The Recombination Principle: The decision and perception (recognition) process as cause and consequence of multiple recombinations, happening from moment to moment
Preliminary note and hint for reading
(NewInfo) is an elementary entry. Especially for readers with mathematical entry knowledge there are precise formulations of the introducing arguments and building up on this quantitative considerations to the nature of proper time(TimePerception). However, it isn't necessary to understand all details at once. Additionally, during first reading the lot of footnotes may be skipped. The table of contents offers a concise outline, also look at the by (***)marked locations within the text[1]. A concise formulary is separately accessible.
{InformationDef} The word "information" resp. "information quantity" plays an important role and is sometimes interpreted variously. Therefore it's emphasized, that here the word "information" is used in the standard sense of technical literature (information theory) [lish1] [2].
1 Infinity resp. infinite diversity is permanently emerging (because of differentiation and decision) and not a priori existing as constant entity (completed in past)
{RealInfinityGrowsWithTime}[3] At the beginning of the 21-th century the (obvious[4]) understanding of the infinite resp. infinite diversity as something, which permanently is emerging together with time (TimeConformApproach) due to differentiations and decisions[5], which cannot be classified as something a priori (i.e. before present in past as completed totality) existing, was still unusual. Basis of mathematical physics were the axioms of traditional set theory[6] which start out of (a priori) existence of infinite sets resp. infinite diversity. But if something exists in the physical meaning, it's already past and thus fixated and naturally restricted (cf. a. [liro]).
To avoid misunderstandings: Of course I don't think, that's all. There is everywhere and always the prerequisite for every distinction, (distinguishable) time and decision. This exceeds the limits of every possible perception and imagination, i.e. it is really unlimited resp. infinite {RealInfinity}[7].
2 To the usage of infinite sets in mathematical physics
Unlike the infinite the (within finite time) measurable (as information perceptible[8]) reality, which is the physical reality, is just characterized by being finite. Particularly it's information content is finite.
Concerning the physically existing reality (resp. physical reality) a scientific consensus should be possible. Even Hilbert comes to the following result ( [lihi] p.165, translated from german):
"Now we have established the finiteness of the reality in two directions: to the infinite small and to the infinite large."
Every physical measuring needs a finite, different from zero measurement time and provides information (cf. a. InfoConcrete) in form of the choice of a measurement result from all possible measurement results. If infinitely many (different) measurement results would be possible, the choice[9] of a measurement result could deliver infinite information (an infinite information quantity [InformationDef]). But the results of physical measurings (of finite[10] duration) never deliver an infinite quantity of information. Therefore the set of all possible measurement results a priori is finite[11].
This (natural) fact has been mentioned in literature already long time ago (cf. e.g. [lipe] p. 195). Nevertheless even in quantum physics analytical models and derived concepts (exponential functions, operators with continuous spectra...) are still usual. - May be that many people (also scientists) cling on the model of continuous reality (as something which already exists) not only because of the macroscopic impression but also because they think that "continuum" is necessary for freedom of decision - because it's subdivisible infinitely often. I think there can be a bridge: Future as the subdivisible together with freedom of decision along proper time as primary axiom, and finer and finer approximation of continuum as consequence - so fine subdivision not a priori, but in the course of time as result of decisions. The order (Order) is important.
In the physical reality only a finite information quantity can be processed within a finite (proper) time[12] interval. For mathematical models whose representation requires a processing of an infinite quantity of information, for example irrational numbers, no (exact) equivalent exists in the physical reality. So mathematical calculations, which have an equivalent in physical reality, can only be rational combinations of rational numbers. Conclusions arise from this for the foundations of mathematical physics.
2.1 The perceptible (physical) reality as that, what is (exactly) conceivable within finite time
Deduced[13] analytic functions like
etc. are frequently used for description of physical (natural) processes. The question arises for a fundamental explanation of the fact, that those functions can be used to make approximative prediction of physical measurement results (that is a limited forecast of perception resp. future). Such explanations should[14] base on as simple as possible axioms.
It is now started out from the assumption, that those axioms permit only a finite number of elementary combinations per time unit, in which an elementary combination {ElementaryCombination} is defined as basic calculating operation, i.e. as addition resp. subtraction or multiplication resp. division of integer quantities resp. numbers. This assumption seems to be justified, because such elementary combinations are within finite time exactly conceivable (comprehensible), at least in the potential sense, for example by counting. That's important, because something, which is perceptible, is also conceivable at least in the potential sense.
There are common mathematic models, e.g. numbers, which aren't conceivable within finite time and therefore aren't perceptible (sometime, in some representation, in complete exactness[15]). Therefore these models deviate from perceptible reality[16] after some time and in principle are unsuitable for an exact elementary approach to it[17], even if these models deliver and further will deliver good approximative results.
Of course also mathematical partial models can be very helpful {helpful} (especially for approximative[18] calculations) and are thus fully justified[19], for example in case of to macroscopic particle numbers greater 10^26 and still much larger number of combinations n per proper time unit (ProperTimeUnit)… We only should guard against over interpretation[20] of our models of thought because they are not equivalent to reality (AnalysisAtBestApproximative). Particularly, if we forget the simplifications contained in our model, we would block[21] a "thinking beyond the model". This problem of course also affects my mathematic suggestions. Also here occasionally approximative considerations are used (as bridging) e.g. the use of the Stirling formula. I hope that it remains clear, where simplification begins. Perhaps sometime there is a possibility that we can speak about this.
At first the following preceding chapter should clarify the mentioned difficulties of current models: They orient themselves too little to our fundamental decision and perception process. Particularly the natural sequence (Order) of the combinations which are (in the large and in the small measure) cause and result of our decisions and perceptions (resp. measurings) isn't taken into account in these models.
For example the axiom of choice (on which basic analytical concepts build up) postulates a priori the existence of infinitely many decisions - from the physical point of view a contradiction in terms, because "a priori" means "before presence" resp. "in past", but physical (measured) past is finite.
3 Model concepts like infinite continuous sets, Hilbert spaces and the axiom of choice permit the disregard of the natural (temporal) order
In mathematical physics analytical approaches and concepts (i.e. approaches and concepts of analysis) are quite common, and with that also the usage of continuous, a priori infinite sets. Most important examples for those sets are the complex and real numbers. They form a metric space equipped with the absolute value norm. Hilbert[22] spaces play a central role in quantum physics. Important quality of these metric Hausdorff-spaces is the completeness i.e. any Cauchy sequence converges towards a limiting value which is contained in the space. This is problematic if used for description of nature, because for the exact description of the limiting value of a Cauchy sequence it's mostly[23] necessary to carry out (isolatedly, before any interaction with the surroundings[24]) an infinite set of approximation steps, if one allows respectively only elementary combinations [ElementaryCombination]. This implicitly means (uncoupled from the natural order[25] (Order)) an infinite number of decisions[26] (application of the axiom of choice[27] [limy]), therefore the processing or production of an infinite large quantity of information[28], which isn't possible under natural conditions within in finite time[29] (at finite availability of free energy (FreeEnergy)).
So from the retrospective point of view the appearance of quantum phenomena in the physical reality [PhysicalReality] would have been predicable (cf. a. [NoAnticipation]): Confirmed is not only the fundamental limitation of the quantity of information, which is perceptible within finite proper time, it also becomes clear, that basic analytic concepts (i.e. continuous sets of numbers) are models, which deviate from reality {AnalysisAtBestApproximative}.[30]
3.1 In principle restricted validity of the used models
It isn't a miracle therefore that mathematical models which are based on the completeness of the underlying metric spaces only can be restrictedly valid for the description of actual natural events[31]. The problem quite similarly also lie in other models which also require the axiom of choice (often indirectly and hidden)[32] or which start out in some other way of the infinite (of infinite diversity) as something which already exists as completed entity [RealInfinityGrowsWithTime].
Specially at calculation models, whose validation is not guaranteed because of missing or only indirect experimental possibilities, there is a strong[33] likelihood that the sequence of calculation steps (implicitly connected to decisions and perceptions[34] within the restricted model) differs from the natural order (Order) of the combinations connected to decisions[35] and perceptions. So nonsensical calculation results are the consequence. The difficulties lying in these results are (more or less) known by many insiders and also should be evident in publications[36].
3.2 Task: Finite approach the existing, physical reality
If our considerations should not be superficial, the mentioned problem is fundamental and severe. Therefore we have to accept, that a mathematical approach, which is faithful to physical reality, also is finite (finite according to http://arXiv.org/abs/quant-ph/0108121 ). There are two possibilities for the method on the way to this approach:
1. We continue to work with the usual mathematical models, which presuppose a priori existence of infinite sets and the axiom of choice, and hope, that sometime the infinities can be in a way reduced, so that the resulting approach finally is finite.
2. We start with plausible approaches, which don't presuppose a priori infinite sets (which are finite from the beginning and so also discrete). This implies that we have to consider initially minimal sets of possibilities for experimental results which are created and enlarged in the course of time (inclusive order). In the borderline case "t_total to infinite" this process should lead to current physics (and more) - a start from the other side to meet in the middle (again).
As far as I know up to the beginning of the 21st century only the first of both possibilities is considered in literature[37]. Due to the diversity of publications it's difficult for me to estimate, how far the in 1. mentioned hope for complete reduction of all infinities is legitimate, but I got the strong impression that lopsided much intellectual capacity is invested in this possibility. It is questionable, whether this is efficient - the method "explore and hope for reduce" allows many wanderings.
Without guideline there are (too) many possibilities {TooManyPossibilities}. Already today there are many special subjects (and special-purpose languages) so that an increasing hindrance of communication is the consequence. Due to the nature of the matter substantial progress on a way to an exact[38](and therefore also finite) approach probably only is possible, if at this research is done without the axiom of choice, continuous number sets and all from it derived model concepts, even if it is difficult[39] at first.
This is one of the reasons for this which caused me to try the 2nd possibility. At this it's comprehensible to work on the principle that the finiteness is reflected in the fact, that the number of combinations (mapped on elementary arithmetic steps) leading from a decision (to measure) to a perception (of a measurement result) is finite, exactly if measuring time is finite. This is the case in the approach discussed below[RecombinationCountFinite], in which the progress of proper time is related to a meeting of (own) pattern with (formerly separated) counter pattern(TimePerception).
3.3 Finite approach by combinatorial inspection of information paths
No doubt, the way of the information from our decisions to our perceptions depends on a physical law. Because of the shown problems this law can not be of analytical and therefore continuous (for example geometric[40]) nature. Primarily it must be a discrete, combinatorial law[41]. Of course in case of a macroscopic measuring many combinations happen, so that the borderline case of the continuous, geometric appearance results.
Abbreviated prehistory: About 1992 I
noticed, that we could only recognize codes, for which we have the counter code
(the decoding code). So before every measurement we have to send away something
from us like an "anti pattern" or "test pattern"
(later I recognized that this action lays in our decision for the measurement)
and the change of the returning pattern (relatively to the original) contains
the information of the measurement result. I studied the probabilities for
return of the test pattern and noticed, that they correspond to the
coefficients of the
One important progress connected with this approach is, that it gives first insight in the (of course finite) ways of information from decision (to measurement) to perception (of measurement result) and that it contains only finitely many arithmetic steps - from start in the present center until return to the center. There is no a priori necessity[42] of analytic models [which implicitly uncouple[43] physical reality from consciousness, which hide the connection between our frames of reference and so can lead to a wrong and restricted philosophy [EgoismIsStupid].
The next chapter should give insight into the nature, way and sequence of the combinations, which are connected to our decision and perception process and make first mathematical suggestions, of course only as far as I could get ideas of it. I hope you then will understand why I have named these combinations "recombinations". I have approached the topic in a similar way in thoughts like the next chapter is written.
4 Geometric appearance as (statistical) result of a discrete combinatorial law
First a recall: We should not forget that all geometrical appearances like "location" and "direction" are only possible, if rest mass exists, i.e. if there is an asymmetrical appearance of matter and antimatter relatively to our present frame of reference {AsymmetryAsPreconditionOfGeo}(***).
4.1 No isolatedly definable metric units, no "smallest particles" resp. "building blocks" of matter
It is already known that the concept "(smallest) particle" is only a model idea which doesn't correspond (exactly) to reality. Of course it's understandable that use of this model concept is frequently made: It is easy to imagine matter as composition of smallest particles or "building blocks”[44] (with firm, absolute sizes) and helpful for calculations. But it isn't consequent to be left with this model (or the mathematically equivalent wave model[45]).
4.2 Special functions as dimensionless conversion factors
When we try to free ourselves from this model idea, first of all the question after the basis of metric[46] sizes arises: What is small, what is big? It is known that measurement results regarding this are dependent on the observer's "point" of view. In this context the functions
play both in geometry and in the physics an important role, e.g. QV(v/c) [47] as factor for relativistic time dilation or QW(v/c) as factor for length contraction (cf. a. [GyroscopeModel]). These functions contain only an approximative approach but they after all are used for calculations which don't start out of absolute scales but of a non-linear[48] connection of the sizes of the observer system and watched system.
4.2.1 Discrete considerations to nonlinear relativistic relations can lead to a bridging from relativity theory to quantum physics
{BridgesToRel} Relativity theory (problematically) presupposes continuous geometry and the frequently used functions QV and QW lead to irrational results. The following discrete considerations (e.g. to finite series expansions of QV and QW, cf. (TaylorQV), (TimeDilation) and (TaylorQW)) avoid this from the beginning and at the same time offer approaches for a bridging from relativity theory to quantum physics.
4.3 Correlation (of sizes) and flow of information
Correlation implies flow of information and this is necessary for every observation (to and for finished perception also back[49](FinishedPerception)) The conservation laws (Cons0Sum)show that calculation is exact in the end {ConservationLawsImplyExactness}.
(To be new the transmitted new information component should lie in (physical) quantities, which are temporarily solved from this connection (correlation coefficient resp. scalar product is 0, geometrically: orthogonal). Probably therefore the individual flow of the information must change the direction[50] {DirectionChanges} at every new observation.)
As generally known, information (initially) is transmitted by photons[51]. These information packets move to the following target point [FollowingMeeting] (where the are absorbed) with light speed and no information exchange is possible in between(otherwise we would regard the "in between” as following target, [WayTimeConstantTillNextMeeting]. Especially in elementary consideration at the moment of the start we don't know, which of two elementary directions[52], which spin the photon will choose resp. which possibility is predestined (by an earlier decision). In abbreviated formulation[53]: We don't have information about the destination of the photon or its next decision.
4.4 Combinatorial considerations to the ways of information (Q0-triangle) (***)
We come to the {combinatorics} now {InformationPath}:
The elementary information unit is a bit which means the information about the choice for one of two possible alternatives. Let us assume that we have no information about the next decision resp. the next direction. No direction is preferred in the beginning {DecisionFreedom}. Thus both alternatives have equal probability p=(1-p)=1/2. Now, each of these possibilities again is starting point for a new decision etc... The so emerging probabilities for the different ramification possibilities resp. recombination points have a symmetrical binomial distribution (cf. [lifa] p. 245-281, [ligr] p. 153-256, also[54][lied] p. 3). We will call the totality of recombination points subsequently "Q0-triangle"[55] :
n k-> -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
¯
0 1' *1/1
1 1 1 *1/2
2 1 2' 1 *1/4
3 1 3 3 1 *1/8
4 1 4 6' 4 1 *1/16
5 1 5 10 10 5 1 *1/32
6 1 6 15 20' 15 6 1 *1/64
7 1 7 21 35 35 21 7 1 *1/128
8 1 8 28 56 70' 56 28 8 1 *1/256
9 1 9 36 84 126 126 84 36 9 1 *1/512
...
The probabilities in the vertical symmetry axis of the triangle are indicated by " ' ". These are the probabilities of meeting events{CentralMeeting}in the center, the "central meeting probabilities” and dependent on the row number n. We shall call them Q0Z (n) and obtain
The resulting table for Q0Z(n) is
On the other hand the Taylor[56] series expansion of QV(x) is
Remember that QV(x) is the factor of relativistic time dilation in case x=v/c {TimeDilation} (cf. e.g. [lifl] p. 26 and 27).
4.5 The middle column as the vertical symmetry axis, the column of the central meeting probabilities of pattern and counter-pattern
{CounterPattern}The (vertical) sum[57] of the "central
meeting probabilities" (probabilities for return) corresponds (because[58] of 4p(1-p)= x^2) to the
4.5.1 Correlation of decision resp. perception (orthogonality as information theoretical concept)
If 0 < x < 1, then we have a correspondence to the case that the probability p of a step to the right differs from the probability of a step to the left, i.e. p is unequal 1/2. This means that we here already have more or less information about the next decision there i.e. already more or less information exchange[59] has been possible between here and there. The proper times[60] of here and there are not orthogonal, i.e. the "correlation coefficient[61] of decision resp. perception" isn't zero, but has a parallel, common component [CommonComponent]. Because of inertia this is valid along a series[62] of steps, in which proper time here seems to run QV(x) times faster then there [TimeDilation], where the factor QV(x) is equivalent to the sum of the own central meeting probabilities [PerceptionInCenter].
4.5.2 Proper time proportional to the sum of the central meeting probabilities; input is descended from former output
In analogous way also from our own (lokal, individual) point of view the central meeting probabilities correspond to the probabilities (per double step n->n+2) that the information resp. pattern, which had been temporarily separated[63] (in form of free energy [FreeEnergy]) by our decisions from us resp. the information starting from us returns to us and is perceived[64] by us again in recombined[65] form. This can be understood indicating, that with our temporal perception {TimePerception}, with each proper time progress necessarily the re-union[66] of our own (lokal, individual) pattern and counter-pattern[67][CounterPattern](which has been separated from us by our former decisions) is connected – that in the end exactly that pattern is perceptible for us, which is descended from us[68] (separated by our own former decisions[69]) [OwnPerception], whereby (outside of present consciousness) in between more or less many recombinations happened[70](***){PTimePropSumQ0}.
May be, that on the second look much of this conclusion is obvious without much arithmetic, already due to the constancy of the light or information speed relatively to us personally.
So the formulae indicate, that all information[71], which we now receive (input) is descended from information, which we formerly have sent (output), and, of course, that also future input will be consequence of former output (***).
4.5.2.1 In the two-dimensional model the total number of steps resp. central meeting points is proportional to t^2
{NPropT2} If we define
then holds
So if we start out from the assumption that (in case of no reference system change, two-dimensional model) the sum the central meeting probabilities Q0Z (return probabilities) is proportional to proper time t, then (for large n) the step resp. row number n increases proportionally to the square of proper time.
4.5.2.2 (Proper time without reference system changes; row number n and distance covered during constant acceleration )
Preliminary note: At last this approach seemed to me less relevant than the one of (small) correlation (GravitationBecauseOfCorrelation).
We know that the length of the distance covered during constant acceleration is proportional to t^2. This may remind to the proportionality of the row number n to t^2 in (NPropT2). Also gravitation conveys the impression of constant acceleration, at "constant" distance (and missing centrifugal force). But the classification of a distance as "constant" is dependent on the own length scale, and with the row number n also squarely the row length and with that the own length scale can increase {DynamicLengthScale}. Of course more exact considerations should take into account also the distance dependence. The probability for way there and way back between recombination points is the greater, the smaller the relative distance between them.)
4.5.3 In case x=E0/E the symmetry center k=0 corresponds to v=0
{xAsE0divEandkAsMomentum}Above(TimeDilation)we set x=v/c and recognized QV(x) as proportionality factor of the relativistic time dilation. But the interpretation possibilities are by no means exhausted. If for example we understand x as quotient E0/E of rest energy and total energy and P as absolute momentum, then because of [QWasExpectationValue]we get
{Ppropkdivn} We see that the momentum is proportional to the expectation value of k/n. So the symmetry center k=0 of the distributions (vgl. (Q0Triangle) und (Q1Triangle)) can be related not only to the case P=mc resp. P=mc but also the case P=0 resp. v=0 (Low Temperature Physics).
4.5.3.1 At this the mean value of k/n is proportional to the quotient wave number/frequency of matter waves
The wave number K of matter waves is
and their angular frequency
.
So in the case x = E0/E because of [Ppropkdivn] follows
.
4.6 The introduced model (Q0-triangle) needs additions
The presented model, which works with the Q0-triangle, surely is incomplete (and also too flat, cf. a. [NotFlat])), questions like "From where comes the source in the start?" cannot be answered so. A modified Q0-triangle shall be introduced now, in which the central meeting probabilities are "flowing out" and therefore in the actual system for the present cannot be sources any more (by the fact that they are put on 0).
They then could flow out to another (orthogonal) direction and come back after[72] and after. Symmetry considerations (attention of conservation laws) can give first hints, where and how this happens. If for example something flows out in the center (concerning both sides exactly symmetrically) then the total effect of the returning must also concern both sides in symmetric way (e.g. output in k=0 <-> input in k=0 or symmetrically round k=0; generally the number of drains isn't necessarily equal to the number of sources. Multiple points (PerceptionOfMultiplicity) of in flow and of out flow (also "back flow"[73]) are possible per proper time unit (ProperTimeUnit).).
4.7 Directed flow of information, centrally "flowing out" probabilities: Q1-triangle
The following "Q1-triangle" bases on the assumption, that during measure resp. perception process all of the central meeting probabilities[74] is taken away, so that they can't be sources (for superposition, interference) in the same triangle any longer (directed flow of information).
So they get incompatible with each other: The definition of incompatibly can be formulated miscellaneously, e.g.:
1. Incompatibly of two events means that they cannot happen[75] simultaneous.
2. Events not appearing at the same time are incompatible with each other if the first excludes the following.
The second definition applies to our case: If a single quantum is "flown out" centrally in the previous row, it cannot flow out in the next but one row again {DistinguishableOrder}.
4.7.1 Orthogonal change of direction in recombination points (multidimensional approach); Separation (of inside/outside, past/future) due to perception (of something separable), due to differentiation
The more central probabilities "flow out" (are set to 0) with increasing row number n, the more disconnected become left and right side of the triangle. Therefore a so described perception resp. measurement {QuantumPhysicalObservation}causes (quantifiable) separation (***) resp. separability (according to the perception[76]), which a decision makes possible in turn, in this model between the left and the right[77], in multidimensional[78] approach perhaps also between "inside" and "outside"[79] or past and future [IOtime]. Geometrical concepts like orthogonality are portable to [information] theoretical concepts, cf.(orthogonal)
If one provides the probabilities in the normal Q0-triangle on the left side with negative[80] sign (cf. [ProbabilityAmplitude]), the following "Q1-triangle" results. It is a modified Q0-triangle with central probabilities[81] set to 0:
n k-> -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
¯
0 ±1 *1/1
1 1 -1 *1/2
2 1 0 -1 *1/4
3 1 1 -1 -1 *1/8
4 1 2 0 -2 -1 *1/16
5 1 3 2 -2 -3 -1 *1/32
6 1 4 5 0 -5 -4 -1 *1/64
7 1 5 9 5 -5 -9 -5 -1 *1/128
8 1 6 14 14 0 -14 -14 -6 -1 *1/256
9 1 7 20 28 14 -14 -28 -20 -7 -1 *1/512
...
4.7.2 Quantitative considerations, symmetries
It is obvious that the amounts of the probabilities decrease (centrally they are flowing out[82]).
For even n the sum of all probabilities in row n equals the central meeting probability Q0Z(n) of the normal Q0-triangle, the sum of all squares of them equals Q0Z(2n). The simple sum yields 0, what matches the conservation laws well {Q1RowSumIs0}.
4.7.3 Differentiation of the Q0-triangle, solutions of the quantum mechanical oscillator, Hermite polynomials
One can arithmetically regard this Q1-triangle as discrete differentiation [DiscreteDiff] of the Q0-triangle along k, the horizontal direction. So the removing (perceiving) of the Q0Z results in a differentiation along the horizontal direction (difference left-right, d/dk). Perception surely also means differentiation along time axis (difference future-past, vertical, d/dn) and the correspondence [TimeSpaceCorrInMiddle] of vertical and horizontal differentiation in the middle is remarkable.
The graph of a multiple (discretely) differentiated Q0 function yields an continuous seeming wave-like picture after a larger number of recombinations [wavelike].There is a far-reaching analogy between those multiple discretely differentiated functions and the solutions of the quantum mechanical harmonic oscillator: By multiple discrete differentiation construction of orthogonal systems is possible, analogous to the Hermite polynomials [HermPolDiscrete]. At this the Hermite polynomials [HermPol] are (except sign) special cases of pre-factors resulting from multiple differentiation in the analytic borderline case. There are further considerations possible concerning integration and differentiation. Some can be found in the download files.
4.7.4 Possible further (open) combination possibilities
Particularly demanding: How can several triangles[83] be combined in different (how much?) directions(NotFlat)? At this the system must remain {open}[84](***). Do multiple application of the Maxwell-equations give partial hints? How can the recombination points (in symmetric way) be connected, to get broad analogies to the physical measure-, distinction and decision process (cf. [PauliMatrices][85])? Which recombination points are (dependent on observer location) differentiable in time[86], which one differentiable in localization, which seem to be an unit (cf. [ElementaryCoordinates])?
4.7.4.1 Strength of interactions
If there is a physical interaction between two systems, there is a way between them over more or less many recombination points. The shorter the passage, the more probable it is in the average (per proper time unit(ProperTimeUnit)), the stronger it appears. For example the strong interaction probably goes over only relatively few recombination points. This also permits more symmetries. The weak interaction however probably needs more recombinations, so that this complex connection hasn't left-right symmetry any longer [DecisionFunnelBorder]. It needs more time, from which the possibility for a comparison with past left-right definition arises.
4.7.5 Neg. sum of central "outflowing probabilities" multiplied by the sum of the central meeting probabilities yields 1
For every even row number n>0 let be the number |Q2Z(n)|=-Q2Z (n) the "outflowing probability", i.e. the probability for flowing out[87] centrally. Q2Z(n) is equivalent to the 1nd (discrete) derivative of Q1(n,k) in k=0 along k, i.e. Q2Z(n) = (Q1(n-1,1)-Q1(n-1,-1))/2; so Q2Z(n) is in k=0 the 2nd derivative of Q0(n,k) along k. It holds:
The resulting table for Q2Z(n) is
On the other hand the Taylor series expansion {TaylorQW}of QW(x) is
The coefficients of the Taylor series expansion of QW(x) correspond to the negative probabilities centrally flowing out. If a system is separated by a potential x^2 (if it moves for example with v/c=x relatively to us), so this expression is proportional to the component {CommonComponent} resp. part of time (reality) which is common between us and observed system, which is also belonging to own proper time[88] and present and therefore will also become own past [ToPast]. It is the greater, the smaller the separation potential x^2 is. One also can think about a matching, more exact description of the initial situation in the Q1-triangle [StartQ1]. The formula of all probabilities in the Q1-triangle can be found in the addendum [FormulaQ1].
A discrete differentiation (DiscreteDiff), here along (proper) time is clearly dependent on the relation subject/object.
4.7.6 QW (x) corresponds to the expectation value of |k/n|
In case of equal probabilities of steps to the right and to the left no direction of coordinate k (cf. (Q0Triangle)(Q1Triangle)) is preferred and the expectation value <k/n> of k/n is zero. <k/n> is equal to the difference of the probability p of a step to the right and (1-p) of a step to the right, i.e. <k/n> = p-(1-p)=2p-1. Because of x^2=4p(1-p) [xAndp] from this results {QWasExpectationValue}
4.7.7 An information theoretical interpretation of the Planck effect quantum h
{hAsConstantProduct}We know, that the Planck effect quantum can be
understood as product t*E of proper time t (of a measurement) and the energy
uncertainty E (of the measurement result). Proper time resp. measuring time is
proportional to the function QV(x) resp. to the sum of the return probabilities
in the Q0-triangle(PTimePropSumQ0). The partial sum of
the
4.8 Many (arithmetical) coherences
Due to the shown coherences of course it is reasonable to examine finite partial sums of the taylor series expansions of QV(x) resp. QW(x) more exactly, also in the case of imaginary[89] x, |x|=1 and even for |x|>1 . The corresponding "probabilities" for steps to the right or to the left wouldn't be limited within the real interval [0,1] any more[90] because of 4p(1-p)=x^2.
The mean vertical reach (1st order momentum) of |Q2Z(n)| from row n=1 on up to the outflow is equivalent to the sum of the Q0Z(n) from row n=2 on (cf. [DefQ2Z]), i.e. the mean reach from row n=0 on is equivalent to the sum of the Q0Z(n) from row n=0 on and therefore approximates QV(x). At more detailed occupation with the topic many coherences stand out (cf. [DeviationQ1Equal1], the formulary in wqm or the concise formulary). A more exact definition of concepts like "simultaneity" resp. "concomitance", a more exact analysis of the process of new creation[91] of information and the process of copying[92] (also parallelizing) information would be necessary. The formation of scalar products (Scalarproduct) in horizontal and vertical (and even sloping) direction in the triangle could serve as bridging to the common mathematical framework of quantum physics[93]. Because of the underlying recombination principle and for study of different branching resp. connection possibilities in the triangle the consultation of specialists in [combinatorics] and graph theory [RGraphTheoreticalResearch] can be helpful (perhaps even of geneticists or in the field of the genetics active mathematicians).
4.8.1 The mean deviation in the Q1-triangle is constant
{DeviationQ1Equal1}An example of arithmetical coherences:
In connection with the constance of the Planck effect quantum
also the constance of the mean deviation (1st order momentum) of the Q1 in horizontal direction is interesting (but the in [hAsConstantProduct]described information theoretical consideration seems more reasonable for me). For example the following (surely abbreviated) interpretation may be a first suggestion for further thinking:
Here the summation goes over both horizontal halves, the impulse was decomposed in time * force. The time ET was interpreted as a not sub-divisible unit, as elementary time between the beginning and end of the summation (the integral) and the force as current probability for flowing out, as probability for leaving within the time ET the relative[94] location with maximum impulse (with maximal speed v=c).
(A possible bridging to quantum
physics: One could consider the Compton effect (***), (the interaction of a photon with
matter; I also think of the generalized
The Plancksche effect quantum hq still was interpreted here quite graphically (as product of physical quantities). [A priori] An information theoretical interpretation is more consistent, though (hAsConstantProduct) [lish1]. One also can understand hq as energy * time and so as information * proper time. On the average much Information can be given [give] resp. transferred only along short time intervals, little information along longer intervals. Of course at this has to be considered, that the conversion factor for energy * time to information * proper time isn't a constant but a function, dependent on the extent of branching depth and on renormalization .
4.8.2 Q1 as finite difference of Q0
The exact formula of the Q1 function is:
Let be n a natural number, k an integer with absolute value smaller or equal n, and p a number contained in the interval [0,1]. If we define
and
then holds
The Q1-triangle results from a superposition of two Q0-triangles with opposite sign, starting in position n=1, k=±1 after multiplication by 1/2. Addition of both means a discrete differentiation {DiscreteDiff}[97] along k. The formula then arises from the difference quotient with minimal dk, i.e. dk=2:
(Q0 (n, k + 2) - Q0 (n, k)) / 2 = Q1 (n + 1, k + 1) .
Analogously one can make m-th order (discrete) differentiation by using row n=m of the Q0-triangle, equipped with along k respectively alternating sign, as starting row[98](BinCoeffDiffMatrix). The initial zigzag of the accompanying function graph flattens in the following rows to m+1 continuous seeming waves {wavelike}. Discrete alternating state functions can also cause wave-like phenomenons (probability distributions), if the initial situation (e.g. the binomial distributed discrete analogue of "phase angle"[PhaseAngle]) is not sharp. The superposition of many rows with even (or odd) row number n and alternating sign respectively yields a wave-like picture.
Still to mention is, that a second order discrete differentiation (finite difference) along k means[99] a first order discrete differentiation (finite difference) along n:
A connection to the (***) Schrödinger equation {Schroedinger} seems reasonable.
Remarkable is the correspondence of vertical and horizontal differentiation in the middle{TimeSpaceCorrInMiddle}:
Q1(n+1,1) = Q0(n+2,0) - Q0(n,0) = (Q0(n,2) - Q0(n,0)) / 2
The middle (the vertical symmetry axis k=0) represents the relativistic borderline case[VisCinMiddle] - also in the common theory arise equal rights of time and location coordinates in the relativistic borderline case.
5 Bridges to current concepts of quantum physics
5.1 Probabilities and probability amplitudes
{ProbabilityAmplitude} [liba][libo][lifi][liko][lipa] In quantum physics usually one talks about complex valued probability amplitudes, the corresponding probabilities are calculated secondarily from the square of the absolute amount, i.e. from the scalar product with the corresponding complex conjugated probability amplitudes. Just also the central meeting probabilities Q0Z(n) (resp. |Q2Z(n)|) correspond to such a scalar product. In the presented example (cf. Scalarproduct) real probability amplitudes are used - even if the currently used probability amplitudes are complex, their scalar product has to be real, if measurable. As bridge to current concepts one also could e.g. identify the absolute value of Q0(n, k) resp. Q1(n, k) as amount of a probability amplitude, whose phase angle is varying with n and k (cf. RowSumAsWave). It has to be remarked here that in context of discrete considerations the phase angle{PhaseAngle} cannot be calculated (exactly) and so doesn't have an equivalent in the reality. The (continuous) trigonometric functions and of course also the (complex valued) exponential function are approximative. However, exact discrete representations[100][DiscreteRepresetations]are possible, which don't imply infinite series expansions.
5.2 Example of a (discrete) scalar product
{Scalarproduct}: In discrete considerations [like] of course integrals have to be replaced by finite sums[101]. Particularly important sums are scalar products. In the compendium of formulas wqm several dependencies are described for different possibilities of scalar product formation in the Q0-triangle. Also in the Q1-triangle there are such. An particularly obvious example, which allows remarkable simplifications[102]