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Topological GeometroDynamics (TGD): an Overall View
TGD
by Dr. Matti Pitkanen
Postal address:
Köydenpunojankatu 2 D 11
10940, Hanko, Finland
e-mail: matpitka@luukku.com
URL-address: http://tgdtheory.com
(former address: http://www.helsinki.fi/~matpitka )
"Blog" forum: http://matpitka.blogspot.com/
Contents
1. Introduction
2. Basic ideas
2.1 Two manners to end up with TGD
2.2 p-Adic mass calculations and generalization of number concept
2.3 Physical states as classical spinor fields in the world of classical worlds
2.4 Magic properties of 3-D light-like surfaces and generalization of super-conformal symmetries
2.5 Quantum TGD as almost topological quantum field theory at parton level
2.6 The properties of infinite-dimensional Clifford algebras as a key to the understanding of the
theory
2.6.1 Does quantum TGD emerge from local version of HFF?
2.6.2 Quantum measurement theory with finite measurement resolution
2.6.3 The generalization of imbedding space concept and hierarchy of Planck constants
2.6.4 M→M/N transition!non-commutativity of imbedding space coordinates → number
theoretic braids
2.7 About the construction of S-matrix
2.7.1 S-matrix as a functor
2.7.2 Zero energy ontology
2.7.3 Quantum S-matrix
2.7.4 Quantum fluctuations and Jones inclusions
2.7.5 N = 4 super-conformal invariance
2.7.6 p-Adic coupling constant evolution at the level of free field theory
2.7.7 Number theoretic universality
2.7.8 Generalized Feynman rules
3. Some applications and predictions
3.1 Astrophysics and cosmology
3.2 p-Adic mass calculations
1
3.3 Hierarchy of scaled variants of standard model physics
4. Theoretical challenges
4.1 Basic mathematical conjectures
4.2 How to predict and calculate?
2
Abstract
A brief summary of the basic ideas of Topological GeometroDynamics (TGD), its recent state, its
applications, and its theoretical challenges is given with a special emphasis on the most recent
developments.
Parton level formulation of Auantum TGD as an almost topological Quantum Field Theory using
light-like 3-surfaces as fundamental objects allows a detailed understanding of generalized superconformal symmetries.
In Zero Energy Ontology, S-matrix can be identified as time-like entanglement coefficients between
positive and negative energy parts of Zero Energy states. Besides super-conformal symmetries number
theoretic universality meaning fusion of real and p-adic physics to single coherent whole forces a
formulation in terms of number theoretic braids. A category theoretical interpretation as a functor is
possible.
Finite-temperature S-matrix can be regarded as genuine quantum state in Zero Energy Ontology.
Hyper-finite factors of Type II1 emerge naturally through Clifford algebra of the ”world of classical
worlds” and allow a formulation of quantum measurement theory with a finite measurement resolution.
A generalization of the notion of imbedding space emerges naturally from the requirement that the
choice of quantization axes has a geometric correlate also at the level of imbedding space. The physical
implication is the identification of dark matter in terms of a hierarchy of phases with quantized values
of Planck constant having arbitrarily large values.
3
1. Introduction
The attribute ”geometro-” in TGD was motivated by the idea that submanifold geometry could allow
to realize the dream of Einstein about geometrization of not only gravitational but also other classical
fields. Later the notion extended to a geometrization program of the entire Quantum Theory identifying
quantum states of the Universe as modes of the classical spinor fields in the ”World of Classical
Worlds” (WCW).
Also, the attribute ”topological” in TGD turned to have a much wider meaning than attached to it
first. The original identification of particles as topological inhomogenities of space-time surface TGD
extended the notion of many-sheeted space-time.
In the beginning of the 1990s, p-adic mass calculations inspired the idea that also the local topology
is dynamical. The outcome was a program involving p-adicization and a fusion of physics associated
with real and p-adic topologies based on the notion of number theoretical universality and generalization
of number concept obtained by gluing real and p-adic numbers fields together along common algebraic
numbers.
Still later, the mathematics of Jones inclusions for von Neumann algebras known as hyper-finite
factors of type II1(HFFs) motivated a further generalization of the notion of imbedding space predicting
quantization of Planck constant and existence of macroscopic quantum phases in all length scales.
These phases were identified as a dark matter hierarchy.
Few remarks about the relationship of TGD to standard model and superstring theories is in order. In
M-theory experimental physics represents a low energy phenomenology happening to prevail at this
particular brane at which we live. It happens to be four-dimensional or effectively 4-dimensional and
happens to possess the symmetries of the standard model.
This low energy phenomenology is coded completely by the Standard Model of particle physics'
Lagrangian above electro-weak length scale. It is not probable that the physics above Planck scale can
give any constraints on the development of the theory or that theory could predict anything.
TGD deduces the basic symmetries of the Standard Model from number theory and extends the
super-conformal symmetries responsible for the amazing mathematical successes of superstring models.
The reductionistic world view is given up.
But thanks due to the fractality and huge symmetries, theory is able to make testable predictions. An
entire fractal hierarchy of copies of Standard Model physics is predicted so that a new era of voyages of
discovery to the worlds of dark matter would be waiting for us if we live in a TGD Universe. What
seem to be anomalies of present-day physics (mention only living matter) indeed are anomalies and have
already been an important guideline in attempts to understand what TGD is and what it predicts.
Furthermore, TGD forces also to extend quantum measurement theory to a theory of Consciousness
by replacing the observer regarded as an outsider with the notion of "self".
4
2. Basic ideas
2.1 Two manners to end up with TGD
One can end up with TGD (for overall view, see A1, A2) as a solution of energy problem of General
Relativity by assuming that space-times are representable as 4-surfaces in certain higher-dimensional
space-time allowing Poincare group as isometries [1].
TGD results also as a generalization of string model obtained by replacing strings with light-like 3surfaces representing partons. The choice H = M4 × CP2 leads to a geometrization of elementary
particle quantum numbers and classical fields if one accepts the topological explanation of family
replication phenomenon of elementary fermions based on genus of partonic 2-surface [F1].
Simple topological considerations lead to the notion of many-sheeted space-time and a general
vision about Quantum TGD. In particular, already Classical considerations involving only the induced
gauge field concept strongly suggests fractality meaning infinite hierarchy of copies of Standard Model
physics in arbitrarily long length and time scales [D1].
The huge vacuum degeneracy of Kähler action implying 4-dimensional spin glass degeneracy for
non-vacuum extremals [D1] means a failure of quantization methods based on path integrals and
canonical quantization and leaves the generalization of the notion of Wheeler’s super-space the only
viable road to Quantum theory. By Quantum-Classical correspondence, 4-D spin glass degeneracy has
an interpretation in terms of quantum critical fluctuations possible in all length and time scales so that
Macroscopic and -temporal quantum coherence are predicted. The implementation of this seems to
require a generalization of the ordinary Quantum theory.
2.2 p-Adic mass calculations and generalization of number concept
The success of p-adic mass calculations based on p-adic thermodynamics (see the first part of [6])
motivates the generalization of the notion of number achieved by gluing reals and various algebraic
extensions of p-adic number fields together along common algebraic numbers.
This also implies a generalization of the notion of the imbedding space. It becomes possible to
speak about real and p-adic space-time sheets whose intersection consists of a discrete set of algebraic
points belonging to the algebraic extension of the p-adic numbers considered.
Mass calculations demonstrate that primes p ≈ 2k, k integer are in special role. In particular, primes
and powers of prime as values of k are preferred. Mersenne primes and the ordinary primes associated
with Gaussian Mersennes (1+i)k −1 as their norm seem to be of special importance. A possible
explanation for prime values of k is based on elementary particle black hole analogy and a generalization
of the area law for black-hole entropy.
One can assign to a particle p-adic entropy proportional to the area of the elementary particle horizon
(i.e., area of wormhole throat) and the size scale of the horizon corresponds to an n-ary p-adic length
scale defined by the power kn of prime k [E5].
Deeper explanations involve number theory and quantum information theory. These special primes
and corresponding elementary particles would be winners in a fight for a number theoretic survival.
These primes could also correspond fixed points of p-adic coupling constant evolution.
5
p-Adic physics is interpreted as physics of cognition and intentionality with p-adic space sheets
defining the correlates of cognition or the "mind stuff" of Descartes [E1, H8]. The hierarchy of p-adic
number fields and their algebraic extensions defines cognitive hierarchies with 2-adic numbers at the
lowest level.
One important implication of the fact that p-adically very small distances correspond to very large
real distances is that the purely local padic physics implies long range correlations of real physics
manifesting as p-adic fractality [E1]. This justifies p-adic mass calculations and seems to imply that
cognition and intentionality are present already at elementary particle level.
2.3 Physical states as classical spinor fields in the world of classical worlds
Generalizing Wheeler’s super-space approach, quantum states are identified as modes of classical
spinor fields in the ”World of Classical Worlds” (call it CH) consisting of light-like 3-surfaces of H.
More precisely: CH  CH   CH  , CH    m CH m , m M 4 .
Here, CH m is the space of light-like 3-surfaces of H   M 4  CP2 . A light-like 3-surface has dual
interpretations as (i) random light-like orbit of a partonic 2-surface or (ii) a basic dynamical unit with the
assumption of light-likeness possibly justified as a gauge choice allowed by the 4-D general coordinate
invariance.
The condition that the "World of Classical Worlds" allows Kähler geometry is highly non-trivial.
The simpler example of loop space geometry [16] suggests that the existence of an infinite-dimensional
isometry group -- naturally identifiable as canonical transformations of ∂H± -- is a necessary
prerequisite. Configuration space would decompose to a union of infinite dimensional symmetric spaces
labeled by zero modes having interpretation as classical dynamical variables essential in quantum
measurement theory. (Without zero modes, space-time sheet of electron would be metrically equivalent
with that of galaxy as a point of CH [B1, B2, B3].)
General coordinate invariance is achieved if space-time surface is identified as a preferred extremal
of Kähler action [B1], which is thus analogous to Bohr orbit so that semi-classical Quantum Theory
emerges already at the level of configuration space geometry.
The only free parameter of the theory is Kähler coupling strength αK associated with the exponent of
the Kähler action defining the vacuum functional of the theory [B1, C5]. This parameter is completely
analogous to temperature and the requirement of quantum criticality fixes the value of αK as an analog of
critical temperature. Physical consideration allow to determine the value of αK rather precisely [C5].
The fundamental approach to quantum dynamics assuming light-like 3-D surfaces identified as
partons are the basic geometric objects. In this approach vacuum functional emerges as an appropriately
defined Dirac determinant and the conjecture is that it equals to the exponent of K¨ahler action for a
preferred extremal containing the light-like partonic 3-surfaces as boundaries or as wormhole throats at
which the signature of the induced metric changes.
6
2.4
Magic properties of 3-D light-like surfaces and generalization of super-conformal
symmetries
The very special conformal properties of both boundary M 4 of 4-D lightcone and of light-like
partonic 3-surfaces X3 imply a generalization and extension of the super-conformal symmetries of superstring models to 3-D context [B2, B3, C1].
Both the Virasoro algebras associated with the light-like coordinate r and to the complex coordinate
z transversal to it define super-conformal algebras so that the structure of conformal symmetries is much
richer than in string models.
(a) The canonical transformations of M 4  CP2 give rise to an infinite dimensional
symplectic/canonical algebra having naturally a structure of Kac-Moody type algebra with
respect to the light-like coordinate of M 4  S 2  R and with finite-dimensional Lie group G
replaced with the canonical group.
The conformal transformations of S2 localized with respect to the light-like coordinate act as
conformal symmetries analogous to those of string models. The super-canonical algebra (call it
SC) made local with respect to partonic 2-surface can be regarded as a Kac-Moody algebra
associated with an infinite-dimensional Lie algebra.
(b) The light-likeness of partonic 3-surfaces is respected by conformal transformations of H made
local with respect to the partonic 3-surface and gives to a generalization of bosonic KacMoody algebra (call it KM) also now the longitudinal and transversal Virasoro algebras
emerge. The commutator [KM, SC] annihilates physical states.
(c) Fermionic Kac-Moody algebras act as algebras of left- and right-handed spinor rotations in M4
and CP2 degrees-of -freedom. Also, the modified Dirac operator allows super-conformal
symmetries as gauge symmetries of its generalized eigen modes.
2.5 Quantum TGD as almost topological Quantum Field Theory at parton level
The light-likeness of partonic 3-surfaces fixes the partonic quantum dynamics uniquely and ChernSimons action for the induced Kähler gauge potential of CP2 determines the classical dynamics of
partonic 3-surfaces [B4]. For the extremals of C-S action, the CP2 projection of surface is at most 2dimensional.
The modified Dirac action obtained as its super-symmetric counterpart fixes the dynamics of the
second quantized free fermionic fields in terms of which configuration space gamma matrices and
configuration space spinors can be constructed. The essential difference to the ordinary massless Dirac
action is that induced gamma matrices are replaced by the contractions of the canonical momentum
densities of Chern-Simons action with imbedding space gamma matrices so that modified Dirac action is
consistent with the symmetries of Chern-Simonas action.
Fermionic statistics is geometrized in terms of spinor geometry of WCW since gamma matrices are
linear combinations of fermionic oscillator operators identifiable also as super-canonical generators
[B4]. Only the light-likeness property involving the notion of induced metric breaks the topological
QFT property of the theory so that the theory is as close to a physically trivial theory as it can be.
7
The resulting generalization of N=4 super-conformal symmetry [28] involves super-canonical
algebra (SC)and super Kac-Moody algebra (SKM) [C1]. There are considerable differences as
compared to string models.
(a) Super generators carry fermion number.
No sparticles are predicted (at least super Poincare invariance is not obtained).
SKM algebra and corresponding Virasoro algebra associated with light-like coordinates of X3
and M 4 M4± do not annihilate physical states which justifies p-adic thermodynamics used
in p-adic mass calculations.
4-momentum does not appear in Virasoro generators so that there are no problems with Lorentz
invariance.And mass-squared is p-adic thermal expectation of conformal weight.
(b) The conformal weights and eigenvalues of modified Dirac operator are complex. And the
conjecture is that they are closely related to zeros of Riemann Zeta [B4, C2]. This means that
positive energy particles propagating into geometric-Future are not equivalent with negative
energy particles propagating in geometric-Past. So that crossing symmetry is broken.
Complex conjugation for the super-canonical conformal weights and eigenvalues of the
modified Dirac operator would transform laser photons to their phase conjugates for which
dissipation seems to occur in a reversed direction of geometric-Time. Hence, irreversibility
would be present already at elementary particle level.
2.6 The properties of infinite-dimensional Clifford algebras as a key to the understanding of the
theory
Infinite-dimensional Clifford algebra of CH can be regarded as a canonical example of a von
Neumann algebra known as a hyper-finite factor of Type II1 [17, 19] (shortly HFF) characterized by the
defining condition that the trace of infinite-dimensional unit matrix equals to unity: Tr(Id) = 1.
In TGD framework, the most obvious implication is the absence of fermionic normal ordering
infinities whereas the absence of bosonic divergences is guaranteed by the basic properties of the
configuration space Kähler geometry (in particular, the non-locality of the K¨ahler function as a
functional of 3-surface).
The special properties of this algebra -- which are very closely related to braid and knot invariants
[18, 27]; quantum groups [20, 19]; non-commutative geometry [24]; spin chains; integrable models [22];
topological quantum field theories [23]; conformal field theories; and at the level of concrete physics to
anyons [21] -- generate several new insights and ideas about the structure of Quantum TGD.
2.6.1 Does Quantum TGD emerge from local version of HFF?
There are reasons to hope that the entire Quantum TGD emerges from a version of HFF made local
with respect to D ≤ 8-dimensional space H whose Clifford algebra Cl(H) raised to an infinite tensor
power defines the infinite-dimensional Clifford algebra.
Bott periodicity meaning that Clifford algebras satisfy the periodicity Cl (n  k 8)  Cl (n)  Cl (8k ) is
an essential notion here [C8, C9]. The points m of Mk can be mapped to elements mkγk of the finitedimensional Clifford algebra Cl(H) appearing as an additional tensor factor in the localized version of
the algebra.
8
The requirement that the local version of HFF is not isomorphic with HFF itself is highly non-trivial.
The only manner to achieve non-triviality is to multiply the algebra with a non-associative tensor factor
representing the space of hyper-octonions M8 identifiable as sub-space of complexified octonions with
tangent space spanned by real unit and octonionic imaginary unit multiplied by commuting imaginary
unit. (For a good review about properties of octonions, see [30] ).
Space-times could be regarded equivalently as surfaces in M8 or in M4×CP2 and the dynamics would
reduce to associativity (hyper-quaternionicity) or co-associativity condition. It is rather remarkable that
CP2 forced by the standard model symmetries has also a purely number theoretic interpretation as
parameterizing hyper-quaternionic 4-planes containing a preferred hyper-octonionic imaginary unit
defining hyper-complex structure in M8.
Physically, this choice corresponds to a choice of Cartan algebra of Poincare algebra for which the
system is at rest so that a connection with quantum measurement theory is suggestive. The color group
is identifiable as a subgroup of octonionic automorphism group G2 respecting this choice.
2.6.2 Quantum measurement theory with finite measurement resolution
Jones inclusions N  M [25, 19] of these algebras lead to quantum measurement theory with a
finite measurement resolution characterized by N [C8, C9]. Quantum Clifford algebra M/N interpreted
as N-module creates physical states modulo measurement resolution. Complex rays of the state space
resulting in the ordinary state function reduction are replaced by N-rays and the notions of unitarity,
hermiticity, and eigenvalue generalize [C9, C2].
Non-commutative physics would be interpreted in terms of a finite measurement resolution rather
than something emerging below Planck length scale. An important implication is that a finite
measurement sequence can never completely reduce quantum entanglement so that entire Universe
would necessarily be an organic whole.
At the level of Conscious experience, the entanglement below measurement resolution would give
rise to a pool of shared and fused mental images giving rise to ”stereo consciousness” (say
stereovision) [H1] so that contents of consciousness would not be something completely private as is
usually believed.
Also, fuzzy logic emerges naturally since ordinary spinors are replaced by quantum spinors for
which the discrete spectrum of the eigenvalues of the moduli of its spinor components can be interpreted
as probabilities that corresponding belief is true is universal [C8].
2.6.3 The generalization of imbedding space concept and hierarchy of Planck constants
Quantum classical correspondence suggests that Jones inclusions [25] have space-time correlates
[C8, C9]. There is a canonical hierarchy of Jones inclusions labeled by finite subgroups of SU(2) [19].
This leads to a generalization of the imbedding space obtained by gluing an infinite number of
copies of H regarded as singular bundles over H / Ga ×Gb where Ga × Gb is a subgroup of
SU (2)  SU (2)  SL(2, C )  SU (3) . Gluing occurs along a factor for which the group is same.
The groups in question define in a natural manner the direction of quantization axes for various
isometry charges. This hierarchy seems to be an essential element of quantum measurement theory.
The ordinary Planck constant(as opposed to Planck constants ђa = na ђ0 and ђb = nb ђ0 appearing in the
9
commutation relations of symmetry algebras assignable to M4 and CP2) is naturally quantized as ђa =
(na/nb) ђ0 where ni is the order of maximal cyclic subgroup of Gi.
The hierarchy of Planck constants is interpreted in terms of dark matter hierarchy [C9]. What is also
important is that (na/nb)2 appear as a scaling factor of M4 metric so that Kähler action via its dependence
on induced metric codes for radiative corrections coming in powers of ordinary Planck constant.
Therefore, quantum criticality and vanishing of radiative corrections to functional integral over WCW
does not mean vanishing of radiative corrections.
Ga would correspond directly to the observed symmetries of visible matter induced by the
underlying dark matter [C9]. For instance, in living matter molecules with 5- and 6-cycles could
directly reflect the fact that free electron pairs associated with these cycles correspond to na = 5 and na =
6 dark matter possibly responsible for anomalous conductivity of DNA [C9, J1] and recently reported
strange properties of graphene [40]. Also, the tedrahedral and icosahedral symmetries of water molecule
clusters could have a similar interpretation [44, F10].
A further fascinating possibility is that the observed indications for Bohr orbit quantization of
planetary orbits [37] could have interpretation in terms of gigantic Planck constant for underlying dark
matter [D6] so that Macroscopic and -temporal quantum coherence would be possible in astrophysicallength scales manifesting itself in many manners (say as preferred directions of quantization axis
(perhaps related to the CMB anomaly) or as anomalously low dissipation rates).
2.6.4 M→M/N transition → non-commutativity of imbedding space coordinates → number
theoretic braids
The transition M→M/N interpreted in terms of finite measurement resolution should have space-time
counterpart. The simplest guess is that the functions (in particular, the generalized eigen values)
appearing in the generalized eigen modes of the modified Dirac operator D become noncommutative.
This would be due the non-commutativity of some H coordinates (most naturally the complex
coordinates associated with the geodesic spheres of CP2 and M 4  S 2  R ).
Stringy picture would suggest that these complex coordinates become quantum fields. Their
appearance in the generalized eigenvalues for the modified Dirac operator would reduce the anticommutativity of the induced spinor fields along 1-dimensional number theoretic string to anticommutativity at the points of the number theoretic braid only. The difficulties related to general
coordinate invariance would be avoided by the special character of the quantized H-coordinates.
The detailed argument runs as follows.
(a) Since anti-commutation relations for the fermionic oscillator operators are not changed, it is the
generalized eigen modes of D (with zero modes included) which must become noncommutative and spoil anti-commutativity except in a finite subset of the number theoretic
string identifiable as a number theoretic braid. This means that also the generalized
eigenvalues become non-commuting numbers and should commute only at the points of the
number theoretic braid. This would provide much deeper justification for the proposed
definition of the Dirac determinant than mere number theoretic arguments.
1
(b) Functions of form p  z ( x ) a ( x ) where a(x) is the ordinary matrix acting on H- spinors appear as
spinor modes. z is the complex coordinate for the geodesic sphere S2 of either CP2 or
M 4  S 2  R . z(x) is obtained as a projection of X3 point x to S2. z is an excellent candidate
10
_
for a noncommutative coordinate. In the reduction process, the classical fields z(x) and z (x)
would transform to N-valued quantum fields quantum fields z(x) and z†(x).
(c) The reduction for the degrees of freedom in M→M/N transition must correspond to the reduction
of number theoretic string to a number theoretic braid belonging to the intersection of the real
and p-adic variants of the partonic 3-surface. At the surviving points of the number theoretic
string, the situation is effectively classical in the sense that quantum states can be chosen to be
eigen states of z(xk) in the set {xk} of points defining the number theoretic braid for which the
commutativity conditions [z(xi), z†(xj)] = 0 hold true by definition. The quantum states in
question would be coherent states for which the description in terms of classical fields makes
sense.
(d) The eigenvalues of z(xi) should be algebraic numbers in the algebraic extension of p-adic
numbers involved. Since the spectrum for coherent states a priori contains all complex
numbers, this condition makes sense. The generalized eigenvalues of Dirac operator at these
points would be complex numbers and Dirac determinant would be well-defined and an
algebraic number of required kind. Number theoretic universality of ζ would fix the
eigenvalue spectrum of z to correspond to ζ(s) at points s = ∑k nksk, nk≥0, ζ(sk = 1/2 + iyk) = 0,
yk>0. (Note, however, that this condition is not absolutely essential for the scenario.)
(e) If a reduction to a finite number of modes defined at the number theoretic string occurs, then
[z(x), z†(y)]= 0 can vanish only in a discrete set of points of the number theoretic string. The
situation is analogous to that resulting when the bosonic field z ( ) defined at circle has a
Fourier expansion z ( )   a m exp( m / n) , m = 0,1,...n−1, am† , an   m,n . z (1 ), z † ( 2 ) is




m
given by
 exp( im / n) ,   
1
  2 and in general is nonvanishing.
m
The commutators vanish at points   k , 0  k  n so that physical states can be chosen
to be eigen states of the quantized coordinate z ( ) at points zk = kπ. One can ask whether n
could be identified as the integer characterizing the quantum phase q=exp(iπ/n). The
realization of a sequence of approximations for Jones inclusion as sequence of braid inclusions
however suggests that all values of n are possible.
(f) This picture conforms with the heuristic idea that the low energy limit of TGD should correspond
to some kind of quantum field theory for some coordinates of imbedding space and provides a
physical interpretation for the quantization of bosonic quantum field theories. M→M/N
reduction is analogous to a construction of quantum field theory with cutoff.
The replacement of complex coordinates of the geodesic spheres associated with CP2 with
time shifted copies of ∂M4+ defining a slicing of M4+ would define the quantization of H
coordinates. This notion of quantum field theory fails at the limit of continuum.
2.7 About the construction of S-matrix
In the construction of S-matrix, standard approaches like path integral formalism cannot be applied
since even at the ontological level, new non-trivial elements are encountered. Hence, only the general
principles behind the construction can be discussed.
11
2.7.1 S-matrix as a functor
The almost topological QFT property of Quantum allows to identify S-matrix as a functor from the
category of generalized Feynman cobordisms to the category of operators mapping the Hilbert space of
positive energy states to that for negative energy states.
These Hilbert spaces are assignable to partonic 2-surfaces. Feynman cobordism is the generalized
Feynman diagram having light-like 3-surfaces as lines glued together along their ends defining vertices
as 2-surfaces. This picture differs dramatically from that of string models.
There is a functional integral over the small deformations of Feynman cobordisms corresponding to
maxima of Kähler function. Functor property generalizes the unitary condition and also allows thermal
S-matrices, which seem to be unavoidable since imbedding space degrees of freedom give rise to a
factor of type I with Tr(Id) = 1 (i.e., for factors of type II1 one has Tr(Id) = 1). Hence, Thermodynamics
becomes a natural part of Quantum Theory.
The most general identification of the time-like entanglement coefficients would be as a ”square
root” of density matrix thus satisfying the condition ρ+=SS†, ρ−= SS†, Tr(ρpm) = 1. ρ± has the
interpretation as a density matrix for positive/negative energy states. Physical intuition suggests that S
can be written as a product of universal unitary matrix and square root of state-dependent density matrix.
Clearly, S-matrix can be seen as a matrix-valued generalization of Schrödinger amplitude. Note that
the ”indices” of the S-matrices correspond to configuration space spinors (i.e., fermions and their bound
states giving rise to gauge bosons and gravitons) and to configuration space degrees of freedom (i.e.,
"World of Classical Worlds"). For hyper-finite factor of II1, it is not strictly speaking possible to speak
about indices since the matrix elements are traces of the S-matrix multiplied by projection operators to
infinite-dimensional subspaces from right and left.
The functor property of S-matrices implies that they form a multiplicative structure analogous (but
not identical) to groupoid [36]. Recall that groupoid has associative product and there exist always right
and left inverses and identity in the sense that ff−1 and f−1f are always defined but not identical. One has
fgg−1 = f and f−1fg = g.
The reason for the groupoid like property is that S-matrix is a map between state spaces associated
with initial and final sets of partonic surfaces. These state spaces are different, so that inverse must be
replaced with right and left inverse. The defining conditions for groupoid are replaced with more
general ones.
Also, now associativity holds. But the role of inverse is taken by hermitian conjugate. Thus one has
the conditions fgg† = fρg,+ and f†fg = ρf,−g. The conditions ff† = ρ+ and f†f = ρ− are satisfied.
Here, ρ± is density matrix associated with positive/negative energy parts of Zero Energy state. If the
inverses of the density matrices exist, groupoid axioms hold true since f L1  f †  f 1, satisfies ff−1L = Id+
and f R1   f 1, f † satisfies f−1R f = Id−.
There are good reasons to believe that also tensor product of its appropriate generalization to the
analog of co-product makes sense with non-triviality characterizing the interaction between the systems
of the tensor product. If so, the S-matrices would form very beautiful mathematical structure bringing in
mind the corresponding structures for 2-tangles and N-tangles. Knowing how incredibly powerful the
12
group-like structures have been in Physics, one has good reasons to hope that groupoid-like structure
might help to deduce a lot of information about the quantum dynamics of TGD.
A word about nomenclature is in order. S has strong associations to unitarity. It might be
appropriate to replace S with some other letter. The interpretation of S-matrix as a generalized
Schrödinger amplitude would suggest Ψ-matrix.
Since the interaction with Kea’s M-theory blog at http://kea-monad.blogspot.com ("M" denotes
Monad or Motif in this context) was led to the realization of the connection with density matrix, also Mmatrix might be considered. S-matrix as a functor from the category of Feynman cobordisms in turn
suggests "C" or "F".
Or could just Matrix denoted by "M" in formulas be enough? Certainly it would inspire feeling of
awe!
2.7.2 Zero Energy Ontology
Zero Energy Ontology [C2] is already suggested already by the fact that Robertson-Walker
cosmologies correspond to vacuum extremals in TGD-inspired Cosmology [D5]. ZEO states that all
physical states have vanishing net quantum numbers and decompose to positive and negative energy
components.
There are arguments supporting the belief that the U-matrix characterizing the unitary process
associated with quantum jump is rather trivial from the point-of-view of particle physics [C2]. U matrix
would be, however, very relevant for understanding of p-adic-to-real transitions serving as correlates for
the transformation of intentions to actions.
The almost triviality for real-real transitions and reduction of U-matrix to tensor product of Umatrices associated with positive and negative energy parts of the zero energy state would explain why
the usual positive energy ontology (i.e., having a clockwork Universe as its extreme version) is so good
an approximation. The properties of U-matrix would also explain why sensory perceptions seem to be
about Reality rather than the change of Reality.
A more natural identification of particle physics S-matrix is as entanglement coefficients for timelike entanglement between positive and negative energy states which at the space-time level are located
at boundaries of Future and Past light-cones with time-like separation [C2]. This hypothesis makes
sense thanks to the condition Tr(SS†) = Tr(Id) = 1 holding true for HFFs.
It seems, however, that one must include also factor of type I -- at least in imbedding space degrees
of freedom. And this forces thermal S-matrix. The condition that the sub-factor N  M defines a
measurement resolution implies that S-matrix is crossing symmetric with respect to the multiplication
with elements of N. The S-matrix modulo measurement resolution is finite-dimensional and defined in
M/N so that quantum groups emerge naturally. Note that N defines an effective symmetry analogous to
gauge symmetry.
2.7.3 Quantum S-matrix
The description of finite measurement resolution in terms of Jones inclusion N  M seems to boil
down to a simple rule. Replace ordinary Quantum Mechanics in complex number field C with that in N.
This means that the notions of unitarity, hermiticity, Hilbert space ray, etc. are replaced with their N
counterparts.
13
The full S-matrix in M should be reducible to a finite-dimensional quantum S-matrix in the state
space generated by quantum Clifford algebra M/N which can be regarded as a finite-dimensional matrix
algebra with non-commuting N-valued matrix elements. This suggests that full S-matrix can be
expressed as S-matrix with N-valued elements satisfying N-unitarity conditions.
Physical intuition also suggests that the transition probabilities defined by quantum S-matrix must be
commuting hermitian N-valued operators inside every row and column. The traces of these operators
give N-averaged transition probabilities. The eigenvalue spectrum of these Hermitian gives more
detailed information about details below experimental resolution. N-hermicity and -commutativity pose
powerful additional restrictions on the S-matrix.
Quantum S-matrix defines N-valued entanglement coefficients between quantum states with Nvalued coefficients. How does this affects the situation? The non-commutativity of quantum spinors
has a natural interpretation in terms of fuzzy state function reduction meaning that quantum spinor
corresponds effectively to a statistical ensemble which cannot correspond to pure state.
Does this mean that predictions for transition probabilities must be averaged over the ensemble
defined by ”quantum quantum states”?
2.7.4 Quantum fluctuations and Jones inclusions
Jones inclusions N  M provide also a first principle description of quantum fluctuations since
quantum fluctuations are -- by definition -- quantum dynamics below measurement resolution. This
gives hopes for articulating precisely what the important phrase "long-range quantum fluctuations
around quantum criticality" really means mathematically.
(a) Phase transitions involve a change of symmetry. One might hope that the change of the
symmetry group Ga×Gb could universally code this aspect of phase transitions. This need not
always mean a change of Planck constant. But it always means a leakage between sectors of
imbedding space. At quantum criticality, 3-surfaces would have regions belonging to at least 2
sectors of H.
(b) The long-range of quantum fluctuations would naturally relate to a partial or total leakage of the
3-surface to a sector of imbedding space with larger Planck constant. Meaning zooming up of
various quantal lengths.
(c) For S-matrix in M/N, quantum criticality would mean a special kind of eigen state for the
transition probability operator defined by the S-matrix. The properties of the number theoretic
braids contributing to the S-matrix should characterize this state. The strands of the critical
braids would correspond to fixed points for Ga×Gb or its subgroup.
(d) Accepting number theoretical vision, quantum criticality would mean that super-canonical
conformal weights and/or generalized eigenvalues of the modified Dirac operator correspond
to zeros of Riemann ζ so that the points of the number theoretic braids would be mapped to
fixed points of Ga and Gb at geodesic spheres of ∂M4+ = S2 × R+ and CP2. Also, weaker
critical points which are fixed points of only subgroup of Ga or Gb can be considered.
14
2.7.5 N = 4 super-conformal invariance
The N=4 super-conformal invariance extended to the 3-D situation and involving both supercanonical and super Kac-Moody type algebras associated with radial light-like coordinate and complex
transversal coordinate must pose very stringent additional conditions on S-matrix elements [C1].
The conserved charges associated with super-conformal symmetries commuting with the Cartan
algebras of rotation and color groups and expressible as 2-D integrals allow a reduction to 1-dimensional
integrals. This sub-algebra can be super-symmetrized if fermions obey stringy anti-commutation
relations (anti-commutators vanish only along 1-D curve rather than for entire partonic 2-surface).
The interpretation would be in terms of quantum measurement theory. Only the maximal subalgebras of full algebras commuting with measured isometry charge would be super-symmetrizable.
Note that p-adic counterparts of various charges might be defined as their real values if they are
algebraic numbers.
N=4 SCA is highest super-conformal algebra which is associative. Since associativity should define
the fundamental dynamical principle in the TGD Universe, one expects that N=4 SCA should allow
interpretation as associative imbedding to N=8 non-associative SCA [32, 31] as the analog for the
identification of space-time surface as associative or co-associative surface of hyper-octonionic space.
2.7.6 p-Adic coupling constant evolution at the level of free field theory
The generalized eigen modes of the modified Dirac operator -- the structure of which is fixed
completely by super-symmetry -- assign to a partonic 3-surface a unique value of p-adic prime p with
log(p) appearing as a scaling factor of eigenvalue spectrum. This allows a first principle formulation of
renormalization group equations for p-adic coupling constant evolution at the level of "free theory"
rather than in terms of radiative corrections.
Also, the Dirac determinant is well-defined and involves a product over sub-determinants defined as
products of eigenvalues at the points of a number theoretic braid defined as a subset in the intersection
of real partonic 3-surface and its p-adic counterpart obeying the same algebraic equations.
Algebraicity is indeed possible at parton level due to the almost topological QFT nature of dynamics.
Algebraicity condition could well imply that the number of eigenvalues belonging to the extension of padic numbers is finite so that Dirac determinant would be a finite algebraic number as required by
padicization program.
Physical intuition suggests that the transition M→M/N replacing spinors of WCW with quantum
spinors implies that the anti-commutators of induced spinor fields vanish only for a discrete point set
defining the number theoretic braids in the algebraic intersection of real and p-adic variants of the
partonic 2-surface.
2.7.7 Number theoretic universality
It is easier to understand number theoretic universality from the fact that U-matrix is a unitary matrix
between zero energy states whereas S-matrix represents time-like entanglement between positive and
negative energy parts of zero energy state. By zero energy property, U-matrix can have matrix elements
between different number fields. For S-matrix, this property is in conflict with quantum classical
correspondence.
15
The non-diagonality of U-matrix with respect to number field makes sense only if U-matrix elements
between different number fields belong to some algebraic extension of rationals. If zero energy states
are created by intentional action, the S-matrix associated with them must be algebraic.
A natural generalization is that general S-matrix is obtained as an algebraic continuation of algebraic
(or perhaps even rational S-matrix) by allowing quantum numbers like momenta to become real- or padic valued. Number-field-diagonal U-matrices would thus represent dispersion from rational momenta
to real resp. p-adic momenta and would also be obtained by algebraic continuation [C2].
Number theoretical universality is achieved naturally if the definition of S-matrix elements involves
only the data associated with the number theoretic braid. This leads naturally to a connection with
braiding S-matrices also in the case of real-to-real transitions.
Also, the concept of the "number theoretic string" emerges. This picture becomes highly predictive
if one accepts number theoretic universality of Riemann Zeta [C2] to be discussed at the end of the
article.
The partonic vertices appearing in S-matrix elements should be expressible in terms of N-point
functions of almost topological N=4 superconformal field theory but with the p-adically questionable Nfold integrals over string replaced with sums over the strands of a braid. Spin chain type string
discretization could be in question [C2]. Propagators (that is, correlations between partonic 2-surfaces)
would be due to the interior dynamics of space-time sheets which means a deviation from super string
theory.
Another function of interior degrees-of-freedom is to provide zero modes of metric of WCW
identifiable as classical degrees-of-freedom of quantum measurement theory entangling with quantal
degrees of freedom at partonic 3-surfaces.
2.7.8 Generalized Feynman rules
It is perhaps wise to close the section with a summary about the construction of S-matrix with
emphasis on some new results concerning the role of hyper-finite factors of type II and also III which
seem to appear at the level of operators but not at level of states [C3].
(a) In TGD framework, functional integral formalism is given up. S-matrix should be constructible
as a generalization of braiding S-matrix in such a manner that the number theoretic braids
assignable to light-like partonic 3-surfaces glued along their ends at 2-dimensional partonic 2surfaces representing reaction vertices replicate in the vertex [A7].
(b) The construction of braiding S-matrices assignable to the incoming and outgoing partonic 2surfaces is not a problem [A7]. The problem is to mathematically express what happens in the
vertex. Here the observation that the tensor product of HFFs of type II is HFF of type II is the
key observation.
Many-parton vertex can be identified as a unitary isomorphism between the tensor product
of incoming resp. outgoing HFFs. A reduction to HFF of type II1 occurs because only a finitedimensional projection of S-matrix in bosonic degrees of freedom defines a normalizable state.
(c) HFFs of type III could also appear at the level of field operators used to create states. But at the
level of quantum states, everything reduces to HFFs of type II1 and their tensor products giving
these factors back. If braiding automorphisms reduce to the purely intrinsic unitary
16
automorphisms of HFFs of type III, then for certain values of the time parameter of
automorphism having interpretation as a scaling parameter these automorphisms are trivial.
These time scales could correspond to p-adic time scales so that p-adic length scale
hypothesis would emerge at the fundamental level. In this kind of situation, the braiding Smatrices associated with the incoming and outgoing partons could be trivial so that everything
would reduce to this unitary isomorphism. (A counterpart for the elimination of external legs
from Feynman diagram in QFT.)
(d) One might hope that all complications related to what happens for space-like 3-surfaces could be
eliminated by quantum classical correspondence stating that space-time view about particle
reaction is only a space-time correlate for what happens in quantum fluctuating degrees of
freedom associated with partonic 2-surfaces. This turns out to be the case only in nonperturbative phase.
The reason is that the arguments of n-point function appear as continuous moduli of Kähler
function. In non-perturbative phases, the dependence of the maximum of Kähler function on
the arguments of n-point function cannot be regarded as negligible and the Kähler function
becomes the key to the understanding of these effects including formation of bound states and
color confinement.
(e) In this picture, light-like 3-surface would take the dual role as a correlate for both state and time
evolution of state. This dual role allows to understand why the restriction of time-like
entanglement to that described by S-matrix must be made.
For fixed values of moduli, each reaction would correspond to a minimal braid diagram
involving exchanges of partons being in one-one correspondence with a maximum of Kähler
function.
By quantum criticality and the requirement of ideal quantum-classical
correspondence, only one such diagram would contribute for given values of moduli. Coupling
constant evolution would not, however, be lost. It would be realized as p-adic coupling
constant at the level of free states via the log(p) scaling of eigen modes of the modified Dirac
operator.
(f) A completely unexpected prediction deserving a special emphasis is that number theoretic braids
replicate in vertices. This is of course the braid counterpart for the introduction of annihilation
and creation of particles in the transition from free QFT to an interacting one.
This means classical replication of the number theoretic information carried by them. This
allows to interpret one of the TGD-inspired models of genetic code [L4] in terms of number
theoretic braids representing at deeper level the information carried by DNA.
This picture also provides further support for the proposal that DNA acts as a topological
quantum computer utilizing braids associated with partonic light-like 3-surfaces (which can
have arbitrary size) [E9]. In the reverse direction, one must conclude that even elementary
particles could be information-processing and -communicating entities in a TGD Universe.
17
3. Some applications and predictions
A brief list of examples about applications of TGD is in order.
3.1 Astrophysics and Cosmology
The first applications of TGD were to Astrophysics [D6] and Cosmology [D5].
In Cosmology, the Lorentz invariance of space-time sheet implies Robertson-Walker cosmology and
the absence of finite horizons. The imbedability required for R-W cosmologies allows cosmologies with
critical or over-critical mass density only for a finite duration of cosmic time, after which a phase
transition to a sub-critical cosmology occurs.
The TGD counterpart for Inflationary Cosmology is a quantum critical cosmology in which quantum
fluctuations in arbitrarily long time and length scales give rise to scaled invariant fractal spectrum of
density fluctuations. Many-sheeted space-time implies many-sheeted cosmology with a "Russian doll"like cosmologies within cosmologies structure.
Dark energy corresponds to magnetic flux quanta carrying dark matter. A fractal hierarchy of
magnetic flux quanta is in a key role. The cosmological constant Λ is predicted to have naturally a
correct order of magnitude. Λ depends on p-adic length scale and is apparently piecewise constant as a
function of cosmic time [D5]. It can change at half octaves of M4+ proper time a.
Dark matter hierarchy predicts also the existence quantization axes in astrophysical and
cosmological length scales. Their presence should be visible and could relate to the anomalous behavior
of cosmic microwave background [43].
Also, anomalously low dissipation rates in astrophysical and cosmological length scales due to large
value of Planck constant are expected. The anomalously low dissipation of solar magnetic field might
be one example of this.
TGD leads also to a model for rotating star predicting dynamo-like structure and concentration of
mass on spherical shells at given space-time sheet [D3]. Dark matter becomes a key player in the
dynamics of planetary systems and dictates the dynamics of visible matter via gravitational binding.
The possibility of huge values of Planck constants for dark matter leads to a Hydrogen atom-like
model for planetary systems suggested first by Nottale [37]. And the curve-fits for the radii of planetary
orbits (also those of exoplanets) and mass ratios of planets in the Solar System -- based on number
theoretically preferred values of Planck constant given in terms of integers n characterizing polygons
constructible only ruler and compass -- are rather accurate [D6].
(It should be emphasized that astrophysical Bohr rules can be formulated in General Coordinate
invariant and Lorentz invariant manner in a TGD framework.)
Quantum criticality of a TGD Universe is mathematically analogous to quantum chaos. This
inspires the idea that chaotic quantum states and scattering could be realized in astrophysical length and
time scales. Dark matter structures consisting of rings and spokes should become visible in (say)
galactic collisions and ring galaxies, cartwheel galaxies, and polar ring galaxies are examples of
predicted structures [D6].
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3.2 p-Adic mass calculations
p-Adic mass calculations represent second application [F2, F1, F3, F4, F5]. Inertial 4-momentum
for a given space-time sheet can be understood in TGD framework as a temporal average of nonconserved gravitational 4-momentum expressible as a Noether charge associated with partons so that
Equivalence Principle is satisfied only in a weak form.
An essential assumption is that CP2 Kähler gauge potential has a pure gauge component Aa =
constant in the direction of light-cone proper time a. This means that M4± and CP2 degrees-of-freedom
are not completely uncorrelated.
This explains the generation of inertial mass and p-adic thermodynamics is justified by the
randomness of the motion of partonic 2-surfaces restricted only by light-likeness of the orbit. It is
essential that the conformal symmetries associated with the light-like coordinates of parton and lightcone boundary are not gauge symmetries but dynamical symmetries.
In p-adic thermodynamics, scaling generator L0 having conformal weights as its eigen values
replaces energy and Boltzmann weight exp(H/T) is re placed by pLo/Tp . The quantization Tp = 1/n of
conformal temperature and thus quantization of mass-squared scale is implied by number theoretical
existence of Boltzmann weights.
p-Adic length scale hypothesis states that primes p  2 k , k integer. A stronger hypothesis is that k is
prime (in particular, Mersenne prime or Gaussian Mersenne) makes the model very predictive and finetuning is not possible.
The basic mystery number of elementary particle physics defined by the ratio of Planck mass and
proton mass follows thus from number theory once CP2 radius is fixed to about 104 Planck lengths.
Mass scale becomes additional discrete variable of particle physics so that there is not more need to
force top quark and neutrinos with mass scales differing by 12 orders of magnitude to the same multiplet
of gauge group. Electron, muon, and tau correspond to Mersenne prime k = 127 (the largest non-superastrophysical Mersenne) and Mersenne primes k = 113, 107. Intermediate gauge bosons and photon
correspond to Mersenne M89 and the graviton to M127.
The value of k for a quark can depend on hadronic environment [F4]. This would produce precise
mass formulas for low energy hadrons. This kind of dependence conforms also with the indications that
neutrino mass scale depends on environment [38]. Amazingly, the biologically most relevant length
scale range between 10nm and 4μm contains 4 Gaussian Mersennes (1+i)n − 1, n = 151, 157, 163, 167 .
Scaled copies of Standard Model physics in cell length scale could be an essential aspect of Macroscopic
quantum coherence prevailing in cell length scale.
p-Adic mass thermodynamics is not quite enough and the Higgs boson is needed. Wormhole contact
carrying fermion and anti-fermion quantum numbers at the light-like wormhole throats is excellent
candidate for the Higgs [F2].
The coupling of the Higgs to fermions can be small and induce only a small shift of fermion mass.
This could explain why Higgs has not been observed. Also, the Higgs contribution to mass-squared can
be understood thermodynamically if identified as the absolute value for the thermal expectation value of
the eigenvalues of the modified Dirac operator having interpretation as complex square root of
conformal weight.
19
The original belief was that only the Higgs corresponds to wormhole contact. The assumption that
fermion fields are free in the conformal field theory applying at parton level forces to identify all gauge
bosons as wormhole contacts connecting positive and negative energy space-time sheets [F2].
Fermions correspond to topologically-condensed CP2 type extremals with single light-like
wormhole throat. Gravitons are identified as string-like structures involving pair of fermions or gauge
bosons connected by a flux tube. Partonic 2-surfaces are characterized by genus which explains family
replication phenomenon and an explanation for why their number is '3' emerges [F1].
Gauge bosons are labeled by pairs (g1,g2) of handle numbers and can be arranged to octet and
singlet representations of the resulting dynamical SU(3) symmetry. Ordinary gauge bosons are SU(3)
singlets. The heaviness of octet bosons explains why higher boson families are effectively absent. The
different character of bosons could also explain why the p-adic temperature for bosons is Tp = 1/n < 1 so
that Higgs contribution to the mass dominates.
3.3 Hierarchy of scaled variants of Standard Model physics
TGD predicts an infinite hierarchy of scaled-up variants of elementary particle physics which means
the failure of reductionism but with precise quantitative predictions made possible by fractal scaling
arguments.
p-Adic length scale hierarchy suggests hierarchy of physics with mass spectra deducible by simple
scaling arguments. Hierarchy of dark matters predicts a hierarchy of zoomed-up variants of ordinary
elementary particles with identical mass spectra. M89-copy of ordinary hadronic physics characterized
byM107 [F5] is an especially interesting possibility concerning LHC. The copies of hadronic physics
(say for M61) could also explain the cosmic rays with anomalously high energies.
The idea about dark matter might have applications already in nuclear physics and even hadron
physics might involve larger values of Planck constant. Valence quarks could correspond to some low - perhaps the lowest level q = exp(inπ/3), n = 3 -- for dark matter hierarchy. Note that the corresponding
group is Z3 and corresponds to SU(3) by McKay correspondence (flavor SU(3) of Gell-Mann perhaps?).
The model of nucleus as entangled nuclear string with nucleons connected by color flux tubes
containing at their ends dark exotic quark and anti-quark with mass scale of electron emerges naturally
[F8, F9]. It predicts with surprising precision the ground state binding energies as well as the energies
associated with giant resonances.
The recently observed dependence of nuclear reaction rates on electronic environment [41] and
claims for cold fusion [42] could be understood in terms of many-sheeted space-time concept with
many-sheeted-ness allowing to circumvent Coulomb barrier [F8, F9].
Many-sheeted space-time and dark matter as phases with a large Planck constant has the most
natural applications to condensed matter physics and biology. The discrete rotational symmetries
associated with given value of M4 Planck constant serve as the unique signature. Copies of atomic and
molecular physics -- but with scaled spectra of atomic energy levels -- are predicted. And this might
explain the so called "hydrino atom" [39] with binding energy scale scaled up by k2, k = 1,2, 3, ... [C9].
A model of high Tc superconductivity as a quantum critical phenomenon involving electrons with
ђ=211ђ0 with zoomed up Compton length emerges. The overlap criterion for the formation of Cooper
pairs is satisfied and gap energy and critical temperature are scaled up correspondingly [J1, J2, J3].
20
Color interactions in zoomed up length scales are essential for the model of exotic Coopers with nonvanishing spin which are present besides large ђ variants of BCS-type Cooper pairs.
Dropping of particles to larger space-time sheets with a liberation of Zero-Point kinetic energy as a
metabolic energy leads to the idea of universal metabolic energy quanta [K6]. The new view about
Energy and Time has important implications. Phase conjugate photons with negative energies
propagating into the geometric-Past provide a mechanism for a communication with the geometric-Past.
The mechanisms of remote metabolism (i.e., ”quantum credit card”) and memory recall as this
communication are possible applications.
Time-like entanglement represented by S-matrix in zero energy ontology with the geometric-Past
provides a second mechanism of long-term memory based on sharing of mental images with the brain
of geometric-Past [H6].
One important implication of many-sheeted space-time concept is the notion of field body. The
classical field configurations associated with a given physical system are topologically quantized so that
it is possible to assign to the system a field identity (i.e., ”field body”). The notion of magnetic body is
in a key role in the theory of living systems (in particular, the model of EEG and its generalization to a
hierarchy of EEGs [M3].
Also GEG corresponding to gluons and ZEG and WEG corresponding to exotic dark Z0 and W
bosons (with the latter making possible charge entanglement in Macroscopic length scales) are possible.
A large value of the Planck constant implying that EEG photons have energies above thermal energy
would explain why EEG photons correlate with brain function and contents of Consciousness.
The strange findings of Libet [45] about time delays of active and passive aspects of Consciousness
could be seen as a support for the notion of magnetic body [M3].
4. Theoretical challenges
The basic theoretical challenges of Quantum TGD relate to the refining the conceptual framework
and the interpretation of the theory to proving the basic mathematical conjectures and the development
of calculational apparatus.
4.1 Basic mathematical conjectures
There are several approaches to TGD. They lead to mathematical conjectures which should hold
true. The basic approach relies on physics as infinite-dimensional geometry with quantum states of
Universe identified as modes of spinor field in WCW and involves several conjectures.
(a) The Kähler function defining the geometry is expressible as a sum of absolute extrema for
regions of space-time surface in which Kähler action density is extremum. Both maximum and
minimum could be allowed.
(b) That Kähler geometry possesses the claimed super-conformal symmetries implying that light-like
partonic surfaces are fundamental objects leading to the formulation of theory as almost
topological field theory.
(c) TGD Universe is quantum critical in the sense that the radiative corrections vanish around
maxima of Kähler function for the critical value of Kähler coupling strength αK so that
functional integral can be calculated exactly (requiring generalization of Duistermaat-Heckman
21
theorem [29] ). This is absolutely essential for the reduction of S-matrix elements to algebraic
numbers required by number theoretic universality.
(d) The vacuum functional defined as a product of appropriately defined Dirac determinants for
light-like partonic 3-surfaces gives exponent of Kähler action for preferred extremals.
TGD as a generalized number theory and p-adicization program relying on number theoretical
universality gives strong constraints on the theory.
(a) Number theoretical universality leads to the generalization of number concept and the notions of
imbedding space and space-time (p-adic spacetime sheets etc.). The approach allows to assign
a prime to partonic 3-surface and leads to the notion of number theoretic braid as a subset of
the intersection of real and p-adic variants of partonic 3-surface [B4, C9, C2].
S-matrix would be expressible in terms of N-point functions in the point set defined by the
number theoretic braid. The transition M→M/N could reduce the anti-commutativity for noncommutative quantal versions of the induced spinor fields along 1-D curve to anticommutativity for their quantum counterparts at points of the number theoretic braid.
(b) Number theoretical approach inspires conjectures related to Riemann Zeta stating that the nontrivial zeros ѕ=1/2+iy of ζ and also the numbers piy for all primes p (and thus for rationals) are
algebraic numbers [E8].
An even stronger conjecture would be that the values of ζ at points s = ∑k nksk, nk≥0, yk>0
(in upper half plane) are algebraic numbers. These conjectures imply that phase transitions
changing value of Planck constant can occur only when the points of number theoretical braid
correspond to zeros of ζ, which has been indeed assigned with criticality.
The failure of some of these conjectures (in particular the third one) does not kill TGD. There is an
entire zoo of ζ functions sharing the basic properties of Riemann zeta (functional equation; zeros at
critical line) [33, 34]. In particular, the so-called "local" zeta functions [35] analogous to the factors
1/(1−p−s) of Riemann zeta in the product decomposition share these properties and are rational functions
of t = p−s.
The values of these ζ functions are automatically algebraic numbers for linear combinations of zeros
if p−s is an algebraic number for zeros. Since the modified Dirac operator assigns to the partonic 2surface a unique p-adic prime p, these zetas would be very natural in TGD framework. Typically, local
zetas code for data about algebraic variety. And in [C1, E3], a local zeta coding for the numbers of
points of partonic 2-surface projecting to a given point of the geodesic sphere of CP2 in the
approximation O(pn) = 0 is proposed.
That the solutions of modified Dirac equation would code algebraic data about partonic 2-surface
would realize quantum classical correspondence very elegantly providing a back-reaction from the
space-time geometry to quantum states essential for the self-referentiality of conscious experience (in
quantum jump, the system can become conscious about what it was conscious of).
The identification of Clifford algebra of "World of Classical Worlds" as a hyper-finite factor of type
II1 (HFF) leads to the following list of beliefs.
(a) TGD Universe emerges from HFF extended to a local algebra with respect to space defining the
finite-dimensional Clifford algebra whose infinite tensor power HFF is. The only local variant
22
of HFF based relies on the localization in hyper-octonionic imbedding space M8 allowing
interpretation in terms of complexified quantum octonions and having hyper-octonions
identifiable as 8-D Minkowski space M8 as a space of eigenvalues for a maximally commuting
set of hermitian quantum coordinates [C8]. Classical number fields define naturally a
dynamical hierarchy.
(b) Dynamics reduces to associativity condition at both quantal (conformal field theories, reduction
of N=8 SCA to N=4 SCA) and at the classical level. One can equivalently regard space-time
surfaces as 4-surfaces in M8± or M4±×CP2 and the dynamics in M8 reduces to the condition that
space-time surface is associative or co-associative or equivalently hyperquaternionic or cohyper-quaternionic. The notion of calibration allowing to understand minimal surfaces
generalizes to Kähler calibration.
(c) Quantum measurement theory with finite measurement resolution emerges from Jones inclusions
N  M and quantum Clifford algebras M/N provide the description of theory modulo
measurement resolution in terms finite-dimensional algebras and state spaces. Number
theoretic braids relate closely to the sequences of inclusions of Temperley-Lieb algebras [26]
associated with N-braids and defining a hierarchy of approximations for inclusions of HFFs.
(d) The proposed quantization of Planck constants inspired by quantum classical correspondence
requiring that Jones inclusions characterized by finite subgroups Ga  Gb of
SU (2)  SU (2)  SL(2, C )  SU (3) have representation as singular bundle structures
H→H/Ga×Gb defines a mathematically-consistent theory and allows interpretation in terms of
quantum measurement theory.
4.2 How to predict and calculate?
Second class of challenges relates to the development of calculational tools. In the recent state, TGD
predicts by using symmetries and scaling arguments justified by fractality (in particular, p-adic length
scale hypothesis and scaling of Planck constant). A good example about how to circumvent these
difficulties is provided by p-adic mass calculations. They should be carried out in full generality.
The basic reason for the state of affairs (besides human limitations) is that the perturbative approach
to Quantum Field Theories relying on functional integrals and Feynman rules fails in TGD framework.
On the other hand, perturbative expansion is not needed since coupling constant evolution by
radiative corrections can be realized as p-adic coupling constant evolution at the level of ”free” theory.
The challenge is to construct a generalization of braiding S-matrix satisfying algebraic universality and
extended super-conformal symmetries and conforming with the notion of quantum measurement theory
with a finite measurement resolution based on Jones inclusions.
(a) The representation theory of 2-dimensional super-conformal algebras should be extended to the
case of light-like 3-surfaces so that the extended super-conformal symmetries could be used to
deduce information about S-matrix elements. The extended super-conformal algebra
decomposes to a direct sum of representations of ordinary super-conformal algebras meaning
that it is possible to apply existing theory to situations like p-adic mass calculations.
(b) One could try to deduce information about S-matrix as a generalization of braiding S-matrix [27]
assuming that S-matrix defined in M can be identified as unitary entanglement coefficients
(Tr(SS†) = 1) between positive and negative energy components of zero energy states. One
23
could also try to deduce the S-matrix modulo measurement resolution (which reduces to a
finite-dimensional S-matrix in non-commutative spinor space) by utilizing the crossing
symmetry of the action of N  M implied by the notion of N ray.
(c) At least for M/N description in terms of non-commutative variants of induced spinor fields, the
notion of number theoretic braid discretizes the construction. But to utilize this powerful
number, theoretic tools would be needed. Algebraic geometers can probably provide this kind
of tools (in particular, theorems about rational and algebraic points of algebraic surfaces).
(d) S-matrix involves also a functional integral over the "World of Classical Worlds" (WCW). The
vanishing of radiative corrections is a necessary condition for algebraicity. Algebraic numbers
result if vacuum functional reduces to a Dirac determinant defined in terms of number theoretic
braids associated with partonic 3-surfaces. Quantum criticality is consistent with p-adic
coupling constant evolution since the scaling of M4 part of imbedding space metric is
proportional to ђ so that Kähler action codes for radiative corrections classically.
Obviously, there is a long way to explicit expressions for S-matrix elements. One could, however,
test the theory by trying to deduce the implications of p-adic coupling constant evolution coded by the
log(p) scaling of eigen modes of modified Dirac operator at the level of ”free” theory rather than as
radiative corrections.
Very strong constraints between electro-weak and color coupling constant evolution are expected on
basis of induced gauge field concept. I have already proposed an explicit formula reducing color
coupling constant evolution to that for electro-weak U(1) coupling strength [C5]. Also, a formula for
gravitational constant has been proposed.
Acknowledgements
I wish to thank the participants of the Unified Theories conference organized by the Institute for
Strategic Research for very stimulating discussions and for the Institute for Strategic Research for a
financial support.
24
References
[StealthSkater note: many of the following are already archived at the "Pitkanen" page at =>
doc pdf URL ]
Online books about TGD
1. M. Pitkanen (2006), Topological Geometrodynamics: Overview.
http://tgdtheory.com/public_html/tgdview/tgdview.html
2. M. Pitkanen (2006), Quantum Physics as Infinite-Dimensional Geometry.
http://tgdtheory.com/public_html/tgdgeom/tgdgeom.html
3. M. Pitkanen (2006), Physics in Many-Sheeted Space-Time.
http://tgdtheory.com/public_html/tgdclass/tgdclass.html
4. M. Pitkanen (2006), Quantum TGD.
http://tgdtheory.com/public_html/tgdquant/tgdquant.html
5. M. Pitkanen (2006), TGD as a Generalized Number Theory.
http://tgdtheory.com/public_html/tgdnumber/tgdnumber.html
6. M. Pitkanen (2006), p-Adic length Scale Hypothesis and Dark Matter Hierarchy.
http://tgdtheory.com/public_html/paddark/paddark.html
7. M. Pitkanen (2006), TGD and Fringe Physics.
http://tgdtheory.com/public_html/freenergy/freenergy.html
Online books about TGD inspired theory of Consciousness and Quantum Biology
8. M. Pitkanen (2006), Bio-Systems as Self-Organizing Quantum Systems.
http://tgdtheory.com/public_html/bioselforg/bioselforg.html
9. M. Pitkanen (2006), Quantum Hardware of Living Matter.
http://tgdtheory.com/public_html/bioware/bioware.html
10. M. Pitkanen (2006), TGD-inspired Theory of Consciousness.
http://tgdtheory.com/public_html/tgdconsc/tgdconsc.html
11. M. Pitkanen (2006), Mathematical Aspects of Consciousness Theory.
http://tgdtheory.com/public_html/genememe/genememe.html
12. M. Pitkanen (2006), TGD and EEG.
http://tgdtheory.com/public_html/tgdeeg/tgdeeg/tgdeeg.html
13. M. Pitkanen (2006), Bio-Systems as Conscious Holograms.
http://tgdtheory.com/public_html/hologram/hologram.html
14. M. Pitkanen (2006), Magnetospheric Consciousness.
http://tgdtheory.com/public_html/magnconsc/magnconsc.html
15. M. Pitkanen (2006), Mathematical Aspects of Consciousness Theory.
http://tgdtheory.com/public_html/magnconsc/mathconsc.html
25
PowerPoint representations
TGD Topological GeometroDynamics.
http://tgdtheory.com/public_html/tgdppt/TGDstartweb.mht
TGD-inspired Theory of Consciousness.
http://tgdtheory.com/tgdppt/TGDconscstartweb.mht
TGD and Quantum Biology.
http://tgdtheory.com/tgdppt/TGDbiostartweb.mht
References to the chapters of books
[A1] The chapter "An Overview about the Evolution of Quantum-TGD" of [1].
http://tgdtheory.com/public_html/tgdview/tgdview.html#evoI
[A2] The chapter "An Overview about Quantum-TGD" of [1].
http://tgdtheory.com/public_html/tgdview/tgdview.html#evoII
[A7] The chapter "Equivalence of Loop Diagrams with Tree Diagrams and Cancellation of Infinities
in Quantum-TGD" of [1].
http://tgdtheory.com/public_html/tgdview/tgdview.html#bialgebra
[B1] The chapter "Identification of the Configuration Space Kähler Function" of [2].
http://tgdtheory.com/public_html/tgdgeom/tgdgeom.html#kahler
[B2] The chapter "Construction of Configuration Space Kähler Geometry from Symmetry
Principles: Part I" of [2].
http://tgdtheory.com/public_html/tgdgeom/tgdgeom.html#compl1
[B3] The chapter "Construction of Configuration Space Kähler Geometry from Symmetry
Principles: Part II" of [2].
http://tgdtheory.com/public_html/tgdgeom/tgdgeom.html#compl2
[B4] The chapter "Configuration Space Spinor Structure" of [2].
http://tgdtheory.com/public_html/tgdgeom/tgdgeom.html#cspin
[C1] The chapter "Construction of Quantum Theory: Symmetries" of [4].
http://tgdtheory.com/public_html/tgdquant/tgdquant.html#quthe
[C2] The chapter "Construction of Quantum Theory: S-matrix" of [4].
http://tgdtheory.com/public_html/tgdquant/tgdquant.html#towards
[C3] The chapter "Hyper-Finite Factors and Construction of S-matrix" of [4].
http://tgdtheory.com/public_html/tgdquant/tgdquant.html#HFSmatrix
[C4] The chapter "Previous Attempts to Construct S-matrix" of [4].
http://tgdtheory.com/public_html/tgdquant/tgdquant.html#smatrix
[C5] The chapter "Is it Possible to Understand Coupling Constant Evolution at Space-Time Level?"
of [4].
http://tgdtheory.com/public_html/tgdquant/tgdquant.html#rgflow
[C8] The chapter "Was von Neumann Right After All?" of [4].
http://tgdtheory.com/public_html/tgdquant/tgdquant.html#vNeumann
[C9] The chapter "Does TGD Predict the Spectrum of Planck Constants?" of [4].
http://tgdtheory.com/public_html/tgdquant/tgdquant.html#Planck
[D1] The chapter "Basic Extremals of Kähler Action" of [3].
http://tgdtheory.com/public_html/tgdclass/tgdclass.html#class
[D3] The chapter "The Relationship Between TGD and GRT" of [3].
26
http://tgdtheory.com/public_html/tgdclass/tgdclass.html#tgdgrt
[D4] The chapter "Cosmic Strings" of [3].
http://tgdtheory.com/public_html/tgdclass/tgdclass.html#cstrings
[D5] The chapter "TGD and Cosmology" of [3].
http://tgdtheory.com/public_html/tgdclass/tgdclass.html#cosmo
[D6] The chapter "TGD and Astrophysics" of [3].
http://tgdtheory.com/public_html/tgdclass/tgdclass.html#astro
[E1] The chapter "TGD as a Generalized Number Theory: p-Adicization Program" of [5].
http://tgdtheory.com/public_html/tgdnumber/tgdnumber.html#visiona
[E2] The chapter "TGD as a Generalized Number Theory: Quaternions, Octonions, and their Hyper
Counterparts" of [5].
http://tgdtheory.com/public_html/tgdnumber/tgdnumber.html#visionb
[E3] The chapter "TGD as a Generalized Number Theory: Infinite Primes" of [5].
http://tgdtheory.com/public_html/tgdnumber/tgdnumber.html#visionc
[E5] The chapter "p-Adic Physics: Physical Ideas" of [5].
http://tgdtheory.com/public_html/tgdnumber/tgdnumber.html#phblocks
[E8] The chapter "Riemann Hypothesis and Physics" of [5].
http://tgdtheory.com/public_html/tgdnumber/tgdnumber.html#riema
[E9] The chapter "Topological Quantum Computation in TGD Universe" of [5].
http://tgdtheory.com/public_html/tgdnumber/tgdnumber.html#tqc
[F1] The chapter "Elementary Particle Vacuum Functionals" of [6].
http://tgdtheory.com/public_html/paddark/paddark.html#elvafu
[F2] The chapter "Massless States and Particle Massivation" of [6].
http://tgdtheory.com/public_html/paddark/paddark.html#mless
[F3] The chapter "p-Adic Particle Massivation: Elementary Particle Masses" of [6].
http://tgdtheory.com/public_html/paddark/paddark.html#padmass2
[F4] The chapter "p-Adic Particle Massivation: Hadron Masses" of [6].
http://tgdtheory.com/public_html/paddark/paddark.html#padmass3
[F5] The chapter "p-Adic Particle Massivation: New Physics" of [6].
http://tgdtheory.com/public_html/paddark/paddark.html#padmass4
[F8] The chapter "TGD and Nuclear Physics" of [6].
http://tgdtheory.com/public_html/paddark/paddark.html#padnucl
[F9] The chapter "Nuclear String Physics" of [6].
http://tgdtheory.com/public_html/paddark/paddark.html#nuclstring
[F10] The chapter "Dark Nuclear Physics and Condensed Matte" of [6].
http://tgdtheory.com/public_html/paddark/paddark.html#exonuclear
[H1] The chapter "Matter, Mind, Quantum" of [10].
http://tgdtheory.com/public_html/tgdconsc/tgdconsc.html#conscic
[H6] The chapter "Quantum Model of Memory" of [10].
http://tgdtheory.com/public_html/tgdconsc/tgdconsc.html#memoryc
[H8] The chapter "p-Adic Physics as Physics of Cognition and Intention" of [10].
http://tgdtheory.com/public_html/tgdconsc/tgdconsc.html#cognic
27
[J1] The chapter "Bio-Systems as Superonductors: Part I" of [9].
http://tgdtheory.com/public_html/bioware/bioware.html#superc1
[J2] The chapter "Bio-Systems as Superconductors: Part II" of [9].
http://tgdtheory.com/public_html/bioware/bioware.html#superc2
[J3] The chapter "Bio-Systems as Superconductors: Part III" of [9].
http://tgdtheory.com/public_html/bioware/bioware.html#superc3
[J6] The chapter "Coherent Dark Matter and Bio-Systems as Macroscopic Quantum Systems" of [9].
http://tgdtheory.com/public_html/bioware/bioware.html#darkbio
[K6] The chapter "Macroscopic Quantum Coherence and Quantum Metabolism as Different Sides of
the Same Coin" of [13].
http://tgdtheory.com/public_html/hologram/hologram.html#metab
[L4] The chapter "Unification of 4 Approaches to the Genetic Code" of [11].
http://tgdtheory.com/public_html/genememe/genememe.html#divicode
[M3] The chapter "Dark Matter Hierarchy and Hierarchy of EEGs" of [12].
http://tgdtheory.com/public_html/tgdeeg/tgdeeg/tgdeeg.html#eegdark
Mathematical references
16. D. S. Freed (1985), "The Geometry of Loop Groups (Thesis)". Berkeley: University of
California.
17. J. Dixmier (1981), Von Neumann Algebras, Amsterdam: North-Holland Publishing Company.
[First published in French in 1957: Les Algebres d’Operateurs dans l’Espace Hilbertien,
Paris: Gauthier-Villars.]
18. V. F. R. Jones (1983), "Braid groups, Hecke algebras, and type II1 factors, Geometric methods in
operator algebras", Proc. of the US-Japan Seminar, Kyoto, July 1983.
19. V. Jones (2003), "In and Around the origin of Quantum Groups", arXiv:math.OA/0309199.
20. C. Gomez, M. Ruiz-Altaba, G. Sierra (1996), Quantum Groups and Two-Dimensional Physics,
Cambridge University Press.
21. F. Wilzek (1990), "Fractional Statistics and Anyon Super-Conductivity", World Scientific. R.B.
Laughlin (1990), Phys. Rev. Lett. 50, 1395.
22. P.Dorey (1998). "Exact S-matrices", arXiv.org:hep-th/9810026.
23. S. Sawin (1995), "Links, Quantum Groups, and TQFT’s", q-alg/9506002.
24. A. Connes (1994), Non-commutative Geometry, San Diego: Academic Press.
25. V. F. R. Jones (1983), "Index for Subfactors", Invent. Math. (72),1-25.
26. N. H. V. Temperley and E. H. Lieb (1971), "Relations between the percolation and colouring
problem and other graph-theoretical problems associated with regular planar lattices:
some exact results for the percolation problem", Proc. Roy. Soc. London 322 (1971), 251280.
27. E. Witten 1989), "Quantum Field Theory and the Jones polynomial", Comm. Math. Phys. 121,
351-399.
28 M. Gunaydin(1993), "N=4 superconformal algebras and gauges Wess-Zumino-Witten models",
Phys. Rev. D, Vol. 47, No 8. A. Ali (2003), "Types of 2-dimensional N=4
superconformal field theories", Pramana, vol. 61, No. 6, pp. 1065-1078.
28
29. Duistermaat, J.J. and Heckmann, G.J. (1982), Inv. Math. 69, 259.
30. John C. Baez (2001), "The Octonions", Bull. Amer. Math. Soc. 39 (2002), 145-205.
http://math.ucr.edu/home/baez/Octonions/octonions.html
31. A. Ali (2003), "An N=8 superaffine Malcev algebra and its N=8 Sugawara",
hep-th/01015313.
32. F. Englert et al (1988), J. Math. Phys. 29, 281.
33. Zeta function, http://en.wikipedia.org/wiki/Zeta−function.
34 A directory of all known zeta functions,
http://secamlocal.ex.ac.uk/mwatkins/zeta/directoryofzetafunctions.htm
35. Weil conjectures, http://en.wikipedia.org/wiki/Weil−conjectures.
36. Groupoid, http://en.wikipedia.org/wiki/Groupoid.
References related to anomalies
37. D. Da Roacha and L. Nottale (2003), "Gravitational Structure Formation in Scale Relativity",
astro-ph/0310036.
38. D. B. Kaplan, A. E. Nelson and N.Weiner (2004), "Neutrino Oscillations as a Probe of Dark
Energy", hep-ph/0401099.
39. R. Mills et al(2003), "Spectroscopic and NMR identification of novel hybrid ions in fractional
quantum energy states formed by an exothermic reaction of atomic hydrogen with certain
catalysts.", http://www.blacklightpower.com/techpapers.html .
40. K. S, Novoselov et al(2005), "2-dimensional gas of massless Dirac fermions in grapheme",
Nature 438, 197-200 (10 November 2005).
Y. Zhang et al (2005), "Experimental observation of the quantum Hall effect and Berry’s phase
in grapheme", Nature 438, 201-204 (10 November 2005).
See also:
B. Dume (2005), "Electrons lose their mass in carbon sheets", Physicsweb, 9. November.
http://physicsweb.org/articles/news/9/11/6/.
C. Rolfs et al (2006), "High-Z electron screening, the cases 50V(p,n)50Cr and
Phys. G: Nuclear. Part Phys. 32 489. Eur. Phys. J. A 28, 251-252.
176
Lu(p,n)", J.
41. C. Rolfs et al (2006), "First hints on a change of the 22Na β decay half-life in the metal Pd", Eur.
Phys. J. A 28, 251.
42. E. Storms (2001), "Cold fusion - an objective assessment".
http://home.netcom.com/ storms2/review8.html.
43. C. L. Bennett et al (2003), "First-Year Wilkinson Microwave Anisotropy Probe (WMAP1)
Observations: Preliminary Maps and Basic Results", Astrophys. J. Suppl. 148, 1-27.
44. M. Chaplin (2005), "Water Structure and Behavior",
http://www.lsbu.ac.uk/water/index.html.
For 41 anomalies, see http://www.lsbu.ac.uk/water/anmlies.html.
For the icosahedral clustering, see http://www.lsbu.ac.uk/water/clusters.html.
45. S. Klein (2002), "Libet’s Research on Timing of Conscious Intention to Act: A Commentary of
Stanley Klein", Consciousness and Cognition 11, 273-279.
29
http://cornea.berkeley.edu/pubs/ccog−2002−0580-Klein-Commentary.pdf.
B. Libet, E. W. Wright Jr., B. Feinstein, and D. K. Pearl (1979),
"Subjective referral of the timing for a conscious sensory experience", Brain 102, 193224.
[more of Dr. Pitkanen's essays have been archived at http://tgdtheory.com/public_html/index.html
http://www.stealthskater.com/Science.htm#Emergent and
http://www.stealthskater.com/Consciousness.htm ]
if on the Internet, Press <BACK> on your browser to return to
the previous page (or go to www.stealthskater.com)
else if accessing these files from the CD in a MS-Word session, simply <CLOSE> this
file's window-session; the previous window-session should still remain 'active'
30