[25] viXra:0908.0112 [pdf] submitted on 30 Aug 2009
Authors: Carlos Castro
Comments: 14 pages, This article appeared in Progress in Physics vol. 2 April (2006) 86-92.
We explore Yang's Noncommutative space-time algebra (involving two length scales) within the context
of QM defined in Noncommutative spacetimes; the Holographic principle and the area-coordinates algebra
in Clifford spaces. Casimir invariant wave equations corresponding to Noncommutative coordinates and
momenta in d-dimensions can be recast in terms of ordinary QM wave equations in d+2-dimensions.
It is conjectured that QM over Noncommutative spacetimes (Noncommutative QM) may be described by
ordinary QM in higher dimensions. Novel Moyal-Yang-Fedosov-Kontsevich star products deformations of
the Noncommutative Poisson Brackets (NCPB) are employed to construct star product deformations of
scalar field theories. Finally, generalizations of the Dirac-Konstant and Klein-Gordon-like equations relevant
to the physics of D-branes and Matrix Models are presented.
Category: Quantum Gravity and String Theory
[24] viXra:0908.0111 [pdf] submitted on 28 Aug 2009
Authors: Carlos Castro
Comments: 22 pages, This article appeared in the Journal of Physics A : Math. Gen 39 (2006) 14205-14229.
A novel Weyl-Heisenberg algebra in Clifford-spaces is constructed that is based on a matrix-valued HAB
extension of Planck's constant. As a result of this modifiedWeyl-Heisenberg algebra one will no longer be able
to measure, simultaneously, the pairs of variables (x, px); (x, py); (x, pz); (y, px), ... with absolute precision.
New Klein-Gordon and Dirac wave equations and dispersion relations in Clifford-spaces are presented. The
latter Dirac equation is a generalization of the Dirac-Lanczos-Barut-Hestenes equation. We display the
explicit isomorphism between Yang's Noncommutative space-time algebra and the area-coordinates algebra
associated with Clifford spaces. The former Yang's algebra involves noncommuting coordinates and momenta
with a minimum Planck scale λ (ultraviolet cutoff) and a minimum momentum p = ℏ/R (maximal length
R, infrared cutoff ). The double-scaling limit of Yang's algebra λ → 0, R → ∞, in conjunction with the
large n → ∞ limit, leads naturally to the area quantization condition λR = L2 = nλ2 ( in Planck area
units ) given in terms of the discrete angular-momentum eigenvalues n. It is shown how Modified Newtonian
dynamics is also a consequence of Yang's algebra resulting from the modified Poisson brackets. Finally,
another noncommutative algebra ( which differs from the Yang's algebra ) and related to the minimal length
uncertainty relations is presented . We conclude with a discussion of the implications of Noncommutative
QM and QFT's in Clifford-spaces.
Category: Quantum Gravity and String Theory
[23] viXra:0908.0108 [pdf] submitted on 29 Aug 2009
Authors: Jack Sarfatti
Comments: 7 pages
Einstein's equivalence principle implies that Newton's gravity force has no local objective meaning.
It is an inertial force, i.e. a contingent artifact of the covariantly tensor accelerating
(non-zero g's) Local Non-Inertial Frame (LNIF) detector. Indeed, Newton's gravity force disappears
in a locally coincident non-accelerating (zero g) Local Inertial Frame (LIF). The presence or
absence of tensor spacetime curvature is completely irrelevant to this fact. In the case of an
extended test body, these remarks apply only to the Center of Mass (COM). Stresses across
separated parts of the test body caused by the local objective tensor curvature are a logically
independent separate issue. Garbling this distinction has generated not-even-wrong critiques of
the equivalence principle among "philosophers of physics" and even among some venerable confused
theoretical physicists. Non-standard terms coupling the spin-connection to the commutator of the
Dirac matrices and to the Lorentz group Lie algebra generators are conjectured.
Category: Quantum Gravity and String Theory
[22] viXra:0908.0107 [pdf] submitted on 28 Aug 2009
Authors: Carlos Castro
Comments: 11 pages, This article appeared in Advanced Studies in Theoretical Physics, 2, no. 17, (2008) 825 - 835.
A novel invariant gauge field theory realization of Kac-Moody extensions
of w∞(w1+∞) algebras and based on a Lie group G is constructed.
The most relevant physical feature of this theory is that it describes
an effective field theory of "colored" internal w∞(w1+∞) strings
when G = SU(3). We conclude with a discussion of how these theories
might provide infinite higher conformal-spins extensions of Grand
Unified Models and the Standard Model in four dimensions.
Category: Quantum Gravity and String Theory
[21] viXra:0908.0106 [pdf] submitted on 28 Aug 2009
Authors: Carlos Castro
Comments: 61 pages, This article appeared in Foundations of Physics vol 35, no.6 (2005) 971.
We construct the Extended Relativity Theory in Born-Clifford-Phase spaces with an
upper R and lower length λ scales (infrared/ultraviolet cutoff ). The invariance symmetry
leads naturally to the real Clifford algebra Cl(2, 6,R) and complexified Clifford
ClC(4) algebra related to Twistors. A unified theory of all Noncommutative branes in
Clifford-spaces is developed based on the Moyal-Yang star product deformation quantization
whose deformation parameter involves the lower/upper scale (ℏλ/R). Previous
work led us to show from first principles why the observed value of the vacuum
energy density (cosmological constant ) is given by a geometric mean relationship
ρ ~ L-2PlanckR-2 = L-4P(LPlanck/R)2 ~ 10-122M4Planck,
and can be obtained when the infrared
scale R is set to be of the order of the present value of the Hubble radius. We
proceed with an extensive review of Smith's 8D model based on the Clifford algebra
Cl(1,7) that reproduces at low energies the physics of the Standard Model and Gravity,
including the derivation of all the coupling constants, particle masses, mixing angles,
....with high precision. Geometric actions are presented like the Clifford-Space extension
of Maxwell's Electrodynamics, and Brandt's action related to the 8D spacetime tangentbundle
involving coordinates and velocities ( Finsler geometries ). Finally we outline
the reasons why a Clifford-Space Geometric Unification of all forces is a very reasonable
avenue to consider and propose an Einstein-Hilbert type action in Clifford-Phase spaces
(associated with the 8D Phase space) as a Unified Field theory action candidate that
should reproduce the physics of the Standard Model plus Gravity in the low energy limit.
Category: Quantum Gravity and String Theory
[20] viXra:0908.0104 [pdf] submitted on 27 Aug 2009
Authors: Carlos Castro
Comments: 49 pages, This article appeared in the Int. J. Mod. Phys. A 23, no. 24 (2008) 3891
It is shown how w∞,w1+∞ Gauge Field Theory actions in 2D emerge
directly from 4D Gravity. Strings and Membranes actions in 2D and 3D
originate as well from 4D Einstein Gravity after recurring to the nonlinear
connection formalism of Lagrange-Finsler and Hamilton-Cartan spaces.
Quantum Gravity in 3D can be described by aW1 Matrix Model in D = 1
that can be solved exactly via the Collective Field Theory method. We describe
why a quantization of 4D Gravity could be attained via a 2D Quantum
W∞ gauge theory coupled to an infinite-component scalar-multiplet.
A proof that non-critical W∞ (super) strings are devoid of BRST anomalies
in dimensions D = 27 (D = 11), respectively, follows and which coincide
with the critical (super) membrane dimensions D = 27 (D = 11). We
establish the correspondence between the states associated with the quasi
finite highest weights irreducible representations of W∞,W∞ algebras and
the quantum states of the continuous Toda molecule. Schroedinger-like
QM wave functional equations are derived and solutions are found in the
zeroth order approximation. Since higher-conformal spin W∞ symmetries
are very relevant in the study of 2D W∞ Gravity, the Quantum Hall effect,
large N QCD, strings, membranes, ...... it is warranted to explore
further the interplay among all these theories.
Category: Quantum Gravity and String Theory
[19] viXra:0908.0102 [pdf] submitted on 27 Aug 2009
Authors: Carlos Castro
Comments: 31 pages, This article appeared in Annals of Physics vol. 321, no.4 (2006) 813-839.
We construct the Clifford-space tensorial-gauge fields generalizations of Yang-Mills
theories and the Standard Model that allows to predict the existence of new particles
(bosons, fermions) and tensor-gauge fields of higher-spins in the 10 Tev regime. We
proceed with a detailed discussion of the unique D4 - D5 - E6 - E7 - E8 model of Smith
based on the underlying Clifford algebraic structures in D = 8, and which furnishes all
the properties of the Standard Model and Gravity in four-dimensions, at low energies.
A generalization and extension of Smith's model to the full Clifford-space is presented
when we write explictly all the terms of the extended Clifford-space Lagrangian. We
conclude by explaining the relevance of multiple-foldings of D = 8 dimensions related
to the modulo 8 periodicity of the real Cliford algebras and display the interplay among
Clifford, Division, Jordan and Exceptional algebras, within the context of D = 26, 27, 28
dimensions, corresponding to bosonic string, M and F theory, respectively, advanced
earlier by Smith. To finalize we describe explicitly how the E8 X E8 Yang-Mills theory
can be obtained from a Gauge Theory based on the Clifford ( 16 ) group.
Category: Quantum Gravity and String Theory
[18] viXra:0908.0100 [pdf] submitted on 26 Aug 2009
Authors: Carlos Castro
Comments: 8 pages, This article appeared in Phys. Letts B 626 (2005) 209.
Noncommutative p-brane actions, for even p+1 = 2n-dimensional world-volumes, are written explicitly
in terms of the novel Moyal-Yang ( Fedosov-Kontsevich ) star product deformations of the Noncommutative
Nambu Poisson Brackets (NCNPB) that are associated with the noncommuting world-volume coordinates
qA, pA for A = 1, 2, 3, ...n. The latter noncommuting coordinates obey the noncommutative Yang algebra
with an ultraviolet LP (Planck) scale and infrared (R ) scale cutoff. It is shown why our p-brane actions in
the "classical" limit ℏeff = ℏLP/R → 0 still acquire nontrivial noncommutative corrections that differ from
ordinary p-brane actions. Super p-branes actions in the light-cone gauge are also amenable to Moyal-Yang
star product deformations as well due to the fact that p-branes moving in flat spacetime backgrounds, in the
light-cone gauge, can be recast as gauge theories of volume-preserving diffeomorphisms. The most general
construction of noncommutative super p-branes actions based on non ( anti ) commuting superspaces and
quantum group methods remains an open problem.
Category: Quantum Gravity and String Theory
[17] viXra:0908.0095 [pdf] submitted on 25 Aug 2009
Authors: Carlos Castro
Comments: 22 pages, This article will appear in the Int J. Geom. Meth in Mod Phys vol 6 , no. 6 (Sept 2009)
A candidate action for an Exceptional E8 gauge theory of gravity in 8D
is constructed. It is obtained by recasting the E8 group as the semi-direct
product of GL(8,R) with a deformed Weyl-Heisenberg group associated
with canonical-conjugate pairs of vectorial and antisymmetric tensorial
generators of rank two and three. Other actions are proposed, like the
quartic E8 group-invariant action in 8D associated with the Chern-Simons
E8 gauge theory defined on the 7-dim boundary of a 8D bulk. To finalize,
it is shown how the E8 gauge theory of gravity can be embedded into a
more general extended gravitational theory in Clifford spaces associated
with the Cl(16) algebra and providing a solid geometrical program of a
grand-unification of gravity with Yang-Mills theories. The key question
remains if this novel gravitational model based on gauging the E8 group
may still be renormalizable without spoiling unitarity at the quantum
level.
Category: Quantum Gravity and String Theory
[16] viXra:0908.0094 [pdf] submitted on 25 Aug 2009
Authors: Carlos Castro
Comments: 11 pages, This article appeared in Phys Letts B 675, (2009) 226-230
Born's reciprocal relativity in flat spacetimes is based on the principle
of a maximal speed limit (speed of light) and a maximal proper force
(which is also compatible with a maximal and minimal length duality)
and where coordinates and momenta are unified on a single footing. We
extend Born's theory to the case of curved spacetimes and construct a
deformed Born reciprocal general relativity theory in curved spacetimes
(without the need to introduce star products) as a local gauge theory of
the deformed Quaplectic group that is given by the semi-direct product
of U(1,3) with the deformed (noncommutative) Weyl-Heisenberg group
corresponding to noncommutative generators [Za,Zb] ≠ 0. The Hermitian
metric is complex-valued with symmetric and nonsymmetric components
and there are two different complex-valued Hermitian Ricci tensors
Rμν, Sμν. The deformed Born's reciprocal gravitational action linear in
the Ricci scalars R, S with Torsion-squared terms and BF terms is presented.
The plausible interpretation of Zμ = Eμa Za as noncommuting
p-brane background complex spacetime coordinates is discussed in the
conclusion, where Eμa is the complex vielbein associated with the Hermitian
metric Gμν = g(μν) + ig[μν] = Eμa Ēνb
This could be one of the underlying reasons why string-theory involves gravity.
Category: Quantum Gravity and String Theory
[15] viXra:0908.0090 [pdf] submitted on 24 Aug 2009
Authors: Carlos Castro
Comments: 23 pages, This article appeared in the J. Math. Phys, 48, no. 7 (2007) 073517.
The Octonionic Geometry (Gravity) developed long ago by Oliveira and Marques
is extended to Noncommutative and Nonassociative Spacetime coordinates
associated with octonionic-valued coordinates and momenta. The octonionic
metric Gμν already encompasses the ordinary spacetime metric gμν, in addition
to the Maxwell U(1) and SU(2) Yang-Mills fields such that implements
the Kaluza-Klein Grand Unification program without introducing extra spacetime
dimensions. The color group SU(3) is a subgroup of the exceptional G2
group which is the automorphism group of the octonion algebra. It is shown
that the flux of the SU(2) Yang-Mills field strength Fμν through the area momentum
Σμν in the internal isospin space yields corrections O(1/M2Planck)
to the energy-momentum dispersion relations without violating Lorentz invariance
as it occurs with Hopf algebraic deformations of the Poincare algebra. The
known Octonionic realizations of the Clifford Cl(8),Cl(4) algebras should permit
the construction of octonionic string actions that should have a correspondence
with ordinary string actions for strings moving in a curved Clifford-space
target background associated with a Cl(3, 1) algebra.
Category: Quantum Gravity and String Theory
[14] viXra:0908.0089 [pdf] submitted on 24 Aug 2009
Authors: Carlos Castro
Comments: 14 pages, This article appeared in the Journal of Geometry and Physics, 57 (2007) 1941-1949
The large N → ∞ limit of the Exceptional F4,E6 Jordan Matrix
Models of Smolin-Ohwashi leads to novel Chern-Simons Membrane Lagrangians
which are suitable candidates for a nonperturbative bosonic
formulation of M Theory in D = 27 real, complex dimensions, respectively.
Freudenthal algebras and triple Freudenthal products permits the
construction of a novel E7 X SU(N) invariant Matrix model whose large
N limit yields generalized nonlinear sigma models actions on 28 complex dimensional
backgrounds associated with a 56 real-dim phase space realization
of the Freudenthal algebra. We argue why the latter Matrix
Model, in the large N limit, might be the proper arena for a bosonic
formulation of F theory. To finalize we display generalized Dirac-Nambu-Goto
membrane actions in terms of 3 X 3 X 3 cubic matrix entries that
match the number of degrees of freedom of the 27-dim exceptional Jordan
algebra J3[0].
Category: Quantum Gravity and String Theory
[13] viXra:0908.0088 [pdf] submitted on 24 Aug 2009
Authors: Carlos Castro
Comments: 13 pages, This article appeared in the International Journal of Geometric Methods in Modern Physics (IJGMMP) vol 3, no. 2 (2006) 187-199.
It is the purpose of this work to pursue a novel physical interpretation of the nontrivial
Riemann zeta zeros and prove why the location of these zeros zn = 1/2+iλn corresponds
physically to tachyonic-resonances/tachyonic-condensates, originating from the scattering
of two on-shell tachyons in bosonic string theory. Namely, we prove that if there were
nontrivial zeta zeros (violating the Riemann hypothesis) outside the critical line Real
z = 1/2 (but inside the critical strip), these putative zeros do not correspond to any
poles of the bosonic open string scattering (Veneziano) amplitude A(s, t, u). The physical
relevance of tachyonic-resonances/tachyonic-condensates in bosonic string theory, establishes
an important connection between string theory and the Riemann Hypothesis. In
addition, one has also a geometrical interpretation of the zeta zeros in the critical line
in terms of very special (degenerate) triangular configurations in the upper-part of the
complex plane.
Category: Quantum Gravity and String Theory
[12] viXra:0908.0084 [pdf] submitted on 22 Aug 2009
Authors: Carlos Castro, M. Pavšič
Comments: 65 pages, This article appeared in Progress in Physics vol 1 (April 2005) 31-64
An introduction to some of the most important features of the Extended Relativity
theory in Clifford-spaces (C-spaces) is presented whose "point" coordinates
are non-commuting Clifford-valued quantities which incorporate lines, areas, volumes,
hyper-volumes.... degrees of freedom associated with the collective particle,
string, membrane, p-brane,... dynamics of p-loops (closed p-branes) in target Ddimensional
spacetime backgrounds. C-space Relativity naturally incorporates the
ideas of an invariant length (Planck scale), maximal acceleration, non-commuting
coordinates, supersymmetry, holography, higher derivative gravity with torsion and
variable dimensions/signatures. It permits to study the dynamics of all (closed)
p-branes, for all values of p, on a unified footing. It resolves the ordering ambiguities
in QFT, the problem of time in Cosmology and admits superluminal propagation
( tachyons ) without violations of causality. A discussion of the maximalacceleration
Relativity principle in phase-spaces follows and the study of the invariance
group of symmetry transformations in phase-space allows to show why
Planck areas are invariant under acceleration-boosts transformations . This invariance
feature suggests that a maximal-string tension principle may be operating in
Nature. We continue by pointing out how the relativity of signatures of the underlying
n-dimensional spacetime results from taking different n-dimensional slices
through C-space. The conformal group in spacetime emerges as a natural subgroup
of the Clifford group and Relativity in C-spaces involves natural scale changes in
the sizes of physical objects without the introduction of forces nor Weyl's gauge
field of dilations. We finalize by constructing the generalization of Maxwell theory
of Electrodynamics of point charges to a theory in C-spaces that involves extended
charges coupled to antisymmetric tensor fields of arbitrary rank. In the concluding
remarks we outline briefly the current promising research programs and their
plausible connections with C-space Relativity.
Category: Quantum Gravity and String Theory
[11] viXra:0908.0083 [pdf] submitted on 22 Aug 2009
Authors: Carlos Castro
Comments: 33 pages, This article appeared in the IJGMMP vol 6, no. 3 (2009) 385.
We continue to study the Chern-Simons E8 Gauge theory of Gravity developed by the
author which is a unified field theory (at the Planck scale) of a Lanczos-Lovelock Gravitational
theory with a E8 Generalized Yang-Mills (GYM) field theory, and is defined
in the 15D boundary of a 16D bulk space. The Exceptional E8 Geometry of the 256-dim
slice of the 256 X 256-dimensional flat Clifford (16) space is explicitly constructed
based on a spin connection ΩMAB, that gauges the generalized Lorentz transformations
in the tangent space of the 256-dim curved slice, and the 256 X 256 components of the
vielbein field EMA, that gauge the nonabelian translations. Thus, in one-scoop, the vielbein
EMA
encodes all of the 248 (nonabelian) E8 generators and 8 additional (abelian)
translations associated with the vectorial parts of the generators of the diagonal subalgebra
[Cl(8) ⊗ Cl(8)]diag ⊂ Cl(16). The generalized curvature, Ricci tensor, Ricci
scalar, torsion, torsion vector and the Einstein-Hilbert-Cartan action is constructed. A
preliminary analysis of how to construct a Clifford Superspace (that is far richer than
ordinary superspace) based on orthogonal and symplectic Clifford algebras is presented.
Finally, it is shown how an E8 ordinary Yang-Mills in 8D, after a sequence of symmetry
breaking processes E8 → E7 → E6 → SO(8, 2), and performing a Kaluza-Klein-Batakis
compactification on CP2, involving a nontrivial torsion, leads to a (Conformal) Gravity
and Yang-Mills theory based on the Standard Model in 4D. The conclusion is devoted
to explaining how Conformal (super) Gravity and (super) Yang-Mills theory in any
dimension can be embedded into a (super) Clifford-algebra-valued gauge field theory.
Category: Quantum Gravity and String Theory
[10] viXra:0908.0081 [pdf] submitted on 21 Aug 2009
Authors: Carlos Castro
Comments: 22 pages, This article will appear in the Int J. of Mod. Phys A.
It is shown how a Conformal Gravity and U(4) X U(4) Yang-Mills Grand Unification
model in four dimensions can be attained from a Clifford Gauge Field
Theory in C-spaces (Clifford spaces) based on the (complex) Clifford Cl(4,C) algebra
underlying a complexified four dimensional spacetime (8 real dimensions).
Upon taking a real slice, and after symmetry breaking, it leads to ordinary Gravity
and the Standard Model in four real dimensions. A brief conclusion about
the Noncommutative star product deformations of this Grand Unified Theory
of Gravity with the other forces of Nature is presented.
Category: Quantum Gravity and String Theory
[9] viXra:0908.0080 [pdf] submitted on 21 Aug 2009
Authors: Carlos Castro
Comments: 44 pages, This article has been submitted to the Journal of Mathematical Physics, Aug 2009
Long ago, Bergshoeff, Sezgin, Tanni and Townsend have shown that the
light-cone gauge-fixed action of a super p-brane belongs to a new kind of supersymmetric
gauge theory of p-volume preserving diffeomorphisms (diffs) associated
with the p-spatial dimensions of the extended object. These authors
conjectured that this new kind of supersymmetric gauge theory must be related
to an infinite-dim nonabelian antisymmetric gauge theory. It is shown in
this work how this new theory should be part of an underlying antisymmetric
nonabelian tensorial gauge field theory of p+1-dimensional diffs (upon supersymmetrization)
associated with the world volume evolution of the p-brane. We
conclude by embedding the latter theory into a more fundamental one based on
the Clifford-space geometry of the p-brane configuration space.
Category: Quantum Gravity and String Theory
[8] viXra:0908.0077 [pdf] submitted on 21 Aug 2009
Authors: Jack Sarfatti
Comments: 4 pages
The argument that virtual photons can be globally gauged away I think is spurious. Indeed, the
inconsistency with the boson commutation rules in Heisenberg and Pauli's historic 1929 attempt to
quantize the electromagnetic field disappears once one uses the recently discovered dark energy density.
Category: Quantum Gravity and String Theory
[7] viXra:0908.0023 [pdf] replaced on 2014-09-28 01:48:24
Authors: Matti Pitkänen
Comments: 658 Pages.
The first part of book discusses the new physics relevant to biology and the vision about Universe as topological quantum computer (tqc).
The third part of the book discusses number theoretical models of the genetic code based on p-adic thermodynamics and maximization of entropy or negentropy. These models reproduce the genetic code but fail to make killer predictions.
Category: Quantum Gravity and String Theory
[6] viXra:0908.0019 [pdf] replaced on 2013-09-09 07:09:54
Authors: Matti Pitkänen
Comments: 1119 Pages.
This book is devoted to a detailed representation of the recent state of quantum TGD.
The first part of the book summarizes quantum TGD in its recent form.
An essential element of the formulation is exact Yangian symmetry obtained by replacing the loci of multilocal symmetry generators of Yangian algebra with partonic 2-surfaces so that conformal algebra of Minkowski space is extened to infinite-dimensional algebra bringing in also the conformal algebra assigned to the partonic 2-surfaces. Yangian symmetry requires the vanishing of both UV and IR divergences achieved if the physical particles are bound states of massless wormhole throats.
Rather general arguments suggest the formulation of TGD in terms of holomorphic 6-surfaces in the product CP3× CP3 of twistor spaces leading to a unique partial differential equations determining these surfaces in terms of homogenous polynomials of the projective complex coordinates of the two twistor spaces.
Second part of the book is devoted to hyper-finite factors and hierarchy of Planck constants.
[5] viXra:0908.0018 [pdf] replaced on 2014-09-26 04:44:22
Authors: Matti Pitkänen
Comments: 856 Pages. Updating
The focus of this book is the number theoretical vision about physics. This vision involves three loosely related parts.
Also a general vision about preferred extremals of Kähler action emerges. The basic idea is that imbedding space allows octonionic structure and that field equations in a given space-time region reduce to the associativity of the tangent space or normal space: space-time regions should be quaternionic or co-quaternionic. The first formulation is in terms of the octonionic representation of the imbedding space Clifford algebra and states that the octonionic gamma "matrices" span a complexified quaternionic sub-algebra. Another formulation is in terms of octonion real-analyticity. Octonion real-analytic function f is expressible as f=q1+Iq2, where qi are quaternions and I is an octonionic imaginary unit analogous to the ordinary imaginary unit. q2 (q1) would vanish for quaternionic (co-quaternionic) space-time regions. The local number field structure of the octonion real-analytic functions with composition of functions as additional operation would be realized as geometric operations for space-time surfaces. The conjecture is that these two formulations are equivalent.
Number theoretical vision suggests that infinite hyper-octonionic or -quaternionic primes could could correspond directly to the quantum numbers of elementary particles and a detailed proposal for this correspondence is made. Furthermore, the generalized eigenvalue spectrum of the Chern-Simons Dirac operator could be expressed in terms of hyper-complex primes in turn defining basic building bricks of infinite hyper-complex primes from which hyper-octonionic primes are obtained by dicrete SU(3) rotations performed for finite hyper-complex primes.
Besides this holy trinity I will discuss in the first part of the book loosely related topics such as the relationship between infinite primes and non-standard numbers.
Second part of the book is devoted to the mathematical formulation of the p-adic TGD. The p-adic counterpart of integration is certainly the basic mathematical challenge. Number theoretical universality and the notion of algebraic continuation from rationals to various continuous number fields is the basic idea behind the attempts to solve the problems. p-Adic integration is also a central problem of modern mathematics and the relationship of TGD approach to motivic integration and cohomology theories in p-adic numberfields is discussed.
The correspondence between real and p-adic numbers is second fundamental problem. The key problem is to understand whether and how this correspondence could be at the same time continuous and respect symmetries at least in discrete sense. The proposed explanation of Shnoll effect suggests that the notion of quantum rational number could tie together p-adic physics and quantum groups and could allow to define real-p-adic correspondence satisfying the basic conditions.
The third part is develoted to possible applications. Included are category theory in TGD framework; TGD inspired considerations related to Riemann hypothesis; topological quantum computation in TGD Universe; and TGD inspired approach to Langlands program.
Category: Quantum Gravity and String Theory
[4] viXra:0908.0017 [pdf] replaced on 2014-09-26 04:40:40
Authors: Matti Pitkänen
Comments: 877 Pages. Updating
This book is devoted to what might be called classical TGD.
The simplest view about the hierarchy of Planck constants is as an effective hierarchy describable in terms of local, singular coverings of the imbedding space. The basic observation is that for Kähler action the time derivatives of the imbedding space coordinates are many-valued functions of canonical momentum densities. If all branches for given values of canonical momentum densities are allowed, one obtains the analogs of many-sheeted Riemann surfaces with each sheet giving same contribution to the Kähler action so that Planck constant is effectively a multiple of the ordinary Planck constant.
The topics of the book are organized as follows.
[3] viXra:0908.0016 [pdf] replaced on 2014-09-26 04:42:43
Authors: Matti Pitkänen
Comments: 480 Pages. Updating
The topics of this book is a vision about physics as infinite-dimensional Kähler geometry of the "world of classical worlds" (WCW), with "classical world" identified either as light-like 3-D surface X3 of a unique Bohr orbit like 4-surface X4(X3) or X4(X3) itself. The non-determinism of Kähler action defining Kähler function forces to generalize the notion of 3-surface. Zero energy ontology allows to formulate this generalization elegantly using a hierarchy of causal diamonds (CDs) defined as intersections of future and past directed light-cones, and a geometric realization of coupling constant evolution and finite measurement resolution emerges.
The general vision about quantum dynamics is that the basis for WCW spinor fields defines in zero energy ontology unitary U-matrix having as orthogonal rows M-matrices. Given M-matrix is expressible as a product of hermitian square root of density matrix and S-matrix. M-matrices define time-like entanglement coefficients between positive and negative energy parts of zero energy states represented by the modes of WCW spinor fields.
One encounters two challenges.
The condition of mathematical existence poses stringent conditions on the construction.
[2] viXra:0908.0015 [pdf] replaced on 2014-09-26 04:33:55
Authors: Matti Pitkänen
Comments: 458 Pages. Updating
The topics of this book could be called fringe physics involving claimed phenomena which do not have explanation in terms of standard physics.
Many-sheeted space-time with p-adic length scale hierarchy, the predicted dark matter hierarchy with levels partially characterized by quantized dynamical Planck constant, and the prediction of long ranged color and weak forces alone predict a vast variety of new physics effects. Zero energy ontology predicts that energy can have both signs and that classical signals can propagate in reversed time direction at negative energy space-time sheets and an attractive identification for negative energy signals would be as generalizations of phase conjugate laser beams. This vision leads to a coherent view about metabolism, memory, and bio-control and it is natural to ask whether the reported anomalies might be explained in terms of the mechanisms giving hopes about understanding the behavior of living matter.
[1] viXra:0908.0014 [pdf] replaced on 2014-09-26 04:51:06
Authors: Matti Pitkänen
Comments: 1157 Pages. Updating
This book tries to give an overall view about quantum TGD as it stands now. The topics of this book are following.