Fundamental Fields as Eigenvectors of the Metric Tensor in a 16-Dimensional Space-Time

An alternative approach to the usual Kaluza-Klein way to field unification is presented which seems conceptually more satisfactory and elegant. The main idea is that of associating each fundamental interaction and matter field with a vector potential which is an eigenvector of the metric tensor of a multidimensional space-time manifold n V (n-dimensional “vierbein”). We deduce a system of field equations involving both Einstein and Maxwell-like equations for the fundamental fields. Confinement of the fields within the observable 4-dimensional space-time and non-vanishing particles’ rest mass problem are shown to be related to the choice of a scalar boson field (Higgs boson) appearing in the theory as a gauge function. Physical interpretation of the results, in order that all the known fundamental interactions may be included within the metric and connection, requires that the extended space-time is 16-dimensional. Fermions are shown to be included within the additional components of the vector potentials arising because of the increased dimensionality of space-time. A cosmological solution is also presented providing a possible explanation both to space-time flatness and to dark matter and dark energy as arising from the field components hidden within the extra space dimensions. Suggestions for gravity quantization are also examined.

field unification based on the usual Kaluza-Klein theory [4] [5] [6] and extension to non-Abelian Yang-Mills fields [7] [8]. Here we develop an alternative way to attack the problem, which appears to be conceptually more satisfactory and elegant. Our proposal is based on the idea of associating each fundamental interaction field with a vector potential which is an eigenvector of the metric tensor of a multidimensional space-time manifold n V (n-dimensional "vierbein"). It is relevant to observe that within an n-dimensional space-time, the metric tensor, when represented onto the basis of its eigenvectors, yields a connection involving a 2-index antisymmetric tensor of the same form as a non-Abelian Maxwell tensor which may be related to the fundamental interaction fields in a quite natural way.
Sec. 2 examines such a formulation of general relativity in n space-time dimensions. Sec. 3 presents a system of field equations involving both Einstein and Maxwell-like equations for the interaction fields.
Sec. 4 is concerned with the problem of confinement of the interaction fields within the observable 4-dimensional space-time and shows that a non-vanishing particles' rest mass arises thanks to a suitable gauge function which can be related to a scalar boson field (Higgs boson).
Sec. 5 is devoted to a physical interpretation of the results and the conditions to be required in order that all the known fundamental interactions, (gravitational, electro-weak and strong) may be included within the metric and connection. As we will see a 16-dimensional space-time is required to fit the standard model of elementary particles [9] [10].
Consideration on the placement of fermions within the theory will be the subject of the Secs. 6-8, while in Sec. 9 we examine applications to cosmology of the theory.
The last two sections propose a new way to gravity quantization starting from a suitable energy-momentum tensor of the gravitational field. A complete presentation of the theory and much more has been proposed in my book [11].

Tensor Representations onto the Basis of the Eigenvectors of the Metric
Let us consider an n-dimensional space-time manifold n V , endowed with a symmetric metric g of signature ( ) ) are reserved to the physically observable components, while the underlined Latin ones ( , , 4,5, , 1 j k n = −   ) identify the non-observable extra components. The connection is given by the Christoffel symbols: (2)

Metric Tensor and Connection Coefficients
The n linearly independent eigenvectors ( ) { } , 0,1, 2, , 1 a n µ σ σ = −  of the metric tensor fulfill the orthonormality conditions: , 0,1, 2, , 1, g a a n µ ν µν σ τ σ τ η σ τ . Then: provides its representation onto the basis of its eigenvectors. These orthonormal eigenvectors are undetermined by an imaginary exponential factor e iθ which leaves unchanged the real metric tensor components, provided that the complex . This degree of freedom allows periodic wave propagating field solutions (see Sec. 4.1). In the following we will drop the * complex conjugation mark, leaving it as understood, to avoid too heavy notations.
The representation of the connection coefficients relative to the basis of the eigenvectors becomes now: with: Significantly the connection includes antisymmetric non-Abelian tensors fields and a symmetric non-tensor field ( ) h σ µν . It is convenient to introduce also the new symbol (reduced connection): Then the complete expression of the connection in n V writes also: Arising of non-Abelian Maxwell-like tensors ( ) f α σ µ within the connection coefficients suggests that the electro-weak and strong interaction fields may be included into the metric tensor in a unified field theory when the space-time dimensionality is greater than four.

Ricci Tensor
The Ricci tensor: is now to be evaluated on the basis of the eigenvectors of the metric. The following auxiliary notations may be useful: from which we obtain: thanks to the symmetries. Calculations leading to an explicit representation of R µν in terms of ( ) and their derivatives are very heavy. But under the covariant Lorentz gauge condition: which can be shown to be equivalent to: relevant simplifications arise, resulting: Then the Ricci tensor assumes the meaningful form: where colon ( : ) denotes here the covariant derivative respect to the reduced connection and the notation: has been introduced. It is convenient to represent also: where µν  is arbitrary, at present, adding vanishing contributions to µν  and R µν because of antisymmetry. As we will see in Sec. 5 the tensor µν  introduces a degree of freedom that will play an important role to fit elementary particle current densities. The Riemannian covariant derivatives will be replaced with non-Abelian covariant ones ( ) D f α α σ µ , when required. Eventually we obtain:

Field Equations
If the Lorentz gauge holds, being equivalent to (13), the Einstein field equations become: Introducing the new variables: we obtain the following system of field equations: Now, if we represent ( ) σ µ λ onto the basis of the vector potentials, as: and the current density: we arrive at a physically relevant form of the field equations: Thanks to the Lorentz gauge and the symmetries the last Equation (28) results also equivalent to: The latter components behave as scalars when observed within 4 V and may be associated with the matter fields governing fermions. The previous equations can be developed as: Physically we need to preserve covariance respect to any transformation of the co-ordinates within 4 V , which are observable, while we may lose covariance respect to transformations involving the extra co-ordinates in the entire n V , since the latter are seen as scalars when observed from the physical space-time : The covariance, which is broken in the extra space, while it is preserved in 4 V , allows different rest mass values which will be attributed respectively to bo-A. Strumia son vectors carrying the fundamental interactions and to fermions characterizing the matter fields. Manifestly a non-vanishing particle rest mass is related to the dependence of the vector potential on the extra co-ordinates i x . So the request that the fields ( ) ( ) , l a a σ σ µ are confined within the physical space-time 4 V so that they depend only on the observable co-ordinates x α , is equivalent to impose that the particles associated with those fields have vanishing rest mass:

Particle Masses and Scalar Boson Gauge Fields
Now we will show how non-vanishing particle rest masses are related to the presence of the n-scalar gauge fields ( ) σ φ , which seem to play a role similar to the one of the Higgs scalar boson field (see [14]- [19]). But here a different mass generating "mechanism" is presented, which is based on a suitable gauge choice which is established in order to ensure the confinement of the physically observable fields.
In fact, as it is well known, the n-vector potential ( ) a σ µ is determined except for a gauge transformation: where the n-scalar field ( ) σ φ is required to fulfill the D'Alembertian equation: so that the Lorentz gauge is preserved. We point out that those scalar fields do not appear in the observable tensors ( ) f σ µν which are gauge invariant, since: into (41) we obtain the Klein-Gordon equation: We remark that while a single solution ( ) σ φ is required for the covariance of the 4-vector ( ) arise for the scalar fields contributing respectively to bosons ( [ ] M σ ) and fer- If we choose a gauge such that that ( ) a σ µ depends only on x α (so that then the gauge transformation (40) implies into (36): which in correspondence to the solution(42) for the field ( ) σ φ may be written as: so preserving the gauge invariance, thanks to (43), resulting: We now perform the following transformation of field variables: the inverse of which is given by: From (50) into (48) we obtain: And from (50) into (43) we have: It follows: being: being: A. Strumia

Gravitational Field
The present section is concerned with the physical interpretation of the components of the vector potentials. We start considering the free gravitational field in ordinary space-time 4 V .

1) 4 n =
In ordinary empty space-time, when no extra dimensions are present (standard general relativity) the metric tensor is interpreted, as usual, as a free gravitational field. Therefore its eigenvectors ( ) a σ µ may be conceived as the vector potentials of the free gravitational field. In fact in this occurrence the metric tensor is given by: and the potentials ( ) a σ µ are clearly responsible of gravitation, according to general relativity. The connection is given by: where: The energy-momentum tensor of the non-embedded into geometry contribution to the gravitational field appears: Journal of Applied Mathematics and Physics together with the gravitational current density: More, an equivalent gravitational energy-momentum embedded into geometry may be defined: So the field equations for the gravitational field can be written in the equivalent energetic form: V are related to the matter fields, i.e., to fermions (leptons, quarks). We observe also that the sectors of the potentials of indices ,l µ :  , and the correspondent additional energy-momentum tensor. We have: is the energy-momentum of the non-gravitational Maxwellian fields as they may be observed within the physical space-time 4 V .
A more familiar form of the Einstein equations, which hides the whole gravitational field into geometry is obtained if we write the metric tensor as: The connection writes now as: where a new partially reduced connection is defined by: In this way the gravitational field is entirely hidden into geometry and only the non-gravitational fields contribute to the energy-momentum tensor. The field equations become: T µν could be considered as possible contributions to dark energy and dark matter emergence.

Electro-Weak Field
In order to describe unified electromagnetic and weak fields we need one Abelian field and three non-Abelian ones. So the indices 4,5, 6, 7 s = will be related to electro-weak interactions and the vector potential components: will be interpreted as electro-weak fields. The space-time dimensionality required, is now raised up to 8 n = . The electromagnetic and weak interaction fields are mixed in the unified electro-weak theory. So the choice of the physical meaning of these vector potentials ( ) s a µ will depend on the standard model representation adopted.
Non-diagonal representation The non-diagonal representation of the electro-weak field involves the vector fields , a B W µ µ , 1, 2, 3 a = , the corresponding strength tensor being given by: where g is one of the electro-weak coupling constants and a bc  is the Levi-Civita symbol. So we are led to associate the components of each potential   Diagonal representation According to the standard model the physical fields: are provided by the diagonal representation, which is obtained thanks to a rotation of 3 , W B µ µ of the Weinberg angle, defined by the relation: g′ being a second electro-weak coupling constant. So that we have the following alternative way to associate our vector potentials with the electro-weak fields: Eventually it results: , , and determine the relations for the structure constants: In terms of the vector potentials the previous equation becomes, in the Lorentz gauge, a Klein-Gordon equation with current density. In Sec. 8 we will examine the current density ( ) s J µ .

Strong Interaction Field
The strong interaction field is carried by massless gluons and we are required to add 8 Identification of (99) with (111) now yields: The Maxwellian field equations are now:

Physical Interpretation of the Field Extra Components
In this subsection we want to show how the 192 extra components of the vector potentials ( ) a σ µ , may be related to the spinor fields associated with the fermions (leptons and quarks) appearing in the standard model of elementary particle theory.
Each spinor is a set of 4 complex valued functions of the observable co-ordinates Then we can associate groups of 4 components to the spinors representing the physical elementary fermions, e.g., as: where l.h., r.h. denote respectively left-hand and right-hand chirality and red, green, blue the quark color. According to this scheme a detailed sketch of the physical meaning of all the components of the vector potentials ( ) a σ µ can be summarized as the following According to the standard model the covariant derivatives are determined in such a way that the gauge invariance conditions in 4 V are preserved even when a gauge choice is fixed in the extra space-time. Such a choice is always possible because of the degrees of freedom provided by the anti-symmetric ten-sor A µν (arbitrary until now). Moreover, thanks to the latter tensor we will be able to obtain also the correct current densities in the r.h.s. of the interaction fields equations. Now we consider (29), with vanishing currents we have when l µ = :

Cosmological Solution
In this section we examine a cosmological solution to the Einstein field equations in Journal of Applied Mathematics and Physics The astonishing result of a negative mass density  provided by (152), Λ being assumed to be positive, suggests that the cosmological constant, due to the extra space-time dimensions, plays the role of a repulsive gravitational source, which is responsible of universe expansion, together with the positive pressure density ℘ given by (153). So the mass-energy density  and ℘ represent the mass-energy and pressure densities of the empty extended space-time 16 V (vacuum energy and pressure) which are seen as matter contributions by an observer living in 4 V .
The matter term includes: 1) The mass-energy and pressure densities of matter/interaction fields (

Conclusion
We have proposed the guidelines of a possible physical interpretation of a model of unified interaction (boson) and matter (fermion) fields within the geometry of a multidimensional space-time manifold 16 V . We have seen how to identify interaction fields with the vector components ( ) , 0,1, 2,3 a with the spinor matter fields. Meaningful consequences of these results have been obtained also in cosmology and a way to quantization of the gravitational field has been examined. All those results have been presented in detail in my book [11].

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.