The generation of mass in a non-linear field theory

Thomas Warmann

doi:10.1515/zna-2022-0018

Article

The generation of mass in a non-linear field theory

Thomas Warmann

Published/Copyright: June 24, 2022

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From the journal Zeitschrift für Naturforschung A Volume 77 Issue 8

Abstract

The mass spectrum of elementary particles is calculated in a new approach, based on B. Heim’s quantum field theory, which manifests in a non-linear eigenvalue equation and merges into the Einstein field equation in the macroscopic limit. The poly-metric of the theory allows spacetime and matter to be described in a unified formalism, representing a radical geometrisation of physics. The calculated mass energies are in very good agreement with the empirical data (error < 1 % on average) if the mass scale is gauged to the electron as lowest mass and the second main parameter, determining the strength of obtained mass hierarchy levels, is close to the half inverse of the fine structure constant, describing the difference in strength between the electromagnetic and the strong interaction. The obtained hierarchy levels are not identical to the particle generations of the Standard Model; however, show a self-similarity typical for non-linear theories. For higher values of the main quantum number N, the calculated mass formula becomes identical to the phenomenological formulae of Nambu, respectively, Mac Gregor.

Keywords: general relativity; geometrisation; mass spectrum of elementary particles; non-linear field theory

Corresponding author: Thomas Warmann, Sopra Steria SE, Hans-Henny-Jahnn-Weg 29, Hamburg 22085, Germany, E-mail: thomas.warmann@mail.de

Private researcher, Ph.D. in theoretical physics from University of Münster, Germany, 1992.

Author contribution: The author has accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The author declares no conflicts of interest regarding this article.

Appendix A: Non-hermitian spacetime structure in the R ₄

We summarise the content of chapter I.3, pages 26–28, in [43]:

We consider as a general case m ≥ 1 non gauge-invariant and n − m (n > m) gauge-invariant interaction fields. For each field a characteristic partial event structure determined by the field symmetries can be defined, given through a geodesic coordinate system in spacetime ξ ⃗ p ( j ) = ϵ ⃗ p ( j ) ξ p ( j ) with 1 ≤ j ≤ n different interactions and 1 ≤ p ≤ 4 coordinates. The unit vectors ϵ ⃗ p ( j ) do not have to be necessarily an orthogonal system. The 4 geodesic coordinates can only be related to the 4 Cartesian coordinates x _k of the empty R ₄ so that transformations ξ ⃗ p ( j ) ( x 1 , x 2 , x 3 , x 4 ) exist which must be those of the global Poincare group.

Regarding the structural properties in the R ₄ the n fields can be split into the subsets of the m non-gauge-invariant and n − m gauge-invariant interactions. The total differentials per coordinate for both subsets can be written as d z ⃗ p + = ∑ j = 1 m d ξ ⃗ p ( j ) and d z ⃗ p − = ∑ j = m + 1 n d ξ ⃗ p ( j ) . So, vectorial line elements appear d s ⃗ ± = ∑ p = 1 4 d z ⃗ p ± with z ⃗ p ± = ϵ ⃗ p ± z p ± .

The vectorial line element in the R ₄ then is d s ⃗ = d s ⃗ + + d s ⃗ − where in general cos ( d s ⃗ + , d s ⃗ − ) ≠ 0 holds.

Thereby the metric for z ⃗ ± = ( z ⃗ ± ) * in the R ₄ becomes d s 2 = d s + 2 + 2 d s ⃗ + d s ⃗ − + d s − 2 , or with the definition of total differentials d z p ± = ∂ z p ± ∂ x i d x i

(72) d s + 2 = ∑ p , q = 1 4 ϵ ⃗ p + ϵ ⃗ q + ∂ z p + ∂ x i ∂ z q + ∂ x k d x i d x k = g i k ( 1 ) d x i d x k , 2 d s ⃗ + d s ⃗ − = 2 ∑ p , q = 1 4 ϵ ⃗ p + ϵ ⃗ q − ∂ z p + ∂ x i ∂ z q − ∂ x k d x i d x k = g i k ( 2 ) d x i d x k , d s − 2 = ∑ p , q = 1 4 ϵ ⃗ p − ϵ ⃗ q − ∂ z p − ∂ x i ∂ z q − ∂ x k d x i d x k = g i k ( 3 ) d x i d x k o r d s 2 = ( g i k ( 1 ) + g i k ( 2 ) + g i k ( 3 ) ) d x i d x k = g i k d x i d x k .

The coefficients of this homogeneous quadratic differential form g i k ( β ) or g _ik are tensor components, namely in relation to the invariance against the global Poincare group. They are field functions of the x _k of an empty R ₄. g _ik(x ₁, x ₂, x ₃, x ₄) is the field of the fundamental metric tensor which designates the allowed event structures of the R ₄ in an invariant form which correspond to that material structure phenomenologically represented by the T _ik ≠ T _ki.

The symmetry of g i k ( 1 ) = g k i ( 1 ) and g i k ( 3 ) = g k i ( 3 ) is immediately evident, while g i k ( 2 ) ≠ g k i ( 2 ) proves to be asymmetric so that g _ik ≠ g _ki overall is asymmetric as well.

So, in the R ₄ a non-hermitian fundamental metric tensor of a general Cartan geometry holds as a consequence of a geometrisation of general interaction fields of matter. Because of their non-hermiticity and being tensors, T _ik and g _ik each can be split into a hermitan (‘+’) and anti-hermitian (‘−’) part, T i k = T i k + + T i k − , g i k = g i k + + g i k − where the index ± of course no longer relates to the z p ± , but to the sign of the hermitian conjugation.^[41]

Appendix B: Derivation of equation (17) according to Heim [43]

Subsequently we reproduce the content of pages 37–39 of chapter II.1 in [43]:

As stated in Subsection 3.3, from the discretisation of the structure field follows that the spacetime R ₄ must be considered as a medium with a Hilbert functional space, i.e. a convergent state function (field) ϕ k m i of the metric state of spacetime must exist and a hermitian state operator C _p acting on ϕ k m i in such a manner that an equivalent to the metric structure term arises C p ϕ k m i → C p Γ k m i = R kmp i . This finding shall be derived hereinafter.

While the Γ k m i do not have to be convergent, for the ϕ k m i convergence can be achieved so that a spacetime integral of the form is J k m i = ∫ Ω ϕ k m i ϕ m k i * d Ω < ∞ , which allows the normalisation J k m i = 1 . But this convergence in general gets lost in the transition ϕ k m i → Γ k m i into the macroscopic continuum. So, the convergent ϕ k m i describe the metric but non-hermitian state of the R ₄ in the microscopic realm and continue according to the principle of correspondence as not necessarily convergent Γ k m i into the macroscopic continuum of the R ₄. Therefore, in the microscopic R ₄ there must be 1 ≤ p ≤ 4 hermitian functional operators C _p which separately act on the ϕ k m i so that for each index p holds

(73) ∫ Ω ( ( ϕ k m ( p ) ) × C ( p ) ϕ k m ( p ) − ϕ k m ( p ) ( C ( p ) ϕ k m ( p ) ) × ) d Ω = 0 .

The functional operators act on the ϕ k m p as C p ϕ k m p so that in the transition into the macroscopic continuum, beside ϕ k m i → Γ k m i , the action of the operators gives C p ϕ k m p → R k m , the metric structure tensor. This transition means at the same time that the discontinuity η _ik from (15) goes over to the steady function w η i k → T i k − 1 2 g i k T (up to a proportionality factor). Then R _ik ∼ wη _ik follows, as R i k = α ( T i k − 1 2 g i k T ) = α W i k . On the other hand C p ϕ k m p → R k m holds, as stated above, so C p ϕ k m p → α W k m can be set.

Now αW _km can be constituted as a sum of 4 parts (correspondent to the summation over p) α W k m = ∑ j = 1 4 G ( j ) k m which differ in their type (j). Ordered in a way so that p = j, we get C ( p ) ϕ k m ( p ) → G ( p ) k m . Due to the quantum principle c. the G ₍ _p ₎ _km can be described by state operators of a state space (as a subspace of the abstract function space) in the same way as the ϕ k m i whose state functions ψ can be interpreted as probabilities (of future possibilities). The respective hermitian linear operators of this state space be H k m ( p ) and L k m ( p ) with the eigenvalues h k m ( p ) = h k m ( p ) * and l k m ( p ) = l k m ( p ) * and the state function ψ, for which the normalisation ∫ψψ*dΩ = 1 holds in terms of the probability interpretation of quantum theory. Therefore, when transferring to the microscopic realm, G ( p ) k m ψ → H k m ( p ) ψ = h k m ( p ) ψ and ϕ k m ( p ) ψ → L k m ( p ) ψ = l k m ( p ) ψ is to be set.

In contrast, in the macroscopic area x ̈ i = − Γ k m i x ̇ k x ̇ m holds and due to Γ k m i → ϕ k m i also x ̈ i → − ϕ k m i x ̇ k x ̇ m . Integrals of the form ∫ x ̈ i d x i are always proportional to energies which holds microscopically, too. Also, a metric structuring of the R ₄ will require energetic effort so that Heim heuristically conceives H k m ( p ) ∼ L k m ( p ) in terms of H k m ( p ) = λ ( p ) ( k , m ) L k m ( p ) . With

(74) H k m ( p ) ψ = λ ( p ) ( k , m ) L k m ( p ) ψ = λ ( p ) ( k , m ) l k m ( p ) ψ one gets 0 = ∫ ( ψ * H k m ( p ) ψ − ψ ( H k m ( p ) ψ ) * ) d Ω = ( λ ( p ) ( k , m ) l k m ( p ) − ( λ ( p ) ( k , m ) l k m ( p ) ) * ) ∫ ψ ψ * d Ω

which because of ∫ψψ*dΩ = 1 and l k m ( p ) = l k m ( p ) * can only be fulfilled by λ ( p ) ( k , m ) = ( λ ( p ) ( k , m ) ) * . So, the λ _p obviously have the property of eigenvalues, too. For the transition of the G ₍ _p ₎ _km into the microscopic realm therefore holds

(75) G ( p ) k m ψ → H k m ( p ) ψ = λ ( p ) ( k , m ) L k m ( p ) ψ = λ ( p ) ( k , m ) ϕ k m ( p ) ψ , t h u s G ( p ) k m → λ ( p ) ( k , m ) ϕ k m ( p ) .

On the other hand the transition G ( p ) k m → C ( p ) ϕ k m ( p ) holds (see above), which provides C ( p ) ϕ k m ( p ) = λ ( p ) ( k , m ) ϕ k m ( p ) , i.e. Eq. (17).

In this system, the λ p = λ p * ≠ 0 are eigenvalues which build discrete point spectra. The indexing (k, m) at the eigenvalue symbol λ _p(k, m) refers to the associated structure (state) function ϕ k m p . So, C ( p ) ϕ k m ( p ) = λ ( p ) ( k , m ) ϕ k m ( p ) indeed fulfils the empirical quantum principle c., but, as stated in Subsection 3.3, the ϕ k m p cannot be interpreted as probability functions, since the operators C _p act (due to C p ϕ k m p → R k m ) as functional operators in terms of non-linear connections so that the solutions (as metric structure functions of the R ₄) do not superpose additively, which is the precondition for the probability interpretation in quantum mechanics.

Appendix C: Dimensions and symmetries

C.1 Heim’s dimension formula

Here we summarise the results of [43], chapter II.1, pages 45–46 and 48:

Heim derives his dimension formula in the following way: From relation (20) for the eigenvalues λ _m(k, m) = λ _m(m, k) = 0 follows that 2n ² values are zero, n being the dimension of the considered space wherein n values are double counted. So 2n ² − n different are zero which have to be subtracted from the n ³ eigenvalues λ _p(k, m) in total. Thus the number of non-empty values amounts to z = n ³ − 2n ² + n = n(n ² − 2n + 1) = n(n − 1)².

Next the further diminution of the non-zero eigenvalues by the property λ _p(m, m) = 0, described in footnote 13 of Subsection 3.3, is taken into account, which means a reduction of n ² − n (no redundant counting of the λ _m(m, m)) so that the overall number of non-zero values becomes

(76) r = z − ( n 2 − n ) = n ( n − 1 ) 2 − n 2 + n = n ( n − 1 ) ( n − 2 ) = 6 n 3 .

To find a mapping R _n → R _N with N ≥ n ≥ 0, r must be equaled to a quantity N ² − αN, where α = const > 1 is a still variable factor. For the resulting quadratic equation N ² − αN = r one gets the solutions N 1,2 = α / 2 ± r + α 2 / 4 and can set α = 2 to ensure a whole-number result for N. Thus

(77) N = 1 ± r + 1 , r = n ( n − 1 ) ( n − 2 ) , 0 ≤ n ≤ N .

In this dimensional law the number r + 1 must be a square number in order to obtain an integer N. For the R ₄, i.e. n = 4, we get r + 1 = 25 and for n = 6, r + 1 = 121 which provides N(n = 4) = 6 and N(n = 6) = 12. Therefore, the R ₄, according to R ₄ → R ₆ → R ₁₂, can only be the subspace of a R ₆ or R ₁₂ while all other possibilities must be excluded. Heim first of all chose and analysed the lower dimensional case, the R ₆, and founded his theory, as elaborated in [43, 44], on it.

C.2 Later expansion of the model space

Later Heim and W. Dröscher also considered the R ₁₂ space and came to the conclusion that the additional 6 coordinates x ₇ − x ₁₂, which do not contain or transport energy (this is restricted to the first 6 coordinates x ₁ − x ₆), can be grouped into two further subspaces, I ₂ of the coordinates x ₇ and x ₈ and G ₄, containing the coordinates x ₉ − x ₁₂. In a longer analysis, published in an additional chapter IV 5. “Background and Sources of the Quantum Principle” (translation by the author) in [43], they concluded that the I ₂ is to be considered as subspace of two dimensions containing information. The remaining subspace G ₄ can only be assigned with a length, no more can be said about it. In their concept, the I ₂ subspace causes that observables in the R ₄ can only detected with uncertainty and probabilities. They speculate that the quantum theory could be interpreted in a way that its sources lie in the G ₄ with timeless structures which become effective via the mapping chain

(78) G 4 ( x 9 , … , x 12 ) → I 2 ( x 7 , x 8 ) → ( x 5 , x 6 ) → x 4 → R 3

so that quantum structures, changing in time, manifest in the physical R ₃ space of the universe. This highly abstract model is based on formal and physical considerations about the dimensions space, including the cosmological part of Heim’s theory, and has been continued in Heim’s third book [71] (with the assistance of W. Dröscher).

In the eyes of the author it is not a priori clear that the extended model of the R ₁₂ or the R ₈ (R ₆ + I ₂(x ₇, x ₈)), which Dröscher/Häuser use in their ‘Extended Heim Theory’ ([45], [46], [47], [48], [49]), is better capable to explain the existence of the quantum world than the R ₆. As stated in Subsection 3.3, already the R ₆ (via the trans-coordinates x ₅, x ₆) leads to a world with an open future and together with the existence of binary alternatives to the abstract quantum theory (see [54] and the further arguments in C.3).

Another question is whether the ‘Extended Heim Theory’ will be able to provide more specific answers with regard to the derivation of the known interactions. With the 8-dimensional EHT Dröscher/Häuser presented an approach which provides 12 [45], respectively, 15 [47] hermetry forms and thus a much bigger set of partial geometries than the original Heim theory. They assign the combinations of 4 subspaces (the three κ ₍ _i ₎ of Subsection 3.4.3 plus a fourth subspace (κ ₍₄₎) of the coordinates x ₇, x ₈) to matter particles, respectively, interaction bosons and suggest that the transformation from the partial geometries to Lagrangians could be accomplished by the non-linear sigma model [72]. However, as far as we know, they have not yet published concrete derivations on this basis, but refer to the known Lagrangians of the gauge theories of the SM [49]. To map to the known forces still seems to be easiest for the electromagnetic interaction. Here the metric can be well approximated by g i k = g i k ( 0 ) + h i k , where g i k ( 0 ) is the flat metric of the Euclidean 4-dimensional space and the h _ik are small quantities whose products are negligible. The derivation in [46] gives a result containing the expected terms with the vector and scalar potential of EM, but also an additional tensor potential which is not present in classical electrodynamics. It remains unclear, how a mapping can be derived for the strong interaction where a similar approximation does not seem to be reasonable. Currently neither Heim’s theory nor the EHT can present an elaborated mapping to the weak and the strong interaction.

C.3 Heim space and gauge symmetries

(Consideration of the author, not by Heim)

The existence of binary alternatives can be modelled mathematically easiest with the SU(2) group. The relevance of this group and symmetry in the physical R ₄ can immediately be seen from the symmetry of the Lorentz group, valid in the Minkowskian R ₄, which is locally isomorphic to a SU(2) ⊗ SU(2) space with 3 generators of spatial rotations and 3 of Lorentz boosts.^[42]

In the 6-dimensional case of the Heim space the symmetry of the R ₆, defined by (22), should be locally isomorphic to SU(2) ⊗ SU(2) ⊗ U(1) ⊗ U(1) if the dimensions x ₅ and x ₆ are distinguishable from each other (which they are in Heim’s approach) and follow a U(1) symmetry, i.e. can be rotated in the complex plane, but without a change of the length (radius).

We compare this overall symmetry to that of the SU(3) which is an eight-dimensional compact group. A parameterisation of the SU(3) can be obtained which matches a duplicated structure of the SU(2) ⊗ U(1) groups, see [73, 74]. Such a structure is called Cartan decomposition:

(79) D ( 3 ) ( α , β , γ , θ , a , b , c , ϕ ) = D ( 2 ) ( α , β , γ ) e i λ 5 θ D ( 2 ) × ( a , b , c ) e i λ 8 ϕ

D ⁽³⁾ denotes an arbitrary element of SU(3), and D ⁽²⁾ is an arbitrary element of SU(2) as a subset of SU(3). The λ _i are 3 × 3-matrices, therefore also D ⁽³⁾ and D ⁽²⁾ are 3 × 3-matrices. The two exponential functions e i λ 5 θ and e i λ 8 ϕ describe the two U(1) manifolds as subgroup and coset in SU(3) [73].

Thus, we have shown the local isomorphism of the 6-dimensional Heim space to a duplicated structure of the SU(2) ⊗ U(1) and, via these, to the SU(3) if the “trans”-dimensions x ₅, x ₆ follow a U(1) symmetry. This finding can be used to conclude that the 6-dimensional Heim space carries a symmetry which contains a SU(2) ⊗ U(1) as a subset, i.e. the gauge symmetry of the electro-weak interaction, and furthermore the SU(3), the gauge symmetry of the strong interaction (according to the QCD).

In a similar way, Görnitz and Schomäcker deduced the gauge groups U(1), SU(2) and SU(3) as structures of these interactions [73]. But their starting point was the abstract quantum space which, according to [54, 73] (and citations therein), carries a U(1) ⊗ SU(2) symmetry. The electro-weak interaction, acting on real particles/quanta, directly can be derived from this. Concerning the strong interaction, they argue [73] that it only acts on “virtual” quanta, the quarks and gluons, never detected as free particles/quanta, and draw the analogy of the transition from real to complex functions in quantum theory, a “duplication” of real functions, to conclude that a duplication of the U(1) ⊗ SU(2) structure is necessary to describe these “virtual” quanta and their interaction, providing the SU(3) (in an analogous way as above).

This derivation now suggests a further connection, viz. that of the 6-dimensional Heim space to the space of abstract quantum theory. A duplication of the symmetry group of the quantum space seems to lead to Heim space – which could mean that the 6-dimensional Heim space includes quantum theory and that it doubles already its space and so accounts for interaction which always “implies a division into separate spaces, …since ultimately the term ‘interaction’ is useful only for things separated from each other” (to quote [73]).

Appendix D: Further contractions of R kmp i

We recap the content of pages 66–67 in chapter II.3 of [43]:

Beside the well-known contractions of the Riemannian curvature tensor, the Ricci tensor R k m = R kmp p and the curvature scalar R = R k k , there is a further possibility to build a trace with i = k,

(80) R kmp k = Γ k p , m k − Γ k m , p k + Γ m s k Γ k p s − Γ p s k Γ k m s ≔ B m p = − B p m

with B _mm = 0. Refering to Eqs. (26)–(28), this antisymmetric tensor arises from the trace

(81) ( C Tr { } ̂ ) m p → B m p o r ( λ ⃗ × Tr { } ̂ ) m p → B m p .

Because of B _mm = 0 and B m p → λ p ( k , m ) k k m also λ m ( k , m ) k k m = 0 holds and thus Tr ( λ ⃗ × Tr { } ̂ ) = λ ⃗ Tr { } ̂ = 0 in Heim’s compact notation. In the hermitian R ₆ this can generally only be met by cos ( λ ⃗ , Tr { } ̂ ) = 0 , i.e. by the orthogonality λ ⃗ ⊥ Tr { } ̂ . Due to the transition Tr { } ̂ m = k k m → Γ k m k in the macroscopic realm this orthogonality can be further interpreted. With the theorem k k m = 1 2 ∂ ∂ x m ln g = ∂ ∂ x m ln w (see [52]), g being the determinant of the metric tensor in the R ₆ and w = − g (compare Subsection 3.3), we can write

(82) λ ⃗ ⊥ Tr { } ̂ → ∇ 6 ln w .

This relation states that the λ ⃗ are always perpendicular to the gradient of the logarithm of the functional determinant w in the R ₆.

Heim additionally considers the contraction R kmp m of the Riemannian curvature tensor, yielding the negative Ricci tensor R kmp m = − R k p . With this contraction another transition, beside the one of Eq. (26) et sqq., C p p k m = λ p ( k , m ) p k m → R k m , can be achieved:

(83) C p m k m = λ p ( k , m ) m k m → − R k p

Heim speculates that this further relation could have a physical meaning, too, and interrelates it to a virtual spectrum of discrete structure levels with a curvature measure (condensation) of opposite sign, instead of the real spectrum, represented by (26) et sqq., with a positive curvature measure (condensation). If this corresponded to the microscopic properties of the R ₆, then real and virtual states of structure levels could interchange as temporal oscillations and establish a dynamic equilibrium of per se non-stationary microscopic structure fluxes (of condensation) in case of stability. Heim relates this possible picture to the perpetual exchange mechanism, which he calls condensation flux, referred to in Subsection 3.5.3. He further speculates whether notions like vacuum polarisation, virtual terms and the empirical Casimir effect could be interpreted by these fluctuations in the microscopic realm.

Appendix E: Structures and metric tensors

E.1 Partial structures from the phenomenology of the R ₄

An analysis by Heim in [43], pages 69–74, of the structure of the energy momentum density tensor T _ik in the R ₆ will give us some insight how T _ik can be composed of the empirical physical fields in the macroscopic realm:

The canonical energy density tensors in the R ₄ can always be written as iterations of field tensors. So, it is obvious to also set T _ik as the iteration of a unified field tensor M _ik in the R ₆ in the macroscopic realm, T i k = ∑ ν = 1 6 M i ν M ν k , and to determine the M _ik from the phenomenologically known T _ik. As T i k = T k i * , it can only be M i k = ± M k i * . In the empirical domain of EM (d1) M is given by the field tensor F ( E ⃗ , H ⃗ ) = − F × with F _mm = 0 in the R ₄. On the other hand, a photon (as gauge particle of EM), although with zero rest mass and imponderable, must be attached with a field mass through its energy. Therefore, the spatial gravitational vector fields G ⃗ and μ ⃗ must exist for a photon, too, and in a form that the spacetime segment of M, M ₍₄₎ → F for G ⃗ → 0 ⃗ and μ ⃗ → 0 ⃗ . From this, it could be concluded that clearly M = −M ^× holds with M _mm = 0. The 6 2 = 15 components of the uniform field tensor M of a photon in the R ₆ are linearly composed of the spatial vector fields E ⃗ , H ⃗ , G ⃗ and μ ⃗ , according to the empirical principles d1 and d2. Here, the field energy=mass of the EM field is to be considered as the source of G ⃗ and μ ⃗ . But field source and field are always a unit so that an additional field K ⃗ must be conceived which accounts for the coupling between the EM and the gravitational field structure. So, M is composed of 5 spatial vector fields whose 3 × 5 components can fill the 15 components of the R ₆ tensor. These 5 phenomenological spatial fields which define the M i k = − M k i * with M _kk = 0 build three phenomenological groups of phenomena, namely ( E ⃗ , H ⃗ ) , ( G ⃗ , μ ⃗ ) and K ⃗ .^[43]

E.2 Composition metric tensors per Hermetry

Referring to chapter VI.1 of [44], pages 80–81, we state the concrete form of the g ^(x) which correlate the partial metric structures per Hermetry, and start with the form d in which all partial structures are active, i.e. “condensed”:

(84) g ( d ) = g ( 11 ) g ( 12 ) g ( 13 ) g ( 21 ) g ( 22 ) g ( 23 ) g ( 31 ) g ( 32 ) g ( 33 )

The columns and rows represent the structure units indexed with μ, ν in (38). On this “super” tensor so-called sieve operators S[n] can act which make the structure units in (84) Euclidian:

(85) g ( c ) = S ( 2 ) g = g ( 11 ) κ ( 1 ) g ( 13 ) κ ( 1 ) E κ ( 3 ) g ( 31 ) κ ( 3 ) g ( 33 )

(86) g ( b ) = S ( 3 ) g = g ( 11 ) g ( 12 ) κ ( 1 ) g ( 21 ) g ( 22 ) κ ( 2 ) κ ( 1 ) κ ( 2 ) E

(87) g ( a ) = S ( 2,3 ) g = g ( 11 ) κ ( 1 ) κ ( 1 ) κ ( 1 ) E E κ ( 1 ) E E

Appendix F: Solution of the hermetric fundamental equation (30) and further relations

We refer to the calculations of chapter IV.2, pages 196–205 and 211 in [43], but apply directly normal differentiation and integration which is valid for Heim’s third area of validity, considered here.

F.1 Solution of equation (30)

Based on the relations for the coefficients a _lm of subsection 3.4.4 and on { } ̂ = { } ̂ × being hermitian, it directly follows i m l = i l m and thus the system of symmetry relations a m l i m m − a l m i l l = 0 or as proportionality i m m = a l m a m l i l l . So we have two systems of proportionalities at hand, the last one and i m l = a m l i m m , already introduced above in the cited subsection. We can use them to substitute in the general equation (30) so that uniformly only the covariant components k, l are related. The mixed-variant sum over s in the quadratic term runs in 1 ≤ s ≤ q, as only in this interval zero factors i s ̃ p = 0 do not occur. So, we get

(88) i k m = a k m i k k = a k m a k l i k l a n d i l s = a l s i l l = a l s a l k i l k = a l s a l k i k l , t h u s i l s s k m = a l s a l k a k m a k l i k l s k l . Similarly follows i m s = a m s i m m = a m s a m k i k m = a m s a m k a k m a k l i k l .

Inserting into (30) gives

(89) a k m a k l ∂ l − ∂ m i k l + a l s a l k a k m a k l − a m s a m k a k m a k l × i k l s k l = λ m ( k , l ) i k l .

With the abbreviations a k m a k l = a m ( k , l ) and a l s a l k a k m a k l − a m s a m k a k m a k l = b m s ( k , l ) Eq. (89) reads

(90) ( a m ( k , l ) ∂ l − ∂ m ) i k l + b m s ( k , l ) i k l s k l = λ m ( k , l ) i k l .

If we now sum over the hermetric index m and use the further abbreviations a ( k , l ) = ∑ m = 1 q a m ( k , l ) , b s ( k , l ) = ∑ m = 1 q b m s ( k , l ) and λ ( k , l ) = ∑ m = 1 q λ m ( k , l ) , then we get with a ( k , l ) ∂ l − ∑ m = 1 q ∂ m = ( a ( k , l ) − 1 ) ∂ l − ∑ m ≠ l ∂ m

(91) ( a ( k , l ) − 1 ) ∂ l − ∑ m ≠ l ∂ m i k l + b s ( k , l ) i k l s k l = λ ( k , l ) i k l .

This equation can be multiplied with the coefficient b _i(k, l) and summed over 1 ≤ i ≤ q, leading to covariant selectors (tensors) ϕ k l = b i ( k , l ) i k l . As further b i ( k , l ) b s ( k , l ) i k l s k l = ( b i ( k , l ) i k l ) ( b s ( k , l ) s k l ) = ϕ k l 2 holds, Eq. (91) becomes

(92) ( a ( k , l ) − 1 ) ∂ l − ∑ m ≠ l ∂ m ϕ k l + ϕ k l 2 = λ ( k , l ) ϕ k l .

With the normalised orthogonal system of the q hermetric coordinates e ⃗ i e ⃗ k = δ i k and a ⃗ k l = e ⃗ l ( a ( k , l ) − 1 ) − ∑ m ≠ l e ⃗ m Eq. (92) can be written as

(93) a ⃗ k l ∇ ⃗ q ϕ k l = λ ( k , l ) ϕ k l − ϕ k l 2 = 1 4 λ 2 ( k , l ) 1 − 2 ϕ k l λ ( k , l ) − 1 2

with ∇ ⃗ q being the gradient in the q coordinates and having reformed the r.h.s. so that a variable u = ± ( 2 ϕ k l λ ( k , l ) − 1 ) can be introduced. If we furthermore apply an inverse of the ( a ⃗ k l ) − 1 = a ⃗ k l − 1 and multiply with the differential d x ⃗ = ∑ i = 1 q e ⃗ i d x i we get

(94) d x ⃗ ∇ ⃗ q u = ± 1 2 q λ ( k , l ) d x ⃗ a ⃗ k l − 1 ( 1 − u 2 ) = ± 1 2 q λ ( k , l ) ( 1 − u 2 ) ∑ i = 1 q d x i a k l − 1 i = ± 1 2 λ ( k , l ) ( 1 − u 2 ) d N k l

with d N k l = 1 q ∑ i = 1 q d x i a k l − 1 i .^[44] Having a total differential on the l.h.s. we can transform to

d u 1 − u 2 = ± 1 2 λ ( k , l ) d N k l ,

or with the primitive on the l.h.s.

(95) 1 2 d ln 1 + u 1 − u = ± 1 2 λ ( k , l ) d N k l .

Integration gives

(96) ln 1 + u 1 − u − ln C k l = ± λ ( k , l ) N k l o r 1 + u 1 − u = C k l e ± λ ( k , l ) N k l

with the integration constant C _kl. With the definition of ψ _kl = ϕ _kl/λ(k, l) and u as above we can write for the negative branch of u

(97) u = − ( 2 ψ k l − 1 ) , which gives 1 + u 1 − u = 1 − 2 ψ k l + 1 1 + 2 ψ k l − 1 = 1 − ψ k l ψ k l = 1 ψ k l − 1 = C k l e − λ ( k , l ) N k l .

This finally gives the result

(98) ψ k l = ( 1 + C k l e − λ ( k , l ) N k l ) − 1

which is already close to Eq. (32), but we still have to evaluate the N _kl and consider the positive branch of u. After an analogous calculation as (97) this branch gives the same result as (98), but only with an inverse constant 1/C _kl in front of the e-function. With a value of C _kl = −1, found in Appendix G, this does not make a difference to (98). For the N _kl holds N k l = 1 q x ⃗ a ⃗ k l − 1 = x 1 q e ⃗ x a ⃗ k l − 1 with x being the absolute value of the q-dimensional “position” vector x ⃗ as in Subsection 3.4.4 ((32) and following). Note that the unit vector e ⃗ x = ( r ⃗ + i ξ ⃗ ) / r 2 − ξ 2 is in general a complex-valued quantity. With the aggregations λ ⃗ k l = 1 q λ ( k , l ) a ⃗ k l − 1 , respectively, ϵ k l = 1 q e ⃗ x a ⃗ k l − 1 = 1 q e ⃗ x ( e ⃗ l ( a ( k , l ) − 1 ) − 1 − ∑ m ≠ l e ⃗ m ) , which contains the angles of the 6-dimensional x ⃗ in the unit vector e ⃗ x , we get

(99) N k l = ϵ k l x , and so ψ k l = 1 + C k l e − λ ⃗ k l x ⃗ − 1 = ( 1 + C k l e − λ ( k , l ) ϵ k l x ) − 1

which provides the result stated in Eq. (32) when denoting λ(k, l)ϵ _kl = λ _kl. We shortly want to note two important properties of the solution:

The extrema of (99) can immediately be derived from Eq. (92) as if the partial derivations become ∂ϕ _kl = 0, then ϕ k l 2 = λ ( k , l ) ϕ k l holds which gives the solutions ϕ _kl = 0 or ϕ _kl = λ(k, l), or expressed with ψ _kl, ψ _kl = 0 or ψ _kl = 1.

The second point is the question whether the general covariance of Eq. (30) and its solution can be shown (which was not explicitly considered by Heim). For this we realise again that the solution of Eq. (92) solves (30) and that we get the expression A(k, l) = λ(k,l)² ψ _kl on both sides of (92) when inserting the solution (99), respectively (98). This expression depends on the coordinates x _i only in scalar form which means that it transforms as ∂ A ∂ x i ′ = ∂ x j ∂ x i ′ ∂ A ∂ x j due to the derivation chain rule, i.e. transforms covariant.

F.2 Calculation of the metric tensor

Next, the metric tensor in the R ₆ shall be calculated from the result ψ _kl. To achieve this, Heim derives from ψ k l = b i ( k , l ) λ ( k , l ) i k l (32) that

(100) ψ k l = ∑ s = 1 q e ⃗ s b s ( k , l ) λ ( k , l ) e ⃗ i i k l = b ⃗ ( k , l ) λ ( k , l ) e ⃗ i i k l

and thus, by summing over the indices k, l, he defines a component

(101) ψ i = ∑ k , l = 1 q λ ( k , l ) b i ( k , l ) ψ k l = ∑ k , l = 1 q i k l .

With the abbreviations ∑ k = 1 q g i k = g i and ∑ i = 1 q g i = g ^[45] and with the well-known relation between the Christoffel symbols and g _ik (see e.g. [52]) we get^[46]

(102) ψ i = g i s ∑ k , l = 1 q s k l = 1 2 g i s ∑ k , l = 1 q ( ∂ k g s l + ∂ l g k s − ∂ s g k l ) = 1 2 g i s ∑ k = 1 q ∂ k g s + ∑ l = 1 q ∂ l g s − ∂ s g = 1 2 g i s 2 ∑ k = 1 q ∂ k g s − ∂ s g .

Summation over i gives (with ∂ ̲ = ∑ k ∂ k )

(103) ∑ i = 1 q 2 ψ i g i = ∑ i , s = 1 q g i s ( 2 ∂ ̲ g s − ∂ s g ) ∑ l = 1 q g i l = ∑ l , s = 1 q δ s l ( 2 ∂ ̲ g s − ∂ s g ) = ∑ s , k = 1 q ( 2 ∂ k g s − ∂ s g k ) = ∑ s , k = 1 q ∂ k g s = ∂ ̲ g = ∑ i = 1 q ∂ ̲ g i , o r ∑ i = 1 q ( 2 ψ i g i − ∂ ̲ g i ) = 0 .

This leads to

(104) 2 ψ i = ∑ k = 1 q ∂ k g i g i = ∑ k = 1 q e ⃗ k ∇ ⃗ g i g i = Z ⃗ ∇ ⃗ ln g i ⇒ ∇ ⃗ ln g i = 2 q Z ⃗ − 1 ψ i

with Z ⃗ = Z ⃗ − 1 = ∑ k e ⃗ k . Here we have made a similar step as in (94), i.e. fragmented into q single equations through the gradient, thus the factor 1/q appears on the r.h.s. Multiplying with d x ⃗ and defining d μ = d x ⃗ Z ⃗ / q = ∑ k d x k / q gives

(105) d x ⃗ ∇ ⃗ ln g i = 2 d μ ψ i , d ln g i = 2 d μ ψ i , and after integration g i = A i exp 2 ∫ d μ ψ i , the A i being constants. The exponent is 2 ∫ d μ ψ i = 2 ∫ d μ ∑ k , l = 1 q i k l = 2 ∑ k , l = 1 q λ ( k , l ) b i ( k , l ) ∫ d μ ψ k l .

The integral in the exponent can be further evaluated as follows, at the same time using the abbreviation λ ⃗ k l = 1 q λ ( k , l ) a ⃗ k l − 1 :

(106) ∫ d μ ψ k l = ∫ ∑ s = 1 q d x s 1 q ψ k l = 1 q ∑ s = 1 q ∫ 1 + C k l e − λ ⃗ k l x ⃗ − 1 d x s , ∫ 1 + C k l e − λ ⃗ k l x ⃗ − 1 d x s = ∫ 1 + C k l e − ∑ p λ p x p − 1 d x s = ∫ 1 + C k l e − ∑ p ≠ s λ p x p − λ s x s − 1 d x s = 1 λ s ∫ ( 1 + e − η ) − 1 d η = 1 λ s ∫ e η e η + 1 d η = 1 λ s ∫ d ( e η + 1 ) e η + 1 = 1 λ s ln ( e η + 1 ) + c = 1 λ s ln B k l e λ ⃗ k l x ⃗ C k l + 1

Here we used the substitution η = −ln C _kl + ∑ _p≠s λ _p x _p + λ _s x _s, introduced the integration constant B _kl and partially suppressed the indices k, l. The integral ∫dμ ψ _i now gives

(107) ∫ d μ ψ i = ∑ k , l = 1 q λ ( k , l ) q b i ( k , l ) ∑ s = 1 q 1 λ s ln B k l e λ ⃗ k l x ⃗ C k l + 1

where the sum over the 1/λ _s and the argument of the ln can be further evaluated as

(108) λ ⃗ k l = λ ( k , l ) q 1 a ( k , l ) − 1 , − 1 , … , − 1 , ⇒ ∑ s = 1 q 1 λ s = q λ ( k , l ) ( a ( k , l ) − 1 − ( q − 1 ) ) = q λ ( k , l ) ( a ( k , l ) − q ) , C k l e − λ ⃗ k l x ⃗ = 1 ψ k l − 1 , ⇒ e λ ⃗ k l x ⃗ C k l + 1 = ( 1 − ψ k l ) − 1 ,

leading to

(109) ∫ d μ ψ i = ∑ k , l = 1 q a ( k , l ) − q b i ( k , l ) ln B k l 1 − ψ k l .

With c _i(k, l) = (a(k, l) − q)/b _i(k, l) the contracted metric tensor becomes^[47]

(110) ∑ k = 1 q g i k = g i = A i e 2 ∑ k , l = 1 q c i ( k , l ) ln B k l 1 − ψ k l = A i ∏ k , l = 1 q 1 − ψ k l B k l − 2 c i ( k , l ) ,

or with the explicit form of the ψ _kl

(111) ∑ k = 1 q g i k = A i ∏ k , l = 1 q B k l e 1 q λ ( k , l ) a ⃗ k l − 1 x ⃗ C k l + 1 2 c i ( k , l ) .

F.3 Relation for a condenser anti-hermetric in the contravariant index

Finally we want to give a relation which Heim derived for a condenser (Christoffel symbol) which is anti-hermetric in the contravariant index. According to the anti-hermetry conditions as given at the beginning of subsection 3.4.4, all condenser components with covariant anti-hermetric indices vanish. This does not hold for a contravariant anti-hermetric index. But due to the cited relations the coefficients b i = r ̃ ( k , l ) are not defined in this case. Nonetheless, a solution for the non-vanishing contravariant anti-hermetric condenser components can be obtained in the following way:

If Eq. (30) is summed over index m, giving (91), and the contravariant index i = r ̃ is anti-hermetric, then due to b s ( k , l ) s k l = ϕ k l = λ ( k , l ) ψ k l (with sum over the hermetric index s) it follows

(112) ( a ( k , l ) − 1 ) ∂ l − ∑ m ≠ l ∂ m r ̃ k l = λ ( k , l ) ( 1 − ψ k l ) r ̃ k l

with ψ _kl being the solution of the hermetric problem according to (32). We use again the vector a ⃗ k l and the transformation to q equations with the gradient

(113) ∇ ⃗ ln r ̃ k l = 1 q λ ( k , l ) a ⃗ k l − 1 ( 1 − ψ k l ) o r d ln r ̃ k l = 1 q λ ( k , l ) a ⃗ k l − 1 ( 1 − ψ k l ) d x ⃗ , t h u s r ̃ k l = B k l r ̃ exp 1 q λ ( k , l ) a ⃗ k l − 1 x ⃗ − ∫ ψ k l d x ⃗ .

Appendix G: Analysis of Hermetry forms a and b

Here we summarise the main content of chapter IV.3, pages 212–225, in [43]:

In case of the form a, which we want to consider first, the general line element x 2 = ∑ i = 1 q x i 2 becomes x 5 2 + x 6 2 = − ζ 2 as x ₅ and x ₆ are imaginary. According to the analysis in Subsection 3.4.4 there must be discrete point spectra λζ for a condensation with a lower limit ɛ > 0 for ζ, as otherwise there would be a continuous spectrum, contrary to the meaning of condensation. As only x ₅ and x ₆ are involved, there are only the condenser components i 55 , i 56 and i 66 which are, as explained in Subsection 3.4.4 and Appendix F, interconnected through proportionalities so that we can define ϕ ̲ i = i 55 , ϕ ̲ i / a = i 56 and ϕ ̲ i b / a = i 66 . If we further set λ ₁ = λ ₆(5, 5) and λ ₂ = λ ₅(6, 6), then the relations a λ 1 ϕ ̲ i = ( ∂ 5 − a ∂ 6 ) ϕ ̲ i − α a ϕ ̲ i ϕ ̲ 6 and b λ 2 ϕ ̲ i = ( ∂ 6 − b ∂ 5 ) ϕ ̲ i − α a ϕ ̲ i ϕ ̲ 5 are obtained with the factor α = ab − 1. Using the abbreviations λ ̲ = a λ 1 + b λ 2 and ϕ ̲ = ϕ ̲ 5 + ϕ ̲ 6 , the in total 4 relations (i = 5, 6) can be added, providing λ ̲ ϕ ̲ = − ( ( b − 1 ) ∂ 5 + ( a − 1 ) ∂ 6 ) ϕ ̲ − α a ϕ ̲ 2 . Now, it can be assumed that ϕ ̲ only depends on the two coordinates in the form ϕ ̲ ( x 5 , x 6 ) = ϕ ̲ ( ξ ) with the new variable ξ, defined via the substitutions x ₅ = x(b − 1), x ₆ = y(a − 1) and d ϕ ̲ d x + d ϕ ̲ d y = d ϕ ̲ d ξ and thus not to be mixed up with ξ ⃗ , as defined in the other chapters of this paper, and with ϕ = α a ϕ ̲ yielding the equation

(114) d ϕ d ξ + ϕ 2 = − λ ̲ ϕ ,

which formally holds for λ = − λ ̲ , too. The equation can easily be solved by separation of variables, leading to the result ϕ = λ ( 1 − 1 C e − λ ξ ) − 1 with C = 1 → C _kl = −1 in Eq. (32), fulfilling the convergence requirements.^[48] As ξ is imaginary, the eigenvalues which provide a maximum of ϕ(= λ) correspond to those of the general problem described in subsection 3.4.4 ([43]). But, as the hermetry form a describes terms in x ₅ and x ₆ outside the R ₄, a physical interpretation seems to be impossible. It is only conceivable that the structure a has an impact on the R ₄ if ξ depends on the coordinates of the R ₄ in any form. This is the case if the structure a fulfils the condition of a null geodesic, because then the R ₄-sector of the fundamental metric tensor is pseudo-Euclidian (due to its anti-hermetry) and the metric quantities of a, for ds ² = 0, depend on the R ₄-coordinates. So, even if a is not directly physically explainable, its effect on the (anti-hermetric) R ₄ can be analysed. Heim therefore undertakes an analysis of the condensers (Christoffel symbols) and the metric tensor components g _ik, i, k = 5, 6, using the algebraic properties g i k = g i k × , d i v 6 g i k = 0 ,^[49] and derives the equation

(115) d 2 ψ d ξ 2 − 1 4 d ψ d ξ 2 + λ 2 d ψ d ξ = 0

with ψ = ln g ₅₅ + ln g ₆₆, giving the solution g 55 g 66 = B ( 1 + A e − λ 2 ξ ) − 4 , A and B being integration constants. Carrying out the step to a null geodesic, this means ds ² = dr ² + dξ ² = 0 with d r 2 = ∑ k = 1 4 d x k 2 or dξ = idr. (115) then becomes an equation in the R ₄ spacetime

(116) d 2 ψ d r 2 − 1 4 d ψ d r 2 + i λ 2 d ψ d r = 0 .

For the further consideration the quadratic term (dψ/dr)² can be neglected (as it describes only interactions between similar null geodesic processes in the R ₄). The remaining equation can be solved by the approximation d 2 / d r 2 → ∑ k = 1 4 ∂ 2 / ∂ x k 2 and by the separation ψ = ϑ(t) w(x _k), 1 ≤ k ≤ 3. After a somewhat longer calculation (see [43], pages 219–222) the following result is obtained with R = ∑ k = 1 3 x k 2 and x ₄ = Ct

(117) w ( R ) ∼ ( 1 + A e 2 β R ) e − i 4 λ R R − β R w i t h β = a − λ R 2 16 , a being the separation constant, ϑ ( t ) ∼ B F ( 1 + F ) e − i λ C t / 4 w i t h F = A e 2 κ t a n d κ = i C λ 2 / 16 + a ,

the remaining constants being constants of integration. As ψ is a state function, it must converge, i.e. lim_t _, _R→∞ ψ = lim_t→∞ ϑ lim_R→∞ w < ∞, which is fulfilled by the spatial function w if Reβ = 0 and Imβ ≥ 0, because only then w becomes a complex periodic function. This provides a constraint for the separation constant a < λ R 2 / 16 . The run of ϑ, however, is determined by C which in the anti-hermetric R ₄ can be C = ic (with c the velocity of light) in the case of an Lorentz invariant Euclidian spacetime R ₋₄ or can be a real value C = ω. The case C = ic gives lim_t→∞ ϑ = ∞ as can be seen from (117), which means that the R ₋₄ cannot be the anti-hermetric spacetime of the a form and the null geodesic process of ψ cannot appear in the R ₄ as an electromagnetic radiation quantum (photon). In the real case C = ω (R ₊₄) ϑ instead becomes a complex oscillation law so that the convergence lim_t→∞ ϑ < ∞ is fulfilled. So, the anti-hermetric realm can only be a R ₊₄ in which only gravitational actions can be present. If the x ₅, x ₆ appear in the connection x 5 2 + x 6 2 = − ζ 2 , and the world lines of the a hermetry form are null geodesics, ψ obviously causes a gravitational disturbance. As the λ are discrete spectra in the a hermetry, too, these advancing gravitational fields must have the character of discrete quantum levels as well, which therefore are identified as gravitons by Heim.

In the Hermetry form b, the coordinates x ₄, x ₅ and x ₆ are hermetric, i.e. all three imaginary coordinates, and this means that an analogous formalism as for the form a exists. So, the respective state function becomes

(118) ψ = ln ( g 44 g 55 g 66 ) = ln B 1 + A e − λ 2 ξ − 4 = ln ⁡ B − 4 ln 1 + A e − λ 2 ξ .

We now calculate^[50] the first and second derivative of ψ, using ξ = x 4 2 + x 5 2 + x 6 2 and d ξ d x 4 = x 4 ξ → 1 , d 2 ξ d x 4 2 = 1 ξ 1 − x 4 2 ξ 2 → 0 for x ₄ → ξ (i.e. x ₅ = x ₆ → 0) and the abbreviations X = A e − λ 2 ξ and f = 1 + X:

(119) ∂ ψ ∂ x 4 = 2 A λ e − λ 2 ξ f − 1 d ξ d x 4 → 2 λ X f − 1 ∂ 2 ψ ∂ x 4 2 = − A λ 2 e − λ 2 ξ f − 2 d ξ d x 4 2 + 2 A λ e − λ 2 ξ f − 1 d 2 ξ d x 4 2 → − λ 2 X f − 2

If we assume X ≪ 1 and therefore expand ψ ≈ − 4 A e − λ 2 ξ = − 4 X linearly, we get f ≈ 1 and the following relation between ψ and its second derivative

(120) ∂ 2 ψ ∂ x 4 2 = − λ 2 X = λ 2 4 ψ .

This relation can now be transferred to the situation in the R ₄ where, because of the anti-hermetric R ₃ coordinates, geodesic null lines exist d s 2 = d r 2 + d x 4 2 = d r 2 − c 2 t 2 = 0 on the light cone. The R ₃ coordinates then occur together with the time coordinate as well-known Lorentz invariant term Δ − 1 c 2 ∂ 2 ∂ t 2 . This leads to the result

(121) Δ − 1 c 2 ∂ 2 ∂ t 2 P = λ 2 4 P

with P = ψ(t)W(x ₁, x ₂, x ₃). If we identify the r.h.s. as a current term,^[51] this differential equation describes the dispersion law of photons in the frame of Quantum Electrodynamics (QED) in which, according to the quantum dualism, the particle character retreats in favour of the wave character for λ ≈ 0. Then Δ P − 1 c 2 ∂ 2 ∂ t 2 P = 0 results approximately. This is the equation of a transversal wave field in the R ₃ which propagates with the velocity of light, identical with the empirical law of electromagnetic waves (principle d1). Therefore, it can be concluded that the imaginary time condensations of the Hermetry form b describe photons, the imponderable quanta of the electromagnetic field ([43]).

Appendix H: Properties and results of the poly-metric geometry

H.1 Properties of g _ik and i k l

In this subsection we present content of chapter VI.2, pages 86–94, of [44]:

The vanishing of a condenser with an anti-hermetric covariant index k ̃ , i k ̃ l ( μ ν ) ( κ λ ) = 0 , means that also the condenser with a completely covariant signature vanishes

(122) i k ̃ l ( p q ) = 1 2 ( ∂ k ̃ g i l ( p q ) + ∂ l g k ̃ i ( p q ) − ∂ i g k ̃ l ( p q ) ) = 0 , t h u s ∂ k ̃ g i l ( p q ) = ∂ i g k ̃ l ( p q ) − ∂ l g k ̃ i ( p q ) ,

leading to results for the hermitian and the anti-hermitian part of the partial metric tensor g ( p q ) = g + ( p q ) + g − ( p q ) :

(123) ∂ k ̃ g + i l ( p q ) = ∂ k ̃ 1 2 ( g i l ( p q ) + g i l ( p q ) × ) = ∂ k ̃ 1 2 ( g i l ( p q ) + g l i ( p q ) * ) = 1 2 ( ∂ i g k ̃ l ( p q ) − ∂ l g k ̃ i ( p q ) + ∂ l g k ̃ i ( p q ) * − ∂ i g k ̃ l ( p q ) * ) = 1 2 ( ∂ i ( g k ̃ l ( p q ) − g k ̃ l ( p q ) * ) − ∂ l ( g k ̃ i ( p q ) − g k ̃ i ( p q ) * ) ) = 0 , ∂ k ̃ g − i l ( p q ) = ∂ k ̃ 1 2 ( g i l ( p q ) − g i l ( p q ) × ) = ∂ k ̃ 1 2 ( g i l ( p q ) − g l i ( p q ) * ) = 1 2 ( ∂ i g k ̃ l ( p q ) − ∂ l g k ̃ i ( p q ) − ∂ l g k ̃ i ( p q ) * + ∂ i g k ̃ l ( p q ) * ) = ∂ i g k ̃ l ( p q ) − ∂ l g k ̃ i ( p q ) ,

as g k l ( p q ) = g k l ( p q ) * is real. This means that g + ( p q ) is constant concerning k ̃ and that g − ( p q ) can be calculated by integration of g k ̃ l ( p q ) over k ̃ , giving a function A l ( p q ) = ∫ g k ̃ l ( p q ) d x k ̃ and thus g − i l ( p q ) = ∂ i A l ( p q ) − ∂ l A i ( p q ) = ( r o t ( x ) A ⃗ ( p q ) ) i l ≔ P i l ( p q ) . So, g − ( p q ) can be expressed through a vector which we identify as a spin field vector and already introduced in Subsection 3.5.1, A ⃗ ( p q ) = ϕ ⃗ ( p q ) .^[52]

Next we can derive a further property of g + ( p q ) with an anti-hermetric coordinate k ̃ : From the defintion of the A l ( p q ) folllows

(124) ∂ l ∫ g k ̃ m ( p q ) d x k ̃ = ∂ l A m ( p q ) ⇒ ∂ l g k ̃ m ( p q ) = ∂ k ̃ ∂ l A m ( p q ) = ∂ l ∂ k ̃ A m ( p q ) o r g k ̃ m ( p q ) = ∂ k ̃ A m ( p q ) and with g k ̃ m ( p q ) = g + k ̃ m ( p q ) + g − k ̃ m ( p q ) = g + k ̃ m ( p q ) + P k ̃ m ( p q ) ⇒ g + k ̃ m ( p q ) + P k ̃ m ( p q ) = ∂ k ̃ ϕ m ( p q ) . W i t h P k ̃ m ( p q ) = ( r o t ( x ) ϕ ⃗ ( p q ) ) k ̃ m we get g + k ̃ m ( p q ) = ∂ k ̃ ϕ m ( p q ) − ( r o t ( x ) ϕ ⃗ ( p q ) ) k ̃ m = ∂ m ϕ k ̃ ( p q ) .

So, we can summarise the results so far as^[53]

(125) g + k ̃ m ( p q ) = const 53 = ∂ m ϕ k ̃ ( p q ) , g − k m ( p q ) = P k m ( p q ) = ( rot ( x ) ϕ ⃗ ( p q ) ) k m .

Now, with g + k ̃ m ( p q ) being hermitian, we can exchange indices and get with (125) ∂ m ϕ k ̃ ( p q ) = g + k ̃ m ( p q ) = g + m k ̃ ( p q ) = ∂ k ̃ ϕ m ( p q ) , thus

(126) ∂ k ̃ ϕ m ( p q ) − ∂ m ϕ k ̃ ( p q ) = ( rot ( x ) ϕ ⃗ ( p q ) ) k ̃ m = P k ̃ m ( p q ) = g − k ̃ m ( p q ) = 0 , ⇒ g + k ̃ m ( p q ) = ∂ k ̃ ϕ m ( p q ) = const.

If we integrate the last relation, we find ϕ m = f ( V r ) x k ̃ , with a function f of V _r, the space of the hermetric coordinates. Doing the same with ∂ m ϕ k ̃ = const , where the constant must be the same as in the previous relation, as both being equal to the hermitian = g + k ̃ m , we get ϕ k ̃ = f ( V r ) x m and, as a generalisation, the expression ϕ l = ∑ m = 1 6 f l m ( V r ) x m . A derivation of this expression with respect to an anti-hermetric coordinate gives a coefficient f l k ̃ , as expected, but a partial derivation to a hermetric coordinate x _k yields not only f _lk, but additionally a second term ∑ m = 1 6 ∂ k ( f l m ) x m , depending also on anti-hermetric coordinates x m = k ̃ . This contradiction to our starting point ∂ k ̃ ϕ m = ∂ m ϕ k ̃ = const with respect to k ̃ can only be solved if the functions f _lm are constants f _lm = a _lm. This leads to

(127) g + k ̃ l = ∂ k ̃ ϕ l = a l k ̃ = a k ̃ l = const

while in the hermetric space V _r always

(128) g + k l = g + k l ( V r ) , b u t g − k l = P k l = ( rot ( x ) ϕ ⃗ ) k l = a k l − a l k = const ≠ 0

holds (we suppressed the indices (pq) of the partial structures in the last expressions), as in this case in general a _kl ≠ a _lk. The obtained relations for the metric tensor in the poly-metric mean that its hermitian part g + ( p q ) is constant in an anti-hermetric sector, but is a structure function in the V _r of the hermetric coordinates. Instead, the anti-hermitian part g − ( p q ) , identified as a spin field tensor P, completely vanishes in the anti-hermetric sector and exists as a constant quantity in the hermetric V _r.

With these results we have derived all relations presented in subsection 3.5.1 in (41) and can conclude from them that the condensers in the poly-metric are hermitian: Their purely covariant form is given as

(129) 2 s k l ( μ ν ) = ∂ k g s l + ∂ l g k s − ∂ s g k l = ∂ k g + s l + ∂ l g + k s − ∂ s g + k l = 2 s k l ( μ ν ) + ,

because always g _−kl = const or g − k ̃ l = 0 holds. But this also means

(130) g ( κ λ ) i s s k l ( μ ν ) = i k l ( μ ν ) + ( κ λ ) o r κ λ μ ν ̂ = κ λ μ ν + ̂ ,

as the covariant symmetry is not effected by an anti-hermitian part in the contravariant signature.

H.2 Solution of equation (43)

We subsequently refer to the content of chapter VI.3, pages 108–113, of [44], but note that we calculate different results than Heim at some important points. Especially the function f(k, l) is not introduced by Heim, i.e. remains equal 1 in his calculation.

As for the λ ̲ m ( l , m ) and λ ̲ l ( m , m ) ^[54] the same identities hold as in the case of the composition field (see Appendix F), we obtain similar relations for the F m l i as for the i m l of the composition field, F k m i = a ̲ m ( k , l ) F k l i as well as F s l i = a ( 1 ) F k l i and F s m i = a ( 2 ) F k l i . This leads to

(131) ( a ̲ m ( k , l ) ∂ l − ∂ m ) F k l i + ( a ( 1 ) s k m − a ( 2 ) s k l ) F k l i = λ ̲ m ( k , l ) F k l i .

Fully analogous proportionalities hold for the components of the so that a ( 1 ) s k m − a ( 2 ) s k l = b ̲ m s ( k , l ) s k l can always be set, which after summing over the hermetric indices 1 ≤ m ≤ q and with the abbreviations a ̲ ( k , l ) = ∑ m = 1 q a ̲ m ( k , l ) , b ̲ s ( k , l ) = ∑ m = 1 q b ̲ m s ( k , l ) and λ ̲ ( k , l ) = ∑ m = 1 q λ ̲ m ( k , l ) provides

(132) ( a ̲ ( k , l ) − 1 ) ∂ l − ∑ m ≠ l ∂ m F k l i + b ̲ s ( k , l ) s k l F k l i = λ ̲ ( k , l ) F k l i .

Note that for simplicity of the notation we have again, as in reference (43), suppressed the indices of the partial structures in the previous expressions, but that in general the underlined terms depend on these indices. Therefore, this will hold for subsequently underlined quantities, too.

With a normalised orthogonal system of the q hermetric coordinates and a ̲ ⃗ = e ⃗ l ( a ̲ ( k , l ) − 1 ) − ∑ m ≠ l e ⃗ m the previous equation can be written as

(133) ∇ ⃗ q F k l i = 1 q a ̲ ⃗ − 1 ( λ ̲ ( k , l ) − b ̲ s ( k , l ) s k l ) F k l i ,

similar as done in Appendix F (94). Also similarly multiplying with the differential d x ⃗ = ∑ i = 1 q e ⃗ i d x i we get

(134) d x ⃗ ∇ ⃗ q F k l i = d F k l i = 1 q ∑ j = 1 q d x j a ̲ j − 1 ( λ ̲ ( k , l ) − b ̲ s ( k , l ) s k l ) F k l i

which can be integrated to

(135) F k l i = A k l i exp 1 q ∫ ∑ j = 1 q d x j a ̲ j − 1 λ ̲ ( k , l ) − b ̲ s ( k , l ) s k l = A k l i exp λ ̲ ⃗ k l x ⃗ − ∫ ∑ j = 1 q d x j c ̲ j s s k l

with the integration constant A k l i and the abbreviations λ ̲ ⃗ k l = 1 q λ ̲ ( k , l ) a ̲ ⃗ − 1 and c ̲ j s = 1 q a ̲ j − 1 b ̲ s ( k , l ) . This result means that the problem of solving the fundamental equation of the poly-metric can be reduced to an integral over the condenser (state function) of the composition field. We therefore insert that result, given in Eq. (32), in the form b s ( k , l ) s k l = b s ( k , l ) s k l = λ ( k , l ) ( 1 − e − 1 q λ ( k , l ) a ⃗ − 1 x ⃗ ) − 1 . To derive a needed expression for a single s k l (instead of the sum over the index s), we again use the fragmentation into q parts on the r.h.s. (as done in references (94), (104) and (133)) and get s k l = λ ( k , l ) q b s ( k , l ) ( 1 − e − 1 q λ ( k , l ) a ⃗ − 1 x ⃗ ) − 1 . The integrand then becomes c ̲ j s s k l = 1 q 2 a ̲ j − 1 b ̲ ⃗ ( k , l ) b ⃗ − 1 ( k , l ) λ ( k , l ) ( 1 − e − 1 q λ ( k , l ) a ⃗ − 1 x ⃗ ) − 1 = ω ̲ j ( k , l ) 1 − e − λ ⃗ k l x ⃗ − 1 , where we have expressed the sum over index s as the scalar product between b ̲ ⃗ and b ⃗ − 1 . In the last step we have introduced the abbreviations λ ⃗ k l = 1 q λ ( k , l ) a ⃗ − 1 and ω ̲ j ( k , l ) = 1 q 2 a ̲ j − 1 b ̲ ⃗ ( k , l ) b ⃗ − 1 ( k , l ) λ ( k , l ) . The integral can be calculated as follows, using the same substitution as in reference (106)

(136) ∫ ∑ j = 1 q d x j c ̲ j s s k l = ∑ j = 1 q ω ̲ j ( k , l ) ∫ d x j 1 − e − λ ⃗ k l x ⃗ − 1 = ln e λ ⃗ k l x ⃗ − 1 ∑ j = 1 q ω ̲ j ( k , l ) λ k l j + const with the factor ∑ j = 1 q ω ̲ j ( k , l ) λ k l j = b ̲ ⃗ ( k , l ) b ⃗ − 1 ( k , l ) × a ̲ ( k , l ) − a ( k , l ) a ( k , l ) − 1 + q − 1 ≔ α ̲ k l

providing^[55]

(137) ∫ ∑ j = 1 q d x j c ̲ j s s k l = ln B k l e λ ⃗ k l x ⃗ − 1 α ̲ k l

with B _kl being a constant. Note that the exponent α ̲ k l can become positive or negative, depending on the coefficients a ̲ ( k , l ) and b ̲ ⃗ ( k , l ) , and that it becomes = 1 if these coefficients become equal to their analogous quantities of the composition field.^[56] We now insert this result into equation (135), merge the constants A k l i B k l − 1 = C k l i and obtain

(138) F k l i = C k l i e λ ̲ ⃗ k l x ⃗ e λ ⃗ k l x ⃗ − 1 − α ̲ k l .

This result can be analysed by identifying the extrema with the condition ∂ F k l i = 0 which is already defined via Eq. (134). It can only be fulfilled by F k l i = 0 or λ ̲ ( k , l ) = b ̲ s ( k , l ) s k l . F k l i = 0 leads again to an eigenvalue problem if x ⃗ contains imaginary parts, which is identical to that of the composition field. The other condition means, as λ ̲ ( k , l ) = const and b ̲ s ( k , l ) = const ≠ 0 , ∂ s k l = 0 , thus λ ̲ ( k , l ) = b ̲ s ( k , l ) s k l ext , but also λ ( k , l ) = b s ( k , l ) s k l ext which directly can be seen from the expression (32) for the composition field and its maxima (ψ = 1). We again have equations with sums over an index s which cannot simply be solved for a single s k l ext . But both equations can be fulfilled if they are fragemented into q parts and the single terms equaled ( b s ( k , l ) s k l ext = λ ( k , l ) / q for a single s, analogical for the underlined terms). This leads to λ ̲ ( k , l ) λ ( k , l ) = b ̲ ( s ) ( k , l ) b ( s ) ( k , l ) for each s, or after summation over s to λ ̲ ( k , l ) = λ ( k , l ) q ∑ s = 1 q b ̲ s ( k , l ) b s ( k , l ) = λ ( k , l ) b ̲ ⃗ ( k , l ) b ⃗ − 1 ( k , l ) q . Thus λ ̲ ( k , l ) = λ ( k , l ) holds if b ̲ s ( k , l ) = b s ( k , l ) .

We now consider the poly-metric state function ψ ̲ k l and transform it as

(139) ψ ̲ k l = e λ ̲ ⃗ k l x ⃗ e λ ⃗ k l x ⃗ − 1 − α ̲ k l = e − λ ̲ ⃗ k l α ̲ k l x ⃗ e λ ⃗ k l x ⃗ − 1 − α ̲ k l = e λ ⃗ k l − λ ̲ ⃗ k l α ̲ k l x ⃗ − e − λ ̲ ⃗ k l α ̲ k l x ⃗ − α ̲ k l .

The analysis of the exponent of the first term gives

(140) λ ⃗ k l − λ ̲ ⃗ k l α ̲ k l = λ ( k , l ) q a ⃗ − 1 − a ̲ ⃗ − 1 1 q a ̲ ( k , l ) − a ( k , l ) a ( k , l ) − 1 + q → λ ( k , l ) a ⃗ − 1 q 1 − q q = 0

if a ̲ ( k , l ) = a ( k , l ) . Note that the dependency on the b ̲ s ( k , l ) b s − 1 ( k , l ) cancels out, so only the a ̲ ( k , l ) need to be = a(k, l) to obtain λ ̲ ⃗ k l = α ̲ k l λ ⃗ k l . Important to see is that this equation does not hold in general, different as Heim specified it. If it holds, then the poly-metric state function can be written as

(141) ψ ̲ k l = 1 − e − λ ⃗ k l x ⃗ − α ̲ k l = ψ k l α ̲ k l .

As next we can derive another important relation by considering again the extrema of our solution. Applying the derivation ∂ F k l i = 0 directly to (138) we find λ ̲ ⃗ k l = e λ ⃗ k l x ⃗ ( λ ̲ ⃗ k l − α ̲ k l λ ⃗ k l ) after a few transformations. If we build the scalar product with the vectors r ⃗ and ξ ⃗ (of x ⃗ = r ⃗ + i ξ ⃗ ) and divide these relations, we get

(142) r ⃗ λ ̲ ⃗ k l r ⃗ λ ̲ ⃗ k l − α ̲ k l r ⃗ λ ⃗ k l = ξ ⃗ λ ̲ ⃗ k l ξ ⃗ λ ̲ ⃗ k l − α ̲ k l ξ ⃗ λ ⃗ k l

as equation which (only) holds at the extrema x ⃗ ext , i.e. r ⃗ ext and ξ ⃗ ext of F k l i . This equation can definitely be fulfilled by ( r ⃗ λ ̲ ⃗ k l ) ext = d ⋅ ( ξ ⃗ λ ̲ ⃗ k l ) ext and ( r ⃗ λ ⃗ k l ) ext = d ⋅ ( ξ ⃗ λ ⃗ k l ) ext with a parameter or function d. The algebraic properties of d are considered in Subsection 4.2.^[57]

We finally can derive an expression for α ̲ k l depending on the eigenvalues β ̲ ± of the poly-metric field and the β _± of the composition field: We consider the arguments of the e-function in ψ ̲ k l and ψ _kl

(143) λ ̲ ⃗ k l x ⃗ λ ⃗ k l x ⃗ = λ ̲ ( k , l ) a ̲ ⃗ − 1 x ⃗ λ ( k , l ) a ⃗ − 1 x ⃗ = λ ̲ ( k , l ) λ ( k , l ) ( a ̲ ( k , l ) − 1 ) − 1 x l − ∑ m ≠ l x m ( a ( k , l ) − 1 ) − 1 x l − ∑ m ≠ l x m ≔ λ ̲ ( k , l ) λ ( k , l ) f ̃ ( k , l )

with the function f ̃ ( k , l ) ≈ ( a ̲ ( k , l ) − 1 ) − 1 − q + 1 ( a ( k , l ) − 1 ) − 1 − q + 1 if the x _l _/ _m can be approximated in average as x _l _/ _m ≈ x, and f ̃ ( k , l ) → 1 if a ̲ ( k , l ) = a ( k , l ) . In this limit also λ ̲ ( k , l ) / λ ( k , l ) = α ̲ k l holds (see above), but in general λ ̲ ( k , l ) / λ ( k , l ) = α ̲ k l c ( k , l ) with c ( k , l ) = a ̲ ( k , l ) − a ( k , l ) q ( a ( k , l ) − 1 ) + 1 (compare reference (136)). This means^[58]

(144) λ ̲ ⃗ k l x ⃗ λ ⃗ k l x ⃗ = α ̲ k l c ( k , l ) f ̃ ( k , l ) ≔ α ̲ k l f ( k , l ) .

On the other hand the ratio of the arguments for the extrema, i.e. eigenvalues (see subsection 3.5.2) is ( λ ̲ ⃗ k l x ⃗ ) ext / ( λ ⃗ k l x ⃗ ) ext = ( a ̲ ext + i β ̲ ± ) / ( a ext + i β ± ) with a ̲ and a being the real parts of the arguments. For the regime of the eigenvalues we can assume y ² = −x ² = ξ ² − r ² > 0 with x ⃗ = r ⃗ + i ξ ⃗ . If we now further apply ( λ ⃗ k l r ⃗ ) ext = d ⋅ ( λ ⃗ k l ξ ⃗ ) ext and the analogous relation for the λ ̲ ⃗ k l , as obtained above, we get

(145) ( λ ̲ ⃗ k l x ⃗ ) ext ( λ ⃗ k l x ⃗ ) ext = ( d + i ) ( λ ̲ ⃗ k l ξ ⃗ ) ext ( d + i ) ( λ ⃗ k l ξ ⃗ ) ext = β ̲ ± β ± ⇒ α ̲ k l = β ̲ ± β ± f ext − 1 ( k , l ) a n d λ ̲ ( k , l ) λ ( k , l ) = β ̲ ± β ± f ̃ ext − 1 ( k , l ) .

These relations become relevant in Section 4.

H.3 Expressions of the metric tensor and the correlation tensor in the poly-metric

Based on result (138), Heim obtains the following expression for the trace of the metric tensor

(146) ∑ k = 1 q g k k ( μ ν ) = C ( μ ν ) ∏ l = 1 q e λ ⃗ l l x ⃗ − 1 α ̲ l l = C ( μ ν ) | e λ ⃗ k l x ⃗ − 1 α ̲ k l δ k l |

with integration constant C ₍ _μν ₎ and the λ ⃗ k l , α ̲ k l as being defined similarly in subsection H.2. | … | denotes the determinant.

The correlation tensor results in the form of an invariant scalar sum as

(147) Q ( μ ν ) ( κ λ ) = ∑ i , s = 1 q Q s ( μ ν ) i ( κ λ ) = A ( μ ν ) ( κ λ ) ∑ l = 1 q C l ( μ ν ) ( κ λ ) 1 − e − λ ⃗ l l x ⃗ − α ̲ l l × ∑ l = 1 q C l ( μ ν ) 1 − e − λ ⃗ l l ( μ ν ) x ⃗ − 1 − 1 × | δ k l e λ ⃗ k l ( κ λ ) x ⃗ − 1 α k l ( κ λ ) | × | δ k l e λ ⃗ k l ( μ ν ) x ⃗ − 1 α k l ( μ ν ) | − 1 − q

with A ( μ ν ) ( κ λ ) = 2 C ( κ λ ) C ( μ ν ) − 1 . Note that these expressions are only valid under the condition that (141) holds. The derivation of these results can be found on pages 113–121 in [44]. We do not present it here, since we do not use the expressions (146) and (147) in our further calculations.

H.4 Calculation of the condenser class [3]

We refer to chapter VII.1 of [44], pages 174–177:

To calculate the function F ₍ ₃ ₎, defined by the general solution (44) and the definition I m F ( 3 ) = I m ( 33 33 ̂ + + Tr ( Q ( 33 ) ( 33 ) × 33 33 ̂ + ) ) = 0 , we introduce the definitions A k l ( 3 ) i = a + i b and ∑ j = 1 3 c ̲ j s ( 3 ) ∫ d x j s k l = f + i F . With the always real λ ̲ ⃗ k l and x ⃗ , d x ⃗ in the R ₃ we get 0 = Im((a + ib)e^ϕ−iF) = e^ϕIm((a + ib)e ^−iF) with ϕ = λ ̲ ⃗ k l x ⃗ − f in the R ₃. With e^−iF = cos F − i sin F it follows (a + ib)e^−iF = a cos F + b sin F + i(b cos F − a sin F), which with the Im() = 0 condition yields tan F = b/a and F ₍₃₎ = e^ϕ(a cos F + b sin F). Using the substitution with tan F we can write cos ⁡ F = ( 1 + tan 2 ⁡ F ) − 1 / 2 = a a 2 + b 2 and sin ⁡ F = ( 1 + cot 2 ⁡ F ) − 1 / 2 = b a 2 + b 2 , thus a cos ⁡ F + b sin ⁡ F = a 2 + b 2 ≔ α k l i = const . Thereby the real-valued solution becomes

(148) F k l ( 3 ) i = α k l i e ϕ w i t h ϕ = λ ̲ ⃗ k l x ⃗ − R e ∫ ∑ j = 1 3 d x j c ̲ j s ( 3 ) s k l

with the coefficients c ̲ j s ( 3 ) being real quantities, as they are composed of real eigenvalues. Similar as already derived below reference (135), the integrand becomes c ̲ j s ( 3 ) s k l = 1 q 2 a ̲ j ( 3 ) − 1 b ̲ ⃗ ( 3 ) b ⃗ − 1 λ ( k , l ) ψ k l = ω j ( 3 ) ( k , l ) ψ k l with the abbreviation ω _j ₍₃₎(k, l) as coefficient and the state function ψ _kl of the composition field. ψ _kl is a function of the 6 coordinates of the R ₆ as ψ k l = 1 − e − λ ⃗ k l x ⃗ − 1 with x ⃗ = r ⃗ + i ξ ⃗ and the λ ⃗ k l , as defined for reference (136). The integral in (148) runs only over the three spatial coordinates r ⃗ so that the imaginary coordinates ξ ⃗ appear as constants in ψ k l = ( 1 − e − λ ⃗ k l r ⃗ ( cos λ ⃗ k l ξ ⃗ − i sin λ ⃗ k l ξ ⃗ ) ) − 1 = ( κ + i η ) − 1 , having defined the quantities κ and η this way. According to this relation, also ψ _kl = α + iβ must split, thus Re ψ _kl = α and f = ∫ ω ⃗ ( 3 ) d r ⃗ α . As κ and η are known, β must be eliminated out of α + iβ = (κ + iη)⁻¹ in order to obtain α. Because of 1 = (α + iβ)(κ + iη) = ακ − βη + i(αη + κβ), thus ακ − βη = 1 and κβ = −αη, we get α = κ ( κ 2 + η 2 ) − 1 = 1 − b e − λ ⃗ k l r ⃗ ( 1 − 2 b e − λ ⃗ k l r ⃗ + e − 2 λ ⃗ k l r ⃗ ) − 1 where we have used the abbreviation b = cos λ ⃗ k l ξ ⃗ . With ω ⃗ ( 3 ) d r ⃗ = ∑ j = 1 3 ω j ( 3 ) d r j and the substitution u = e λ ⃗ k l r ⃗ − b / 1 − b 2 , d r j = d u 1 − b 2 λ k l j e − λ ⃗ k l r ⃗ we can calculate the integral for f after some steps as

(149) f = ∑ j = 1 3 ω j ( 3 ) λ k l j ∫ d u u u 2 + 1 = ω ⃗ ( 3 ) λ ⃗ k l − 1 ( ln 1 + u 2 + C ) = ω ⃗ ( 3 ) λ ⃗ k l − 1 1 2 ln ( 1 + e λ ⃗ k l r ⃗ − b 2 / ( 1 − b 2 ) ) + C .

The integration constant C can be determined by choosing u ( r ⃗ = 0 ) as lower limit of the integral above. Then C becomes

(150) C = − 1 2 ln 1 + ( 1 − b ) 2 / ( 1 − b 2 ) = − 1 2 ln 1 + ( 1 − b ) / ( 1 + b ) = 1 2 ln 1 2 ( 1 + b ) .

This gives

(151) f = ω ⃗ ( 3 ) λ ⃗ k l − 1 1 2 ln 1 2 ( 1 + b ) ( 1 + e λ ⃗ k l r ⃗ − b 2 / ( 1 − b 2 ) )

and for the factor

(152) ω ⃗ ( 3 ) λ ⃗ k l − 1 = 1 q a ⃗ a ̲ ⃗ ( 3 ) − 1 b ̲ ⃗ ( 3 ) b ⃗ − 1 = b ̲ ⃗ ( 3 ) b ⃗ − 1 a ̲ ( 3 ) ( k , l ) − a ( k , l ) a ( k , l ) − 1 + q − 1 ≔ α ̲ ( 3 ) .

in analogy to reference (136). The condenser function then becomes

(153) F k l ( 3 ) i = α k l i e λ ̲ ⃗ k l r ⃗ 1 2 ( 1 + b ) 1 + e λ ⃗ k l r ⃗ − b 2 1 − b 2 − α ̲ ( 3 ) 2

with b = cos λ ⃗ k l ξ ⃗ (b not to be confused with b ⃗ = b ⃗ ( k , l ) ).

H.5 Further classes of condensations

The first part of this subsection refers to chapter VII.1 of [44], pages 179–186.

Analogue to Subsection H.2 we can evaluate the expression of reference (54) by inserting the function of the composition field in the form c ̲ j s s k l = 1 q q v a ̲ j − 1 b ̲ ⃗ ( k , l ) b ⃗ − 1 ( k , l ) λ ( k , l ) ( 1 − e − 1 q λ ( k , l ) a ⃗ − 1 Q ⃗ ) − 1 ≔ ω ̲ j ( k , l ) ( 1 − e − λ ⃗ k l ( P ⃗ + V ⃗ ) ) − 1 . The integral in reference (54) then can be calculated as follows, using the same substitution as in reference (106) and taking into account V ⃗ = 0 at the lower limit of the integral over the V _j

(154) ∫ ∑ j = 1 q v d V j c ̲ j s s k l = ∑ j = 1 q v ω ̲ j ( k , l ) ∫ d V j ( 1 − e − λ ⃗ k l ( P ⃗ + V ⃗ ) ) − 1 = ∑ j = 1 q v ω ̲ j ( k , l ) λ k l j ln ( e λ ⃗ k l ( P ⃗ + V ⃗ ) − 1 ) − ln e λ ⃗ k l P ⃗ − 1 with the factor ∑ j = 1 q v ω ̲ j ( k , l ) λ k l j = b ̲ ⃗ ( k , l ) b ⃗ − 1 ( k , l ) × a ̲ ( k , l ) − a ( k , l ) a ( k , l ) − 1 + q − 1 ≔ α ̲ v ,

providing

(155) ∫ ∑ j = 1 q v d V j c ̲ j s s k l = α ̲ v ⁡ ln e λ ⃗ k l ( P ⃗ + V ⃗ ) − 1 e λ ⃗ k l P ⃗ − 1 = ln e λ ⃗ k l V ⃗ − e − λ ⃗ k l P ⃗ 1 − e − λ ⃗ k l P ⃗ α ̲ v

Inserting this expression into reference (54), the general result for classes of condensations of reference (55) is obtained.

We finally want to give the concrete expressions for the possible classes of condensations according to the definition of the hermetry forms in subsection 3.4.3:

In the hermetry form a only the partial structure 1 with the coordinates x ₅ and x ₆ is hermetric, so we have V ⃗ = x ⃗ 5 + x ⃗ 6 ≔ i T ⃗ , P ⃗ = 0 and only one condenser function according to (45)

(156) F ( 1 ) ( a ) = e i λ ̲ ⃗ T ⃗ e i λ ⃗ T ⃗ − 1 − α ̲ = e i ( λ ⃗ − λ ̲ ⃗ α ̲ − 1 ) T ⃗ − e − i λ ̲ ⃗ α ̲ − 1 T ⃗ − α ̲ = e i λ ⃗ ( 1 − f ) T ⃗ − e − i λ ⃗ f T ⃗ − α ̲ → 1 − e − i λ ⃗ T ⃗ − α ̲ i f f → 1

where we have used, as in (56), λ ̲ ⃗ α ̲ − 1 V ⃗ = λ ⃗ V ⃗ f . To keep the notation simple, we have suppressed the coordinate indices k, l at λ ̲ ⃗ , λ ⃗ , α ̲ and f = f(k, l).

For the hermetry form b there are three possible combinations of the hermetric coordinates x ⃗ 4 = i c t e ⃗ 4 ≔ i c t ⃗ and x ⃗ 5 + x ⃗ 6 ≔ i T ⃗ which yield (with (56), respectively (45))

(157) F ( 1 ) ( b ) = e i λ ⃗ ( 1 − f ) T ⃗ − e − i λ ⃗ ( c t ⃗ + f T ⃗ ) 1 − e − i λ ⃗ c t ⃗ − α ̲ → 1 − e − i λ ⃗ ( c t ⃗ + T ⃗ ) 1 − e − i λ ⃗ c t ⃗ − α ̲ F ( 2 ) ( b ) = e i λ ⃗ ( 1 − f ) c t ⃗ − e − i λ ⃗ ( T ⃗ + f c t ⃗ ) 1 − e − i λ ⃗ T ⃗ − α ̲ → 1 − e − i λ ⃗ ( c t ⃗ + T ⃗ ) 1 − e − i λ ⃗ T ⃗ − α ̲ F ( 12 ) ( b ) = e i λ ⃗ ( 1 − f ) ( c t ⃗ + T ⃗ ) − e − i λ ⃗ f ( c t ⃗ + T ⃗ ) − α ̲ → 1 − e − i λ ⃗ ( c t ⃗ + T ⃗ ) − α ̲

including the results for the limit f → 1.

For the form c with the hermetric coordinates r ⃗ = ∑ k = 1 3 x k e ⃗ k and again x ⃗ 5 + x ⃗ 6 = i T ⃗ , which describes neutral ponderable particles as complex-valued space condensations, there is only one class with a leading imaginary form (F ₍₁₎(c)) and the class according to (45):

(158) F ( 1 ) ( c ) = e i λ ⃗ ( 1 − f ) T ⃗ − e − λ ⃗ ( r ⃗ + i f T ⃗ ) 1 − e − λ ⃗ r ⃗ − α ̲ → 1 − e − λ ⃗ ( r ⃗ + i T ⃗ ) 1 − e − λ ⃗ r ⃗ − α ̲ F ( 13 ) ( c ) = e λ ⃗ ( 1 − f ) ( r ⃗ + i T ⃗ ) − e − λ ⃗ f ( r ⃗ + i T ⃗ ) − α ̲ → 1 − e − λ ⃗ ( r ⃗ + i T ⃗ ) − α ̲

The hermetry form d with the hermetric coordinates r ⃗ , i c t ⃗ and i T ⃗ , describing charged ponderable particles as complex-valued space condensations, provides six classes:

(159) F α ( d ) = e λ ⃗ ( 1 − f ) V ⃗ α − e − λ ⃗ ( P ⃗ α + f V ⃗ α ) 1 − e − λ ⃗ P ⃗ α − α ̲ → 1 − e − λ ⃗ ( P ⃗ α + V ⃗ α ) 1 − e − λ ⃗ P ⃗ α − α ̲ w i t h V ⃗ α = i T ⃗ , P ⃗ α = r ⃗ + i c t ⃗ f o r α = ( 1 ) V ⃗ α = i c t ⃗ , P ⃗ α = r ⃗ + i T ⃗ f o r α = ( 2 ) V ⃗ α = i T ⃗ + i c t ⃗ , P ⃗ α = r ⃗ f o r α = ( 12 ) V ⃗ α = r ⃗ + i T ⃗ , P ⃗ α = i c t ⃗ f o r α = ( 13 ) V ⃗ α = r ⃗ + i c t ⃗ , P ⃗ α = i T ⃗ f o r α = ( 23 ) a n d F ( 123 ) ( d ) = e λ ⃗ ( 1 − f ) ( r ⃗ + i c t ⃗ + i T ⃗ ) − e − λ ⃗ f ( r ⃗ + i c t ⃗ + i T ⃗ ) − α ̲ → 1 − e − λ ⃗ ( r ⃗ + i c t ⃗ + i T ⃗ ) − α ̲

These results correspond to those of Heim in [44], page 186 (his reference 80), but are slightly different, as Heim evaluates λ ̲ ⃗ = α ̲ λ ⃗ exactly, which we do not, compare reference (140) above. Only if the function f → 1, the results become identical to Heim’s. Note that, although we have abstained here from marking them with indices, the exponent α ̲ and the function f in general differ between the single results, as according to their general expressions (136), (143) and (144), they depend on the particular hermetric coordinate structure of each class.

Finally, we analyse the condenser functions of the form as given in reference (55) concerning their dependency on the variable P ⃗ .^[59] We are interested in the functions F ₍₁₃₎(d) and F ₍₂₃₎(d) in which one of the variables P ⃗ = i c t ⃗ or P ⃗ = i T ⃗ , in the following generalised as P ⃗ = i z ⃗ , appears in the term e − λ ⃗ P ⃗ , and the other variable is part of the vector V ⃗ = r ⃗ + i ξ ⃗ − i z ⃗ :

(160) F ( 1 / 23 ) ( d ) = e λ ̲ ⃗ ( r ⃗ + i ξ ⃗ − i z ⃗ ) e λ ⃗ ( r ⃗ + i ξ ⃗ − i z ⃗ ) − e − i λ ⃗ z ⃗ 1 − e − i λ ⃗ z ⃗ − α ̲ = e λ ̲ ⃗ ( r ⃗ + i ξ ⃗ ) e λ ⃗ ( r ⃗ + i ξ ⃗ ) − 1 − α ̲ e − i λ ̲ ⃗ z ⃗ e i α ̲ λ ⃗ z ⃗ × 1 − e − i λ ⃗ z ⃗ α ̲ = F ( 123 ) ( d ) e − i λ ̲ ⃗ z ⃗ e i λ ⃗ z ⃗ − 1 α ̲

Here the definition of F ₍ ₁₂₃ ₎(d) in its initial form (45) was used. We continue with the abbreviation x = λ ⃗ z ⃗ and the definition λ ̲ ⃗ z ⃗ = α ̲ f z λ ⃗ z ⃗ in analogy to (144) (i.e. f _z depends on the indices k, l and in general on z ⃗ as well) and use the substitution e i λ ⃗ z ⃗ − 1 = cos x − 1 + i sin x ≔ r e i ϕ , meaning r ² = (cos x − 1)² + sin² x, ϕ = arctan ( sin x cos x − 1 ) and ( r e i ϕ ) α ̲ = r α ̲ e i ϕ α ̲ = ( 2 − 2 ⁡ cos x ) α ̲ / 2 ( cos α ̲ ϕ + i sin α ̲ ϕ ) :

(161) F ( 1 / 23 ) ( d ) = F ( 123 ) ( d ) ( cos α ̲ f z x − i sin α ̲ f z x ) × ( cos α ̲ ϕ + i sin α ̲ ϕ ) ( 2 ( 1 − cos x ) ) α ̲ / 2 = F ( 123 ) ( d ) ( cos α ̲ ( f z x − ϕ ) − i sin α ̲ ( f z x − ϕ ) ) × ( 2 ( 1 − cos x ) ) α ̲ / 2

We now assume that f _z hardly varies with z ⃗ , i.e. x, and at the (for our purposes) relevant index pair k, l = 4 takes a numerical value similar to f _ext(4, 4) (see Section 4). Then the dependence on the variable z ⃗ is defined by x and ϕ (see above) and is periodic in form of the cos-, respectively, sin-function, and ( 2 ( 1 − cos x ) ) α ̲ / 2 is a modulating factor. In the area of data given in Section 4 (for f _ext and thus α ̲ ) this factor only creates small peaks at the cos x = 1 points, but does not change the overall numerical result that the arithmetic mean of F _(1/23)(d) over a whole period (depending on z ⃗ ) is essentially zero.

Appendix I: Simplification of equation (64)

To simplify the term in the square bracket of (64) it is set = Re^iϕ and the abbreviation K = K(n) used so that R 2 = ( e K d ⁡ cos ⁡ K − 1 ) 2 + e 2 Kd ⁡ sin 2 ⁡ K = e 2 Kd − 2 e K d ⁡ cos ⁡ K + 1 = e 2 Kd + 1 for m = 1 and = e 2 Kd ± 2 e K d + 1 = ( e K d ± 1 ) 2 for m = 2. For both R → e ^Kd holds if e ^Kd ≫ 1, which is true in our data range. The angle ϕ is given by ϕ = arctan ( e K d ⁡ sin ⁡ K e K d ⁡ cos ⁡ K − 1 ) and ( R e i ϕ ) − α ̲ = R − α ̲ ( cos α ̲ ϕ − i sin α ̲ ϕ ) → e − α ̲ K d ( cos α ̲ ϕ − i sin α ̲ ϕ ) = e − π 2 n ̲ d f ext − 1 ( cos π n ̲ 2 K f ext − 1 ϕ − i sin π n ̲ 2 K f ext − 1 ϕ ) . The angle ϕ can be evaluated in the range e ^Kd ≫ 1 via cos ⁡ K = cos ( π 2 ( 2 n + 1 ) ) = 0 and sin ⁡ K = sin ( π 2 ( 2 n + 1 ) ) = ( − 1 ) n for m = 1 and cos(πn) = (−1)ⁿ and sin(πn) = 0 for m = 2. This gives ϕ = arctan ( ( − 1 ) n + 1 e π 2 ( 2 n + 1 ) d ) → ( − 1 ) n + 1 π 2 for m = 1 and n > 0 and ϕ = arctan(0) = 0 for m = 2.

With these results the expressions of reference (65) are easily obtained.

Appendix J: Linear approximation, Dirac and Maxwell equations

We study the linear limit of the fundamental equations of Heim’s theory (according to [44], chapter VIII.5, pages 358–363). We can expect that in case of the b hermetry, a linear approximation should lead to the known equations which determine the electromagnetic field and the particles interacting with this field, i.e. the equations of Maxwell in the classical limit and Dirac’s equation for a relativistic fermion field (particle).

The b hermetry contains 6 possible signatures in terms of combinations of partial structures, namely the elements (1), (11), (2), (22), (12) and (21), which can occur in the co- and the contravariant signatures. If we now consider Eq. (43), neglect the non-linear terms in it and assume a stationary status in which only the components of the R ₄ are non-zero and all field components and also eigenvalues λ _i for the trans-dimensions x ₅, x ₆ become zero, and finally sum over two indices F m = ∑ k F k m k , we get

(162) ∂ m F p − ∂ p F m = λ ̲ p F m ≔ i λ p F m

where we have re-defined the λ ̲ p . The relevant partial structures of the b hermetry lead to two different condenser functions according to Appendix H if we take into account the limit x ₅, x ₆ = 0, thus T ⃗ = 0 , namely the functions F ₍₁₎(b) and F ₍₁₂₎(b) according to (157). This means that there is a second equation for the second function, which we name G:

(163) ∂ m G p − ∂ p G m = λ ̲ p G m ≔ i λ p G m

(Heim uses the notation of functions g _m and h _m, see [44] page 359 bottom.) Summation over 1 ≤ p ≤ 4 leads to

(164) ∑ p ∂ m F p − ∂ p F m = i λ F m a n d ∑ p ∂ m G p − ∂ p G m = i λ G m

with ∑ _p λ _p = λ, or in matrix representation

(165) ∂ 1 − ∑ p ∂ p ∂ 1 ∂ 1 ∂ 1 ∂ 2 ∂ 2 − ∑ p ∂ p ∂ 2 ∂ 2 ∂ 3 ∂ 3 ∂ 3 − ∑ p ∂ p ∂ 3 ∂ 4 ∂ 4 ∂ 4 ∂ 4 − ∑ p ∂ p F ⃗ ( G ⃗ ) = i λ F ⃗ ( G ⃗ ) .

These two linear matrix equations for F ⃗ and G ⃗ can be transformed to one matrix equation with two other quadruple vectors a ⃗ and b ⃗ which are linearly coupled:^[60]

(166) ± i ∂ 4 0 ∂ 3 ∂ 1 0 0 0 ∂ 2 0 ± i ∂ 4 ∂ 1 − ∂ 3 0 0 − ∂ 2 0 ∂ 3 ∂ 1 ± i ∂ 4 0 0 ∂ 2 0 0 ∂ 1 − ∂ 3 0 ± i ∂ 4 − ∂ 2 0 0 0 0 0 0 − ∂ 2 ± i ∂ 4 0 ∂ 3 ∂ 1 0 0 ∂ 2 0 0 ± i ∂ 4 ∂ 1 − ∂ 3 0 − ∂ 2 0 0 ∂ 3 ∂ 1 ± i ∂ 4 0 ∂ 2 0 0 0 ∂ 1 − ∂ 3 0 ± i ∂ 4 × a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 = i λ − a 1 − a 2 a 3 a 4 − b 1 − b 2 b 3 b 4

This system can be written as a four-dimensional equation of a complex vector field ϕ ⃗ = a ⃗ + i b ⃗

(167) ± i ∂ 4 0 ∂ 3 ∂ 1 − i ∂ 2 0 ± i ∂ 4 ∂ 1 + i ∂ 2 − ∂ 3 ∂ 3 ∂ 1 − i ∂ 2 ± i ∂ 4 0 ∂ 1 + i ∂ 2 − ∂ 3 0 ± i ∂ 4 ϕ ⃗ = i λ − ϕ 1 − ϕ 2 ϕ 3 ϕ 4

which in turn can be expressed through the well-known four-dimensional matrices of the Dirac equation

(168) ± i ∂ 4 + ∂ 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 + ∂ 2 0 0 0 − i 0 0 i 0 0 − i 0 0 i 0 0 0 + ∂ 3 0 0 1 0 0 0 0 − 1 1 0 0 0 0 − 1 0 0 ϕ ⃗ = − i λ 1 0 0 0 0 1 0 0 0 0 − 1 0 0 0 0 − 1 ϕ ⃗

or compact (and with partial derivatives according to x ^k written out)

(169) ± i ∂ ∂ x 4 + α 1 ∂ ∂ x 1 + α 2 ∂ ∂ x 2 + α 3 ∂ ∂ x 3 ϕ ⃗ = − i λ β ϕ ⃗ w i t h α i = 0 σ i σ i 0 , i = 1,2,3 , β = 1 0 0 − 1 ,

the σ _i being the Pauli matrices. The α _i, β fulfil the anti-commutator relations [α _i, α _k] = 2δ _ik, [α _i, β] = 0 and α i 2 = β 2 = 1 . The eigenvalues λ have the dimension meter⁻¹ and should be proportional to the mass of an electric charge carrier if the photon field of the b hermetry interacts with the field of this charge carrier. On the other hand it is always a matter of quantum structure so that λ can be regarded as ratio of the momentum mc to ℏ, thus λ = mc/ℏ. With x ₄ = ict this gives

(170) ± i ℏ ∂ ∂ t ϕ ⃗ = ℏ c i ∑ k = 1 3 α k ∂ ∂ x k ϕ ⃗ + m c 2 β ϕ ⃗

which is identical to the Dirac equation if the four-dimensional complex vector field ϕ ⃗ is equalised with a four-dimensional spinor ψ and the branch with negative sign in front of the energy operator ℏ ∂ ∂ t is omitted.^[61] The usual form of the equation which explicitly shows the Lorentz invariance can be obtained by multiplying with the matrix β/c on both sides and introducing the gamma matrices γ ⁰ = β, γ ⁱ = βα _i, i = 1, 2, 3 and μ = 0, 1, 2, 3 [76]:

(171) i ℏ γ μ ∂ ∂ x μ − m c ψ = 0

Hence, it is possible to obtain the Dirac equation, the linear wave equation of relativistic quantum mechanics, from Heim’s theory (basic equation (43)) as a possible result in the linear limit of hermetry b. However, only its structure as linear partial differential equation of first order, acting on a complex 4-dimensional function, can be deduced unambiguously, but not its Clifford algebra, given by the γ matrices. Equation (165) does not provide them automatically, but allows only a linear transformation to them.

We now consider the macroscopic domain (i.e. Heim’s fourth area of validity) and an empty R ₃ space which means mass m → 0, thus λ → 0. Going back to the system of equations (166) and setting the components of the vectors a ⃗ and b ⃗ as a ₁ ∼ E ₃, a ₂ ∼ E ₁, a ₃ = 0, a ₄ ∼ −B ₂, b ₁ = 0, b ₂ ∼ E ₂, b ₃ ∼ B ₃, b ₄ ∼ B ₁, we obtain the set of equations

(172) ± i ∂ 4 E 3 − ∂ 1 B 2 + ∂ 2 B 1 = 0 , ± i ∂ 4 E 1 + ∂ 3 B 2 − ∂ 2 B 3 = 0 , ∂ 3 E 3 + ∂ 1 E 1 + ∂ 2 E 2 = 0 , ∂ 1 E 3 − ∂ 3 E 1 − ± i ∂ 4 B 2 = 0 , ∂ 2 B 2 + ∂ 3 B 3 + ∂ 1 B 1 = 0 , ± i ∂ 4 E 2 + ∂ 1 B 3 − ∂ 3 B 1 = 0 , − ∂ 2 E 1 + ∂ 1 E 2 + ± i ∂ 4 B 3 = 0 , ∂ 2 E 3 − ∂ 3 E 2 + ± i ∂ 4 B 1 = 0

which, when using again x ₄ = ict, can be written in the compact form

(173) rot B ⃗ = ± 1 c ∂ E ⃗ ∂ t , d i v E ⃗ = 0 , rot E ⃗ = − ± 1 c ∂ B ⃗ ∂ t , d i v B ⃗ = 0

and obviously corresponds to the microscopic Maxwell equations in the vacuum. An exact identity is obtained with the positive branch of the ± signs and when transforming B ⃗ → c B ⃗ .

So, Heim’s theory allows connecting to the classical theory of electromagnetism on a mathematical level.

Appendix K: The spin in Heim’s theory

We have seen in subsection 3.5.1 and Appendix H, subsection H.1 that a spin field tensor can be derived from the properties of the g _ik and i k l , meaning that in a hermetric space V _r always g − i k ≔ P i k = ( rot ( x ) ϕ ⃗ ) i k = const ≠ 0 holds for the spin field tensor P _ik, with ϕ ⃗ being a so-called spin field vector.

We now want to connect these quantities to the properties of a spin in 3-dimensional space as an angular momentum-like quantity (see chapter VII.3 in [44]). In analogy to the electromagnetic field we construct the angular momentum density from the energy momentum tensor T ^kl as

(174) L ikl = 1 w ( x i T k l − x k T i l )

with w being the absolute value of the imaginary part of the 6-dimensional world velocity ( Y ⃗ = v ⃗ + i w ⃗ , w 2 = c 2 + ϵ ̇ 2 + η ̇ 2 ) . The divergence of L becomes

(175) ∂ l L ikl = 1 w ∂ l x i T k l + x i ∂ l T k l − ∂ l x k T i l − x k ∂ l T i l = 1 w δ l i T k l − δ l k T i l = 1 w ( T k i − T i k ) = 1 w ( T k i − ( T k i ) × ) = 1 w ( T k i ) − ,

having used that the divergence of T vanishes. For a hermitian T, i.e. T ⁻ = 0, this means that the angular momentum L is conserved. Vice versa a conservation of angular momentum forces T ⁻ = 0, which with (18) and (28) leads to { } ̂ = { } ̂ × and g = g ^×. In the metronised poly-metric case, despite g ( κ λ ) ≠ g ( κ λ ) × , it remains κ λ μ ν ̂ − = 0 (see (42)) which leads to the constant metron spin g − ( κ λ ) ≔ P ( κ λ ) = const , compare above.

A spatial spin density vector in 6 dimensions is now defined by a trace over the angular momentum density tensor

(176) σ i = 1 w ∑ j ε jkl x k T l i = 1 w ∑ j x k ε klj T l i ≔ x ⃗ w ⋅ P ̃ ⃗ i

with the Levi-Civita symbol ɛ _jkl and sums over identical indices. The tensor P ̃ k i = ∑ j ε klj T l i is in general not identical to P ^ki of above, but structurally similar. It has the dimension of an energy density, so it can be written as a sum of energy density contributions P ̃ = ∑ α W α where the W _α depend on the partial geometry of the respective hermetry form, denoted by the usual indices, and can be expressed as W α = W ( μ ν ) ( κ λ ) ∼ Tr C κ λ μ ν ̂ + = λ ⃗ ( μ ν ) ( κ λ ) κ λ μ ν ̂ + according to (57). If we now integrate over the 3-dimensional space in the metron calculus (compare Section 4) we get a spin vector

(177) s i = ∫ ∂ V σ i = ∫ ∂ V x ⃗ w ⋅ P ̃ ⃗ i = ∫ ∂ V x k w ∑ α W α k i = ∑ α ∫ ∂ E α k i x k w

where i denotes the index of the vector and α stands for the hermetric partial structure. For the differential ∂ E α k i the quantisation of energy must be taken into account E = hν so that the differential becomes ∂ E α k i = h ν k ∂ m α k i , with m α k i being selectors of whole quantum numbers when the metron calculus is applied.^[62] This provides

(178) s i = h ∑ α ∫ ∂ m α k i x k w ν k .

In Heim’s model the velocity w and the frequencies ν ^k can be related, as the spin is considered as a result of the cyclical flux of condensations (compare Subsection 3.5.3). The world vector then lies on a circular path, x ⃗ = e ⃗ x u 2 π , with the perimeter u, in Cartesian coordinates x k = e ⃗ k ⋅ e ⃗ x u 2 π ≔ u k 2 π . Heim now relates w = 2u _l ν ^l which means that the circulation velocities u _l ν ^l (perimeter times frequency) shall be determined by w. The factor 2 obviously corresponds to the property of the quantum mechanical spin that the spin state vector is identically reproduced only after a rotation of a 4π angle. After inserting in (178) we obtain

(179) s i = h ∑ α ∫ ∂ m α k i u k ν k 4 π u l ν l → ℏ 2 ∑ α ∫ ∂ m α i = ℏ 2 m i

if the selectors m α k i → m α i do not depend on index k, i.e. simply form a vector, and m i = ∑ α m α i , being natural numbers. Thus we find the well-known quantitative result of the quantum mechanical spin.

We can write the result as a vector and take into account that in the 6-dimensional space the sector of the coordinates x ₄ − x ₆ is imaginary:

(180) s ⃗ = ℏ 2 ∑ i = 1 6 m i e ⃗ i ≔ ℏ ( σ ⃗ r + i σ ⃗ t )

The real part ℏ σ ⃗ r must be the spin in 3-dimensional space, the second imaginary part is identified by Heim as the isospin.

Appendix L: Qualitative derivation of an interaction potential

(Consideration of the author, not by Heim)

In Sections 3 and 4, we have derived state functions for partial geometries and calculated mass-energies for related particle states which are associated with the hermetry forms c and d in Heim’s theory. An important question is also, how the interaction between the particles can be described, which must be related to the strong interaction. We can use the fundamental geodesic equation (12) to derive an ansatz for an interaction potential:

In the nonrelativistic limit for velocities ≪c we may neglect d x ⃗ / d τ (with the proper time dτ) with respect to dt/dτ and write (12) as^[63]

(181) d 2 x i d τ 2 + Γ 44 i d t d τ 2 = 0 ⇒ d 2 x i d t 2 + Γ 44 i = 0 .

If we now define Γ 44 i = − ∂ ∂ x i Φ , we get the classical equation of motion d 2 x i d t 2 = ∂ ∂ x i Φ with the potential Φ ( x ⃗ ) . This means that a nonrelativistic potential can be determined from the relation

(182) ∂ ∂ x i Φ ( x ⃗ ) = − Γ 44 i → − F 44 i

when the classical Christoffel symbol is transformed to Heim’s poly-metric state function F, as defined in subsection 3.5. We are interested in the course of the Φ in the 3-dimensional space and consider therefore only the F 44 i at their extrema with respect to the remaining coordinates ξ ⃗ . Hence, we can use the expressions of (51), (52) and (159) for F ₍₃₎ _ext and F ₍ ₁₂₃ ₎ _ext. To these expressions we apply the same approximation as in (53), which is not only valid in the limit of big r, which was considered in (53), but also for middle to small r if α ̲ 44 = ( β ̲ ± / β ± ) f ext − 1 is determined by the value of f _ext ≈ −2.16, as specified in Subsection 4.3 (i.e. r in the same order of magnitude as given by |f _ext|). The terms in front of the e-function are constant (C ⁱ) with regard to r ⃗ , and so we get

(183) ∂ ∂ x i Φ ( r ⃗ ) = − C i e − α ̲ 44 λ ⃗ r ⃗ ( 1 − f ( r ⃗ ) ) = − C i e − α ̲ 44 λ r ⁡ cos ⁡ θ ( 1 − f ( r ⃗ ) )

with θ as angle between the λ ⃗ and r ⃗ . Now we have to model the regarding its exact course unknown function f ( r ⃗ ) , respectively, 1 − f ( r ⃗ ) . First of all, we assume a radial symmetry, i.e. f = f(r). Next, we know that 1 − f has to be approximately constant at a value of 1 − f _ext ≈ 3.16 for small r and that it has to change sign in the course to bigger r, see Subsection 4.2. This can be modelled by the approach

(184) 1 − f ( r ) = a 0 ≈ 3.16 f o r r ≤ r 0 , = − b r + c f o r r 0 < r < r 1 , = − a 1 < 0 f o r r ≥ r 1

with r ₁ ≫ r ₀ and requiring that the function is steady which defines the constants b > 0 and c > 0 in dependence of the other parameters. To come to a simply assessable relation, we make a restriction and hide the dependency on θ, so consider only that on r. Inserting 1 − f(r) as defined by (184) and assuming C ⁱ < 0, we obtain a function on the r.h.s. of (185) with the course of a shifted Gaussian function

(185) ∂ ∂ r Φ ( r ) ∼ e A r ( 1 − f ( r ) ) → e − A ( b r 2 − c r ) ∼ e − A b r − c 2 b 2 f o r r 0 < r < r 1

with the abbreviation A = − α ̲ 44 λ ⁡ cos ⁡ θ . A is > 0 , as f ext − 1 < 0 (cos θ > 0 assumed). Its width depends via b, c on the unknown parameters r ₀, r ₁. With this function, Eq. (185) cannot be integrated analytically, but the course of Φ(r) can be detected approximately. A Taylor expansion of the Gaussian up to sixth order and subsequent integration yields a result which can be well approximated by a linear function Φ ( r ) ≈ k r + c ̃ in the area r ₀ < r < c/b. The term c/b marks the zero of 1 − f(r) = −br + c. For r ≥ c/b Φ flattens out and approximates a constant value c ₁. In the sectors r ≤ r ₀ and r ≥ r ₁ the function 1 − f is constant in our model, so here (185) can easily be integrated and Φ be determined as Φ ( r ) ∼ e A a 0 r A a 0 + c 0 , respectively, ∼ − e − A a 1 r A a 1 + c 1 . The integration constants c ̃ , c 0 , c 1 have to be set so that Φ(r) is steady.

We conclude that we have derived a potential from the hermetry forms c, d in Heim’s theory by means of the resulting state functions and from an a priori unknown function f, modelled in a way that empirical data are met – the fit of f in the inner region so that the mass spectrum comes out, and its necessary behaviour for r → ∞. We obtain a result with an approximately linearly increasing course in a region which we assume as decisive for the question whether free particles can “escape” at higher energies and in which particle shape. It is interesting to notice that we find this linear course ∼ k r + c ̃ which is also the typical potential for the confinement in the quark model. As we currently cannot determine the quantities r ₀, r ₁ and the further introduced constants, we cannot make any quantitative statements, but consider our qualitative result as encouragement for further research.

Appendix M: Heim’s minimum particle mass

Here we shortly outline Heim’s approach to derive a smallest particle mass, which turns out to lie very close to the electron mass. We cannot reproduce his entire derivation due to the limited space of this paper and refer to [43], chapter II.4 and chapter IV.3 pages 226–248, and to [44], chapter V.1, for the complete description, but present some essential parts.

M.1 Gravitational space structures and their extrema

From his approach to take into account the field mass in the overall energy–mass balance of gravity (compare Section 3.1) Heim derives a non-linear differential equation (see page 85 in [43]) with an implicit solution for the gravitational potential ϕ(r)

(186) r q e − q = A ( 1 − r / ρ ) 2 w i t h q = 1 − 1 − ε ϕ , M 0 = L m 0 , ε = 3 8 c 2

which can be calculated approximately after expansion of qe^−q ≈ ɛϕ/2 to first order and fixing of the integration constant A so that the empirical Newtonian law is reproduced for r ≪ ρ:

(187) A = ε γ M 0 2 , ϕ ( r ) → γ L m ( r ) r ( 1 − r / ρ ) 2

γ is the gravitational constant, M ₀ the macroscopic field source, consisting of L microscopic elementary masses m ₀. This means ϕ > 0 and accelleration g < 0 for r < ρ, i.e. an attractive gravitational field which is limited at r = ρ with ϕ(ρ) = 0 and g(ρ) = 0. For r > ρ in contrast, ϕ > 0 increases again, and with g > 0 a weak repelling field appears. To determine the distance ρ one realises that ρ should be related to a quantum wavelength λ′ = w/ν _w with ϵ _g = hν _w, as λ′ = 2ρ if we consider L = 1, thus only an atomic mass unit m ₀ of M ₀. An approximation in a Taylor series of (186) (qe ^−q to second order) gives ( 1 − r / ρ ) 2 ∼ r q e − q ≈ 1 2 r ε ϕ ( 1 − ε ϕ / 2 ) or m(r)(1 − ɛϕ/2) = const due to (187), thus m(r)(1 − ɛϕ(r)/2) = m ₀(1 − ɛϕ(r ₀)/2) or m(r) − m ₀ = m(r)ɛϕ(r)/2 − m ₀ ɛϕ(r ₀)/2. With ϕ(ρ) = 0 this leads to a field mass of μ g = m ( ρ ) − m 0 = − ε m 0 ϕ ( r 0 ) / 2 = − ε m 0 2 γ 2 r 0 ( 1 − r 0 / ρ ) 2 . On the other hand μ _g = −h/(2wρ) follows from ϵ g = − μ g w 2 = h ν w = h w 2 ρ . If the length 2r ₀ is identified with the Compton wavelength of the energy m ₀ c ², λ = h m 0 c = 2 r 0 , the equation ρ ( 1 − h 2 m 0 c ρ ) 2 = h 2 γ m 0 3 is obtained^[64] and thereby the approximate solution

(188) ρ = h 2 γ m 0 3

for sufficiently weak fields.^[65] Reference (186)–(187) is not yet a structure-theoretical relation, but a phenomenological extension of the basic laws d1, d2 and c.

Now one can derive extrema of the space structure (186) with help of the fact that the r.h.s. of (186) is always real, Im(qe ^−q) = 0 → 1 − ɛϕ ≥ 0. Is r = R > ρ the position of this border of reality, this means ϕ(R) = ϕ _ext = 1/ɛ or ɛγM(R) = R(1 − R/ρ)⁻². On the other hand q(R) = 1 holds (from (186)) which provides ɛγM(R) = eA and thus M(R) = M ₀ e/2. This leads to a quadratic equation for x = R/ρ:

( 1 − x ) 2 = 2 ρ e ε γ M 0 x . With the abbreviation α = ρ e ε γ M 0 = 8 c 2 ρ 3 e γ M 0 the two solutions appear

(189) R ± = ρ ( α + 1 ) ( 1 ± 1 − ( α + 1 ) − 2 ) , f o r α ≫ 1 → ρ α ( 1 ± 1 − α − 2 ) → ρ α ( 1 ± ( 1 − 1 / ( 2 α 2 ) ) ) .

The last convergence is valid if all terms ( 1 / α 2 ) ν ≈ 0 for ν > 1 which always is fulfilled approximately. Then, the very simplified approximation can be derived

(190) R + = 2 α ρ , R − = ρ 2 α , R + R − = ρ 2

from which R ₊ < ∞ follows. With α an obviously microscopic value is obtained for the other limit

(191) R − = e 2 ε γ M 0 = 3 e 16 γ M 0 c 2

which is largely identical to the Schwarzschild radius of general relativity, which in turn is a measure for the extension of a black hole.

Now the existence of only one single mass system M ₀ = Lm ₀ in the R ₃ is considered, with L = 1 being reduced to one atomic element with mass m _M(R). This gives

(192) R − = ρ M ( α M + 1 ) ( 1 − 1 − ( α M + 1 ) − 2 )

where α _M is the corresponding value of α with m _M(R) = M ₀ e/2. According to (187), R ₋ fulfills the reality border 1 − ɛϕ(R ₋) = 0 as per R − / ε = γ m M ( R − ) ( 1 − R − / ρ M ) 2 . With the Compton wavelength λ = h/(m _M c) this becomes λ R − = γ h 2 w c 2 ( 1 − R − / ρ M ) 2 (expressions for ɛ and w used). For empty space λ → ∞ the limits lim_λ→∞ m _M(R ₋) = lim_λ→∞1/ρ _M = lim_λ→∞ R ₋ = 0 are reached (due to (188) and (191)), for lim λ → ∞ 1 / α M ∼ lim λ → ∞ R − ρ M = 0 , too. With these results the limit of the improper product λR ₋ can be determined as well:

(193) τ = lim λ → ∞ ( λ R − ) = γ h 2 w c 2 = 3 γ h 8 c 3 = const > 0

τ obviously is a universal constant with the dimension of an area that only depends on other fundamental physical constants which determine the laws of nature d1, d2 and c. This constant, which Heim calls “metron” and which we already introduced in Section 3.3, (24), is obviously independent of whether the space is empty or not. Since the relation (15) cannot be executed to an infinitesimal R ₄-element, the term (193) must be extended to the R ₄, but to the R ₆ as well, as such differences also arise for R ₆-volumes due to Eq. (28).

The results above serve Heim as ingredients to derive further cosmological theorems which lie beyond the scope of this work and can be found in [44].

M.2 Mass formula from elementary hermetric structures

The next step towards a calculation of a minimum mass consists of an analysis of the structural properties of the c hermetry. In subsection 3.4.5 we found the relation (36) between the coordinates r and ξ for the respective eigenvalue spectra. ξ ² = ϵ ² + η ² and squaring Eq. (36) yields r 2 − ( 2 n r + 1 2 n ξ + 1 ) 2 ϵ 2 = ( 2 n r + 1 2 n ξ + 1 ) 2 η 2 (n _ξ is identical to n ₋ of (36)). The two quantum numbers describe the course of the metronic eigenvalues in the range 0 ≤ r ² < ξ ² where n _r < n _ξ must hold. For r = ϵ, i.e. on the singular area in case of η ≠ 0, η/r becomes η / r = ± ( 2 n ξ + 1 2 n r + 1 ) 2 − 1 = ± 2 ( 2 n r + 1 ) − 1 ( n ξ + n r + 1 ) ( n ξ − n r ) . With the definitions n _ξ = n and n _r = n _ξ − j we get η / r = ± 2 ( 2 n + 1 − 2 j ) − 1 j ( 2 n + 1 − j ) and for j _min = 1 which provides the closest approximation n _r → n _ξ and hence minimal value for η/r, η / r = ± 2 ( 2 n − 1 ) − 1 2 n ≔ ± 2 f ( n ) .

Heim’s strategy is now to find a relation of two fundamental lengths which could be equated with η/r and thereby define a mass spectrum. He considers the characteristic distances R ₋, λ _C/2, ρ and R ₊. λ _C = h/(mc) is the Compton wavelength of a matter quantum. It can be seen that only the relations R ₋/(λ _C/2), R ₋/ρ and λ _C/(2ρ) could serve for this purpose, amongst others as the relation | η / r | ≤ 2 2 must be fulfilled. Since the mass terms lie in the subatomic range, the approximations of (188) and (189) can be used, (189) in the approximate form ρ = 2(1 + α)R ₋. With (188) and the definition μ = c h / γ the term α = 8 c 2 ρ 3 e γ m can be written as α = 8 3 e ( μ / m ) 4 . Inserting supplies R − / ( λ C / 2 ) = ( μ / m ) 2 ( 1 + α ) − 1 = ( m / μ ) 2 ( ( m / μ ) 4 + 8 3 e ) − 1 = 2 f ( n ) , R − / ρ = 1 2 ( 1 + α ) − 1 = 1 2 ( m / μ ) 4 ( ( m / μ ) 4 + 8 3 e ) − 1 = 2 f ( n ) and λ C / ( 2 ρ ) = 1 2 ( m / μ ) 2 = 2 f ( n ) .

Solving the first two equations for the mass m shows that m becomes complex (Im(m) ≠ 0) for 1 ≤ n ≤ 8, which contradicts physical reality. The third solution (m/μ)² = 4f(n) in contrast remains real for all n > 0 and m = 0 for n = 0. This means that this solution is the only one which provides a physically relevant expression for the mass. The mass spectrum for neutral quanta of the c hermetry, which implicitly includes all possible gravitational and photonic condensations and therefore cannot yet serve as a discrete partial spectrum of matter particles, thus is to be set as

(194) m ( n ) = 2 c h γ 2 n 4 2 n − 1 .

Next Heim analyses how the obtained result must be expanded for the d hermetry, which has the same complex coordinate structure x ⃗ = r ⃗ + i ξ ⃗ , but with ξ ⃗ containing also the time coordinate ct. We learned in Subsection 3.4.5 that the Hermetry form d describes charged particles. Thus the influence of the coupling of the electromagnetic field on the energy and hence mass terms has to be developed. After a somewhat longer heuristic derivation, which as another result provides a formula for the fine structure constant,^[66] Heim obtains the result

(195) m ( n , q ) = 1 2 m ( n ) η q , η q = π 4 q 4 + π 4 4

where q is the electric charge quantum number. Note that Heim adjusted the result by a factor 1/2 also for neutral masses (q = 0). From (194) can be seen that there is an upper limit m max = c h / γ 2 4 η q (for n = 1) which is (regarding the order of magnitude) identical to the maximons once conceived in particle physics [77].^[67]

M.3 The minimal mass

In chapter V.1 “The minimal complex condensation” of [44] Heim derives a minimal mass for matter quanta, i.e. for the c and d hermetries, based on the results described in the two previous subsections. Starting from the relations (190), one can derive ρ ² = eR ₊ A, with A defined as in (187) which can also be expressed through the metron and the Compton wavelengh as A = τ 2 λ . The last two expressions combined give eR ₊ τ = 2λρ ².

Referring to formula (194), Heim now reasons that a minimal mass is reached for a maximum number of metrons n = n _N by which the distance R ₊ is represented. In the metron picture he makes the ansatz that the volume 2λρ ² is projected onto the meridian plane of the matter quantum m by a grid selector C which operates on the number of metrons C n = p τ n . Here p is a projective factor.^[68] Then the field source is expressed by this metron number n along the distance R ₊: e R + τ = 2 λ ρ 2 = F C n = p F τ n where F is the unit area that, according to a projection of all spatial potential areas of the field structure in the plane R ₂, is limited by a contour line: F = π s 0 2 for spherical potential areas and p 2 F = s 0 = 1 [ m ] , thus p = π ^−1/4.^[69] Then

(196) e τ R + = p F n = π 3 / 4 n

results. From this relation and from (188), which is an optimal approximation for a minimum mass, n = 16 h 2 μ 4 3 π 3 / 4 γ m 7 τ can be obtained. For this minimum and thus a maximum n = n _N, (194) and (195) can be approximated as m μ η q 4 = 2 n N ( 2 n N − 1 ) 2 ≈ 1 2 n N , hence n N ≈ 1 2 μ η q m 4 . Equating both expressions for n, n _N yields 32 h 2 3 π 3 / 4 γ m 3 τ = η q 4 and thus, with the expression (193) for τ, the minimum mass results as

(197) m = 4 π 4 c η q η q 3 3 π γ ℏ 3 c ℏ 3 γ .

Inserting the fundamental physical constants c, ℏ and γ gives m ₁ = 0.5137 MeV for charge number q = 1, which is close to the mass of the electron. For q = 0 formula (197) yields m ₀ = 0.5069 MeV. Finally, Heim considers the value m ₍₀₎ = m ₁/η ₁ = 0.5189 MeV which can be considered as neutral complement of m ₁ according to (195). In the frame of his full mass spectrum theory (which we do not follow in this paper due to the reasons mentioned in Subsection 4.1), Heim derives a more complex formula which reproduces the electron mass even better, and he associates m ₀ with a speculative neutral electron which, however, has never been found in experiments so far (but cannot be easily detected either).

Appendix N: Complete calculated mass spectrum

Table 3:

Calculation of rest energies (masses) in MeV for scenario 2: Only even numbers of mass units (m = 2) for bosons and only odd numbers of mass units (m = 1) for fermions allowed.

Particle	Mass (MeV)	m	n ̲	N	mN	E ₁ C = 0.9^*	Error %	E ₂ C = 1.0^*	Error %	E ₃	Error %	E ₄	Error %
Leptons:
e	0.511	1	0	1	1	0.511	0.00	0.511	0.00	0.511	0.00	0.511	0.00
μ	105.66	1	2	3	3	106.26	0.57	106.43	0.73	104.51	1.09	105.04	0.59
τ	1776.86	1	2	51	51	1779.65	0.16	1778.60	0.10	1776.70	0.01	1785.65	0.49
Quarks:
u, d	315.00^**	1	2	9	9	316.16	0.37	315.44	0.14	313.54	0.46	315.11	0.04
s	525.00^**	1	2	15	15	525.34	0.07	524.46	0.10	522.56	0.46	525.19	0.04
c	1270.00	1	2	37	37	1291.90	1.72	1290.88	1.64	1288.98	1.49	1295.47	2.01
b	4180.00	1	2	119	119	4148.62	0.75	4147.53	0.78	4145.63	0.82	4166.51	0.32
t	172,760.0	1	4	73	73	173,774.9	0.59	173,635.4	0.51	173,376.5	0.36	175,127.0	1.37
Mesons:
unflavored:
π ^±	139.57	2	2	2	4	139.35	0.16	139.35	0.16	139.35	0.16	140.05	0.34
π ⁰	134.98	2	2	2	4	139.35	3.24	139.35	3.24	139.35	3.24	140.05	3.76
η	547.86	2	2	8	16	557.40	1.74	557.40	1.74	557.40	1.74	560.20	2.25
f ₀(500)	400–550	2	2	7	14	493.84		491.53		487.72		490.18
ρ(770)	775.26	2	2	11	22	772.51	0.36	770.22	0.65	766.42	1.14	770.28	0.64
ω(782)	782.65	2	2	11	22	772.51	1.30	770.22	1.59	766.42	2.07	770.28	1.58
η′(958)	957.78	2	2	14	28	975.44	1.84	975.44	1.84	975.44	1.84	980.36	2.36
f ₀(980)	990.00	2	2	14	28	975.44	1.47	975.44	1.47	975.44	1.47	980.36	0.97
a ₀(980)	980.00	2	2	14	28	975.44	0.46	975.44	0.46	975.44	0.46	980.36	0.04
ϕ(1020)	1019.46	2	2	15	30	1051.19	3.11	1048.92	2.89	1045.12	2.52	1050.38	3.03
h ₁(1170)	1166.00	2	2	17	34	1190.54	2.10	1188.27	1.91	1184.47	1.58	1190.43	2.10
b ₁(1235)	1229.50	2	2	18	36	1254.14	2.00	1254.14	2.00	1254.14	2.00	1260.46	2.52
a ₁(1260)	1230.00	2	2	18	36	1254.14	1.96	1254.14	1.96	1254.14	1.96	1260.46	2.48
f ₂(1270)	1275.50	2	2	18	36	1254.14	1.67	1254.14	1.67	1254.14	1.67	1260.46	1.18
f ₁(1285)	1281.90	2	2	18	36	1254.14	2.17	1254.14	2.17	1254.14	2.17	1260.46	1.67
η(1295)	1294.00	2	2	19	38	1329.88	2.77	1327.62	2.60	1323.82	2.30	1330.48	2.82
π(1300)	1300.00	2	2	19	38	1329.88	2.30	1327.62	2.12	1323.82	1.83	1330.48	2.34
a ₂(1320)	1316.90	2	2	19	38	1329.88	0.99	1327.62	0.81	1323.82	0.53	1330.48	1.03
f ₀(1370)	1200–1500	2	2	19	38	1329.88		1327.62		1323.82		1330.48
π ₁(1400)	1354.00	2	2	19	38	1329.88	1.78	1327.62	1.95	1323.82	2.23	1330.48	1.74
η(1405)	1408.80	2	2	20	40	1393.49	1.09	1393.49	1.09	1393.49	1.09	1400.51	0.59
h ₁(1415)	1416.00	2	2	20	40	1393.49	1.59	1393.49	1.59	1393.49	1.59	1400.51	1.09
a ₁(1420)	1411.00	2	2	20	40	1393.49	1.24	1393.49	1.24	1393.49	1.24	1400.51	0.74
f ₁(1420)	1426.30	2	2	20	40	1393.49	2.30	1393.49	2.30	1393.49	2.30	1400.51	1.81
ω(1420)	1410.00	2	2	20	40	1393.49	1.17	1393.49	1.17	1393.49	1.17	1400.51	0.67
f ₂(1430)	1430.00	2	2	20	40	1393.49	2.55	1393.49	2.55	1393.49	2.55	1400.51	2.06
a ₀(1450)	1474.00	2	2	21	42	1469.23	0.32	1466.96	0.48	1463.17	0.74	1470.53	0.24
ρ(1450)	1465.00	2	2	21	42	1469.23	0.29	1466.96	0.13	1463.17	0.13	1470.53	0.38
η(1475)	1475.00	2	2	21	42	1469.23	0.39	1466.96	0.54	1463.17	0.80	1470.53	0.30
f ₀(1500)	1506.00	2	2	22	44	1532.84	1.78	1532.84	1.78	1532.84	1.78	1540.56	2.29
f ₁(1510)	1518.00	2	2	22	44	1532.84	0.98	1532.84	0.98	1532.84	0.98	1540.56	1.49
f 2 ′ (1525)	1517.40	2	2	22	44	1532.84	1.02	1532.84	1.02	1532.84	1.02	1540.56	1.53
f ₂(1565)	1542.00	2	2	22	44	1532.84	0.59	1532.84	0.59	1532.84	0.59	1540.56	0.09
ρ(1570)	1570.00	2	2	22	44	1532.84	2.37	1532.84	2.37	1532.84	2.37	1540.56	1.88
h ₁(1595)	1594.00	2	2	23	46	1608.57	0.91	1606.31	0.77	1602.51	0.53	1610.58	1.04
π ₁(1600)	1660.00	2	2	24	48	1672.19	0.73	1672.19	0.73	1672.19	0.73	1680.61	1.24
a ₁(1640)	1655.00	2	2	24	48	1672.19	1.04	1672.19	1.04	1672.19	1.04	1680.61	1.55
f ₂(1640)	1639.00	2	2	23	46	1608.57	1.86	1606.31	1.99	1602.51	2.23	1610.58	1.73
η ₂(1645)	1617.00	2	2	23	46	1608.57	0.52	1606.31	0.66	1602.51	0.90	1610.58	0.40
ω(1650)	1670.00	2	2	24	48	1672.19	0.13	1672.19	0.13	1672.19	0.13	1680.61	0.64
ω ₃(1670)	1667.00	2	2	24	48	1672.19	0.31	1672.19	0.31	1672.19	0.31	1680.61	0.82
π ₂(1670)	1670.60	2	2	24	48	1672.19	0.10	1672.19	0.10	1672.19	0.10	1680.61	0.60
ϕ(1680)	1680.00	2	2	24	48	1672.19	0.46	1672.19	0.46	1672.19	0.46	1680.61	0.04
ρ ₃(1690)	1688.80	2	2	24	48	1672.19	0.98	1672.19	0.98	1672.19	0.98	1680.61	0.48
ρ(1700)	1720.00	2	2	25	50	1747.92	1.62	1745.66	1.49	1741.86	1.27	1750.63	1.78
a ₂(1700)	1705.00	2	2	24	48	1672.19	1.92	1672.19	1.92	1672.19	1.92	1680.61	1.43
f ₀(1710)	1704.00	2	2	24	48	1672.19	1.87	1672.19	1.87	1672.19	1.87	1680.61	1.37
η(1760)	1751.00	2	2	25	50	1747.92	0.18	1745.66	0.30	1741.86	0.52	1750.63	0.02
π(1800)	1810.00	2	2	26	52	1811.54	0.08	1811.54	0.08	1811.54	0.08	1820.66	0.59
f ₂(1810)	1815.00	2	2	26	52	1811.54	0.19	1811.54	0.19	1811.54	0.19	1820.66	0.31
X(1835)	1826.50	2	2	26	52	1811.54	0.82	1811.54	0.82	1811.54	0.82	1820.66	0.32
ϕ ₃(1850)	1854.00	2	2	27	54	1887.27	1.79	1885.01	1.67	1881.21	1.47	1890.69	1.98
η ₂(1870)	1842.00	2	2	26	52	1811.54	1.65	1811.54	1.65	1811.54	1.65	1820.66	1.16
π ₂(1880)	1874.00	2	2	27	54	1887.27	0.71	1885.01	0.59	1881.21	0.38	1890.69	0.89
ρ(1900)	1900.00	2	2	27	54	1887.27	0.67	1885.01	0.79	1881.21	0.99	1890.69	0.49
f ₂(1910)	1900.00	2	2	27	54	1887.27	0.67	1885.01	0.79	1881.21	0.99	1890.69	0.49
a ₀(1950)	1931.00	2	2	28	56	1950.89	1.03	1950.89	1.03	1950.89	1.03	1960.71	1.54
f ₂(1950)	1936.00	2	2	28	56	1950.89	0.77	1950.89	0.77	1950.89	0.77	1960.71	1.28
a ₄(1970)	1967.00	2	2	28	56	1950.89	0.82	1950.89	0.82	1950.89	0.82	1960.71	0.32
ρ ₃(1990)	1990.00	2	2	29	58	2026.62	1.84	2024.36	1.73	2020.56	1.54	2030.74	2.05
π ₂(2005)	1963.00	2	2	28	56	1950.89	0.62	1950.89	0.62	1950.89	0.62	1960.71	0.12
f ₂(2010)	2011.00	2	2	29	58	2026.62	0.78	2024.36	0.66	2020.56	0.48	2030.74	0.98
f ₀(2020)	1992.00	2	2	29	58	2026.62	1.74	2024.36	1.62	2020.56	1.43	2030.74	1.94
f ₄(2050)	2018.00	2	2	29	58	2026.62	0.43	2024.36	0.32	2020.56	0.13	2030.74	0.63
π ₂(2100)	2090.00	2	2	30	60	2090.24	0.01	2090.24	0.01	2090.24	0.01	2100.76	0.51
f ₀(2100)	2086.00	2	2	30	60	2090.24	0.20	2090.24	0.20	2090.24	0.20	2100.76	0.71
f ₂(2150)	2157.00	2	2	31	62	2165.96	0.42	2163.71	0.31	2159.91	0.13	2170.79	0.64
ρ(2150)	2150.00	2	2	31	62	2165.96	0.74	2163.71	0.64	2159.91	0.46	2170.79	0.97
ϕ(2170)	2160.00	2	2	31	62	2165.96	0.28	2163.71	0.17	2159.91	0.00	2170.79	0.50
f ₀(2200)	2187.00	2	2	31	62	2165.96	0.96	2163.71	1.06	2159.91	1.24	2170.79	0.74
f _J(2220)	2231.10	2	2	32	64	2229.59	0.07	2229.59	0.07	2229.59	0.07	2240.81	0.44
η(2225)	2221.00	2	2	32	64	2229.59	0.39	2229.59	0.39	2229.59	0.39	2240.81	0.89
ρ ₃(2250)	2250.00	2	2	32	64	2229.59	0.91	2229.59	0.91	2229.59	0.91	2240.81	0.41
f ₂(2300)	2297.00	2	2	33	66	2305.31	0.36	2303.06	0.26	2299.26	0.10	2310.84	0.60
f ₄(2300)	2300.00	2	2	33	66	2305.31	0.23	2303.06	0.13	2299.26	0.03	2310.84	0.47
f ₀(2330)	2330.00	2	2	33	66	2305.31	1.06	2303.06	1.16	2299.26	1.32	2310.84	0.82
f ₂(2340)	2345.00	2	2	34	68	2368.93	1.02	2368.93	1.02	2368.93	1.02	2380.86	1.53
ρ ₅(2350)	2330.00	2	2	33	66	2305.31	1.06	2303.06	1.16	2299.26	1.32	2310.84	0.82
f ₆(2510)	2465.00	2	2	35	70	2444.66	0.83	2442.41	0.92	2438.61	1.07	2450.89	0.57
strange:
K ^±	493.68	2	2	7	14	493.84	0.03	491.53	0.43	487.72	1.21	490.18	0.71
K ⁰	497.61	2	2	7	14	493.84	0.76	491.53	1.22	487.72	1.99	490.18	1.49
K 0 ⋆ (700)	630–730	2	2	10	20	696.75		696.75		696.75		700.25
K ^⋆(892)	894.24	2	2	13	26	911.85	1.97	909.57	1.71	905.77	1.29	910.33	1.80
K ₁(1270)	1253.00	2	2	18	36	1254.14	0.09	1254.14	0.09	1254.14	0.09	1260.46	0.60
K ₁(1400)	1403.00	2	2	20	40	1393.49	0.68	1393.49	0.68	1393.49	0.68	1400.51	0.18
K ^⋆(1410)	1414.00	2	2	20	40	1393.49	1.45	1393.49	1.45	1393.49	1.45	1400.51	0.95
K 0 ⋆ (1430)	1425.00	2	2	20	40	1393.49	2.21	1393.49	2.21	1393.49	2.21	1400.51	1.72
K 2 ⋆ (1430)	1429.85	2	2	20	40	1393.49	2.54	1393.49	2.54	1393.49	2.54	1400.51	2.05
K(1460)	1460.00	2	2	21	42	1469.23	0.63	1466.96	0.48	1463.17	0.22	1470.53	0.72
K ₂(1580)	1580.00	2	2	23	46	1608.57	1.81	1606.31	1.67	1602.51	1.42	1610.58	1.94
K(1630)	1629.00	2	2	23	46	1608.57	1.25	1606.31	1.39	1602.51	1.63	1610.58	1.13
K ₁(1650)	1672.00	2	2	24	48	1672.19	0.01	1672.19	0.01	1672.19	0.01	1680.61	0.51
K ^⋆(1680)	1718.00	2	2	25	50	1747.92	1.74	1745.66	1.61	1741.86	1.39	1750.63	1.90
K ₂(1770)	1773.00	2	2	25	50	1747.92	1.41	1745.66	1.54	1741.86	1.76	1750.63	1.26
K 3 ⋆ (1780)	1776.00	2	2	25	50	1747.92	1.58	1745.66	1.71	1741.86	1.92	1750.63	1.43
K ₂(1820)	1819.00	2	2	26	52	1811.54	0.41	1811.54	0.41	1811.54	0.41	1820.66	0.09
K(1830)	1874.00	2	2	27	54	1887.27	0.71	1885.01	0.59	1881.21	0.38	1890.69	0.89
K 0 ⋆ (1950)	1945.00	2	2	28	56	1950.89	0.30	1950.89	0.30	1950.89	0.30	1960.71	0.81
K 2 ⋆ (1980)	1943.00	2	2	28	56	1950.89	0.41	1950.89	0.41	1950.89	0.41	1960.71	0.91
K 4 ⋆ (2045)	2048.00	2	2	29	58	2026.62	1.04	2024.36	1.15	2020.56	1.34	2030.74	0.84
K ₂(2250)	2247.00	2	2	32	64	2229.59	0.78	2229.59	0.78	2229.59	0.78	2240.81	0.28
K ₃(2320)	2324.00	2	2	33	66	2305.31	0.80	2303.06	0.90	2299.26	1.06	2310.84	0.57
K 5 ⋆ (2380)	2382.00	2	2	34	68	2368.93	0.55	2368.93	0.55	2368.93	0.55	2380.86	0.05
K ₄(2500)	2490.00	2	2	36	72	2508.28	0.73	2508.28	0.73	2508.28	0.73	2520.91	1.24
K(3100)	3100.00	2	2	44	88	3065.68	1.11	3065.68	1.11	3065.68	1.11	3081.12	0.61
charmed:
D ^±	1869.65	2	2	27	54	1887.27	0.94	1885.01	0.82	1881.21	0.62	1890.69	1.13
D ⁰	1864.83	2	2	27	54	1887.27	1.20	1885.01	1.08	1881.21	0.88	1890.69	1.39
D ^⋆(2007)⁰	2006.85	2	2	29	58	2026.62	0.98	2024.36	0.87	2020.56	0.68	2030.74	1.19
D ^⋆(2010)^±	2010.26	2	2	29	58	2026.62	0.81	2024.36	0.70	2020.56	0.51	2030.74	1.02
D 0 ⋆ ( 2300 ) 0	2300.00	2	2	33	66	2305.31	0.23	2303.06	0.13	2299.26	0.03	2310.84	0.47
D 0 ⋆ ( 2300 ) ±	2349.00	2	2	34	68	2368.93	0.85	2368.93	0.85	2368.93	0.85	2380.86	1.36
D ₁(2420)⁰	2420.50	2	2	35	70	2444.66	1.00	2442.41	0.91	2438.61	0.75	2450.89	1.26
D ₁(2420)^±	2423.20	2	2	35	70	2444.66	0.89	2442.41	0.79	2438.61	0.64	2450.89	1.14
D ₁(2430)⁰	2427.00	2	2	35	70	2444.66	0.73	2442.41	0.63	2438.61	0.48	2450.89	0.98
D 2 ⋆ ( 2460 ) 0	2460.56	2	2	35	70	2444.66	0.65	2442.41	0.74	2438.61	0.89	2450.89	0.39
D 2 ⋆ ( 2460 ) ±	2465.40	2	2	35	70	2444.66	0.84	2442.41	0.93	2438.61	1.09	2450.89	0.59
D(2550)⁰	2564.00	2	2	37	74	2584.01	0.78	2581.76	0.69	2577.96	0.54	2590.94	1.05
D J ⋆ (2600)	2623.00	2	2	38	76	2647.63	0.94	2647.63	0.94	2647.63	0.94	2660.97	1.45
D ^⋆(2640)^±	2637.00	2	2	38	76	2647.63	0.40	2647.63	0.40	2647.63	0.40	2660.97	0.91
D(2740)⁰	2737.00	2	2	39	78	2723.36	0.50	2721.10	0.58	2717.31	0.72	2730.99	0.22
D 3 ⋆ (2750)	2763.50	2	2	40	80	2786.98	0.85	2786.98	0.85	2786.98	0.85	2801.02	1.36
D(3000)⁰	3214.00	2	2	46	92	3205.03	0.28	3205.03	0.28	3205.03	0.28	3221.17	0.22
charmed, strange:
D s ±	1968.34	2	2	28	56	1950.89	0.89	1950.89	0.89	1950.89	0.89	1960.71	0.39
D s ⋆ ±	2112.20	2	2	30	60	2090.24	1.04	2090.24	1.04	2090.24	1.04	2100.76	0.54
D s 0 ⋆ ( 2317 ) ±	2317.80	2	2	33	66	2305.31	0.54	2303.06	0.64	2299.26	0.80	2310.84	0.30
D _s1(2460)^±	2459.50	2	2	35	70	2444.66	0.60	2442.41	0.69	2438.61	0.85	2450.89	0.35
D _s1(2536)^±	2535.11	2	2	36	72	2508.28	1.06	2508.28	1.06	2508.28	1.06	2520.91	0.56
D s 2 ⋆ (2573)	2569.10	2	2	37	74	2584.01	0.58	2581.76	0.49	2577.96	0.34	2590.94	0.85
D s 1 ⋆ ( 2700 ) ±	2708.30	2	2	39	78	2723.36	0.56	2721.10	0.47	2717.31	0.33	2730.99	0.84
D s 1 ⋆ ( 2860 ) ±	2859.00	2	2	41	82	2862.70	0.13	2860.45	0.05	2856.66	0.08	2871.04	0.42
D s 3 ⋆ ( 2860 ) ±	2860.50	2	2	41	82	2862.70	0.08	2860.45	0.00	2856.66	0.13	2871.04	0.37
D _sJ(3040)^±	3044.00	2	2	44	88	3065.68	0.71	3065.68	0.71	3065.68	0.71	3081.12	1.22
bottom:
B ^±	5279.34	2	2	76	152	5295.26	0.30	5295.26	0.30	5295.26	0.30	5321.93	0.81
B ⁰	5279.65	2	2	76	152	5295.26	0.30	5295.26	0.30	5295.26	0.30	5321.93	0.80
B ^⋆	5324.70	2	2	76	152	5295.26	0.55	5295.26	0.55	5295.26	0.55	5321.93	0.05
B ₁(5721)⁺	5725.90	2	2	82	164	5713.31	0.22	5713.31	0.22	5713.31	0.22	5742.08	0.28
B ₁(5721)⁰	5726.10	2	2	82	164	5713.31	0.22	5713.31	0.22	5713.31	0.22	5742.08	0.28
B J ⋆ (5732)	5698.00	2	2	82	164	5713.31	0.27	5713.31	0.27	5713.31	0.27	5742.08	0.77
B 2 ⋆ ( 5747 ) +	5737.20	2	2	82	164	5713.31	0.42	5713.31	0.42	5713.31	0.42	5742.08	0.09
B 2 ⋆ ( 5747 ) 0	5739.50	2	2	82	164	5713.31	0.46	5713.31	0.46	5713.31	0.46	5742.08	0.04
B _J(5840)⁺	5851.00	2	2	84	168	5852.66	0.03	5852.66	0.03	5852.66	0.03	5882.13	0.53
B _J(5840)⁰	5863.00	2	2	84	168	5852.66	0.18	5852.66	0.18	5852.66	0.18	5882.13	0.33
B _J(5970)⁺	5964.00	2	2	86	172	5992.01	0.47	5992.01	0.47	5992.01	0.47	6022.18	0.98
B _J(5970)⁰	5971.00	2	2	86	172	5992.01	0.35	5992.01	0.35	5992.01	0.35	6022.18	0.86
bottom, strange:
B s 0	5366.88	2	2	77	154	5370.98	0.08	5368.74	0.03	5364.94	0.04	5391.96	0.47
B s ⋆	5415.40	2	2	78	156	5434.61	0.35	5434.61	0.35	5434.61	0.35	5461.98	0.86
X(5568)^±	5566.90	2	2	80	160	5573.96	0.13	5573.96	0.13	5573.96	0.13	5602.03	0.63
B _s1(5830)⁰	5828.70	2	2	84	168	5852.66	0.41	5852.66	0.41	5852.66	0.41	5882.13	0.92
B s 2 ⋆ ( 5840 ) 0	5839.86	2	2	84	168	5852.66	0.22	5852.66	0.22	5852.66	0.22	5882.13	0.72
B s J ⋆ (5850)	5853.00	2	2	84	168	5852.66	0.01	5852.66	0.01	5852.66	0.01	5882.13	0.50
bottom, charmed:
B c +	6274.90	2	2	90	180	6270.71	0.07	6270.71	0.07	6270.71	0.07	6302.29	0.44
B _c(2S)^±	6871.60	2	2	99	198	6903.82	0.47	6901.57	0.44	6897.78	0.38	6932.51	0.89
c c ̄ : ^***
η _c(1S)	2983.90	2	2	43	86	3002.05	0.61	2999.80	0.53	2996.01	0.41	3011.09	0.91
J/ψ(1S)	3096.90	2	2	44	88	3065.68	1.01	3065.68	1.01	3065.68	1.01	3081.12	0.51
χ _c0(1P)	3414.71	2	2	49	98	3420.10	0.16	3417.85	0.09	3414.05	0.02	3431.24	0.48
χ _c1(1P)	3510.67	2	2	50	100	3483.73	0.77	3483.73	0.77	3483.73	0.77	3501.27	0.27
h _c(1P)	3525.38	2	2	51	102	3559.45	0.97	3557.20	0.90	3553.40	0.79	3571.30	1.30
χ _c2(1P)	3556.17	2	2	51	102	3559.45	0.09	3557.20	0.03	3553.40	0.08	3571.30	0.43
η _c(2S)	3637.50	2	2	52	104	3623.08	0.40	3623.08	0.40	3623.08	0.40	3641.32	0.11
ψ(2S)	3686.10	2	2	53	106	3698.79	0.34	3696.55	0.28	3692.75	0.18	3711.35	0.68
ψ(3770)	3773.70	2	2	54	108	3762.42	0.30	3762.42	0.30	3762.42	0.30	3781.37	0.20
ψ ₂(3823)	3822.20	2	2	55	110	3838.14	0.42	3835.90	0.36	3832.10	0.26	3851.40	0.76
ψ ₃(3842)	3842.71	2	2	55	110	3838.14	0.12	3835.90	0.18	3832.10	0.28	3851.40	0.23
χ _c0(3860)	3862.00	2	2	55	110	3838.14	0.62	3835.90	0.68	3832.10	0.77	3851.40	0.27
χ _c1(3872)	3871.69	2	2	56	112	3901.77	0.78	3901.77	0.78	3901.77	0.78	3921.42	1.28
Z _c(3900)	3888.40	2	2	56	112	3901.77	0.34	3901.77	0.34	3901.77	0.34	3921.42	0.85
X(3915)	3918.40	2	2	56	112	3901.77	0.42	3901.77	0.42	3901.77	0.42	3921.42	0.08
χ _c2(3930)	3922.20	2	2	56	112	3901.77	0.52	3901.77	0.52	3901.77	0.52	3921.42	0.02
X(3940)	3942.00	2	2	57	114	3977.49	0.90	3975.25	0.84	3971.45	0.75	3991.45	1.25
X(4020)^±	4024.10	2	2	58	116	4041.12	0.42	4041.12	0.42	4041.12	0.42	4061.47	0.93
ψ(4040)	4039.00	2	2	58	116	4041.12	0.05	4041.12	0.05	4041.12	0.05	4061.47	0.56
X(4050)^±	4051.00	2	2	58	116	4041.12	0.24	4041.12	0.24	4041.12	0.24	4061.47	0.26
X(4055)^±	4054.00	2	2	58	116	4041.12	0.32	4041.12	0.32	4041.12	0.32	4061.47	0.18
X(4100)^±	4096.00	2	2	59	118	4116.84	0.51	4114.59	0.45	4110.80	0.36	4131.50	0.87
χ _c1(4140)	4146.80	2	2	59	118	4116.84	0.72	4114.59	0.78	4110.80	0.87	4131.50	0.37
ψ(4160)	4191.00	2	2	60	120	4180.47	0.25	4180.47	0.25	4180.47	0.25	4201.52	0.25
X(4160)	4156.00	2	2	60	120	4180.47	0.59	4180.47	0.59	4180.47	0.59	4201.52	1.10
Z _c(4200)	4196.00	2	2	60	120	4180.47	0.37	4180.47	0.37	4180.47	0.37	4201.52	0.13
ψ(4230)	4220.00	2	2	61	122	4256.19	0.86	4253.94	0.80	4250.15	0.71	4271.55	1.22
R _c0(4240)	4239.00	2	2	61	122	4256.19	0.41	4253.94	0.35	4250.15	0.26	4271.55	0.77
X(4250)^±	4248.00	2	2	61	122	4256.19	0.19	4253.94	0.14	4250.15	0.05	4271.55	0.55
ψ(4260)	4220.00	2	2	61	122	4256.19	0.86	4253.94	0.80	4250.15	0.71	4271.55	1.22
χ _c1(4274)	4274.00	2	2	61	122	4256.19	0.42	4253.94	0.47	4250.15	0.56	4271.55	0.06
X(4350)	4350.60	2	2	62	124	4319.82	0.71	4319.82	0.71	4319.82	0.71	4341.57	0.21
ψ(4360)	4368.00	2	2	63	126	4395.54	0.63	4393.29	0.58	4389.50	0.49	4411.60	1.00
ψ(4390)	4391.50	2	2	63	126	4395.54	0.09	4393.29	0.04	4389.50	0.05	4411.60	0.46
ψ(4415)	4421.00	2	2	63	126	4395.54	0.58	4393.29	0.63	4389.50	0.71	4411.60	0.21
Z _c(4430)	4478.00	2	2	64	128	4459.17	0.42	4459.17	0.42	4459.17	0.42	4481.63	0.08
χ _c0(4500)	4506.00	2	2	65	130	4534.89	0.64	4532.64	0.59	4528.84	0.51	4551.65	1.01
ψ(4660)	4633.00	2	2	66	132	4598.52	0.74	4598.52	0.74	4598.52	0.74	4621.68	0.24
χ _c0(4700)	4704.00	2	2	67	134	4674.24	0.63	4671.99	0.68	4668.19	0.76	4691.70	0.26
b b ̄ : ^***
η _b(1S)	9398.70	2	2	135	270	9412.10	0.14	9409.86	0.12	9406.06	0.08	9453.43	0.58
ϒ(1S)	9460.30	2	2	136	272	9475.74	0.16	9475.74	0.16	9475.74	0.16	9523.45	0.67
χ _b0(1P)	9859.44	2	2	141	282	9830.15	0.30	9827.91	0.32	9824.11	0.36	9873.58	0.14
χ _b1(1P)	9892.78	2	2	142	284	9893.78	0.01	9893.78	0.01	9893.78	0.01	9943.61	0.51
h _b(1P)	9899.30	2	2	142	284	9893.78	0.06	9893.78	0.06	9893.78	0.06	9943.61	0.45
χ _b2(1P)	9912.21	2	2	142	284	9893.78	0.19	9893.78	0.19	9893.78	0.19	9943.61	0.32
η _b(2S)	9999.00	2	2	144	288	10,033.13	0.34	10,033.13	0.34	10,033.13	0.34	10,083.66	0.85
ϒ(2S)	10,023.26	2	2	144	288	10,033.13	0.10	10,033.13	0.10	10,033.13	0.10	10,083.66	0.60
ϒ₂(1D)	10,163.70	2	2	146	292	10,172.48	0.09	10,172.48	0.09	10,172.48	0.09	10,223.71	0.59
χ _b0(2P)	10,232.50	2	2	147	294	10,248.19	0.15	10,245.95	0.13	10,242.16	0.09	10,293.73	0.60
χ _b1(2P)	10,255.46	2	2	147	294	10,248.19	0.07	10,245.95	0.09	10,242.16	0.13	10,293.73	0.37
h _b(2P)	10,259.80	2	2	147	294	10,248.19	0.11	10,245.95	0.13	10,242.16	0.17	10,293.73	0.33
χ _b2(2P)	10,268.65	2	2	147	294	10,248.19	0.20	10,245.95	0.22	10,242.16	0.26	10,293.73	0.24
ϒ(3S)	10,355.20	2	2	149	298	10,387.54	0.31	10,385.30	0.29	10,381.51	0.25	10,433.78	0.76
χ _b1(3P)	10,513.40	2	2	151	302	10,526.89	0.13	10,524.65	0.11	10,520.85	0.07	10,573.83	0.57
χ _b2(3P)	10,524.00	2	2	151	302	10,526.89	0.03	10,524.65	0.01	10,520.85	0.03	10,573.83	0.47
ϒ(4S)	10,579.40	2	2	152	304	10,590.53	0.11	10,590.53	0.11	10,590.53	0.11	10,643.86	0.61
Z _b(10610)	10,607.20	2	2	152	304	10,590.53	0.16	10,590.53	0.16	10,590.53	0.16	10,643.86	0.35
Z _b(10650)	10,652.20	2	2	153	306	10,666.24	0.13	10,664.00	0.11	10,660.20	0.08	10,713.89	0.58
ϒ(10753)	10,752.70	2	2	154	308	10,729.88	0.21	10,729.88	0.21	10,729.88	0.21	10,783.91	0.29
ϒ(10860)	10,885.20	2	2	156	312	10,869.23	0.15	10,869.23	0.15	10,869.23	0.15	10,923.96	0.36
ϒ(11020)	11,000.00	2	2	158	316	11,008.58	0.08	11,008.58	0.08	11,008.58	0.08	11,064.01	0.58
Baryons:
p	938.27	1	2	27	27	943.50	0.56	942.51	0.45	940.61	0.25	945.34	0.75
n	939.57	1	2	27	27	943.50	0.42	942.51	0.31	940.61	0.11	945.34	0.61
N(1440)	1440.00	1	2	41	41	1431.26	0.61	1430.23	0.68	1428.33	0.81	1435.52	0.31
N(1520)	1515.00	1	2	43	43	1500.94	0.93	1499.90	1.00	1498.00	1.12	1505.55	0.62
N(1535)	1530.00	1	2	43	43	1500.94	1.90	1499.90	1.97	1498.00	2.09	1505.55	1.60
N(1650)	1650.00	1	2	47	47	1640.30	0.59	1639.25	0.65	1637.35	0.77	1645.60	0.27
N(1675)	1675.00	1	2	47	47	1640.30	2.07	1639.25	2.13	1637.35	2.25	1645.60	1.76
N(1680)	1685.00	1	2	49	49	1709.97	1.48	1708.92	1.42	1707.03	1.31	1715.62	1.82
N(1700)	1720.00	1	2	49	49	1709.97	0.58	1708.92	0.64	1707.03	0.75	1715.62	0.25
N(1710)	1710.00	1	2	49	49	1709.97	0.00	1708.92	0.06	1707.03	0.17	1715.62	0.33
N(1720)	1720.00	1	2	49	49	1709.97	0.58	1708.92	0.64	1707.03	0.75	1715.62	0.25
N(1875)	1875.00	1	2	53	53	1849.33	1.37	1848.27	1.43	1846.38	1.53	1855.67	1.03
N(1880)	1880.00	1	2	53	53	1849.33	1.63	1848.27	1.69	1846.38	1.79	1855.67	1.29
N(1895)	1895.00	1	2	55	55	1919.00	1.27	1917.95	1.21	1916.05	1.11	1925.70	1.62
N(1900)	1920.00	1	2	55	55	1919.00	0.05	1917.95	0.11	1916.05	0.21	1925.70	0.30
N(1990)	2020.00	1	2	57	57	1988.68	1.55	1987.62	1.60	1985.72	1.70	1995.72	1.20
N(2000)	2000.00	1	2	57	57	1988.68	0.57	1987.62	0.62	1985.72	0.71	1995.72	0.21
N(2060)	2100.00	1	2	61	61	2128.03	1.33	2126.97	1.28	2125.07	1.19	2135.77	1.70
N(2100)	2100.00	1	2	61	61	2128.03	1.33	2126.97	1.28	2125.07	1.19	2135.77	1.70
N(2120)	2120.00	1	2	61	61	2128.03	0.38	2126.97	0.33	2125.07	0.24	2135.77	0.74
N(2190)	2180.00	1	2	63	63	2197.71	0.81	2196.65	0.76	2194.75	0.68	2205.80	1.18
N(2220)	2250.00	1	2	65	65	2267.39	0.77	2266.32	0.73	2264.42	0.64	2275.83	1.15
N(2250)	2280.00	1	2	65	65	2267.39	0.55	2266.32	0.60	2264.42	0.68	2275.83	0.18
N(2600)	2600.00	1	2	75	75	2615.77	0.61	2614.69	0.57	2612.80	0.49	2625.95	1.00
N(2700)	2612.00	1	2	75	75	2615.77	0.14	2614.69	0.10	2612.80	0.03	2625.95	0.53
Δ	1232.00	1	2	35	35	1222.22	0.79	1221.20	0.88	1219.30	1.03	1225.44	0.53
Δ(1600)	1570.00	1	2	45	45	1570.62	0.04	1569.58	0.03	1567.68	0.15	1575.57	0.35
Δ(1620)	1610.00	1	2	47	47	1640.30	1.88	1639.25	1.82	1637.35	1.70	1645.60	2.21
Δ(1700)	1710.00	1	2	49	49	1709.97	0.00	1708.92	0.06	1707.03	0.17	1715.62	0.33
Δ(1750)	n/a
Δ(1900)	1860.00	1	2	53	53	1849.33	0.57	1848.27	0.63	1846.38	0.73	1855.67	0.23
Δ(1905)	1880.00	1	2	53	53	1849.33	1.63	1848.27	1.69	1846.38	1.79	1855.67	1.29
Δ(1910)	1900.00	1	2	55	55	1919.00	1.00	1917.95	0.94	1916.05	0.84	1925.70	1.35
Δ(1920)	1920.00	1	2	55	55	1919.00	0.05	1917.95	0.11	1916.05	0.21	1925.70	0.30
Δ(1930)	1950.00	1	2	55	55	1919.00	1.59	1917.95	1.64	1916.05	1.74	1925.70	1.25
Δ(1940)	2000.00	1	2	57	57	1988.68	0.57	1987.62	0.62	1985.72	0.71	1995.72	0.21
Δ(1950)	1930.00	1	2	55	55	1919.00	0.57	1917.95	0.62	1916.05	0.72	1925.70	0.22
Δ(2000)	2015.00	1	2	57	57	1988.68	1.31	1987.62	1.36	1985.72	1.45	1995.72	0.96
Δ(2150)	2150.00	1	2	61	61	2128.03	1.02	2126.97	1.07	2125.07	1.16	2135.77	0.66
Δ(2200)	2200.00	1	2	63	63	2197.71	0.10	2196.65	0.15	2194.75	0.24	2205.80	0.26
Δ(2300)	2300.00	1	2	65	65	2267.39	1.42	2266.32	1.46	2264.42	1.55	2275.83	1.05
Δ(2350)	2350.00	1	2	67	67	2337.06	0.55	2336.00	0.60	2334.10	0.68	2345.85	0.18
Δ(2390)	2390.00	1	2	69	69	2406.74	0.70	2405.67	0.66	2403.77	0.58	2415.88	1.08
Δ(2400)	2400.00	1	2	69	69	2406.74	0.28	2405.67	0.24	2403.77	0.16	2415.88	0.66
Δ(2420)	2450.00	1	2	71	71	2476.41	1.08	2475.34	1.03	2473.45	0.96	2485.90	1.47
Δ(2750)	2794.00	1	2	81	81	2824.79	1.10	2823.72	1.06	2821.82	1.00	2836.03	1.50
Δ(2950)	2990.00	1	2	85	85	2964.14	0.86	2963.07	0.90	2961.17	0.96	2976.08	0.47
Λ	1115.68	1	2	31	31	1082.86	2.94	1081.85	3.03	1079.96	3.20	1085.39	2.71
Λ(1405)	1405.10	1	2	41	41	1431.26	1.86	1430.23	1.79	1428.33	1.65	1435.52	2.17
Λ(1520)	1519.00	1	2	43	43	1500.94	1.19	1499.90	1.26	1498.00	1.38	1505.55	0.89
Λ(1600)	1600.00	1	2	45	45	1570.62	1.84	1569.58	1.90	1567.68	2.02	1575.57	1.53
Λ(1670)	1674.00	1	2	47	47	1640.30	2.01	1639.25	2.08	1637.35	2.19	1645.60	1.70
Λ(1690)	1690.00	1	2	49	49	1709.97	1.18	1708.92	1.12	1707.03	1.01	1715.62	1.52
Λ(1800)	1800.00	1	2	51	51	1779.65	1.13	1778.60	1.19	1776.70	1.29	1785.65	0.80
Λ(1810)	1790.00	1	2	51	51	1779.65	0.58	1778.60	0.64	1776.70	0.74	1785.65	0.24
Λ(1820)	1820.00	1	2	53	53	1849.33	1.61	1848.27	1.55	1846.38	1.45	1855.67	1.96
Λ(1830)	1825.00	1	2	53	53	1849.33	1.33	1848.27	1.28	1846.38	1.17	1855.67	1.68
Λ(1890)	1890.00	1	2	55	55	1919.00	1.53	1917.95	1.48	1916.05	1.38	1925.70	1.89
Λ(2000)	2000.00	1	2	57	57	1988.68	0.57	1987.62	0.62	1985.72	0.71	1995.72	0.21
Λ(2050)	2056.00	1	2	59	59	2058.36	0.11	2057.30	0.06	2055.40	0.03	2065.75	0.47
Λ(2070)	2070.00	1	2	59	59	2058.36	0.56	2057.30	0.61	2055.40	0.71	2065.75	0.21
Λ(2080)	2082.00	1	2	59	59	2058.36	1.14	2057.30	1.19	2055.40	1.28	2065.75	0.78
Λ(2085)	2020.00	1	2	57	57	1988.68	1.55	1987.62	1.60	1985.72	1.70	1995.72	1.20
Λ(2100)	2100.00	1	2	61	61	2128.03	1.33	2126.97	1.28	2125.07	1.19	2135.77	1.70
Λ(2110)	2090.00	1	2	59	59	2058.36	1.51	2057.30	1.56	2055.40	1.66	2065.75	1.16
Λ(2325)	2325.00	1	2	67	67	2337.06	0.52	2336.00	0.47	2334.10	0.39	2345.85	0.90
Λ(2350)	2350.00	1	2	67	67	2337.06	0.55	2336.00	0.60	2334.10	0.68	2345.85	0.18
Λ(2585)	2585.00	1	2	75	75	2615.77	1.19	2614.69	1.15	2612.80	1.08	2625.95	1.58
Σ⁺	1189.38	1	2	35	35	1222.22	2.76	1221.20	2.68	1219.30	2.52	1225.44	3.03
Σ⁰	1192.64	1	2	35	35	1222.22	2.48	1221.20	2.39	1219.30	2.24	1225.44	2.75
Σ⁻	1197.45	1	2	35	35	1222.22	2.07	1221.20	1.98	1219.30	1.83	1225.44	2.34
Σ(1385)⁺	1382.80	1	2	39	39	1361.58	1.53	1360.55	1.61	1358.65	1.75	1365.50	1.25
Σ(1385)⁰	1383.70	1	2	39	39	1361.58	1.60	1360.55	1.67	1358.65	1.81	1365.50	1.32
Σ(1385)⁻	1387.20	1	2	39	39	1361.58	1.85	1360.55	1.92	1358.65	2.06	1365.50	1.56
Σ(1660)	1660.00	1	2	47	47	1640.30	1.19	1639.25	1.25	1637.35	1.36	1645.60	0.87
Σ(1670)	1675.00	1	2	47	47	1640.30	2.07	1639.25	2.13	1637.35	2.25	1645.60	1.76
Σ(1750)	1750.00	1	2	51	51	1779.65	1.69	1778.60	1.63	1776.70	1.53	1785.65	2.04
Σ(1775)	1775.00	1	2	51	51	1779.65	0.26	1778.60	0.20	1776.70	0.10	1785.65	0.60
Σ(1910)	1910.00	1	2	55	55	1919.00	0.47	1917.95	0.42	1916.05	0.32	1925.70	0.82
Σ(1915)	1915.00	1	2	55	55	1919.00	0.21	1917.95	0.15	1916.05	0.05	1925.70	0.56
Σ(2030)	2030.00	1	2	59	59	2058.36	1.40	2057.30	1.34	2055.40	1.25	2065.75	1.76
Σ(2250)	2250.00	1	2	65	65	2267.39	0.77	2266.32	0.73	2264.42	0.64	2275.83	1.15
Ξ⁰	1314.86	1	2	37	37	1291.90	1.75	1290.88	1.82	1288.98	1.97	1295.47	1.47
Ξ⁻	1321.72	1	2	37	37	1291.90	2.26	1290.88	2.33	1288.98	2.48	1295.47	1.99
Ξ(1530)⁰	1531.80	1	2	43	43	1500.94	2.01	1499.90	2.08	1498.00	2.21	1505.55	1.71
Ξ(1530)⁻	1535.00	1	2	43	43	1500.94	2.22	1499.90	2.29	1498.00	2.41	1505.55	1.92
Ξ(1690)	1690.00	1	2	49	49	1709.97	1.18	1708.92	1.12	1707.03	1.01	1715.62	1.52
Ξ(1820)	1823.00	1	2	53	53	1849.33	1.44	1848.27	1.39	1846.38	1.28	1855.67	1.79
Ξ(1950)	1950.00	1	2	55	55	1919.00	1.59	1917.95	1.64	1916.05	1.74	1925.70	1.25
Ξ(2030)	2025.00	1	2	59	59	2058.36	1.65	2057.30	1.59	2055.40	1.50	2065.75	2.01
Ω(1672)⁻	1672.45	1	2	47	47	1640.30	1.92	1639.25	1.99	1637.35	2.10	1645.60	1.61
Ω(2012)⁻	2012.40	1	2	57	57	1988.68	1.18	1987.62	1.23	1985.72	1.33	1995.72	0.83
Ω(2250)⁻	2252.00	1	2	65	65	2267.39	0.68	2266.32	0.64	2264.42	0.55	2275.83	1.06
charmed, bottom:
Λ c +	2286.46	1	2	65	65	2267.39	0.83	2266.32	0.88	2264.42	0.96	2275.83	0.47
Σ_c(2455)	2453.75	1	2	71	71	2476.41	0.92	2475.34	0.88	2473.45	0.80	2485.90	1.31
Ξ c 0	2470.91	1	2	71	71	2476.41	0.22	2475.34	0.18	2473.45	0.10	2485.90	0.61
Ω c 0	2695.20	1	2	77	77	2685.44	0.36	2684.37	0.40	2682.47	0.47	2695.98	0.03
Λ b 0	5619.60	1	2	161	161	5611.80	0.14	5610.70	0.16	5608.80	0.19	5637.04	0.31
Σ_b	5813.10	1	2	167	167	5820.82	0.13	5819.72	0.11	5817.82	0.08	5847.12	0.59
Ξ b 0	5791.90	1	2	167	167	5820.82	0.50	5819.72	0.48	5817.82	0.45	5847.12	0.95
Ω b −	6046.10	1	2	173	173	6029.84	0.27	6028.75	0.29	6026.85	0.32	6057.20	0.18
Heavy Bosons:
m _gb	42891.7#	2	4	9	18	43,591.2	1.63	43,270.9	0.88	42,750.4	0.33	43,182.0	0.68
W	80,379.0	2	4	17	34	81,582.8	1.50	81,269.8	1.11	80,750.7	0.46	81,566.0	1.48
Z	91,187.6	2	4	19	38	91,081.9	0.12	90,769.7	0.46	90,250.8	1.03	91,162.0	0.03
H ⁰	125,100.0	2	4	26	52	123,501.0	1.28	123,501.0	1.28	123,501.0	1.28	124,748.0	0.28
Average error Δ:							0.85		0.84		0.85		0.91
Ratio with ‘statistical’ error Δ/Δ_s:							0.86		0.85		0.86		0.92

Column 2 contains the empirical masses, E ₁ and E ₂ are calculated according to Eq. (70), E ₃ according to Eq. (71), all three with parameters f _ext = −2.1573, s = 1.089. E ₄ according to Eq. (71) et sqq. with exp(πa)≔1/2α = 68.518, i.e. an approximation which corresponds to the phenomenological approach of Section 2. The error columns display the percentaged discrepancies between calculation and empirical mass. ^* For even terms (N = 2n) always C = 0 holds (no contribution of F ₍₃₎). ^**: Masses in the constituent quark model. ^***: + possibly non q q ̄ states. #: 0.5 (m _W + m _Z)/2 ≈ Mac Gregor’s gauge boson mass unit m _gb = m _u _, _d/α.

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Received: 2022-01-23

Accepted: 2022-05-03

Published Online: 2022-06-24

Published in Print: 2022-08-26

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Articles in the same Issue

Frontmatter
General
The generation of mass in a non-linear field theory
Gravitation & Cosmology
Towards a self-interacting complex scalar field boson-star model
Beyond semiclassical time
Solid State Physics & Materials Science
Performance of a tunable photoconductive graphene plasmonic photodetector

https://doi.org/10.1515/zna-2022-0018

Keywords for this article

general relativity; geometrisation; mass spectrum of elementary particles; non-linear field theory

Articles in the same Issue

Frontmatter
General
The generation of mass in a non-linear field theory
Gravitation & Cosmology
Towards a self-interacting complex scalar field boson-star model
Beyond semiclassical time
Solid State Physics & Materials Science
Performance of a tunable photoconductive graphene plasmonic photodetector

The generation of mass in a non-linear field theory

Article

Abstract

Appendix A: Non-hermitian spacetime structure in the R 4

Appendix B: Derivation of equation (17) according to Heim [43]

Appendix C: Dimensions and symmetries

C.1 Heim’s dimension formula

C.2 Later expansion of the model space

C.3 Heim space and gauge symmetries

Appendix D: Further contractions of R kmp i

Appendix E: Structures and metric tensors

E.1 Partial structures from the phenomenology of the R 4

E.2 Composition metric tensors per Hermetry

Appendix F: Solution of the hermetric fundamental equation (30) and further relations

F.1 Solution of equation (30)

F.2 Calculation of the metric tensor

F.3 Relation for a condenser anti-hermetric in the contravariant index

Appendix G: Analysis of Hermetry forms a and b

Appendix H: Properties and results of the poly-metric geometry

H.1 Properties of g ik and i k l

H.2 Solution of equation (43)

H.3 Expressions of the metric tensor and the correlation tensor in the poly-metric

H.4 Calculation of the condenser class [3]

H.5 Further classes of condensations

Appendix I: Simplification of equation (64)

Appendix J: Linear approximation, Dirac and Maxwell equations

Appendix K: The spin in Heim’s theory

Appendix L: Qualitative derivation of an interaction potential

Appendix M: Heim’s minimum particle mass

M.1 Gravitational space structures and their extrema

M.2 Mass formula from elementary hermetric structures

M.3 The minimal mass

Appendix N: Complete calculated mass spectrum

References

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Articles in the same Issue

Articles in the same Issue

Appendix A: Non-hermitian spacetime structure in the R ₄

E.1 Partial structures from the phenomenology of the R ₄

H.1 Properties of g _ik and i k l