Summary
Newton's Calculus has not changed for 3 to 4 centuries. The same is true with complex mathematics. I believe that this mathematical impasse is one of the main reasons for the lack of progress in quantum mechanics.
Newton defined the differential particle as the smallest possible particle. I redefine the differential particle as the smallest possible particle IN REAL SPACE.
This means that if complex space is valid and exists a sma;;er particle can exist (the complex particle).
The Nature of the Complex Particle
The exact size of the differential particle is not officially agreed upon. I assume it is between 10^-30 and 10^-40 metres and take the value of
10^-30 metres. If so then the complex particle is defined as having a size of 10^-900 metres.
If we keep magnifying a smaller and smaller region of space we eventually are left with a single point (the differential particle).
I would say that at this point 3D space has not lost any integrity and is not distorted.
I also define the complex particle in 3D as circular or spherical. If we now proceed to a smaller distance with greater magnification what would we see?
I
n Classical Quantum Mechanics this region is full of quantum foam. The concept of quantum foam allows for normal 3D space to break down but not completely.
There is always a tenuous connection between the quantum bubbles. However it is hard to argue that there is no loss in integrity.
The complex particle provides a better explanation as the fabric of space does not lose it's integrity. That integrity simply changes.
I have already defined 3D space has being spherical and 4D space has being cubic. The question is what kind of geometry can allow such a drastic transformation while keeping the integrity of space? Any such geometry must be seamless, continuous, flexible with both 3D and 4D while maintaining a sustainable geometry of it's own in between 3D and 4D.
In the figure "Geometry of the Complex Particle" we see that four squares enclosing a circle each can provide a shape I define as a "ztar" or complex star
shape shown in yellow. In this figure the ztar has an inverse radius to that of the circle and allows for the geometry of the ztar and thus the complex particle to be hidden from our normal experience.
The ztar can be any combination of complex particles naturally combining into this shape (as spherical particles do in 3D) while the complex
particle is the smallest possible particle in undistorted complex space.
In the figure "1 - 1/(y^2) = 1/(x^2)" an approximation to the inverse of the circle 1 = 1/(y^2) + 1/(x^2) is given by 1 - 1/(y^2) = 1/(x^2) or
1 = 1/(x^2) + 1/(y^2). This equation is not the same as 1 = 1/[(x^2) + (y^2)] though. The equation y = 1/x or y^2 = 1/(x^2) is a close match and I believe the true
shape of the complex particle although it is asymptotic to the X and Y axes. The curve 1 = 1/(x^2 + y^2) is actually the same as the unit circle!
The rectangular hyperbola y^2 = 1/(x^2) will fulfil our requirements and we will see subsequently allow for a perfect intersection with a complex
plane where distances vary not linearly but logarithmically.
It is debatable wether the feometry of the ztar y^2 = 1/(x^2) distorts as we approach 3D into the equation 1 = 1/[(x^2) + (y^2)] but at least
raises the possibility.
This definition for the ztar allows complex numbers and complex space to exist between any two neighbouring spatial dimensions.
The Arms of the Ztar
I envisage the length of the arms as being variable depending upon the inclination of the complex plae and it's distance from real space (e.g. 3D). There can be any number of orthogonal arms depending upon which two neighbouring dimensions it bridges.
