# A Principle of Directional Invariance

1. Jul 5, 2004

### Antonio Lao

This principle is needed for a theory on the quantization of spacetime at the infinitesimal region.

The properties of this principle are:

1. Top-right-front 2. Top-left-front 3. Top-right-back 4. Top-left-back
5. Down-right-front 6. Down-left-front 7. Down-right-back 8. Down-left-back

All physical objects must have all these properties in order to appear as 3D objects.

If some of these are missing, the objects will appear as 2D or even 1D.

For 2D objects, the missing properties could be the ones that is associated with top/down or right/left or front/back.

1. top-right 2. top-left 3. down-right 4. down-left

For 1D objects, only one of either of these is used.

1. front-back 2. back-front.

For 1D time motion, these are the properties of forward-backward and backward-forward. These imply that time has two directions

For 1D space motion, these are the properties of right-left and left-right.

For the combined properties of space and time become the following

1. RL-BF 2. RL-FB 3. LR-BF 4. LR-FB

2. Jul 5, 2004

### Antonio Lao

If we were given a task to write a book about the universe, we start with page 1. For a mortal writer, this book can never be finished. So the next generation of science writers will continue to write this book. And then the next generation after that, so on and so forth. This book is a testament to the progress of science as witnessed by human beings. As time progresses, more and more pages are added to this book. The book seems to be progressing in a time directional invariance, from past to present to future.

Each page of the book will have two page numbers. Since we started with page 1, the 1st page will have the numbers 1 and 2, the 2nd page will be 3-4, the next 5-6, then 7-8, etc. the odd number of the page is always smaller than the even number. Each page is a quantum of the entire book. So there are as many quanta as there are pages. By doing this numbering, we are really imposing a numerical direction into the page, odd < even system. But there is another way of numbering the page, by even < odd system which will start with the number 0. So the pages have 0-1, 2-3, 4-5, 6-7, etc. The information still the same in all the pages but there are now two separately distinct books. The quantum of one book can never be inserted into the other book and vice versa. In reality, by just renumbering the page, we have created two distinct directional invariance.

3. Jul 6, 2004

### Antonio Lao

Using Hadamard matrices, two types of n by n square matrices can be formulated with each matrix associated with a directional invariance property under matrix operations of addition and multiplication. The two distinct matrices are denoted by $H^{+}_n$ and $H^{-}_n$ where n is the order or level of existence (LOE) of the particular matrix.

The elments of $H^{-}_n$ is given by

$$e^{-}_{ij} = \left( -1 \right)^{i+j+1}$$

and the elments of $H^{+}_n$ is given by

$$e^{+}_{ij} = \left( -1 \right)^{i+j}$$

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

Last edited: Jul 6, 2004
4. Jul 6, 2004

### Antonio Lao

These two matrices can interact and their algebra are the following:

$$H^{+} \oplus H^{-} = 0$$
$$H^{+} \oplus H^{+} = a H^{+}$$
$$H^{-} \oplus H^{-} = b H^{-}$$

$$H^{-} \otimes H^{-} = c H^{+}$$
$$H^{+} \otimes H^{+} = d H^{+}$$
$$H^{+} \otimes H^{-} = e H^{-}$$

where a, b are values of electric charge and c, d, e are values of mass.

5. Jul 13, 2004

### Antonio Lao

A continuation of post#2: These two distinct books of the universe which almost contain all the mathematical principles and physical laws, can be loosen and separated with all the pages mix up. But if one investigate each loosed page, the order of the books can be recovered. For each page contains the information for determining of which of the two books the page belong to. The directional invariance becomes a conserved property of each page.

6. Jul 13, 2004

### Antonio Lao

It is the mix up states of the pages that gives rise to the statistical and random nature of the universe.