The structure of leptons

Weinberg angle, neutrino mixing matrix, PMNS matrix, weak force, neutrinos

Idea: This chapter examines the local, microscopic anisotropy of space-time originating from the space-time lattice. The three independent energy states – which are degenerate in the isotropic case – thereby split up and are interpreted as particle generations. This relationship between the particle generations provides the Weinberg angle and the neutrino-mixing-matrix (PMNS-matrix) as results.

 

Point defects in the elastic universe theory

In chapters one to three, the basis was laid for the description of spacetime as a four dimensional elastic lattice with matter particles as defects in it. The framework of this theory can now be tested by deducing conclusions that can then be verified or falsified.

Idea

A point defect is a deviation in the lattice at a single location. In Chapter 2 it was shown that the equations of motion and the interactions with the surrounding lattice correspond precisely to the equations for a charged lepton (electron, muon, tauon). The point defect was assumed to be actually point-like (zero dimensional) in a homogeneous elastic four-dimensional body.

The idea is that this corresponds to the view of a distant observer, but that it is possible to "zoom in" until the finite lattice structure finally becomes visible.

Diagram: On the left, the far view: a single point defect in isotropic spacetime. When zooming in one starts seeing the partly structured, not anymore isotropic spacetime and possibly the volume of the defect.

The hypothesis is that three effects become visible:

1. There is – due to the properties of the lattice – a smallest possible unit defect.
2. The defect is not point-like, it occupies a very small volume.
3. The body becomes inhomogeneous through the influence of the surrounding lattice. Certain directions are now more elastic than others.

This case will now be examined and described mathematically. For the sake of simplicity only the static case will be treated, i.e. only the rest energy of such a defect is considered.

 

What is expected from the study of this case?
The finite size of the defect obliges it to take a certain shape. This in turn means that the defect can no longer have either purely isotropic or purely directed components; it rather exists as a mixture of both; partly as an isotropic-volume-change and partly as a volume-preserving-shear.

This separation corresponds to a standard decomposition into homogeneous and trace-free parts of tensors, which are complementary to each other.

The anisotropy of the surroundings forces the possible states of the defect to actually differ energetically. Depending on the orientation of the shear terms, its configuration contains more or less energy.

Assuming the homogeneous terms remain the same, the possible configurations of the shear terms in the four dimensional body form a three dimensional sub-manifold and can thus be described by three linearly independent eigenmodes. A specific defect can therefore have three different eigen-shear configurations.

The conversion between these configurations occurs through changes in the shear terms, while keeping the homogeneous terms constant.

 

Interpretation

This allows the following, astonishing interpretation:

Such a unitary point defect is the model of a lepton.

• The isotropic terms correspond to the electric charge. The electromagnetic interaction only acts onto these isotropic parts (in accordance with Chapters 2 & 3).
• The three eigenstates of the shear terms have different total energies and correspond to each generation (= flavour) of the particle, so whether it is measured as electron, muon or tauon.
• To switch between the shear eigenstates, the shear components must be changed. A conversion by means of electromagnetic interaction is thus impossible.
• However, the conversion is possible by subtracting the shear components from the initial configuration and then re-adding them to the final configuration. The neutrinos then correspond to the negative shear components of the corresponding charged leptons. Example:

Feynman diagram of a muon decay: In this interpretation, a muon and an electron only differ in their shear terms. The neutrinos consist only of shear terms. The muon releases its shear terms in the form of a muon neutrino, while the electron captures its shear terms in the form of an anti-electron neutrino. The vertices are thus each in a pure isotropic state, which is the same for muon and electron, and is transmitted through a W boson.

• Neutrinos have no volume, only shear terms.
• The sum of isotropic and shear energy corresponds to the rest energy of a defect and thus the mass, or the measure with which the defect distorts the surrounding lattice.

Further implications regarding lepton number conservation, flavour conservation under electromagnetic interaction, charge conservation, and a model for W and Z bosons are to be expected.

 

Approach

First, an expression for the rest energy of a hypothetical unitary defect in the isotropic medium is derived.

The state is then sheared while preserving its volume. The focus lies in finding a practical parametrization of the shear. Since the three charged leptons (electron, muon, tauon) determine the energies of three individual states, a system of equations which describes the total shear by means of three parameters may be created.

Based on the chosen parametrization, this result can be re-elaborated and further interpreted:

On the one hand, the strength of the shear can be determined and transformed in such a way that it can be directly compared to a parameter of today's model: the Weinberg angle, which gives a measure for the influence of symmetry breaking on the energy differences of the W and Z bosons.

Furthermore, the shear components can be parameterized equivalently in the complementary space [SOURCE]. If the original and the complementary parameterization are projected into the three-dimensional instantaneous space, a transformation matrix can be specified between the two configurations, which is compared with the neutrino mixing matrix. This leads to a proposal for the mechanism of neutrino oscillation and the calculation of the neutrino mixing parameters.

 

Existence of a unitary defect

A consequence of there being a spacetime lattice is that there must be a smallest unitary defect. A smallest element of finite spacial extension, which can be added or removed from the lattice. Since nothing has yet been assumed about the precise structure of the lattice, there may also be several unitary defects in case the lattice contains a more complex unit cell of different masses. Such defects – where only one of the elements of the lattice is changed – are called point defects.

Diagram: harmonic lattice (left) and point defect (right) with a single anomalous element. Further treatment in Chapter 2.

Such a unitary defect will now be inserted as a 4-sphere with the radius of a four-dimensional unitary cell:

<equation id="eqn:su3.01" shownumber> $\mathbf {u} (\mathbf {r} )=\mathbf {\Theta } ^{4}(\mathbf {r_{0}} -\mathbf {r} )$ </equation>

Where $\mathbf {\Theta } ^{4}$ is the four dimensional Heaviside function. The deformation tensor corresponds to the derivative of the deflection function:

<equation id="eqn:su3.01" shownumber> Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \mathbf{ε}(\mathbf{r})=\frac{\partial}{\partial r}\mathbf{u}(\mathbf{r})=\mathbf{δ}^4(\mathbf{r_0}-\mathbf{r}) } </equation>

and is therefore only unequal to zero on the surface of the ball (the 3-sphere $S_{3}$), where all angle dependent components disappear due to isotropy. For the same reason, the delta function can be expressed as a sole function of the radius $r$ (analogously to the case in three dimensions ):

<equation id="eqn:su3.01" shownumber> Failed to parse (syntax error): {\displaystyle \displaystyle \mathbf{δ}^4(r, \eta, \xi_1, \xi_2)=\frac{1}{2\pi^2r^3}\delta(r) } </equation>

And therfore:

<equation id="eqn:su3.01" shownumber> Failed to parse (syntax error): {\displaystyle \displaystyle \mathbf{ε}(r)=\frac{1}{2\pi^2r_0^3}\delta(r_0-r) } </equation>

The radius is normalized in the microscopic view:

<equation id="eqn:su3.01" shownumber> $r_{0}=1$ </equation>

And therefore:

<equation id="eqn:su3.01" shownumber> Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \mathbf{ε}(r)=\frac{1}{2\pi^2}\delta(1-r) } </equation>

The corresponding microscopic stress tensor is (Chapter 3):

<equation id="eqn:su3.01" shownumber> Failed to parse (syntax error): {\displaystyle \displaystyle \mathbf{σ}(r) = k \cdot \frac{\mathbf{ε}(r)}{(2\mu^\prime+4\lambda^\prime)} =k \cdot \frac{1}{2\pi^2(2\mu^\prime+4\lambda^\prime)}\delta(1-r) =\frac{4\mu^\prime c^2}{2\pi^2}\delta(1-r) } </equation>

The conversion factor $k=k_{1}\cdot k_{2}=4\mu ^{\prime }c^{2}\ \cdot \left(2\mu ^{\prime }+4\lambda ^{\prime }\right)\cdot e^{\ast }$ from the macroscopic to the microscopic spring constant is introduced inversely proportional to the Voigt coefficients.

STIMMT DIESER SATZ IM DEUTSCHEN? Der Umrechnungsfaktor k=... von der makroskopischen in die mikroskopische Federkonstante wird umgekehrt proportional zu den Voigt-Koeffizienten eingeführt.

Thus it holds true for the energy:

<equation id="eqn:su3.01" shownumber> Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle V=V_{\parallel}=-\frac{1}{2}\cdot \int_\mathbf{V} \mathrm{tr}(\mathbf{εσ})\mathbf{dV} =-\frac{1}{2}\cdot 4\mu^\prime c^2 \frac{1}{{\left(2{\pi{}}^2\right)}^2} \int_{\mathbf{∂V}} \delta (1-r)\mathbf{dA} =-\frac{1}{2}\cdot 4\mu^\prime c^2 \frac{1}{{\left(2{\pi{}}^2\right)}^2}S_3=-\frac{\mu^\prime c^2}{{\pi}^2} } </equation>

Where $S_{3}=2\pi ^{2}$ is the surface of the unit 3-sphere. The Lagrange density, as a function of the deflection field and by assuming independent fields (analogously to Chapter 2), is then:

<equation id="eqn:su3.01" shownumber> $\displaystyle {\mathcal {L}}=-V=-V_{\parallel }={\frac {2\mu ^{\prime }c^{2}}{{\pi }^{2}}}\mathbf {w_{\alpha }^{\dagger }w_{\alpha }}$ </equation>

Where $\int {\mathbf {w_{\alpha }^{\dagger }w_{\alpha }} }dV=c^{2}$. The prefactor $c^{2}$ appeares in the transition from Euclidean space to the measurement coordinates. An additional prefactor $c$ hereby appears from both fields respectively, through the change of the coordinate $x_{0}$ to the measurement as eigen-time $\tau$ where ${\frac {dx_{0}}{d\tau }}=c$.

The measured rest energy of the unitary defect in the isotropic medium is therefore:

<equation id="eqn:su3.01" shownumber> $\displaystyle E_{\parallel }=\int {\mathcal {L}}dV={\frac {2\mu ^{\prime }c^{4}}{{\pi }^{2}}}$ </equation>

And

<equation id="eqn:su3.01" shownumber> $\displaystyle m_{\parallel }={\frac {E_{\parallel }}{c^{2}}}={\frac {2\mu ^{\prime }c^{2}}{{\pi }^{2}}}=1.890\,742\,34(9)\cdot {10}^{-27}\,kg$ </equation>

The 3σ-error interval is given, where :

<equation id="eqn:su3.01" shownumber> $c=299\ 792\ 458\,{\frac {m}{s}}$ </equation>

And $\mu ^{\prime }$ from the earlier chapters:

<equation id="eqn:su3.01" shownumber> $\mu ^{\prime }=1.038\,151\,40(5)\cdot {10}^{-43}\,N^{-1}$ </equation>

Yet now the deformation is not isotropic anymore:

${\overset {\sim }{x}}$

Symmetry breaking

Consider a unit volume of the grid without defect. Symmetry breaking means that an isotropic stress no longer causes an isotropic deformation, but it causes instead different degrees of deformation in different directions.

This can be introduced into the system of equations by letting the proportionality constant between stress and deformation tensors in the material law take a tensorial form and become the fourth stage elasticity tensor ${\overset {\sim }{c}}_{ijkl}$:

<equation id="eqn:su3.01" shownumber> $\varepsilon _{ij}={\overset {\sim }{c}}_{ijkl}\cdot \sigma _{kl}$ </equation>

Without loss of generality, the deformation tensor Failed to parse (syntax error): {\displaystyle \mathbf{ε}} and the stress tensor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{σ}} remain symmetrical and can be diagonalized by real eigenvalues.

Approximation:
Furthermore, the following assumption, which simplifies the problem considerably, is made:

The stress tensor and the deformation tensor are simultaneously diagonalizable. This condition should be fulfilled especially when averaging over several cells.

This approximation is equivalent to saying that the two tensors commute, i.e. that a system of orthonormal eigenvectors $\mathbf {e} _{i}$ exists, which are eigenvectors of both tensors, each with different (real) eigenvalues $\lambda _{i}$ and ${\overset {\sim }{c}}_{i}\lambda _{i}$.

<equation id="eqn:su3.01" shownumber> Failed to parse (syntax error): {\displaystyle \mathbf{σe}_i=\lambda_i\mathbf{e}_i \qquad \qquad \mathbf{εe}_i=\overset{\sim}{c}_i\lambda_i\mathbf{e}_i } </equation>

Thus, a volume-preserving deformation, since it merely changes the radii, has the effect of deforming the initial isotropic state into an ellipse.

This process can be parameterized by tracking the movement of a (red) point on the unit sphere and by specifying the hyperbolic angle $A_{hyp}$ between the initial and final state:

This is the analogous principle for parameterizing a rotation: in order to parameterize rotations, it is customary to select a unit vector in the initial state and to track its movement to the final state. The strength of the rotation is characterized by the angle between the vectors.

However, in the analogous approach to volume-preserving shears, the observed point does not move on a unit sphere but on a hyperbola (red line in the illustration below), and the strength of the shear is characterized by a hyperbolic angle:

Example of a weaker and stronger volume-preserving shear. The red point, in the undeformed state lying on the x-axis moves along a hyperbola toward the upper right. The hyperbolic angle $A$ corresponds to twice the area beneath the position vector (green) and is a measure for the strength of the shear.

Essentially, however, the principle of rotation or shear is the same; one can even convert the hyperbolic angle into a trigonometric angle $\alpha ^{\prime }$ by considering the projection of the shear state onto the unit sphere:

On the left the situation with the hyperbolic angle $A$ and the depicted proportional projections on the coordinate axes $\mathrm {sinh} \ A$ and $\mathrm {cosh} \,A$. The triangle $PQR$ is similar to the triangle $PST$ in the figure to the right, which highlights the projection on the unit circle and the trigonometric angle $\alpha ^{\prime }$. The similarity implies that the slope of the line $PQ$ on the left and $PS$ on the right are the same. The slopes of the two lines, given, respectively, by $\mathrm {tanh} \,A={\frac {\mathrm {sinh} \,A}{\mathrm {cosh} \,A}}$ und $\mathrm {tan} \,\alpha ^{\prime }\ ={\frac {\mathrm {sin} \,\alpha ^{\prime }\ }{\mathrm {cos} \,\alpha ^{\prime }\ }}$ can be equated, which means that:

$\alpha ^{\prime }=\mathrm {atan} (\mathrm {tanh} \,A)$

When mapping onto the unit circle, therefore, the hyperbolic angle is converted into the trigonometric angle and the hyperbolic functions are replaced by trigonometric functions.

However, although the corresponding mapping is bijective, it is not a homomorphism, which means that different shears can only be added to each other (or manipulated by any other operation) in the hyperbolic parameterization. Further details on the parameterization are found in article 2 [REF]. A real parameterization of the shear of a unit 3-sphere in four dimensions looks like this:

<equation id="eqn:su3.01" shownumber> $\mathbf {v} _{i}=\left({\begin{array}{cccc}0\\0\\0\\1\end{array}}\right)\qquad \qquad \mathbf {v} _{f}=\left({\begin{array}{cccc}x_{0}\\x_{1}\\x_{2}\\x_{3}\end{array}}\right)=\left({\begin{array}{cccc}\mathrm {cosh} \,\eta \ \mathrm {sinh} \,{\xi }_{1}\\\mathrm {sinh} \,\eta \ \mathrm {cosh} \,{\xi }_{2}\\\mathrm {sinh} \,\eta \ \mathrm {sinh} \,{\xi }_{2}\\\mathrm {cosh} \,\eta \ \mathrm {cosh} \,{\xi }_{1}\end{array}}\right)$ </equation>

This parameterization is closely related to the Hopf coordinates in Euclidean space. An introduction of the special properties of these coordinates can be found e.g. in . An overview of the utilized parametrizations is shown in Table 1.

The real shear parts are therefore:

<equation id="eqn:su3.01" shownumber> $\mathbf {v} _{f}-\mathbf {v} _{i}=\left({\begin{array}{cccc}\mathrm {cosh} \,\eta \ \mathrm {sinh} \,{\xi }_{1}\\\mathrm {sinh} \,\eta \ \mathrm {cosh} \,{\xi }_{2}\\\mathrm {sinh} \,\eta \ \mathrm {sinh} \,{\xi }_{2}\\\mathrm {cosh} \,\eta \ \mathrm {cosh} \,{\xi }_{1}-1\end{array}}\right)$ </equation>

The above parameterization ensures that the sheared points are always to be found on a unit hyperbola, where $-{x_{0}}^{2}-{x_{1}}^{2}+{x_{2}}^{2}+{x_{3}}^{2}=1$. This is the unit manifold in hyperbolic space $\mathbb {H} ^{2}$, again analogous to the unit sphere ${x_{0}}^{2}+{x_{1}}^{2}+{x_{2}}^{2}+{x_{3}}^{2}=1$, which is the unit manifold in $\mathbb {R} ^{4}$ on which unit vectors move when rotated. The corresponding Lie-groups are $SU(1,1)$ for shears and $SU(2)$ for rotations.

The change of sign of the metric causes to switch between volume preserving shear and rotation parts.

This insight is important, since in the case at hand a further complication arises:
The measurement of the symmetry breaking (of the shear) is not carried through in the Euclidean space but in the Minkowski space with the metric signature (---+). This problem can be solved with help of the above considerations: also in the Minkowski space one can parametrize a unit manifold, whose components can then be transferred to the hyperbolic space.

The three parameters $\eta ,\ \xi _{1},\ \xi _{2}$ can be determined if three different, allowed states are known. The three charged leptons correspond accordingly with the model to four-dimensional point defects. The shear part of their energy corresponds therefore precisely to the three allowed shear states.

The isotropic part of the energy $E_{\parallel }={\frac {2\mu ^{\prime }c^{4}}{\pi ^{2}}}$ was already calculated, so the shear part $E_{\bot }$ must then corresponds to the difference $E_{\bot }=E_{tot}-E_{\parallel }$ between the individual states. The energy of the undeformed state can be parametrized as a vector with magnitude $E_{\parallel }$.

In summary, this then gives the following system of equations:

<equation id="eqn:su3.01" shownumber> $E_{\parallel }\cdot \left({\begin{array}{cccc}\mathrm {cosh} \,\eta \ \mathrm {sinh} \,{\xi }_{1}\\\mathrm {sinh} \,\eta \ \mathrm {cos} \,{\xi }_{2}^{\prime }\\\mathrm {sinh} \,\eta \ \mathrm {sin} \,{\xi }_{2}^{\prime }\\\mathrm {cosh} \,\eta \ \mathrm {cosh} \,{\xi }_{1}-1\end{array}}\right)=\left({\begin{array}{cccc}E_{e}-E_{\parallel }\\E_{\mu }-E_{\parallel }\\E_{\tau }-E_{\parallel }\\E_{4}-E_{\parallel }\end{array}}\right)$ </equation>

With the result:

<equation id="eqn:su3.01" shownumber> $\eta =0.9674(2)$ </equation>

<equation id="eqn:su3.01" shownumber> $\xi _{1}=-0.62284(8)$ </equation>

<equation id="eqn:su3.01" shownumber> $\xi _{2}=\mathrm {atanh} \left(\mathrm {tan} \,\xi _{2}^{\prime }\right)=0.9729(12)$ </equation>

<equation id="eqn:su3.01" shownumber> $\left(m_{4}={\frac {E_{4}}{c^{2}}}=3.4168(6)\cdot {10}^{-27}\ kg\right)$ </equation>

Remarks:

• The 3σ-error interval is given, the error calculation is in the appendix.
• The parametrization is done in the Minkowski space. An overview on the utilized parametrization is given in table 1.
• The parametrization is not unique, other parametrizations are also allowed. The assignement of coordinates can especially be permuted. This parametrization was chosen because it gives real and almost isoclinal results ($\xi _{2}\sim \eta$ and also $\xi _{2}^{\prime }\ \sim \left|\xi _{1}\right|$).
• The system of equations has multiple solutions. The one which maps all angles in the interval $\left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)$ and has as many positive values as possible is chosen.
• The fourth elementary particle with energy $E_{4}$ can be expressed as a linear combination of the three main particles and is not measured independently.

The input quantities were taken from . The standard uncertainty is given in parentheses:

<equation id="eqn:su3.01" shownumber> $m_{e}={\frac {E_{e}}{c^{2}}}=9.109\,383\,56(11)\cdot {10}^{-31}\ kg$ </equation>

<equation id="eqn:su3.01" shownumber> $m_{\mu }={\frac {E_{\mu }}{c^{2}}}=1.883\,531\,594(48)\cdot {10}^{-28}\ kg$ </equation>

<equation id="eqn:su3.01" shownumber> $m_{\tau }={\frac {E_{\tau }}{c^{2}}}=3.167\,47(31)\cdot {10}^{-27}\ kg$ </equation>

 

Table 1: The utilized spaces, parametrizations and relations. Sources: hyperbolic space: , Minkowski space: , Euclidean space: .

Remarks on the parametrization:

• Complex valued parametrizations are convenient for the groups $SU(1,1)$ and $SU(2)$. Two possibilities are presented here.
• For complex parametrizations half angles are used to underline the double covering property of $SO(3)$. Yet the bisection of the angles is carried out again in hyperbolic space.
• Normally the transition from $SU(1,1)$ to $SU(2)$ is done simply by complexifying the field on which each respective group operates in. Yet by doing this one loses information and the difference between rotation and shear is not clear anymore in this representation. This is why an alternative approach by projection on the unit manifold is outlined here. (The Lie-algebras $su(1,1)$ and $su(2)$ are both subalgebras of the algebra $sl(2,\mathbb {C} )$ of the 2x2 matrices having determinant equal to 1 in the field $\mathbb {C}$. Both subalgebras differ solely through the complexification of individual elements, see e.g. .)

 

Weinberg angle

Idea

The discovered parametrization of $SU(1,1)$ can, with help of the above considerations, be projected on the unit 3-sphere and hence on a parametrization of $SU(2)$.

The thus determined projection corresponds to the effect of symmetry breaking through volume preserving shears on objects with $SU(2)$ symmetry. The strength of the breaking of symmetry is a measure for the degree of fragmentation of the otherwise degenerate energy states in $SU(2)$.

The gauge bosons of the weak interaction are objects with $SU(2)$ symmetry. The splitting of their energy levels is given by the Weinberg angle $\theta _{W}$:

<equation id="eqn:su3.01" shownumber> $\cos ^{2}\theta _{W}=\left({\frac {m_{W}}{m_{Z}}}\right)^{2}\qquad$ or $\qquad \sin ^{2}\theta _{W}=1-\left({\frac {m_{W}}{m_{Z}}}\right)^{2}$ </equation>

Where $m_{W}$ and $m_{Z}$ are the rest masses of the W and Z bosons.

So:
The projection of the parametrized symmetry breaking is:
Initial state:

<equation id="eqn:su3.01" shownumber> $z_{1i}=0\qquad \qquad z_{2i}=1$ </equation>

End state:

<equation id="eqn:su3.01" shownumber> $z_{1f}=\sin \left(\eta _{h}^{\prime }\right)\cdot \exp \left(i\cdot \xi _{h2}^{\prime }\right)\qquad \qquad z_{2f}=\cos \left(\eta _{h}^{\prime }\right)\cdot \exp \left(i\cdot \xi _{h1}^{\prime }\right)$ </equation>

Where

<equation id="eqn:su3.01" shownumber> $\displaystyle \eta _{h}^{\prime }=\mathrm {atan} \left(\tanh {\frac {\eta }{2}}\right)=0.42218(6)$ </equation>

<equation id="eqn:su3.01" shownumber> $\displaystyle \xi _{h1}^{\prime }=\mathrm {atan} \left(\tanh {\frac {\xi _{1}}{2}}\right)=-0.29304(3)$ </equation>

<equation id="eqn:su3.01" shownumber> $\displaystyle \xi _{h2}^{\prime }=\mathrm {atan} \left(\tanh {\frac {\xi _{2}}{2}}\right)=0.4240(4)$ </equation>

All values are in radian, the 3σ-error interval is given, the error calculation is in the appendix.

The switch to the complex representation allows compatibility with the standard parametrization of $SU(2)$.

Result

The strength of the symmetry break corresponds to the angle between these two vectors, which can be found with the scalar product. This angle corresponds to the Weinberg angle:

<equation id="eqn:su3.01" shownumber> $\cos \theta _{W}=\langle \left({\begin{matrix}z_{1i}\\z_{2i}\\\end{matrix}}\right),\left({\begin{matrix}z_{1f}\\z_{2f}\\\end{matrix}}\right)\rangle =\left(\mathrm {Re} ({\bar {z}}_{1i}),\mathrm {Im} ({\bar {z}}_{1i}),\mathrm {Re} ({\bar {z}}_{2i}),\mathrm {Im} ({\bar {z}}_{2i})\right)\cdot \left({\begin{matrix}\mathrm {Re} (z_{1f})\\\mathrm {Im} (z_{1f})\\\mathrm {Re} (z_{2f})\\\mathrm {Im} (z_{2f})\\\end{matrix}}\right)=\cos \left(\eta _{h}^{\prime }\right)\cdot \cos \left(\xi _{h1}^{\prime }\right)$ </equation>

and therefore

<equation id="eqn:su3.01" shownumber> $\displaystyle \theta _{W}=\mathrm {acos} \left[\cos \left(\eta _{h}^{\prime }\right)\cdot \cos \left(\xi _{h1}^{\prime }\right)\right]=29.154(4)^{\circ }$ </equation> <equation id="eqn:su3.01" shownumber> $\displaystyle {\sin }^{2}\theta _{W}=0.23733(6)$ </equation> The 3σ-error interval is given. This error calculation can be found in the appendix.

 

Neutrino mixing matrix (PMNS-Matrix)

For the parameterization of shears

A shear state can be parameterized in several ways. As stated in Article 2 [SOURCE], the vector $\mathbf {{\overset {\smallsmile }{v}}{}_{sp}^{\prime }}$ parametrizes the same state of shear, measured in the complementary Minkowski space $\mathbb {\overset {\smallsmile }{M}} {}^{4}=\mathbb {R} (3,1)$, as the vector $\mathbf {v} ^{\prime }$ measured in the original Minkowski space $\mathbb {M} ^{4}=\mathbb {R} (1,3)$. However, with the angle-preserving mapping of these two states into the $\mathbb {R} ^{4}$, the direction of rotation and thus its handedness changes:

Representation of a shear state $\mathbf {v} ^{\prime }$ (left), and its equivalent configuration in the complementary space $\mathbf {{\overset {\smallsmile }{v}}{}_{sp}^{\prime }}$, shown in the $x_{0}x_{3}$-plane of the $\mathbb {R} ^{4}$. Both states describe the same shear state (striped gray shape), however, the parameterization is mirrored on the 45° axis and the direction of the shear angle is reversed.

The original configuration $\mathbf {v} ^{\prime }$ is right-handed $\left({\mathrm {sgn} (\xi }_{1})\neq {\mathrm {sgn} (\xi }_{2})\right)$, the complementary configuration changes the sign of the shear angle in the $x_{0}x_{3}$-plane and is therefore left-handed $\left({\mathrm {sgn} (\xi }_{1})={\mathrm {sgn} (\xi }_{2})\right)$.

 

Interpretation and thesis

For pure shear defects without volume parts (as in the postulated model for neutrinos), there are two equivalent realizable parametrizations.

Thesis:
It is possible for pure shear states (neutrinos) to switch between the two equivalent parametrizations given by $\mathbf {v} ^{\prime }$ and $\mathbf {{\overset {\smallsmile }{v}}{}_{sp}^{\prime }}$.

The angle-preserving projection of this transition into the three-dimensional momentaneous space by the Hopf-map is measured as the neutrino-mixing-matrix (PMNS-Matrix).

 

What is the Hopf-map?

One feature of the rotation group in four dimensions is its relationship to rotations in three dimensions. This property is well represented by the Hopf coordinates and is called Hopf map (Hopf Map).

This map projects the 3-sphere onto the 2-sphere. Since it is a projection, some points of the 3-sphere are projected onto the same point of the 2-sphere. The set of all points that are projected onto the same point are located on a circle. These are called the Hopf-fibration.

The same principle also works in hyperbolic space by means of the hyperbolic Hopf map.

 

Apporach

1. Substitution of the coordinates for the Hopf map in the $\mathbb {H} ^{2}$.
2. Mapping of the original R-parametrization onto $\mathbb {R} ^{4}$ and subsequent projection onto $\mathbb {R} ^{3}$ using the Hopf-map.
3. The same procedure for the complementary L-parameterization, given by interchanging two axes (mirroring at the 45° line between the two axes).
4. Treatment of the Hopf-fibration as a phase.
5. Determining of the transformation matrix between R and L state in $\mathbb {R} ^{3}$.
6. Comparing of the squares of the absolute values with the PMNS matrix. 

(The approach would also be possible using the hyperbolic Hopf map, but this approach requires more extensive consideration of the underlying Lie-algebras and is significantly longer. The calculation is in the appendix.)

Calculation and Result

1. & 2. Substitution in $\mathbb {H} ^{2}$ and projection onto $\mathbb {R} ^{3}$:

Table 2: Short form of the Hopf map. The angles are added in hyperbolic space (arrow).

The result in momentaneous space is:

<equation id="eqn:su3.01" shownumber> $\mathbf {v^{\prime }} =\left({\begin{array}{c}\sin \left({\eta }^{\prime }\right)\cdot \cos \left({\xi }_{-}^{\prime }\right)\\\sin \left({\eta }^{\prime }\right)\cdot \sin \left({\xi }_{-}^{\prime }\right)\\\cos \left({\eta }^{\prime }\right)\end{array}}\right)$ </equation>

 

3. The same for the complementary parameterization:
The mirroring is along the 45° axis in the plane spanned by $x_{0}$ and $x_{3}$. In the complex parametrization this corresponds to swapping complex and real parts of the complex conjugate coordinate, i.e. $z_{2}^{\ast \prime }=i\cdot {\bar {z}}_{2}^{\prime }$.

Table 3: Short form of the Hopf map after space reflection. The angles are added in hyperbolic space (arrow).

The effect of the reflection in $\mathbb {R} ^{3}$ is therefore a change from ${\xi ^{\prime }}_{-}$ to $\xi _{+}^{\prime }$ and an exchange of the x and y axes. The mirrored configuration is:

<equation id="eqn:su3.01" shownumber> $\mathbf {v^{\ast \prime }} =\left({\begin{array}{c}\sin \left({\eta }^{\prime }\right)\cdot \sin \left({\xi }_{+}^{\prime }\right)\\\sin \left({\eta }^{\prime }\right)\cdot \cos \left({\xi }_{+}^{\prime }\right)\\\cos \left({\eta }^{\prime }\right)\end{array}}\right)$ </equation>

 

4. Treatment of the Hopf-Fibration as a phase:
The comparison between the two configurations $\mathbf {v} ^{\prime }=(x^{\prime },y^{\prime },z^{\prime })$ and $\mathbf {v} ^{\ast \prime }=(x^{\ast \prime },y^{\ast \prime },z^{\ast \prime })$ already provides quite interesting results. However, it must still be considered that, starting with the same initial configuration, the two configurations are not projected in the same momentaneous space.

The two projected states differ by the phase in the Hopf fibration. This phase cannot be absolutely determined, but the relative difference between the two can:

<equation id="eqn:su3.01" shownumber> $\displaystyle \delta =\xi _{-}^{\prime }-\xi _{+}^{\prime }={-\xi }_{2}^{\prime }=-36.87(2)^{\circ }$ </equation> Or, transferred to the interval $\left[0,\ 360^{\circ }\right)$: <equation id="eqn:su3.01" shownumber> $\displaystyle \delta =323.13(2)^{\circ }$ </equation> The 3σ-error interval is given. The error calculation is carried out in the appendix.

The phase difference in the fibration means that if one configuration is completely projected onto momentaneous space, the other configuration still contains time components and is therefore measured to be shorter.

Tracing the mirrored configuration back to its initial state, this results in it having being rotated in the time plane by a phase $\exp \left(i\cdot \delta \right)$ compared to the unmirrored initial state. The projection of the state onto momentaneous space (i.e., only the real part of the phase) can be determined as follows:

<equation id="eqn:su3.01" shownumber> $\mathbf {v_{phase}^{\prime }} =\mathrm {Re} \left[\mathbf {R_{phase}} \cdot \mathbf {v^{\prime }} \right]=\mathrm {Re} \left[\left({\begin{array}{ccc}1&0&0\\0&1&0\\0&0&\exp \left(i\cdot \delta {}\right)\end{array}}\right)\cdot \mathbf {v^{\prime }} \right]=\left({\begin{array}{ccc}\sin \left({\eta }^{\prime }\right)\cdot \cos \left({\xi }_{-}^{\prime }\right)\\\sin \left({\eta }^{\prime }\right)\cdot \sin \left({\xi }_{-}^{\prime }\right)\\\cos \left(\delta \right)\cdot \cos \left({\eta }^{\prime }\right)\end{array}}\right)$ </equation>

With a diminished value in real momentaneous space.

 

5. Determination of the transformation matrix between R and L state in $\mathbb {R} ^{3}$
The change to the complementary space not only causes a reflection of the shear, but also reverses the role of base and vectors. As stated in article 1 [REF], changing the sign in the metric causes co-contravariant and contravariant objects to interchange the roles.

In this case it means that when $\mathbf {v_{phase}^{\prime }}$ is a state (contravariant vector), the version in the complementary space $\mathbf {{\overset {\smallsmile }{v}}{}_{sp}^{\prime }}$ takes on the transformation properties of a basis (covariant vector).

Consider now the matrix $\mathbf {R}$ which transforms $\mathbf {v_{phase}^{\prime }}$ into the mirrored state $\mathbf {v} ^{\ast \prime }$:

<equation id="eqn:su3.01" shownumber> $\displaystyle \mathbf {v} ^{\ast \prime }=\mathbf {R} \cdot {\frac {\mathbf {v_{phase}^{\prime }} }{\left|\mathbf {v_{phase}} ^{\prime }\right|}}$ </equation>

Then this is not yet the transformation matrix to the state in the complementary space, but instead the transformation to the basis of the complementary space. To get to the complementary state $\mathbf {{\overset {\smallsmile }{v}}{}_{sp}^{\prime }}$ one must consider the transposed matrix (proof in article 2 [REF]):

<equation id="eqn:su3.01" shownumber> $\displaystyle \mathbf {{\overset {\smallsmile }{v}}{}_{sp}^{\prime }} =\mathbf {R^{T}} \cdot {\frac {\mathbf {v_{phase}^{\prime }} }{\left|\mathbf {v_{phase}^{\prime }} \right|}}$ </equation>

The transformation matrix $\mathbf {R}$ corresponds to the standard rotation matrix, and its transpose is:

<equation id="eqn:su3.01" shownumber> $\mathbf {R^{T}} =\left({\begin{array}{ccc}0.799&0.571\pm {}0.001&0.188\pm {}0.001\\-0.178&0.523&-0.834\pm {}0.001\\-0.574&0.633&0.519\end{array}}\right)$ </equation>

Where $\mathrm {det} \left(\mathbf {R^{T}} \right)=1$. (rounded values, 3σ-error interval only calculated for terms used further on)

 

6. Comparison with the PMNS matrix:
Rotation and phase are combined to be compared with experimental results of the PMNS matrix:

<equation id="eqn:su3.01" shownumber> $\left\vert \mathbf {v_{phase}^{\prime }} \right\vert \cdot \mathbf {{\overset {\smallsmile }{v}}{}_{sp}^{\prime }} =\left(\mathbf {R^{T}} \cdot \mathrm {Re} [\mathbf {R_{phase}} ]\right)\cdot \mathbf {v^{\prime }}$ </equation>

Where

<equation id="eqn:su3.01" shownumber> $\mathbf {B} =\left(\mathbf {R^{T}} \cdot \mathrm {Re} [\mathbf {R_{phase}} ]\right)=\left({\begin{array}{ccc}0.799&0.571\pm {}0.001&0.1504\pm {}0.0007\\-0.178&0.523&-0.667\pm {}0.001\\-0.574&0.633&0.416\end{array}}\right)$ </equation>

On the other hand, the standard parametrization for the PMNS matrix $\mathbf {U}$ is:

<equation id="eqn:su3.01" shownumber> $\mathbf {U} =\left({\begin{array}{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta {}}\\{-s}_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta {}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta {}}&s_{23}c_{13}\\s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta {}}&{-c}_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta {}}&c_{23}c_{13}\end{array}}\right)$ </equation>

Where $c_{ij}=\cos \theta _{ij}$ and $s_{ij}=\sin \theta _{ij}$.

Since the phase was introduced differently and only the real component was considered in the calculated case, the two matrices do not correspond directly. Yet by squaring of the absolute values, one gets rid of the phase, and the two matrices can then be compared:

<equation id="eqn:su3.01" shownumber> $\left|b_{ij}\right|^{2}=\left|u_{ij}\right|^{2}$ </equation>

The squared absolute values of each element can further be interpreted as transition probabilities. The neutrino mixing angle can then be found directly:

<equation id="eqn:su3.01" shownumber> $\displaystyle \theta _{13}=\arcsin \left({\sqrt {\left|b_{13}\right|^{2}}}\right)=\arcsin \left(0.1504\right)=8.65(4)^{\circ }$ </equation> <equation id="eqn:su3.01" shownumber> $\displaystyle \theta _{12}=\arcsin \left({\frac {\sqrt {\left|b_{12}\right|^{2}}}{\cos \theta _{13}}}\right)=\arcsin \left({\frac {0.571}{\cos \theta _{13}}}\right)=35.28(8)^{\circ }$ </equation> <equation id="eqn:su3.01" shownumber> $\displaystyle \theta _{23}=\arcsin \left({\frac {\sqrt {\left|b_{23}\right|^{2}}}{\cos \theta _{13}}}\right)=\arcsin \left({\frac {0.667}{\cos \theta _{13}}}\right)=42.42(8)^{\circ }$ </equation> The 3σ-error interval is given. The error calculation is in the appendix.

 

Open questions

There are countless open questions. Nearly every point in this chapter needs to be further and more thoroughly investigated. Likewise, the wider consequences of the presented thesis. A few topics are briefly mentioned here:

Structure of leptons, homogeneous and non-marking parts

• Elaborate further conclusions regarding lepton number conservation, flavor conservation under electromagnetic interaction, charge conservation, CP symmetry breaking, interpretation of handedness and spin.
• Creating a model for W and Z bosons.
• Where does the $c^{2}$ factor come from exactly in the energy consideration of the unitary defect in isotropic space?

Strain eigenstates as lepton masses

• Is the interpretation correct and complete? Further consequences?
• The m4 particle is not measurable according to these explanations, but disappears as a particle as well as regarding interactions completely as a linear combination of the known three leptons.
• It is unclear whether the particle does not exist at all or whether effects from it could somehow still be measured e.g. by terms of higher order which were neglected so far. Does it have any influence? Would it be a candidate for dark matter with mass but without its own interaction?
• Interesting in this regard could be results that may indicate an increased decay frequency of mesons over tau leptons. .

Symmetry breaking and Weinberg angle

• The calculated Weinberg angle corresponds with measurements at low kinetic energies Q (SLAC E158 & APV (Cs)). But these measurements are not very precise.
• The data from accelerator experiments at high kinetic energies Q can be extrapolated to Q = 0 using several assumptions (for example, ), but the result is not exact. How can this be interpreted?
• More low-energy experiments are planned with higher precision or in progress (MOLLER, P2).

Parameterization of the shears

• What happens at the extremal points of the parametrization?
• Transition to finite transformations, curvilinear coordinates & curved spaces.
• Influence by (non-commuting) terms of higher order, consideration of the approximations, possibly related to mechanisms for particle decay.

 

References

 

Appendix
Appendix 4A: Error analysis
Appendix 4B: Hyperbolic Hopf-map

 

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