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A unified field theory of gravity, electromagnetism, and the Yang-Mills gauge field.

In this work, we attempt at constructing a comprehensive four-dimensional unified field theory of gravity, electromagnetism, and the non-Abelian Yang-Mills gauge field in which the gravitational, electromagnetic, and material spin fields are unified as intrinsic geometric objects of the space-time manifold S4 via the connection, with the generalized non-Abelian Yang-Mills gauge field appearing in particular as a sub-field of the geometrized electromagnetic interaction.

1 Introduction

In our previous work [1], we developed a semi-classical conformal theory of quantum gravity and electromagnetism in which both gravity and electromagnetism were successfully unified and linked to each other through an "external" quantum space-time deformation on the fundamental Planck scale. Herein we wish to further explore the geometrization of the electromagnetic field in [1] which was achieved by linking the electromagnetic field strength to the torsion tensor built by means of a conformal mapping in the evolution (configuration) space. In so doing, we shall in general disregard the conformal mapping used in [1] and consider an arbitrary, very general torsion field expressible as a linear combination of the electromagnetic and material spin fields.

Herein we shall find that the completely geometrized Yang-Mills field of standard model elementary particle physics, which roughly corresponds to the electromagnetic, weak, and strong nuclear interactions, has a more general form than that given in the so-called rigid, local isospace. We shall not simply describe our theory in terms of a Lagrangian functional due to our unease with the Lagrangian approach (despite its versatility) as a truly fundamental physical approach towards unification. While the meaning of a particular energy functional (to be extremized) is clear in Newtonian physics, in present-day space-time physics the choice of a Lagrangian functional often appears to be non-unique (as it may be concocted arbitrarily) and hence devoid of straightforward, intuitive physical meaning. We shall instead, as in our previous works [1-3], build the edifice of our unified field theory by carefully determining the explicit form of the connection.

2 The determination of the explicit form of the connection for the unification of the gravitational, electromagnetic, and material spin fields

We shall work in an affine-metric space-time manifold [S.sup.4] (with coordinates [x.sup.u]) endowed with both curvature and torsion. As usual, if we denote the symmetric, non-singular, fundamental metric tensor of [S.sup.4] by g, then [g[micro][lambda][g.sup.v[lambda] = [[delta.sup.v.sub.u], where [delta] is the Kronecker delta. The world-line s is then given by the quadratic differential form [ds.sup.2] = g[micro][dx.sup.[micro]] [dx.sup.v]. (The Einstein summation convention employed throughout this work.) As in [1], for reasons that will be clear later, we define the electromagnetic field tensor F via the torsion tensor of spacetime (the anti-symmetric part of the connection [GAMMA]) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCTIBLE IN ASCII.]

wheremis the mass (of the electron), c is the speed of light in vacuum, and e is the electric charge, and where [u.sup.[micro] = [dx.sup.[micro] are the components of the tangent world-velocity vector whose magnitude is unity. Solving for the torsion tensor, we may write, under very general conditions,

[MATHEMATICAL EXPRESSION NOT REPRODUCTIBLE IN ASCII.]

where the components of the third-rank material spin (chirality) tensor 3S are herein given via the second-rank antisymmetric tensor 2S as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCTIBLE IN ASCII.]

As can be seen, it is necessary that we specify the following orthogonality condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCTIBLE IN ASCII.]

We note that 3S may be taken as the intrinsic angular momentum tensor for microscopic physical objects which may be seen as the points in the space-time continuum itself. This way, 3S may be regarded as a microspin tensor describing the internal rotation of the space-time points themselves [2]. Alternatively, 3S may be taken as being "purely material" (entirely non-electromagnetic).

The covariant derivative of an arbitrary tensor field T is given via the asymmetric connection [GAMMA] by

[MATHEMATICAL EXPRESSION NOT REPRODUCTIBLE IN ASCII.]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Then, as usual, the metricity condition [[gradient].sub.[lambda]g[micro]v] =0, or, equivalently, [[partial derivative].sub.[micro]v[lambda]] = [[GAMMA].sub.[micro]v[lambda]] + [[GAMMA].sub.v[micro][lambda]] (where [[GAMMA].sub.[lambda]g[micro]v] = [gradient.sub.[micro][rho]][[gradient].sup.[rho].sub.v[lambda]]), gives us the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence we obtain, for the connection of our unified field theory, the following explicit form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

are the components of the usual symmetric Levi-Civita connection, and where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

are the components of the contorsion tensor in our unified field theory.

The above expression for the connection can actually be written alternatively in a somewhat simpler form as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

At this point, we see that the geometric structure of our space-time continuum is also determined by the electromagnetic field tensor as well as the material spin tensor, in addition to the gravitational (metrical) field.

As a consequence, we obtain the following relations (where the round brackets on indices, in contrast to the square ones, indicate symmetrization):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We also have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

in addition to the usual relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

At this point, we may note that the spin vector ' is always orthogonal to the world-velocity vector as

[[phi].sub.[micro]][u.sup.[micro]] = 0 .

In terms of the four-potentialA, if we take the electromagnetic field tensor to be a pure curl as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [bar.[gradient]] represents the covariant derivative with respect to the symmetric Levi-Civita connection alone, then we have the following general identities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The electromagnetic current density vector is then given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Its fully covariant divergence is then given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If we further take [J.sup.[micro]] = [[rho].sub.em][u.sup.[micro]], where [[rho].sub.em] represents the electromagnetic charge density (taking into account the possibility of a magnetic charge), we see immediately that our electromagnetic current is conserved if and only if [[bar.[gradient].sub.[micro]][J.sup.[micro]] = 0, as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In other words, for the electromagnetic current density to be conserved in our theory, the following conditions must be satisfied (for an arbitrary scalar field [PHI]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

These relations are reminiscent of those in [1]. Note that we have made use of the relation ([[gradient].sub.[micro]][[gradient].sub.v] - [[gradient].sub.v][[gradient].sub.[micro]]) [PHI] = =2[[GAMMA.sup.[lambda].sub.[[micro]v][[gradient].sub.[lambda]]][PHI].

Now, corresponding to our desired conservation law for electromagnetic currents, we can alternatively express the connection as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Contracting the above relation, we obtain the simple relation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. On the other hand, we also have the relation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Hence we see that [PHI] is a constant of motion as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

These two conditions uniquely determine the conservation of electromagnetic currents in our theory.

Furthermore, not allowing for external forces, the geodesic equation of motion in [S.sub.4], namely,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

must hold in [S.sub.4] in order for the gravitational, electromagnetic, and material spin fields to be genuine intrinsic geometric objects that uniquely and completely build the structure of the space-time continuum.

Recalling the relation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we obtain the equation of motion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which is none other than the equation of motion for a charged particle moving in a gravitational field. This simply means that our relation [F.sub.[micro]v] =2[mc.sup.2]/e[[GAMMA.sup.[lambda].sub.[[micro]v]][u.sub.[lambda]] does indeed indicate a valid geometrization of the electromagnetic field.

In the case of conserved electromagnetic currents, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3 The field equations of the unified field theory The (intrinsic) curvature tensor R of [S.sub.4] is of course given by the usual relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where V is an arbitrary vector field. For an arbitrary tensor field T, we have the more general relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Of course,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If we define the following contractions:

[R.sub.[micro]v] = [R.sup.[lambda].sub.[micro][lambda]v], R = [R.sup.[micro].sub.[micro],

then, as usual,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where C is the Weyl tensor. Note that the generalized Ricci tensor (given by its components [R.sub.[micro]v]) is generally asymmetric.

Let us denote the usual Riemann-Christoffel curvature tensor by [bar.R], i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The symmetric Ricci tensor and the Ricci scalar are then given respectively by [[bar.R].sub.[micro]v] = [[bar.R].sup.[lambda].sub.[micro][lambda]v] and [bar.R] = [[bar.R].sup.[micro].sub.[micro]].

Furthermore, we obtain the following decomposition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence, recalling that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We then obtain the following generalized Bianchi identities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

in addition to the standard Bianchi identities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(See [2-4] for instance.)

Furthermore, we can now obtain the following explicit expression for the curvature tensor R:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the tensor [OMEGA] consists of the remaining terms containing the material spin tensor [sup.2]S (or [sup.3]S).

Now, keeping in mind that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], and also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], and decomposing the components of the generalized Ricci tensor as [R.sub.[micro]v] = [R.sub.([micro]v)] + [R.sub.[[micro]v]], we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In particular, we note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence we obtain the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where D[F.sub.[micro]v]/[D.sub.s] = [u.sup.[lambda]][[gradient].sub.[lambda]][F.sub.[micro]v]. More explicitly, we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It is therefore seen that, in general, the special identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

holds only when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We are now in a position to generalize Einstein's field equation in the standard theory of general relativity. The usual Einstein's field equation is of course given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [bar.G] is the symmetric Einstein tensor, T is the energy-momentum tensor, and k = [+ or -] 8[pi]G/[c.sup.4] is Einstein's coupling constant in terms of the Newtonian gravitational constant G. Taking c=1 for convenience, in the absence of pressure, traditionally we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [[rho].sub.m] is the material density and where the second term on the right-hand-side of the equation is widely regarded as representing the electromagnetic energy-momentum tensor.

Now, with the generalized Bianchi identity for the electromagnetic field, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], at hand, and assuming the "isochoric" condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In other words,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This is our first generalization of the standard Einstein's field equation, following the traditional ad hoc way of arbitrarily adding the electromagnetic contribution to the purely material part of the energy-momentum tensor.

Now, more generally and more naturally, using the generalized Bianchi identity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we can obtain the following fundamental relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Alternatively, we can also write this as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Now, as a special consideration, let [SIGMA] be the "area" of a three-dimensional space-like hypersurface representing matter in [S.sub.4]. Then, if we make the following traditional choice for the third-rank material spin tensor [sup.3]S:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where now T is the total asymmetric energy-momentum tensor in our theory, we see that, in the presence of matter, the condition [S.sup.[micro]v[lambda]] = 0 implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In this special case, we obtain the simplified expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If we further assume that the sectional curvature [PSI] = 1/12 R of [S.sub.4] is everywhere constant in a space-time region where the electromagnetic field (and hence the torsion) is absent, we may consider writing [R.sub.[micro]v[rho][sigma]] = [PSI] ([g.sub.[micro][rho]][g.sub.v[sigma]] - [g.sub.[micro][sigma]][g.sub.v[rho]]) such that [S.sub.4] is conformally flat ([C.sub.[micro]v[rho][sigma]] = 0), and hence [R.sub.[micro]v] = = 3[PSI][g.sub.[micro]v]] and [R.sub.[[micro]v]] =0. In this case, we are left with the simple expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This is equivalent to the equation of motion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

4 The minimal lagrangian density of the theory

Using the general results from the preceding section, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for the curvature scalar of [S.sub.4]. Here [f.sup.[micro]] = [F.sup.[micro].sub.v][u.sup.v] can be said to be the components of the so-called Lorentz force.

Furthermore, we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The last two terms on the right-hand-side of the expression can then be grouped into a single scalar source as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Assuming that [phi] accounts for both the total (materialelectromagnetic) charge density as well as the total energy density, our unified field theory may be described by the following action integral (where the L=R [square root of (det(g))] is the minimal Lagrangian density):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In this minimal fashion, gravity (described by [bar.R]) appears as an emergent phenomenon whose intrinsic nature is of electromagnetic and purely material origin since, in our theory, the electromagnetic and material spin fields are nothing but components of a single torsion field.

5 The non-Abelian Yang-Mills gauge field as a subtorsion field in [S.sub.4]

In [S.sub.4], let there exist a space-like three-dimensional hypersurface [[THETA].sub.3], with local coordinates [X.sup.i] (Latin indices shall run from 1 to 3). From the point of view of projective differential geometry alone, we may say that [[THETA].sub.3]i s embedded (immersed) in [S.sub.4]. Then, the tetrad linking the embedded space [[THETA].sub.3] to the enveloping space-time [S.sub.4] is readily given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Furthermore, let N be a unit vector normal to the hypersurface [[THETA].sub.3]. We may write the parametric equation of the hypersurface [[THETA].sub.3] as H ([x.sup.[micro]]; d) =0, where d is constant. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In terms of the axial unit vectors a, b, and c spanning the hypersurface [[THETA].sub.3], we may write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [[epsilon].sub.[micro]v[rho][sigma]] are the components of the completely antisymmetric four-dimensional Levi-Civita permutation tensor density.

Now, the tetrad satisfies the following projective relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If we denote the local metric tensor of [[THETA].sub.3] by h, we obtain the following relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Furthermore, in the hypersurface [[THETA].sub.3], let us set [[gradient].sub.i] = = [[omega].sup.[micro].sub.i][[gradient].sub.[micro]] and [[partial derivative].sub.i] = [partial derivative]/[partial derivative] [X.sup.i] = [[omega].sup.[micro].sub.i][[partial derivative].sub.[micro]]. Then we have the following fundamental expressions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where Z is the extrinsic curvature tensor of the hypersurface [[THETA].sub.3], which is generally asymmetric in our theory.

The connection of the hypersurface [[THETA].sub.3] is linked to that of the space-time [S.sub.4] via

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

After some algebra, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The fundamental geometric relations describing our embedding theory are then given by the following expressions (see [4] for instance):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Actually, these relations are just manifestations of the following single expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We may note that [[GAMMA].sup.p.sub.[ik]] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

are the components of the torsion tensor and the intrinsic curvature tensor of the hypersurface [[THETA].sub.3], respectively.

Now, let us observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence letting

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

we arrive at the expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In addition, we also see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Now, with respect to the local coordinate transformation given by [X.sup.i] = [X.sup.i] [[bar.X].sup.A] in [[THETA].sub.3], let us invoke the following Cartan-Lie algebra:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [e.sub.i] = [e.sup.A.sub.i] [partial derivative]/[partial derivative][[bar.X].sup.A] are the elements of the basis vector spanning [[THETA].sub.3], [C.sup.p.sub.ik] are the spin coefficients, i= [square root of (-1)] [??] is a coupling constant, and [[member of].sub.ikl] = [square root of (det (h))][[epsilon].sub.ikl] (where [[epsilon].sub.ikl] are the components of the completely anti-symmetric three-dimensional Levi-Civita permutation tensor density).

Hence we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

At this point, our key insight is to define the gauge field potential as the tetrad itself, i.e.,

[B.sup.i.sub.[micro]] = [[omega].sup.i.sub.[micro]].

Hence, at last, we arrive at the following important expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Clearly, [F.sup.i.sub.[micro]v] are the components of the generalized Yang-Mills gauge field strength. To show this, consider the hypersurface [E.sub.3] of rigid frames (where the metric tensor is strictly constant) which is a reduction (or, in a way, local infinitesimal representation) of the more general hypersurface [[THETA].sub.3]. We shall call this an "isospace". In it, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence we arrive at the familiar expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In other words, setting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Finally, let us define the gauge field potential of the second kind via

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

such that

[B.sup.i.sub.[micro]] = 1/2[[epsilon].sub.ikl][[omega].sub.ikl].

Let us then define the gauge field strength of the second kind via

[R.sub.ik[micro]v] = [[member of].sub.ikp][F.sup.p.sub.[micro]v],

such that

[F.sup.p.sub.[micro]v] = 1/2 [[member of].sub.pik][R.sub.ik[micro]v].

Hence we obtain the general expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We may regard the object given by this expression as the curvature of the local gauge spin connection of the hypersurface [[THETA].sub.3].

Again, if we refer this to the isospace [E.sub.3] instead of the more general hypersurface [[THETA].sub.3], we arrive at the familiar relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

6 Conclusion

We have just completed our program of building the structure of a unified field theory in which gravity, electromagnetism, material spin, and the non-Abelian Yang-Mills gauge field (which is also capable of describing the weak force in the standard model particle physics) are all geometrized only in four dimensions. As we have seen, we have also generalized the expression for the Yang-Mills gauge field strength.

In our theory, the (generalized) Yang-Mills gauge field strength is linked to the electromagnetic field tensor via the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [u.sup.i] = [[omega].sup.i.sub.[micro]][u.sup.[micro]]. This enables us to express the connection in terms of the Yang-Mills gauge field strength instead of the electromagnetic field tensor as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

i.e., the Yang-Mills gauge field is nothing but a sub-torsion field in the space-time manifold [S.sub.4].

The results which we have obtained in this work may subsequently be quantized simply by following the method given in our previous work [1] since, in a sense, the present work is but a further in-depth classical consideration of the fundamental method of geometrization outlined in the previous theory.

Dedication

I dedicate this work to my patron and source of inspiration, Albert Einstein (1879-1955), from whose passion for the search of the ultimate physical truth I have learned something truly fundamental of the meaning of being a true scientist and independent, original thinker, even amidst the adversities often imposed upon him by the world and its act of scientific institutionalization.

Submitted on November 22, 2007

Accepted on November 29, 2007

References

[1.] Suhendro I. A new conformal theory of semi-classical Quantum General Relativity. Progress in Physics, 2007, v. 4, 96-103.

[2.] Suhendro I. A four-dimensional continuum theory of spacetime and the classical physical fields. Progress in Physics, 2007, v. 4, 34-46.

[3.] Suhendro I. A new semi-symmetric unified field theory of the classical fields of gravity and electromagnetism. Progress in Physics, 2007, v. 4, 47-62.

[4.] Suhendro I. Spin-curvature and the unification of fields in a twisted space. Ph. D. Thesis, 2003. The Swedish Physics Archive, Stockholm, 2008 (in print).

Indranu Suhendro

Department of Physics, Karlstad University, Karlstad 651 88, Sweden

E-mail: spherical symmetry@yahoo.com
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