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Unification of Quantum Theory and General Relativity

  1. Jul 6, 2004 #1
    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.] Language is descriptive.

    [3.] Language 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.
     
  2. jcsd
  3. Jul 6, 2004 #2
    Then please show us where can we find a Physical point that has exactly 0-dimension.
     
  4. Jul 6, 2004 #3
    Who said the exact abstract representation must be a continuum?
     
  5. Jul 6, 2004 #4

    Chronos

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    Gold Member
    2015 Award

    Quantization of space is one of the tenets of quantum loop gravity theory, which appears to be the most promising approach to unifying quantum theory with GR. I also agree defining entropy in Shannon as opposed to thermodynamic terms is more precise.
     
  6. Jul 7, 2004 #5
    Can we understand that continuum is not a part of your system?
     
  7. Jul 7, 2004 #6
    Maybe relativity space does not even exist. This explains photon probability waves in a double slit experiment. The space between the slits does not exist.
     
  8. Jul 7, 2004 #7
    A wave-particle duality appears to be not exclusively continuum, and not exclusively quantum. It is a complementary logic. both continuum AND quantum.

    A causally connected system with a big question mark.
     
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