E Heims Mass Formula 1982
13/05/2014 22:55
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
1
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
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
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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.
e so-called “time-helicity“. Refering to the R4 e = +1 or e = -1 decides whether it
concerns an R4 - structure or the mirror-symmetrical anti-structure (e = -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 .
k(l) “doublet-number“, which distinguishes between several doublets by
k(l) = 0 or k(l) = 1 .
L Upper limit of k-interval 1 £ l £ L .
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 isospinmultiplet.
q amount of charge quantum number q = ½qx ½.
Uniforme Description of Quantum Numbers by k und e
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
k(l) = (1 - d1l ) d1P , 1 £ l £ L = 4 - k
C = 2(PeP + QeQ)(k - 1 + k)/(1 + k)
eP,Q = e cos aP,Q
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aP = pQ(k + ( ) 2
P )
aQ = pQ[Q(k - 1)+ ( ) 2
P ] } (II)
2qx = (P - 2x)[1 - kQ(2 - k)] + e[k - 1 - (1 + k)Q(2 - k)] + C , 0 £ x £ P , q = ½qx½
Possible configurations k = 1, k = 2 with e = ± 1
Possible Multiplets of Basic States
Multiplet xn of serial number n for e = +1 and anti-multiplet xn with e = -1.
General Representation: xn (eB,eP,eQ,ek)eC(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, L(k=1) = 3, k(1) = 0, k(2) = k(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, L(k=2) = 2, k(1) = 0, k(2) = 1
____________________
Possible multipletts for e = +1:
k = 1: x1 (0000)0(0) º (h)
x2 (0110)0(0,-1) º (e0,e-), (is the existence of e0 possible ? )
x3 (0111)0(-1,-1) º x3 (0111)0(-1) º (m-) pseudo-singlet }(III)
x4 (0101)+1(+1,0) º (K+, K0)
x5 (0200)0(+1,0,-1) º x5 (0200)0(±1,0) º (p±, p0) anti-triplet to itself
k = 2: x6 (1010)-1(0) º (L)
x7 (1030)-3(-1) º ( W-)
x8 (1110)0(+1,0) º (p,n)
x9 (1111)-2(0,-1) º (X0,X-) }(IV)
x10 (1210)-1(+1,0,-1) º (S+,S0,S-)
x11 (1310)-2(+1,0,-1,-2) º (o+,o0,o-,o--), (existence possible ?)
x12 (1330)0(+2,+1,0,-1) º (D++, D+, D0, D-), (thinkable as a basic state ?)
______________________
Abbreviations:
h = p/(p4 + 4)1/4
hkq = p/[p4 + (4+k)q4]1/4
J = 5 h + 2 Öh + 1 }(V)
A1 = Öh11 (1 - Öh11)/ (1 + Öh11)
A2 = Öh12 (1 - Öh12)/ (1 + Öh12)
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Planck’s constant: h = h/2p, light-velocity: c = (e0m0)-1/2 , wave-resistance of empty space
R3 (electro-magnetic): R - = cm0 , with e0 and m0 constants of influence and induction.
Elektrical elementary charge: e± = 3C± with
C± = ± 2Jh / R- /(4 p)2 (possibly electr. quark-charge ?)
Finestructure-constant: aÖ(1- a2) = 9J (1 - A1A2) / (2p)5 , a > 0 .
Solution: a(+) (positive branch) and a(-) (negative branch).
Numerical: a(+)
- 1 = 137,03596147
a(-)
- 1 = 1,00001363
[A better formula, 1992, yields a(+) = 1/137,0360085 and a(-)_ = 1/1,000026627]
What is the meaning of that strong coupling a(-) ?
Abbreviation: a(+) = a , a(-) = ß » 137 a .
B) Mass-Spectrum of Basic Patterns and its Resonances
Used constants of nature and pure numbers:
Planck’s constant: h = h/2p = 1,0545887 x 10-34 J s,
light-velocity: c = 2,99792458 x 108 m s-1,
Newton’s constant of gravitation: g = 6,6732 x 1011 N m2 kg-2
constant of influence e0 = 8,8542 x 10-12 A sV-1 m-1,
constant of induction m0 = 1,2566 x 10-6 A-1 s V m-1,
vacuum wave-resistance R- = (m0/e0)1/2 = 376,73037659 V A-1
derived constants of nature (mass-element):
m = 4 p pg g -
0
3
0
3 hs h / 3c s 1 , s0 = 1 [m] (gauge factor) (VI)
Basis of natural logarithms: e = 2,71828183
number p = 3,1415926535
geometrical constant: x = 1,61803399
[Limes of the “creation-selector“] limn®¥ an : an-1 = x by the series an = an-1 + an-2 .
(till the 8th decimal place, represented by x = (1 + Ö5)/2).
Auxiliary functions:
h = p/(p4 + 4)1/4 (VII)
t = 1 - 2/3 x h2 (1 - Öh)
a+ = t (h2 h1/3 )-1 - 1} (VIII)
a- = t (hh1/3 )-1 - 1
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Quantum numbers by (A):
hqk = p/[p4 + (4+k)q4]1/4
N1 = a1
N2 = (2/3) a2 ,
N3 = 2 a3 ,
with
a1 = ½ (1 + Öhqk ) ,
a2 = 1/ hqk , }(IX)
a3 = e(k-1) /k - q {a/3 [( 1 + Öhqk ) (x/hqk
2)](2k +1) hqk
3 +
+ [h(1,1)/ e hqk] (2 Ö xhqk)k [(1 - Öhqk) /(1 + Öhqk)]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 ma+ .
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
F = 3 P/(pÖhqk) (1 - a-/a+)(P+Q)(-1)P+Q[1-a/3+p/2 (k-1) 31-q/2 ]
*{1+2 k k/(3 h2) x[1 + x²(P-Q)(p 2-q)]} [1 +( 4 x( ) 2
P /k)(x /6)q] - 1
*[ 2 Öh11Öhqk + qh2 (k - 1)] (1+4pa/hÖh)(1+Q(1-k)(2-k)n1/Q1]
+ 4 (1 - a-/a+)a(P+Q)/x2 + 4 qa-/a+
Uniform Mass spectrum:
M = ma+ (K + G + H + F) (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), s(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 s selective.
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Principle of increase of the configurations zones:
n4+Q4 £ (n3+Q3)a3 £ (n2+Q2)² a2 £ (n1+Q1)3a3 (XIII)
Selection rule for the Occupation of Configuration Zones
(n1+Q1)3a1 + (n2+Q2)² a2 + (n3+Q3)a3 + exp[1-2k(n4+Q4)/3Q4] + iF(G) = (XIV)
= Wnx{1 + [1-Q(2-k)(1-k)][anxN/(N+2) + bnx N(N - 2) ]}.
Wnx = g(qk) wnx ,
Basis rise: g(qk) = Q1
3a1 + Q2² a2 + Q3a3 + exp[(1-2k)/3] for nj = 0. (XV)
Structure power of the discussed state wnx = (kPQk)eC(qx) as component x of multiplets n
is:
wnx = {(1-Q)[A11-P(A12+A13qk/hqk) - ( ) 2
P (A14-A15q/hqk)] + kQhqkA16}2 - k +
+ {(q-1)A21 + (1-P)A22 + ( ) 2
P [A23-qxhqk(1+A24(+qx))- 1A25] + }(XVI)
+ k(A26+qhqk²A31) + ( ) 3
Q hqkA32 + ( ) 3
P [A33q3(qx - (-1)q)/(3-q) +
+
e( )h( ) /
( )
P Q
A
q q
q q
-
-
+
-
1 4
66
8 1
(1 - q(2-q)A34
1 - q
xA35/hqk) hqk/h² - A36]}k - 1 .
w(1) = (1-Q)[A11 - P(A12+ A13qk/hqk ) - ( ) 2
P (A14 - A15q/hqk)] + kQhqkA16 (XVII)
and
w(2) = (q-1)A21 + (1-P)A22 + ( ) 2
P [A23 - A25qxhqk(1 + A24(1+qx)) - 1] +
+ k(A26 + qhqk²A31) + ( ) 3
Q hqkA32 + ( ) 3
P {A33q3[qx - (-1)q)/(3-q)] + (XVIII)
+
e( )h( ) /
( )
P Q
A
q q
q q
-
-
+
-
1 4
66
8 1
[1 - q(2-q)A34
1 - q
xA35/hqk] hqk/h² - A36}
in wnx = [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 -1 and w(2) ¹ -1
remain, but only k=1 or k=2 is possible, the actually terms in the expression
wnx(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 wnx (k=1) =
1 + w(1) and for barionical structures wnx (k=2) = 1 + w(2) holds.
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
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As a basis of resonance holds anx = A41 (1 + anaq)/k (XX)
with an = PA42 [1 - kA43 (1 + A44 (-a)2 - k A45
k - 1)*
*(1 - kQA46(2-k)) - A51(k-1)(1-k)] (XXI)
and aq = 1 -qA52(1 - 2A53
k)[1 + qx(3-qx)(k-1)(1-k)/6] (XXII)
Resonance grid is
bnx = {A54A55
k - 1 [1 - PA56(1-kA61A62
1 - k)(1 + qA63(1 + kA64))] * (XXIII)
* (1-k- 1 (A65(q+k-1))2 - k ( ) 2
P (1 - ( ) 3
P )}/[kP(1+P+Q+kh2 - q)] .
The coefficients Ars can be seen as elements of the quadratic coefficient matrix $A = (Ars)6
with Ars ¹ Asr and ImArs = 0 .
Proposal for the determination of matrix elements (reduction to p, e and x):
A11 = (x² p e)² (1 - 4 p a² ) / 2 h² ,
A12 = 2 p x² (J/24 - e p h a² / 9)
A13 = 3 (4 + h a)[1 - (h²/5)((1 - Öh)² /(1 + Öh)² ]
A14 = [1 + 3 h (2 h a - e²x(1 - Öh)²/(1 + Öh)²)/4x]/ a
A15 = e²(1 - 2ea²/h)/3
A16 = (pe)²[1 + a(1+6a/p)/5h]
A21 = 2(ea/2h)²(1 - a/2x²)
A22 = x[1 - x(ax/h²)²]/12
A23 = (h² + 6xa²)/e
A24 = 2x²/3h
A25 = x(pe)²(1 - ß2)
A26 = 2{1 - [p(exa)²Öh]/2}/ex2
A31 = (pea)²[1 - (pe)²(1 - ß²)]
A32 = x²[1 + (2ea/h)²]/6
A33 = (pexa)²[1 - 2p(ex)²(1 - ß²)]
A34 = h 2ph
(XXIV)
A35 = 3a/ex²
A36 = [1 - pe(xe)²(1 - ß²)] - 1
A41 = {x[2 + (xa)²] - 2ß}/(2ß - a)
A42 = [px²h(ß - 3a)]/2
A43 = x/2
A44 = 2(h/x)²
A45 = (3ß - a)/6x
A46 = pe/xh - eh²a/2
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A51 = (2a + 1)²
A52 = 6a/h²
A53 = (x/h)3
A54 = a(ß -a)Ö(3/2)
A55 = x²
A56 = (x/h)4
A61 = px(2ß - a)/12ß
A62 = p²(ß - 2a)/12
A63 = (Öh)/9
A64 = p/3h
A65 = p/3x
A66 = xh
The order of resonance N ³ 0 (positive integer) selects the admitted quadruple nj with
1 £ j £ 4 . With
f(N) = [1 - Q(2 - k)(1 - k)][anx N/(N+2) + bnx N(N - 2) ] (XXV)
follows that the unknown function F(G) remains 0 for all N ¹ 1 (right side is real).
In the case of N = 0 is f = 0 , so that
(n1 + Q1)3a1 + (n2 + Q2)² a2 + (n3 + Q3) a3 + exp[(1-2k)(n4+Q4)/3Q4] = Wnx (XXVI)
describes the nj of the state xnx and hence the mass M0(nx) of the component x of the
multiplet xn . The N ³ 2 assign xnx to a spectrum of occupation-parameter quadruples and
with that, according to the mass-formula, resonance-masses MN(nx) (for each component xnx
a spectrum of masses). In the case of N = 1 no spectral term. Here is not f(N ) ³ 0, f(1) is
complex.
Real part: (n1+Q1)3a1 + (n2+Q2)² a2 + (n3+Q3) a3 + exp[(1-2k)(n4+Q4)/3Q4] =
= Wnx{1+[1-Q(2-k)(1-k)]anx/3} (XXVII)
Imaginary part F(G) = Wnx[1-Q(2-k)(1-k)]bnx . (XXVIII)
The nj and F(G) are somehow related with N to the complete bandwidths G . 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)3a1 + (n2 + Q2)² a2 + (n3 + Q3) a3 + exp[(1-2k)(n4+Q4)/3Q4] = Wnx (1+f) (XXIX)
has to be discussed. Generally f > 0 for N ³ 2 and f = 0 for N = 0. But in the case of the
multiplets x2 f = 0 for all N ³ 0, since only here is Q(2-k)(1-k) = 1 . Electrons according
to this image can not be stimulated !
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For a numerical evaluation of Wnx , anx , bnx and Fnx (quantum number function in mass
spectrum M) not QN = Q(N) , but use Q = Q(0) of xn . For the evaluation of nj the
principle of increase of the occupations of configuration zones is considered. First determine
the right side Wnx (1+f(N)) = W1 numerically for an order of resonance N = 0 or N ³ 2 .
Determine according to the selection rule the maximal cubic number K1
3 whose product
with a1 is contained in W1 . Then insert W1 - a1K1
3 = W2 ³ 0 into
(n2 + Q2)² a2 + (n3 + Q3) a3 + exp[(1-2k)(n4+Q4)/3Q4] = W2 . (XXX)
Now maximal quadratic number K2² such, that a2K2
2 is still a factor of W2 , i.e.
W2 - a2K2
2 = W3 ³ 0 . Accordingly in
(n3 + Q3) a3 + exp[(1-2k)(n4+Q4)/3Q4] = W3 (XXXI)
Determine maximal number K3 in the way W3 - a3K3 = W4 ³ 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 ³ 0 has to
be.
According to n4+Q4 £ (n3+Q3)a3 of the principle of rise K3 will be lowered by 1 and
a3K3 is added to K4 < 0 , so that a new value K4 ³ 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 xnx (forbidden term).
In the case (a) W4 ® 0 would have as a consequence the divergence K4 ® ¥ , but this is
impossible according to K4 £ a3K3 (particularly there are no diverging self-potentials). For
that reason will be calculated in case of (a) the maximal value K4 = a3K3 . From the
computed Kj it follows nj = Kj - Qj . Beside nj ³ 0 also nj < 0 is possible , but it holds
always Kj ³ 0 , i.e. nj ³ -Qj . The quadruple nj determined in that way will be inserted with
Fnx in the spectrum of masses, which numerically yields MN(nx) as a spectral-term of
mass-spectrum at xnx .
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.
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Limits of Resonance Spectra
General construction-principle of configuration-zones
n4+Q4 £ (n3+Q3)a3 ,
a3 (n3+Q3)(1+n3+Q3) £ 2a2(n2+Q2)² , (XXXII)
a2 (n2+Q2)[2(n2+Q2)² + 3(n2+Q2) + 1] £ 6a1 (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 ³ 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(nx) = M0
holds
4ma+a1 (L1+Q1)³ = [2(P+1)]2 - kM0G (XXXIII)
with G = k+1 and from that by the construction-principle
a2 (L2+Q2)[2(L2+Q2)² + 3(L2+Q2) + 1] £ 6a1 (L1+Q1)³ ,
a3 (L3+Q3)(1+L3+Q3) £ 2a2(L2+Q2)² , (XXXIV)
L4+Q4 £ (L3+Q3)a3 .
For L implicitly the resonance-order is
(L1 + Q1)3a1 + (L2 + Q2)² a2 + (L3 + Q3) a3 + exp[(1-2k)(L4+Q4)/3Q4] =
=Wnx [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.
