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                         Atoms of Time and Space

Whether we recognize it or not Quantum Theory is based on two untested
axioms.

1/ Matter is comprised of discrete elemental units.
2/ Space and time are continuous.

The mathematics of Quantum Theory works fine. However when we try to
interpret what that mathematics is telling us we are faced with a sea of
confusion. Matter is in some incomprehensible way made up of continuous
waves which are at the same time minute points of matter.

If we try to pinpoint exactly where a particular elemental unit of matter
is we lose all information about its energy.

This paper represents an attempt to make a simple change of perspective.
The two fundamental axioms of Quantum Theory mentioned above are reversed,
The mathematics is reconstructed on the axioms.

1/ Matter is a continuous field.
2/ Space and time are made up of discrete elemental units.



                   Conventions

In order to transmit this paper as a text file the following conventions
are adopted.

U represents the Greek letter lower case mu.
W represents the Greek letter lower case omega.
xSy represents the summation sign indicating summation from x to y.

In a paper of 1927 Louis De Broglie postulated that for each continuous
solution U of Schrodinger's wave equation there existed a solution W with a
singularity representing the particle1. In this way he sought to overcome
the conceptual problems which had, even then, become apparent in Quantum
Theory.

He became discouraged with the idea following its poor reception at the
1927 Solvay Congress2. After embracing the alternative probability
interpretation for some twenty five years he returned to his earlier idea
in the early fifties doing further work in collaboration with Bohm and
Einstein3. At that time the mathematical difficulties seemed however
insuperable. 

As shown below if the matter is approached from a different postulate
solutions approximating to those he suggested 1927 can be found. Using
these solutions it is possible to explain the photoelectric effect without
recourse to wave/particle dualism.

Born and others have suggested the possibility of absolute minimum values
of length, time and mass4.  

The notions of absolute minimum time and absolute minimum length imply that
any length or time can be expressed by multiplying the absolute minimum
value by an integer.

Huygen's principle has long been used to explain refraction etc. as a wave
phenomena. According to the principle, as rays of light pass through
imaginary points of space secondary wavelets are generated. These spherical
wavelets form wave fronts. The rays of light are considered to be
orthogonal to these wave fronts5.

The above ideas suggest the following postulate:

Any region of spacetime can be regarded as being comprised of N discrete
but contiguous elements, N being an integer. These discrete elements are
real counterparts of the imaginary space points of Huygen's principle and
are points of origin of real secondary wavelets.

A region of space-time such as that postulated can be thought of as a
matrix of cells. A spherical wave originating at a cell in the region will
spread out generating spherical secondary waves as it passes through each
cell. Each of these secondary waves will generate spherical tertiary waves
which will generate further waves and so on.

Since each new wave generated will be spherical in form it can be seen that
the resultant complex of waves will trace out every possible wave path
leading from the cell of origin. The number of possible paths will be large
but, given the postulated discrete structure of spacetime, it will be
finite in a finite region. Each individual photon will thus be comprised of
a complex of waves which, all having the same frequency, will form an
interference pattern. 

The wave functions of both classical and wave mechanics are continuous. In
a space-time region such as that postulated wave functions must perforce be
discontinuous, or at best piecewise continuous, since they can only be
defined at the discrete contiguous cells of the region.

It is necessary to define a suitable coordinate system in such a space as that postulated. this can be conveniently done by numbering the cells each axis of the coordinate system passes through beginning with zero at the origin and numbering the cells with integers. 

Given the postulate, power series representing functions of the type
forseen by De Broglie, can be found. Considering a single dimension which
we will call x then the following function can be written.


  Ux  =    (nx - 1)(nx -2)......(nx-(px-1))(nx-(ux+1)....(nx - (Nx - 1)) 
             ------------------------------------------------------------ (1)
                           (-1)^(nx-px)(px - 1)!(nx-ux)!
 Where:  
     
     N is the number of space-time cells lying along the axis.

     n is an integer         0 > n < N

     px is the location of a cell on the x axis.         
     
     the x axis is one of a set of four mutually
     perpendicular axes defining positions in the space-time
     region by numbering the discrete cells along each of the
     individual axes. These numbers are always integers and,
     while they can be negative, for simplicity it is
     convenient to consider that the origin of the coordinates
     is chosen so that they are always positive.
    
Where Ux is the sum of the series for each value of n on the axis.
  
The derivation of equation (1) is discussed in Appendix A.

On inspection it is found that (1) has a value of zero at every cell on the
x axis except px where it has a value of unity.

As a function of x,y,z,t this becomes:
           Uxyzt = Ux.Uy.Uz.Ut                                     (2)
  Where

      Ux Uy Uz Ut are respectively functions of x, y, z and t
      similar to (2).

On inspection it will be seen that (2) is a function which is only defined
at the discrete elements of the space-time region under consideration. It
has a value of unity at pxyzt and a value of zero everywhere else. that is
it has a singularity at px. 

It is now necessary to consider the relationship between (2) and the W
function of quantum mechanics.

Note that the right hand sides of (1) and (2) reduce to power series.

The W function wave mechanics is a wave function. It is usually interpreted
in terms of the paths of many particles. The square of its amplitude gives
the probability of finding a particle at any given cell6.
 
We can extend (2) so that it can describe many singularities quite easily
by summing over a range of values for px giving (using the convention
listed on the title page):

             U = 1Sp Ux.Uy.Uz.Ut               (3) 

where p is the number of particles under consideration.

If the particles are scattered randomly through the region of space-time
then the probability of finding a particle at any given cell in a region of
space-time which contains many cells and many particles is given by the
mean amplitude of (3) in that region.

According to the probability interpretation of quantum mechanics the square
of the amplitude of W is also the probability of finding a particle at a
given cell in a region of space-time.

That is for an extended region containing many space-time cells and many
particles:
                   
                     (U)^1/2 = W.                       (4)

Since W is a solution of Schrodinger's equation (U)^1/2 is also a
solution7.

(U)^1/2 can be substituted for W any time it appears in quantum mechanics.
.                    
         
Thus all the results of quantum mechanics are recovered intact when (3) is
used in a form which describes many particles. 

(3) however can describe a singularity which represents a single particle.
Suppose we carry out an experiment in which a particle is located at a
specific point px. (3) tells us that the probability of finding the
particle at px is unity which is in perfect agreement with the experimental
result.

Note that (3) contains no information other than the probability of finding
the particle at px so the uncertainty principle remains intact.

W then is a special case of the more general expression U. U may describe a
single photon while W is really restricted to describing the probabilty of
finding a photon at a given point.

The advantage of U is that many of the apparent ambiguities of quantum
mechanics vanish since it gives a simpler view of what a photon is. 

Appendix B contains a discussion of the physical interpretation of U and a
discussion of various ambiguities and paradoxws which have been raised in
discussions of quanta. 

In order that full use may be made of graphics Appendix B is in the form of
executable file which can be run from the DOS prompt. It is still under
development and will be uploaded later.

                     Appendix A

As mentioned earlier the photon wave takes every possible path through the
matrix of space-time cells. For mathematical convenience we can define an
idealized space-time in which every possible path has its own dimension,
all of the dimensions being mutually perpendicular. 

In a single discrete unit of time the wave moves forward one discrete unit
of space. 

This gives immediately an important result. The ratio of the unit of length
to the unit of time has the dimensions of a velocity. It is a velocity. It
is the velocity of light.

Considering just one dimension of our idealized space-time. The wave
originates in a discrete cell of space-time. It then moves through a number
of space-time cells along the axis of the dimension corresponding to that
path. 

In any given time interval t the number of space-time cells passed though
along each of the paths is constant for all paths since the velocity of
light is constant. In a vacuum the photon wave will occupy a sphere of
space with a radius ct.

The photon may travel to almost any cell of space-time within that sphere
provided the length of the path it follows is always ct. The one exception
is the cell of origin. While the photon can travel back to the its cell of
origin in space it would reach there at a later time. 

The number of secondary waves generated is constant for any given time
interval since the path length depends only on the length of the time
interval. Letting n be the number of cells along a particular path the
number of secondary waves it generates along each path is constant and is
given by n-1, not n as the photon cannot return to and pass through its
cell of origin.

Similarly the number of tertiary waves generated by each secondary wave is
n-2. These tertiary waves will generate n-3 further waves and so on.

This suggests that an expression for the total number of waves generated
along each possible path will have the general form:
             (n-)(n-2).....(1)                  (A1)

This is the form in the idealized space of multiple dimensions. We however
live in a space of the four dimensions xyzt. 

In our space the paths are not separated. They are convoluted together.
They interact with each other forming an interference pattern.

Our experiments can only reveal a projection of the idealized space
structure onto our xyzt space.

The results of countless experiments suggest empirically that the actual
form in our space is:

 (nx - 1)(nx -2)........(nx-(px-1))(nx-(px+1)....(nx - (Nx -1))   (A2)      

This expression has the property that its amplitude is zero  everywhere
except at the cell px where it has a very large value.

(A1) can be normalized by dividing by it by its value at px giving: 
                      
Ux = (nx - 1)(nx -2)........(nx-(px-1))(nx-(ux+1)....(nx - (Nx-1))  
     -------------------------------------------------------------                
                 (-1)^(nx-px)(px - 1)!(nx-ux)!

         Which is the equation (1) above.



                             Bibliography


       1   Louis de Broglie, New Perspectives in Physics, Basic
           Books Inc., New York, 1962. p92-94 

       2   Ibid p96-97

       3   Ibid p101-103
       
       4   Max Born, Physics in My Generation, Springer-Verlag
           New York Inc., New York, 1969.p76.

       5   Most elementary physics text books would contain a
           description of Huygen's Principle. Specifically a
           description of the principle and many of the other topics
           discussed in this paper can be found in:            
           Kane and Sternheim, Physics SI Version, John Wiley and 
           Sons, New York, 1980.    Chapter 25.
  
       6   Ibid p535 and p548

       7    For a lucid description of the properties of power series 
            see:
            Philip Gillett, Calculus and Analytic Geometry, D. C. 
            Heath & Coy Lexington Massachusetts, 1981. Chapter 13.