burkhard heim mass formula

Introduction to Heim's Mass Formula

Abstract

A unified 6-dimensional polymetric structure quantum theory by Burkhard Heim (1925-2001) will be described, which yields remarkably exact theoretical values for the masses, the resonances, and the mean lifetimes of elementary particles, as well as the Sommerfeld finestructure constant. Since this paper is not an original contribution, the overview of the derivation of the mass formula within Heim’s structure theory will not be printed in a journal, but published in the Internet. This paper is an attempt to present Heim’s nearly 700 pages on a semi-classical unified field theory of elementary particles and gravitation in a more understandable form, because the results of this theory should be brought to the attention of the international scientific community. In the beginning of the 1950s, Heim discovered the existence of a smallest area (the square of the Planck’s length) as a natural constant, which requires calculations with area differences (called metrons) instead of the differential calculus in microscopic domains. Here we use selector calculus, which Heim employs exclusively in his books, only when its use is indispensable and maintain the general tensor calculus otherwise. For comparison with the work of Heim, in the introduction we discuss briefly the state of the art in the domains of elementary particles and in structure theory. Heim begins by adapting Einstein’s field equations to the microscopic domain, where they become eigenvalue equations. The Ricci tensor in the microscopic domain corresponds to a scalar influence of a non-linear operator Cp on mixed variant tensor components of 3rd degree ϕp

kl (corresponding to the Christoffel-symbols Γpkl in the macroscopic domain). In the

microscopic domain the phenomenological part will become a scalar product of a vector consisting of the eigen values λp(k,l) with mixed variant tensorial field-functions. These terms are energy densities proportional:

Ci ϕikl = λi(k,l) ϕi

kl (i, k, l = 1,...,4) The non-linear structure relation describes „metrical steps of structure“ because of the quantum principle. 28 of these 64 tensorial differential equations remain identical to zero. The remaining 36 equations can be written in a scheme of 6 × 6 elements of a tensor, whose rows and columns are vectors and therefore define an R6 for the representation of the world. The two new coordinates x5 and x6 are interpreted by a collection of values which are organising events, since they can change the distributions of probabilities of micro states in space-time. The 6 coordinates will be unified in three semantic units which do not commute: s1 = (x5, x6), s2 = (x4), s3 = (x1, x2, x3) , where s1 and s2 are imaginary and s3 is real. The metrical tensors which can be construed from these sµ are partial structures κik

(µ) (with µ = 1,2,3). The matrix trace of the tensorial product of the 9 elements, which each are construed by 2 of these lattice cores κ µm

i( )

gik mi

mm

k( ) ( ) ( )µν µ νκ κ=

=∑

1

6

,

constitute a quadratic hyper-matrix, called „correlator“ , where x = 1, ...,4 , depending on the kind of non-euclidian („hermetrical“) groups of coordinates involved: : a = s

$ ( )g xikµν

1, b = (s1 s2), c = (s1 s3), d = (s1 s2 s3). This polymetry corresponds to a Riemannian geometry with a double dependency on coordinates. The solution of the eigen value equations for each of the 4 groups of coordinates (“hermetry-forms“) can be interpreted physically in such a way that the self condensations a are gravitons, the time-condensations b are photons, the space-

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condensations c are neutral particles, and the space-time-condensations d are electrical by charged particles. The correspondences of the Christoffel-symbols in microscopical domains are tensorial functions, “condensors“, of the 6 coordiantes i, k, l and of the µ partial structures:

[ ]ϕ∂∂

∂∂

∂∂

µνκλ

µν µν µν

µ νκ λµν

κλ

kli is sk

msmk

kmsg

gx

gx

gx

( )( )

( ) ( ) ( )

,,/ ( ) $= + −

=

==

∩

∑∑ −+1 21

3

1

3

.

The law of variance steps for the destination of mixed variant forms holds only if the same correlator element is used. Otherwise the analogy to the Kronecker tensor will be described by the „correlation-tensor“ . The condensor must be complemented by this part, since it is also possible to perform affine displacements with it:

Q g gki il

lk( ) ( ) ( )µνκλ

µν κλ=

[ ] [ ]∩ ∩

= +∑ −+( )( )( )1 spQkiµνκλ

κλµνµν

κλ

If is the „structure compressor“, which corresponds to the Riemannian curvature tensor, then Heim’s field equations (after forming traces) read:

ρ κλµν

klmi

( )( )

[ ] [ ]ρ λκλµν

κλµν

κλµν

kl kl klK( )( )

( )( )

( )( )= =

∩ ∩

with the operator Kkl , which constitutes the first derivatives and products of the [ , respectively, as well as additionally a tensor which denotes the correlations, and which is set up by squares of the Q

]∩

ik and of the condensors.

By this extension of the Riemannian geometry a very large manifold of solutions arises. Since the phenomenological part which appears in Einstein’s field equations now is totally geometrizised, there is, according to Heim, no “big bang“ with an infinitely dense energy. Instead, matter appears only after very long evolution of a world without any physical measurable objects, which only consists of a dynamics of geometrical area quanta. In the solutions the exponential function ϕkl = f ( e ) with y² = xy kl− λ

1²+x2²+x3² or y² = (x4²+x5²+x6²) i.e. appear. For real y static exponentially fading fields arise. In the case of imaginary y there will be periodically appearing maximal and minimal condensations of metrons, or structure curvatures, respectively. The maxima of structure deformations ϕkl

(µν)max coincide with the minima of internal correlations: Q . The extrema

exchange with each other periodically. With the possible combinations of the four partial structures for the fundamental tensors, several correlation-tensors as extrema can be united each in a group. For gravitons only two groups of couplings exist; for photons and neutral particles there exist 6 groups with 30 condensors, and for charged particles there are 9 groups of couplings with 72 condensors. Between the groups there are “condensor bridges“, which form complicate dynamical systems of networks.

ki ( )µν

κλ = 0

For the minimum as well as for the maximum of condensations there exists a spin tensor. It is based on the non-hermitic part of the fundamental tensor, which forms an orientation of

spins of the hyperstructure in the region of the involved condensor [ as a “field-activation“. There “fluxes of condensors“ can be formed when 2 neighbouring condensors are such that the contra-signature of one and the basic-signature of the other are identical (i.e.

and [ ), since then both condensor-minima have a joint maximum of couplings, and the joint field-activator activates the proto-field in the correlating basic-signature of the other condensor. That results in a movement of the condensor around the maximum of coupling in

]

] ]

µνκλ∩

[ µνκλ∩

κλµν∩

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the sense of an exchange process. The structure condensations (condensor fluxes), which exchange periodically act against the principle of balance of the compressor, so that a balanced position arises (compressor-isostasy). The structures of couplings of the possible hermetry-forms form 6 different classes of condensor fluxes in the possible subspaces of R6 , which can generate flux aggregates, whose structure depends on the order of flux classes. Therefore, for a structure of coupling there exist at most 1956 structure-isomers. The cyclical fluxes always generate a spin. This ambigious condensor-spin additionally leads to spin-isomers. A condensor flux is stable in time only if an initial condition for the involved condensor signature in the structure of coupling alters to a final state after a distinct time, which is identical with the initial condition. Such a condensor flux circles around the diameter of the aggregate (λ = h/cm) with a certain frequency. The masses are proportional to the eigenvalues of the composite condensation levels λm(k,l). It is found that only such flux aggregates can exist for which the cyclic flux directions of condensation-levels are orthogonal to the so-called world-velocity Y (that is the sum of vectors of temporal changes of all R6-directions): λm(k,l) ⊥ Y , while the vectors of eigen values are parallel to each other. Each alteration of the constant relative velocity in space has the effect that the λm(k,l) must adjust themselves, which presents a complex rotation in R4 (corresponding to the Lorentz matrix). The reactive resistance which is connected herewith acts as a pseudo-power, which appears as inertia. Therefore all condensor- and corresponding energy-terms behave inertially. Since all the hermetry-forms contain the condensor [1

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1] , which consists of the s1 , they are sources of gravitation. Only gravitation fields can be transformed away, since in this condensor only one single partial structure occurs. The 6 flux classes consist of the combinations of the hermetry-forms [s1], [s2], [s1 s2], [s1 s3], [s2 s3], [s1 s2 s3], for each of which the field equations have to be solved. They yield prototypical basic flux courses (prototrope) and appear in the heteronomous case (basic-signature different from contra-signature in a condensor) as basic fluxes of the flux-unit, a „flucton“, in the underlying hermetry-space or as a spectrum of structure-levels in the stationary homonomous case, which are called “shielding fields“ and are enveloping fluctons. Such a primordially simple structure consisting of a flucton and ashielding field, called “protosimplex,“ is a structural primordial form of material objects. By correlation of several such prototropes by which the fluctonic elements of the protosimplexes will be joined to cyclic flux aggregates (conjunctives), material properties arise. Prototropes with the condensor which is built up from s3 take on ponderability. Those in which combinations from s2 and s3 are contained have an electric charge, too. The λm(k,l) assign to each protosimplex an inertial action as mass. The spin number in R6 (related to the action-quantum) is composed of the spin in the imaginary sub-space of R6 and of the spatial spin in R3. The imaginary spin component changes with integers P according to P/2 and shows how many spin-isomorphic matter field quanta of the involved hermetry form constitute a isospin family. The spatial spin is characterised by the integers Q and counts in the form Q/2 also imaginary but it appears with the factor of parity multiplied, i.e. by the number -1 in the power Q/2 . If Q is even, i.e. Q/2 is an integer number, then the tensor terms are bosons, which can superimpose in the same volume. If Q is odd, then the parity will be an imaginary factor, and the spatial spin of such matterfield quanta will be half-integers. Terms of this kind are fermions or spinor terms, respectively, which exclude each other in the same R3-volume. The integral total-spin of an R6 flux aggregate defines a screw-sense with respect to time. This axial vector take a parallel or anti-parallel direction with respect to the arrow of time. The two settings of the spin vector are two enantio-stereoisomeric forms of the same aggregate in R4 , and each represents the anti-structure of the other one.

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The determination of the particle masses means that a dynamical system has to be projected onto an algebraic structure. Heim restricts himself to the special case of the state condition of a dynamical equilibrium. The polymetric tensor relations are all defined on the the field of complex numbers and therefore can be split into a real and an imaginary part. Heim only analyses the real part, since in this case the restricted condition of a stationary state of dynamical equilibria can be used. It was found that the physical R3 of a c- or d- hermetry form has a fourfold contouring by correlating condensor fluxes or protosimplexes, respectively, which are ordered in 4 “configuration levels“ (n, m, p, σ) of different density. In the practically impenetrable central zone n the density grows with the cube of the occupation of protosimplexes; in the likewise dense zone m the density grows quadratically, and in the “mesozone“ p it grows linearly. From this mesozone the outwards directed interactions go out. For mesons there exist 2 quasi-corpuscular regions. For baryons there are three, which justifies an interpretation as quarks. The kind of occupations of the zones in case of the underlying unit structures always depends on the invariants which determine the complex hermetry, and which as quantum numbers determine the basic dynamics of the internal correlating aggregates of condensor fluxes and thus represent invariant basic pattern. The basic patterns correspond to a set of quantum numbers (kPQκ)C(qx), where k is a “configuration number“, P is the double isospin, Q is the double spatial-spin, κ is the “doublet number“, C is the “structure distributor“ (strangeness) and qx is the quantum number of charge. According to this scheme there should exist a spin-isomorphic neutral counterpart of the electron. The masses of the basic states of the elementary particles with mean life times > 10 -

16 sec agree very well with the empirical values. Some particle masses (e, p, n, π+, Λ, K+, K0, Σ+ und Ξ 0) only deviate from the measured values relatively by nearly 10 -6, but the particle µ only by 10 - 7, and the other by 10 - 5 (η is known to three places only). Also the mean life times of these basic states agree well with experimental data ( the particles π±, K 0 and Σ+ show a relative deviation by 10 - 5 , the remaining correspond to the third or fourth place, respectively, with measured values). The masses of the excitation states (resonances) are located at the position or rather close to the measured values. But the theoretical values still follow each other too tightly (with distances going down to 20 MeV/c²), since a selection rule is still missing. The theory predicts a new particle o+ (omicron), whose mass is about 1540 MeV/c². One of the resonances of the omicron is located at 2317.4 MeV/c², which is exactly the value for the particle DSJ

*(2317), which recently was detected by the Barbar Collaboration experiment at SLAC (2003). An energetic excitation of a unit structure happens stepwise from of the external zone via the two internal to the central zone and lets the occupations of protosimplexes raise. In this case the quantity of the “protosimplex-generator“, which describes the invariant quadruple of the occupations parameters of all 4 zones and which is built up from quantum numbers, must be multiplied by an stimulation function, which depends on the integer numbers N. Each value N > 0 in relation to a basic pattern always generates a quadruple of numbers of occupation parameters of cunfiguration zones, whose energy-masses thus represented are interpreted as resonance stimulations of the pattern N = 0. If in the unit mass spectrum the particular frame structures provided with negative sign are inserted, then the protosimplexes will be extinguished, which would correspond to an empty-space condition. Nevertheless a non-zero mass term remains, which only depends on the involved basic patterns. These ponderable structures are neither defined by a coupling structure nor by any flux aggregate. These “field catalytes“ represent the “identity“ of an isospin family, which consists of P + 1 components, and can be identified with neutrino states. For k = 2 there are 4 neutrinos. For instance, the ß-neutrino has the mass m(νß) = 0,003818 eV.

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For further empirical tests Heim investigated proton-electron interaction in H-atoms. On this occasion a relation for the finestructure constant α could be derived, in which a correction must be performed, which is required by the existence of R3-celles due to metrons, and which yields the numerical value: 1/α = 137,03603953 . An excellent confirmation of Heim’s structure theory was established in 2002, when we computed Heim’s mass formula anew. If of the three natural constants h, c, G which enter this theory the most recent values for the gravitation constant G are inserted, then some of the masses of basic states will become more exact (e, p and n up to 7 places, for instance), as would be expected for a correct theory. Illobrand von Ludwiger, July 2003 IGW Innsbruck

Remarks on the Physicist Burkhard Heim

The physicist Burkhard Heim (Febr.9,1925 - Jan.14, 2001) is today mostly unknown amongst physicists. In the 1950s on the other hand, Heim became an international celebrity, when at an international congress on space flight he discussed the theoretical possibility of “field propulsions“ for space vehicles for the first time.

In 1944 Heim lost both hands, his eyesight and his sense of hearing in an explosion

accident. With the help of his father, Burkhard Heim studied in Goettingen and got his diploma-degree (M.Sc.) in physics. For several months in 1952 he was employed at the Max-Planck-Institute for Astrophysics in Goettingen, where the famous physicist C. F. von Weizsaecker had called him. Very soon it was found that it was impossible for him to work within a team, because of his handicap, and he left the MPI and after that continued to work alone and privately on a unified theory of matter and gravitation. In the year of Einstein’s death (1955) Heim informed Einstein on his work on a unified field theory. (Regrettably, only the mathematician W. Hlávaty could answer his letter.)

In close collaboration with the relativity theorist Pascual Jordan, Heim wanted to carry

out experiments on gravitation, but success eluded him, as the necessary budget was not available. Instead, L. Boelkow, director of the leading aero-space company in Germany, MBB/DASA, gave some financial support, since he was interested in the field propulsion system which Heim had proposed. (In a letter to Heim, Wernher von Braun enquired about progress in the development of such a field propulsion system since otherwise he could not accept responsibility for the enormous cost of the moon-landing project. Heim answered in the negative.)

The scientific community awaited publications by B. Heim. However, financially

Heim was absolutely independent. He was not pressed to publish papers or to give lectures at congresses in the physical field. Also, Heim declared to colleagues that he would publish only if he could present a confirmation for the correctness of his theory. Therefore Heim became more and more unknown to the new generation of physicists.

Already in the seventies Heim reached his goal, i.e., a confirmation of his structure

theory (a quantum-geometric 6-dimensional polymetric unified field theory, with which the internal structure of elementary particles could be understood purely geometrically) by comparison with experimental particle data. Now Heim wanted to publish, but he no longer had the necessary lobby. The director of the MPI for Elementary Particles in Munich, H. P. Duerr (who succeeded to the chair of W. Heisenberg) proposed to Heim to write an overview of his theory in the MPI publication organ “Z. f. Naturforschung,“ which Heim did (32a, 1977). Since the readers’ resonance to it was great, and many desired to read in greater detail about this theory, Heim began to publish his theory in two books (“Elementarstrukturen der Materie und Gravitation,“ Innsbruck: Resch; 1984, 1989), with a total of 694 pages.

The reception of the results of his investigations was extremely hesitant from the

beginning, since Heim was not as a member of an institute or a university or involved in a group of known scientists, and therefore he lacked advocates in the scientific community. In the beginning famous German physicists accused Heim of pursuing a “space flight fantasy,“ which was despised by theoretical physicists at that time.

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Further reasons can be found for the lack of reception: 1. Scientists are not inclined to study about 700 pages of very difficult content by an author who is not yet known. Instead, such readers await judgement on the books by a respected authority. Therefore, unfortunately the head of DESY was silent, when Heim’s mass formula had been programmed and calculated there, in 1982. Although the results were assessed as outstanding (by the DESY co-workers Schmid and Ribgen), it was thought desirable to wait for an assessment by structure theorists. 2. After his manuscript had experienced a more than one year delay with a renowned German publishing company, Heim eventually published his books in a publishing house that was not specialised in mathematics and physics. 3. Heim’s books contain some vagueness - beside the correct results - what is not astonishing for such a difficult matter, which was worked on by only one author, without the help of academically trained colleagues in a team and without checks from outside. Thus it becomes more difficult to understand for the reader. 4. The text did not appear simultaneously in German and English, so that international physicists, who perhaps could invest more time and effort, were excluded as possible readers. When, however, the importance of the work will be measured by the results, it follows

that the principles and the theory structure on which the theory based are far-reaching and therefore should be kept in mind in future works! This theory should be noticed by the scientific community, since it yields testable results, corresponding to empirical data in all regions, which no other physical theory can supply.

Heim’s theory, which yields in a totally geometrical way the spectrum of masses and

the mean lifetimes of the known and not yet discovered elementary particles, as well as masses of neutrinos, claims that the world requires a 6-dimensional continuum (otherwise particles could not be described), which has very far-reaching philosophical consequences.

Working Team Heim’s Theory IGW Innsbruck, Juni 2003

Introduction to Heim's Mass Formula IGW Innsbruck, 2003

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Heim’s Mass Formula (1982)

Original Text by Burkhard Heim for the Programming of his Mass Formula

Reproduction by Research Group

Heim's Theory IGW Innsbruck,2002


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On the Description of Elementary Particles (Selected Results)

by Burkhard Heim Northeim, Schillerstraße 2,

2-25-1982

A) Invariants of Possible Basic Patterns (Multiplets) Symbols: k Configuration number, k = 0 : no ponderable particle (no rest mass). For ponderable particles only k = 1 and k = 2 possible, not k > 2. k is a metrical

index number. ε so-called “time-helicity“. Refering to the R4 ε = +1 or ε = -1 decides whether it concerns an R4 - structure or the mirror-symmetrical anti-structure (ε = -1). G the number of quasi-corpuscular internal sub-constituents of structural kind. bi symbol for these 1 ≤ i ≤ G internal sub-constituents of an elementary particle. B baryonnumber P double isospin P = 2s . P1,2 locations in P-interval, where multiplets appear multiplied (doubled). I number of components x of an isospin-multiplet, i.e. 1 ≤ x ≤ I . Q double space-spin Q = 2J . Q value of Q at P1,2 . κ(λ) “doublet-number“, which distinguishes between several doublets by κ(λ) = 0 or κ(λ) = 1 . Λ Upper limit of κ-interval 1 ≤ λ ≤ Λ . C structure-distributor, identical with sign of charge of the strangeness quantum number. qx electrical charge quantum number with sign of the component x of the isospin-multiplet. q amount of charge quantum number q = qx . Uniforme Description of Quantum Numbers by k und ε G = k + 1 B = k - 1 P1 = 2 - k P2 = 2k - 1 I = P + 1 , 0 ≤ P ≤ G (I) Q(P) = k - 1 Q(P) = 2k - 1 κ(λ) = (1 - δ1λ ) δ1P , 1 ≤ λ ≤ Λ = 4 - k C = 2(PεP + QεQ)(k - 1 + κ)/(1 + κ) εP,Q = ε cos αP,Q


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αP = πQ(κ + ( )2

P )

αQ = πQ[Q(k - 1)+ ( )2P ] (II)

2qx = (P - 2x)[1 - κQ(2 - k)] + ε[k - 1 - (1 + κ)Q(2 - k)] + C , 0 ≤ x ≤ P , q = qx Possible configurations k = 1, k = 2 with ε = ± 1

Possible Multiplets of Basic States Multiplet xν of serial number ν for ε = +1 and anti-multiplet xν with ε = -1. General Representation: xν (εB,εP,εQ,εκ)εC(q0,...,qP)

Mesons: k = 1, G = 2 (quark?), B = 0, 0 ≤ P ≤ 2, i.e from singlet I = 1 to triplet I = 3. Q = 0, Q = 1, Λ(k=1) = 3, κ(1) = 0, κ(2) = κ(3) = 1 Baryons: k = 2, G = 3 (quark?), B = 1, 0 ≤ P ≤ 3 from singulett I = 1 to quartet I = 4, Q = 1, P1 = 0, P2 = 3, Q = 3, Λ(k=2) = 2, κ(1) = 0, κ(2) = 1

____________________ Possible multipletts for ε = +1: k = 1: x1 (0000)0(0) ≡ (η) x2 (0110)0(0,-1) ≡ (e0,e-), (is the existence of e0 possible ? ) x3 (0111)0(-1,-1) ≡ x3 (0111)0(-1) ≡ (µ-) pseudo-singlet (III) x4 (0101)+1(+1,0) ≡ (K+, K0) x5 (0200)0(+1,0,-1) ≡ x5 (0200)0(±1,0) ≡ (π±, π0) anti-triplet to itself k = 2: x6 (1010)-1(0) ≡ (Λ) x7 (1030)-3(-1) ≡ ( Ω-) x8 (1110)0(+1,0) ≡ (p,n) x9 (1111)-2(0,-1) ≡ (Ξ0,Ξ-) (IV) x10 (1210)-1(+1,0,-1) ≡ (Σ+,Σ0,Σ-) x11 (1310)-2(+1,0,-1,-2) ≡ (o+,o0,o-,o--), (existence possible ?) x12 (1330)0(+2,+1,0,-1) ≡ (∆++, ∆+, ∆0, ∆-), (thinkable as a basic state ?)

______________________ Abbreviations: η = π/(π4 + 4)1/4 ηkq = π/[π4 + (4+k)q4]1/4 ϑ = 5 η + 2 √η + 1 (V) A1 = √η11 (1 - √η11)/ (1 + √η11) A2 = √η12

(1 - √η12)/ (1 + √η12)


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Planck’s constant: h = h/2π, light-velocity: c = (ε0µ0)-1/2 , wave-resistance of empty space R3 (electro-magnetic): R - = cµ0 , with ε0 and µ0 constants of influence and induction. Elektrical elementary charge: e± = 3C± with

C± = ± 2ϑh / R− /(4 π)2 (possibly electr. quark-charge ?) Finestructure-constant: α√(1- α2) = 9ϑ (1 - A1A2) / (2π)5 , α > 0 . Solution: α(+) (positive branch) and α(-) (negative branch). Numerical: α(+) - 1 = 137,03596147 α(-) - 1 = 1,00001363 [A better formula, 1992, yields α(+) = 1/137,0360085 and α(-)_ = 1/1,000026627] What is the meaning of that strong coupling α(-) ? Abbreviation: α(+) = α , α(-) = ß ≈ 137 α .

B) Mass-Spectrum of Basic Patterns and its Resonances

Used constants of nature and pure numbers: Planck’s constant: h = h/2π = 1,0545887 x 10-34 J s, light-velocity: c = 2,99792458 x 108 m s-1, Newton’s constant of gravitation: γ = 6,6732 x 1011 N m2 kg-2 constant of influence ε0 = 8,8542 x 10-12 A sV-1 m-1, constant of induction µ0 = 1,2566 x 10-6 A-1 s V m-1, vacuum wave-resistance R- = (µ0/ε0)1/2 = 376,73037659 V A-1 derived constants of nature (mass-element): µ π πγ γ= −4

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013 3h hs c s/ , s0 = 1 [m] (gauge factor) (VI)

Basis of natural logarithms: e = 2,71828183 number π = 3,1415926535 geometrical constant: ξ = 1,61803399 [Limes of the “creation-selector“] limn→∞ an : an-1 = ξ by the series an = an-1 + an-2 . (till the 8th decimal place, represented by ξ = (1 + √5)/2). Auxiliary functions: η = π/(π4 + 4)1/4 (VII) t = 1 - 2/3 ξ η2 (1 - √η) α+ = t (η2 η1/3 )-1 - 1 (VIII) α- = t (ηη1/3 )-1 - 1


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Quantum numbers by (A): ηqk = π/[π4 + (4+k)q4]1/4 N1 = α1 N2 = (2/3) α2 , N3 = 2 α3 , with α1 = ½ (1 + √ηqk ) , α2 = 1/ ηqk , (IX) α3 = e(k-1) /k - q α/3 [( 1 + √ηqk ) (ξ/ηqk

2)](2k +1) ηqk3 +

+ [η(1,1)/ e ηqk] (2 √ ξηqk)k [(1 - √ηqk) /(1 + √ηqk)]2 Invariants of metrical steps-structure (abbreviation s = k² + 1): Q1 = 3 ⋅ 2 s - 2 , Q2 = 2s - 1 , (X) Q3 = 2s + 2(-1)k , Q4 = 2s - 1 - 1 . Fourfold R3-construct 1≤ j ≤ 4 . Qj = const. with respect to time t. Parameter of occupation nj = nj(t) caused radioactive decay. Mass elements of occupations of the configurations zones j are µα+ . Further auxiliary functions of zones occupations: K = n1

2 (1+n1)²N1 + n2 (2n2²+3n2+1)N2 + n3 (1+n3)N3 + 4n4 , G = Q1²(1+Q1)²N1 + Q2(2Q2²+3Q2+1)N2 + Q3(1+Q3)N3 + 4Q4 , (XI) H = 2n1Q1[1+3(n1+Q1+n1Q1) + 2(n1²+Q1²)]N1 + 6n2Q2(1+n2+Q2)N2 + 2n3Q3N3 Φ = 3 P/(π√ηqk) (1 - α-/α+)(P+Q)(-1)P+Q[1-α/3+π/2 (k-1) 31-q/2 ] ∗1+2 k κ/(3 η2) ξ[1 + ξ²(P-Q)(π 2-q)] [1 +( 4 ξ( )2

P /k)(ξ /6)q] - 1 ∗[ 2 √η11√ηqk + qη2 (k - 1)] (1+4πα/η√η)(1+Q(1-κ)(2-k)n1/Q1] + 4 (1 - α-/α+)α(P+Q)/ξ2 + 4 qα-/α+

Uniform Mass spectrum: M = µα+ (K + G + H + Φ) (XII) Not each quadruple nj yields a real mass! To the selection rule: in the fourfold R3-construct 1≤j≤4 configurations zones n(j=1), m(j=2), p(j=3), σ(j=4). Increase of occupation with metrical structure elements: central zone n cubic, internal zone m quadratic, meso-zone p linear (continuation to the empty space R3), external zone σ selective.


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Principle of increase of the configurations zones: n4+Q4 ≤ (n3+Q3)α3 ≤ (n2+Q2)² α2 ≤ (n1+Q1)3α3 (XIII)

Selection rule for the Occupation of Configuration Zones (n1+Q1)3α1 + (n2+Q2)² α2 + (n3+Q3)α3 + exp[1-2k(n4+Q4)/3Q4] + iF(Γ) = (XIV) = Wνx1 + [1-Q(2-k)(1-κ)][aνxN/(N+2) + bνx N N( )− 2 ]. Wνx = g(qk) wνx ,

Basis rise: g(qk) = Q13α1 + Q2² α2 + Q3α3 + exp[(1-2k)/3] for nj = 0. (XV)

Structure power of the discussed state wνx = (kPQκ)εC(qx) as component x of multiplets ν is: wνx = (1-Q)[A11-P(A12+A13qκ/ηqk) - ( )2

P (A14-A15q/ηqk)] + κQηqkA162 - k +

+ (q-1)A21 + (1-P)A22 + ( )2P [A23-qxηqk(1+A24(+qx))- 1A25] + (XVI)

+ κ(A26+qηqk²A31) + ( )3Q ηqkA32 + ( )3

P [A33q3(qx - (-1)q)/(3-q) +

+ ε η( ) ( ) /

( )

P QA

q q

q q

−−

+

−

1 4

6618

(1 - q(2-q)A341 - q

xA35/ηqk) ηqk/η² - A36]k - 1 .

w(1) = (1-Q)[A11 - P(A12+ A13qκ/ηqk ) - ( )2

P (A14 - A15q/ηqk)] + κQ ηqkA16 (XVII) and w(2) = (q-1)A21 + (1-P)A22 + ( )2

P [A23 - A25qxηqk(1 + A24(1+qx)) - 1] +

+ κ(A26 + qηqk²A31) + ( )3Q ηqkA32 + ( )3

P A33q3[qx - (-1)q)/(3-q)] + (XVIII)

+ ε η( ) ( ) /

( )

P QA

q q

q q

−−

+

−

1 4

6618

[1 - q(2-q)A341 - q

xA35/ηqk] ηqk/η² - A36

in wνx = [w(1)]2 - k

+ [w(2)]k - 1 (XIX) can become w(2) = 0 for single sets of quantum numbers at k = 1 or w(1) = 0 at k = 2 , which leads to terms 00 , which but must have always have the value 1 as parts of structure power. Therefore it is recommended for programming to complete w(1) and w(2) by the numerical non-relevant summands k-1 and 2-k . Since always w(1) ≠ -1 and w(2) ≠ -1 remain, but only k=1 or k=2 is possible, the actually terms in the expression wνx(k) = [k-1+w(1)]2 - k

+ [2-k+w(2)]k - 1 do no more appear. By this correction it is evident that for mesonical structures wνx (k=1) = 1 + w(1) and for barionical structures wνx (k=2) = 1 + w(2) holds.


7

As a basis of resonance holds aνx = A41 (1 + anaq)/k (XX) with an = PA42 [1 - κA43 (1 + A44 (-α)2 - k A45

k - 1)* *(1 - κQA46(2-k)) - A51(k-1)(1-κ)] (XXI) and aq = 1 -qA52(1 - 2A53

k)[1 + qx(3-qx)(k-1)(1-κ)/6] (XXII) Resonance grid is bνx = A54A55

k - 1 [1 - PA56(1-κA61A621 - k)(1 + qA63(1 + κA64))] ∗ (XXIII)

∗ (1-k- 1 (A65(q+k-1))2 - k ( )2P (1 - ( )3

P )/[kP(1+P+Q+κη2 - q)] . The coefficients Ars can be seen as elements of the quadratic coefficient matrix $A = (Ars)6 with Ars ≠ Asr and ImArs = 0 . Proposal for the determination of matrix elements (reduction to π, e and ξ): A11 = (ξ² π e)² (1 - 4 π α² ) / 2 η² , A12 = 2 π ξ² (ϑ/24 - e π η α² / 9) A13 = 3 (4 + η α)[1 - (η²/5)((1 - √η)² /(1 + √η)² ] A14 = [1 + 3 η (2 η α - e²ξ(1 - √η)²/(1 + √η)²)/4ξ]/ α

A15 = e²(1 - 2eα²/η)/3 A16 = (πe)²[1 + α(1+6α/π)/5η] A21 = 2(eα/2η)²(1 - α/2ξ²) A22 = ξ[1 - ξ(αξ/η²)²]/12 A23 = (η² + 6ξα²)/e A24 = 2ξ²/3η A25 = ξ(πe)²(1 - ß2) A26 = 21 - [π(eξα)²√η]/2/eξ2 A31 = (πeα)²[1 - (πe)²(1 - ß²)] A32 = ξ²[1 + (2eα/η)²]/6 A33 = (πeξα)²[1 - 2π(eξ)²(1 - ß²)] A34 = η 2πη (XXIV) A35 = 3α/eξ² A36 = [1 - πe(ξe)²(1 - ß²)] - 1 A41 = ξ[2 + (ξα)²] - 2ß/(2ß - α) A42 = [πξ²η(ß - 3α)]/2 A43 = ξ/2 A44 = 2(η/ξ)² A45 = (3ß - α)/6ξ A46 = πe/ξη - eη²α/2


8

A51 = (2α + 1)² A52 = 6α/η² A53 = (ξ/η)3 A54 = α(ß -α)√(3/2) A55 = ξ²

A56 = (ξ/η)4 A61 = πξ(2ß - α)/12ß A62 = π²(ß - 2α)/12 A63 = (√η)/9 A64 = π/3η A65 = π/3ξ A66 = ξη

The order of resonance N ≥ 0 (positive integer) selects the admitted quadruple nj with 1 ≤ j ≤ 4 . With f(N) = [1 - Q(2 - k)(1 - κ)][aνx N/(N+2) + bνx N N( )− 2 ] (XXV) follows that the unknown function F(Γ) remains 0 for all N ≠ 1 (right side is real). In the case of N = 0 is f = 0 , so that (n1 + Q1)3α1 + (n2 + Q2)² α2 + (n3 + Q3) α3 + exp[(1-2k)(n4+Q4)/3Q4] = Wνx (XXVI) describes the nj of the state xνx and hence the mass M0(νx) of the component x of the multiplet xν . The N ≥ 2 assign xνx to a spectrum of occupation-parameter quadruples and with that, according to the mass-formula, resonance-masses MN(νx) (for each component xνx a spectrum of masses). In the case of N = 1 no spectral term. Here is not f(N ) ≥ 0, f(1) is complex. Real part: (n1+Q1)3α1 + (n2+Q2)² α2 + (n3+Q3) α3 + exp[(1-2k)(n4+Q4)/3Q4] = = Wνx1+[1-Q(2-k)(1-κ)]aνx/3 (XXVII) Imaginary part F(Γ) = Wνx[1-Q(2-k)(1-κ)]bνx . (XXVIII) The nj and F(Γ) are somehow related with N to the complete bandwidths Γ . Also there must be a connection QN = Q(N) between doubled spin quantum-number Q and N . How could this connection be like? If N = 1 is excluded, then F = 0 , and the real relationship (n1 + Q1)3α1 + (n2 + Q2)² α2 + (n3 + Q3) α3 + exp[(1-2k)(n4+Q4)/3Q4] = Wνx (1+f) (XXIX) has to be discussed. Generally f > 0 for N ≥ 2 and f = 0 for N = 0. But in the case of the multiplets x2 f = 0 for all N ≥ 0, since only here is Q(2-k)(1-κ) = 1 . Electrons according to this image can not be stimulated !


9

For a numerical evaluation of Wνx , aνx , bνx and Φνx (quantum number function in mass spectrum M) not QN = Q(N) , but use Q = Q(0) of xν . For the evaluation of nj the principle of increase of the occupations of configuration zones is considered. First determine the right side Wνx (1+f(N)) = W1 numerically for an order of resonance N = 0 or N ≥ 2 . Determine according to the selection rule the maximal cubic number K1

3 whose product with α1 is contained in W1 . Then insert W1 - α1K1

3 = W2 ≥ 0 into (n2 + Q2)² α2 + (n3 + Q3) α3 + exp[(1-2k)(n4+Q4)/3Q4] = W2 . (XXX) Now maximal quadratic number K2² such, that α2K2

2 is still a factor of W2 , i.e. W2 - α2K2

2 = W3 ≥ 0 . Accordingly in (n3 + Q3) α3 + exp[(1-2k)(n4+Q4)/3Q4] = W3 (XXXI) Determine maximal number K3 in the way W3 - α3K3 = W4 ≥ 0 . Three possibilities for W4 : (a): W4 = 0 , (b): 0 < W4 ≤ 1 , (c): W4 > 1 . General case (b): lnW4 ≤ 0 and K4(2k-1) = -3Q4lnW4 . In case of (c) it is lnW4 > 0 and K < 0 . This is impossible, since always nj+Qj ≥ 0 has to be. According to n4+Q4 ≤ (n3+Q3)α3 of the principle of rise K3 will be lowered by 1 and α3K3 is added to K4 < 0 , so that a new value K4 ≥ 0 will be generated., which requires K3 > 0, since in that case K3 = 0. This dilatation can not happen because of the quadratic rise of j = 2 , so that this order of resonance N does not exist for xνx (forbidden term). In the case (a) W4 → 0 would have as a consequence the divergence K4 → ∞ , but this is impossible according to K4 ≤ α3K3 (particularly there are no diverging self-potentials). For that reason will be calculated in case of (a) the maximal value K4 = α3K3 . From the computed Kj it follows nj = Kj - Qj . Beside nj ≥ 0 also nj < 0 is possible , but it holds always Kj ≥ 0 , i.e. nj ≥ -Qj . The quadruple nj determined in that way will be inserted with Φνx in the spectrum of masses, which numerically yields MN(νx) as a spectral-term of mass-spectrum at xνx . Note: The Kj are always integers. But in the case of the evaluation of K4 generally decimal figures will occur. In case of the decimal places ,99... 99 one has to use the identity ,99... 99 = 1 . But if the series of decimal places is different from this value, then one has not to round up. The decimal places are to cut off , since the Kj are the numbers of structure entities.


10

Limits of Resonance Spectra

General construction-principle of configuration-zones n4+Q4 ≤ (n3+Q3)α3 , α3 (n3+Q3)(1+n3+Q3) ≤ 2α2(n2+Q2)² , (XXXII) α2 (n2+Q2)[2(n2+Q2)² + 3(n2+Q2) + 1] ≤ 6α1 (n1+Q1)³ . If by the increase of N between two zones equality is reached, then nj+Qj → 0 in j, while j-1 will be raised by 1 to nj-1 + Qj-1 + 1 . The stimulation takes place “from outside to the interior“. Always nj+Qj ≥ 0 is an integer, since they are the numbers of structure entities. Empty-space-condition: nj = -Qj , but (nj)max = Lj < ∞ (no diverging self-energy potentials). Intervals -Qj ≤ nj ≤ Lj < ∞ cause 0 ≤ N ≤ L < ∞ of resonance-order. With M0(νx) = M0 holds 4µα+α1 (L1+Q1)³ = [2(P+1)]2 - kM0G (XXXIII) with G = k+1 and from that by the construction-principle α2 (L2+Q2)[2(L2+Q2)² + 3(L2+Q2) + 1] ≤ 6α1 (L1+Q1)³ , α3 (L3+Q3)(1+L3+Q3) ≤ 2α2(L2+Q2)² , (XXXIV) L4+Q4 ≤ (L3+Q3)α3 . For L implicitly the resonance-order is (L1 + Q1)3α1 + (L2 + Q2)² α2 + (L3 + Q3) α3 + exp[(1-2k)(L4+Q4)/3Q4] = =Wνx [1+f(L)] (XXXV) Also in the evaluation of Lj and L do not round up, but cut off decimal digits! The Lj which are obtained by the construction-principle, yield the absolute maximal masses Mmax , and the quadruples, which are obtained from the L, yield the real limit-terms ML < Mmax , which are to stimulate secondaryly with (Mmax - ML)c² and then reach Mmax . Northeim, Schillerstraße 2 gez. (Heim) 2-25-1982 Distributed to: Deutsches Elektronen-Synchrotron (DESY) Hamburg, Eidgenössische Technische Hochschule (ETH) Zürich, Max-Planck-Institut für Theoretische Physik, München, Messerschmitt-Bölkow-Blohm GmbH (MBB), Ottobrunn bei München: Dr. G. Emde, Dr. W. Kroy, Dipl.-Phys. I. v. Ludwiger. Staatsanwalt G. Sefkow, Berlin, und H. Trosiner, Hamburg.


10

Heim’s Mass Formula (1989)

According to a Report by Burkhard Heim

Prepared by the Research Group Heim's Theory

IGW Innsbruck,2002

Content

• Introduction

• Mass of Basic States and of the Excited States of Elementary Particles

• The Average Life Times of the Basic States

• The Sommerfeld Finestructure Constant

• The Masses of Neutrino States

• Concluding Remarks


11

Introduction After DESY physicists in 1982 had programmed and calculated the mass formula which was published in the book Elementarstrukturen der Materie (Heim 1984), the mentioned formula by B. Heim was extended and in 1989 a 57 pages report with a new formula and the results of the calculations were sent to the company MBB/DASA. Unfortunately this later code could no more be recovered today. Parts of these formulae have now been programmed again by the research group „Heim Theory“ (by Dr. A. Mueller). It was found that in the manuscript some brackets in very long equations were lost during the process of writing; this had to be corrected at best estimate. The code covers the masses of basic states only and no lifetimes. Other than the program written in 1982, Heim’s 1989 computation also includes the life times of the basic states, the neutrino masses, and the finestructure constant. Therefore, these equations shall be given here, as far as they deviate from those given in the manuscript in 1982. The structure distributor C (i.e. strangeness) given in eq. (I) of chapt. E has to be divided by k. One of the angles by which the time helicity ε is defined must read αQ = π Q [Q + ( )2

P ] (B1) The expression for the quantum number of charge other than in (II) now reads: qx = ½ [ (P - 2x + 2) [1 - κQ(2 - k)] + ε[k - 1 - (1 + κ)Q(2 - k)] + C ] (B2) All other constants are defined by eq.(I). 1. Mass of Basic States and of the Excited States of Elementary Particles The modified mass formula of elementary particles is built up - other that in eq.(XII) - by the following parts: M = µα+ [(G + S + F + Φ) + 4 q α - ] (B3) The parts G and S are the same as G and K in eq.(XII) (now using n, m, p instead of n1, n2, n3); µ is the mass element as in eq.(VI). The constants α± have the form:

α+ = η

ηϑ

η

η ηη

62

12 1

12

²( )

( )−

−

+

- 1 , α - = (α+ + 1)η - 1 (B4)

The calculated results for α+ and α - in (B4) are shown in a table VI/chapterG. The abbreviations for F and Φ, which depend on the quantum numbers, read: F = 2 n Qn [1 + 3(n + Qn + n Qn) + 2(n² + Qn²)] + (B5) + 6 m Qm (1 + m + Qm)N2 + 2 p Qp N3 + ϕ (p,σ)∗δ(N)


12

Φ = P(-1)P+Q (P + Q) N5 + Q(P + 1) N6 (B6) ϕ = ϕ (p,σ) , δ(0) = 1 (0 for N ≠ 0) (B7) with

ϕ = N p

p4

1²²+

σ

σσ+

+− =

−QBUWN1

2 440

1

²( ) +P(P - 2)²(1 + κ(1 - q)/2αϑ )(π/e)²√η12(Qm-Qn) -

- (P + 1) ( )3Q /α , (comp. with eq. B49)

U = 2Z [P² + 3/2 (P - Q) + P(1 -q) + 4κB (1 -Q)/(3 - 2q) +

+ (k - 1)P + 2Q - 4π(P - Q)(1 - q)/ 24 ] ηqk - ² (comp. with eq. B50)

and Z = k + P + Q + κ (comp. with eq. B51) ϕ is a term of self-couplings, which depends on p and σ and essentially determines the life time of a basic state. ϕ appears only in the basic states; therefore the symbol δ(N) as a unit element is used. The functions Qi from eq. (X) remain unchanged. For n1, n2, n3, n4 in eq. (B5) here n, m, p, σ will be written. The constants ηq,k, ϑ and η (with η10 = η, and ϑ 1,0 =ϑ ), as well as the functions N1 and N2 read as in eq.(IX). The remaining Ni with i > 2 are:

ln (N3 k/2) = (k -1) [1 - π1

11 1 1

1

1

1

−

+− − −− +

η

η

η

ϑα α ηq k

q

q

q

u,

,

,

,

( / )( )² ] -

- 2/(3π e) (1 - η )² (6 π²e²/ϑ 1

11+

−

η

ηq ,

- 1) (B8)

N4 = (4/k) [1 + q(k - 1)] (B9)

N5 = A[1 + k(k - 1) 2k²+3 N(k) A 1

1

2−

+

η

ηq k

q k

,

,

] (B10)

A = (8/η)(1 - α-/α+)(1 - 3η/4) (B11) N(k) = Qn + Qm + Qp + Qσ + k(-1)k 2k²-1 (B12)

N6 = 2k/(π eϑ ) [ k (k² - 1) N k

k

( )

,η1

q - (1 - q) N k

Qn k

' ( )

,η1

+

+ (-1)k+1 ] η(1 - α α− +/ ) 41

1

2−

+

η

ηQσ (B13)

N’(k) = Qn + Qm + Qp + Qσ - 2k -1 (B14) The calculated results for B8, B9, B10 and B13 can be found in a table VII/chapter G.


13

Let L be the upper barrier such that as soon as it is reached the filling of the zone x disappears and the foregoing zone filling of next higher order is raised by 1. With the symbols L(x) (x - 1) for this barrier and M0 = M(N=0) the limits of the fillings of structure zones corresponding to eq.(XXXIII) are given by:

- Qn ≤ n ≤ L(n) = ( )P M

NQn

+−

+

12

03

1µα (B15)

since in the case of the central region there are no further fillings. For the series of numbers m the limitation holds: - Qm ≤ m ≤ L(m)(n) (B16) with 2(Qm + L(m)(n))³ + 3 (Qm + L(m)(n))² + Qm + Lm(n) = 4 N1(n + Qn)³/N2 (B17) Correspondingly, we have - Qp ≤ p ≤ L(p)(m) (B18)

with 2 L(p)(m) = 24 12

3

NN

m Qm( )²+ + - 2Qp - 1 (B19)

and - Qσ ≤ σ ≤ L(σ)(p) (B20) with 2 L(σ)(p) = N3(p + Qp) - 2 Qσ (B21) The calculated results for B15 can be found in a table IX/chapter G. The selection rule which expresses the n, m, p, σ by the quantum numbers k, P, Q, κ, q and N, is described by eq.(XXIX). In that f(N) is the excitation function for N > 0. For the factor Wνx ≡ WN=0 , which is independent of the exciting state, holds: WN=0 = A ex (1 - η)L + (P - Q)(1 - ( )2

P )(1 - ( )3Q )(1 - η )² √2 (B22)

with A = 8 g H[2 - k + 8H (k - 1)] - 1 (B23) H = Qn + Qm + Qp + Qσ (B24) g = Qn² + Qm² + (Qp²/k) ek-1 + exp[(1- 2k)/3] - H(k - 1) (B25) L = (1 - κ) Q (2 - k) (B26) x = [1 - Q - ( )2

P ](2 - k) + 1/4B [a1 + k³/(4H)(a2 + a3/(4B))] (B27) B = 3 H [k² (2k - 1)] - 1 (B28) The calculated results for B23, B24, and B28 can be found in a table VI/chapter G.


14

For the three parameters a1, a2, and a3 the following combinatorical relations hold: a1 = 1 + B+ k(Q² + 1) ( )3

Q - κ[(B - 1)(2 - k) - 3H - 2(1 + q)(P - Q) + 1] -

- (1 - κ) [(3(2 - q) ( )2P - Q3(P + Q) + q)(2 - k) + [k(P + 1) ( )2

P +

+ 1 + B/k (k + P - Q)(1 - ( )2P )(1 - ( )3

Q ) - q(1 - q) ( )3Q ] (k - 1)] (B30)

a2 = B [1 - ( )3

Q (1 - ( )3P )] + 6/k - κ[Q/2 (B - 7k) - (3q -1)(k - 1) +

+ ½ (P - Q)4 + (B + 1)(1 - q)] - (1 - κ) [(P(B/2 + 2 + q) - - QB/2 + 1 - 4(1 + 4q)) (2 - k) + ( ¼ (B - 2)1 + 3/2(P - Q) - (B29) - B/2 (1 - q) - ( )2

P [ ½ (B + q - εqx) + 3 εqx(2 - εqx) -

- ¼ (B + 2)(1 -q)]) (1 - ( )3Q )(k - 1) - ( )3

P [2 (1 + εqx) + + ½ (2 - q)3(1 - q) + εqx - q - q/4 (1 - q)(B - 4) - ¼ (B - 2) + + B/2 (1 - q)]] a3 = 4 B y’/(y’+1) - (B + 4) - 1 (B31) with y’ 2 B = κ[ η /k 4 (2 - √η) - π e (1 - η) η k + e η (k - 1) +

+ 5 1

2 1( )

( )−

+ −q

k k (4B + P + Q)] + (1 - κ)[(P - 1)(P - 2)2/k² (H + 2) +

+ (2-k)/(2π) + ( )2P (1 - ( )3

Q )(q B/2 B + 2(P - Q) + P (P + 2)B + + (P + 1)² - q(1 + εqx) [k(P² + 1)(B + 2) + ¼ (P² + P + 1)] - - q (1 - εqx)(B + P² + 1) (k - 1) + (P - Q)(H + 2) + + P[5 B (1 + q) Q + k (k - 1) k(P + Q)²(H + 3k + 1)(1 - q) - - ½ (B + 6k)](1 - ( )2

P )(1 - ( )3Q ) + ( )3

P (2 - q) Q εqx(B + 2Q + 1) +

+ q/(2k)(1 - εqx)(2k + 1) + (1 - q)(Q² + 1 + 2B)] The calculated results for B29, B30, B31 and B22 can be found in a table VIII/chapter G. For the excitation function f from eq.(XXXV) Heim got the expression f (N) = a N/(N+1) + b N (B32) with the substitutions (α is the finestructure constant):

a = P

kX q k q k

²

, ,η η2 (1 - k/4) + (k -1)π/4 ( )3P - η1,1η1,2 ( )2

P (B33)

X = κ[4α( )

( )( )( )

[ /( )( )

]B k

qe+ +

−+−

−

− − −

+−

11

1 51 5

234

2 11

2 1 62

2

2

22

2ααα

απ

π απ ηϑ α

+ 1 (B34)


15

b = 1

2 2η ηqk qk

[αϑ /8 (P² + 1)[ ½ (1 + η )(1 + η1,1 η1,2 (3/4) ( )3P (k - 1)] +

+ (k - 1) ϑ 1,2/ϑ - 8 ( )2P (P² + 1 ) - 1] - C (B35)

C = π (1 - η )² [1 + π (k - 1) + P/k³ (3/e + q(8 + ηqk) +

+ (4 πe/ η )(1 - κ)[1 - q 3

5πηηe qk

] - 2(k - 1) ( )2P (3 - P)2 e (η + ηqk) (B36)

+ εqx πe/(3 η ) + 8 1π κ

η ηe k e q

e( )−

−

)] + (2 e κ q/ η² )(2 - k)(1 - η)²

The excitations can lead to a change of angular momentum. Since Q is the double quantum number of angular momentum, Q(N = 0) could change additive by an even number 2z with the integer function z(N), such that: Q (N) = Q (N = 0) + 2 z (N), (B37) where z(N) is yet unknown. One has to hold in mind, that the σ-fillings of the external region of a term M(N) can get an additional excitation because of their external character. If the zones nN, mN, pN, and σN are occupied and if L(σ)(p) = ½ N3 (p + Qp) - 2 Qσ with - Qσ ≤ σ ≤ L(σ)(p) , (B38) is the complete occupation of the external region related to pN , then KB = L(σ)(p) - σN (B39) describes a real number, which as a bandwidth determines the number of the possible excitations of the external field of an excitation state M(N). For KB ≤ 0 there is no possibility of an external field excitation. If L(N) describes the maximal occupation of all the four structure zones 0 ≤ N ≤ L(N) < ∞, then the equation of the excitation limit is given by eq.(XXXV) and eq.(B32) with N = L(N). If the quantum numbers k, P, Q, κ, and qx , as well as the excitation N, are given for a basic state, then the right-hand side of eq. (XXXV), i.e. (n + Qn)³α1

+ (m + Q m)² α2 + (p + Qp) α3 + exp[-(2k - 1) /3Qσ(σ + Qσ)] =

= WN=0(1 + f(N)) (B40) with α1 = N1 , α2 = 3/2 N2 , α3 = ½ N3, and eq.(B22) to eq.(B36) can be calculated numerically. By an exhaustion process based on


16

w = WN=0(1 + f) (B41) n, m, p, and σ can be determined using eq.(B15) to eq.(B21) and (B40). Let be K ≥ 1 the series of natural numbers. Then, first of all, w - K³α1 ≥ 0 will be formed. K will be raised as long as K = Kn changes its sign. Then Kn will reduced by 1, which results in: w - (Kn - 1)³ α1 = w1 (B42) The process will be repeated with w1 in the form w1 - K² α2 ≥ 0 . With K = Km w1 - (Km - 1)² α2 = w2 (B43) will be generated. In the same way w2 - K α3 ≥ 0 yields the relation w2 - (Kp - 1) α3 = w3 (B44) and with the abbreviation ß = (2k-1)/3Qσ w3 - e -ßK ≤ 0 (B45) is determined, which changes its sign for K = Kσ . Next, Kσ will be reduced by 1.With the limits now known, Kn to Kσ , the n, m, p, σ can be calculated: n = Kn - 1 - Qn m = Km - 1 - Qm (B46) p = Kp - 1 - Qp σ = Kσ - 1 - Qσ With these quantum numbers the mass formula (B3) with its parts eq.(B4) to eq.(B14) can be calculated. 2. The Average Life Times of the Basic States Let be T the average life time of the masses of elementary particles determined by eq. (B3). If TN = T(N) << T is a function depending on N, so that T0 = 0 for N = 0, then according to Heim the unified relation for the times of existence is: (T - TN) =

= 192

1 1 12 2 1 1 1 2 0

hHy

Mc H n m p n m p ß²[ ( )²( )²( )²]( )( ), , , ( )η η η η σ− − − + + + + + +δ

(B47) where δ = δ(N) is as in eq.(B7) .M is taken from eq.(B3), and H from eq.(B24). The substitution y is given by:


17

y = F [ϕ + (-1)s (1 + ϕ)(b1 + b2/WN=0)] (B48) with

ϕ = N p

p4

1²²+

σ

σσ+

+− =

−QBUWN1

2 440

1

²( ) + P (P - 2)²(1 + κ(1 - q)/(2αϑ ))

(π/e)²√η12(Qm-Qn) -- (P + 1) ( )3Q /α , (B49)

U = 2Z [P² + 3/2 (P - Q) + P(1 -q) + 4κB (1 -Q)/(3 - 2q) + (k - 1)P + 2Q - - 4π(P - Q)(1 - q)/ 24 ] /ηqk² (B50) and Z = k + P + Q + κ (B51) The calculated results for B48 and B49 can be found in a table IX. B will be calculated from eq.(B28). It is(B52) F= 1 - 1/3 (1 - q)(P - 1)²(3 - P)(1 + P - Q - ε C P/2)(1 + ß(0)(-1)k) - ( )3

P (1 + D), (B52) s= 2 - k + ε C + (2kQ - κP) + ( )3

Q : 1/k (P-1)(P-2)(P-3) (B53) b1= [P 7 + 6(1 - q)(C - ( )2

P ) - 2q (1 - ( )2P ) + κ Q(3 Z - 1) B + 1] (2 - k) +

+ ½ (1 - κ)(q - εqx - 2) Q + ε C P + 2 (P + 1) - (B54)

- (1 - q) P P

P P( )

( ² )−

+ −3

1 1 (4 B - 6 + P)(k - 1) - ( )3

P (q - ε qx)

b2= B(5B+3) + 2 3

1HP

−+

+ CkB( 3B+2(H+1)) + H + ½ (1 - q) - Q B(2(B+H) - 1) +

+ H/2 + 3 + κ q B (3B + 1) - 5/2(k-Q) - ( )2P P²(P + Q)²[8B+1 -

-5B - (2H+1)(1 + 2 ( )3P - Q) + 2 q] - ( )2

P H(1-q) - (B -3/4)²(P-1)(P-2)(P-3)(-1)k-1 + + (Q-q)(1-q + Bq)3(H+B) + πe/η - q/4(P+1)³(k-1) + κ(-1)1-q [7HB+3(H+B)-5/2 + + (1-q)H(3B-4) + B+7/2](k-1) + Q ( )2

P (2 -q)(1 + ε qx)[B/2(H+2) + ¾ ] +5/2HB +

+ 3H - BP

++

51

- 5/2 H² ( )3P q (1+π/3(2-q) η2,2) B - (2-q)(1 - q) (B55)

with ß(0) = 2απe

1

1

2−

+

η

η (B56)

and D = [1 + 4 q²(q - 1)(2q + 1)] - 1 η ß(0) (1-√η)4 P2+εq (P - 1)(q-1)q/2/(3√2) (B57) With the systems eq.(B3) to eq. (B14) and with the quantum numbers (Table I) the particular masses M can be calculated, and from eq.(B47) to eq.(B57) the life times T of all the multiplet components for N = 0 can be determined numerically and compared with empirical values (Table II). The life times T are shown in multiples of 10 - 8 seconds.


18

3. The Sommerfeld Finestructure Constant: In ϕ and ß(0) the finestructure constant α is contained. The value in chapter D, section 8 is calculated only approximately. Heim now also gives the exact formula for α: According to eq.(8.21) we get:

α 192

15− = −αϑπ

²( )

( ' )C (B58)

with 1 - C’ = 1 - 1 1

12 2

1 1 1 2

2+ −

+

ηηη η

η

η,

, ,

= Kα (B59)

With the abbreviation

D’ = ( )29

5πϑ αK

(B60)

it follows for the reciprocal square of these solutions: α(±)

-² = ½ D’²(1 ± 1 4− / '²)D (B61) With eq. (V/chapter E) the values for both branches are calculated: α+ = 0.72973525 × 10 - 2 and α - = 0.99998589 (B62) 1/α(+) = 137,03601 1/α( ) = 1,0000142 which, compared with the empirical value (Nistler & Weirauch 2002) for the finestructure constant, 1/α(+) = 137,0360114 ± 3.4 .10 - 8 yields a value which falls into the tolerance region of measurement. The negative branch shows an extremely strong interaction, which probably is based on the inner connections of the four zones in an elementary particle. But Heim did not investigate this further. 4. The Masses of Neutrino States Supposing that in the central region of an elementary particle an euclidian metric rules, i.e. that there is no structure element, than that means: L(n) = - Qn . According to eq.(B15) it means that there also is no ponderable mass M0 . According to eq.(B16) to eq.(B21) it follows, that also the remaining structure zones are governed by an euclidian metric. In eq.(B3) then we must substitute n = - Qn , m = - Qm , p = - Qp und σ = - Qσ , (B63) from which follows: G + F + S = ϕ (B64)


19

According to eq.(B49) generally ϕ ≠ 0 holds, in spite of σ + Qσ = 0, and also Φ ≠ 0 is not affected by the lower barrier of the n. m, p, σ . If Φ + ϕ ≠ 0, since P > 0 or Q > 0, then eq.(B49) yields a field mass unequal zero, in spite of eq.(B63). This field mass is not interpretable as a ponderable particle, but is - according to Heim - a kind of „spin-potence“ which as a „field catalyst“ permits transmutations of elementary particles or enforces the validity of certain conservation principles (angular momentum). This behaviour is equivalent to those properties which made the definition of neutrinos necessary by empirical reasons. If according to eq.(B3) one substitutes for the mass of neutrinos in whole generality Mν = µα+ (Φ + ϕ0) (B65) where ϕ0 relates eq.(B49) to the lower bounds of n, m, p, σ, than it follows, that Mν is determined only by the quantum numbers k, κ, P, and Q . For Mν(kPQκ) > 0 the following possibilities result: Mν (1110) = Mν (1111) and Mν (1200) in the mesonic region, and Mν (2110) and Mν (2111) in the baryonic region. In addition there is another neutrino, which only transfers the angular momentum Q = 1 and which is required by the ß-transfer. For this neutrino only the two possibilities exist: Mν (2010) or Mν (1010). Since in the case (2010) Mν < 0 would be, only Mν ( 1010) remains as a possibility for the ß-neutrino. With i = 1,...,5 the possible neutrino states νi are: for k = 1: ν1 (1010) , ν2 (1110), ν3(1200) for k = 2: ν4 (2110) , ν5 (2111). For each νi there exists the mirror-symmetrical anti-structure νi . From eq.(B3) with the possibly non-zero quantum numbers the neutrino-masses may be determined. The calculated results are collected in table II. The masses are given in electron volt. The empirical ß-neutrino can be interpreted by ν1 and the empirical µ-neutrino by ν2. For the time being it cannot be decided whether the rest of the neutrinos also are implemented in nature or whether it concerns merely logical possibilities. 5. Concluding Remarks For the numerical investigation of the states N > 0 the system (B32) must be used, which is uncertain because of the uncertain relations eq.(B33) to eq.(B36). The function z(N) in eq.(B37)


20

must still be determined. Since z is not given, also Q(N) for N > 0 remains unknown. The mass values of the spectra N > 0 which belong to the basic states therefore still have an approximate character. Also the life times TN of such states cannot be described yet. In eq.(B49) the free eligible parameters for the expression ϕ with eq.(B50) were fitted by empirical facts [i.e. 24 2, ( / )π e and 4 1 24π / ] . The error Q(N) = Q(0) = Q based on the approximation z = 0 for all of the N only causes an approximation error less than 0.1 MeV. In spite of the mentioned uncertainties the numerical calculation of the relations eq.(B22) to eq.(B36) and eq.(B3) yields a spectrum of excitations for each basic state, whose limits are given by eq.(XXXV) with eq.(B32), and whose finestructure is described by eq.(B39). In these spectra of excitation all empirical masses of short living resonances fit which were available to Heim at that time (CERN - Particle Properties - 1973). But there are much more theoretical excitation terms than were found empirically. That could be caused either by the existence of a yet unknown selection rule for N, or the selection rule is only pretended since the terms are not yet recordable by measurements. In the tables IV and V Heim listed only such states N > 0 which seem to be identical with empirical resonances. The N-description in the third column differs between N and N , where the underlining means that a term is addressed which does not fit the selection rule for N of the masses M(NB) - M(NA) > 0 with NB > NA . The values put in brackets in the 3rd and 4th column (with KB from eq.(B39) ) are related to possible electrically charged components. For the ∆ - states, q = 2 was used. In the 5th column, the theoretical masses in MeV are indicated. Here also the brackets are related to electrically charged components. The resonance states in general are represented very well, in spite of the approximate character (because ofz(N) = 0), but the uncertainty appears for k = 1 in the particles ω(783) and η’(958), as well as for k = 2 in the particle N(1688). While the functions z(N) and TN yet have been searched for by Heim, he already possessed an ansatz for a unified description of magnetic spin moments of particles with Q ≠ 0, which was not yet published. After discovering z and TN, Heim wanted to calculate the cross sections of interaction, which regrettably could not more be done. Apart from the above-mentioned incompleteness, it can be stated that on the basis of the far-reaching correspondence with the empirical data Heim’s structure theory meets all requirements to be fulfilled by a mathematical scheme for a unified theory, and there is no other unified structure theory which allows for more exact or much better confirmed descriptions of the geometro-dynamical processes within the microregion.


1

Selected Results

Research Group Heim's Theory, IGW Innsbruck, 2003 Pay attention also to Heim,B. 1979/89/98, 1984

IGW Innsbruck

Content

• Quantum Numbers of Basic States (N=0) 2

• Theoretical Data of Elementary Particles 3 with Mean Lives > 10-16 sec Calculated by B. Heim 1989

• Experimental Data of Elementary Particles 4

with Mean Lives > 10-16 sec

• Approximated Meson Resonances 5

• Approximated Baryon Resonances 6-8

• Numerical Evaluations of Different Equations 9-11

• Relative Deviations of the Theoretically Determined 12 Particle Masses from the Experimental Mean Values for Different Values of the Gravity Constant

• Relative Deviations of the Theoretically Determined 13

Particle Lifetimes from the Corresponding Experimental Mean Values


2

Table I

Quantum Numbers of Basic States (N=0)

Partikel k n m p σ P Q εqx εC ℜ +− ee , 1 0 0 0 0 1 1 -1 0 0 00 ,ee 1 0 0 0 1 1 1 0 0 0

+− µµ , 1 11 6 11 6 1 1 -1 0 1 ηη, 1 18 22 17 14 0 0 0 0 0

−+ KK , 1 17 26 30 28 1 0 1 1 1 00 , KK 1 18 5 5 2 1 0 0 1 1

±π mπ, 1 12 9 2 3 2 0 ±1 0 0 00 ,ππ 1 12 3 6 4 2 0 0 0 0

ΛΛ, 2 1 3 0 -11 0 1 0 -1 0 +− ΩΩ , 2 4 4 -1 -15 0 3 -1 -3 0

pp, 2 0 0 0 0 1 1 1 0 0 nn, 2 0 0 -2 17 1 1 0 0 0

+− ΞΞ , 2 2 7 -17 2 1 1 -1 -2 1 00 , ΞΞ 2 2 6 -1 6 1 1 0 -2 1 −+ ΣΣ , 2 2 -7 -12 10 2 1 1 -1 0 00 , ΣΣ 2 2 -7 -14 -2 2 1 0 -1 0 +− ΣΣ , 2 2 -6 -5 -8 2 1 -1 -1 0

−−++ οο , 2 2 1 9 4 3 3 2 0 0 −+ οο , 2 2 -1 -1 -6 3 3 1 0 0 00 ,οο 2 2 -1 -10 2 3 3 0 0 0 +− οο , 2 2 -1 -16 -15 3 3 -1 0 0


3

Tabelle II

Theoretical Data of Elementary Particles with Mean Lives > 10-16 sec Calculated by B. Heim 1989

(J = spin, P = parity, I = isospin, S = strangeness, B = baryon number)

Type Symbol Mass MeV

J P I S B Mean Life 10-8 sec

Photons γ 0 1 -1 - - 0 ∞ νe 0.00381 × 10-6 1/2 - - - 0 ∞ νµ 0.00537 1/2 - - - 0 ∞ ντ 0.010752 1/2 - - - 0 ∞ ν4 0.021059 1/2 - - - 0 ∞ ν5 0.207001 1/2 - - - 0 ∞ e 0.51100343 1/2 ±1 - - 0 ∞ e0 0.51617049 1/2 1 - - 0 ∞

Leptons

µ 105.65948493 1/2 ±1 - - 0 219.94237553 π± 139.56837088 0 -1 1 0 0 2.60282911 π0 134.96004114 0 -1 1 0 0 0.84016427 × 10-8 η 548.80002432 0 -1 0 0 0 0.00233820 × 10-8 K± 493.71425074 0 -1 1/2 ±1 0 1.23709835 K0 497.72299959 0 -1 1/2 1 0 5.17900027

Mesons

0Κ 497.72299959 0 -1 1/2 -1 0 0.00887666 p 938.27959246 1/2 1 1/2 0 1 ∞ n 939.57336128 1/2 1 1/2 0 1 917.33526856 × 10 8 Λ 1115.59979064 1/2 1 0 0 1 0.02578198 Σ+ 1189.37409717 1/2 1 1 1 1 0.00800714 Σ- 1197.30443002 1/2 1 1 1 1 0.01481729 Σ0 1192.47794854 1/2 1 1 1 1 0.42908026 × 10-10 Ξ- 1321.29326013 1/2 1 1/2 -2 1 0.01653050 Ξ0 1314.90206200 1/2 1 1/2 -2 1 0.02961947 Ω- 1672.17518902 3/2 1 0 -3 1 0.01317650

o++,o--- 1232.91663788 3/2 1 3/2 0 1 5.99071759 × 10-16 o+,o-- 1234.60981181 3/2 1 3/2 0 1 5.72954997 × 10-16 o-,ο + 1229.99529979 3/2 1 3/2 0 1 6.74230244 × 10-16

Baryons

o0,ο 0 1237.06132359 3/2 1 3/2 0 1 5.08526841 × 10-16


4

Tabelle III

Experimental Data of Elementary Particles with Mean Lives > 10-16 sec

(J = spin, P = parity, I = isospin, S = strangeness, B = baryon number)

Type Symbol Mass/MeV (PDG,CERN 2002)

J P I S B Mean Life 10-8 sec

Photons γ 0 1 -1 - - 0 ∞ νe ≤ 5 × 10-8 1/2 - - - 0 ∞ νµ < 0.17 1/2 - - - 0 ∞ ντ <18.2 1/2 - - - 0 ∞ e 0.51099907±0.00000015 1/2 ±1 - - 0 ∞

Leptons

µ 105.658389±0.000034 1/2 ±1 - - 0 219.703±0.004 π± 139.57018±0.000351 0 -1 1 0 0 2.6033±0.0005 π0 134.9766±0.0006 0 -1 1 0 0 (0.84±0.06)×10-8 η 547.30±0.12 0 -1 0 0 0 K± 493.677±0.016 0 -1 1/2 ±1 0 1.2384±0.0024 K0 497.672±0.031 0 -1 1/2 1 0 5.2±0.5(Rohlf1994)

Mesons

0Κ 497.672±0.031 0 -1 1/2 -1 0 0.0089±0.0002 ( " ) p 938.27231±0.00026 1/2 1 1/2 0 1 ∞ n 939.56563±0.00028 1/2 1 1/2 0 1 (886.7±1.9)×10 8 Λ 1115.683±0.006 1/2 1 0 0 1 0.02632±0.0002 Σ+ 1189.37±0.07 1/2 1 1 1 1 0.00799±0.00004 Σ- 1197.449±0.03 1/2 1 1 1 1 0.01479±0.00011 Σ0 1192.642±0.024 1/2 1 1 1 1 (7.4±0.7)×10-12 Ξ- 1321.32±0.13 1/2 1 1/2 -2 1 0.01639±0.00015 Ξ0 1314.9±0.6 1/2 1 1/2 -2 1 0.029±0.0009 Ω- 1672.45±0.29 3/2 1 0 -3 1 0.00822±0.00012

∆++ ≈1232 3/2 1 3/2 0 1 ∆+ ≈1232 3/2 1 3/2 0 1 ∆0 ≈1232 3/2 1 3/2 0 1

Baryons

∆- ≈1232 3/2 1 3/2 0 1

The data are taken from the Particle Data Group homepage http://pdg.lbl.gov , CERN, (2002), except for the life times of K0 and 0Κ , which are taken from J.W. Rohlf 1994: Modern Physics from α to Z0, New York: John Wiley & Sons.


5

Table IV

Approximated Meson Resonances (k=1)

Partikel P N(N)± KB(KB)± Theoretische Masse in MeV

ε 0 49 10 691,7094 ( )783ω 0 64 51 783,9033

)958´(η 0 144 28 956,8400 ( )993*S 0 170 -1 992,6142

( )1019Φ 0 153 63 1019,6306 ( )1270f 0 253 26 1274,5452 ( )1285D 0 255 27 1286,1728 ( )1420E 0 272 82 1414,1873 ( )1514'f 0 323 2 1517,8602 ( )1675ω 0 342 71 1664,0125 ( )892*

−K 1 23(11)- 29(3) 891,1955(892,2211) ( )1240AK 1 83(69) 6(15) 1241,1180(1239,9767) ( )1420*K 1 98(101) 25(23) 1420,2213(1414,4956)

( )1770L 1 161(164) 65(11) 1775,2145(1764,9862) ( )770p 2 8(5) 30(34) 769,9833(769,3101) ( )970δ 2 39(21) 19(5) 976,4931(973,6704) ( )11001A 2 76(48) 41(5) 1106,9780(1106,7462) ( )1235B 2 93(79) 27(10) 1239,5340(1239,1994) ( )13102A 2 127(86) 22(59) 1310,4695(1309,6730) ( )15401F 2 182(145) 37(4) 1539,5100(1537,9095) ( )1600'p 2 215(156) 43(29) 1604,8640(1605,1008) ( )16403A 2 221(160) 4(7) 1637,2669(1634,2138) ( )1680g 2 228(165) 28(5) 1686,0154(1678,6425)


6

Table Va

Approximated Baryon Resonances

(k=2)

Partikel P N(N)± KB(K)± theoretische Masse in MeV N(1470) 1 13(12) 10(38) 1470,4888(1480,1770) N(1520) 1 14(13) 29(8) 1509,6087(1515,7293) N(1535) 1 18(17) -2(8) 1533,9788(1535,3254) N(1670) 1 23(22) 8(0) 1657,9536(1679,5754) N(1688) 1 24(23) -23(11) 1694,3687(1719,4898) N(1700) 1 25(27) 63(-12) 1734,6717(1751,2494) N(1770) 1 26(24) 14(65) 1771,8218(1769,0721) N(1780) 1 31(29) -9(0) 1784,3644(1782,2884) N(1810) 1 32(30) 38(40) 1808,3795(1808,5253) N(1990) 1 37(35) 60(50) 1974,9129(1989,7028) N(2000) 1 42(39) -3(-37) 2011,0552(2001,9706) N(2040) 1 44(41) 7(30) 2044,8079(2034,6322) N(2100) 1 40(44) 78(25) 2107,8085(2120,5890) N(2190) 1 49(46) -14(21) 2200,5168(2195,5259) N(2220) 1 50(47) 66(43) 2244,1911(2245,4563) N(2650) 1 73(69) 2(-9) 2653,5304(2652,4071) N(3030) 1 90(85) 41(54) 3036,2404(3033,5279) N(3245) 1 95(90) 61(28) 3234,0166(3231,8730) N(3690) 1 119(113) 3(4) 3689,8085(3684,1957) N(3755) 1 113(115) 37(31) 3751,7230(3728,0808) Λ(1330) 0 25 10 1329,8831 Λ(1405) 0 22 79 1403,3999 Λ(1520) 0 37 36 1516,3419 Λ(1670) 0 54 4 1669,9762 Λ(1690) 0 55 61 1693,2832 Λ(1750) 0 58 25 1754,7613 Λ(1815) 0 70 -10 1815,4961


7

Table Vb

Approximated Baryon Resonances (k=2 continuation)

Partikel P N(N)± KB(K)± Theoretische Masse in MeV Λ(1830) 0 71 11 1830,4081 Λ(1860) 0 73 -5 1864,6313 Λ(1870) 0 74 1 1884,4529 Λ(2010) 0 87 17 2010,5372 Λ(2020) 0 88 18 2018,1998 Λ(2100) 0 94 0 2095,9533 Λ(2110) 0 84 34 2113,6593 Λ(2350) 0 116 30 2344,7465 Λ(2585) 0 136 5 2591,7184 Ξ(1530) 1 4(2) 9(5) 1531,5487(1534,7628) Ξ(1630) 1 7(4) 30(20) 1621,5840(1661,1690) Ξ(1820) 1 16(10) 35(9) 1828,9065(1810,8367) Ξ(1940) 1 19(13) 59(27) 1944,8454((1945,2579) Ξ(2030) 1 25(19) -4(-3) 2027,8157(2037,5528) Ξ(2250) 1 31(24) 65(-4) 2247,4841(2241,9080) Ξ(2500) 1 42(35) 42(13) 2481,8202(2517,9008) ∆(1650) 3 44 11 1651,0807 ∆(1670) 3 48 44 1678,6242 ∆(1690) 3 71 0 1690,0383 ∆(1890) 3 124 1 1887,9876 ∆(1900) 3 125 56 1900,8602 ∆(1910) 3 129 -27 1915,2764 ∆(1950) 3 134 59 1949,2695 ∆(1960) 3 137 38 1965,3571 ∆(2160) 3 211 33 2153,9221 ∆(2420) 3 302 12 2422,5186 ∆(2850) 3 419 63 2856,6694 ∆(3230) 3 572 34 3229,6911


8

Table Vc

Approximated Baryon Resonances (k=2 continuation)

Partikel P (N)+N(N)- (KB)+KB(KB)- Theoretische Masse in MeV Σ(1385) 2 (13)+6(13)- (11)+59(22)- (1383)+1382(1386)- Σ(1440) 2 (16)8(16) (9)71(-5) (1441)1434(1441) Σ(1480) 2 (18)20(18) (64)12(52) (1492)1490(1489) Σ(1620) 2 (32)35(32) (18)10(20) (1624)1622(1616) Σ(1670) 2 (34)27(35) (8)15(-23) (1664)1660 (1678) Σ(1690) 2 (35)38(36) (-10)43(57) (1691)1683((1705) Σ(1750) 2 (43)41(38) (-25)34(5) (1752)1747(1750) Σ(1765) 2 (45)49(46) (9)10(-2) (1769)1766(1770) Σ(1840) 2 (50)45(51) (19)11(47) (1847)1844(1848) Σ(1880) 2 (42)57(43) (65)61(7) (1884)1887(1885) Σ(1915) 2 (53)59(54) (28)16(24) (1909)1923(1908) Σ(1940) 2 (54)60(55) (23)44(-10) (1932)1951(1931) Σ(2000) 2 (63)70(64) (8)1(-45) (2003)2012(2002) Σ(2030) 2 (66)72(59) (21)12(5) (2035)2031(2031) Σ(2070) 2 (68)75(69) (2)38(40) (2066)2071(2064) Σ(2080) 2 (69)76(70) (9)29(10) (2083)2089(2074) Σ(2100) 2 (70)77(71) (31)52(6) (2103)2106(2093) Σ(2250) 2 (76)84(78) (-12)33(35) (2243)2250(2252) Σ(2455) 2 (94)104(85) (18)56(3) (2444)2458(2455) Σ(2620) 2 (110)121(103) (27)-12(26) (2624)2625(2621) Σ(3000) 2 (136)150(140) (-85)38(12) (2994)3001(3003)


9

Tables VI

Numerical Evaluations of the Equations V and VIII (chapter E)

symbol numerical value symbol numerical value η 0,98998964 ϑ 7,93991266

η1,1 0,98756399 ϑ1,1 7,92534503 η1,2 0,98516776 ϑ1,2 7,91095114 η2,2 0,84242385 ϑ2,2 7,04779227 α+ 0,01832211 α- 0,00812835

Numerical Evaluations of the Equations

X and B23,B24,B28 (chapter E+F)

k Qn Qm Qp Qσ B H A 1 3 3 2 1 27 9 2787,59025432 2 24 31 34 15 26 104 14727,57867072

Table VII

Numerical Evaluations of the Equations IX and B8,B9,B10,B13 (chapter E+F)

Ni(k,q ) numerical value Ni(k,q ) numerical value N1(1,1 ) 0,99688127 N4(1,1 ) 4 N1(1,0 ) 1 N4(1,0 ) 4 N1(2,1 ) 0,99627809 N4(2,1 ) 4 N1(2,0 ) 1 N4(2,0 ) 2 N1(2,2 ) 0,95891826 N4(2,2 ) 6 N2(1,1) 0,67506174 N5(1,1) 1,15773470 N2(1,0) 0,66666667 N5(1,0) 1,15773470 N2( 2,1) 0,67670370 N5( 2,1) 1,73247496 N2(2,0) 0,66666667 N5(2,0) 1,15773470 N2(2,2) 0,79136728 N5(2,2) 76,73214581 N3(1,1) 1,95731764 N6(1,1) 0.00000164 N3(1,0) 2 N6(1,0) 0,00000164 N3(2,1) 2,59881924 N6(2,1) 0,02518725 N3(2,0) 2,71828183 N6(2,0) -0.10493009 N3(2,2) 2,12190443 N6(2,2) 0,15580107


10

Table VIII

Numerical Evaluations of the Equations B22,B29,B30,B31 (Chapter F)

Particle a1 a2 a3 WN=0

e- 35 11 89,96774158 38,70294226 e0 34 28 77,11059862 38,51308957 µ 1 23 7,26891022 2830,2632345 π± 25 0 95,62488526 3514,46294316 Κ+ 16 31 7,26891022 8857,95769020 π0 22 2 -0,03225806 3419,16217346 Κ0 22 17 98,29474138 9332,35821820 η 28 33 48,65020426 9905,00599107 p 0 23 84,22944059 14792,56308050 Σ+ 21 30 26,15371691 18124,03136129 Σ− 21 47 94,49556347 18183,30294347 Ξ− 26 25 15,61504747 18998, 73451193 Ω− 47 3 69,73881899 23157,61451004 o++ 23 27 82,92386515 18115,38391620 o+ 23 22 22,64335811 18467,56082305 o- 21 27 69,73881899 18448,51703290 n 0 36 101,15000035 14828,61089116 Λ 13 45 -0,033333333 16827,97671482 Σ0 21 46 83,86257747 18179,59733741 Ξ0 26 22 71,62409771 18990,08927597 o0 23 39 93,76289283 18508,94119539


11

Table IX Numerical Evaluations of the Equations

XXXV and B48,B49 (Chapter E+F)

Particle Y ϕ L(N) +− ee , -408,54063248 0 1021 00 ,ee -53,97104336 0 1373

+− µµ , 1086,93016693 2,57120915 2340 ηη, 0,26273140⋅10-8 5,06612007 3236

−+ KK , 184,84508008 -40,78574065 3258 0K 147,94249859 -12,73395842 3166 0K 0,25356917 -12,73395842 3166

±π mπ, 17,08389288 -2,32863274 1485 00 ,ππ 3,70004027⋅10-8 -5,12094079 1833

ΛΛ, 0.06178705 0 1964 +− ΩΩ , 0,09369559 -137,03604095 2062

pp, 17,31698079 9,28034058 1841 nn, 1228,02191382 11,16885467 1932

+− ΞΞ , 0,10666692 23,44132266 2247 00 , ΞΞ 0,20184712 90,44612205 2382 −+ ΣΣ , 0,04603481 -6,00947753 5785 00 , ΣΣ 211,63404729⋅10-8 11,78154008 6375 +− ΣΣ , 0.06836890 -2,01125294 5991

−−++ οο , 14,72282381⋅10-16 -1364,07751672 35510 −+ οο , 11,51525605⋅10-16 -623,74523006 5115 00 ,οο 10,13617609⋅10-16 -985,00227539 5551 +− οο , 10,19390807⋅10-16 -548,14408156 5102


12

Relative Deviations of the Theoretically Determined Particle Masses from the Experimental Meanvalues forDifferent Values of the Gravity Constant G

v


13

Relative Deviations of the Theoretically Determined Particle Lifetimesfrom the Corresponding Experimental Mean Values Heim 1989, ? : an experimental value could not be found at the PDG data set, Heim 1998, measuring uncertainty

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Content

A Abstract 1-4 B Remarks on the Physicist Burkhard Heim* 1-2 C About the State of the Elementary Particle- and Geometrised Physics* 1. The Fields of Interaction and the Masses of Elementary Particles 1-6 in the Standard-Model of Particle Physics 2. Theories with Geometrically Structured Particles 7-9 3. Problems in Elementary Particle Physics and the Requirement for a 10 Structure Theory of Particles D On the Derivation of Heim's Mass Formula* 1. Gravitation in the Microscopic Domain 1-6 2. Solution of the 6-dimensional Field Equations for the Microscopic Domain 2.1 The Three Structure Units of the World 7-9 2.2 Solutions of the Field Equations for the Four Hermetry-Forms 10-12 2.3 Theoretical Fixing of the Elementary Charge and of the Fine-structure-Constant 13-16 3. The Polymetric Geometry 3.1 The Polymetric Field Equations 17-21 3.2 Correlations of the Partial Structures and their Extrema 22-24 3.3 Groups of Couplings and Condensor Fluxes 25-27 4. Microscopic Structure Dynamic: the Reason of Inertia 4.1 Condensor Fluxes 28-31 4.2 The Inertia of all Hermetry Forms 32-33 5. Prototypical Basic Fluxes and Prototrope Conjunctors 34-38 6. The Geometrical Reasons of Spin, Isospin, Helicity, and Anti-Structures 39-43 7. Determination of the Sum of Partial Masses in an Elementary Structure 44-51 8. Fine-Structure Constant and the Electromagnetic Field 52-60 9. Basic States of the Elementary Particles and "Quarks" 61-66 10. Limits of Excitation of Resonance and Masses of the Neutrino States 67-72 11. Experimental Confirmations of Heim's Structure Theory 73-74 E Heim's Mass Formula (1982) 1-9 F Heim's Mass Formula (1989) 10-18 G Selected Results: 1-13 Theoretical Values of the Masses of Elementary Particles (Basic States and Resonances), Mean Lifetimes of Basic States, Masses of Neutrinos, Sommerfeld Fine-structure Constant, Influence of the Value of the Gravitational Constant on the Masses of Basic States H References * Not jet available in English

burkhard heim mass formula

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