5. Robson quantum theory of gravity

The generation model (GM) [23] of particle physics has been developed over many years as a viable alternative to the standard model (SM) [12] of particle physics. The SM is considered by the majority of physicists to be incomplete in the sense that it provides no understanding of many empirical observations including the existence of three families or generations of leptons and quarks, which apart from mass have similar properties, a nonunified description of the origin of mass, and the nature of the gravitational interaction.

The GM overcomes the incompleteness inherent in the SM by introducing three important different assumptions [24]: (1) a simplified unified classification scheme of the leptons and quarks in terms of additive quantum numbers, (2) an alternative version of quark mixing for hadronic processes, and (iii) the weak interactions are not fundamental interactions.

The development of the GM, primarily to describe the three generations of leptons and quarks of the SM [25], employing a unified classification scheme involving only three conserved additive quantum numbers, led to a composite model of the leptons and quarks and also the weak bosons, W and Z, mediating the weak interactions [23, 24].

Thus, the essential difference between the GM and the SM is that in the GM the leptons, quarks, and weak bosons are composite particles rather than elementary particles as in the SM.

In the GM, the leptons, quarks, and weak bosons consist of massless spin-1/2 particles called rishons and/or their antiparticles (antirishons). Each rishon carries a single color charge—red, green or blue—and each antirishon carries an anticolor charge—antired, antigreen or antiblue.

The first generation of leptons and quarks comprising the electron, the electron neutrino, the up quark, and the down quark are composed of two kinds of rishons: a T-rishon with electric charge <sup>Q</sup> ¼ þ <sup>1</sup> and <sup>a</sup> <sup>V</sup>-rishon with <sup>Q</sup> <sup>¼</sup> <sup>0</sup> and/or their antiparticles: <sup>a</sup> <sup>T</sup>-antirishon with <sup>3</sup> electric charge <sup>Q</sup> ¼ � <sup>1</sup> and <sup>a</sup> <sup>V</sup>-antirishon with <sup>Q</sup> <sup>¼</sup> 0. Both the <sup>T</sup>-rishon and the <sup>V</sup>-rishon <sup>3</sup> were introduced in 1979 by Harari [26] in his schematic model of the first generation of leptons and quarks describing their electric charge states.

The second and third generations of leptons and quarks are composed of the same "core" rishons and/or antirishons as the first generation plus the addition of one and two rishonantirishon pair(s), Π, respectively, where

$$
\Pi = \left[ \left( \overline{U} V \right) + \left( \overline{V} U \right) \right] / \sqrt{2} \tag{5}
$$

and the U-rishon has Q ¼ 0 and carries a single color charge [23].

The constituents of the leptons and quarks are bound together by a strong QCD color-type interaction [27], corresponding to a local gauged SUð Þ3 field (analogous to QCD in the SM) mediated by massless hypergluons (analogous to gluons in the SM).

The nature of the hypergluon fields acting between the rishons and/or antirishons of the composite leptons and quarks are analogous to the gluon fields acting between quarks and/or antiquarks in the SM. In particular, the nature of the hypergluon fields is such that they lead to a runaway growth of the fields surrounding an isolated color charge, implying that an isolated rishon or antirishon would have an infinite energy associated with it [28]. Nature requires such infinities to be essentially canceled or at least made finite. It does this for the composite systems of rishons and/or antirishons by requiring that the composite particle be colorless. However, quantum mechanics prevents these color charges from occupying exactly the same place so that the color fields are not exactly canceled although sufficiently to remove the infinities associated with isolated rishons or antirishons.

In the GM, each lepton of the first generation is colorless being composed of three antirishons carrying different anticolors. On the other hand, each quark of the first generation is colored being composed of one rishon and one colorless rishon-antirishon pair [23].

The second and third generations are identical to the first generation plus one and two colorless rishon-antirishon pairs, respectively, so that all leptons are colorless and all quarks are colored. Consequently, leptons do not combine to form more complex systems, while the quarks form hadrons that consist of two families: colorless baryons, made of three quarks with different color charges, and colorless mesons, made of one quark and one antiquark with opposite color charges [23].

Within the framework of the GM, the assumption that the elementary particles of the SM—the six leptons, six quarks, and three weak bosons—are all composite particles has led to a unified origin of mass [29] and a quantum theory of gravity [30].

In 1905, Einstein concluded [31] that the mass of a body m is a measure of its energy content E <sup>2</sup> and is given by <sup>m</sup> <sup>¼</sup> <sup>E</sup>=<sup>c</sup> , where <sup>c</sup> is the speed of light in <sup>a</sup> vacuum. Recently, this relationship has been verified [32] to within 0:00004% for atomic systems.

In the SM, the mass of a hadron arises mainly from the energy content of its constituent quarks and gluons, in agreement with Einstein's conclusion. However, the masses of the elementary particles—the leptons, quarks, and weak bosons—are interpreted [33] in a completely different way involving a Higgs field [34, 35]. Thus, the SM does not provide a unified origin of mass, contrary to Einstein's conclusion. Furthermore, the so-called Higgs mechanism does not provide any physical explanation for the origin of the masses of the leptons, quarks, and weak bosons, as pointed out by Lyre [36].

In the GM, the elementary particles of the SM are composite particles. Since the mass of a hadron originates mainly from the energy of its constituents, the GM postulates that the mass of a lepton, quark, or weak boson arises from a characteristic energy associated with its constituent massless rishons, antirishons, and hypergluons. The mass of each of these composite particles arises from the energy stored in the motion of the rishons and/or antirishons and the energy of the color hypergluon fields, <sup>E</sup>, according to Einstein'<sup>s</sup> equation <sup>m</sup> <sup>¼</sup> <sup>E</sup>=c2. Thus, unlike the SM, the GM provides a unified description of the origin of all mass and hence has no requirement for a Higgs field to generate the mass of any particle.

Since, to date, there is no direct evidence for any substructure of leptons or quarks, it is expected that the rishons and/or antirishons of each lepton or quark are localized within a very small volume of space by the strong "intrafermion" color interactions, acting between the colored rishons and/or antirishons.

In the GM, the mass hierarchy of the three generations arises from the substructures of the leptons and quarks. The mass of a composite particle will be greater if the degree of localization of its constituents is smaller, as a consequence of the nature of the strong intrafermion color interactions possessing the property of asymptotic freedom [37, 38], whereby the color interactions become stronger for larger separations of the color charges, as a result of antiscreening effects. In addition, particles with two or more like electrically charged rishons or antirishons will have larger structures due to electric repulsion. Ref. [23] presents a qualitative understanding of the mass hierarchy of the three generations of leptons and quarks: a quantitative calculation of the mass hierarchy requires very sophisticated computations.

On the other hand, the SM, involving the Higgs field to generate the masses of its elementary leptons and quarks, is unable to provide any understanding of the mass hierarchy of the three generations. As Lyre [36] has pointed out, the introduction of the Higgs field into the SM simply corresponds mathematically to putting in "by hand" the masses of the elementary particles of the SM.

The GM also provides a quantum theory of gravity. Gravitational interaction acts between particles with mass. Such particles are composed of rishons and/or antirishons that carry colored or anticolored charges and hence are required to be colorless in order to avoid infinite energies within their systems.

In the GM, the constituent electrons, neutrons, and protons of ordinary matter are all composite and colorless fermion particles. Between any two such fermion particles, there exists a residual interaction arising from the color interactions acting between the rishons and/or antirishons of one fermion and the color-charged constituents of the other fermion. Robson proposed [29, 30] that such "interfermion" color interactions could be identified with the usual gravitational interaction.

In the GM, gravity essentially emerges from the residual color forces between all electrons, neutrons, and protons. This leads [22, 39] to a new law of gravity: the residual color interactions between any two bodies of masses m<sup>1</sup> and m2, separated by a distance r, lead to a universal law of gravitation, which closely resembles Newton's original law given by:

$$F = H(r)m\_1m\_2/r^2,\tag{6}$$

where Newton's gravitational constant is replaced by a function of r, H rð Þ.

Both the fundamental intrafermion and the residual interfermion color interactions possess two properties arising from the self-interactions of the hypergluons mediating these interactions: (1) asymptotic freedom and (2) color confinement [39].

The antiscreening effects arising from the self-interactions of the hypergluons cause the color interactions to become stronger for larger separations of the color charges. In the case of the fundamental intrafermion interactions, this results in an increase in the characteristic energy and hence the mass of a composite particle that is less localized, as discussed earlier. In the case of the residual interfermion (gravitational) interactions acting between two masses, it leads to an increase in the strength of the gravitational interaction for larger separations so that H rð Þ becomes an increasing function of r.

It is known from particle physics that the strong color interactions tend to increase with the separation of color charges, and for large separations, this increase is approximately a linear function of r [40], in agreement with the flat rotation curves observed for spiral galaxies. Thus, H rð Þ is expected to be approximately a linear function of r:

$$H(r) = G(1 + kr/r\_S)\_\prime \tag{7}$$

where G is Newton's gravitational constant, k represents the relative strengths of the modified and Newtonian gravitational fields, and rS is a radial length scale dependent upon the radial mass distribution of the spiral galaxy so that rS varies from galaxy to galaxy.

Thus, the modified law of gravity in the GM may be written as:

$$\mathbf{g} = \mathbf{G}\mathbf{M}/r^2 + \mathbf{G}\mathbf{M}k/(rr\_{\mathcal{S}}).\tag{8}$$

Eq. (8) is very similar to Eq. (3) of the MOND theory and one can relate the modified terms in the two gravitational acceleration expressions to obtain:

$$a\_0 = \text{GM}(k/r\_{\mathbb{S}})^2. \tag{9}$$

Thus, the scale length rS may be regarded as the radial parameter beyond which weak acceleration takes place. Eq. (9) implies that the physical basis of the critical weak acceleration a<sup>0</sup> of the MOND theory is the existence of a radial parameter rS that defines a region beyond which the gravitational field behaves essentially as 1=r.

To summarize: gravity in the GM is identified with the very weak, universal, and attractive residual color interactions acting between the particles of ordinary matter. This gravitational interaction is mediated by hypergluons, which self-interact, leading to a significant modification of Newton's universal law of gravitation, especially at galactic distance scales. However, the self-interactions of the hypergluons cease at a sufficiently large distance as a consequence of the color confinement property associated with the QCD-like gravitational interaction. This leads to a finite range of the gravitational interaction for very large cosmological distances, estimated to be ≈ several billion light years [39].

Eq. (8) describes both the flat rotation curves of spiral galaxies and also the Tully-Fisher relation [22]. This modification of Newton's universal law of gravitation is essentially equivalent to that of the MOND theory, in that both describe these two overarching observational facts. However, the GM is based upon a quantum field theory of gravity, which provides a general underlying theory of gravity and hence a more physical understanding of the MOND results. Furthermore, unlike the MOND theory, the quantum theory of gravity provides a possible understanding of the observed "accelerating" expansion of the universe [41, 42].
