Co-written with Nahid Sahel Gozin

 

 

 

 

 

 

 

 

Classical physics has been developed over the centuries and has been quite effective as a base for science and technology. But many puzzles and paradoxes exist in quantum mechanics (the sub-atomic arena) and astrophysics which cannot be solved in the context of the classical physics. Consciousness also remains a mystery. In fact, the findings of the past century defy the principles of contemporary physics. New revelations have painted a more detailed universe and present an alternative reality that we cannot explain with the traditional axiom. A brand new physics is needed to explain the newfound reality. That is why new fundamental theories are being introduced to explain the unexplained.
In this book, I will present an alternative physical model for the universe and offer explanations for existing paradoxes based on this new concept. In this model, the space-time universe is embedded in an unconventionally defined singularity. In order to better follow this model, familiarity with the concept of complex numbers is helpful. I will try to explain the concept in layman’s terms for the reader who is not keen in mathematics.  Alternatively, the reader may choose to skip the math equations and just look over the assertions made. Doing so will not prevent comprehending the concept.
First, I am going to explain the basic principle of complex numbers. Our physical interpretation of different elements in complex number mathematics will be followed in this chapter and ensuing chapters as needed. The interpretations and assertions made do not necessarily apply or accepted in contemporary physics. The analysis is derived and defined on the context of a new model.
Complex number system can be defined as a Cartesian system where the x-axis represents the real value and the y-axis denotes an imaginary part of a parameter.

Imaginary Numbers
What are imaginary numbers? In the sixteenth century, Italian mathematicians frequently encountered the square root of negative numbers (√-n) in their calculations. Known mathematics could not offer a solution for this problem because number could be squared to a negative number (like -1). in another words, these numbers does not point to any objective physical element. Therefore they where called imaginary numbers and is shown by the symbol i. The square of this imaginary number would be -1.
In today’s physics we attribute imaginary numbers to qualities that are in a sense hidden from us. Or better qualities that do not possess objectivity. However these numbers denote potentiality. Because if we square them they turn to a real number that would affect the equations.
Later on, it was noticed that a combination of real number and imaginary number is essential to explain the fundamentals of mathematics. This combination is called the complex number system. Therefore,

Complex number = [ x (real number) + iy (imaginary portion)]

In 1806, Jean-Robert Argand, trying to give geometrical visualization to complex numbers suggested the diagram below:

Argand Diagram

This configuration demonstrates the imaginary dimension embedded in any real number.
In 1799, Gauss proved the fundamental theorem of algebra using complex numbers. Nowadays, the use of complex numbers pervades a major portion of mathematics and its applications in science. Even pure real numbers are rightfully reintroduced as:

N = x + 0i
Where N denotes a parameter, x is the real value and i conveys the imaginary aspect of the parameter.

Modified Complex Plane

 

The second and third quarter of the above diagram is not defined in the current complex number system

Underneath, polar version of Argand diagram is shown where r = |z|, called the absolute value or modulus, and

a = arg(z), called the complex argument of z. 

Every point in the diagram above can be shown as:

Z =x +iy =r (cos,a +i   sin,a)
  
x= r cos a, is called the  real part
  
iy = i sin a is called the imaginary part.

Does it sound like gibberish? It simply says that complex numbers are a combination of purely real and purely imaginary numbers.
 
Do not be disappointed if the definitions presented in this chapter do not exactly match the conventional definitions for the complex number system. The descriptions and assumptions made in this chapter are defined within the context of the proposed model in this book. As long as the assertions are mathematically sound, we should be able to rely on them and refer to them in future quotations.
One would assume if imaginary numbers are so fundamental in mathematics, they would have to represent a physical reality. As Roger Penrose points out,
“These strange numbers also play an extraordinary and very basic role in the operation of the physical universe at its tiniest scales.” ⁵⁶
In this context, I take the real numbers to represent observables and interpret imaginary numbers as non-observables and qualitative values of physical elements. As mentioned, we measure the elements of space-time by real numbers. The notion of complex numbers implies that any of these elements should have an imaginary dimension in their nature. In other words:
Assumption C1: Any computable in the universe also contains a qualitative non-measurable aspect embedded in it.
Therefore, I conclude that at a profound level, just dealing with objective reality is not enough. To get the whole picture we have to open our scope to include non-observable aspect of physical elements as well.
In 1707, Abraham De Moivre found a similarity between complex numbers and trigonometry. These numbers follow the same rules applied to trigonometric calculations. For example, when we square a complex number we double its phase (angle).

Z² = r [cos a+i sin a]× r[cos a+i sin a]
Z² = r²[cos ²a - sin ² a+2 i cos a sin a]
Z² = r² [(cos² a+1)/2 +i sin ² a ]

Here, Z² is a complex number.  [r² [(cos² 2a+1)/2] is its real part and (sin² a) is its imaginary part.
In this text, we take point zero in the Argand diagram to represent singularity and the imaginary portion of the diagram as the effect of the proposed singularity on the observables.
Imaginary numbers are sometimes called magic numbers. One of the strange characteristics of these numbers is the fact that in De Moivre diagram, any real number coupled with (multiplied by) them will be reduced to zero.

 

 

 

As shown in the diagram above, when we multiply any real quantity by i, its real value (its real number coordinate value) is reduced to zero. Algebraically this can be written as:

(X+0i) i = Xi + 0(ii) = Xi = 0

In trigonometry we can show this with;

X = r Cos a, if we take a = 90, then Cos a = 0, therefore x = 0.

Assumption C2: Although the real number field may create the illusion of continuity, the more accurate complex number version shows us that the continuity of real number breaks down periodically.
Therefore, I conclude that physical elements (like space, time and matter) are discrete and not continuous. We can also show this fact by an evaluation of the function of (x) in any equation. We take y = x IxI as an example.

                                  y=f(x)=x│x│ 
Function

                 

 

                            y / =f / (x) =2│x│

First Derivative

y// = f /  / (x) =2 +4teta(x)
The second derivative exhibits discontinuity around point zero. A lack of smoothness and continuity in derivatives of real number functions.56

For any other function of finite real numbers, we can come to a derivative which shows a lack of smoothness and continuity.

We can take y = 1/x as another example.

 

Plot of 1/x

Although the plot is infinitely differentiable, it lacks continuity. The continuity breaks down as it approaches point zero. So real number quantities inherently are not smooth or continuous (holomorphic). If continuity is desired we have to incorporate the imaginary number and adopt the concept of complex numbers into the equation and rewrite the y = 1/x equation as y = 1/z. where z is a complex number shown as: z = x + ib, where b is any number. The above plots also reveal that discontinuity of real numbers occurs as the curve approaches the y-axis. Therefore:
Assumption C3: The real numbers are discrete and continuity always breaks around point zero.
Here, I conclude that as physical elements aproach singularity their continuity break down and therefore they are discrete.
On the other hand, we can choose any other point in the domain and shift the point zero to that point and use Cauchy formula in the origin shifted form.

 n!/2pi∫ f(z)/(z-p)n+1dz = f(p),

And the nth-derivative expression would be:
       
n!/2pi∫ f(z)/(z-p)n+1dz = f (n)(p),
Roger Penrose writes: “Thus complex smoothness implies analyticity (holomorphicity) at every point of the domain.”56
Assumption C4: The mathematics of complex numbers also indicates that any point in the domain can be considered point zero (cross section of coordinates).
This is important for us when we define the proposed singularity and its relation with space-time in the next chapter. Taking every point of the domain as zero is called blowing up the origin.
The complex number equation Z = R [cos a + i sin a] indicates that these numbers have a periodic nature. So they loose their real number value and hit zero twice in each period. We take the periodic nature and intermittent appearance and disappearance of real value of measurables the basis for our fifth assertion.
Assumption C5: Any measurable in our universe has a discrete nature. This will include matter, time and space.
For example in the diagram at page 22, if x-coordinate denotes the mass of particles, somewhere in its endeavor the tangible mass gradually looses its value and disappears. With the same token, if x indicates dimension and distance, because of the periodic function of complex system, they have to disappear and reappear during each period. This is the basis for our assumption that space and time are not continuum. They have to be discrete.
Another interesting characteristic of imaginary numbers is the fact that although they are influencing the real numbers in equations, they normally do not mix up with them. In a complex number, we normally have to deal with each portion separately. For example for addition we write the equation as:

( 6  + 3 i ) + ( 5  + 2 i ) = 11 + 5 i

Assumption C6: The real numbers and imaginary numbers represent two separate domains.
On the other hand, pure real numbers are represented by just one line of the field (X Axis), whereas anything else in the field has a complex nature. This concept can be shown by using the Riemann Sphere. Riemann sphere is an extended complex plane that includes infinity. If we let the sphere represent a physical object, Just one circle plots the pure quantitative value of that object (blue circle for instance). However, there are infinite circles on the sphere that demonstrate the combined quantitative and qualitative measures of the object.

Riemann Sphere

So we may come to conclusion that,
Assumption C7: The real value (computable value) of an object is numbered whereas, non-computable (quality) of things are infinite.
As mentioned above, in this model, we take the imaginary number (i) as a factor, which represents the singularity effect on different phenomenon.
In addition, in Cartesian coordinate system if we take x, y and z to represent different values in space-time like, distance, permeability, temperature, weight and so on, we notice that zero is located at the center of all of those values (At the center of the line representing each value) But its not any of them (they are measured zero at point zero).

X-axis can represent time while Y-axis represents mass and z - axis may represent volume. Zero is in the center of all of these values.

Assumption C8: While point zero is at the center of any value in space-time, it does not represent any of them (values turn to zero at point zero).
Here, I assume the point zero to represent singularity.
On the other hand if any real value is engulfed by an imaginary domain, then any point of each observable is separated from other points by a kind of environment. Therefore, any measurable is in a so-called, Hausdorff space. The last assumption therefore will be,
Assumption C9: any measurable in the universe is quantized.
 

Summary
Although the mathematics of complex numbers is highly developed, the physical interpretation of the complex system is not fully understood.
The concept of complex numbers opens our eyes to the combination of the qualitative and quantitative characteristics of the elements present in the universe. In reality, this combination is the through meaning of any element. Quantitative aspect alone does not convey the whole entity. The book in front of you is not just its size or the number of pages, words or letters. It is much more than that.
In this model, I take the lead and intuition from the concept of complex numbers system and enter consciousness as the imaginary and effective element in the physical world. I further examine if this extra factor have the ability to take theoretical physics out of the current dead-end and can offer solutions to existing paradoxes.
I hypothesize that; zero is the center of the universe and represents an entity here in after we call singularity. In following chapters we examine the similarities between the singularity and consciousness. In addition, I take zero as a separate entity because in it, the value of space-time elements is zero. 
Zero affects the elements of space-time and the whole existence revolves around it (unit circle concept). Furthermore, zero exists at any point at any domain and field. This assumption defies the concept of continuity and suggests that every physical element in our universe is discrete.

The arguments presented are open for debate. The reader is encouraged to email his/her inputs to correct, modify or develop the contents. Please send your emails to; zpfields@yahoo.ca