What is the geometry of a unified field theory?

[PFanalog57]

Antisymmetric tensors combine with symmetric tensors to give the thermodynamic arrow of time, which is really a continual densification of spacelike surfaces?

More random thoughts on the unified field theory:

symmetric tensor: A^uv = A^vu

antisymmetric tensor: A^uv = -A^vu

asymmetric tensor: A^uv does not equal A^vu

Asymmetric tensor(A^uv) = (1/2)[A^uv + A^vu] + (1/2)[A^uv - A^vu]

The gravity tensor should be able to rotate into the electromagnetic tensor and the electromagnetic tensor should be able to rotate into the gravity tensor.

Time
^
|
|
|
|-------------->space



G
^
|
|
|
|-------------->EM


Distance is a property between objects in space. Space is a structure,
which is constructed of discrete units. The structure of space is
possibly a distributive lattice? A lattice is a partially ordered
set, closed under least upper and greatest lower bounds. Any lattice
which is isomorphic to a collection of sets, closed under
complementation and intersection, is a Boolean algebra.

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.

The continual intersection and collapse of probability
distributions, also known as quantum phase entanglement, is a
continual increasing of the "total" combined information of the
universal wavefunction itself. Information density. With more
information, more complex structures can be created.

The information density of the universal system must be increasing.
The increase of information density is analogous to a pressure
gradient.


[density 1]--->[density 2]--->[density 3]---> ... --->[density n]


[<-[->[<-[-><-]->]<-]->]

Intersecting wavefronts = increasing density of spacelike slices

As the wavefronts intersect, it becomes a mathematical computation:

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...2^n





The area of a spacetime surface and the maximum amount of information contained in a finite region of space, cannot be greater than one quarter of the area in Planck units. Spin networks can describe the quantum geometry of space at the intersection of horizon boundaries, where the spin networks intersect with the boundary at a finite number of points.

There is a finite amount of energy contained by a given region of spacetime. A finite amount of information. A finite number of quantum phase entanglements and random fluctuations.

A superconductor is a system of discrete units operating in two dimensions along a standing wave. By definition, a superconductor is a material that is precisely in balance, such, that it will not allow any external magnetic fields inside the superconductive domain.

The inner[scalar] product of two vectors, a and b, is given by a*b .

The outer product of two vectors a and b, is given by a /\ b .

a*b = b*a

a /\ b = -b /\ a

the geometric product ab is given by ab = a*b + a /\ b

a*b = [ab + ba]/2

a /\ b = [ab - ba]/2

The electromagnetic field in terms of the four potential A:

F = grad /\ A = grad A - div A


The electromagnetic field bivector, F, in terms of the orthonormal basis vectors v:

Electric field E

Magnetic field B

F = E + iB

E = [F - vFv]/2

iB = [F + vFv]/2


grad F = 4 current = J


F = inverse grad J

 

[member 11137]

a*b (the inner scalar product of two vectors) is a scalar
a ^ b (the wedge product of two vectors) is a vector
How can you calculate a*b + a ^ b ? Blackforest

 

[Antonio Lao]

$$\vec{a} \cdot \vec{b} + \vec{c} \times \vec{d}$$

is a higher hypercomplex number called a quaternion.

 

[PFanalog57]

a*b (the inner scalar product of two vectors) is a scalar
a ^ b (the wedge product of two vectors) is a vector
How can you calculate a*b + a ^ b ? Blackforest

Geometric Algebra, the Geometric Product:


http://planetmath.org/encyclopedia/GeometricAlgebra.html

The geometric product is related to the inner product a*b and the exterior product a /\ b by

ab = a*b + a /\ b = b*a - b /\ a = 2a*b - ba

 

[PFanalog57]

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!

 

[member 11137]

First of all, just want to say thank you to antonio Lao and to Russel E. R. for the help. I have visited the link and I will try to start a confrontation between what I have seen and my small mathematical problem: that is the systematic decomposition of a wedge product u ^ w (E x E; dimE = N; E built on R -real number- for example) in M(N,N) x E where M(N,N) is the collection of square matrix (N ligns- N...) with real components. It looks like an amazing problem (at least for me) but I get the sensation that it must be quite more difficult as it looks like and could lead to introduce the topology of E in the discussion. Do you have some other good adresses for the topology? Blackforest

 

[Antonio Lao]

Do you have some other good adresses for the topology?

For my works, the topology that keep popping out is the Hopf ring similar to a doubly twisted Moebius strip cut along in the middle. This ring is symbolized by $\psi_i \times \phi_i \cdot \psi_j \times \phi_j$.