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I consider the standard model, together with a preon version of it, to search for unifying principles between quantum particles and general relativity. Argument is given for unified field theory being based on gravitational and electromagnetic interactions alone. Conformal symmetry is introduced in the action of gravity with the Weyl tensor. Electromagnetism is geometrized to conform with gravity. Conformal symmetry is seen to improve quantization in loop quantum gravity. The Einstein-Cartan theory with torsion is analyzed suggesting structure in spacetime below the Cartan scale. A toy model for black hole constituents is proposed. Higgs metastability hints at cyclic conformal cosmology.

The purpose of this article is to search for unifying principles for quantum models of matter and spacetime on all possible length scales: from the tiniest distances of high energy accelerators and colliders up to galaxies and towards the radius of the universe. Even the Planck scale Gedanken experiments are con- sidered. Cosmological developments are included with a lesser emphasis― though not less important. This task is motivated by a large number of theore- tical results on the various sectors of the subject, not necessarily on the most fashionable areas of main stream research, but they are unfortunately scattered around widely in the literature. The author feels therefore that trying to collect some of the pieces of the puzzle together is well justified. Evidence for mathe- matical unity of matter and spacetime structure is indeed found, even though this work is bound to require much more effort in the future. Some of material of this note is of this author but mostly what follows is a mini review based on a personal, and partly random, selection of papers.

In the last fifty years, or more, symmetry has been the leading principle in classifying particles and their interactions. Specifically, the gauge symmetry is the basis of particle phenomenology and theory. All known particles belong to a presentation of some group, be it e.g. the Lorentz, Poincaré or an SU(N) group. The standard model (SM) of particles with its some twenty parameters describes all measured accelerator data available today. There are known limitations and problems with the standard model. Bigger problems occur when one considers astrophysical and cosmological measurements, like dark energy and dark matter. Gravity has its own known problems with quantum theory in general. String theory was a promising candidate for unifying the standard model with gravity. Unfortunately, not much progress has taken place in about fortyfive years, apart from experimentally discovering the top quark, the Higgs boson and the acce- lerating expansion of the universe. Have we considered in depth all possible symmetries now? Perhaps not.

One specific symmetry has long been known in special circumstances: scale, or conformal, invariance in deep inelastic scattering, magnetism and the primor- dial cosmic microwave background (CMB) fluctuations.^{1} Here I use the term conformal symmetry. The standard model is conformal symmetric if one leaves out the Higgs sector. The standard model is stable towards Planck scale but the Higgs sector is metastable. This metastability is a problem to inflationary cosmo- logy but it fits well to conformal cyclic cosmology where the decay of the current vacuum is a prediction ending the current cycle and beginning the next one [

Together with several other people, I have gone quite some time ago one step further down to preon level, for a review of early work see e.g. [

Unification of physics based on particle internal symmetry has been successful until recently when geometry has taken a significant role in the form of the local conformal symmetry. Gravity itself is not limited to Einstein gravity (EG). Several extra terms in the gravitational action have been studied with more and less success. In fact, a good old starting point is the gravity based on the Weyl tensor [

Quantization of geometry has been pursued for a few decades within a theory called loop quantum gravity (LQG) [

This note is organized as follows. In Section 2 I briefly recall the preon model, which is discussed partly for historical reasons. In Subsection 3.1 conformal gravity and in Subsection 3.2 loop quantum gravity are summarized. In Section 4 electromagnetism is geometrized. The conformal standard model is discussed in Section 5. Section 6 is on the outer edge of this study. The Dirac field in the presence of torsion is introduced in Subsection 6.1. The massive Dirac field is discussed in Subsection 6.2. Cosmology is reviewed in Section 7. Conformal symmetry and black holes are treated in Section 8. Finally, conclusions are given in Section 9.

The presentation of the material is concise and goal oriented rather than comprehensive but an attempt is made for it to be reasonably self contained. The author feels he had to go through all the material presented in this note but the reader may find it better to start with the first and last section on first reading and save Sections 2 and 6 for later.

The virtue of conformal symmetry is that the action for conformal gravity is defined uniquely by the Weyl tensor, described in Subsection 3.1. All particles in conformal theory are massless. Other properties of conformal theory include renormalizability, unitarity, and the theory is ghost-free. It has been shown to explain dark matter and energy [

Requiring charge quantization

A binding interaction between preons is needed to make the quark and lepton bound states possible. I have at the moment no detailed form for this interaction. Its details are not expected to be of primary importance at this preliminary stage. I suppose this attractive, non-confining interaction is strong enough to keep together the charged preons but weak enough to liberate the preons at high temperature. Some more thoughts are indicated in Subsection 2.2.

A useful feature in (2.1) with two identical preons^{2} is that the construction provides a three-valued subindex for quark SU(3) color, as it was originally discovered [

One may now propose that, as far as there is an ultimate unified theory, it is a preon theory with only gravitational and electromagnetic interactions operating between preons. The strong and weak forces are generated in the early universe later when massless preons combine into quarks and leptons at lower tempera- ture and they operate only with short range interaction within nuclei making atoms, molecules and chemistry possible. In a contracting phase of the universe processes take place in the opposite order.

The unification picture is supposed to hold in the present scheme up to the energy of about 10^{16} GeV. The electroweak interaction has the spontaneously broken symmetry phase below an energy of the order of 100 GeV and symmetric phase above it. The electromagnetic and weak forces take separate ways at higher energies (

The proton, neutron, electron and

The baryon number (B) is not conserved [

Unification of gravity and electromagnetism is discussed in Section 4.

The preon model described in [

The preons have to be kept inside the quarks and leptons using some elegant, preferably non-confining mechanism. I also want to exclude scalar, vector and spinorial self-interactions. Same sign charges should be kept inside the bound states. Therefore a possibility could be that the zero mass black hole preons would form together one single non-spherical horizon around the quark or lepton.

Unification of black hole particles (i.e. preons) and spacetime is discussed in Subsection 3.2.

Weyl introduced, while working on the geometrization of electromagnetism, a new tensor called the Weyl or conformal tensor [

(3.1)

where

the Weyl tensor transforms simply as

The pure local conformal gravity is based on the following action

where

Conformal gravity is power-counting renormalizable and unitary [

invariant. The cosmological constant will appear later when conformal symme- try is dynamically broken and gives the dimensionful

The reason for local conformal invariance is that massless particles move on the light cone which is invariant under the 15 parameter conformal group SO(4,2) [

Functional variation of

where

By defining the left hand side of (3.6) as the energy-momentum tensor

We see that gravity and matter sectors are on equal footing and the total energy-momentum tensor of the universe is zero.

The connection between Einstein gravity and conformal gravity is that the solutions of the former are solutions of the latter [

As a final piece of support to conformal invariance it can be mentioned that the high energy limit of all non-trivial renormalizable field theories is comfor- mally invariant [

A statistical physics model for quantum black holes has been presented in [

where the sum is over punctures

Among the problems in LQG is finding quantization without quantum anomalies. A second difficulty is making contact with the semiclassical physical picture of gravity. The existence of Planck scale sets restrictions in going to the continuum limit. If one adds points to the spin network to refine it, the conti- nuum approximation of volumes and areas does not get better, one just adds volume to the spacetime as the area eigenvalue has a minimum value. In a con- formal theory there is no length scale available and it is possible to improve the situation. Thirdly, though the theory is discrete and therefore finite, a finite renormalization is needed to separate the lower energy physics from the Planck scale features [

Conformal invariance helps in all the above difficulties. In a conformal theory spin networks can be defined which can be indefinitely refined to arbitrary precision. In the renormalization problem no counter terms are needed in spacing dependent renormalization.

The geometric operators of area of a surface and the volume of a region can be generalized to their conformal invariant counterparts which are the same as before but now without factors of Planck length to the relevant power. Thus a conformal geometry of a spin network can be defined [

Conformal invariance is also important for understanding the geometrization of other interactions. Metrication of electromagnetism coupled to a Dirac field

The generalized Dirac action is

where the

which is obtained from the generalized connection

where

where

Secondly, the Weyl connection drops out from the generalized Dirac action and therefore does not provide geometrization of electromagnetism. Replacing

and

where

The action

One sees that

It is possible to couple massless particles to conformal gravity. One can also couple the massless standard model, or even the complete standard model in a way in which the Higgs boson acquires mass in the gauge fixed conformal theory [

The SM total Lagrangian can be written as a sum of the gravitational and matter terms

where

(5.2)

where the

Choosing the gauge

where

This is a brief summary of a conformal invariant theory of gravity coupled to the SM that can be quantized by the LQG techniques [

In this section I want to emphasize another kind of treatment of conformal gravity and the Dirac field, which can be a quark, lepton, or preon. In GR, one has to discuss torsion arising from rotations and translations of the Poincaré group, just as energy gives rise to curvature [

The Riemann-Cartan geometry with metric and torsion is defined in terms of the metric tensor

The torsion tensor is antisymmetric in its lower indices. A symmetric connec- tion is known as torsion-free.

The most general conformal transformation for the metric and torsion are with

where

The Dirac field conformal transformation is

Let us introduce the modified metric-torsional curvature tensor with the Riemann curvature tensor

whose irreducible part is

and it is conformally covariant. The commutator of covariant derivatives obeys the equation

The conformal transformation for torsion is not uniquely defined [

with the parameters

(6.9) is antisymmetric in the first and second pair of indices, irreducible and conformally covariant. This reduces to the form

with k the gravitational constant. By variation one gets

where

The Dirac action is

where

and the massless matter field equations are

To see the effects of the complete antisymmetry of the spin on the structure of the field equations one should rewrite the field equations as follows

with the massless matter field equations (6.16).

In Weyl gravity there is no more a completely antisymmetric torsion and there are additional constrictions on the curvature tensor. This happens because both field equations for the spin and energy couple to both torsion and curva- ture so that the complete antisymmetry of the spin is partly imposed on torsion and partly on the curvature. We may decompose torsional terms away from the torsionless ones in all curvatures and derivatives. Thereafter all curvatures and derivatives are written in terms of purely metric curvature and derivatives given by the Weyl conformal tensor

where

In Weyl-type of gravity there is no possibility to substitute torsion with the spin of the spinors and there are no longer non-linear self-interactions in the spinor field equations. The Dirac equation is linear even in the presence of torsion.

The conclusion from all the above analysis of torsion in GR is that the non- linear self-interactions of Dirac matter fields are absent.

A Dirac field with mass is a non-conformal theory, but it is a very interesting case and is discussed briefly. The Einstein-Cartan (EC) [

The dynamical variables in Einstein-Cartan theory are the vierbein

where

The variation with respect to spin connection

The ECKS Lagrangian density is

where

where

where

where () denotes symmetrization, is quadratic in

The Cartan equation (6.25) is a linear relation and torsion is proportional to spin density. Therefore the torsion is zero outside material bodies. This makes detection of torsion difficult. The torsion field of ECKS theory does not propagate, unlike curvature.

The relativistic Dirac Lagrangian density in curved spacetime is

where the

This spin density (6.29) does not depend on

If one assumes the simplest possible fermion system, namely a point particle or a system of point particles, it turns out that there exist no solutions for the spinor field, i.e.

For an electron,

These results imply that the Dirac wave function of an electron forms a non- singular form of spacetime structure of a toroid which has the outer radius of the electron Compton wave length and the inner radius of its Cartan radius. This is valid both for charged and uncharged leptons. The weak interactions do not change the situation in any significant amount. The toroid structure works also for quarks for which asymptotic freedom holds at distances^{3}

Finally, for the cosmological constant it is derived in [

This

The running standard model quartic Higgs coupling ^{10-12} GeV, assuming that no new physics below the Planck scale changes the situation [

But for the metastable Higgs there is a better solution, cyclic cosmology [

In [

The guiding principle of the model is conformal symmetry. The Weyl invariant action

The term

The action (7.1) is invariant under Weyl transformations by a local function

where

In the gauge

The action (7.1) defines a conformally invariant homogenous and isotropic Friedman-Robertson-Walker (FRW) universe [

where

Cosmological variables and gauges are briefly discussed, in

where

The authors are now able to conclude having found a band of continuous solutions that undergo acceptable repeated cycles of expansion and contraction as illustrated in detail in [

The conformal action is (3.4). The general static, spherically symmetric solutions of (3.8) is [

where the function

where

which gives

where

For certain values of the parameters (8.4) is a black hole line element. The metric on the spacelike surfaces of constant

This metric describes a non-compact hyperbolic two-space

There are several possibilities to find black holes with non-trivial topologies. With

which is singular at

This spacetime is similar to the Schwarzschild-de Sitter solution but with non- trivial topology. As

In the limit

with

Depending on the parameter values (8.10) may represent a black hole. The angular sector has a flat metric

on gets a compact orientable surface, a torus, with a topology

This is for

For different genus values one has:

I propose a toy model for generic black hole structure, or constituents, in which the hole consist of tori of decreasing sizes starting from the radius of the hole. The next torus radius is the previous torus tube radius increasing the complexity of the hole topology. The scale dependence of this spacetime structure should be studied by this scale method.

The elegance and power of general relativity are realized when the basic Einstein equations are generalized to the largest local symmetry groups including the Weyl conformal symmetry and the full Poincaré symmetry with torsion of spacetime. The statement “to modify it [EG] without destroying the whole structure seems to be impossible” did not turn out to be true, if the modification is done properly. The main conclusion of this study is that local conformal symmetry allows us to obtain a unified description of gravity and the standard model. All interactions are described in geometrical or geometrized formalism which contains the familiar SM quantum particles. A possible model for matter- spacetime unification was reviewed in Subsection 3.2. I proposed in Section 8 a toy model for black hole structure, or constituents, in which the holes consist of tori of decreasing sizes starting from the radius of the hole.

With conformal symmetry the applicability of GR is greatly expanded. The cosmological picture of the universe is changed substantially as became clear several decades later [

I thank Dr. William Straub for correspondence and comments on the manu- script.

Raitio, R. (2017) On the Conformal Unity between Quan- tum Particles and General Relativity. Open Access Library Journal, 4: e3342. http://dx.doi.org/10.4236/oalib.1103342