Can 2-Brane Structure Describe Universe?

[Antonio Lao]

Can the complete physical structure of the universe be described by a two-brane matrix?

First, we need some key words taken from the glossary of Brian Greene's book "The Elegant Universe."

Brane. Any of the extended objects that arise in string theory. A one-brane is a string, a two-brane is a membrane, a three-brane has three extended dimensions, etc. More generally, a p-brane has p spatial dimensions.

Closed string. A type of string that is in the shape of a loop.

Open string. A type of string with two free ends.

Curled-up dimension. A spatial dimension that does not have an observably large spatial extent; a spatial dimension that is crumpled, wrapped, or curled up into a tiny size, therby evading direct detection.

Extended dimension. A space (and spactime) dimension that is large and directly apparent; a dimension with which we are ordinarily familiar, as opposed to a curled-up dimension.

Dimension. An independent axis or direction in space or of spacetime. The familiar space around us has three dimensions (left-right, back-forth, up-down) and the familiar spacetime has four (the previous three axes plus the past-future axis).

Note: Although superstring theory requires the universe to have additional spatial dimensions, it is the purpose of this thread to show otherwise.

 

[Antonio Lao]

A two-brane matrix can be defined as a special type of tensor with only two covariant indices.

[tex]T_{ij}[/tex]

The elements of this matrix are 1 and -1. The element 1 represents symmetrical transformation. The element -1 represents non-symmetrical transformation. The product of any number of [1] transformation are always symmetrical, while the product of only even [-1] transformation are symmetrical.

 

[Antonio Lao]

Two lowest order 2 (2 by 2) symmetrically square matrices can be formed:

[tex]H^{+} = \left(\begin{array}{cc}+1 & -1\\-1 & +1\end{array}\right)[/tex]

and

[tex]H^{-} = \left(\begin{array}{cc}-1 & +1\\+1 & -1\end{array}\right)[/tex]

An algebra can be made for these Hadamard matrices with a table for addition and a table for multiplication.

 

[Antonio Lao]

For order n

[tex] H^{+}_{n} [/tex]

has element

[tex] e_{ij} = (-1)^{i+j} [/tex]

and

[tex] H^{-}_{n} [/tex]

has element

[tex] e_{ij} = (-1)^{i+j-1} [/tex]

where i=1,2,3,..,n and j=1,2,3,...,n

All these matrices are singular. They do not have inverses. Their determinant are all zeros. Matrix multiplication is always commutative.

 

[Antonio Lao]

Matrix additions define electric charge and matrix multiplications define mass.

Only matrices of the same order can be added or multiplied.

All particles in nature can be composed of H+ and H-. The electron is made up of 7H- and 1H+, while the photon is made up of 4H+ and 4H-.
The neutrino is made of 1H+ and 1H-. The up-quark is 5H+ and 1H-, while the down-quark is made of 1H+ and 3H-. The particle generations are made of higher order matrices.

H+ of any order represents bosons. H- of any order represents fermions. H+ can also be defined as the kinetic mass, while H- can be defined as the potential mass or gravitational mass.

The entire two-brane universe can be represented as

[tex]H^{-}_{n} [/tex]

where n approaches infinity.

 

[Antonio Lao]

To create a graviton, two two-brane universes are needed.
They must have the same order.

For example when the order is 2: One H- from one two-brane and one from the other two-brane. Matrix multiplication gives 2H+. This is a boson of mass zero and made from H-'s from both universes. Each two-brane defines three spatial dimensions. The interaction of two two-brane universes defines a spacetime structure. One two-brane structure is static in time, while the interaction of two two-brane structures is dynamic in time. One two-brane structure contains all the matter and the other contains all the anti-matter. The time direction of each brane is the exact "opposite" of the other.

 

[Antonio Lao]

One two-brane structure by itself is representable by a given Euclidean plane. But when this brane interacts with another brane, the 3 spatial structures are transformed into a spacetime structure. This interaction of two two-brane universes implies that the energy-momentum tensor of Einstein's field equations of general relativity is that of the other interacting brane. The mass in one brane cannot have enough force to curve itself. It will take the total mass-energy of another brane to do this. The two-brane structures meet at a point called the temporal intersect. The geometry does allow multiple intersects. This intersect can be called the "singularity" of the big bang or that of any location of a black hole.

 

[Antonio Lao]

It can be theorized that particles such as gravitons, Higgs bosons and magnetic monopoles all share something in common. By analogy, if these particles are bipeds, each would have one foot in one brane and another foot in the anti-brane.

 

[Antonio Lao]

The supersymmetry of H+ and H- can be derived from higher order (much higher even values of the order) of the Hadamard matrices.

For example, if the electron is in order 6, and the up and down quarks are also in order 6.

[tex]H^{+}_6 [/tex]

[tex]H^{-}_6 [/tex]

Then the mass ratio of the proton (composed of 2 up's and 1 down) to electron is 1832. The margin of error is less than 1 percent of the experimental value of 1836.

 

[Antonio Lao]

The distinction between dimension and the order of the matrix will now be made.

Traditionally, the order of the matrix is the same as the number of components or independent variables of the physical equations. In this thread, the order is used to describe the number of "direction" in the physical pattern under investigation.

The element 1 signifies one "direction" and the element -1 signifies the "opposite" direction. In the 2 by 2 matrices, one H+ signifies one quantum of "2-direction" and H- signifies the "opposite" to this given quantum. The same can be carried into higher order matrices. Each order has its own unique "opposite."

 

[Antonio Lao]

A better picture of "direction" and its "opposite" can be described as the following:

The element 1 of the matrix implies a string "direction" analogous to the number line. It signifies an increasing value of number to the right of zero. The element -1 implies a string "direction" to the left of zero and signifies decreasing value of number. If this number line is joined end to end into a closed loop, only one of the two "direction" can be kept. These create two loops, one with a "direction" of 1 and the other with a "direction" of -1. In each loop, although the "direction" can still be preserved, it numeric values is lost forever. One loop is the same as a unit vector wherein its magnitude is always 1. An infinitely extended number line with zero at the middle can help quantify physical quantities, a loop cannot. A loop is a quantum of "direction" of a unit vector.