The Dual Space Concept

Welcome to the site of Hans-Dieter Herrmann,
Physicist, Dr. rer.nat.
Email: Dieter.Herrmann@tonsa.de
Natural Systems and State Spaces
A joint model of particles and spacetime
Ein gemeinsames Modell für Teilchen und Raumzeit
Compositeness and spatial extension of fundamental particles in a circular extra space (DPG – Conference on Particle Physics Göttingen April 2025)
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Particle models established in cylindrical eigenspaces with D = 6, 10 and 26 dimensions (DPG – Conference on Particle Physics Karlsruhe March 2024)
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Extra dimensions vs. extra space
Extra dimensions vs. extra space
Fig. 1: Superstring theories and the dual space concept are compared for the example D = 10. In string theories, the total number of 10 dimensions is divided into four large space-time dimensions and six compactified dimensions, also embedded in space-time, however too small for observation. In the dual space models, the four dimensions of space-time remain unchanged, however in addition to space-time a 10 dimensional eigenspace exists, independently and outside of space-time. The eigenspace as part of the “basic space” is considered to establish a new level of reality “below” space-time.
Fig. 2. Particles can be composited and extended in basic space, while they are fundamental and point-like in space-time. The circular motion with the velocity of light is force-free in basic space. The so called ‘intrinsic’ properties, such as mass, spin, magnetic moment, and sometimes the Compton wave length, have its origin in basic space.
The history of extra dimensions
Fig. 3: Historical data on the required dimensions for consistent theories with D > 4 dimensions. The conditions of consistency do not depend on the existence of strings and do not contain a separation of 4 large from D – 4 compactified dimensions. Therefore basic space models having the critical dimensions are considered to be consistent as well.
Photon, lepton, meson, baryon models
Fig. 4: Body fixed and lab fixed coordinate systems of a spinning top (left panel). The fundamental particles cannot be modelled in a similar way, a big number of such circular or helical models had no success. Dual space representation of a photon (right panel). The eigenspace of the photon (a part of the basic space) is fixed to the photon’s structure, but invisible and inaccessible to observers. The wavelength of the photon in space-time equals the circumference in basic space.
4b. A biroton is assumed as a composited particle model living in basic space. The biroton has different properties of a Dirac – particle, for instance a 4π – cycle of the internal roton. The two partial spins ħ and –ħ/2 of the rotons add up to the total spin ħ/2 of the biroton. The two par-tial masses of the rotons consist of mass quanta. The circular velocities +c and –c of the roton masses correspond to the eigenvalues in the Dirac – theory belonging to the ‘Zitterbewegung’. The symmetry of partial masses and the asymmetry of spin components of the rotons reveal some kind of internal su-persymmetry between them. The circulation planes of the rotons represent two separate anticommutative two-dimensional vector spaces. Gamma-matrices are used as unit vectors. The two-dimensional circulation planes, the common spin axis and the time coordinate result in a total of six dimensions of the biroton. The length of one cycle, that means the circumference of the external roton, is equal to the Compton wave length.
Fig. 5: Three models show the critical dimensions of different string theories, however without strings. The eigenspaces of particles are determined by rotons. These spaces consist of a two-dimensional circulation plane (colored ellipse), a spin axis and a time coordinate. The lepton model consists of one biroton with the total spin ħ/2, the sum of the roton spins +ħ, -ħ/2. The biroton has 6 dimensions, such as the so called “little string theories”. The meson model consists of a biroton (spin ħ/2) and an anti-biroton (spin –ħ/2), shown is a mass-symmetric pseudoscalar meson. It has 10 dimensions, known from superstring-theories. Baryon models need 6 birotons and 26 dimensions, the critical number for bosonic string theories. Each baryonic biroton such as the proton model presented in the figure has a quarter valued spin, this is forced by the magnetic moment anomalies of the nucleons and other baryons.
Fig.6: Four states of a biroton or an anti-biroton correspond to four states of a Dirac spinor in rest frame. A biroton has a 4π – cycle of the internal roton and a 2π-cycle of the external roton. The left – right asymmetry between the two states of a biroton (or an anti-biroton) represents a geometric picture of the weak parity violation. Doubly charged rings belonging to the weak charge group are depicted. The two charges of a double ring overlap at the right handed variant of the model, not at the left handed; vice versa for anti-birotons (bottom row). The overlap of charges causes the unability of weak coupling.
Four fundamental constants
7. Charges circulate together with masses or independently. The energy conservation between basic space and space-time leads to different mass relations for circulating charges or masses. Masses are halved if rotational energy (in basic space) is conserved as invariant (rest) energy in space-time. However, if an electromagnetic (em) charge circulates independently of the masses, then the mass equivalent of its Coulomb self-energy is conserved.
8. The table summarizes the role of fundamental constants and the conservation of energy quantities at the transition from basic space to space-time. The four fundamental physical constants c, h, e, and MQ are sufficient for the formulation of basic space models, if the laws of model construction are known. The mass quantum MQ can be positive or negative at the roton level. The mass quantum mQ = ½ |MQ|at particle level in space-time is always positive. The small number of four fundamental constants would render particle physics more compact in com-parison to the Standard Model.
Physical laws in a circular space
9. The definitions of physical quantities are adapted to a circular space. Rotational energy defined by classical point-mass mechanics translates into invariant energy in space-time. Angular momentum translates into spin. The quantities for the rotons 1 and 2 of a biroton seem to correspond to an unbroken supersymmtry between the rotons, see formulas with blue background. The Coulomb energy, classicaly the energy of one charge relative to another charge in the linear distance d, changes in basic space into a self-energy. The distance d becomes the self-distance of a single charge along its circle, it is an arc length. This unusual assumption is one of the cornerstones of the basic space models.
Mass generation: Self energy of circulating charges
10. Energy factors determine the contribution of the self-energy of circulating charges to the particle mass. The factors differ depending on the number of cycles contributing to self-energy. Single electromagnetic (em) charges circulate infinitely and generate the maximum self-energy. The infinite row wk has a simple sum formula.
11. The structural diagrams of three lepton models show the skeletons (circulating mass quanta) and the dresses (circulating charges). The lepton masses result in the sum of both contributions. The resulting skeleton mass of the electron is zero, the model contains an equal number of positive and negative mass quanta. The mass quantum MQ ≈ mµ/16 was derived from measured muon data mµ and aµ as in-put. MQ (in basic space) and mQ = MQ/2 (in space-time) are used as universal constants for the construc-tion of models of other leptons, mesons and baryons.
12. The masses of leptons can be derived from a general formula (blue colored). D1(k) means the dress of a roton, k the ring parameter. The electron model has equal numbers of positive and negative mass quanta, its skeleton mass is zero. The self-energy of an em charge added to one roton of the skeleton provides the main contribtion (≈ 0.502 MeV) to the measured electron mass. The neutrino masses represent also the vanishing difference between large positive and negative roton masses. Therefore the introduction of negative mass quanta is essential, unavoidable, for the construction of lepton models.
Mass generation: Mass quanta and charge clusters
12. The skeleton masses of leptons in different generations are determined by the square number of mass quanta. The skeleton masses of hadrons contain mass quanta enhanced by charge clusters. The energy of charge clusters increases steeply with the number l of charges per cluster. The product of both increasing functions determines the scale of skeleton masses of mesons.
13. The formulas for meson masses lead to the assumption of four mass generations, where the first generation contains only the partons of pions and the fourths generation starts with the W particle and reaches up to the top quark. Predictions of particle masses in the Tev region and above are possible. The mass of a “full electron” as shown is not realized in nature. The full mass is given for comparison with the muon and tauon masses and means the sum of both roton masses instead of its difference.
14. A table of numerical values shows the steep increase of mass values from partons of π – mesons to the top-quark. Values of higher masses can be predicted, however, they are not realized in nature as particles. In the sixth generation we arrive at the region of the Planck mass. There is no prediction of particle masses in this region, however, a dimensionless coupling constant of gravitation can be derived. One can use the mass quantum or the ring mass of the sixth generation as the fundamental mass.
Matter, anti-matter, mixed and dark matter
15. An overview on particle types and the dimensions of their eigenspaces shows the relationship between nearly massless leptons like neutrinos and the candidates of dark matter and dark energy. Mono-rotons appear as isolated “half-neutrinos”. We remark, that the eigenspaces of mono-rotons have four dimensions, the same as space-time.

A dual space concept for natural systems at different levels of reality

DPG-Conference on Philosophy of Physics, Bonn 2025

https://www.dpg-verhandlungen.de/year/2025/conference/bonn/part/agphil/session/1/contribution/4?lang=en

DPG – Conference on Philosophy of Physics, Berln 2024

https://www.dpg-verhandlungen.de/year/2024/conference/berlin/part/agphil/session/17/contribution/1

Vertical and horizontal evolution
Historical periods and levels of organization in the evolution of natural systems. The last column shows eight levels, each level is characterized by newly emergent systems called „units“.
Units (with red frames) and compounds (between the units) represent the main system types in the molecular level L3 and the macro-molecular level L4.
Units (with red frames) and compounds (between the units) represent the main system types in the micro-organismic level L5 and the organismic level L6.
General view of the two-dimensional character of evolution at level n.
The transition to the next higher level of reality
The transition from one level to the next higher level needs the development of a special type of systems, ready for this transition. In the non-living world we have at the subatomic level L1 stable particles such as proton, neutron and electron, which are ready for the transition to level L2. At the atomic level L2 develop stable atoms capable of chemical bonds. At the molecular level L3, stable molecules emerge such as amino acids, sugars or nucletids, capable to build chains or to polymerize. At the macro-molecular level L4, stable RNA / DNA – protein complexes take the role of systems ready for the big step into the living world, they can form viruses and other micro-organisms.
In the world of living systems, the leading role for the transitions to the next higher level have the emergence of cells, human beings and stable, cooperative nations. We identify 8 levels and 7 transitions between them. This simple presentation concentrates on vertical developments towards the higher levels and does not reflect the horizontal evolution within a level.
Units and compounds, eigenspaces and common spaces
A more complete description is demonstrated at the molecular level L3 as a typical example. At this level the simple relation holds “molecules consist of atoms”. We consider two system types (brown colours) and two space types (green colours). The term “units” is used for the building stones of evolution, at level L3 for the atoms. The term “compound” is defined for the different composite systems such as molecules and radicals. The spaces are “state spaces”. They consist of rotational and vibrational states of a molecule. Only multi-atomic systems have such states. We call this the “upper state space” of a molecule. In contrast, the states of electronic excitation are called the “lower state space” of molecules. The lower state space is accessible to units (atoms) as well as to molecules (compounds).
Curved arrows show the development of systems as well as of their state spaces. New types of matter and new types of state spaces are the result of one and the same evolutionary process.
We attempt now to apply the same general terms such as “units” and “compounds” at the subatomic level L1 as well. We consider space-time no longer as a universal vessel of all things and processes. In contrast, space-time appears at the subatomic level L1 as the state space of the stable and unstable particles, the compounds of this level. If the compounds vanish, their state spaces disappear too. That means, at the sub-particle level below L1 we do not expect to find particles and also space-time does not exist.
The “twofold existence of natural systems” and the “dual space concept” describe the picture with four fields of knowledge at each level.
The whole story of the evolution of narural systems from level L0 to L8. Only a few keywords per level are displayed. The angle arrows are smooth, where the transitions are more or less well understood. The angle arrows are dotted, where the knowledge is incomplete. If we look at the upper end of the evolution chain: Stable nations are needed for a permanent cooperation within the solar system. It is unknown, whether mankind reaches the next higher level L8 of a multi-planetary civilization. The alternative would be the self-destruction of our civilzation. At the lower termination of the chain, the transition from L0 to L1 has the character of a conjecture. One has to cross two “red lines of thinking” to arrive the ultimate origin of the evolution chain. The first (vertical) red line means the assumption of compositeness of fundamental particles. In crossing this line, we arrive the column of unobservable, sub-particle entities. The second (horizontal) red line represents the idea of unbound, “free rotons”, that are no longer components of particles. In crossing the second line, we enter the basement of the column L0 of non-observable mono-rotons. To cross these lines could be a step towards the understanding of dark matter and dark energy.

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