transition):  Regarding 10-generator Poincare-group symmetries for many (but not all elementary particles), I see this matter as somewhat (but not completely) unrelated to the Standard Model symmetries.

2. (Details) and 3. (Overview):  In sections 2.7 and 2.8, I show "extended representations" for particles.  (Technically, some people might say that the extended representations pertain to fields related to the particles.)  These representations extend the core representations.  I interpret the extensions as correlating with numbers of generations (for elementary fermions), number of instances of particles, and symmetries related to the Poincare group.

3. (Details): (Regarding the 3 points below, see sections 2.7 and 2.8.), …:

* Perhaps people would say I provide a new math model that (starts from some assumed 'fundamentals' and) correlates with numbers of generations for fermions.

* Perhaps people would say that the question of 'number of instances' is new.  I show that the math model correlates with the possibility of 48 instances of each free-ranging Standard Model particle.  (Note: the numbers of instances for each of quarks and gluons would be less than 48.  But the number of instances of composite particles made from quarks and gluons would be 48.)  I interpret (via other aspects of the math) that people can group these 48 instances into 8 sets, each of 6 instances.  Each set would have its own graviton.  From the perspective of physics-savvy beings correlating with any 1 instance, the beings would consider the other 5 instances in the same set to be dark matter and the 42 remaining instances to be dark-energy stuff.  (For more about dark energy and dark matter, see sections 3.1 and 3.4.)

* Regarding the question of Poincare-group symmetries, I found (as I looked into the math related to generations and instances) "1 generator plus 3 times SU(2)" math that seemed to correlate with many, but not all, of the elementary particles.  So, based on 10 = 1+3+3+3 generators and some interpretations of the relevance of this construct to special relativity, I decided to see where 'counting generators for such symmetries' might lead.  For me, it leads to the findings I reprise in the findings section of the question we are discussing.

4. (Other point):  We have been focusing on some (of what I would consider) key aspects of my attempted research.  I think such is good, especially in the context of the question I asked.  Just permit me to note here that I think the overall attempted research [a] may point to ways to narrow various gaps between known data and theoretical 'predictions' and [b] may have enough cohesion that people would consider (much of this) to be (at least) a nucleus for new theory.  (See sections 0.1 through 0.4.)

Again, thanks for showing interest in this.  Your thoughts?

Mohamed:

Thanks for providing perspective and encouragement.  I appreciate your willingness to look at my attempted research.  I look forward to your comments, questions, suggestions, … .

- Tom

Thomas,

according to Noether:

"The symmetry group of special relativity, the Poincar´e group, is a Lie subgroup of the group of general coordinate transformations. It has a finite number (7) of independent infinitesimal generators"

"What if no more than 7 Poincare-group generators correlate with behavior of the Z and W bosons?"

According to Noether the Poincarè Group has 7 generators..why should it have more??

Stefano:

I was trying to refer to the 10-generator group that the following page discusses:  http://en.wikipedia.org/wiki/Poincar%C3%A9_group

Curiously, there seem to be 7 generators for which the 'explanations' have not changed and 3 for which the discussion has changed.  The first 7 are: 1 time-translation, 3 space-translations, and 3 boosts.  The other 3 generators had been described as correlated with rotations and seem now to be described as correlating with reflections.  But, 'perhaps not to worry:' the new description links that concept of rotation to the concept of reflection.

I asked my question because, I think, my attempted research links the following for each known and each of some 'possible' elementary particles: spins, some information about interactions, some masses, numbers of generations (for fermions), and relevant numbers of generators out of a collection of 10 generators.  From the patterns in the math that underlies all this, I think it possible that the 10 generators correlate with the "Poincare group" (and, hence, with "special relativity") in ways that correlate with concepts the above page presents.  (See the attachment, especially sections 2/7 and 2.8.)

The "findings" section of my question (above) summarizes possibilities regarding those "10 generators."  To the extent 'all the above correlates within itself,' people might say that [a] most free-ranging elementary particles and all composite particles correlate well with special relativity; [b] there may be a lack of complete correlation between weak-interaction bosons, gravitons, and some other 'possible' elementary particles and special relativity; [c] quarks and gluons do not correlate well with special relativity.

In any event, I would appreciate your (and other people's) helping regarding the possible studying, improving (or disproving), and usefulness of such concepts.

- Tom

Book Theory of Particles plus the Cosmos: Small Things and Vast E...

Tom,

usually everything is quite clear (using Noether):

The symmetries of the Poincare group are related to conservation laws. One has 10 generators and one has 10 laws (energy, momentum, angular momentum and center of mass). The translations correspond to momentum and energy. The rotations correspond to the angular momentum and center of mass.

For the classification of particles you have to include the charges. The corresponding Noether symmetry comes from quantum mechanics (QM). It is the (local) phase shift leading to gauge invariance (and the appearance of gauge fields). Or, charges are connected with the complex structure of the wave function in QM (and the choice of the phase). The Poincare group has nothing to do with charges. Therefore I don't see why one should relate the generators of the Poincare group to particles. According to the representation theory of Wigner, you will get the mass (continuous) and the spin (in units of 1/2) from the representation of Poincare group.

Torsten