last update 99-04-18

Elementary Structures of Matter

The following article gives a short overview about
the content of the first part of the Heim Theory (as described in
"Elementary structures of matter" Vol 1 and 2, see
references). After a short reference to other common theories the way is
demonstrated what Heim did to make unification of field theories
possible. This leads to an uniform field theory as a geometrical theory
of 6-dimensional space wich is completely quantized. Combinations of the
3 basic groups of coordinates provide 4 different kinds of elementary
structures which represent the 4 different kinds of physical
phenomenas.

Analysis of time and metronic structures gives a picture about history
of space and time (cosmology).

While describing internal dynamics of 6-dimensional space it is possible
to calculate a complete mass spectra of any mass which is possible in
our physical world.

Introduction

Unified field theory

Elementary structures

Cosmology

Elementary masses

In physics a number of approaches are possible in order to arrive at a corn-prehensive, unified theory describing a maximum number of observable phenomena.

One such possibility would be the geometrization of physical
structures described in the two volumes "*EIementarstrukturen
der Materie"* (Elementary Structures of Matter) [1] and [2].
Its evident advantage is the fact that in this treatment space and
object are no longer foreign to each other. Instead, an object now
appears as a specific metric structure of space. A consequence of this
is the unity of field and field source. The first attempt at such a
geometrization, applied to the gravitational field, was carried out by
A. Einstein in his general theory of relativity, later on extended by
Kaluza, Klein, and Penrose, but also by P. Jordan and others.

The theory of supergravity and superstring theory may be regarded as
successors to the Kaluza-Kiem model. At present, superstring theory is
being further developed in the hope of attaining a unified description
in a 10-dimensional space, R_{10}, of the 4 empirically known
interactions. However, it is not clear how the predictions of this
theory, involving particle masses of some 10^{18} GeV, can be
verified in view of the limitations of present-day high energy
experiments. Here a radical geometrization of space as described in [1]
and [2] appears to lead to more suitable results that may readily be
compared with observation.

An entirely different approach to the unified description of nature by the use of group theoretical arguments has been applied with some success to the unification of the 4 known elementary forces. So far it has not been possible to verify these theories experimentally, especially since the resulting mass spectrum of elementary particles involves much too high energies. In addition, there is a lack of predictions concerning quantum numbers of resonance spectra representing these masses and about their upper limits.

Extending the ideas of Einstein, Kaluza, Klein and Jordan the theory
described in this report shows how to geometrize in principle not only
the gravitational field but the other force fields as well. They appear
as geometrical structures of spacetime, R_{4} (a Minkowski space
with x_{4} = ict) subject to the usual conservation laws, and
lead to a general non-Hermitian geometry in R_{4}. The covariant
components of the corresponding triple index symbols, representing
generalized Christoffel symbols, can be split into a Hermitian and an
anti-Hermitian part, but, in contrast to Riemannian geometry, they
cannot in general be expressed explicitly in terms of derivatives of the
fundamental metric tensor unless additional conditions are introduced.
However, since such conditions reduce the generality it is not known
whether they are at all admissible from a physical point of view. Thus,
the -symbols are to be treated as components of a unified field of
metric structures.

A transition to the microscopic region by use of the correspondence principle turns the unified field components into true tensor components with respect to the group of one-to-one, continuous, and non-singular coordinate transformations (Poincare-group), with mixed covariant and contravariant indices. This also applies to the macroscopic range, where a unique set of metric structures corresponds to each type of field (gravitational, e.m.). Different fields, characterized by different conservation laws, lead to different geodetic conditions.

On passing over to the microscopic range the -symbols, which are
non-Hermitian in their covariant indices, transform into
components of a tensor field with mixed indices. The condition = 0 can only
be satisfied if the corresponding R_{4} is completely
unstructured. In the microscopic range the phenomenological 'energy
density leads to a discrete set of eigenvalues and
eigen-functions, characterizing a discrete spectrum of metric structures
(parantheses around denote suspension of the summation convention). The
concept of energy density appears in geometrized form because
phenomenological energies are time derivatives of actions, so that
spatial energy densities ultimately are spacetime-dependent action
densities. Hence, due to the classical quantization of actions it is
impossible in the limit to make the transition to differential
quotients. This implies the necessary existence of a smallest geometric
unit.

The nonlinear and non-Hermitian equations of state satisfied by the
R_{4}-structures exhibit algebraic symmetries such that 28 out of
the total of 4^{3} = 64 relations for always remain
empty. Thus, 36 equations lead to relations /= 0, while the empty level
spectra of geometrical structures with = 0 require that = 0
because =/ 0. The fact that geometrical structures in the microscopic
range vary in discrete steps is the equivalent of the well-known
discrete level structure of energy densities, so that the 36 nonzero
components must of necessity form a tensor scheme having 6 rows and
columns. In accordance with tensor algebra this requires R_{4}
to be a subspace of a 6-climensional reference space, R_{6}. The
new coordinates, x_{5} and x_{61} are imaginary, like
x_{4}.

The appearance of two new coordinates indicates that the limitation
of quantum theory to 4 coordinates may be too restrictive. After all,
space and time are only aspects of human perception (I. Kant). It should
be mentioned that, since the R -coordinates are obtained on the basis of
conservation laws and energy relations, R_{6} must be regarded
as referring to the material world with x_{5} and x_{6}
having to be interpreted as * organizational
coordinates* of material structures in R

A proof of existence shows that the eigenvalues actually exist, but there appears a further symmetry, indicating the vanishing of an additional 12 of the remaining 36 components of the macroscopic' energy density tensor. These are the 4*3 space like elements occupying the upper half of columns 5 and 6 and the left half of rows 5 and 6 of the tensor. This is to be expected on the basis of macroscopic physics. Furthermore, it has led to the following law governing the number of possible dimensions in hyperspace:

If p >= 0 dimensions of an R_{p} are empirically
given,then there exists a reference space R_{n} containing
R_{p}, n being an integer, _{}such that the
condition

is satisfied. For p = 0, 1, and 2 the positive branch of this equation gives n = 2, while the negative branch gives n = 0. However, the latter is irrelevant for all p > 2. Finally, the equation is not satisfied for p = 3, p = 5, and all p > 6. For p = 4, on the other hand, i.e. for spacetime, n actually turns out to equal 6.

Furthermore, for the material world, R

In order to derive the smallest geometric unit mentioned above it is
necessary to consider a universal background phenomenon. A suitable
quantity is the general inertia of all masses, which always is
equivalent to gravitational phenomena according to the principle of
equivalence. A phenomenological dynamics of gravitation is derived in
[1], which, together with a self-consistent treatment of the
mass-equivalent of the gravitational field energy, leads to the
description of a scalar field function by means of a non-linear system
of equations, which formally agrees with the nonlinear structural
relations of R_{6}. For constant x_{4}, x_{5}
and x_{6} the function (x_{1}...,x_{6}) then
becomes (x_{1}, x_{2}, x_{3}) of
R_{3}.

is a
real, positive gravitational potential (potential energy per unit mass),
satisfying a nonlinear differential equation, which can be solved in
spherical geometry and results in a transcendental algebraic equation
for .
The solution shows that remains real only between the limits R_{-}
and R_{+}, where R corresponds to the Schwarzschild radius and
R_{+} to the Hubble radius. In the range of relatively small
distances (planetary systems) is almost exactly proportional to hr and
hence practically identical to Newton's law of gravitation. This
changes, however, in the range of very large distances, because there
exists a limit, , of the attractive gravitational field, lying between R_
and R_{+}, at which goes to zero. This limit depends on the cube
of the mean atomic weight of the field source according to A^{3} = ca. 46
Mpc. Beyond the field becomes weakly repulsive before definitively going
to zero at R_{+}.

A single elementary particle is characterized not only by and the
limiting distances R_{+-} of its gravitational field, but also
by its Compton wavelength. R_{-} vanishes in empty space when
the mass of the field source approaches zero, while R+, , and the Compton
wavelength all diverge. However, since the smallest geometrical unit
must be a real number and a property of empty space its value has to
remain finite. As shown in [1], only a single product having this
property can be formed from the 4 characteristic lengths above. The
result is an area, , bounded on all sides by geodesics, whose present
numerical value is = ca. 6.15x10^{-70} m^{2}. This
quantity, called a * metron,* represents the
smallest area existing in empty space and requires the differential
calculus to be replaced by a calculus of finite areas. Accordingly, a
whole chapter in [1] is devoted to the development of a difference
calculus considering the finite area of . This enables any differential
expression to be metronized. It follows that in any subspace
R

While the are always bounded by geodesics, their area remains constant
in a deformed lattice. The metronized state function then describes the
projection of a deformed R_{6}-lattice into any Euclidian
reference space, where the metrons now appear in distorted or
"condensed" form, in analogy to the projection of a curved
lattice onto a plane sheet, or to lines of constant altitude on a map
providing information on the level structure of a mountain range. In
this respect there seems to exist a certain analogy to Regge poles. The
metronic system of equations itself has the character of a selection
principle, selecting out of a multiply infinite manifold of possible
R_{6}-structures the ones whose projections into R_{4}
describe elementary material processes of the physical world. The
operator performing this selection is called the "world
selector".

Further analysis shows that the world selector separates out 4 sets
of solutions, denoted by a, *b, c,* and *d,* involving 3
subspaces: A 2-dimensional sub-space
**S _{2}**(x

A different set of coordinates is involved in each of the 4
structures a-*d* mentioned above: *a* depends on
(x_{5}, x_{6}), *b* on x_{4} and
(x_{5}, x_{6}), *c* on (x_{1},
x_{2}, x_{3}) and (x_{5}, x_{6}), and
*d* depends on all 6 coordinates x_{1} ... x_{6}.
In every one of these combinations the coordinates are always grouped
into subspaces S_{2}**,** T_{1} and
R_{3}. Note that the organizational coordinates x_{5}
and x_{6} constituting S_{2} appear in all elementary
structures.

A sort of hermeneutics (from *hermeneuo:* to interpret) of the
world geometry, or "*hermetry"* for short, is required
for interpreting the forms *a* to *d : a* represents
structures outside of R_{4}, which do not in general have a
physical interpretation. However, when projected into R_{4} they
appear as graviton fields.

The world lines belonging to elements of *b* all lie on the
two-fold light cone in R4. For this reason they always travel with the
velocity of light in R_{3} and are to be interpreted as
photons.

Hermetry forms *c *and *d* are
characterized by the inclusion of the real subspace R_{3},
leading to inertia and hence to rest mass, in contrast to *a* and
*b. c *is interpreted as referring to neutral
particles, while *d* refers to charged ones. In the case of
*d* a coupling appearing between *b* and ccharacterizes
the charged condition. It was possible to derive a simple relationship
for an elementary charge determined by the quantum principle, whose
numerical value deviates by 0.125% from the experimental electron
charge.

In addition, a general mass spectrum can be derived, whose terms turn
out to lie so close together that for all practical purposes they
approach a continuum. This is entirely due to the fact that the spectrum
is a superposition of energy terms of *all* hermetry forms. Thus,
the practically continuous spectrum of the massless *a-* and
b-terms is superimposed on the discrete spectrum of hermetry forms
c**and ***d.* For this reason a "term
selector" is required for separating out the discrete mass spectra.
Independently of this it is possible to derive the lower bounds of
spectra ** C** and

As shown in [2], the upper reality bound, R_{+}, of the
gravitational field increases with diminishing field source, i.e. the
largest value of R, R_{max} results from the smallest rest mass.
Thus, 2R_{max} = D is the greatest possible distance in
R_{3}. It is defined in [2] as the diameter of the universe and
depends entirely on natural constants. These constants disappear if the
expression for is substituted into the formula for R, and there results
a higher order algebraic expression for the dependence of D on , referred to
in [2] as the cosmological relation. Astrophysical reasons
require >0 , where is the time derivative, which in turn results
in
< 0. Thus, as cosmic time progresses the metronic mesh size shrinks,
while the universe expands.

Going back in time, D decreases while increases. This ends
when
encompasses a "proto universe", whose diameter, D_{0},
is given by . Since cannot become smaller, this represents an initial event
in R_{4}, beyond which there is no past. This instant has,
therefore, been defined as the moment t = 0 of the cosmogonic origin. By
employing an appropriate substitution D() becomes the solution of a 7th order
algebraic equation. At t = 0 and at the end of time, , the equation has
3 real positive, 3 real negative, and one complex solution for D.

The positive solutions are interpreted as the diameters of 3 primordial
spheres emerging and expanding, one after another, after t = 0. The
spheres mark the boundaries of the expanding universe, their calculated
separations in time shrinking to very small values, but never to zero,
as cosmic time advances. After D reaches a maximum value, contraction
sets in. Finally, at the end of time the trinity of spheres, now having
diameters corresponding to the 3 negative solutions, disappear one after
another.

A long time after the initiation of cosmic motion at t =0, presumably
after the appearance of matter, a symmetry break of global groups leads
to the development of 3 geometric units k_{i} =/
k_{j}^{*}, i = 1, 2, 3, (k_{i} is a 6x6 tensor)
in the sense of tensorial integrands of integral operators. Tensor
multiplication and taking the trace according to the metronic difference
calculus results in generally non-Hermitian partial metric tensors
g_{ij} = T_{r}(k_{i}x k_{j}) forming the elements of the
full metric tensors, .**Since the 3 tensors
k**_{i} go back to the 3 primordial spheres, every elementary
particle retains a memory of the cosmic origin.

The metric units k_{i} depend on the hermetric subspaces of
R_{6} according to k_{1}(S_{2}),
k_{2}(T_{1}), and k_{3}(R_{3}).
Corresponding to the 4 hermetric forms *a-d* one can now form 4
generalized metric tensors from the partial tensors g defined above. If
k_{1}(S_{2}), k_{2}(T_{1}) and
k_{3}(R_{3}) all differ from E (unit matrix) the
resulting _{d} (S_{2},T_{1} , R_{3})
depends on all 6 coordinates and represents hermetry form
*d.*

_{c} (S_{2}, R_{3}), belonging to the
spacelike hermetry form c, is obtained by putting
k_{2}(T_{1}) = E. In similar manner the timelike metric
tensor, _{b}(S_{2},T_{l}), for hermetry form
*b* is obtained by putting k_{3}(R_{3}) = E.
Finally, _{a} (S2) for hermetry form *a* results from
k_{2}(T_{1}) = k_{3}(R_{3}) = E.

_{d}has 9 independent elements, _{c}and _{b}both have 6, and _{a}has 2,
representing respectively 9-fold, 6-fold, and 2-fold metrics. These
polymetric structures combine to form the Hermitian metric field of the
"condensor", which is also Hermitian. The condensor is an
operator projecting a deformation in the 6-dimensional metronic lattice
of R_{6} into R_{4}, where it appears as an intricate,
geometrically structured, compressed or "cordensed" lattice
configuration. This condensed, structured region is what we call matter
constituting an elementary particle, as described in more detail
below.

Thus, both the world selector and the condensor describing the
internal particle structure can be split up in a manner allowing a
system of partial metrics to be specified for each hermetric form. An
appropriate choice of indices (ij) then results in a solution of the
general energy spectrum corresponding to a separation of the discrete
spectra c and *d.* These solutions actually yield discrete
spectra of inertial masses, showing good agreement with measured
particle and resonance spectra.

The following picture regarding the spectrum of elementary particles found in high energy experiments emerges from the theoretical analysis above:

Elementary particles having rest mass constitute self-couplings of
free energy. They are indeed elementary as far as their property of
having rest mass is concerned, but internally they possess a very
subtle, _{}dynamic structure. For this reason they are
"elementary" only in a relative sense.

Actually, such a particle appears as an elementary flow system in
R_{6} (equivalent to energy flows) of primitive dynamic units
called * protosimplexes,* which combine to form
flux aggregates. The protosimplex flow is a circulatory, periodic motion
similar to an oscillation. A particle can only exist if the flux period
comprises at least one full cycle, so that the duration of a
particle's stability is always expressible as an integer multiple of
the flux period. Every dynamical R

The physically relevant parts of an R_{6}-flux aggregate are
its k + 1 components in the physical space, R_{3}, which are
enveloped by a metric field. Thus, mesons contain two and baryons three
components. Evidently, there exists an analogy to the empirically
formulated concept of quarks. If this is true, then quarks are not
fundamental particles but non-separable, quasi-corpuscular
subconstituents in R_{3} of a mesonic or baryonic elementary
particle. In this picture the condition of quark "confinement"
is unnecessary. The significance of a possible "quantum
chromodynamics" will have to be derived on the basis of a unified
description of possible interactions. This problem is being investigated
in [3].

Responsible for the inertial mass are the protosimplexes, i.e. the
basic building blocks of flux aggregates, which form the structures of
the k + 1 subconstituents in R_{3}. They compose 4 concentric
spherical shell-like configuration zones maintaining a dynamical
equilibrium, during whose existence there appears a measurable particle
mass. However, an attempt to measure the mass of a subconstituent part
by scattering experiments will result in a very broad, variable
bandwidth of measurements, because such a mass depends on the
instantaneous flux phase. The sum of the k + 1 subconstituent masses, on
the other hand, is constant and gives in essence the measurable particle
mass. The relevant quantity in this connection is the degree to which
the 4 configuration zones in R_{3} are occupied by dynamic flux
elements.

For k = 1 and k = 2 there are altogether 25 sets of 6 quantum numbers each, characterizing the occupation of configuration zones and the corresponding invariant rest masses. The particles belonging to these invariant basic patterns are in turn combined into several families of spin isomorphisms,in which the spatial flux dynamics of the configuration zones is in dynamic equilibrium.

In all these terms there exists a single basic invariant framework of
occupied zones, _{}depending only on whether k = 1 or k = 2.
Substituted into the mass formula derived in [2] this reproduces the
masses of electron and proton to very good accuracy. The masses of all
other ground states are produced in similar quality. However, the mass
formula contains ratios of coupling constants, which could not at the
time be derived theoretically. and therefore had to be adjusted to fit
experiments carried out at CERN in 1974. Only in [3] has it become
possible to derive the coupling constants from first principles, but a
revised set of particle masses has not yet been calculated.

It seems that the lifetime of a state depends on the deviation of its
configuration zone occupation from the framework structure mentioned
above. It is conceivable, in analogy to the optically active antipodes
of organic chemistry, that there exist isomers with spatial reflection
symmetry also in the area of flux aggregates, giving rise to variations
in lifetime. Perhaps the two equal-mass components of the
K^{0}-meson, K^{0}_{S} and
K^{0}_{L} are to be interpreted in this way.

Finally, the requirement that empty space be characterized by vanishing zonal occupations and electric charge states leads to some masses in the case of k =1, which may be interpreted as neutrino states. However, these refer neither to rest masses nor to free field energies (in analogy to photons), but to quantum-like "field catalysts", i.e. particles able to catalyse nuclear reactions that otherwise would not take place. They transfer group theoretical properties, arising from the sets of quantum numbers, through physical space.

The formula derived in [2] for the spectrum of elementary particles
also depends on an integer N >= 0, where N = 0 refers to the 25
ground state masses. For N > 0 the sets of quantum numbers again
yield masses, which now denote resonance excitations of the basic
structural patterns to states of higher energy. According to the
dynamics of configuration zones only a single set of zonal occupations
is possible for each N. Evidently, the corresponding masses represent
short-lived resonance states, for all measured resonances appear among
these spectra. In each case N is limited, since for every set x of
quantum numbers there exists a finite resonance limit, G_{x}
< , such that the closed intervals 0<= N<= G < apply to every
resonance order N, including the ground state.

Out of the relatively large number of logically possible particle masses present-day high energy accelerator experiments only record the small subset of particles whose probabilities of formation (depending on experimental conditions) are sufficiently large. What evidently still is lacking is a general mathematical expression relating these probabilities of formation to particle properties and experimental boundary conditions.

The theory developed in (1] and [2] represents a semi-classical
investigation. A third volume _{"}*Strukturen der
physikallschen Welt und ihrer nichtmateriellen Seite"*
(Structures of the Physical World and its Non-Material Aspect), [3],
leads beyond the semi-classical domain to a dynamics in the hyperspace
of R_{12}.

BURKHARD HEIM

Institut für Kraftfeldphysik u. Kosmologie

D-37154 Northeim

*[1] Heim, B. EIementarstrukturen der Materie, Vol. 1 (revised),
Resch Verlag, Innsbruck, Austria, 1998: x + 313 S.; ISBN 3-85 382-008-5;
DM 166,00*

*[2] Heim, B. EIementarstrukturen der Materie, Vol. 2, Resch Verlag,
Innsbruck, Austria, 1984,1996: 2; xii + 385 S.; ISBN 3-85 382-036-0; DM
166,00*

*[3] Dröscher, W., and Heim, B. Strukturen der physikalischen
Welt und ihrer nichtmateiellen Seite, Resch Verlag; Innsbruck; 1996; 3;
163 + 16 S.; ISBN 3-85 382-059-X; DM 95,00*

original text by Burkhard Heim

translation by T. Auerbach

*1999 html, little corrections and structured by
Olaf Posdzech*

© Olaf Posdzech, 1999