Why do Fermions form a Cube?
Mark William Hopkins
structure:
nu_e
/ | \
/ | \
d_R d_G d_B
| \ / \ / |
| / \ / \ |
u_B u_G u_R
\ | /
\ | /
e
which clearly suggests a connection to the cube.
In fact, consider the discrete function space formed by the vertex-valued
functions over the cube. Essentially, this is the discrete analogue of the
function space of a sphere. The analogous form of the Laplace-Beltrami
operator is the discrete operator:
3V - E
D = ------
6
where V = identity operator and E is the operator which adds up the sums
of the values on each of the 3 vertices that sit on opposite sides of
edges from the given vertex.
Then the eigenstates of D form the discrete analogue of the spherical
harmonics. These are depicted graphically below:
+ denotes +1 value on vertex
- denotes -1 value on vertex.
S-state + + Eigenvalue: 0
+ +
+ +
+ +
P-states + + + + + - Eigenvalue: 1/3
- - + + + -
+ + - - + -
- - - - + -
D-states + - + - + + Eigenvalue: 2/3
+ - - + - -
- + + - - -
- + - + + +
F-state + - Eigenvalue: 1
- +
- +
+ -
One also has the "sign" operator
C(x) = value of vertex on opposite side of cube from x
that yields eigenvalues of +1 for the S and D states, -1 for the P and
F states, so that the charge operator is Q = e C D.
The patterns get even more interesting when you start to write down the
Laplacian for the Standard Model in "Cube Space". In particular, the
W+ and W- particles couple (for left-handed fermions) to operators that
look suspiciously like the creation and annihilator operators for the
eigenstates of C.
The mystery is: where is this regularity coming from?
Matt McIrvin
Hopkins) wrote:
>The fermions in each generation arrange themselves naturally into a
>structure:
> nu_e
> / | \
> / | \
> d_R d_G d_B
> | \ / \ / |
> | / \ / \ |
> u_B u_G u_R
> \ | /
> \ | /
> e
>
>which clearly suggests a connection to the cube.
[...]
>The mystery is: where is this regularity coming from?
Here's a guess:
In Grand Unified Theories the fermions of each generation fall into some
representation of the GUT gauge group. If the symmetry group of the cube
were a subgroup of this gauge group, the cubical structure might fall out
quite naturally.
Maybe somebody here has the inclination to figure out whether this is the
case in the simplest GUTs...
--
Matt McIrvin http://world.std.com/~mmcirvin/
ba...@galaxy.ucr.edu
Mark William Hopkins <hu...@alpha1.csd.uwm.edu> wrote:
>The fermions in each generation arrange themselves naturally into a
>structure:
> nu_e
> / | \
> / | \
> d_R d_G d_B
> | \ / \ / |
> | / \ / \ |
> u_B u_G u_R
> \ | /
> \ | /
> e
>
>which clearly suggests a connection to the cube.
Nice picture!
In Geoffrey Dixon's theory this is "explained" using the octonions:
if we ignore spin, we can think of the fermions in each generation
as forming a copy of the octonions, which we may think of as functions
on a set with 2^3 elements - e.g. the vertices of a cube. The symmetry
group of the octonions is G2, but if we look at the elements of G2 that
fix a unit imaginary octonion, we get the smaller group SU(3). This is
the gauge group of the strong force. The representation of SU(3) on the
octonions splits into a direct sum of a 2-dimensional trivial representation
(corresponding to nu_e and e, which don't feel the strong force) and a
6-dimensional representation (corresponding to the 6 colors of quarks,
which do). SU(3) contains Z/3 as a subgroup in a natural way, and this
corresponds to the 3-fold symmetry of your cube about the vertical axis.
For more on Dixon's theory try:
Geoffrey M. Dixon, Division Algebras: Octonions, Quaternions,
Complex Numbers and the Algebraic Design of Physics, Kluwer Press,
ISBN 0-7923-2880-6.
Whether or not one believes this theory, it's full of interesting
ideas for anyone interested in why elementary particles are the way
they are.
Alejandro Rivero
ba...@galaxy.ucr.edu wrote:
> In article <7m2qmu$irn$1...@uwm.edu>,
> Mark William Hopkins <hu...@alpha1.csd.uwm.edu> wrote:
> >The fermions in each generation arrange themselves naturally into a
> >structure:
> > nu_e
> > / | \
> > / | \
> > d_R d_G d_B
> > | \ / \ / |
> > | / \ / \ |
> > u_B u_G u_R
> > \ | /
> > \ | /
> > e
> >
> >which clearly suggests a connection to the cube.
>
> Nice picture!
If my pictographical memory does not fail me, you can copy a full page
version of that picture form the Scientific American recopilation of
articles on Elementary Particles. Some classic 70-early 80 article
Modernly, it seems that vectorial SU(3) color is related somehow to
Poincare Duality, the property that let us to slice space and perform
integrations.
Alejandro Rivero
>
> In Geoffrey Dixon's theory this is "explained" using the octonions:
> ...
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Share what you know. Learn what you don't.
Mike Vaughn
> In article <7m2qmu$irn$1...@uwm.edu>,
> Mark William Hopkins <hu...@alpha1.csd.uwm.edu> wrote:
> >The fermions in each generation arrange themselves naturally into a
> >structure:
> > nu_e
> > / | \
> > / | \
> > d_R d_G d_B
> > | \ / \ / |
> > | / \ / \ |
> > u_B u_G u_R
> > \ | /
> > \ | /
> > e
> >
> >which clearly suggests a connection to the cube.
Wait a minute! If you are suggesting 8 Dirac fermions per generation, then
you have to account for the invisibility of the right handed neutrino.
Apart from that, with 16 left-handed fermions, you have the standard
16-domensional representation of SO(10) -- one of the early 'grand unified'
groups.
> Nice picture!
>
> In Geoffrey Dixon's theory this is "explained" using the octonions:
> if we ignore spin, we can think of the fermions in each generation
> as forming a copy of the octonions, which we may think of as functions
> on a set with 2^3 elements - e.g. the vertices of a cube. The symmetry
> group of the octonions is G2, but if we look at the elements of G2 that
> fix a unit imaginary octonion, we get the smaller group SU(3). This is
> the gauge group of the strong force. The representation of SU(3) on the
> octonions splits into a direct sum of a 2-dimensional trivial representation
> (corresponding to nu_e and e, which don't feel the strong force) and a
> 6-dimensional representation (corresponding to the 6 colors of quarks,
> which do). SU(3) contains Z/3 as a subgroup in a natural way, and this
> corresponds to the 3-fold symmetry of your cube about the vertical axis.
Of course the decomposition of the 7-dimensional representation of G2
on reduction to SU(3) is
7 -> 3 + 3* +1
so you need quark + antiquark together in one multiplet, and then you
separate multiplets for u and d, wich isn't very unified.
On an historical note: G2 as a (flavor) symmetry group dates back
to a paper by Ralph Behrends in 1959 or 1960 -- you can look it up
in the famous Behrends, Dreitlein, Fronsdal and Lee article in
Revs. Mod. Phys. (1962).
> For more on Dixon's theory try:
>
> Geoffrey M. Dixon, Division Algebras: Octonions, Quaternions,
> Complex Numbers and the Algebraic Design of Physics, Kluwer Press,
> ISBN 0-7923-2880-6.
I haven't seen this book, but I know the late Feza Gursey was trying
hard for many years to find a use for octonionic structures in
physics.
Aaron Bergman
>
>Modernly, it seems that vectorial SU(3) color is related somehow to
>Poincare Duality, the property that let us to slice space and perform
>integrations.
Could you explain that further? I'm trying to have any idea what
that could be, but I'm completely lost. I assume that by
Poincare' duality, you mean
H^q(M) = H_c^{n-q}(M)^*
Aaron
--
Aaron Bergman
<http://www.princeton.edu/~abergman/>
ba...@galaxy.ucr.edu
Mike Vaughn <mtva...@neu.NOSPAM+-.edu> wrote:
>In article <7m7n3j$r...@charity.ucr.edu>, ba...@galaxy.ucr.edu wrote:
>> In article <7m2qmu$irn$1...@uwm.edu>,
>> Mark William Hopkins <hu...@alpha1.csd.uwm.edu> wrote:
>> >The fermions in each generation arrange themselves naturally into a
>> >structure:
>> > nu_e
>> > / | \
>> > / | \
>> > d_R d_G d_B
>> > | \ / \ / |
>> > | / \ / \ |
>> > u_B u_G u_R
>> > \ | /
>> > \ | /
>> > e
>> >
>> >which clearly suggests a connection to the cube.
>Wait a minute! If you are suggesting 8 Dirac fermions per generation, then
>you have to account for the invisibility of the right handed neutrino.
Hi! Nice to see you back here.
I don't know about Mark, but in my discussion of Geoffrey Dixon's scheme,
I was only talking about left-handed fermions. I don't know how he
handles the right-handed ones. I'll be seeing him in a few weeks in
Cambridge (England), so I'll bug him about this.
>> In Geoffrey Dixon's theory this is "explained" using the octonions:
>> if we ignore spin, we can think of the fermions in each generation
>> as forming a copy of the octonions, which we may think of as functions
>> on a set with 2^3 elements - e.g. the vertices of a cube. The symmetry
>> group of the octonions is G2, but if we look at the elements of G2 that
>> fix a unit imaginary octonion, we get the smaller group SU(3). This is
>> the gauge group of the strong force. The representation of SU(3) on the
>> octonions splits into a direct sum of a 2-dimensional trivial representation
>> (corresponding to nu_e and e, which don't feel the strong force) and a
>> 6-dimensional representation (corresponding to the 6 colors of quarks,
>> which do). SU(3) contains Z/3 as a subgroup in a natural way, and this
>> corresponds to the 3-fold symmetry of your cube about the vertical axis.
>
>Of course the decomposition of the 7-dimensional representation of G2
>on reduction to SU(3) is
>
> 7 -> 3 + 3* +1
>
>so you need quark + antiquark together in one multiplet, and then you
>separate multiplets for u and d, wich isn't very unified.
I think you're talking about something slightly different than I
am - though I may be confused. I'm starting with the octonions,
whose automorphism group is G2. This gives us an 8-dimensional
*real* representation of G2. Now I define SU(3) as the subgroup
of G2 that preserves a given unit imaginary octonion, say i. We
thus have an 8-dimensional *real* representation of SU(3). But
by definition the action of this SU(3) on the octonions commutes
with the action of i by left multiplication, so we can use the
action of i to make the octonions into a 4-dimensional *complex*
representation of SU(3). (You might worry that the nonassociativity
of the octonions would prevent left multiplication by i from
being an operation whose square is -1, but the octonions are
alternative so i(ix) = (ii)x= -x for all x.) And this 4-dimensional
complex rep of SU(3) splits as 3 + 1.
Hmm, but maybe this is precisely your point. I have to go now, but
I should think about this more.
Mark William Hopkins
is still moderating singlehandedly, and is having shortages of
moderating time... -MM]
In article <mtvaughn-ya0240800...@nntp.neu.edu> mtva...@neu.NOSPAM+-.edu (Mike Vaughn) writes:
>In article <7m7n3j$r...@charity.ucr.edu>, ba...@galaxy.ucr.edu wrote:
>
>> In article <7m2qmu$irn$1...@uwm.edu>,
>> Mark William Hopkins <hu...@alpha1.csd.uwm.edu> wrote:
>> >The fermions in each generation arrange themselves naturally into a
>> >structure:
>> > nu_e
>> > / | \
>> > / | \
>> > d_R d_G d_B
>> > | \ / \ / |
>> > | / \ / \ |
>> > u_B u_G u_R
>> > \ | /
>> > \ | /
>> > e
>> >
>> >which clearly suggests a connection to the cube.
>
>Wait a minute! If you are suggesting 8 Dirac fermions per generation, then
>you have to account for the invisibility of the right handed neutrino.
It's not. Perhaps you're not aware, but one of the biggest discoveries
of the century was made in 1998 with the discovery that the neutrino has
mass. You'll see the relevant articles at the Particle Data Group's web
site.
In any case, the standard model already includes the right-handed neutrino
in its Laplacian: it's 0, since it has zero coupling to everything else
(except gravity).
That, of course, explains why you don't normally see it and why its only
visible effect is, in all likelihood, Dark Matter.
The only extra field equation that comes out is the 0-mass free field
neutrino equation for the right-handed component.
Even if it weren't known to exist, you'd still almost certainly have to
extrapolate its existence just from the symmetry of the structure alone
(in analogy to the way Maxwell extrapolated the existence of the
displacement current before it was known to exist). So count it as a
prediction that's just been confirmed.
Alejandro Rivero
aber...@princeton.edu (Aaron Bergman) wrote:
> In article <7mb70s$78h$1...@nnrp1.deja.com>, Alejandro Rivero wrote:
> >
> >Modernly, it seems that vectorial SU(3) color is related somehow to
> >Poincare Duality, the property that let us to slice space and
perform
> >integrations.
>
> Could you explain that further? I'm trying to have any idea what
> that could be, but I'm completely lost. I assume that by
> Poincare' duality, you mean
>
> H^q(M) = H_c^{n-q}(M)^*
Yes. Or also, the existence of a "cup" product and a volume form
(which are used to stablish such duality).
Point was, when you try to make a space that becomes discrete at
high energies, using the Formalism of Connes et al., you find
that in orther for poincare duality to survive at that level,
either the electroweak or the colour group must be vectorial. As
the electroweak has axial part, the other one must can not have
any axial bias. It is a effect of this kind of models, well
documented (taking in accound the small number of researchers
in that field :-)
Yours,
Alejandro Rivero
Mark William Hopkins
>The fermions in each generation arrange themselves naturally into a
>structure:
[Picture]
>which clearly suggests a connection to the cube.
In article <7m7n3j$r...@charity.ucr.edu> ba...@galaxy.ucr.edu writes:
>Nice picture!
>
>In Geoffrey Dixon's theory this is "explained" using the octonions:
>if we ignore spin, we can think of the fermions in each generation
>as forming a copy of the octonions, which we may think of as functions
>on a set with 2^3 elements - e.g. the vertices of a cube...
[Goes on to explain the 3+1 splitting of SU(3) on the octonions]
But...
the real questions lay directly underneath the figure:
(A) Why is the charge operator equal to the discrete Laplace-Beltrami
operator (apart from sign) in the complementary cube space?
(B) Why do the fermions look like discrete analogues of the spherical
harmonics in that space with the identifications below (and depicted
in the diagrams in the prior article)?
nu = S-state
d_R, d_G, d_B = P_x, P_y, P_z states
u_R, u_G, u_B = D_yz, D_zx, D_xy states
e = F_xyz state
The almost looks like a repeat of history when the source of the
regularity in the periodic table was found to be the spectrum
associated with spherical harmonics.
(C) The particle quantum numbers essentially coincide with the [eigenvalues
of the] isometric operators in the complementary space (of which there
are 4 independent ones). Why are the particle states linked directly
to the symmetry group in the complementary space?
(D) Color is directly related to the 3 principle directions of the cube in
that space. What's the link between isometry in the complementary space
and color / color confinement?
(E) In the cube-space representation, the action for the QED sector looks
like:
integral (psi' e A C D psi)
where D is the discrete Laplace-Beltrami operator, C the "sign" operator
described in the article (the operator corresponding to transposing
vertices on opposite sides in the complenmentary space), A the
electromagnetic field, and psi the 8x4 component combined fermion
wavefunction.
Where could this come from? This combination looks very familiar.