C.                   THE QUANTIZATION OF SPACE TIME

       It is possible to justify, or perhaps to rationalize, the quantization of space-time from prior work.  The basic definition of force is

                               F º m  a                                                        eq 22

and the basic definition of energy is

                               dE º F  • ds                                                  eq 23

thus                        dE = m a  • ds  = (E/C²) a  • ds               eq 24

or                            dE/E =  a • ds / C²                                   eq 25a

                               E/E_o =  _e ò(d²s /dt² )•ds /C ²                                                           eq 25b

Notice that there is energy and only that on the left side of eq 25a or 25b and space time variables and only those variables on the right.  If energy is quantized, then that implies that space-time also is quantized.

As an induction it is proposed that the very fabric of space-time itself is quantized. This assumption implies that there is a fundamental smallest unit of space-time itself which determines the position of an event. "Position" implies the four dimensional coordinates of the event, including the x,y,z, and the time, t, of the change.  This smallest unit of space-time will be called the SPASON.

The thing we call a vacuum is not "nothing".  Indeed it does not have either mass or energy,  and it does not exhibit momentum or in fact any mass related property.  The most important thing is does have is the ability to convey position.  In addition it has a capacitance,  dielectric constant, e = -1,  a speed of light (when paired with the attenuation per radian or the absorptive index, k, this defines the complex refractive index  n* =  n(1-ik) for other materials),  permitivity¹⁴⁷ (8.85418782 x 10⁸ F/m)    and  permeability (4 p x 10^-7 H/m  or 1.2566370614 ... x 10^-6  H/m)

       The concept of the Spason is intended to subsume the older, and now semi-forgotten concepts of the HODON and the CHRONON previously proposed ca 1932 as the quanta of space and the quanta of time.  The concepts of Hodons and Chronons fell into disrepute after their initial proposal since the early workers tried to apply energy quanta equations to them, and this failed.  It failed since the concept of energy is irrelevant to these more fundamental quantum concepts.  But there is a more basic flaw in the logic of having both separate quanta which caused the creation of the one new name Spason. Space and time are not really independent, but different aspects of a unified space-time continuum.  Just like the electro-magnetic fields should not really be separated in the general case, space and time should not be separated in the general case.

        TIME is CHANGE. To have change implies that there must be something there to change, that space itself must exist before time can exist. Time can not exist without space. Space might  conceivably exist without time, but it would be unchanging and therefore rather un-interesting. Thus the concepts of space and time should not be separated, but the quantum of space and the quantum of time are linked, and the one word Spason is meant to remind us of this linkage.  Where the spason exhibits mostly time like behavior the word chronon, which has priority,  will be preserved to indicate this, and where the spason exhibits mostly distance like behavior, or the positional component is to be emphasized over the time like part, the word hodon will be preserved.

       The basic quantum of the spason has been  measured either in time units or in space (distance) units.  But since it is one and only one concept,  it is improper to measure it in both sets of arbitrary units at the same time.  If the spason is measured in time units the chronon, T, would be expressed in seconds.  If the spason is measured in distance units then the hodon, x, would be measured in centimeters or cm (or meters, but there will appear a reason for use of cm. as there is a coincidence that makes the cgs system of units very easy to use,  and despite the recent domination of SI a return to cgs is here warranted).

       The least possible change in distance, x,  divided into the least change in time, T, yields a constant ratio for this universe (or any universe, but it may change for other universes).  This least possible change ratio is C, the speed of the photon in that universe.

       The speed of light  in a vacuum, C, is defined as:

                               C= x/T cm/sec                                                           eq 26

1.                  THE FORMULATION  OF A BASIS SET OF UNITS

       The units  used in "C" above violates the key statement that the spason should be measured only in either "time like" or "space like" units but not a mixture since this involves BOTH arbitrary units of time in seconds and arbitrary units in distance in centimeters. Here the basis will be taken in time like units alone and a new basis distance,  x_b , equivalent to one second will be found.  This equation will then be reformulated into a dimensionless set of BASIS units where the speed of light expressed only in consistent time like units is D and the distance in this basis frame of units, x_b, is expressed in units equivalent to seconds.  The normalizing factor, b, that allows this is such that 

                               x/b = x_b                                                                       eq 27

                               C/b  = D                                                                      eq 28

                               D=  x_b/T                                                                      eq 29

Looking ahead the normalization factor b will be very close to one (in fact  b = 1.04609857356)  if time is in seconds converting to distance in centimeters. Thus this is the reason for reversion to the cgs system of units, since D for example comes out very near C in quantitative measure. It allows easy conceptualization of the system.

a)                  A BASIS VOLUME

       In order to evaluate b it is necessary to introduce the concept of a "basis volume".  The basis volume, B, is a small cell (this will turn out to be approximately one cc- in fact exactly 1.144768919  cc ) such that if all the mass and all the energy in the whole universe were divided by the total volume of the whole universe each photon would occupy one and only one basis volume.  This implies that there will be one and only one set of changes happening in this basis volume. The photons thus are assumed to be the place where changes are occurring,  i.e. as an induction the photon poles are being linked to space time itself as the place where changes occur.  Since the poles change in two opposing ways these pairs of changes should not be de-coupled and a pair of changes is actually linked in the basis volume. [This pair of changes in fact are a sort of conservation of motion, one rotating left handed and the other right handed when the directly forward or the front option is blocked, thus this gives rise to several conservation laws at higher levels of interaction].  

       The basis volume, B, is related to the normalizing factor, b, such that

                               B = b³                                                                                     eq 30

       Let the total energy density of the universe, E_r , be defined such that if all the mass, m_u, and all the energy, E_u’, of the universe were converted to energy via equation 6

                               m_u C² = E_u"                                                                          eq  6b

and if the total volume of the Universe, V_u , is divided into the sum of all types of mass or  energy expressed in energy units, E_t  then:

                               E_t= E_u’ +  E_u"            (= m_t C² )                                           eq 31a

                               r =  m_t/V_u      (gm/cc)                                                           eq 31b

and in a basis cell    r/ B = r_b     gm/basis volume                                         eq 31c

or                             r /C² = E_r     erg/cc                                                                         eq 31d

then the energy density (also the total energy since there is one and only one photon involved) in the basis volume, E_b , will be

                               E_r /B = E_b      erg/basis volume                                           eq 32

and  the number of disturbances,  N_d  , or the number of changes that take place, in this volume for the basis time of one second will require that the number times the duration of each quantum expressed in time, T be equal to 1 (second).

                               N_d T = 1 sec      or   N_d = 1/T                                               eq 33

       The next concept is that the number of spasons along one edge of this basis volume,  N_e , can be given in terms of the quantum of distance, x,  or more properly the quantum expressed in basis units, x_b,  such that

                                N_e   = b/ x  = 1/ x_b                                                                 eq 34

And the total possible number of spason in the basis volume, N, must also be the number along any edge cubed

                                N_e³  = N = 1/ ( x_b )³                                                             eq 35

Note that these all must be in integer values, i.e. N,  N_e ,   N_d ,  T and   x_b all must be related by integer powers of some constant.

It is now possible to state all of these variables in terms of one key variable, the dimensionless speed of light, D.   We already know that  x_b

is related to T this way;  x_b/T = D.  These other variables must also be related by some integer power of D, and it is proposed that they must be related such that:

                                 x_b  = 1/D²                                                                             eq 36

                                 T   = 1/D³                                                                             eq 37

                                 N_d  = D³                                                                              eq 38

                                 N   = D⁶                                                                                eq 39

                                 N_e  = D²                                                                               eq 40

       Note that the proportionality constants all are unity, since the events being described are also integer quantum events. There was early consideration that the quantum of time or the quantum of distance might be variable in differing parts of the universe,  and might vary with, for example, photon energy.  This was rejected since the photon energy is irrelevant and while the space-time quanta no doubt control the photon, it is less basic and can not control the more fundamental entity.  Each change in space - time results in ONE - precisely one - no more no less - unit of time thus there is a one to one relationship in the number of disturbances,  N_d,  involved.  What ever the number of possible spasons on the edge,  N_e,  of a basis cell is the total number clearly is N_e cubed.

       Thus since  N_d  and T are inverse quantities,  we must have some combination of integers that has  powers of 3 and 2 in them and the lowest possible number for N is  D⁶ . Since 6 works with  N_e  coming out to the power 2, the whole system is the least power possible, and by Occam’s razor we cut the problem.

       D is related to L₁ and the interpole distance to wavelength ratio implied in equation 3 

                   l  = L₁ d =  6 p⁶         d                                              eq 3(repeated)

 where

                   L₁ = 5768.3351 = 6 p⁶                                               eq 4(repeated)

       The ratio of mass of electron to mass of proton has previously been stated  in eq 13 as  L₁/p.  There are five poles involved and the space volume disturbed in the proton thus must be corrected to reach the linear portion of the motion by 5/3 for each photon.  Thus the value of D should be

                   [(5/3) L₁/p ]³  = D                                                                eq 41

or in another form:

                   [(5/3) M_p/m_e]³  = D                                                             eq 42

or                D= 1000 p¹⁵ = 2.8658145969 x 10 ¹⁰                           eq 43

                                 =  28,658,145,969   possibly as an exact  integer.

       C also can be calculated from theory, and is  given by the equation below relating the mass of electron in a hydrogen atom plus the mass of the proton (6 p⁶  + p ) to the photon linear motion content involved as if the 30 photon poles,  correcting for the restricted motion because of the electron- proton interaction accounting for  1/6750 total as if the poles were allowed to move in a linear (non- rotary or mass like) motion.

       C= [ (30 + 1/ 2(15) ³ )( 6 p ⁶ + p )]²=                                        eq 44

       2.99792456193 x 10 ¹⁰ =29,979,245,619.3  compared to

       29,979,245,620   (±110)  =  2.997924562  x 10 ¹⁰   observed.

With the evaluation of D and C it is possible to evaluate the rest of the values.

       b = C/D  (eq 28)  = 1.04609857356                                                   eq 45

       B = b³   (eq 30)  = 1.144768919                                                    eq 46

       x_b =1/D² (eq 36)  =1.2175977039 x 10 ^-21 (pure number)           eq 47

       T  =1/D³ (eq 37)  = 4.2486966942 x 10 ^-32 (pure number)               eq 48

       N_d = D³ (eq 38)   = 2.35366295117 x 10 ³¹ (pure number)             eq 49

       N   = D⁶ (eq 39)  = 5.53972928467 x 10 ⁶² (pure number)               eq 50

       N_e = D² (eq 40)   = 8.212893218   x 10 ²⁰ (pure number)                eq 51

       x = b( x_b ) = 1.2737272175 x 10 ^-21 cm equivalent to 1 sec               eq 52

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