(Reprinted with permission)

Mathematically speaking, the existence of a "dual" vector space, abstractly reflects the relationship between row vectors (1?) and column vectors (n?). 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:

Quote:

Wave-Like Form of the Scalar Equation It was discovered only recently by me that the scalar equation naturally derived from the tensor equation for vacuum, particularly in the case of 4 space-time dimensions, has a form extremely suggestive of waves. The scalar derived equation can be obtained by formally contracting the general vacuum equation with the metric tensor. This results at first in an equation involving G (the scalar derived from the Einstein tensor) and the Ricci tensor and the scalar curvature R. And G, being the scalar trace of the Einstein tensor, can be expressed in term of R but this expression involves the number of dimensions, n.

And now two things are notable about the form of this resulting scalar equation: (1): If n = 2 there is a singularity and this simply corresponds to the fact that the Einstein G-tensor is identically vanishing if n = 2, so there isnt any derived scalar equation of this type for two dimensions. (2): For n = 4 we find the nice surprise that the scalar equation entirely simplifies and then asserts simply that the scalar curvature satisfies the wave operator [], (which is a d'Alembertian if we think in terms of 3 + 1 dimensions). So the scalar equation is

[]R = 0 PROVIDED that n = 4

end quote.

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