F Heims Mass Formula 1989
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Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
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
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
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 e is defined must read
aQ = p 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 - kQ(2 - k)] + e[k - 1 - (1 + k)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 = ma+ [(G + S + F + F) + 4 q a- ] (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);
m is the mass element as in eq.(VI). The constants a± have the form:
a+ =
h
h
J
h
h h
h
6
2
1
2 1
1
2
²
( )
( )
-
-
+
é
ë êê
ù
û úú
æ
è
çç
ö
ø
÷÷
- 1 , a- = (a+ + 1)h - 1 (B4)
The calculated results for a+ and a- in (B4) are shown in a table VI/chapterG.
The abbreviations for F and F, 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 + j (p,s)*d(N)
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
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F = P(-1)P+Q (P + Q) N5 + Q(P + 1) N6 (B6)
j = j (p,s) , d(0) = 1 (0 for N ¹ 0) (B7)
with
j =
N p
p
4
1
²
+ ²
s
s
s +
+
- =
- Q
BUWN 1
4 2 4
0
1
²
( ) +P(P - 2)²(1 + k(1 - q)/2aJ )(p/e)²Öh12(Qm-Qn) -
- (P + 1) ( ) 3
Q /a , (comp. with eq. B49)
U = 2Z [P² + 3/2 (P - Q) + P(1 -q) + 4kB (1 -Q)/(3 - 2q) +
+ (k - 1){P + 2Q - 4p(P - Q)(1 - q)/ 4 2 }] hqk
- ² (comp. with eq. B50)
and Z = k + P + Q + k (comp. with eq. B51)
j is a term of self-couplings, which depends on p and s and essentially determines the life time
of a basic state. j appears only in the basic states; therefore the symbol d(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, s will be written. The constants hq,k, J and h (with h10 = h, and J 1,0 =J ), 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 - p
1
1
1 1 1
1
1
1
-
+
- - - - +
h
h
h
J
a a h q k
q
q
q
u ,
,
,
,
{ ( / )( )²}] -
- 2/(3p e) (1 - h )² (6 p²e²/J
1
1
1 +
-
h
h
q, - 1) (B8)
N4 = (4/k) [1 + q(k - 1)] (B9)
N5 = A[1 + k(k - 1) 2k²+3 N(k) A
1
1
2 -
+
æ
è
çç
ö
ø
÷÷
h
h
q k
q k
,
,
] (B10)
A = (8/h)(1 - a-/a+)(1 - 3h/4) (B11)
N(k) = Qn + Qm + Qp + Qs + k(-1)k 2k²-1 (B12)
N6 = 2k/(p eJ ) [ k (k² - 1)
N k
k
( )
, h1
{q - (1 - q)
N k
Qn k
'( )
, h1
} +
+ (-1)k+1 ] h(1 - a a - + / ) 4
1
1
2 -
+
æ
è
çç
ö
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÷÷
h
h
Qs (B13)
N’(k) = Qn + Qm + Qp + Qs - 2k -1 (B14)
The calculated results for B8, B9, B10 and B13 can be found in a table VII/chapter G.
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
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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
N
Qn
+
-
+
1
2
0
3
1 ma
(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 2 1
3
N
N
m Qm ( + )² + - 2Qp - 1 (B19)
and - Qs £ s £ L(s)(p) (B20)
with 2 L(s)(p) = N3(p + Qp) - 2 Qs (B21)
The calculated results for B15 can be found in a table IX/chapter G.
The selection rule which expresses the n, m, p, s by the quantum numbers
k, P, Q, k, q and N, is described by eq.(XXIX).
In that f(N) is the excitation function for N > 0. For the factor Wnx º WN=0 , which is
independent of the exciting state, holds:
WN=0 = A ex (1 - h)L + (P - Q)(1 - ( ) 2
P )(1 - ( ) 3
Q )(1 - h )² Ö2 (B22)
with
A = 8 g H[2 - k + 8H (k - 1)] - 1 (B23)
H = Qn + Qm + Qp + Qs (B24)
g = Qn² + Qm² + (Qp²/k) ek-1 + exp[(1- 2k)/3] - H(k - 1) (B25)
L = (1 - k) 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.
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For the three parameters a1, a2, and a3 the following combinatorical relations hold:
a1 = 1 + B+ k(Q² + 1) ( ) 3
Q - k[(B - 1)(2 - k) - 3{H - 2(1 + q)}(P - Q) + 1] -
- (1 - k) [(3(2 - q) ( ) 2
P - Q{3(P + Q) + q})(2 - k) + [k(P + 1) ( ) 2
P +
+ {1 + B/k (k + P - Q)}(1 - ( ) 2
P )(1 - ( ) 3
Q ) - q(1 - q) ( ) 3
Q ] (k - 1)] (B30)
a2 = B [1 - ( ) 3
Q (1 - ( ) 3
P )] + 6/k - k[Q/2 (B - 7k) - (3q -1)(k - 1) +
+ ½ (P - Q){4 + (B + 1)(1 - q)}] - (1 - k) [(P(B/2 + 2 + q) -
- Q{B/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 - eqx) + 3 eqx}(2 - eqx) -
- ¼ (B + 2)(1 -q)]) (1 - ( ) 3
Q )(k - 1) - ( ) 3
P [2 (1 + eqx) +
+ ½ (2 - q){3(1 - q) + eqx - 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 = k[ h /k {4 (2 - Öh) - p e (1 - h) h }{k + e h (k - 1)} +
+
5 1
2 1
( )
( )
-
+ -
q
k k (4B + P + Q)] + (1 - k)[(P - 1)(P - 2){2/k² (H + 2) +
+ (2-k)/(2p)} + ( ) 2
P (1 - ( ) 3
Q )(q B/2 {B + 2(P - Q)} + {P (P + 2)B +
+ (P + 1)² - q(1 + eqx) [k(P² + 1)(B + 2) + ¼ (P² + P + 1)] -
- q (1 - eqx)(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 - ( ) 3
Q ) + ( ) 3
P (2 - q) Q {eqx(B + 2Q + 1) +
+ q/(2k)(1 - eqx)(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 (a is the finestructure constant):
a =
P
kX q k q k
²
, , h2 h (1 - k/4) + (k -1){p/4 ( ) 3
P - h1,1h1,2 ( ) 2
P } (B33)
X = k[4a
( )
( )
( )
( )
[ /
( )
( )
]
B k
q
+ + e
-
+
-
-
æè ç
öø ÷
- - -
+
-
1
1
1 5
1 5
2
3
4
2 1
1
2 2 1 6
2
2
2
2
a 2
a
a
a
p
p a
p h
J a
+ 1 (B34)
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Ó IGW Innsbruck, 2003
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b =
1
2h2 h qk qk
[aJ /8 (P² + 1)[ ½ (1 + h )(1 + h1,1 h1,2 (3/4) ( ) 3
P (k - 1)] +
+ (k - 1) {J 1,2/J - 8 ( ) 2
P (P² + 1 ) - 1}] - C (B35)
C = p (1 - h )² [1 + p (k - 1) + P/k³ (3/e + q(8 + hqk) +
+ (4 pe/ h )(1 - k)[1 - q
3
5
ph
ehqk
] - 2(k - 1) ( ) 2
P (3 - P){2 e (h + hqk) }(B36)
+ eqx pe/(3 h ) } +
8p k 1
h h
e k e q
e
( - )
-
æ
è çç
ö
ø ÷÷
)] + (2 e k q/ h² )(2 - k)(1 - h)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 s-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 sN are
occupied and if
L(s)(p) = ½ N3 (p + Qp) - 2 Qs with - Qs £ s £ L(s)(p) , (B38)
is the complete occupation of the external region related to pN , then
KB = L(s)(p) - sN (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, k, 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)³a1 + (m + Q m)² a2 + (p + Qp) a3 + exp[-(2k - 1) /3Qs(s + Qs)] =
= WN=0(1 + f(N)) (B40)
with a1 = N1 , a2 = 3/2 N2 , a3 = ½ N3, and eq.(B22) to eq.(B36) can be calculated numerically.
By an exhaustion process based on
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Ó IGW Innsbruck, 2003
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w = WN=0(1 + f) (B41)
n, m, p, and s can be determined using eq.(B15) to eq.(B21) and (B40).
Let be K ³ 1 the series of natural numbers. Then, first of all, w - K³a1 ³ 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 a1 = w1 (B42)
The process will be repeated with w1 in the form w1 - K² a2 ³ 0 . With K = Km
w1 - (Km - 1)² a2 = w2 (B43)
will be generated. In the same way w2 - K a3 ³ 0 yields the relation
w2 - (Kp - 1) a3 = w3 (B44)
and with the abbreviation ß = (2k-1)/3Qs
w3 - e -ßK £ 0 (B45)
is determined, which changes its sign for K = Ks . Next, Ks will be reduced by 1.With the limits
now known, Kn to Ks , the n, m, p, s can be calculated:
n = Kn - 1 - Qn m = Km - 1 - Qm
(B46)
p = Kp - 1 - Qp s = Ks - 1 - Qs
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 1 2 2 1 1 1 2 0
hHy
Mc²[ ( )²( )²( )²](H n m p )(n m p ß ) , , , ( ) h - h - h - h + + + + s + +
d
(B47)
where d = d(N) is as in eq.(B7) .M is taken from eq.(B3), and H from eq.(B24). The substitution
y is given by:
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
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y = F [j + (-1)s (1 + j)(b1 + b2/WN=0)] (B48)
with
j =
N p
p
4
1
²
+ ²
s
s
s +
+
- =
- Q
BUWN 1
4 2 4
0
1
²
( ) + P (P - 2)²(1 + k(1 - q)/(2aJ ))
(p/e)²Öh12(Qm-Qn) -- (P + 1) ( ) 3
Q /a , (B49)
U = 2Z [P² + 3/2 (P - Q) + P(1 -q) + 4kB (1 -Q)/(3 - 2q) + (k - 1){P + 2Q -
- 4p(P - Q)(1 - q)/ 4 2 }] /hqk² (B50)
and Z = k + P + Q + k (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 - e C P/2)(1 + ß(0)(-1)k) - ( ) 3
P (1 + D), (B52)
s= 2 - k + e C + (2kQ - kP) + ( ) 3
Q : 1/k (P-1)(P-2)(P-3) (B53)
b1= [P { 7 + 6(1 - q)(C - ( ) 2
P ) - 2q (1 - ( ) 2
P )} + k Q{(3 Z - 1) B + 1}] (2 - k) +
+ ½ (1 - k){(q - eqx - 2) Q + e 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 - e qx)
b2= B(5B+3) +
2 3
1
H
P
-
+
+ Ck{B( 3B+2(H+1)) + H + ½ }(1 - q) - Q {B(2(B+H) - 1) +
+ H/2 + 3} + k q {B (3B + 1) - 5/2}(k-Q) - ( ) 2
P P²(P + Q)²[8B+1 -
-{5B - (2H+1)(1 + 2 ( ) 3
P - 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) + pe/h - q/4}(P+1)³(k-1) + k{(-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 + e qx)[B/2(H+2) + ¾ ] +5/2HB +
+ 3H -
B
P
+
+
5
1
} - 5/2 H² ( ) 3
P {q (1+p/3(2-q) h2,2) B - (2-q)(1 - q)} (B55)
with ß(0) =
2a
pe
1
1
2 -
+
æ
è çç
ö
ø ÷÷h
h
(B56)
and D = [1 + 4 q²(q - 1)(2q + 1)] - 1 h ß(0) (1-Öh)4 P2+eq (P - 1)(q-1)q/2/(3Ö2) (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.
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Ó IGW Innsbruck, 2003
18
3. The Sommerfeld Finestructure Constant:
In j and ß(0) the finestructure constant a is contained. The value in chapter D, section 8 is
calculated only approximately. Heim now also gives the exact formula for a:
According to eq.(8.21) we get:
a 1
9
2
- a = 5 1-
J
p
²
( )
( C') (B58)
with 1 - C’ = 1 -
1 1
1
2 2
1 1 1 2
2 + -
+
æ
è çç
ö
ø ÷÷
h
hh h
h
h
,
, ,
= Ka (B59)
With the abbreviation
D’ =
(2 )
9
p 5
J a K
(B60)
it follows for the reciprocal square of these solutions:
a(±)
-² = ½ D’²(1 ± 1- 4 / D'²) (B61)
With eq. (V/chapter E) the values for both branches are calculated:
a+ = 0.72973525 ´ 10 - 2 and a- = 0.99998589 (B62)
1/a(+) = 137,03601 1/a( ) = 1,0000142
which, compared with the empirical value (Nistler & Weirauch 2002) for the finestructure
constant,
1/a(+) = 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 s = - Qs , (B63)
from which follows:
G + F + S = j (B64)
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19
According to eq.(B49) generally j ¹ 0 holds, in spite of s + Qs = 0, and also F ¹ 0 is not
affected by the lower barrier of the n. m, p, s . If F + j ¹ 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
Mn = ma+ (F + j0) (B65)
where j0 relates eq.(B49) to the lower bounds of n, m, p, s, than it follows, that Mn is
determined only by the quantum numbers k, k, P, and Q .
For Mn(kPQk) > 0 the following possibilities result:
Mn (1110) = Mn (1111) and Mn (1200) in the mesonic region, and
Mn (2110) and Mn (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:
Mn (2010) or Mn (1010).
Since in the case (2010) Mn < 0 would be, only Mn ( 1010) remains as a possibility for the ßneutrino.
With i = 1,...,5 the possible neutrino states ni are:
for k = 1: n1 (1010) , n2 (1110), n3(1200)
for k = 2: n4 (2110) , n5 (2111).
For each ni there exists the mirror-symmetrical anti-structure ni . 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 n1 and the empirical m-neutrino by n2.
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)
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
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 j with eq.(B50) were fitted by empirical
facts [i.e. 4 2,(p / e)2 and 4p 4 1/ 2 ] .
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 D -
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 w(783) and h’(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 ¹ 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 farreaching
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.
