Although the D4-D5-E6-E7-E8 VoDou Physics model is NOT a SuperString Theory, the nice math involved in String Theory is relevant, although not as a "theory of elementary particles". In the D4-D5-E6-E7-E8 VoDou Physics model, closed strings represent the world-lines of fermion particle-antiparticle pairs ( the pair of fermions acting as a boson so that the entire string is bosonic ) from the time of their creation to their eventual mutual annihilation (perhaps with lots of interactions with lots of other particles/antiparticles of other world-lines in the meantime, and with mutual creation/annihilation guaranteed by considering String Theory spacetime to be a compactified 25+1 dimensional Minkowski spacetime).
The D4-D5-E6-E7-E8 VoDou Physics model has:
whose Geometric and Physical Interpretations are:
Algebraic formulation of Bohm-many-worlds string quantum theory, timelike brane M-theory, and spacelike brane F-theory may be useful in sum-over-histories interpretations, particularly with respect to prime numbers and zeta functions.
Such String, M, and F Theories are used in my Quantum Consciousness paper contributed to Quantum Mind 2003.
26-dimensional String Theory and 27-dimensional M-theory can be represented by 3x3 Hermitian octonion matrices of the form:
a Y X Y* b Z X* Z* c
where X, Y, Z are 8-dim octonions, * is conjugation, and a, b, c are real numbers, which are independent for 27-dim M-theory, in which case they form the 27-dim Jordan algebra J3(O) and which sum to zero for 26-dim string theory in which case they form the 26-dim traceless part J3(O)o of that Jordan algebra. THE X, Y, Z COORDINATES FORM A CONFIGURATION SPACE for 1-particle states, in that, for a given 1 fermion particle: the octonion X determines a position in 4-dim spacetime and in 4-dim internal symmetry space; the octonion Y determines an identity as a fermion particle; the octonion Z determines an identity as a fermion antiparticle. The real numbers a, b, c are Auxiliary variables. 27-dimensional M-theory and 28-dimensional F-theory can be represented by 4x4 quaternion Hermitian matrices of the form:
a U T R U* b S V T* S* c W R* V* W* d
Here R, S, T, U, V, W are quaternions, * is conjugation, and a, b, c, d, are real numbers, and THE R, S, T, U, V, W COORDINATES FORM A CONFIGURATION SPACE for 1-particle states, in that, for a given 1 fermion particle: the quaternion R determines a position in 4-dim spacetime; the quaternion S determines in 4-dim internal symmetry space; the quaternion pair T, U determines an identity as a fermion particle; the quaternion pair V, W determines an identity as a fermion antiparticle. The real numbers a, b, c, d are Auxiliary variables. In this case, the 4x4 quaternion Hermitian matrices form the Jordan algebra J4(Q) (where I use Q to denote quaternion), and the theory is like A 28-DIM F-THEORY (if you recall, in the string theory community a few years ago F-theory was popular as a "generalization" of M-theory, with 1 more dimension than M-theory. 27-DIM M-theory itself can also be seen in terms of quaternions, by using the traceless J4(Q)o instead of J3(O).
Here are the Geometrical and Physical interpretations of:
Jordan algebra Lie algebra Sphere structure symmetry structure
26-dim Strings J3(O)o E6 / F4 Real S0 = {-1,+1} is boundary 78-52 = 26 of String Interval [-1,+1] ( closed string if you identify -1 = +1 ) (Compare Graded Lie Algebra structure.)
In the D4-D5-E6-E7-E8 VoDou Physics model, closed strings represent the world-lines of fermion particle-antiparticle pairs ( the pair of fermions acting as a boson so that the entire string is bosonic ) from the time of their creation to their eventual mutual annihilation,
* / \ ... / \ / / | (The illustrated closed string is red. \ | It interacts with a partially shown gray string.) \ / \... \ / *
perhaps with lots of interactions with lots of other particles/antiparticles of other world-lines in the meantime, so that part of each string might represent a photon or other particle of any type formed by interaction of one of the particle/antiparticle pair.Note that since our Universe began with a Big Bang, all its particles originate from pair creation since then. For pairs that do not appear to reconnect for mutual annihilation within the volume of 26-dimensional spacetime being considered in working with the String Theory,
**************** \ ... / \ \ / \ | (The illustrated string is red. \ | It interacts with a partially shown gray string. \ / \... A perfect absorber in the future \ / is indicated by ******* ). *
the string is closed by considering the 26-dimensional spacetime to be a compactified 25+1 dimensional Minkowski spacetime due to considering the Universe to "... be a perfect absorber in the future ...[as in]... the Wheeler-Feynman ... absorber theory of radiation ..." described by Narlikar in his book Introduction to Cosmology (Cambridge 1997) (Section 8.8.1) and related to the Collective Electrodynamics of Carver Mead. For most of the matter in our Galactic Cluster, such an absorber could be a Black Hole of the Black Hole Era. Such a compactification is also similar to the conformally compactified 3+1 dimensional Minkowski spacetime M# used by Penrose and Rindler in their book Spinors and Space-Time, Volume 2 (Cambridge 1986) (particularly Chapter 9). ]]
These Strings look like "world-line paths" of fermions in "configuration space-time" and their String Theory is the way to calculate the Generalized Bohm Quantum Potential that "tells" the fermion how to "move around" in configuration space is itself acted back on by the Fermion Configuration.
The Quantum Theory of the D4-D5-E6-E7-E8 VoDou Physics model has several ( in my opinion equivalent ) formulations, including Many-Worlds, Nelson's Stochastic Theory, and generalized Bohm Quantum Theory. In the generalized Bohmian point of view
"states of the physics model" are viewed as being completely determined by "configurations of fermions in configuration space",
which is consistent with the lattice physics view that
fermions live on vertices of the lattice, and gauge bosons live on links of the lattice,
so that gauge bosons can be represented in terms of fermions by the "state" of one fermion at the beginning of a link and of another fermion on a vertex at the end of a link, and composite bosons (like pions, which are pairs of quarks) can be represented in terms of their constituent fermions.
In other words, 26-dim String Theory can be interpreted as being the theory of movement of a "Bohm Point" in configuration space. ( Since I like to use a Lagrangian formulation using spacetime as opposed to a Hamiltonian spatial formulation with time as an "outside" variable, the configuration space is not particle positions (points) in fermion representation spaces, internal symmetry space, and spatial space but is particle world-lines (strings) in fermion representation spaces, internal symmetry space, and spacetime including both spatial dimensions and time dimension. )
If you look at String Theory as being analogous to General Relativity of a 26-dimensional SpaceTime, then you see that
so that the Quantum Theory is, like General Relativity, NonLinear in that the Generalized Bohm Quantum Potential (compare Gravitation) acts on the Configurations (compare Matter Fields) and the Configurations (compare Matter Fields) also act (by what Jack Sarfatti calls Post-Quantum Back-Reaction) on the Generalized Bohm Quantum Potential (compare Gravitation).
Jordan algebra Lie algebra Sphere structure symmetry structure
27-dim M-Theory J3(O) E7 / E6xU(1) Complex S1 is boundary J4(Q)o 133-78-1 = 54 = 2x27 of Unit Disk = 2-ball B2 Time Part of (2,4) vector space of Conformal Spin(2,4) (Compare Graded Lie Algebra structure.)
The (2,4) Conformal vector space corresponds to 6 of the 10 dimensions of the vector representation of the D5 Lie algebra of the D4-D5-E6-E7-E8 VoDou Physics Model. On those 6 dimensions, the Conformal Group Spin(2,4) = SU(2,2) acts linearly. The corresponding subspace of the 8 dimensions of the vector representation of the D4 Lie algebra of the D4-D5-E6-E7-E8 VoDou Physics Model is 4-dimensional Physical SpaceTime, on which the Conformal Group acts non-linearly.
The 2-ball B2 corresponds to the 2 Timelike dimensions of (2,4) Conformally Linear Physical SpaceTime.
The S1 boundary of B2 corresponds to the 1 Timelike dimension of (1,3) 4-dimensional ( Conformally non-linear) Physical SpaceTime.
These Timelike spaces look like p-Brane Membranes and their M-theory describes interactions among the Timelike parts of parallel material Brane Universes that Jack Sarfatti describes (in Decenber 2001 e-mail correspondence) as being "... next door to each other across thin Josephson tunnel "weak link" junctions in which Star Gates form ...".
Jordan algebra Lie algebra Sphere structure symmetry structure
28-dim F-Theory J4(Q) E8 / E7xSU(2) Quaternion S3 is boundary 248-133-3 = 112 = 4x28 of 4-ball B4 Spatial Part of (2,4) vector space of Conformal Spin(2,4) (Compare Graded Lie Algebra structure.)
The 4-ball B4 corresponds to the 4 Spacelike dimensions of (2,4) Conformally Linear Physical SpaceTime.
The S3 boundary of B4 corresponds to the 3 Spacelike dimension of (1,3) 4-dimensional ( Conformally non-linear) Physical SpaceTime.
These Spacelike spaces look like p-Brane Membranes and their F-theory describes interactions among the Spacelike parts of parallel material Brane Universes that Jack Sarfatti describes (in Decenber 2001 e-mail correspondence) as being "... next door to each other across thin Josephson tunnel "weak link" junctions in which Star Gates form ...".
Jack Sarfatti describes (in Decenber 2001 e-mail correspondence) "... The parallel material "brane universes", next door to each other across thin Josephson tunnel "weak link" junctions in which Star Gates form ...". If you consider that new Universes might form from Quantum Fluctuations in older Universes, you can see that it might be reasonable to expect that many Parallel Material Universes ( one of which might be Our Universe ) might exist very close to each other, or intersect with each other at Intersection Star Gates.
Consider the following illustration, adapted from an Ann Feild STScI illustration of a Model of Expanding Universe:
The illustration shows only 3 levels of Universes, and so is much simpler than reality.
The illustration is a 2-dimensional projection of an embedding into 3 dimensions of a 2-dimensional ( 1 timelike, 1 spacelike ) universe, with only 2 spacelike dimensions of 4-dimensional Physical SpaceTime being suppressed, and so is probably not unreasonably unrealistic ( just replace a S1 circle "horizontal" cross-section of each universe with an S3 3-sphere ) if you restrict your viewpoint to only the 4 dimensions of 4-dimensional Physical SpaceTime.
However, for 27-dimensional J4(Q)o M-theory and 28-dimensional J4(Q) F-theory, another 23 (for J4(Q)o M-theory) and 24 (for J4(Q) F-theory) dimensions are suppressed in the illustration.
Could those 23 or 24 degrees of freedom provide enough separation among the Universes so that they are not really very close to, or intersecting with, each other? To answer that question, consider the physical nature of those 23 or 24 degrees of freedom within the structure of the Jordan algebra J4(Q) of 4x4 Quaternionic Hermitian matrices:
a U T R U* b S V T* S* c W R* V* W* d
Here R, S, T, U, V, W are quaternions, * is conjugation, and a, b, c, d, are real numbers, and:
The dimensions of the 4-dimensional Physical SpaceTime that form the viewpoint of the above illustration of close neighbor/intersecting Universes are Large ( with respect to human experience ) Dimensions.
If the other 24 ( or 23 if a, b, c, d are not independent, as in 27-dimensional J4(Q)o M-theory ) dimensions are also Large, then maybe they could provide enough room for separation among the Universes so that they are not really very close to, or intersecting with, each other. Therefore, we need to ask: How Large are those other 24 dimensions?
It has compact geometric structure of CP2, on which color and electroweak gauge bosons are represented, whose size must be no greater than
If BU were N-dimensional, compactified and smooth, then by a theorem of John F. Nash, Jr., (Ann. Math. 56 (1952) 405-421) it could be realized as a sheet of a real algebraic variety in R^(2N+1).
For an embedding to be isometric, if BU3 is smooth, a dimensionality of n = 3+(1/2)3(3+1) = 9 of an embedding target manifold Mn = M9 would be sufficient, according to the Isometric Embedding Theorem of John F. Nash, Jr., (Ann. Math. 63 (1956) 20-63; Bull. AMS 60 (1954) 480), with a correction noted by Robert M. Solovay in 1998, and improvements as to the required number of dimensions as described by Deane Yang discussing work of Matthias Gunther (Matthias Gunther, Proceedings ICM (Kyoto 1990), Math. Soc. Japan, 1991, pp. 1137-1143).
Marcel Berger says in his book (28 December 2001 version being proof-read by Benjamin McKay) Riemannian Geometry Today Introduction and Panorama at page 173: "... (Nash, 1956, and ... Various authors [who] have since improved N ... ) Every smooth Riemannian manifold of dimension n can be smoothly isometrically embedded in E^N where N = (n + 2)(n + 3)/2 ... We now know that abstract Riemannian manifolds are no more general than submanifolds of the various E^N . ... In very low differentiability, Nash in 1954 and Kuiper in 1955 obtained surprising results ... Any continuous embedding of a Riemannian manifold can be deformed into a C1 isometric embedding. In particular, any n dimensional Riemannian manifold embeds C1 isometrically into E^(2n+1). ...".
Y. Eliashberg N. Mishachev, in their book Introduction to the h-Principle, say: "... A C1-map f: V -> W is called strictly short if f*h < g ... It is well known from classical differential geometry that for r > 1 the Cr -smooth isometric immersions of two-dimensional Riemannian C1-manifolds into R^3 are very specific and rigid maps. For example, any isometric C2 -immersions of the standard sphere S^2 in R^3 into R^3 is congruent to the standard embedding S2 -> R^3 . Till the middle of 1950's mathematicians mostly believed that C1 -smooth isometric immersions V^n -> W^q are also rigid and hard to construct, and, in particular, the aforementioned uniqueness survives also for isometric immersions S2 -> R^3 which are only C1-smooth. It was discovered by J. Nash in 1954 that the situation is, in fact, drastically different when one passes to C1-smooth immersions. On contrast with C2 -immersions they appeared to be extremely flexible: ... (Nash-Kuiper) If n < q then any strictly short immersion f: (V^n, g) ->( R^q, h), where h is the standard metric on R^q , can be C0 -approximated by isometric C1-smooth immersions. Moreover, if the initial immersion f is an embedding then f can be approximated by isometric C1-embeddings. For example there exists a C1-isometric embedding of the standard sphere S^2 and the standard disk D^2 into an arbitrarily small ball in R^3 . Nash proved in ...[1954]... this theorem for n < q - 2 and later Kuiper in ...[1955]... extended the theorem to the case n = q - 1. The parametric version of the theorem is also true and implies ( together with the Example ... Given an arbitrarily C1-map f: (V, g) -> R^q , the composition Ha o f: (V, g) -> R^q , where Ha(x) = ax is a homothety centered at the origin, is strictly short for all sufficiently small a > 0. ... ) the following ... Isometric C -immersions V^n -> R^q , n < q, satisfy the parametric h-principle for all Riemannian manifolds V = (V, g). ... Homotopy principle (h-principle). We say that a differential relation R satisfies the h-principle, or that the h-principle holds for solutions of R , if every formal solution of R is homotopic in Sec R to a genuine solution of R. ... R satisfies the one-parametric h-principle if every family of formal solutions ... of R which joins two genuine solutions ... can be deformed inside Sec R , keeping ...[the two genuine solutions]... fixed, into a family ... of genuine solutions of R. ... A partial differential relation R is any condition imposed on the partial derivatives of an unknown function. A solution of R is any function which satisfies this relation. ... It is customary to visualize a map f: R^n -> R^q as its graph ... in R^n x R^q. ... Mathematicians call this map a section, while Physicists prefer to call it a field (or an R^q -valued field). ...".
Nima Arkani-Hamed, Savas Dimopoulos, Gia Dvali, and Nemanja Kaloper, in hep-ph/9911386, say: "...We propose that our world is a brane folded ...[ or branched ]... many times inside the sub-millimeter extra dimensions. The folding ...[ or branching ]... produces many connected parallel branes or folds with identical microphysics ...[as in this illustration adapted from their paper:
with branching used instead of folding to produce ] ... - a Manyfold. Nearby matter on other folds can be detected gravitationally as dark matter...".
For an embedding of a pseudo-Riemannian manifold, such as the (1+3)-dim SpaceTime Manifold of All the Many Generations of Brane-Universes BU(1,3), Chris Clarke (Proc. Roy. Soc. A314 (1970) 417-428) gave a sufficient isometric embedding target manifold dimensionality of (2,89), or, if BU(1,3) were globally hyperbolic (1,88).
( auxiliary a, b, c, d and neutrino S1 and antineutrino S1 )
of the other 24 ( or 5 of the other 23 if a, b, c, d are not independent, as in 27-dimensional J4(Q)o M-theory ) dimensions might be Larger than 10^(-16) cm.
Therefore, it seems to me that the Casimir force calculations and experimental data, by requiring either an unrealistic UXD size of R = 50 nm or no UXD at all, rule out the existence of UXDs in which not only Gravity, but also Standard Model particles and forces, propagate.
where the constant of proportionality depends not only on the value of n but upon the geometry of the compactified dimensions. Interestingly, if Ms is near a TeV then R = 10^( (30/n) - 19 ) meters; for separations between two masses less than R the gravitational force law becomes 1 / r^(2+n) . For n = 1, R = 10^11 meters and is thus obviously excluded, but, for n = 2 one obtains R = 1 mm, which is at the edge of the sensitivity for existing experiments ... For 2 < n ... the value of R is further reduced and thus we may conclude that the range 2 < n is of phenomenological interest. Astrophysical arguments ...
... suggest that Ms > 50 TeV for n = 2, but allow Ms = 1 TeV for n > 2. ...".
... A second brane near our brane would prevent gravity from spreading far into the extra dimensions and would mean that at distances greater than the brane separation, gravity would fall off at the rate one would expect for four dimensions. ...
... On the other hand, for distances less than the separation of the branes, gravity would vary more rapidly. The very small gravitational force between heavy objects has been measured accurately in the lab but the experiments so far would not have detected the effects of branes separated by less than a few millimeters. ...
... A black hole on a brane will extend to a black hole in the extra dimensions. If the black hole is small, it will be almost round; that is, it will reach about as far into the extra dimensions as its size on the brane. On the other hand, a large black hole will extend to a "black pancake", which is confined to a vicinity of the brane and which is much less thick ( in the extra dimensions ) than it is wide ( on the brane ) ...
... quantum theory means that black holes won't be completely black; they will emit particles and radiation of all kinds like hot bodies. The particles and radiation-like light would be emitted along the brane because matter and nongravitational forces like electricity would be confined to the brane. However, black holes also emit gravitational waves. These would not be confined to the brane but would travel in the extra dimensions as well. If the black hole [were] large and pancake-like, the gravitational waves would stay near the brane. This would mean that the black hole would lose energy ... at the rate one would expect for a black hole in four-dimensional spacetime. ...".
If two near-neighbor branes both had large pancake-like black holes at nearby spacetime regions, as in this illustration adapted from Hawking's illustration above,
then perhaps the gravitational interactions in the extra dimensions between the two black holes would create
a StarGate between the two brane-Universes.
......