- #1
Russell E. Rierson
- 384
- 0
Mathematically speaking, the existence of a "dual" vector
space, abstractly reflects the relationship between row
vectors (1×n) and column vectors (n×1). The construction
can also take place for infinite-dimensional spaces and
gives rise to important ways of looking at different
distributions and Hilbert space. The use of the dual space can be a characteristic of functional analysis. It is also built into the Fourier transform.
Because the tangent space and the cotangent space at a
given point are both real vector spaces of the same
dimension, they are isomorphic to each other. But they are
not "naturally isomorphic", since, for an arbitrary tangent
covector, there is no canonical tangent vector
associated with it. With the introduction of a symplectic
form, the additional structure gives a "natural
isomorphism". Longitudinal compression waves agree with
thermodynamic, and Shannon, entropies.
In the general relativistic curved spacetime, there is no
preferred definition for the concept of particles, it
seems. Representations of canonical commutation relations
will be unitarily inequivalent, correspondingly, in both
the asymptotic past and the asymptotic future, for a
"natural notion of particles", analogously to the
"infrared catastrophie" of quantum electrodynamics.
The solution?
Derive quantum theory in terms of general relativity
tensors, using cotangent bundles.
If the universe is closed, the "information" or entangled
quantum states cannot leak out of the closed system. So the
density of entangled quantum states, continually increases,
as the entropy must always increase. While to us, it is
interpreted as entropy or lost information, it is actually
recombined information, to the universe.
Shannon entropy.
What is needed is a tensor equation which is parallel to
"wave" equations described in terms of a covariant
d'Alembertian operator... An alternative description for
the general relativistic space-time continuum that allows
for parallel "compressional" waves, rather than allowing
only "transverse" waves.
Interesting...
By quantizing spacetime geometry, it seems that the
wavefunctions/waveforms aren't based on a background space.
The wavefunction space, can be thought of as the space of
square-integrable wavefunctions over classical
configuration space.
In ordinary quantum mechanics, configuration space is space
itself {i.e.,to describe the configuration of a particle,
location in space is specified}. In general relativity,
there is a more general kind of
configuration space: taken to be the space of 3-metrics
{"superspace", not to be confused with supersymmetric
space} in the geometrodynamics formulation,{or the space of
connections of an appropriate gauge group)in the
Ashtekar/loop formulation. So the wavefunctions will be
functions over these abstract spaces, not space itself--
the wavefunction/algorithm defines "space itself".
The resultant metric spaces are thus defined as being
diffeomorphism invariant. Intersecting cotangent
bundles{manifolds} are the set of all possible
configurations of a system, i.e. they describe the phase
space of the system. When the "wave-functions/forms"
intersect/entangle, and are "in phase", they are at
"resonance", giving what is called the "wave-function
collapse" of the Schrodinger equation. the action principle
is a necessary consequence of the
resonance principle.
Here is mathematician John Nash's "Einstein field equation"
where he talks about gravity "compression" waves:
http://www.stat.psu.edu/~babu/nash/intereq.pdf
A phonon is a quantized mode of vibration occurring in a rigid crystal
lattice, e.g. as in an atomic lattice of a solid.
Could it be that reality surfs on the universal standing wave of
spacetime, emerging out of a "solid block" of nothingness? Standing
wave resonance i.e. "spacetime phonons". The present moment is thus
created and recreated constantly - like a continuous image…
originating deep in twistor space. The Heisenberg Uncertainty Relation
provides both a boundary and the fabric for a translation between
twistor[Planck scale] space and experiential reality, and it is
quantum phase compactification that provides the color electric
superconductive "bricks" for the boundary. Unstable or chaotic states
at a given level are always "compactified" (stabilized and bounded by
a finite number of eigenstates) into higher dimensions at the next
level. The organic analogues of quantum attractors are translated into
quantized fractal exitation modes onto the classical domain via
compactification, while events on the classical domain influence the
collapse or transition of these attractors on the quantum-nano level
via feedback oscillations. The state vector becomes an interactive
participant.
Background independence!
The description of any entity inside the real universe can only be
with reference to other things in the universe. Space is then
relational, and the universe, self referential. For example, if an
object has a momentum, that momentum can only be explained with
respect to another object within the universe. Space then becomes an
aspect of the relationships between things in reality.
Physicist Lee Smolin says that space becomes analogous to a sentence,
and it is absurd to say that a sentence has no words in it. So the
grammatical structure of each sentence[space] is defined by the
relationships that hold between the words in it.
For example, relationships like object-subject or adjective-noun. So
there are many different grammatical structures composed of different
arrangements of words, and the varied relationships between them.
If the universe is closed, the "information" or entangled quantum
states cannot leak out of the closed system. So the density of
entangled quantum states, continually increases, as the entropy must
always increase. While to us, it is interpreted as entropy or lost
information, it is actually recombined information, to the universe.
Since entropy can also be defined as the number of states within a
region of space, and the entropy of the universe must always
increase, the next logical step is to realize that the spacetime
density, i.e. the information encoded within a circumscribed region
of space, must be increasing in the thermodynamic direction of time.
The entropy of thermodynamics and entropy of Shannon, are equivalent
concepts, because the number of arrangements that are counted by
Boltzmann entropy reflects the amount of Shannon information needed
to implement any particular combination, or arrangement. The two
entropies also appear to have superficial differences.
Thermodynamic entropy is interpreted in units of energy divided by
temperature, while, the Shannon entropy is interpreted in terms of
dimensionless bits. This seems to point towards a computational/language or "cognitive-mind" structure for reality.
[1.] Mathematics is a meta language.
[2.] Langauge is descriptive.
[3.] Langauge must be free of contradiction. Mathematics is also
defined as a descriptive system that has "freedom from contradiction".
[4.] Mathematics describes physical existence/processes/events.
[5.] Observation is a physical process.
[6.] Mathematics describes observations.
[7.]A description of an observation must be free of
contradiction-following from [3.]
[8.] Observation must be free of contradiction.
[8.] A description is an abstract representation of a physical system.
The description must be as exact as possible.
[9.] An exact description[as an algorithmic limit] implies equivalence between abstract structures and physical[natural] systems.
[10.] If the exact description exists, then physical existence is a computational-algorithmic meta-language. A self descriptive entity, free of contradiction. The universe is equivalent to its[exact] abstract representation.
space, abstractly reflects the relationship between row
vectors (1×n) and column vectors (n×1). The construction
can also take place for infinite-dimensional spaces and
gives rise to important ways of looking at different
distributions and Hilbert space. The use of the dual space can be a characteristic of functional analysis. It is also built into the Fourier transform.
Because the tangent space and the cotangent space at a
given point are both real vector spaces of the same
dimension, they are isomorphic to each other. But they are
not "naturally isomorphic", since, for an arbitrary tangent
covector, there is no canonical tangent vector
associated with it. With the introduction of a symplectic
form, the additional structure gives a "natural
isomorphism". Longitudinal compression waves agree with
thermodynamic, and Shannon, entropies.
In the general relativistic curved spacetime, there is no
preferred definition for the concept of particles, it
seems. Representations of canonical commutation relations
will be unitarily inequivalent, correspondingly, in both
the asymptotic past and the asymptotic future, for a
"natural notion of particles", analogously to the
"infrared catastrophie" of quantum electrodynamics.
The solution?
Derive quantum theory in terms of general relativity
tensors, using cotangent bundles.
If the universe is closed, the "information" or entangled
quantum states cannot leak out of the closed system. So the
density of entangled quantum states, continually increases,
as the entropy must always increase. While to us, it is
interpreted as entropy or lost information, it is actually
recombined information, to the universe.
Shannon entropy.
What is needed is a tensor equation which is parallel to
"wave" equations described in terms of a covariant
d'Alembertian operator... An alternative description for
the general relativistic space-time continuum that allows
for parallel "compressional" waves, rather than allowing
only "transverse" waves.
Interesting...
By quantizing spacetime geometry, it seems that the
wavefunctions/waveforms aren't based on a background space.
The wavefunction space, can be thought of as the space of
square-integrable wavefunctions over classical
configuration space.
In ordinary quantum mechanics, configuration space is space
itself {i.e.,to describe the configuration of a particle,
location in space is specified}. In general relativity,
there is a more general kind of
configuration space: taken to be the space of 3-metrics
{"superspace", not to be confused with supersymmetric
space} in the geometrodynamics formulation,{or the space of
connections of an appropriate gauge group)in the
Ashtekar/loop formulation. So the wavefunctions will be
functions over these abstract spaces, not space itself--
the wavefunction/algorithm defines "space itself".
The resultant metric spaces are thus defined as being
diffeomorphism invariant. Intersecting cotangent
bundles{manifolds} are the set of all possible
configurations of a system, i.e. they describe the phase
space of the system. When the "wave-functions/forms"
intersect/entangle, and are "in phase", they are at
"resonance", giving what is called the "wave-function
collapse" of the Schrodinger equation. the action principle
is a necessary consequence of the
resonance principle.
Here is mathematician John Nash's "Einstein field equation"
where he talks about gravity "compression" waves:
http://www.stat.psu.edu/~babu/nash/intereq.pdf
A phonon is a quantized mode of vibration occurring in a rigid crystal
lattice, e.g. as in an atomic lattice of a solid.
Could it be that reality surfs on the universal standing wave of
spacetime, emerging out of a "solid block" of nothingness? Standing
wave resonance i.e. "spacetime phonons". The present moment is thus
created and recreated constantly - like a continuous image…
originating deep in twistor space. The Heisenberg Uncertainty Relation
provides both a boundary and the fabric for a translation between
twistor[Planck scale] space and experiential reality, and it is
quantum phase compactification that provides the color electric
superconductive "bricks" for the boundary. Unstable or chaotic states
at a given level are always "compactified" (stabilized and bounded by
a finite number of eigenstates) into higher dimensions at the next
level. The organic analogues of quantum attractors are translated into
quantized fractal exitation modes onto the classical domain via
compactification, while events on the classical domain influence the
collapse or transition of these attractors on the quantum-nano level
via feedback oscillations. The state vector becomes an interactive
participant.
Background independence!
The description of any entity inside the real universe can only be
with reference to other things in the universe. Space is then
relational, and the universe, self referential. For example, if an
object has a momentum, that momentum can only be explained with
respect to another object within the universe. Space then becomes an
aspect of the relationships between things in reality.
Physicist Lee Smolin says that space becomes analogous to a sentence,
and it is absurd to say that a sentence has no words in it. So the
grammatical structure of each sentence[space] is defined by the
relationships that hold between the words in it.
For example, relationships like object-subject or adjective-noun. So
there are many different grammatical structures composed of different
arrangements of words, and the varied relationships between them.
If the universe is closed, the "information" or entangled quantum
states cannot leak out of the closed system. So the density of
entangled quantum states, continually increases, as the entropy must
always increase. While to us, it is interpreted as entropy or lost
information, it is actually recombined information, to the universe.
Since entropy can also be defined as the number of states within a
region of space, and the entropy of the universe must always
increase, the next logical step is to realize that the spacetime
density, i.e. the information encoded within a circumscribed region
of space, must be increasing in the thermodynamic direction of time.
The entropy of thermodynamics and entropy of Shannon, are equivalent
concepts, because the number of arrangements that are counted by
Boltzmann entropy reflects the amount of Shannon information needed
to implement any particular combination, or arrangement. The two
entropies also appear to have superficial differences.
Thermodynamic entropy is interpreted in units of energy divided by
temperature, while, the Shannon entropy is interpreted in terms of
dimensionless bits. This seems to point towards a computational/language or "cognitive-mind" structure for reality.
[1.] Mathematics is a meta language.
[2.] Langauge is descriptive.
[3.] Langauge must be free of contradiction. Mathematics is also
defined as a descriptive system that has "freedom from contradiction".
[4.] Mathematics describes physical existence/processes/events.
[5.] Observation is a physical process.
[6.] Mathematics describes observations.
[7.]A description of an observation must be free of
contradiction-following from [3.]
[8.] Observation must be free of contradiction.
[8.] A description is an abstract representation of a physical system.
The description must be as exact as possible.
[9.] An exact description[as an algorithmic limit] implies equivalence between abstract structures and physical[natural] systems.
[10.] If the exact description exists, then physical existence is a computational-algorithmic meta-language. A self descriptive entity, free of contradiction. The universe is equivalent to its[exact] abstract representation.