E-gravity theory as a Yang-Mills theory

Introduction

The most plausible approach for making a model for fundamental elementary particle interaction is the Yang-Mills theory. In this theoretical framework, a local invariance of a physical action under a gauge group action is assumed. This invariance imposes the existence of a gauge connection that is also called “gauge boson field”. Electromagnetic, electroweak and strong interaction can be described very well in terms of Yang-Mills theory. Electroweak interaction is generated by imposing a  gauge group invariance and strong interaction can be obtained if the action functional is invariant under  group transformations. The Standard model of particle physics is based on the gauge group . May be  a gauge connection in spacetime direction  with generator index ,  the structure constants of the gauge group,  the coupling constant and  the associated field strength. Then the Yang-Mills Lagrangian density  has the form:

The additional Lagrangian density  is arising from the measure factor of the quantum path integral and contains the Faddeev-Popov ghost fields  that are Grassmannian variables. It holds:

However, gravity, also another fundamental force, cannot be described in terms of Yang-Mills theory. The only well-known consistent way for describing gravity is General Relativity, which is a classical theory of gravity. An open question is whether gravity can be formulated as a Yang-Mills theory. Several research papers show that there is existing a relationship between gravity and Yang-Mills theory (Bern, Z. et al. 1998) (Hsu, J.P. 2006).

There are existing theories that quantize gravity, e.g. Loop Quantum Gravity. The derivation of Loop Quantum Gravity is based on a reformulation of General Relativity by introducing  gauge fields (Ashtekar,A. 1985). However, Loop Quantum Gravity does not coincide with Yang-Mills theory, e.g. due to the use of constraint terms and the quantization in terms of Wilson loops. Another recent approach to quantum gravity is E-gravity theory (Linker, P. 2016). This theory provides a discretization of spacetime with simplices. Simplices can have two or more simplex corners that are identified (in mathematical terms: there exists a nontrivial equaliser). Because of this additional property, a curvature measure can be assigned to each simplex. General Relativity is rewritten in this discrete form and a path integral over all possible spacetime geometries is performed.
In this research paper, an alternative derivation of E-gravity theory is shown. At first, spacetime is discretized by finite elements which are the simplices. Finite element methods are typically used for engineering problems, but are also applied to quantum field theories (Sopuerta, Carlos F. et al. 2005). Instead of using assumptions about the discretization of the Einstein-Hilbert action, a Yang-Mills theory with the diffeomorphism operator as the gauge group is worked out. Also, the field strength tensor, which is different from the smooth fiber bundle curvature form, is derived. With proper projection of the Yang-Mills theory to physical states, E-gravity theory is recovered.

The Derivation of E-gravity theory by Yang-Mills theory

The theory of gravity is assumed to be a gauge theory with respect to the Poincaré group. If General Relativity is transformed locally by the Poincaré group, the action remains the same. Generators of the Poincaré group are local spacetime translations and local Lorentz transformations. Since the translation generators  corresponding to a 4-vector  are operators and not finite-dimensional matrices, Yang-Mills theory in the form (1) cannot be applied. The goal is to find a matrix-like gauge group for gravity. The first step to find a proper gauge group is to use a finite element discretization of spacetime. In general, a physical field  (can be scalar, spinor, vector or tensor field) on a spacetime point  is discretized as follows:

Here,  are the field values at simplex corner (or node)  and  the corresponding form functions that have the value 1 at simplex point  and the value 0 elsewhere. These form functions generate a discretized spacetime without self-intersection of simplices. Additionally, if two simplex nodes are identified (a key feature of E-gravity theory), the sum in equation (3) runs from  to  with the number of identified nodes . Denotes  the set of all possible form functions that discretize the spacetime and  the set of all nodes in the spacetime, one obtains the path integral:

Clearly, the set  is quite different from the simplex node equaliser set. However, the path integral (4) can be reduced to E-gravity theory by an appropiate gauge group. May the gauge operator  acting on a function  with the entire spacetime region  be defined as follows:

Because the operator (5) modifies the physical fields globally, the field  is a global gauge transformation. The operator (5) can be turned into a local gauge transformation by replacing .
For a plausible model of gravity, the invariance of the action under the modified Poincaré group  with group elements  acting on physical states in 4-dimensional spacetime is imposed. This group is called “modified”, because all elements of the group acting like the convolution operator (5); these groups are “infinite-dimensional matrices”. Moreover it holds , i.e.  is an unitary transformation. Due to linearity of  and the invariance of the action  under group transformations in , the choice of form functions is arbitrary. If the deformation of simplices by  is diffeomorphic, one can set without loss of generality , i.e. local and global gauge transformations are the same. This can be easily seen if a differential operator  with respect to  is performed: By chain rule,

and equation (6) is only true if and only if . Hence, the gauge connection vanishes in case of any diffeomorphic deformations.
If two or more simplex nodes are identified, one performs a non-diffeomorphic deformation. The left hand side of equation (6) does not exist in general. By switching the gauge transformation from a global to a local transformation, one can generalize the gauge group  to a non-diffeomorphic gauge group. This implies the existence of a non-smooth (!) gauge connection .

Gauge group  ensures that physical laws are unchanged not only if local diffeomorphic deformations are present as it is the case in General Relativity. Local non-diffeomorphic deformations also let physical laws be the same. Diffeomorphic spacetime deformations of General Relativity can be assumed as a very dilute collection of non-diffeomorphic deformations on Planck scales in average. An important condition for this gauge group is that given the map  between form functions  for arbitrary , the resulting form function  generates also a simplex element.

Clearly, a gauge connection which is not differentiable, makes not sense in a physical field theory. If  on a certain spacetime point , then  is singular, since  is necessary only for non-diffeomorphic deformations. As a consequence, physical states can only be observed in regions with  for all spacetime directions . Field strengths  are not vanishing necessarily, because derivatives of  are not zero in general even if .
The path integral (4) can be reduced with the assumptions made above. Introducing a parameter , where  denotes the index of the topological simplex structure (i.e. it denotes which nodes are identified). This parameter runs over all different states of a simplex with same topology and therefore it is called the “diffeomorphic deformation parameter”; in diffeomorphic deformation this parameter changes. For non-diffeomorphic deformations, the gauge connection  is changing. Therefore:

The set  is the set of all possible simplex topologies, i.e. the set of all identified simplex nodes. Physical action does not change under diffeomorphic deformations (including Lorentz transformations) and hence, the integral over  for each  is constant. The integration over all gauge connections makes only sense for . All contributions with  can be neglected; only physical states are taken into account, while the number of unphysical states is vanishing small. Hence, the Faddeev-Popov factor in the path integral factorizes in the path integral and can be omitted. One is left with the following path integral:

The path integral (8) is the path integral of E-gravity theory if the field strength term  corresponds to the Einstein-Hilbert action in vacuum. It holds  for a 1-form gauge connection  and a 2-form field strength tensor  for . Using theorem of Stokes over a 2-dimensional area , one obtains . If  is sufficiently small, i.e.  ( denotes the measure of a set), one can define the field strength corresponding to the area :

From (9) it follows that if  and the path integration is not performed around a singularity, it holds . This is the case if there are no identified simplex nodes in a simplex element. Every node which is identified with another node causes a nonvanishing value of  due to the change of spacetime structure by making topological "shortcuts" in it. The statement  with coboundary map , E-semigroup elements  and indicator function  (that is 1 if there are no identified simplex nodes and 0 otherwise) where the  are pairwise different is proven. If there are 2 or more identified simplex nodes (these induce singularities), one can choose closed paths in arbitrary spacetime directions such that all 3 or more singularities are contained in the path. Since  is small, one has  everywhere on  and in this case .
It has to be shown that if exactly one simplex node  is identified with , it holds  if  is even and  is odd or  is even and  is odd,  if  and  are odd and  if  and  are even; the value  is a specific gravitational constant. At first, the following decomposition of the gauge field action (which is known from electromagnetic theory) can be performed:

The indices  are running from 1 to 3 (all space directions). The term  is the “gravito-electric” field strength and the term  is the “gravito-magnetic” field strength. By Lorentz transformations, the time direction can be replaced by a direction pointing anywhere in spacetime, where all other indices denote all other directions perpendicular to the direction with index 0; action functional does not change after any Lorentz transformation.

Simplex elements have to be oriented properly while discretizing spacetime. Non-diffeomorphic deformations have singular values of the connection across the 0-direction, if the simplex node generators  or  are identified with another simplex node (by convention of the coboundary map, all nodes with odd index are negatively oriented), and have singular values of the connection perpendicular to this 0-direction, if all other simplex node generators  and  are identified with another simplex node (by convention of the coboundary map, all nodes with even index are positively oriented). Every singularity has the same magnitude of the value  given by . This assumption relies on the choice of infinite hypercomplex numbers  with ordinary finite complex numbers  for the gauge connection field; indeed, it holds  by nonstandard analysis. If  is even and  is odd or  is even and  is odd, the path integration passing through 0-direction encounters one singularity and the path integration passing through directions only perpendicular to 0-direction encounters also one singularity; hence  and .

If  and  are odd, the path integration in arbitrary 0-direction encounters two singularities of  with equal magnitude, while path integration in directions perpendicular to the 0-direction have no singularities; hence: ,  and  with a constant . Finally, if  and  are even, the path integration in directions perpendicular to 0-direction encounters two singularities of  with equal magnitude, while path integration in directions in 0-direction have no singularities; hence: ,  and  with a constant . The values of  are also infinite values, but with renormalization of the Yang-Mills coupling constant  this infinity can be removed.

Discussion

In four spacetime dimensions, the Ricci scalar defined by  (averaged over a volume) can be also recovered by a Yang-Mills theory. Some idealizations like the equal values of a topological singularity are assumed, which might lead to small inaccuracies if the field strength is computed by more advanced concepts (not treated in this research paper). The equivalence between Yang-Mills theory and discretized Einstein-Hilbert action with curvature measure  is only valid for 3+1-dimensional spacetime (one direction generated by 2 simplex nodes makes the pure “electric” field strength and all three other directions generated by all other 3 simplex nodes make the pure “magnetic” field strength).  Extra dimensions can play a role if E-gravity theory is embed in a certain plausible framework of superstring theory or M-theory. However, there is still not found a way to do this. However, there is evidence that the Standard model of particle physics with gravity is recovered if superstring theories are compactified on the Calabi-Yau manifolds (Candelas et al. 1985). Getting the Standard model combined with E-gravity theory from superstring theory or M-theory by suitable compactification is still not clear even since the gauge group  has different structure compared with gauge groups that impose all other fundamental interactions.

Inhomogenity in spacetime, which is also a feature of E-gravity theory, is caused exactly by the field  which is nonvanishing between the simplex elements, where physical states can occur. This gauge connection separates simplex elements which are different. E-gravity theory in terms of a Yang-Mills theory would conserve energy, momentum and spin-angular momentum, but if the 1-form field  is “cut out”, energy and momentum has to be created and annihilated to include effects performed by the gauge connection.
Gravitational interactions are able to materialize multiple particles. From the Yang-Mills theory, gravity couples on the currents (here, the fermionic case with fermion fields  and coupling constant )

where  is the generator of diffeomorphic deformations. It is a convolution operator and from (11) it holds that gravity is induced by correlations of particles in the entire spacetime. A simple example is the correlation of an apple that correlates with the earth; the apple falls towards earth because gravity is triggered by the apple-earth correlation. In systems where quantum entanglement can occur, physical states can correlate; therefore gravity is also produced in entangled systems. Assuming that the generators  of the simplex modification are acting only on a small, bounded region, the correlation current (11) involves only correlations of states that are very close to each other. Moreover, due to the very small support of these generators, one can assume that the commutators of these are neglectible; an abelian gauge theory arises. Therefore, gravity has an “electromagnetic” character and the force is transmitted with speed of light.

The fact that entangled states produce gravitational energy can be understood as follows: If two particles are interacting with each other, the structure of quantum uncertainty between these particles is modified by gravity. Two different independent random motions that are interacting with each other, will combine to one correlated random motion. But such a process would lead to decreasing entropy . Gravity that is caused by correlation avoids a violation of the second law of Thermodynamics, it increases entropy by   and therefore, the energy that is produced by gravity has to be of order  for a given temperature . Energy has to be bounded by Planck energy which implies that the entropy change due to gravitational interaction  has to be finite.     

Quantum Gravity theories are still not proven by experiments. Some other research was performed to discuss effects predicted by quantum gravity (Takeuchi et al. 2016). However, experiments for effective gravity theories (e.g. General Relativity) were performed. A recent experimental verification of General Relativity (Abbott 2016) is the detection of gravitational waves from binary Black Holes. In terms of E-gravity theory, two Black Holes are correlating with each other and therefore these would induce a gravitational current given by (11). By using the Yang-Mills form of E-gravity theory one would obtain equations similar to Maxwell’s equations to electromagnetism with a correlation current instead of a electric current. The fields of the gauge connection have large values on small length and time scales. If the action of E-gravity theory is treated as a mean-field theory for large length and time scales, General Relativity can be recovered; moreover, since the regions of spacetime where large values of the nonvanishing gauge connection occur are very little, one can treat these gauge connections also as small values on large length and time scales. This implies that E-gravity theory predicts also waves induced by gravitational correlations.  The largest amplitudes of gravitational waves occurring when the two Black Holes are collided; the Black Holes have the largest correlation in this case.

Another experimental result, where predictions of E-gravity theory agree, is the entropy of a Black Hole. Black Hole entropy computed with E-gravity theory is proportional to the event horizon area of the Black Hole (Linker 2016).

Gravity is a very weak force in comparison with all other fundamental forces of nature. In E-gravity theories the value of the curvature measure vanishes at the most of the possible simplex configurations; making gravitational effects nearly irrelevant for particle scattering at low energies and low particle densities. Therefore, E-gravity provides an explaination of the weakness of gravity in comparison with all other fundamental forces.

Conclusions

An interesting correspondence between Yang-Mills theory and General Relativity is shown in this research paper. Yang-Mills theory with another version of the Poincaré group, called  with special assumptions about simplex topology turns out to coincide with the Einstein-Hilbert action that uses a generalized curvature measure based on chain complexes of simplices. Formulating fundamental interactions in terms of Yang-Mills theory makes it easier to find unified descriptions of fundamental forces. Indeed, the gauge group of Standard model of particle physics combined with E-gravity theory can be regarded as the gauge group  that looks like a unified description of fundamental forces. Unfortunately, a plausible Grand Unified Theory is still not found; only experimentally unverified models are proposed. However, E-gravity theory also applies to effects, where not all other fundamental forces are unified.

References

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