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TOPOLOGICAL GEOMETRODYNAMICS:
What Might be the First Principles?
Article: http://www.helsinki.fi/~matpitka/articles/tgd.pdf
Slides as pdf:http:///www.helsinki.fi/~matpitka/tgdppat/vision.pdf
Matti Pitkänen
http://www.helsinki.fi/~matpitka/
Books about TGD
[email protected]
Two approaches to physics: constructive and
analytic
Einstein:
To the beginning
We can distinguish various kind of theories in physics. Most of them are constructive. They
attempt to build up a picture of the more complex phenomena out of the materials of a
relatively simple formal scheme from which they start out. Thus the kinetic theory of gases
seeks to reduce mechanical, thermal, and diffusional processes to movements of molecules i.e., to build them up out of the hypothesis of molecular motion. When we say that we have
succeeded in understanding a group of natural processes, we invariably mean that a
constructive theory has been found which covers the processes in question.
Along with this most important class of theories there exists a second, which I will call
"principle-theories." These employ the analytic, not the synthetic, method. The elements
which form their basis and starting-point are not hypothetically constructed but empirically
discovered ones, general characteristics of natural processes, principles that give rise to
mathematically formulated criteria which the separate processes or the theoretical
representations of them have to satisfy. Thus the science of thermodynamics seeks by
analytical means to deduce necessary conditions, which separate events have to satisfy,
from the universally experienced fact that perpetual motion is impossible.
The advantages of the constructive theory are completeness, adaptability, and clearness,
those of the principle theory are logical perfection and security of the foundations. The
theory of relativity belongs to the latter class.
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Problem: some basic principle might be
wrong!
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Landscape problem of M theory. Is reductionism extrapolated to Planck
length scale might be the reason?
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Perhaps one must tolerate the tension between constructive and
analytic approaches.
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In TGD this tension has been present all the time. Path integral
formalism failed. The attempt to construct S-matrix led to a long process
forcing to modify existing visions about ontology of physics.
New view about time, the notion of many-sheeted space-time, zero
energy ontology, p-adic physics as physics of intentionality and
cognition, hierarchy of Planck constants, role of von Neumann algebras
known as hyperfinite factors of type II1 in the construction of theory,....
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Plan of the talk
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Represent various visions about basic principles behind quantum TGD
keeping in mind the tension present also between them and trying to
tolerate uncertainty.
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From Equivalence Principle to zero energy ontology ?
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The tree strands of physics as generalized number theory vision: fusion of p-adic
and real physics, classical number fields and physics, infinite primes and physics.
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Quantum TGD from hyperoctonionic generalization of conformal field with values
in HFF of type II?
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Physics as spinor geometry of the world of classical worlds?
Hierarchy of Planck constants and generalization of imbedding space H= M4xCP2
Finite measurement resolution as basic dynamical principle of quantum theory?
Hyperfinite factors of type II1 (HFFs) as a basic building block.
Should quantum physics be extended to a theory of consciousness?
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To the beginning
From Equivalence Principle to Zero Energy
Ontology
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In Einstein's GRT the tension has been present from the beginning.
Equivalence Principle (EP): gravitational and inertial masses are
identical.
Einstein's equations identify gravitational and inertial energy
momentum tensors. Purely local statement. No global variant because
gravitational and inertial four-momenta are not defined at all unless
space-time has translations as symmetries. Something wrong?
Cure: space-time as a 4-D surface in H=M4xS, S=CP2. CP2 from
standard model symmetries and geometrization of classical gauge fields.
Inertial energy momentum as conserved Noether charge from Poincare
symmetries of H rather than space-time. Gravitational four- momentum
defined but not conserved unless the action is curvature scalar, in
which case Equivalence Principle as identity.
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Curvature scalar would predict conserved gravitational four-momentum:
not consistent with cosmology.
Kähler action (Maxwell action for induce Kähler form of CP2) more
natural physically. Gravitational momentum exists but not conserved.
Robertson Walker cosmologies vacuum extremals (vanishing energy
momentum and other conserved charges) but non-vanishing and nonconserved gravitational four-momentum.
Does Equivalence Principle fail? Or is the interpretation wrong?
The problem might have resolution in zero energy ontology. All physical
states have vanishing net conserved quantum numbers. Positive and
negative energy parts of states correspond to initial and final states of
scattering event in positive energy ontology. Quantum states as
events.
Basic principle: every quantum state is creatable from vacuum.
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E<0
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M 4-
Causal diiamond as
pair of lightcones
T/2
T
M 4+
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E>0
 Physical state can be visualized in terms of causal diamond. Lower
lightcone boundary contains 3-D surfaces with positive energy and upper
lightcone boundary those with negative energy.
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To the beginning
Equivalence Principle in TGD
Elementary partides as CP2 type vacuum extremals of Kähler action .
Random lightlike curve as M4 projection: virtual particles. For
extremals of curvature scalar projection light-like geodesic: on mass
shell particles.
EP at 4-D level: Topologically condensed CP2 type vacuum extremal
creates a non-vacuum region around around it. The resulting inertial
four-momentum equals to the gravitational four-momentum.
EP at macro level: Space-time surfaces containing gravitons must have
double sheeted structure (positive and negative energy space-time
sheets). Near the vicinity of wormhole contacts (pieces of CP2 type
vacuum extremals) vacuum extremals are deformed to non-vacuum
extremals and carry inertial four-momentum equal to the gravitational
four- momentum of the vacuum extremal.
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To the beginning
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EP at parton level: Gravitational momentum of CP2 type extremal =
inertial Chern-Simons four-momentum, which non-vanishing only if
Kähler gauge potential has pure gauge part in M4+/- equal to Aa=
constant (a is lightcone proper time). Aa expressible in terms of G/R^2.
Quantum criticality fixes this ratio. No breaking of Poincare
invariance if configuration space union of configuration spaces
assignable to lightcones.
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Physics as classical physics for
configuration space spinor fields
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Generalization of Einstein's geometrization program of classical
physics to quantum physics.
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The world of classical worlds (configuration space) consisting of
3-surfaces of H=M4xCP2 arena of quantum dynamics. Quantum
states classical spinor fields in this space.
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Geometrize configuration space.
Kähler geometry required in order to geometrize Hermitian
conjugation. Kahler function K. General coordinate invariance:
definition of K assigns to 3-surface more or less unique 4surface analogous to Bohr orbit. Classical physics exact part of
quantum theory.
Mathematical existence of geometry extremely powerful
constraint. For loop spaces Kähler geometry unique. InReturn
3-D case
even stronger constraints from existence of Riemann connection
and finiteness. H=M4xCP2 be the only possible choice? InfiniteD Kähler geometric existence unique?
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Quantum dynamics from quantum criticality. Exponent of Kähler
function defines a universal vacuum functional analogous to
expoment of Hamiltonian. Kähler couplings strength analogous
to critical temperature and quantized. All coupling parameters as
predictions of the theory.
Quantum states as classical spinor fields of configuration space.
Only quantum jumps genuinely quantal element of theory.
Geometrization of Fermi statistics. Configuration space spinors as
Fock states for fermions. Anticommuting configuration space
gamma matrices linear combinations of fermionic oscillator
operators for free second quantized induced spinor fields at 3surface.
Configuration space gamma matrices generate infinite-D Clifford
algebra. Canonical representation for von Neumann algebra
known as hyper-finite factor of type II1 (HFF). Encodes quantum
group and non-commutative physics.
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Quantum TGD as almost topological quantum
field
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beginning
theory?
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General coordinate invariance again! 3-surfaces can be
chosen to be light-like 3-surfaces.
Gauge bosons and Higgs as wormhole contacts: wormhole
throats at which induced metric changes signature are lightlike. Carry fermionic and antifermionic quantum numbers.
Fermions as topologically condensed CP2 type extremals.
Lightlike 3-surrfaces look locally random orbits of partonic 2surfaces with light velocity.
Metric 2-dimensionality implies generalization of the superconformal invariance of string models. Kac-Moody symmetries
as isometries of the Kähler metric. Lightlike boundary of
M4+/- gives rise to additional superconformal symmetries.
Light-likeness: parton level formulation of TGD as
almost topological quantum field theory. Chern-Simons action
and its fermionic counterpart. “Almost” means that notions of
energy and momentum make sense although theory relies on
the formalism of TQFT:s.
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To the beginning
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TGD does not reduce to string model although causal
determinants 1-dimensional! D= 4,3,2,1-dimensionality in
discretized sense. Locally 3,2,1,0 dimensionality.
At the bottom discrete “number theoretical braids” replace
strings. Braids are basic objects of TQFTs. Interpretation in
terms of finite measurement resolution.
Tension: Kähler function of configuration space expressible in
terms of data associated with modified Dirac operator at
points of number theoretic braid.
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The three threads of “Physics as
generalized number theory” program”To the beginning
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First thread: p-adic physics.
Reals and p-adic numbers, p=2,3,5,7... completions of rationals .
p-Adic mass calculations.
What is the interpretation of p-adic physics? P-Adic space-time
sheets define correlates for cognitions and intentions . Mind
stuff of Descartes.
How to glue real and various p-adic physics p=2,3,5,7,... to
single coherent whole?
Basic principle: Real and p-adic physics completions of rational
physics .
Generalization of number concept. Glue all these number fields
together along rationals and common algebraics to form book
like structure.
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P-Adic and real space-time sheets intersect along common
rational and algebraic points. Interpretation in terms of
cognitive representations and discreteness of mathematical
cognition and finite measurement and sensory resolution .
P-Adic transcendentals literally infinite as real numbers.
Most points of p-adic space-time sheets and spatial and
temporal infinity. Cognition and intentionality cosmic
phenomena not localizable to finite space-time volume.
P-Adic topology induces effective p-adic topology of real
space-time sheet when the number of common points large.
P-Adic fractality predicted and explains the success of p-adic
mass calculations. Cognition present already at elementary
particle level.
Tension:The discretization required by p-adicization implies
the notion of number theoretic braid needed in almost TQFT
approach.
Tension: Exponent of the configuration space Kähler function
as Dirac determinant using only the data defined by braid.
Construction assigns a preferred space-time surface to a Return
given collection of light-like 3-surfaces.
Second thread: classical number fields
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Imbedding space, space-time surface, partonic 2-surface, and
causal determinant of conformal field theory (“strings”) have
dimensions of classical number fields.
Could one also formulate QTGD by replacing M4xCP2 wth M8
=HO: space of hyper-octonions identifiable as subspace of
complefixiifed octonions?
Could associativity and commutativity fix dynamics?
Observations about subspaces of M8:
Associative subspaces are hyperquaternionic planes M4 subset
M8. Commutative subspaces are hypercomplex planes M2.
M2 space of non-physical polarizations selected also in gauge
theories and string models. Gauge choice would have purely
number theoretical meaning.
Subspaces M4 containing M2 parameterized by CP2!
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HO-H duality(“number theoretic compactification”) .
Hyper-quaternionic space-time surfaces of M8 for which hyperquaternionic tangent planes M4 contain fixed hypercomplex
plane M2 can be mapped to M4xCP2.
Bohr orbitology: at light-like 3 surface X3 M2 defines the
direction of tangent space and one can assign unique 4-surface
to X3 .
Are these 4-surfaces of M8 mapped to preferred extremals of
Kähler action?
M4xCP2 and standard model symmetries have purely number
theoretic origin! Also classical TGD!
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Third thread: infinite primes and further
generalization of number concept
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Take product of all primes and add to it 1. Prime results.
Construction generalizes. Infinite hierarchy of infinite
primes.
Procedure equivalent with repeated second quantization of
arithmetic quantum field theory with bosons and fermions
labelled by primes of given level. All Fock states and bound
states constructed at first level become elementary particles
at the next level. Continue indefinitely.
The hierarchical structure of many-sheeted space-time could
correspond to that for infinite primes. Proton- although
manyquark state - would be elementary fermion at
appropriate level of hierarchy.
Hierarchy could also correspond to the hierarchy of n:th
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order logics.
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Also infinite integers and rationals exist. Infinite number of
real units defined as ratios of infinite integers. Not To the beginning
equivalent number theoretically. Infinite-dimensional space
U of real units.
Point of imbedding space replaced with infinite-D space U8 of
points equivalent in real sense. What Schrödinger
amplitudes in this space could mean?
Number theoretic Brahman =Atman identity. Is the world of
classical worlds and even the space of configuration space
spinor fields representing quantum states of Universe
representable in this generalized imbedding space which is
8-D in real sense? Algebraic holography.
Can one map configuration space spinor fields to
“Schrödinger amplitudes” in U8 in natural manner?
Good hopes for mapping the basis of configuration space
spinor fields associated with zero energy states inside given
causal diamond and appearing as zero energy insertions to
Schrodinger amplitudes in U8 !
Stuff below measurement resolution experienced
subjectively?
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Extension of Clifford algebra of CH to
hyper-octonionic conformal fields in
HO?
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Configuration space Clifford algebra correspond to
vibrational degrees of freedom of 3-surface with cm degrees
of freedom fixed. How to include cm degrees of freedom in
H?
In string models gamma fields define conformal fields.
Generalization: replace complex argument z with hyperoctonionic argument. HFF valued conformal fields.
Could local variant of HFF analogous to local gauge algebra
provide first principle approach to quantum and classical
TGD?
By non-associativity of hyper-octonions the resulting
algebra does not reduce to HFF as in case of other number
fields. Could this unique local version of HFF underlie
quantum TGD?
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Associativity: n-point functions involving local Clifford
algebra elements must be associative at least. Arguments
restricted to 4-D hyper-quaternionic subspace M4 subset M8.
Commutatitivity: fix preferred imaginary unit by choosing
subspace HC=M2 subset M4 subset M8.
Assume that partonic 2-surfaces X2 belong to the boundary
of lightcone (causal diamond) of M8,, not only M4xCP2! !
Nontrivial condition.
Stronger assumption: also pre-images of lightlike 3-surfaces
X3 belong to M4+/- subset M8.
These assumptions would explain the preferred role of
partonic 2-surfaces and of their lightlike orbits as being due
to their associativity. Associativity not true in the interior of
space-time surface in general.
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Commutativity of n-point functions at partonic 2-surfaces
if
2
arguments belong to the intersection of X with lightlike
ray. Integral over rays would give maximal information.
Unique discrete set of points identifiable in terms number
theoretic braids. With additional assumptions about X2
rational or algebraic points of H.
If also light-like 3 surfaces in M4+/- then commutative
submanifolds 1-D curves and define the strands of braids.
Conclusion: Very many speculative “must-be-trues” of
quantum TGD follow as a consequence!
Critical question: Octonions would be number theoretically
more elegant. Could one replace hyper-octonions with
octonions and interpret duality in terms of generalization of
Wick rotation?
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Hierarchy of Planck constants
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Is Planck constant really constant? Planetary orbits obey in
reasonable approximation Bohr rules with gigantic Planck
constant (see this and this). Effects of ELF em fields on
vertebrate brain quantal although energies of the photons
extremely small as compared to thermal energy for ordinary
value of Planck constant (see this). Could hbar be quantized
and have arbitrarily large values?
Possible only if the notion of imbedding space is generalized.
Imbedding space has a book like structure. Each page
covering or factor space of original imbedding space. The
pages are glued together like pages of book.
The matters at different pages of book dark relative to each
other in sense that only particles from same page can appear
in vertices of Feynman diagrams. Classical interactions
between different pages possible. Also exchange of particles
if phase transition changing hbar occurs (particle moves
from page to another one).
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To the beginning
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Dark matter as matter with nonstandard value of Planck
constant. Macroscopic quantum phases possible even in
astrophysical length scales. Condensation of visible matter
around dark matter with large Planck constant explains Bohr
orbits for planets.
This picture consistent with what is known about dark
matter since classical em fields are not assumed to be of
importance in astrophysical length scales.
The anomalies associated with living cell understood if
considerable fraction of biologically important ions dark and
in macroscopic quantum phase.
Tension: Is the hierarchy of Planck constant necessary for
quantum TGD. The generalization of imbedding space
essential for the realization of quantum criticality and
construction of exponent of Kähler function of configuration
space.
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Could finite measurement resolutionTo the beginning
fix the quantum dynamics?
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Clifford algebra for configuration space HFF of type II1
algebraic fractal. Infinite hierarchies of inclusions of HFF to
itself as subalgebra. What could be the interpretation?
Subalgebra N defines finite measurement resolution.
Complex rays of state space replaced with N rays.
In zero energy ontology elements of N insert to the positive
or negative energy part of state zero energy part. The added
part corresponds to time scale T/2, or more generally, T/2n.
Interpretation in terms of radiative corrections which
correspond to the increase of measurement resolution.
Coupling constant evolution not continuous but comes in
powers of 2.
p-Adic length scale hypothesis as a consequence: p-adic
primes near powers of 2 preferred.
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Additional prediction: the temporal distances between tips of
causal diamond macroscopic time scale for elementary
particles. Argument goes as follows.
Light-like 3-surfaces locally as random light-like orbits of
partonic 2-surfaces. Brownian motions.
During time T particles moves average distance given by padic length scale Lp. Analogy with Brownian motion implies
T= Lp2/cR, R CP2 length.
T corresponds to secondary time p-adic time scale
T = Tp,2 =sqrt(p)Lp/c
For electron T = .1 seconds. Fundamental biorhythm!
Zero energy ontology necessary in order to understand
biology.
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Definition of generalized S-matrix, M-matrix, as time-like
entanglement coefficients between positive and negative
energy parts of zero energy state interpreted as initial and
final states of particle scattering experiment.
Unitarity not necessary.
M-matrix complex square root of
density matrix. Product of positive square root of density
matrix and of unitary S-matrix. Matrix generalizations of
modulus and phase of Schrödinger amplitude. Quantum
physics as a square root of thermodynamics.
M-matrix must commute with hermitian elements of algebra
N defining
measurement resolution. N acts as infinite-D
symmetry algebra. Connection with integrable QFT:s and
conformal invariance.
M-matrix identifable in terms of Connes tensor product and
is highly unique. Decomposition of HFF to direct summands
implies non-uniquess which corresponds to thermal states.
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Tension: How closely this vision relates to other visions?
Configuration space Clifford algebra as HFF: this is essential.
Finite measurement resolution means discretization at
space-time level. Conformal field theories and TQFTs relate
closely to HFFs and inclusions. Connection with geometric,
almost TQFT, and number theoretic visions.
Hyper- octonionic analog of HFF might imply generalized
imbedding space as emergent structure.
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Should one extend quantum physics to
a quantum theory of consciousness? To the beginning
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Quantum measurement theory the black sheet of quantum
physics. Before one can speak of TOE one must be able to
make observer a genuine part of Universe.
Quantum jump and self basic notions of TGD inspired theory
of consciousness.
Quantum jump has complex anatomy: unitary process, state
function reduction, state preparation.
Unitary process corresponds to unitary matrix U between
zero energy states. M-matrix property of zero energy state.
U-matrix can be assigned with the non-determinism of
volition and intentional action. U-matrix tells also the
probabilities for p-adic to real transitions characterizing
intentional action. U-matrix probably almost trivial.
Self as a system able to remain unentangled. Self results as a
fusion of quantum jumps. Self relates to quantum jumps like
composite particle to elementary particles from which it
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Negative energy ontology means that also negativeToenergy
signals to geometric past are possible. Mechanisms of
memory, intentional action, and remote metabolism.
New view about relationship between geometric and
experienced time. (see this and this) Quantum classical
correspondence: quantal time evolution by quantum jumps
must have space-time correlate. Failure of strict determinism
for Kähler action makes this possible.
Attention of self directed to a fixed volume of imbedding
space H=M4xCP2: in standard picture it would gradually shift
towards future.
In quantum jump configuration space spinor fieldsuperposition of space-time surface shifts backwards in
geometric time in a good approximation: this realizes
quantum classical correspondence. Observer experiences
that geometric future of space-time sheet emerges to the
perceptive field from future. Time flows. Only
approximation in question: dissipative effects.
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Negentropy Maximization Principle the basic variational
principle of quantum consciousness theory. Information
content of conscious experience measured by the reduction
of the entanglement entropy in state function reduction.
For standard definition of entanglement entropy in terms of
Shannon entropy pure state would always result in state
function reduction.
For rational or even algebraic entanglement probabilities
one can define number theoretical variants of Shannon
entropy by replacing logarithm of probabily with the
logarithms of p-adic norm of probability . This entropy can
be negative. Entanglement can carry genuine information.
State function reduction can generate entanglement!
Possible interpretation in terms of formation of
macrosopically quantum coherent systems. Synergy.
Experience of understanding associated with this kind of
quantum jumps?
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Tension: Relationship with other approaches
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Consciousness theory generalization of quantum
measurement theory.
Consciousness theory not alternative: brings in genuinely
new elements. Solves problems of other approaches.
Difficulties of quantum measurement theory. The problem of
time. The notion of information.
The hierarchy of conscious entities and evolutionary
hierarchy corresponds naturally to the hierarchy of Planck
constants. Evolution as growth of largest Planck constant
assignable to “personal magnetic body” having onionlike
structure.
P-Adic physics forced by the necessity to describe properly
cognition and intentionality .
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Philosophical comment about number theoretic To the beginning
Brahman=Atman
Real units correspond to zero energy states of arithmetic
QFT. Physical zero energy states must be mapped to these.
Must correspond to zero energy insertions to positive or
negative part of zero energy state. States below
measurement resolution mapped to the points of number
theoretic braid in the resolution used.
Conformal hyperoctonionic fields O(hi) at points of number
theoretic braid create these zero energy states. In better
resolution nonlocal operators formed from Cliffod algebra.
Small causal diamond inside bigger causal diamond.
Physical measurements give information about universe
above measurement resolution. The Schrödinger amplitudes
in the space of real units below it. Subjective experience
gives information about quantum state below resolution.
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Technical comment about number theoretic Brahman=Atman
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Operators O(hi) constructed using generators of supercanonical
and super KM algebras . Map basis to basis. Enumerability:
mathematician would stop here.
Infinite primes <--> hierarchy of space-time sheets. Map of CH
spinor fields to real units induced at the lowest level from the
map of qnumbers of superalgebra generators to finite primes.
Find natural ordering of these quantum numbers/states. Map to
primes must respect this ordering.
Order states of irrep of given group by values of quantum
numbers assuming ordering of quantum numbers.
Order irreps of given group by the dimension of representation or
by the highest weight of irrep.
Order the groups involved: first SU(3) xSO(3) associated with
Hamiltonians of S2xCP2 first, then KM color group, KM ew group,
and KM rotation group (say). This allows highly unique map of
the labels of super generators to finite primes.
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