Space and Family (short version)

A geometrical picture for the family structure of fundamental particles is developed. It indicates that there might be a relation between the family repetition structure and the number of space dimensions.


With the discovery of the top quark all in all 12 fundamental fermions (6 leptons and 6 quarks) are known today. At least a decade ago it has become clear that these particles can be organized into three "families" each containing 2 quarks and 2 leptons. These particle families (or generations) behave identically under the electroweak and strong interactions and do not differ by anything else than their masses. The number of generations will probably be restricted to three forever, because it has been shown experimentally that at most 3 species of light neutrinos exist. A fourth family,if it exists at all, would necessarily contain a heavy neutrino and would therefore be different in nature from the known families.

Over the last 20 years there has been one outstanding puzzle in elementary particle physics. This is the question whether the variety of "elementary" particles, the quarks and leptons, can be derived from some more fundamental principle. To answer this question is quite difficult because up to now no experimental indications exist of which might be the nature of this principle. Present models usually lead to an inflation of new particles (like supersymmetry) at higher energies and/or tend to shift the basic problems to higher energy scales where they reappear in slightly modified form (like technicolor).

Preon models avoid these deficiencies but have severe problems of other kinds, like the smallness of fermion masses as compared to the binding scale. Still, I want to follow in this work a preon type idea, that the quarks and leptons have a spatial extension, and contain sub--constitutents sitting on a cubic lattice. My guideline will be that the spatial dimensions correspond to a sort of shells which are successively filled up by the generations. The third shell -- corresponding in some sense to the third dimension -- becomes closed with the top quark. I shall make use of some discrete, nonabelean subgroups of O(3), in particular the symmetric group $S_4$ which is closely related to the symmetry transformations of a cube. I shall also address the question of how to understand the vector bosons and the mass hierarchy within this model.

I would like to warn the reader that in this article I am mostly doing simple minded geometry without really clarifying the mathematics behind it. There are all sorts of unanswered questions concerning the dynamics of the model. My hope is that I can motivate readers to do more refined work on the basis of these suggestions.

The Family Repetition

The known quarks and leptons within one family are: \begin{equation} \nu_{L,R} \qquad e_{L,R} \qquad u^{1,2,3}_{L,R} \qquad d^{1,2,3}_{L,R} \label{eq1} \end{equation} where the upper index denotes the quark color degrees of freedom and the lower index the helicity. I have made the reasonable assumption a righthanded neutrinos $\nu_R$ exist.

There are $ 2_{spin} \times 2_{isospin} \times (1+3_{color})=16$ degrees of freedom in one family (32, if antiparticles are counted as well) and altogether 48 degrees of freedom in all three generations. This will be related to a property of 3--dimensional space, namely that there are 48 symmetry transformations which leave the 3--dimensional cube invariant.

One might think that some information on the nature of the fermions can be obtained from their measured mass spectrum. However, the fermion masses are running, i.e. energy dependent, and it is not really known which dynamics governs this energy dependence. In other words, their renormalization group equations are not known precisely and therefore no complete knowledge of the fermion mass spectrum exists. For instance, the masses could be running according to some SUSY--GUT theory. However, apart from the fact that the SUSY breaking scale is not known precisely, new physics may set in at some point and modify the RG equations. Furthermore, the masses are extremely tiny as compared to the binding scale, so that in any preon model they will arise as remnants of cancellations between large numbers. Therefore it is not to be expected that the fermion mass spectrum gives direct insight into the dynamics of the preons. The fermion mass spectrum is at most a qualitative guideline to understand the family structure.

t'Hooft has suggested to decree chiral invariance as the principle which suppresses the fermion masses, and from this has derived conditions on the anomaly structure of the preon model. In the present work this is not a necessary condition. The point is that the extension of the observed fermions is not fixed by the binding energy of a superstrong force but {\it by the structure of space}. In my model, space is essentially discrete with massless preons sitting on the sites of a cubic lattice. I shall stick to the notion of 'binding energy', though, to mean the inverse extension of the 'bound states'. Actually, the binding energy is the scale at which the fermion masses should be defined.

The Model

The basic assumption in this work goes as follows: the fermions in the first family define a planar structure in space composed of a 'shell' which is successively filled up when one goes from the electron--neutrino to the up--quark. The closed first family shell survives in the second family where another shell/plane orthogonal to the first is beginning to fill. Finally, the third family fills 3-dimensional space completely. This qualitative picture is depicted in fig \ref{fig4} where the 'first', 'second' and 'third' family plane are drawn. To account for the rather large mass difference between the overall family masses (factors $\sim 100$) one might speculate that they arise from "exciting" the successive dimensions.

More precisely, my model rests on the symmetry groups of cubes and tetrads, which are discrete subgroups of O(3). Of particular relevance will be the group of permutations $S_4$ which is isomorphic to the symmetry group of the tetrad. The discussion will concentrate on the subgroups themselves and not on their representations. This is a somewhat unusual approach, because normally in physics particle multiplets are identified with the representation spaces and not with the symmetry groups themselves. In contrast, the philosophy here is that by applying the symmetry transformations on the ground state ($\nu_e$) one can generate all other fermion states. This is only possible if there is a symmetry {\it breaking} which distinguishes the generated states. Such a symmetry breaking will be realized by geometrical means in the following section. To show how this can happen, I have visualized in fig. \ref{figt} an element of $S_3$, the permutation symmetry of the three sides of an equilateral triangle in 3 dimensions, assuming the existence of distinct preons A,B,C on the sites and distinct binding forces on the links of the triangle. If the $S_3$--transformation is applied to the sites but not to the links of the triangle, a completely different state is generated. In this case there are 6 such states corresponding to the 6 elements of $S_3$.

The symmetry group of the cube is the octahedral group $O_h$. It has a direct product form $O_h\equiv P \oplus S_4$ where $P$ is the parity transformation and $S_4$ is the group of symmetry transformations of the tetrad, sometimes also called $T_h$ or $O$ (where the $O$ stands for the octahedron whose transformations it also describes). $O_h$ contains exactly the 48=2x24 elements needed for a one to one correspondence with the particles of the 3 families. The philosophy here is that by applying the 48 symmetry transformations of the cube on one of the 48 fermion states one can generate all the other 48 fermion states. This indicates that the fermions should have some 3 dimensional substructure which completely breaks the cubic symmetry. Such a symmetry breaking can be realized in a variety of different ways as will be shown below.

The symmetry group $S_4$ of a tetrad is isomorphic to the group of permutations of 4 objects. This way the 24 symmetry tansformations on a tetrad can be viewed as the set of all (directed, open) paths that connect the 4 corner points 1,2,3,4 of the tetrad. From the tetrad a cube can be generated by applying the parity transformation. In fact there are two tetrads, a 'lefthanded' (with corner points 1,2,3,4) and a 'righthanded' (with corner points 1',2',3',4') embedded in a cube. (cf. fig. \ref{fig5}) related by P.

From now on I will follow the philosophy that the spatial structure of a fermion (quark or lepton) is that of a tetrad. Furthermore, I assume that by applying the 24 symmetry transformations of the tetrad state one can create all 24 fermion states of the three families out of one of these states. In order to guarantee that all of the symmetry transformations yield different states, the tetrad cannot be completely symmetric. For example, one may assume that there are 4 different preons sitting on the 4 corners of the tetrad. Another possibility is that the preons are identical, but the binding forces between them are different. In the following we shall pursue this latter option. More specifically, we shall assume that the bindings between the 4 preons are given according to the 24 permutations of the set {1,2,3,4}. One of the 24 permutations (namely the identity element) is visualized in fig. \ref{fig5} by the 3 arrows $1 \rightarrow 2 \rightarrow 3 \rightarrow 4$. Any other element $1234 \rightarrow abcd$ of $S_4$, denoted by $\overline{abcd}$ in the following, could be drawn as the path $a \rightarrow b \rightarrow c \rightarrow d$ in fig. \ref{fig5}.

Starting with the 'lefthanded' tetrad, one can construct all the 24 lefthanded fermion states of the 3 generations. By applying the parity transformation $(x,y,z) \rightarrow (-x,-y,-z)$, righthanded fermions can be obtained. Any such righthanded state will be denoted by $\overline{a'b'c'd'}$ in the following. %As an example, $\overline{3'4'2'1'}$ %is shown in fig. \ref{fig6} %I have not undertaken to explicitly produce Dirac fermions out of %this construction. So admittedly, at the moment, only qualtitativ. As well known, a (Dirac) fermion $f$ has four degrees of freedom, of which only two, $f_L$ and $f_R$, have been described so far. The way to obtain antiparticles $\bar f_L$ and $\bar f_R$ is as follows: $\bar f_L$ is a righthanded object and its preons should therefore form a righthanded tetrad $\overline{a'b'c'd'}$ with field values corresponding not to $f_R$ but to the complex conjugate of $f_L$. Similarly, $\bar f_R=\overline{abcd}$ with field values which are complex conjugate to $f_R$.

As a side remark note that there might be something in the tetrad's centre, but this is not modified by $S_4$ nor parity transformations (cf. footnote 1 below).

Let us now explicitly relate the elements of $S_4$ to the various members of the 3 generations. The geometrical model fig. \ref{fig5} naturally suggests a separation of the 24 permutations into 3 subsets. To see this, look at the figures \ref{fig8}, \ref{fig9} and \ref{fig10}, where the 3 possible closed paths which connects the points 1,2,3 and 4 are shown. These closed paths consist of 4 links. Permutations lying on the path I (fig. \ref{fig8}) will be attributed to the members of the first family, permutations on path II (fig. \ref{fig9}) to the second family and path III (fig. \ref{fig10}) to the third family. For example, look at the lefthanded states of the first family \begin{eqnarray} \nonumber \nu_L &=& \overline{1234} \qquad u^1_L= \overline{2341} \qquad u^2_L= \overline{3412} \qquad u^3_L= \overline{4123} \\ e_L &=& \overline{4321} \qquad d^1_L= \overline{1432} \qquad d^2_L= \overline{2143} \qquad d^3_L= \overline{3214} \label{eq8} \end{eqnarray} More precisely, these fermion states correspond to the various (open) paths consisting of 3 links which one can lay on the closed path fig. \ref{fig8}. A typical example of an open path (representing $u^1_L$) is shown in fig. \ref{fig55}.

Using the assignments eq. (\ref{eq8}) one sees that a weak isospin transformation corresponds to reversing a permutation, i.e. reversing all 3 arrows in a figure like fig. \ref{fig55}. This way weak isospin is not any more a quantum number carried by a fundamental constituent but is determined by the binding of the state. Analogously, constructions for color and elctric charge can be envisaged.

For completeness let us write down the $S_4$--assignment of the second and third generation. They correspond to the various paths consisting of 3 links which one can draw into the closed loops depicted into figs. \ref{fig9} and \ref{fig10}. \begin{eqnarray} \nonumber \nu_{\mu L} &=& \overline{2134} \qquad c^1_L= \overline{1342} \qquad c^2_L= \overline{3421} \qquad c^3_L= \overline{4213} \\ \mu_L &=& \overline{4312} \qquad s^1_L= \overline{2431} \qquad s^2_L= \overline{1243} \qquad s^3_L= \overline{3124} \label{eq9} \end{eqnarray} \begin{eqnarray} \nonumber \nu_{\tau L} &=& \overline{4231} \qquad t^1_L= \overline{2314} \qquad t^2_L= \overline{3142} \qquad t^3_L= \overline{1423} \\ \tau_L &=& \overline{1324} \qquad b^1_L= \overline{4132} \qquad b^2_L= \overline{2413} \qquad b^3_L= \overline{3241} \label{eq10} \end{eqnarray} It should be noted that there is an intimate connection between the 3 closed paths figs. \ref{fig8}, \ref{fig9} and \ref{fig10} and the 3 planes in fig. \ref{fig4}, i.e. the dimensionality of space. Fig. \ref{fig8} corresponds to the first family plane, Fig. \ref{fig9} to the second family plane and Fig. \ref{fig10} to the third family plane. This can be seen easily by drawing the octahedron with corners given by the middle points of the cube's face diagonals.

Vector Bosons

Now that we have constructed all states of the fermion generations the most important question is how to understand their interactions. As is well known the interactions of fermions proceed through left-- and right--handed currents with the vector bosons, more precisely the lefthanded currents $\bar{F}_L \times f_L$ : $= \bar{F}_L \gamma_{\mu} f_L$ interact with the weak bosons and the sum of left- and righthanded currents interact with photons and gluons. The strength of the photonic and gluonic interaction is given by the electric and color charge, respectively.

The picture to be developed is that of the vector bosons as a sort of fermion--antifermion bound state. However, it will be constructed in such a way that the vector bosons do not 'remember' the flavor of the fermion--antifermion pair from which they were originally formed. The way to obtain antiparticles (and Dirac fermions) is as follows: I have already shown that left-- and right--handed fields are interpreted as permutations of corners 1,2,3,4 and 1',2',3',4' in the cube (cf. fig. \ref{fig11}), for example $e_L=\overline{4321}$ and $e_R=\overline{4'3'2'1'}$. The antiparticle of a lefthanded fermion is a righthanded object and its preons should form a righthanded tetrad. Therefore, the $\bar e_L$ is defined to live on the righthanded tetrad ($\overline{4'3'2'1'}$) but with field values corresponding to the complex conjugate of $e_L$. Similarly $\bar e_R=\overline{4321}$, with field values which are complex conjugate to $e_R$. Antifield configurations are denoted by open circles in fig. \ref{fig11}. \footnote{ I do not know whether the preons at the corners are real or complex, or whether one should prefer the bindings between them as the more fundamental objects. There are various disadvantages as to the existence of antipreons, both on the conceptual and on the explanation side. On the conceptual side the main disadvantage is that the preons themselves become more complicated than just real scalar pointlike particles without any further property than their simple superstrong interaction with neighbouring preons. On the explanation side I have found it difficult to accomodate parity violation -- everything is so unpleasantly P--symmetric in the pictures so far. An alternative one may follow is to do without antipreons, and to put the information of a quark or lepton being particle or antiparticle into the centre of the cube. More precisely, assume there is some nucleus $M_L$ at the centre of the lefthanded tetrads $ e_L, \nu_L, d^1_L, \dots$ and $\overline M_L$ at the centre of the righthanded tetrads $ \bar e_L, \bar \nu_L,\bar d^1_L, \dots$. Note that we do not assume the existence of a nucleus $M_R$ for the righthanded states $ e_R, \nu_R, d^1_R, \dots$ nor for their (lefthanded) antiparticles. We might assume its existence but for the sake of parity violation we must demand that $M_L$ and $M_R$ behave differently. In the following I shall assume for simplicity no $M_R$ at all. When a lefthanded current $\bar f_L \times f_L$ is formed, the $M_L$ and $\overline M_L$in the centre of the corresponding cube either annihilate or encircle each other. If they annihilate each other, a state is formed which cannot be distinguished from the corresponding right handed current $\bar f_R \times f_R$. That corresponds to the formation of a photon or a gluon. If they keep encircling each other, a Z or a W is formed, decaying very quickly after their short lifetime back to a fermion antifermion pair. The probability by which all these processes happen, is dictated by the various charges. A neutrino--antineutrino pair cannot form a photon nor a gluon, because it does not have an electric nor a color charge.} %I can think of two %ways to implement antiparticles in the present picture. %One possibility is to assume the existence of antipreons (white circles %in fig. \ref{fig11}) %in addition to the ordinary preons (black circles), so that %the corresponding antiparticle states can be formed. %The antiparticle of a lefthanded fermion is a righthanded object %and its preons should therefore form a righthanded tetrad. More precisely, the combinations of a lefthanded fermion of the first family and their righthanded antiparticles are shown in fig. \ref{fig11}. This way all the corners of the cube are filled. % by this combination. Fig. \ref{fig11} more or less represents how vector bosons should be imagined in this model.

Fig. \ref{fig11} is a rather characteristic picture of a fermion--antifermion bound state. The point is that the vector boson interactions always take place within one family, and fig. \ref{fig11} corresponds to interactions within the first family. One sees that the bindings between the links join together to form bindings along the plaquettes. Altogether, the bindings form an oriented closed circle of plaquettes. In the case of the second family interactions there is also a closed circle and it lies in the second family plane (cf. fig. \ref{fig4}) and similarly for interactions between members of the third family. The three planes can be rotated into each other to make the corresponding vector bosons identical. The difference to fermions will be understood better in a group representation approach to be discussed below.

In my model, vector bosons are superpositions of fermion--antifermion states $\bar F \times f$ with the appropriate quantum numbers. The $\bar F \times f$ binding arises from interactions along the four body diagonals of the cube defined by fermions (1234) and antifermions (1'2'3'4'), i.e. interactions between the full and open circles in fig. \ref{fig11}. I shall come back to the body diagonals later.

For finite groups the number of irreducible representations (IR's) is equal to the number of conjugacy classes. In the present case the IR's are usually called $A_1$, $A_2$, $E$, $T_1$ and $T_2$ with dimensions 1, 1, 2, 3 and 3, respectively, and their characters are shown in table 1. $A_1$ is the identity representation. $A_2$ differs from $A_1$ by having a negative value for odd permutations. $T_1$ is the representation induced by the permutations of the corner points 1,2,3,4 of a tetrad in three dimensions. Its representation space is therefore the three dimensional space, in which the fermions live, i.e. {\it $T_1$ konstituiert den Anschauungsraum}. $T_2$ is obtained from $T_1$ by changing the sign of the representation matrices for the odd permutations. Finally, $E$ is induced by a representation of $S_3$ on the corners of a triangle, as discussed in connection with fig. \ref{figt}, for example $E(\overline{2134})\bf{a}=\bf{b}$, $E(\overline{2134})\bf{b}=\bf{a}$, $E(\overline{2134})\bf{c}=\bf{c}=-\bf{a}-\bf{b}$, $E(\overline{1243})\bf{a}=\bf{b}$, $E(\overline{1243})\bf{b}=\bf{a}$, $E(\overline{1243})\bf{c}=\bf{c}=-\bf{a}-\bf{b}$, etc.

In order to obtain the vector bosons $\bar F \times f$, one should take the 9--dimensional product representation \begin{equation} T_1 \times T_1 =A_1+E+T_1+T_2 \label{eqzut} \end{equation} On the right hand side, the term $T_1$ corresponds to arbitrary rotations of the closed loops of plaquettes, as claimed in connection with fig. \ref{fig11}. %The interpretation of eq. (\ref{eqzut}) is as follows: $A_1$ corresponds to the photon, the totally symmetric singlet configuration, where all tetrad--antitetrad combinations contribute in the same way. $T_2$ is induced by 24 permutations of some objects I,II,III,IV (much like the $T_1$ on the left hand side of equation (\ref{eqzut}) was induced by the 24 permutations of 1,2,3,4). Finally, $E$ is induced by the 6 permutations on the triangle (fig. \ref{figt}).

A possible interpretation of $T_2$ is as follows \footnote {Alternatively, $T_2$ could represent 24 gluons which would then differ for the 3 families. The six permutations of the triangle might be the weak bosons $W^{1,2,3}$ and $W^{1,2,3}_{R}$.} : By definition, the different vector bosons correspond to permutations of the cube's four body diagonals called I, II , III and IV, which define another group $S_4$. It is ordered not as in the case of fermions, equations (\ref{eq8})--(\ref{eq10}), but according to its conjugacy classes. In fact, the 24 elements of $S_4$ can be ordered in 5 conjugacy classes with 1, 3, 8, 6 and 6 elements. They are given as follows: \begin{itemize} \item identity \\ $\overline{I,II,III,IV}$ \\ the U(1) gauge boson \item 3 $C_2$ rotations by $\pi$ about the coordinates axes x, y and z \\ $\overline{II,I,IV,III}$, \qquad $\overline{III,IV,I,II}$, \qquad $\overline{IV,III,II,I}$ \\ the SU(2) gauge bosons \item 8 $C_3$ rotations by $\pm {2 \over 3}\pi$ about the cube diagonals (like x=y=z) \\ $\overline{II,III,I,IV}$, \qquad $\overline{III,I,II,IV}$, \qquad $\overline{II,IV,III,I}$, \qquad $\overline{IV,I,III,II}$, \\ $\overline{III,II,IV,I}$, \qquad $\overline{IV,II,I,III}$, \qquad $\overline{I,III,IV,II}$, \qquad $\overline{I,IV,II,III}$ \\ the gluons \item 6 $C_4$ rotations by $\pm {\pi \over 2}$ about the coordinate axes \\ $\overline{II,I,III,IV}$, \qquad $\overline{III,II,I,IV}$, \qquad $\overline{IV,II,III,I}$, \\ $\overline{I,III,II,IV}$, \qquad $\overline{I,IV,III,II}$, \qquad $\overline{I,II,IV,III}$ \\ leptoquarks \item 6 $C_2 '$ rotations by $\pi$ about axes parallel to the face diagonals (like x=y, z=0) \\ $\overline{II,III,IV,I}$, \qquad $\overline{II,IV,I,III}$, \qquad $\overline{III,IV,II,I}$, \\ $\overline{III,I,IV,II}$, \qquad $\overline{IV,III,I,II}$, \qquad $\overline{IV,I,II,III}$ \\ leptoquarks \end{itemize} where reference is made to the cartesian coordinates x,y and z with origin at the cube's centre. This ordering is reminiscent of the ordering of gauge bosons in grand unified theories where there are leptoquarks in addition to the 8 gluons and the four electroweak gauge fields. The elements of the first two classes form Klein's 4--group (an abelean subgroup of $S_4$), whereas the elements of the first three classes form the nonabelean group of even permutations.

In summary, the interpretation of eq. (\ref{eqzut}) is as follows: As discussed before, the fermions constitute ordinary three--dimensional space. As soon as two fermions approach each other to form a vector boson, space opens up to 9 dimensions. Three of them correspond to ordinary space, whereas the remaining six decompose into 1+2+3 dimensional representation spaces $A_1$, $E$ and $T_2$ of $S_4$. They become fibers to ordinary space. It remains to be shown how the complex structure of a $U(1)\times SU(2)\times SU(3)$ Lie algebra arises.

Since parity violation is not present in these pictures, I want to add an alternative related to the observation that there are two 1--dimensional and two 3--dimensional, but only one 2--dimensional IR of $S_4$. One could relate parity transformations to even--odd transitions between permutations by modifying the assignments made in equations (\ref{eq8})--(\ref{eq10}), namely \begin{eqnarray} \nonumber \nu_L &=& \overline{1234} \qquad u^1_R= \overline{2341} \qquad u^2_L= \overline{3412} \qquad u^3_R= \overline{4123} \\ e_L &=& \overline{4321} \qquad d^1_R= \overline{1432} \qquad d^2_L= \overline{2143} \qquad d^3_R= \overline{3214} \label{eq8s} \end{eqnarray} i.e. assigning odd permutations to righthanded states. According to the character table 1 the character of $E$ vanishes for odd permutations. Therefore, there is no action of $E$ on righthanded fermions. In contrast, the products $A_1\times A_2$ and $T_1\times T_2$ act like $-\gamma_5$ on left and righthanded fermions. {\bf Conclusions}

According to present ideas the elementary particles (leptons, quarks and vector bosons) are pointlike and their mathematical description follows this philosophy (Dirac theory, Yang--Mills theory). They certainly receive an effective extension by means of quantum effects, but these are fluctuations and do not affect the primary idea of pointlike objects.

In contrast, in my model the observed fermions naturally have an extension right from the beginning. This seems to be difficult to accomodate because their radius should be of the order of their inverse masses. Following t'Hooft one may assume that there is a symmetry principle which leaves the masses small.

In its present stage, the model does not allow to make quantitative predictions of fermion masses, although some qualitative statements about fermion masses have been made in the previous sections. As compared to the binding energy, all fermion masses (including $m_t$) are tiny perturbations which might be induced by some radiative mechanism of the 'effective' standard model interactions leading to masses $\sim \alpha^F$ for the F--th family. The 'textures' of those masses have been discussed by Ramond. Within the present model one may argue that the fermions are basically massless by some symmetry and that there are small symmetry breaking effects within the family planes leading to different family masses.

Whatever this symmetry principle may be, there is still the question how large the radius R of the quarks and leptons is. In principle, there are three possibilities, it may be large ($\sim 1$ TeV$^{-1}$), small ($\sim M_{Planck}^{-1}$) or somewhere in between. In the first case there will be experimental signals for compositeness very soon. In the second case there will never be direct experimental indications and it will be difficult to verify the preon idea. Furthermore, in that case one would have the GUT theories as correct effective theories whose particle content would have to be explained. In addition, it may be necessary to modify the theory of relativity. In fact, the superstring models are a realization of this idea, the 'preons' being strings instead of point particles.

Personally, I like the scenario $R \sim M_{Planck}^{-1}$ reasonably well. In the model presented in this paper, the preons are pointlike and sitting on a cubic lattice. This lattice would have to fluctuate in some sense to reconstitute Lorentz invariance. This certainly raises many questions which go beyond the scope of this article. For example, the renormalization of gravity would be modified because high energies ($> M_{Planck}$) would be cut away by the lattice spacing.

As for the third possibility 1 TeV $<< R << M_{Planck}$, gravity and its problems play no role and my models are just a more or less consistent picture of particle physics phenomena. Since no attempt was made to explicitly construct the states of quarks and leptons in their known complex representations they are at best a qualitative guideline for understanding. I did not write a Lagrangian for the preons and just speculated about their interactions. The ultimate aim would be to construct a Lagrangian and derive from it an effective interaction between Dirac fermions and gauge fields.

© Copyright: B. Lampe, 2002

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