# A new conformal theory of semi-classical quantum general relativity.

We consider a new four-dimensional formulation of semi-classical
quantum general relativity in which the classical space-time manifold,
whose intrinsic geometric properties give rise to the effects of
gravitation, is allowed to evolve microscopically by means of a
conformal function which is assumed to depend on some quantum mechanical
wave function. As a result, the theory presented here produces a unified
field theory of gravitation and (microscopic) electromagnetism in a
somewhat simple, effective manner. In the process, it is seen that
electromagnetism is actually an emergent quantum field originating in
some kind of stochastic smooth extension (evolution) of the
gravitational field in the general theory of relativity.

1 Introduction

We shall show that the introduction of an external parameter, the Planck displacement vector field, that deforms ("maps") the standard general relativistic space-time [S.sub.1] into an evolved space-time [S.sub.2] yields a theory of general relativity whose space-time structure obeys the semi-classical quantum mechanical law of evolution. In addition, an "already quantized" electromagnetic field arises from our schematic evolution process and automatically appears as an intrinsic geometric object in the space-time [S.sub.2]. In the process of evolution, it is seen that from the point of view of the classical space-time [S.sub.1] alone, an external deformation takes place, since, by definition, the Planck constant does not belong to its structure. In other words, relative to [S.sub.1], the Planck constant is an external parameter. However from the global point of view of the universal (enveloping) evolution space [M.sub.4], the Planck constant is intrinsic to itself and therefore defines the dynamical evolution of [S.sub.1] into [S.sub.2]. In this sense, a point in [M.sub.4] is not strictly single-valued. Rather, a point in [M.sub.4] has a "dimension" depending on the Planck length. Therefore, it belongs to both the space-time [S.sub.1] and the space-time [S.sub.2].

2 Construction of a four-dimensional metric-compatible evolution manifold [M.sub.4]

We first consider the notion of a four-dimensional, universal enveloping manifoldM4 with coordinates [x.sup.[mu]] endowed with microscopic deformation structure represented by an exterior vector field [empty set] ([x.sup.[mu]]) which maps the enveloped space-time manifold [S.sub.1] [member of] [M.sub.4] at a certain initial point [P.sub.0] onto a new enveloped space-time manifold [S.sub.2] [member of] [M.sub.4] at a certain point [P.sub.1] through the diffeomorphism

[x.sup.[mu]] ([P.sub.1]) = [x.sup.[mu]] ([P.sub.0]) + l [[xi].sup.[mu]],

where l = [square root of G[??]/[c.sup.3] [approximately equal to] [10.sup.-33] cm is the Planck length expressed in terms of the Newtonian gravitational constant G, the Dirac-Planck constant [??], and the speed of light in vacuum c, in such way that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From its diffeomorphic structure, we therefore see that [M.sub.4] is a kind of strain space. In general, the space-time [S.sub.2] evolves from the space-time [S.sub.1] through the non-linear mapping

P([member of]) : [S.sub.1] [right arrow] [S.sub.2].

Note that the exterior vector field [member of] can be expressed as [member of] = [[member of].sup.[mu]] [h.sub.[mu]] = [[??].sup.[mu]] [g.sub.[mu]] (the Einstein summation convention is employed throughout this work) where [h.sub.[mu]] and [g.sub.[mu]] are the sets of basis vectors of the space-times [S.sub.1] and [S.sub.2], respectively (likewise for [xi]). We remark that [S.sub.1] and [S.sub.2] are both endowed with metricity through their immersion in [M.sub.4], which we shall now call the evolution manifold. Then, the two sets of basis vectors are related by

[g.sub.[mu]] = ([[delta].sup.v.sub.[mu]] + [[nabla].sub.[mu]] [[xi].sup.v]) [h.sub.v]

or, alternatively, by

[g.sub.[mu]] = ([[delta].sup.v.sub.[mu]] + l [[bar.[nabla]].sub.[mu]] [[bar.[xi]].sup.v]) [h.sub.v]

where [[delta].sup.v.sub.[mu]] are the components of the Kronecker delta.

At this point, we have defined the two covariant derivatives with respect to the connections [omega] of [S.sub.1] and [GAMMA] of [S.sub.2] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for arbitrary tensor fields A and B, respectively. Here [[partial derivative].sub.[mu]] = [partial derivative]/[partial derivative][x.sup.[mu]], as usual. The two covariant derivatives above are equal only in the limit [??][right arrow] 0.

Furthermore, we assume that the connections [omega] and [GAMMA] are generally asymmetric, and can be decomposed into their symmetric and anti-symmetric parts, respectively, as

[[omega].sup.[lambda].sub.[mu]v] = (h[lambda], [[partial derivative].sub.[upsilon]] [h.sub.[mu]]) = [omega][lambda]([mu]v) + [omega][[lambda].sub.[mu,v]]

and

[[GAMMA].sup.[lambda].sub.[mu]v] = (g[lambda], [[partial derivative].sub.[upsilon]][g.sub.[mu]]) = [[GAMMA].sup.[lambda].sub.([mu]v) + [[GAMMA].sup.[lambda].sub.[mu]v].

Here, by (a; b) we shall mean the inner product between the arbitrary vector fields a and b.

Furthermore, by direct calculation we obtain the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, setting

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we may simply write

[[partial derivative].sub.v][g.sub.[mu]] = [F.sup.[lambda].sub.[mu]v] [h.sub.[lambda]]

Meanwhile, we also have the following inverse relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using the relation [[partial derivative].sub.v] [g.sub.[mu]] = [[GAMMA].sup.[lambda].sub.[mu]v][g.sub.[lambda]] (similarly, [[partial derivative].sub.v]h[mu] = [[omega].sup.[lambda].sub.[mu]v] [h.sub.[lambda]]), we obtain the relation between the two connections [GAMMA] and [omega] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a general non-linear relation in the components of the exterior displacement field [xi]. We may now write

[[GAMMA].sup.[lambda].sub.[mu]v] = [F.sup.[lambda].sub.[mu]v] + [G.sup.[lambda].sub.[mu]v]

where, recalling the previous definition of [F.sup.[lambda].sub.[mu]v], it can be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

At this point, the intrinsic curvature tensors of the spacetimes [S.sub.1] and [S.sub.2] are respectively given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We may also define the following quantities built from the connections [[omega].sup.[lambda].sub.[mu]v] and [[GAMMA].sup.[lambda].sub.[mu]v]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

from which we may define two additional "curvatures" X and P by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that [K.sup.[sigma].sub.[rho][mu]v] = 2[X.sup.[sigma].sub.[rho][mu]v] and [R.sup.[sigma].sub.[rho][mu]v]] = 2 [P.sup.[sigma].sub.[rho][mu,v]]

Now, we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In addition, we also have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, the metric tensor g of the space-time [S.sub.1] and the metric tensor h of the space-time [S.sub.2] are respectively given by

[h.sub.[mu]v] = ([h.sub.[mu]], [h.sub.v])

and

[g.sub.[mu]v] = ([g.sub.[mu]], [g.sub.v])

where the following relations hold:

[h.sub.[mu][sigma]][h.sup.v[sigma]] = [[delta].sup.v.sub.[mu]]

[g.sub.[mu][sigma]][g.sup.v[sigma]] = [[delta].sup.v.sub.[mu]]

In general, the two conditions [h.sub.[mu][sigma]] [g.sup.v[sigma]] [not equal to] [[delta].sup.v.sub.[mu]] and [g.sub.[mu][sigma]] [h.sub.v[sigma]] [not equal to] [[delta].sup.v.sub.[mu]] must be fulfilled unless l=0 (in the limit [??] [right arrow] 0). Furthermore, we have the metricity conditions

[[nabla].sub.[lambda]] [h.sub.[mu]v] [not equal to] 0 and [[nabla].sub.[lambda]g[mu]v] [not equal to] 0,

and

[bar.[nabla]] [g.sub.[mu]v] = 0.

However, note that in general, [[bar.nabla].sub.[lambda]] [h.sub.[mu]v] [not equal to] 0 and [[nabla].sub.[lambda]g[mu]v] [not equal to] 0.

Hence, it is straightforward to see that in general, the metric tensor g is related to the metric tensor h by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which in the linear approximation reads

[g.sub.[mu]v] = [h.sub.[mu]v] + 2l [nabla]([mu][[xi].sub.v]).

The formal structure of our underlying geometric framework clearly implies that the same structure holds in n dimensions as well.

3 The conformal theory

We are now in the position to extract a physical theory of quantum gravity from the geometric framework in the preceding section by considering the following linear conformal mapping:

[g.sub.[mu]] = [e.sup.[phi]][h.sub.[mu]]

where the continuously differentiable scalar function [phi] ([x.sup.[mu]]) is the generator of the quantum displacement field in the evolution space [M.sub.4] and therefore connects the two space-times [S.sub.1] and [S.sub.2].

Now, for reasons that will be apparent soon, we shall define the generator [psi] in terms of the canonical quantum mechanical wave function [psi] ([x.sup.[mu]]) as

[psi] = ln [(1 + M[psi]).sup.1/2]

where

M = [+ or -] 1/2 l [(i [m.sub.0]c/[??]).sup.2].

Here [m.sub.0] is the rest mass of the electron. Note that the sign [+ or -] signifies the signature of the space-time used.

Now, we also have the following relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as well as the conformal transformation

[g.sub.[mu]v] = [e.sub.2[psi]][h.sub.[mu]v].

Hence

[g.sub.[mu]v] = [e.sub.-2[psi]][h.sub.[mu]v].

We immediately see that

[g.sub.[mu][sigma]][h.sup.v[sigma]] = [e.sup.2[phi]][[delta].sup.v.sub.[mu]],

[h.sub.[mu][sigma]] [g.sup.v[sigma]] = [e.sup.-2[psi]] [[delta].sup.v.sub.[mu]].

At this point, we see that the world-line of the space-time [S.sub.2], [sigma] = f [square root of [g.sub.[mu]v] [dx.sup.u][dx.sup.v]] is connected to that of the spacetime [S.sub.1], s = [integral] [square root of [h.sub.[mu]v] [dx.sup.[mu]] [dx.sup.v], through

d[sigma] = [e.sup.2[phi]]ds.

Furthermore, from the relation

[g.sub.[mu]] = ([[delta].sup.v.sub.[mu]] + l [[nabla].sub.[mu]][[xi].sup.v]) [h.sub.v] = [e.sup.[phi]]h[mu]

we obtain the important relation

l [[nabla].sub.v][[xi].sub.[mu]] = ([e.sup.[psi]] - 1) [h.sub.[mu]v],

which means that

[[PHI].sub.[mu]v] = l [[nabla].sub.v][[xi].sub.[mu]] = [[PHI].sub.v[mu]],

i.e., the quantum displacement gradient tensor field [PHI] is symmetric. Hence we may simply call [PHI] the quantum strain tensor field. We also see that the components of the quantum displacement field, [[member of].sub.[mu]] = l [[xi].sup.u], can now be described by the wave function [psi] as

[[phi].sub.[mu]] = l [[partial derivative].sub.u] [psi]

i.e.,

[psi] = [[psi].sub.0] + 1/l [integral] [[phi].sub.[mu]][dx.sup.[mu]]

for an arbitrary initial value [[psi].sub.0] (which, most conveniently, can be chosen to be 0).

Furthermore, we note that the integrability condition [[PHI].sub.[mu]v] = [[PHI].sub.v[mu]] means that the space-time [S.sub.1] must now possess a symmetric, linear connection, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which are just the Christoffel symbols {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} in the space-time [S.sub.1]. Hence [omega] is now none other than the symmetric Levi-Civita (Riemannian) connection. Using the metricity condition [[partial derivative].sub.[lambda]g[mu]v] = [[GAMMA].sub.[mu]v[lambda]] + [[GAMMA].sub.v[mu][lambda]], i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we obtain the mixed form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It may be noted that we have used the customary convention in which [[GAMMA].sub.[lambda][mu]v] = [g.sub.[lambda][rho]] [[GAMMA].sup.[rho].sub.[mu]v] and [h.sub.[lambda][rho]][[omega].sup.[rho].sub.[mu]v].

Now we shall see why we have made the particular choice [psi] = 1n [(1+ M [psi]).sup.1/2]. In order to explicitly show that it now possess a stochastic part, let us rewrite the components of the metric tensor of the space-time [S.sub.2] as

[g.sub.[mu]v] = (1 + M[phi]) [h.sub.[mu]v].

Combining this relation with the linearized relation [g[mu].sub.v] = [h.sub.[mu]v] +2l [nabla]([mu], [[xi].sub.v]) and contracting the resulting relation, we obtain

l[D.sup.2][psi] = 2 ([e.sup.2][psi]- 1) = 2M[psi],

where we have defined the differential operator [D.sup.2] = = [h.sub.[mu]v][[nabla].sub.[mu]][[nabla].sub.v] such that

l[D.sup.2] [psi] = [h.sup.[mu]v] ([[partial derivative].sub.[mu]] [[partial derivative].sub.v] [psi] - [[omega].sup.[rho].sub.[mu]v] [[partial derivative].sub.[rho]][phi]).

Expressing M explicitly, we obtain [D.sup.2] [phi] = [??] [([m.sub.0]c/[??]).sup.2] [psi], i.e.,

([D.sup.2] [+ or -] [[m.sub.0]c/[??]).sup.2]) [psi] = 0

which is precisely the Klein-Gordon equation in the presence of gravitation.

We may note that, had we combined the relation [g.sub.[mu]v] = = (1 + M [phi]) [h.sub.[mu]v] with the fully non-linear relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we would have obtained the following non-linear Klein-Gordon equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, from the general relation between the connections [GAMMA] and [omega] given in Section 2, we obtain the following important relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which not only connects the torsion of the space-time [S.sub.2] with the curvature of the space-time [S.sub.1], but also describes the torsion as an intrinsic (geometric) quantum phenomenon. Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are now the components of the Riemann-Christoffel curvature tensor describing the curvature of space-time in the standard general relativity theory.

Furthermore, using the relation between the two sets of basis vectors [g.sub.[mu]] and [h.sub.[mu]], it is easy to see that the connection [GAMMA] is semi-symmetric as

[[GAMMA].sup.[lambda].sub.[mu]v] = [[omega].sup.[lambda].sub.[mu]v] + [[delta].sup.[lambda].sub.[mu]] [[partial derivative].sub.v][psi]

or, written somewhat more explicitly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We immediately obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Additionally, using the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we may now define two semi-vectors by the following contractions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or, written somewhat more explicitly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We now define the torsion vector by

[[tau].sub.[mu]] = [[GAMMA].sup.v.sub.[v[mu]]] = 3/2 [partial derivative][mu][psi].

In other words,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Furthermore, it is easy to show that the curvature tensors of our two space-times [S.sub.1] and [S.sub.2] are now identical:

[R.sup.[sigma].sub.[rho][mu]v] = [K.sup.[sigma].sub.[rho][mu]v]

which is another way of saying that the conformal transformation g[mu] = [e.sup.[psi]] [h.sub.[mu]] leaves the curvature tensor of the spacetime [S.sub.1] invariant. As an immediate consequence, we obtain the ordinary expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence the following cyclic symmetry in Riemannian geometry:

[R.sub.[rho][sigma][mu]v] + [R.sub.[rho][mu]v[sigma]] + [R.sub.[rho][mu]v[sigma]] = 0

is preserved in the presence of torsion. In addition, besides the obvious symmetry [R.sub.[rho][sigma][mu]v] = -[R.sub.[rho][sigma]v[mu]], we also have the symmetry

[R.sub.[rho][sigma][mu]v] = -[R.sub.[sigma][rho][mu]v]

which is due to the metricity condition of the space-times [S.sub.1] and [S.sub.2]. This implies the vanishing of the so-called Homothetic curvature as

[H.sub.[mu]v] = [R.sup.[sigma].sub.[sigma][mu]v] = 0.

The Weyl tensor is given in the usual manner by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [R.sub.[mu]v] = [R.sup.[sigma].sub.[mu][sigma]v] are the components of the symmetric Ricci tensor and R = [R.sup.[mu].sub.[mu]] is the Ricci scalar.

Now, by means of the conformal relation [g.sub.[mu]v] = [e.sup.2[psi]] [h.sub.[mu]v] we obtain the expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that despite the fact that the curvature tensor of the space-time [S.sub.2] is identical to that of the space-time [S.sub.1] and that both curvature tensors share common algebraic symmetries, the Bianchi identity in [S.sub.2] is not the same as the ordinary Bianchi identity in the torsion-free space-time [S.sub.1]. Instead, we have the following generalized Bianchi identity:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Contracting the above relation, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Combining the two generalized Bianchi identities above with the relation [[GAMMA].sup.[lambda].sub.[mu]v] = 1/2 ([[partial derivative].sup.[lambda].sub.[mu]] [[partial derivative].sub.v][phi] - [[delta].sup.[lambda].sub.v] [[partial derivative].sub.[mu]] [phi], as well as recalling the definition of the torsion vector, and taking into account the symmetry of the Ricci tensor, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[bar.[nabla].sub.[upsilon]] ([R.sup.[mu][upsilon]] - 1/2 [g.sup.[mu][uspilon]]R) = -2([R.sup.[mu][upsilon]] - 1/2 [g.sup.[mu][uspilon]]R)[[partial derivative].sub.[upsilon]][psi]

which, upon recalling the definition of the torsion vector, may be expressed as

[bar.[nabla].sub.[upsilon]] ([R.sup.[mu][upsilon]] - 1/2 [g.sup.[mu][uspilon]]R) = -4/3([R.sup.[mu][upsilon]] - 1/2 [g.sup.[mu][uspilon]]R)[[tau].sub.[upsilon]].

Apart from the above generalized identities, we may also give the ordinary Bianchi identities as

[[nabla].sub.[lambda]][R.sub.[rho][sigma][upsilon][lambda]] + [[nabla].sub.[mu]][R.sub.[rho][sigma][upsilon][lambda]] + [[nabla].sub.[upsilon]][R.sub.[rho][sigma][lambda][mu]].

and

[[nabla].sub.[upsilon]]([R.sup.[mu][upsilon]] - 1/2[h.sup.[mu][upsilon]]R) = 0.

4 The electromagnetic sector of the conformal theory. The fundamental equations of motion

Based on the results obtained in the preceding section, let us now take the generator [psi] as describing the (quantum) electromagnetic field. Then, consequently, the space-time [S.sub.1] is understood as being devoid of electromagnetic interaction. As we will see, in our present theory, it is the quantum evolution of the gravitational field that gives rise to electromagnetism. In this sense, the electromagnetic field is but an emergent quantum phenomenon in the evolution space [M.sub.4].

Whereas the space-time [S.sub.1] is purely gravitational, the evolved space-time [S.sub.2] does contain an electromagnetic field. In our present theory, for reasonsthat will be clear soon, we shall define the electromagnetic field F [member of] [S.sub.2] [member of] [M.sub.4] in terms of the torsion of the space-time [S.sub.2] by

[F.sub.[mu][upsilon]] = 2 [m.sub.0][c.sup.2]/[bar.e] [[GAMMA].sup.[lambda].sub.[mu][upsilon]][u.sub.[lambda]],

where [bar.e] is the (elementary) charge of the electron and

[u.sub.[mu]] = [g.sub.[mu][upsilon]] d[x.sup.[upsilon]]/d[sigma] = [e.sup.2[psi]][h.sub.[mu][upsilon]] d[x.sup.[upsilon]]/d[sigma],

are the covariant components of the tangent velocity vector field satisfying [u.sub.mu] [u.sup.[mu]] = 1.

We have seen that the space-time [S.sub.2] possesses a manifest quantum structure through its evolution from the purely gravitational space-time [S.sub.1]. This means that [bar.e] may be defined in terms of the fundamental Planck charge [??] as follows:

[bar.e] = N[??] = N [square root of 4[pi][[epsilon].sub.0][??]c,

where N is a positive constant and "0 is the permitivity of free space. Further investigation shows that N = [square root [alpha]] where [[alpha].sup.-1] [approximately equal to] 137 is the conventional fine structure constant.

Let us now proceed to show that the geodesic equation of motion in the space-time [S.sub.2] gives the (generalized) Lorentz equation of motion for the electron. The result of parallel-transferring the velocity vector field u along the world-line (in the direction of motion of the electron) yields

[bar.D][u.sup.[mu]]/ds = ([bar.[nabla].sub.v] [u.sup.[mu]]) [u.sup.v] = 0,

i.e.,

[du.sup.[mu]]/ds + [[GAMMA].sup.[mu].sub.[rho][sigma]] [u.sup.[rho]][u.sup.[sigma]] = 0,

where, in general,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Recalling our expression for the components of the torsion tensor in the preceding section, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is completely equivalent to the previously obtained relation

[[GAMMA].sup.[lambda].sub.[mu][upsilon]] = [[omega].sup.[lambda].sub.[mu][upsilon]] + [[delta].sup.[lambda].sub.[mu]] [[partial derivative].sub.[upsilon]][psi].

Note that

[[DELTA].sup.[lambda].sub.[mu][upsilon]] = 1/2 [g.sup.[sigma][lambda]] ([[partial derivative].sub.[sigma]] [g.sub.[mu][upsilon]] + [[partial derivative].sub.[mu]] [g.sub.[upsilon][sigma]]

are the Christoffel symbols in the space-time [S.sub.2]. These are not to be confused with the Christoffel symbols in the spacetime [S.sub.1] given by [[omega].sup.[lambda].sub.[mu]v].

Furthermore, we have

d[u.sup.[mu]]/ds + 1/2 [[DELTA].sup.[mu].sub.[rho][sigma]][u.sup.[rho]] = 2[g.sup.[mu][rho]] + [[GAMMA].sup.[lambda].sub.[rho][sigma]] [u.sub.[lambda]] [u.sub.[sigma]].

Now, since we have set [F.sub.[mu][upsilon]] = 2[m.sub.0] [c.sup.2]/e [[GAMMA].sup.[lambda].sub.[[mu]v]] U[lambda], we obtain the equation of motion

[m.sup.0][c.sup.2](d[u.sup.[mu]]/ds + [[DELTA].sup.[mu].sub.[rho][sigma]][u.sup.[rho]][u.sup.[sigma]] = [bar.e] [F.sup.[mu].sub[upsilon]][u.sup.[upsilon]],

which is none other than the Lorentz equation of motion for the electron in the presence of gravitation. Hence, it turns out that the electromagnetic field, which is non-existent in the space-time [S.sub.1], is an intrinsic geometric object in the spacetime [S.sub.2]. In other words, the space-time structure of [S.sub.2] inherently contains both gravitation and electromagnetism.

Now, we see that

[F.sub.[mu][upsilon]] = [m.sub.0][c.sup.2]/[bar.e] ([u.sub.[mu]][[partial derivative].sub.[upsilon]][psi] - [u.sub.[upsilon]][[partial derivative].sub.[mu]][psi].

In other words,

[bar.e][F.sub.[mu][upsilon]][u.sub.[upsilon]] = [m.sub.0][c.sup.2] ([u.sub.[mu]]d[psi]]/ds - [g.sub.[mu][upsilon]] [[partial derivative].sub.[upsilon]][psi].

Consequently, we can rewrite the electron's equation of motion as

d[u.sup.[mu]]/ds + [[DELTA].sup.[mu].sub.[rho][sigma]][u.sup.[rho]][u.sup.[sigma]] = [u.sup.[mu]]d[psi]/ds - [g.sup.[mu][upsilon]][[partial derivative].sub.[upsilon]]][psi].

We may therefore define an asymmetric fundamental tensor of the gravoelectromagnetic manifold [S.sub.2] by

[[??].sub.[mu][upsilon]] = [g.sub.[mu][upsilon]]d[psi]/ds - [bar.e][m.sup.0][c.sup.2][F.sub.[mu][upsilon]].

satisfying

[[??].sub.[mu][upsilon]] [u.sup.v] = [[partial derivative].sub.u][psi].

It follows immediately that

([[delta].sup.[mu].sub.[upsilon]] d[psi]/ds - [bar.e][m.sup.0][c.sup.2][F.sup.[mu].sub.[upsilon]]) [u.sup.[upsilon]] = [g.sup.[mu][upsilon]][[partial derivative].sub.[upsilon]][psi].

which, when expressed in terms of the wave function [psi] gives the Schrodinger-like equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We may now proceed to show that the electromagnetic current density given by the covariant expression

[j.sup.[mu]] = - c/4[pi] [bar.[nabla].sub.v] [F.sup.[mu]v]

is conserved in the present theory.

Let us first call the following expression for the covariant components of the electromagnetic field tensor in terms of the covariant components of the canonical electromagnetic four-potential A:

[F.sub.[mu]v] = [bar.[nabla].sub.v][A.sub.[mu]] - [[bar.nabla].sub.[mu]][A.sub.v]

such that [bar.e] [bar.[nabla]] [A.sub.[mu]] = [m.sub.0] [c.sup.2] [u.sub.[mu]] [[partial derivative].sub.v] [psi]i.e.,

[m.sub.0][c.sup.2][[partial derivative].sub.[mu]][psi] = [bar.e][[mu].sup.v][bar.[nabla].sub.[mu]][A.sub.v]

which directly gives the equation of motion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, we obtain the following equation of state:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Another alternative expression for the electromagnetic field tensor is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the particular case in which the field-lines of the electromagnetic four-potential propagate in the direction of the electron's motion, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [LAMBDA] is a proportionality constant and [beta] = [+ or -] [bar.e] [square root of [LAMBDA]/[[m.sub.0]]. Then, we may define a vortical velocity field, i.e., a spin field, through the vorticity tensor which is given by

[[omega].sub.[mu]v] = 1/2 ([[partial derivative].sub.v] [partial derivative].sub.u] -[partial derivative].sub.u] [u.sub.v])

and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which describes an electrically charged spinning region in the space-time continuum [S.sub.2].

Furthermore, we have the following generalized identity for the electromagnetic field tensor:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which, in the present theory, takes the particular form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Contracting, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We therefore expect that the expression in the brackets indeed vanishes. For this purpose, we may set

[j.sup.[mu]] = -c/4[pi] [[GAMMA].sup.[mu].sub.[rho][sigma]] [F.sup.[rho][sigma]]

and hence, again, using the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we immediately see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e.,

[[bar.[nabla]].sub.[mu]][j.sup.[mu]] = 0.

At this point, we may note the following: the fact that our theory employs torsion, from which the electromagnetic field is extracted, and at the same time guarantees electromagnetic charge conservation (in the form of the above continuity equation) in a natural manner is a remarkable property.

Now, let us call the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

obtained in Section 3 of this work (in which [R.sup.[sigma].sub.[rho][mu]v] = [K.sup.[sigma].sub.[rho][mu]v]). This can simply be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, we obtain the elegant result

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or, in terms of the components of the (dimensionless) microscopic displacement field [xi],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which further reveals how the electromagnetic field originates in the gravitational field in the space-time [S.sub.2] as a quantum field. Hence, at last, we see a complete picture of the electromagnetic field as an emergent phenomenon. This completes the long-cherished hypothesis that the electromagnetic field itself is caused by a massive charged particle, i.e., when [m.sub.0] = 0 neither gravity nor electromagnetism can exist. Finally, with this result at hand, we obtain the following equation of motion for the electron in the gravitational field:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In addition, we note that the torsion tensor is now seen to be given by

[[tau].sub.[mu]] = - 1/2 [le.sup.-[psi]][R.sub.[mu]v][[xi].sup.v]

or, alternatively,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In other words,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, the second generalized Bianchi identity finally takes the somewhat more transparent form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

5 Final remarks

The present theory, in its current form, is still in an elementary state of development. However, as we have seen, the emergence of the electromagnetic field from the quantum evolution of the gravitational field is a remarkable achievement which deserves special attention. On another occasion, we shall expect to expound the structure of the generalized Einstein's equation in the present theory with a generally non-conservative energy-momentum tensor given by

[T.sub.[mu][upsilon]] = [+ or -] [c.sup.4]/8[pi]G ([r.sub.[mu][upsilon]] - 1/2 [g.sub.[mu][upsilon]]R)

which, like in the case of self-creation cosmology, seems to allow us to attribute the creation and annihilation of matter directly to the scalar generator of the quantum evolution process, and hence the wave function alone, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

6 Acknowledgements

The author is dearly indebted to Dmitri Rabounski and Stephen J. Crothers for their continuous support and sincere assistance.

Submitted on August 23, 2007

Accepted on September 24, 2007

References

[1.] Thiemann T. Introduction to Modern Canonical Quantum General Relativity. arXiv: gr-qc/0110034.

[2.] Barber G. A. The principles of self-creation cosmology and its comparison with General Relativity. arXiv: gr-qc/0212111.

[3.] Brans C. H. Consistency of field equations in "self-creation" cosmologies. Gen. Rel. Grav., 1987, v. 19, 949-952.

Indranu Suhendro

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

E-mail: spherical symmetry@yahoo.com

1 Introduction

We shall show that the introduction of an external parameter, the Planck displacement vector field, that deforms ("maps") the standard general relativistic space-time [S.sub.1] into an evolved space-time [S.sub.2] yields a theory of general relativity whose space-time structure obeys the semi-classical quantum mechanical law of evolution. In addition, an "already quantized" electromagnetic field arises from our schematic evolution process and automatically appears as an intrinsic geometric object in the space-time [S.sub.2]. In the process of evolution, it is seen that from the point of view of the classical space-time [S.sub.1] alone, an external deformation takes place, since, by definition, the Planck constant does not belong to its structure. In other words, relative to [S.sub.1], the Planck constant is an external parameter. However from the global point of view of the universal (enveloping) evolution space [M.sub.4], the Planck constant is intrinsic to itself and therefore defines the dynamical evolution of [S.sub.1] into [S.sub.2]. In this sense, a point in [M.sub.4] is not strictly single-valued. Rather, a point in [M.sub.4] has a "dimension" depending on the Planck length. Therefore, it belongs to both the space-time [S.sub.1] and the space-time [S.sub.2].

2 Construction of a four-dimensional metric-compatible evolution manifold [M.sub.4]

We first consider the notion of a four-dimensional, universal enveloping manifoldM4 with coordinates [x.sup.[mu]] endowed with microscopic deformation structure represented by an exterior vector field [empty set] ([x.sup.[mu]]) which maps the enveloped space-time manifold [S.sub.1] [member of] [M.sub.4] at a certain initial point [P.sub.0] onto a new enveloped space-time manifold [S.sub.2] [member of] [M.sub.4] at a certain point [P.sub.1] through the diffeomorphism

[x.sup.[mu]] ([P.sub.1]) = [x.sup.[mu]] ([P.sub.0]) + l [[xi].sup.[mu]],

where l = [square root of G[??]/[c.sup.3] [approximately equal to] [10.sup.-33] cm is the Planck length expressed in terms of the Newtonian gravitational constant G, the Dirac-Planck constant [??], and the speed of light in vacuum c, in such way that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From its diffeomorphic structure, we therefore see that [M.sub.4] is a kind of strain space. In general, the space-time [S.sub.2] evolves from the space-time [S.sub.1] through the non-linear mapping

P([member of]) : [S.sub.1] [right arrow] [S.sub.2].

Note that the exterior vector field [member of] can be expressed as [member of] = [[member of].sup.[mu]] [h.sub.[mu]] = [[??].sup.[mu]] [g.sub.[mu]] (the Einstein summation convention is employed throughout this work) where [h.sub.[mu]] and [g.sub.[mu]] are the sets of basis vectors of the space-times [S.sub.1] and [S.sub.2], respectively (likewise for [xi]). We remark that [S.sub.1] and [S.sub.2] are both endowed with metricity through their immersion in [M.sub.4], which we shall now call the evolution manifold. Then, the two sets of basis vectors are related by

[g.sub.[mu]] = ([[delta].sup.v.sub.[mu]] + [[nabla].sub.[mu]] [[xi].sup.v]) [h.sub.v]

or, alternatively, by

[g.sub.[mu]] = ([[delta].sup.v.sub.[mu]] + l [[bar.[nabla]].sub.[mu]] [[bar.[xi]].sup.v]) [h.sub.v]

where [[delta].sup.v.sub.[mu]] are the components of the Kronecker delta.

At this point, we have defined the two covariant derivatives with respect to the connections [omega] of [S.sub.1] and [GAMMA] of [S.sub.2] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for arbitrary tensor fields A and B, respectively. Here [[partial derivative].sub.[mu]] = [partial derivative]/[partial derivative][x.sup.[mu]], as usual. The two covariant derivatives above are equal only in the limit [??][right arrow] 0.

Furthermore, we assume that the connections [omega] and [GAMMA] are generally asymmetric, and can be decomposed into their symmetric and anti-symmetric parts, respectively, as

[[omega].sup.[lambda].sub.[mu]v] = (h[lambda], [[partial derivative].sub.[upsilon]] [h.sub.[mu]]) = [omega][lambda]([mu]v) + [omega][[lambda].sub.[mu,v]]

and

[[GAMMA].sup.[lambda].sub.[mu]v] = (g[lambda], [[partial derivative].sub.[upsilon]][g.sub.[mu]]) = [[GAMMA].sup.[lambda].sub.([mu]v) + [[GAMMA].sup.[lambda].sub.[mu]v].

Here, by (a; b) we shall mean the inner product between the arbitrary vector fields a and b.

Furthermore, by direct calculation we obtain the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, setting

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we may simply write

[[partial derivative].sub.v][g.sub.[mu]] = [F.sup.[lambda].sub.[mu]v] [h.sub.[lambda]]

Meanwhile, we also have the following inverse relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using the relation [[partial derivative].sub.v] [g.sub.[mu]] = [[GAMMA].sup.[lambda].sub.[mu]v][g.sub.[lambda]] (similarly, [[partial derivative].sub.v]h[mu] = [[omega].sup.[lambda].sub.[mu]v] [h.sub.[lambda]]), we obtain the relation between the two connections [GAMMA] and [omega] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a general non-linear relation in the components of the exterior displacement field [xi]. We may now write

[[GAMMA].sup.[lambda].sub.[mu]v] = [F.sup.[lambda].sub.[mu]v] + [G.sup.[lambda].sub.[mu]v]

where, recalling the previous definition of [F.sup.[lambda].sub.[mu]v], it can be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

At this point, the intrinsic curvature tensors of the spacetimes [S.sub.1] and [S.sub.2] are respectively given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We may also define the following quantities built from the connections [[omega].sup.[lambda].sub.[mu]v] and [[GAMMA].sup.[lambda].sub.[mu]v]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

from which we may define two additional "curvatures" X and P by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that [K.sup.[sigma].sub.[rho][mu]v] = 2[X.sup.[sigma].sub.[rho][mu]v] and [R.sup.[sigma].sub.[rho][mu]v]] = 2 [P.sup.[sigma].sub.[rho][mu,v]]

Now, we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In addition, we also have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, the metric tensor g of the space-time [S.sub.1] and the metric tensor h of the space-time [S.sub.2] are respectively given by

[h.sub.[mu]v] = ([h.sub.[mu]], [h.sub.v])

and

[g.sub.[mu]v] = ([g.sub.[mu]], [g.sub.v])

where the following relations hold:

[h.sub.[mu][sigma]][h.sup.v[sigma]] = [[delta].sup.v.sub.[mu]]

[g.sub.[mu][sigma]][g.sup.v[sigma]] = [[delta].sup.v.sub.[mu]]

In general, the two conditions [h.sub.[mu][sigma]] [g.sup.v[sigma]] [not equal to] [[delta].sup.v.sub.[mu]] and [g.sub.[mu][sigma]] [h.sub.v[sigma]] [not equal to] [[delta].sup.v.sub.[mu]] must be fulfilled unless l=0 (in the limit [??] [right arrow] 0). Furthermore, we have the metricity conditions

[[nabla].sub.[lambda]] [h.sub.[mu]v] [not equal to] 0 and [[nabla].sub.[lambda]g[mu]v] [not equal to] 0,

and

[bar.[nabla]] [g.sub.[mu]v] = 0.

However, note that in general, [[bar.nabla].sub.[lambda]] [h.sub.[mu]v] [not equal to] 0 and [[nabla].sub.[lambda]g[mu]v] [not equal to] 0.

Hence, it is straightforward to see that in general, the metric tensor g is related to the metric tensor h by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which in the linear approximation reads

[g.sub.[mu]v] = [h.sub.[mu]v] + 2l [nabla]([mu][[xi].sub.v]).

The formal structure of our underlying geometric framework clearly implies that the same structure holds in n dimensions as well.

3 The conformal theory

We are now in the position to extract a physical theory of quantum gravity from the geometric framework in the preceding section by considering the following linear conformal mapping:

[g.sub.[mu]] = [e.sup.[phi]][h.sub.[mu]]

where the continuously differentiable scalar function [phi] ([x.sup.[mu]]) is the generator of the quantum displacement field in the evolution space [M.sub.4] and therefore connects the two space-times [S.sub.1] and [S.sub.2].

Now, for reasons that will be apparent soon, we shall define the generator [psi] in terms of the canonical quantum mechanical wave function [psi] ([x.sup.[mu]]) as

[psi] = ln [(1 + M[psi]).sup.1/2]

where

M = [+ or -] 1/2 l [(i [m.sub.0]c/[??]).sup.2].

Here [m.sub.0] is the rest mass of the electron. Note that the sign [+ or -] signifies the signature of the space-time used.

Now, we also have the following relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as well as the conformal transformation

[g.sub.[mu]v] = [e.sub.2[psi]][h.sub.[mu]v].

Hence

[g.sub.[mu]v] = [e.sub.-2[psi]][h.sub.[mu]v].

We immediately see that

[g.sub.[mu][sigma]][h.sup.v[sigma]] = [e.sup.2[phi]][[delta].sup.v.sub.[mu]],

[h.sub.[mu][sigma]] [g.sup.v[sigma]] = [e.sup.-2[psi]] [[delta].sup.v.sub.[mu]].

At this point, we see that the world-line of the space-time [S.sub.2], [sigma] = f [square root of [g.sub.[mu]v] [dx.sup.u][dx.sup.v]] is connected to that of the spacetime [S.sub.1], s = [integral] [square root of [h.sub.[mu]v] [dx.sup.[mu]] [dx.sup.v], through

d[sigma] = [e.sup.2[phi]]ds.

Furthermore, from the relation

[g.sub.[mu]] = ([[delta].sup.v.sub.[mu]] + l [[nabla].sub.[mu]][[xi].sup.v]) [h.sub.v] = [e.sup.[phi]]h[mu]

we obtain the important relation

l [[nabla].sub.v][[xi].sub.[mu]] = ([e.sup.[psi]] - 1) [h.sub.[mu]v],

which means that

[[PHI].sub.[mu]v] = l [[nabla].sub.v][[xi].sub.[mu]] = [[PHI].sub.v[mu]],

i.e., the quantum displacement gradient tensor field [PHI] is symmetric. Hence we may simply call [PHI] the quantum strain tensor field. We also see that the components of the quantum displacement field, [[member of].sub.[mu]] = l [[xi].sup.u], can now be described by the wave function [psi] as

[[phi].sub.[mu]] = l [[partial derivative].sub.u] [psi]

i.e.,

[psi] = [[psi].sub.0] + 1/l [integral] [[phi].sub.[mu]][dx.sup.[mu]]

for an arbitrary initial value [[psi].sub.0] (which, most conveniently, can be chosen to be 0).

Furthermore, we note that the integrability condition [[PHI].sub.[mu]v] = [[PHI].sub.v[mu]] means that the space-time [S.sub.1] must now possess a symmetric, linear connection, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which are just the Christoffel symbols {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} in the space-time [S.sub.1]. Hence [omega] is now none other than the symmetric Levi-Civita (Riemannian) connection. Using the metricity condition [[partial derivative].sub.[lambda]g[mu]v] = [[GAMMA].sub.[mu]v[lambda]] + [[GAMMA].sub.v[mu][lambda]], i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we obtain the mixed form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It may be noted that we have used the customary convention in which [[GAMMA].sub.[lambda][mu]v] = [g.sub.[lambda][rho]] [[GAMMA].sup.[rho].sub.[mu]v] and [h.sub.[lambda][rho]][[omega].sup.[rho].sub.[mu]v].

Now we shall see why we have made the particular choice [psi] = 1n [(1+ M [psi]).sup.1/2]. In order to explicitly show that it now possess a stochastic part, let us rewrite the components of the metric tensor of the space-time [S.sub.2] as

[g.sub.[mu]v] = (1 + M[phi]) [h.sub.[mu]v].

Combining this relation with the linearized relation [g[mu].sub.v] = [h.sub.[mu]v] +2l [nabla]([mu], [[xi].sub.v]) and contracting the resulting relation, we obtain

l[D.sup.2][psi] = 2 ([e.sup.2][psi]- 1) = 2M[psi],

where we have defined the differential operator [D.sup.2] = = [h.sub.[mu]v][[nabla].sub.[mu]][[nabla].sub.v] such that

l[D.sup.2] [psi] = [h.sup.[mu]v] ([[partial derivative].sub.[mu]] [[partial derivative].sub.v] [psi] - [[omega].sup.[rho].sub.[mu]v] [[partial derivative].sub.[rho]][phi]).

Expressing M explicitly, we obtain [D.sup.2] [phi] = [??] [([m.sub.0]c/[??]).sup.2] [psi], i.e.,

([D.sup.2] [+ or -] [[m.sub.0]c/[??]).sup.2]) [psi] = 0

which is precisely the Klein-Gordon equation in the presence of gravitation.

We may note that, had we combined the relation [g.sub.[mu]v] = = (1 + M [phi]) [h.sub.[mu]v] with the fully non-linear relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we would have obtained the following non-linear Klein-Gordon equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, from the general relation between the connections [GAMMA] and [omega] given in Section 2, we obtain the following important relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which not only connects the torsion of the space-time [S.sub.2] with the curvature of the space-time [S.sub.1], but also describes the torsion as an intrinsic (geometric) quantum phenomenon. Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are now the components of the Riemann-Christoffel curvature tensor describing the curvature of space-time in the standard general relativity theory.

Furthermore, using the relation between the two sets of basis vectors [g.sub.[mu]] and [h.sub.[mu]], it is easy to see that the connection [GAMMA] is semi-symmetric as

[[GAMMA].sup.[lambda].sub.[mu]v] = [[omega].sup.[lambda].sub.[mu]v] + [[delta].sup.[lambda].sub.[mu]] [[partial derivative].sub.v][psi]

or, written somewhat more explicitly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We immediately obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Additionally, using the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we may now define two semi-vectors by the following contractions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or, written somewhat more explicitly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We now define the torsion vector by

[[tau].sub.[mu]] = [[GAMMA].sup.v.sub.[v[mu]]] = 3/2 [partial derivative][mu][psi].

In other words,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Furthermore, it is easy to show that the curvature tensors of our two space-times [S.sub.1] and [S.sub.2] are now identical:

[R.sup.[sigma].sub.[rho][mu]v] = [K.sup.[sigma].sub.[rho][mu]v]

which is another way of saying that the conformal transformation g[mu] = [e.sup.[psi]] [h.sub.[mu]] leaves the curvature tensor of the spacetime [S.sub.1] invariant. As an immediate consequence, we obtain the ordinary expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence the following cyclic symmetry in Riemannian geometry:

[R.sub.[rho][sigma][mu]v] + [R.sub.[rho][mu]v[sigma]] + [R.sub.[rho][mu]v[sigma]] = 0

is preserved in the presence of torsion. In addition, besides the obvious symmetry [R.sub.[rho][sigma][mu]v] = -[R.sub.[rho][sigma]v[mu]], we also have the symmetry

[R.sub.[rho][sigma][mu]v] = -[R.sub.[sigma][rho][mu]v]

which is due to the metricity condition of the space-times [S.sub.1] and [S.sub.2]. This implies the vanishing of the so-called Homothetic curvature as

[H.sub.[mu]v] = [R.sup.[sigma].sub.[sigma][mu]v] = 0.

The Weyl tensor is given in the usual manner by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [R.sub.[mu]v] = [R.sup.[sigma].sub.[mu][sigma]v] are the components of the symmetric Ricci tensor and R = [R.sup.[mu].sub.[mu]] is the Ricci scalar.

Now, by means of the conformal relation [g.sub.[mu]v] = [e.sup.2[psi]] [h.sub.[mu]v] we obtain the expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that despite the fact that the curvature tensor of the space-time [S.sub.2] is identical to that of the space-time [S.sub.1] and that both curvature tensors share common algebraic symmetries, the Bianchi identity in [S.sub.2] is not the same as the ordinary Bianchi identity in the torsion-free space-time [S.sub.1]. Instead, we have the following generalized Bianchi identity:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Contracting the above relation, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Combining the two generalized Bianchi identities above with the relation [[GAMMA].sup.[lambda].sub.[mu]v] = 1/2 ([[partial derivative].sup.[lambda].sub.[mu]] [[partial derivative].sub.v][phi] - [[delta].sup.[lambda].sub.v] [[partial derivative].sub.[mu]] [phi], as well as recalling the definition of the torsion vector, and taking into account the symmetry of the Ricci tensor, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[bar.[nabla].sub.[upsilon]] ([R.sup.[mu][upsilon]] - 1/2 [g.sup.[mu][uspilon]]R) = -2([R.sup.[mu][upsilon]] - 1/2 [g.sup.[mu][uspilon]]R)[[partial derivative].sub.[upsilon]][psi]

which, upon recalling the definition of the torsion vector, may be expressed as

[bar.[nabla].sub.[upsilon]] ([R.sup.[mu][upsilon]] - 1/2 [g.sup.[mu][uspilon]]R) = -4/3([R.sup.[mu][upsilon]] - 1/2 [g.sup.[mu][uspilon]]R)[[tau].sub.[upsilon]].

Apart from the above generalized identities, we may also give the ordinary Bianchi identities as

[[nabla].sub.[lambda]][R.sub.[rho][sigma][upsilon][lambda]] + [[nabla].sub.[mu]][R.sub.[rho][sigma][upsilon][lambda]] + [[nabla].sub.[upsilon]][R.sub.[rho][sigma][lambda][mu]].

and

[[nabla].sub.[upsilon]]([R.sup.[mu][upsilon]] - 1/2[h.sup.[mu][upsilon]]R) = 0.

4 The electromagnetic sector of the conformal theory. The fundamental equations of motion

Based on the results obtained in the preceding section, let us now take the generator [psi] as describing the (quantum) electromagnetic field. Then, consequently, the space-time [S.sub.1] is understood as being devoid of electromagnetic interaction. As we will see, in our present theory, it is the quantum evolution of the gravitational field that gives rise to electromagnetism. In this sense, the electromagnetic field is but an emergent quantum phenomenon in the evolution space [M.sub.4].

Whereas the space-time [S.sub.1] is purely gravitational, the evolved space-time [S.sub.2] does contain an electromagnetic field. In our present theory, for reasonsthat will be clear soon, we shall define the electromagnetic field F [member of] [S.sub.2] [member of] [M.sub.4] in terms of the torsion of the space-time [S.sub.2] by

[F.sub.[mu][upsilon]] = 2 [m.sub.0][c.sup.2]/[bar.e] [[GAMMA].sup.[lambda].sub.[mu][upsilon]][u.sub.[lambda]],

where [bar.e] is the (elementary) charge of the electron and

[u.sub.[mu]] = [g.sub.[mu][upsilon]] d[x.sup.[upsilon]]/d[sigma] = [e.sup.2[psi]][h.sub.[mu][upsilon]] d[x.sup.[upsilon]]/d[sigma],

are the covariant components of the tangent velocity vector field satisfying [u.sub.mu] [u.sup.[mu]] = 1.

We have seen that the space-time [S.sub.2] possesses a manifest quantum structure through its evolution from the purely gravitational space-time [S.sub.1]. This means that [bar.e] may be defined in terms of the fundamental Planck charge [??] as follows:

[bar.e] = N[??] = N [square root of 4[pi][[epsilon].sub.0][??]c,

where N is a positive constant and "0 is the permitivity of free space. Further investigation shows that N = [square root [alpha]] where [[alpha].sup.-1] [approximately equal to] 137 is the conventional fine structure constant.

Let us now proceed to show that the geodesic equation of motion in the space-time [S.sub.2] gives the (generalized) Lorentz equation of motion for the electron. The result of parallel-transferring the velocity vector field u along the world-line (in the direction of motion of the electron) yields

[bar.D][u.sup.[mu]]/ds = ([bar.[nabla].sub.v] [u.sup.[mu]]) [u.sup.v] = 0,

i.e.,

[du.sup.[mu]]/ds + [[GAMMA].sup.[mu].sub.[rho][sigma]] [u.sup.[rho]][u.sup.[sigma]] = 0,

where, in general,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Recalling our expression for the components of the torsion tensor in the preceding section, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is completely equivalent to the previously obtained relation

[[GAMMA].sup.[lambda].sub.[mu][upsilon]] = [[omega].sup.[lambda].sub.[mu][upsilon]] + [[delta].sup.[lambda].sub.[mu]] [[partial derivative].sub.[upsilon]][psi].

Note that

[[DELTA].sup.[lambda].sub.[mu][upsilon]] = 1/2 [g.sup.[sigma][lambda]] ([[partial derivative].sub.[sigma]] [g.sub.[mu][upsilon]] + [[partial derivative].sub.[mu]] [g.sub.[upsilon][sigma]]

are the Christoffel symbols in the space-time [S.sub.2]. These are not to be confused with the Christoffel symbols in the spacetime [S.sub.1] given by [[omega].sup.[lambda].sub.[mu]v].

Furthermore, we have

d[u.sup.[mu]]/ds + 1/2 [[DELTA].sup.[mu].sub.[rho][sigma]][u.sup.[rho]] = 2[g.sup.[mu][rho]] + [[GAMMA].sup.[lambda].sub.[rho][sigma]] [u.sub.[lambda]] [u.sub.[sigma]].

Now, since we have set [F.sub.[mu][upsilon]] = 2[m.sub.0] [c.sup.2]/e [[GAMMA].sup.[lambda].sub.[[mu]v]] U[lambda], we obtain the equation of motion

[m.sup.0][c.sup.2](d[u.sup.[mu]]/ds + [[DELTA].sup.[mu].sub.[rho][sigma]][u.sup.[rho]][u.sup.[sigma]] = [bar.e] [F.sup.[mu].sub[upsilon]][u.sup.[upsilon]],

which is none other than the Lorentz equation of motion for the electron in the presence of gravitation. Hence, it turns out that the electromagnetic field, which is non-existent in the space-time [S.sub.1], is an intrinsic geometric object in the spacetime [S.sub.2]. In other words, the space-time structure of [S.sub.2] inherently contains both gravitation and electromagnetism.

Now, we see that

[F.sub.[mu][upsilon]] = [m.sub.0][c.sup.2]/[bar.e] ([u.sub.[mu]][[partial derivative].sub.[upsilon]][psi] - [u.sub.[upsilon]][[partial derivative].sub.[mu]][psi].

In other words,

[bar.e][F.sub.[mu][upsilon]][u.sub.[upsilon]] = [m.sub.0][c.sup.2] ([u.sub.[mu]]d[psi]]/ds - [g.sub.[mu][upsilon]] [[partial derivative].sub.[upsilon]][psi].

Consequently, we can rewrite the electron's equation of motion as

d[u.sup.[mu]]/ds + [[DELTA].sup.[mu].sub.[rho][sigma]][u.sup.[rho]][u.sup.[sigma]] = [u.sup.[mu]]d[psi]/ds - [g.sup.[mu][upsilon]][[partial derivative].sub.[upsilon]]][psi].

We may therefore define an asymmetric fundamental tensor of the gravoelectromagnetic manifold [S.sub.2] by

[[??].sub.[mu][upsilon]] = [g.sub.[mu][upsilon]]d[psi]/ds - [bar.e][m.sup.0][c.sup.2][F.sub.[mu][upsilon]].

satisfying

[[??].sub.[mu][upsilon]] [u.sup.v] = [[partial derivative].sub.u][psi].

It follows immediately that

([[delta].sup.[mu].sub.[upsilon]] d[psi]/ds - [bar.e][m.sup.0][c.sup.2][F.sup.[mu].sub.[upsilon]]) [u.sup.[upsilon]] = [g.sup.[mu][upsilon]][[partial derivative].sub.[upsilon]][psi].

which, when expressed in terms of the wave function [psi] gives the Schrodinger-like equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We may now proceed to show that the electromagnetic current density given by the covariant expression

[j.sup.[mu]] = - c/4[pi] [bar.[nabla].sub.v] [F.sup.[mu]v]

is conserved in the present theory.

Let us first call the following expression for the covariant components of the electromagnetic field tensor in terms of the covariant components of the canonical electromagnetic four-potential A:

[F.sub.[mu]v] = [bar.[nabla].sub.v][A.sub.[mu]] - [[bar.nabla].sub.[mu]][A.sub.v]

such that [bar.e] [bar.[nabla]] [A.sub.[mu]] = [m.sub.0] [c.sup.2] [u.sub.[mu]] [[partial derivative].sub.v] [psi]i.e.,

[m.sub.0][c.sup.2][[partial derivative].sub.[mu]][psi] = [bar.e][[mu].sup.v][bar.[nabla].sub.[mu]][A.sub.v]

which directly gives the equation of motion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, we obtain the following equation of state:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Another alternative expression for the electromagnetic field tensor is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the particular case in which the field-lines of the electromagnetic four-potential propagate in the direction of the electron's motion, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [LAMBDA] is a proportionality constant and [beta] = [+ or -] [bar.e] [square root of [LAMBDA]/[[m.sub.0]]. Then, we may define a vortical velocity field, i.e., a spin field, through the vorticity tensor which is given by

[[omega].sub.[mu]v] = 1/2 ([[partial derivative].sub.v] [partial derivative].sub.u] -[partial derivative].sub.u] [u.sub.v])

and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which describes an electrically charged spinning region in the space-time continuum [S.sub.2].

Furthermore, we have the following generalized identity for the electromagnetic field tensor:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which, in the present theory, takes the particular form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Contracting, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We therefore expect that the expression in the brackets indeed vanishes. For this purpose, we may set

[j.sup.[mu]] = -c/4[pi] [[GAMMA].sup.[mu].sub.[rho][sigma]] [F.sup.[rho][sigma]]

and hence, again, using the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we immediately see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e.,

[[bar.[nabla]].sub.[mu]][j.sup.[mu]] = 0.

At this point, we may note the following: the fact that our theory employs torsion, from which the electromagnetic field is extracted, and at the same time guarantees electromagnetic charge conservation (in the form of the above continuity equation) in a natural manner is a remarkable property.

Now, let us call the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

obtained in Section 3 of this work (in which [R.sup.[sigma].sub.[rho][mu]v] = [K.sup.[sigma].sub.[rho][mu]v]). This can simply be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, we obtain the elegant result

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or, in terms of the components of the (dimensionless) microscopic displacement field [xi],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which further reveals how the electromagnetic field originates in the gravitational field in the space-time [S.sub.2] as a quantum field. Hence, at last, we see a complete picture of the electromagnetic field as an emergent phenomenon. This completes the long-cherished hypothesis that the electromagnetic field itself is caused by a massive charged particle, i.e., when [m.sub.0] = 0 neither gravity nor electromagnetism can exist. Finally, with this result at hand, we obtain the following equation of motion for the electron in the gravitational field:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In addition, we note that the torsion tensor is now seen to be given by

[[tau].sub.[mu]] = - 1/2 [le.sup.-[psi]][R.sub.[mu]v][[xi].sup.v]

or, alternatively,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In other words,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, the second generalized Bianchi identity finally takes the somewhat more transparent form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

5 Final remarks

The present theory, in its current form, is still in an elementary state of development. However, as we have seen, the emergence of the electromagnetic field from the quantum evolution of the gravitational field is a remarkable achievement which deserves special attention. On another occasion, we shall expect to expound the structure of the generalized Einstein's equation in the present theory with a generally non-conservative energy-momentum tensor given by

[T.sub.[mu][upsilon]] = [+ or -] [c.sup.4]/8[pi]G ([r.sub.[mu][upsilon]] - 1/2 [g.sub.[mu][upsilon]]R)

which, like in the case of self-creation cosmology, seems to allow us to attribute the creation and annihilation of matter directly to the scalar generator of the quantum evolution process, and hence the wave function alone, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

6 Acknowledgements

The author is dearly indebted to Dmitri Rabounski and Stephen J. Crothers for their continuous support and sincere assistance.

Submitted on August 23, 2007

Accepted on September 24, 2007

References

[1.] Thiemann T. Introduction to Modern Canonical Quantum General Relativity. arXiv: gr-qc/0110034.

[2.] Barber G. A. The principles of self-creation cosmology and its comparison with General Relativity. arXiv: gr-qc/0212111.

[3.] Brans C. H. Consistency of field equations in "self-creation" cosmologies. Gen. Rel. Grav., 1987, v. 19, 949-952.

Indranu Suhendro

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

E-mail: spherical symmetry@yahoo.com

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Author: | Suhendro, Indranu |
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Publication: | Progress in Physics |

Geographic Code: | 1USA |

Date: | Oct 1, 2007 |

Words: | 4935 |

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