Non-Interval Probability
and its possible relationship to particle masses

his piece is a continuation of the last page. Though I doubt that this will pan out in the long run, I must nevertheless pursue it to its logical conclusion. It may well be the last plausible (or semi-plausible) idea I shall ever have in regard to the subject of particle masses. New ideas are getting fewer and farther between.

In the previous discussion, "normal" probabilistic calculations are based upon the concept of the "interval". As we have seen, without such an interval no statistical calculation is possible ... until now.

Non-interval statistics

What I wish to obtain is a set of discreet (but seemingly random) numbers consistent with simple rules which somehow miraculously turn out to be equal to all the particle masses discovered or ever to be discovered. "Pheeeewwww ! Can he do it?"

I am going to proceed on the assumption that the theory (unstated) of compound mass values is unworkable and must be replaced by a theory of "discrete, acausal, holistic values" independent of the particle's quantum constituents. By compound mass, I mean the theory that there are specific mass values related to each quantum parameter present in every particle, i.e. you just add up all the mass contributions from each parameter and that is the mass of the particle.

By discreet, acausal and holistic, I mean that the mass value of a particle is independent of its quantum parameters and is applied to that particle by reference to a (non-continuous) spread of discreet possibilities obtained from a master equation of probability keyed by integer denominators rather than integer numerators as is the case with common interval probability. Since this "spread" is non-continuous, we cannot take an interval in the normal way. We cannot take pieces and put them together summing to a specific value.

Thus, instead of having units of mass/energy like 1 unit of charge or 1/2 unit of angular momentum, and adding them together to get a "sum", we just have this odd value which at first sight seems to make no sense relative to other values. The values chosen won't be composites of other values. They are what they are and are not divisible. They do have a relationship (purely mathematical) to other values but that relationship is not a physically, causal one. It is "holistic".

These values are subject to some normal rules of physics. For instance, we can't create a particle of a given mass value if there is insufficient mass-energy available for its production. The standard rules of thermodynamics apply and must supercede any mass-specifying principle.

The Fundamental Rule of Quantum Mechanics

As I have given it before, this rule requires that no two particles of differing quantum properties can exist relative to one another and at the same time relative to an infinite. This produces contradictions, e.g. if a particle possessed one unit of mass and another non-composite particle had a mass of seven units, then, they cannot simultaneously exist relative to one another and to an infinite space with an infinite number of particles in it ... for if so, we should have to say that 1 is to infinity in the same manner as 7 is to infinity and therefore 1=7 which is false in all finite situations (a contradiction at the finite level - for two things cannot both be and not be at the same time).

Since it is known that there are a multitude of dissimilar particles existing relative to one another, some sort of explanation is in order. It is this.

Because space is not as yet infinite relative to any given particle, they may co-exist temporarily. And, in fact, all particles decay in a finite time except the electron and proton. But these two also differ in mass so that another explanation is required.
It is this.
As the electron-proton mass ratio increases, the electron mass approaches "0" at infinity. Therefore, when both electron and proton exist relative to an "infinite", the electron mass will have collapsed to zero leaving just the proton with a mass value of "one". And, as all protons are identical, it is logically acceptable.

As an initial equation of non-interval probability, I have chosen ...

its simplest representative. Namely,

y = xe / ex

Wherein, "e" is Euler's number 2.71828...
This produces the quintessential "Maxwellian-type" distribution with a unique point at value [x=1, y=1] ... (which is to represent our "unit" mass value, i.e. either the proton or neutron mass).

Our graph will look like this ...

Maxwellian-type Distribution - 3kb

This distribution allows for any mass to be possible ... but not equally possible. If all values to infinity were equally possible, there would be no logical way to choose any one of them by any probabilistic means whatsoever. In an unbounded field there is no way to set an interval because all intervals are identical relative to that infinite field (1/infinity = 6/infinity). More specifically, the area under the curve must be finite in order to calculate probability ... not infinite, as in an unbounded field of equal probability.

If we were calculating most probable masses in the "interval way", we would make an interval and calculate its area as a fraction of the area under the curve and that would be the probability of finding a particle in that mass spread. However, this continuous spread of masses is not found in nature. We find specific, discreet masses ... always the same. We must therefore pick such values off the graph at specific places.

And there is no "fixed" interval dependence. That is, we can't obtain points on the graph by drawing a vertical line up from the x-axis to intersect the graph ... and ... put those vertical lines a "unit" distance apart on the x-axis. This is not found in nature. We have to do something which is graph dependent and x-axis independent ... something "weird" or "different".

An Interval - 3kb

We might also try to cut up the "y" axis into finite intervals. In this case, we would have to choose some number (which number?) of intervals such that they would fit into the given finite length a whole number of times. Here I've cut it into 21 intervals ... why not 27 or 168 or 5684653 ??

pmass3e.gif - 3kb

Suppose we cut up "y" into all whole number intervals and let probability be determined by those which are more likely, i.e. the prime numbers? Thus, if we cut "y" into 44 pieces, we've covered the line numbered 6 of 44 as 6/44=3/22 ... i.e. that line was already accounted for when we cut "y" up into 22 intervals. The larger the number the greater the likelyhood that its intervals are already covered.

So, we will cut "y" up (denoting discreet mass values) in this manner. Low numbers first, prime numbers cutting off the bulk of the possible intervals (actually ALL of them).

Discreet Values at Intercepts - 3kb

Then, we will redefine the zero point mass as that of the electron. Hence, [0,0] will be where the electron is placed ... or ... the spectrum of probability values will extend from "1" unit mass to (presently) 1/1836 unit masses such that no particle can be denoted at less than the electron mass. Thusly,

Electron correction - 3kb

This process must necessarily produce an abundance of possible mass values from a single, simple logical mechanism. Note that the "y" axis does not denote standard interval probability but rather a different discreet function. The area under the curve therefore needn't be "1" unit.

A particle may then have any one of these values provided that there is enough available energy in a given interaction. Which value is more likely is determined by the "logic" of quantum mechanics along with a prime number check-off system wherein the "2" checks off all further fractions divisible by two ... "3" checks off all further integers divisible by three ... and so on so that the most likely mass value is denoted by the smallest integral denominator.

Since the mass is hypothesized to be independent of other quantum parameters, a fair question would be ... Why can't one mass value denote more than one particle in violation of experimental observation? The answer must be that the given array of sub-atomic particles is the only consistent one possible given the above restrictions (which admittedly at this point appear to be too "loose" to hold up to the evidence).

Why do this in this manner?

Because I demand an extremely simple explanation for something so elementary.

Because I am committed to the view that the electron-proton mass ratio is given by probability applied to the hydrogen bound state in conjunction with the fine structure constant as given in the page

"The e/P Mass Ratio and Fine Structure Constant".

Because it fits the presently known general layout of mass values as given in the previous page even though it may be dissimilar in specifics. We know we must have a mass of "one" unit and that there are many other masses on the same general scale as that unit. The likelyhood of producing an extremely aberrant mass value will be extremely small or non-existent. The spread of mass values will be more than the electron yet no common mass value can be very close to that of unit mass (proton) as is known to be the case. Etc.

In the next page, I shall generate values for hypothetical masses. I can guarantee that they will appear to be a random set yet be simply derived. Whether they will correspond to known masses values is dubious at best. But I will (as I say) ... try.

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