c, where e is the electron charge, 
is Planck's constant
divided by 2
, and c is the speed of
light. THEORY OF QUANTUM GRAVITY
James Constant
grav@ccoolissues.com
A Theory of Quantum Gravity Based on the Ideas of Quantum Electrodynamics
Introduction
The gravitational strength between objects is extremely small: it is a force that is weaker, by 30 to 40 orders of magnitude, than the electrical force between two electrons. While electrical forces hold atoms together, gravitational forces hold planets, galaxies, and perhaps the cosmos together. Few experiments are possible and only observations can be made of gravitational effects. There are, however, a number of theories of gravitation that involve "gravitons" as the intermediaries between objects but no theory has the precision of a quantum theory of gravitation to explain the effect of gravity. So not only have we no experiments with which to check a quantum theory of gravitation, we also have no reasonable theory. This work intends to provide a reasonable theory of gravitation based on the ideas of Quantum Electrodynamics (QED).1
The need for a QED emerged from the problem of the interaction of light and matter: Maxwell's theory of electromagnetism had to be changed to be in accord with the new principles of quantum mechanics that had then been developed and was a tremendous success in explaining all of chemistry and properties of substances. It was known that electrons interact with light but the computations were increasingly impossible beyond a certain accuracy. QED now fully developed offers no significant difference between experiment and theory.2 The need for a quantum theory for gravitation is less justified because we have no reasonable dynamic theory for gravitation and only a handful of experiments and observations to proceed with.
QED rests on the idea that
charged particles (e.g., electrons and positrons) interact by
emitting and absorbing photons, the particles of light that
transmit electromagnetic forces. These photons are virtual; that
is, they cannot be seen or detected in any way because their
existence violates the conservation of energy and momentum. The
particle exchange is merely the "force" of the
interaction, because the interacting particles change their speed
and direction of travel as they release or absorb the energy of a
photon. Photons also can be emitted in a free state, in which
case they may be observed. The QED theory states that the more
complex the process (i.e., the presence of additional virtual
photons), the smaller the probability of its occurrence. For each
level of complexity, a factor of (1/137)2
decreases the contribution of the process, and thus, after a few
levels the contribution is negligible. This factor, symbolized by
a, is called the fine-structure constant and serves as a
measure of the strength of the electromagnetic interaction. It
equals e2/c, where e is the electron charge, is Planck's constant
divided by 2, and c is the speed of
light.
QED is often called a
perturbation theory because of the smallness of the
fine-structure constant and the resultant decreasing size of
higher order contributions. This relative simplicity and the
success of QED have made it a model for other quantum field
theories. Finally, the picture of electromagnetic interactions as
the exchange of virtual particles has been carried over to the
theories of the strong, weak, and gravitational forces. The
present work rests on the idea that masses interact by emitting
and absorbing gravitons, particles of matter that transmit
gravitational forces. These gravitons are virtual; that is, they
cannot be seen or detected in any way and perhaps because their
existence violates the conservation of energy and momentum. The
exchange of gravitons is merely the "force" of the
gravitational interaction, because the interacting masses change
their speed and direction of travel as they emit or absorb a
graviton. However, in the present theory, gravitons are not only
virtual, they also have imaginary proper masses and, therefor,
they cannot exist in a free state. The present Quantum Gravity
Theory (QGT) states that the more complex the process (i.e., the
presence of additional virtual gravitons), the smaller the
probability of its occurrence. For each level of complexity, a
factor decreases the contribution of the process, and thus, after
a few levels the contribution is negligible. This factor,
symbolized by ag, is called the gravitational
fine-structure constant and serves as a measure of the strength
of the gravitational interaction. The factor is Gm/
c, where G is the Newtonian constant, m
is mass, 
is Planck's constant divided by
2,
and c is the speed of light.
In what follows, I first summarize the well known Quantum Electrodynamics (QED)3 and then I set forth my Quantum Gravity Theory (QGT).
A relativistic wave equation can be obtained by writing the relativistic energy equation
E2 = p2 c2+ mp2c4 . . . . . . . . . .p2 =px2 +py2 +pz2
and replacing the total energy and the momentum components by the associated operators by
E
. . . . . px
. . . . . py
. . . . . pz
and
then allowing the operator equation thereby obtained to operate
on the function 
. The
result is
for photons, the quanta of the electromagnetic field.
The classical wave equation has a static solution of the form
as can easily be verified by substitution, using the relation
for
=(r). For
mp
the Klein-Gordon equation has a
static solution of the form
. . . . . . . . . . r>0 . . . . . . . . . . 
The
constant g2 determines the strength of
the Yukawa potential, just as the constant e2
(the square of the electron charge) determines the strength of
the coulomb potential. The dimensionless quantity g2/
c
has a value about 
15
whereas the dimensionless quantity e2/4pe
c (the
fine structure constant) has the value about 
1/137.
This is an indication of the strength of the nuclear force
compared to the electric force.
. . . . . . . . . . r>0 . . . . .
. . . . . 
as
can also easily be verified by substitution. The solution
for imaginary rest mass quanta gives the interaction potential of
the gravitational field. Unlike the Newtonian potential,
as r
the gravitational interaction potential
spatially oscillates between +Gm2/r
and -Gm2/r with a period 1/ri'=2
Since
the value for mi
is not known, values for the distance ri'
and spatial wavelength 
- .are unknown. The interaction
force for the static gravitational field is obtained as F=-
which is
a modified oscillatory Newtonian force.
F=
. . . . . . . . . . ri'>>r>0
. . . . . . . . . . 
For
masses m of interacting electrons, the dimensionless
quantity Gm2/
c
has a value about 
10-27
which indicates the extremely small value of the gravitational
force compared to the electric and nuclear forces discussed
previously. However, while all forces fall off with distances,
the electric and nuclear forces are short range forces but the
gravitational force alone increases as the product of interacting
masses increases.
In the foregoing, the sole discussion concentrates on static gravitation, i.e., the static solution of the Klein-Gordon equation. A static solution means that gravitons travel at infinite speed. For dynamic gravitation, i. e., for a dynamic solution of the Klein-Gordon equation, gravitons travel at speeds v>c but less than infinite speed. Theories of dynamic gravitation 4 and quantum gravity 5 have been proposed by many theorists, including theories based on Einstein's theory of gravitation and theories based on Maxwell's theory of electromagnetism. Such theories have serious defects, namely, they predict that gravitons travel at the speed of light and that virtual or free state gravitons can be detected. In contrast, the present QGT predicts that gravitons travel at speed v>c and that virtual gravitons cannot be detected. And, since gravitons have imaginary rest mass mp=imi, they cannot be emitted or absorbed in a free state. Looking for gravity waves with energy detectors is both a waste of time and money. Gravity effects can be detected only with spatial distortions such as tides and, perhaps, anomalous orbits which deviate from predictions using Newton's law of gravity.
which, in many cases, can be reduced to the form of a relativistic energy equation, and corresponding Klein-Gordon equations can be obtained. Note that the conservation of energy and momentum is violated in these equations.
In
summary, the relativistic energy equation tells us that
detectable mass travels at speeds less than the speed of light v<c
and that detectable light travels at speed equal to light v=c.
QED tells us that, based on the relativistic energy equation and
quantum theory, undetectable virtual mass (rest mass mp
) and
undetectable virtual light (rest mass mp=0)
interact between nuclear and atomic objects at speeds v
c
but violate the conservation of energy and momentum. QGT tells us
that, based on the relativistic energy equation and quantum
theory, undetectable virtual mass (imaginary rest mass mp=imi)
interacts between masses at speeds v
c
and pending experiments may violate the conservation of energy
and momentum.
The
identities of energy, mass and frequency, E=mc2=
,
with their numerical
constants c and h are the basis of all laws of
physics. They are the points of departure, not the results of any
theory.6 The
relativistic wave equation, QED and QGT are theories based on
these identities which are beyond our comprehension. Apparently,
the sole criterion of a theory is whether it works. The fact that
a theory introduces virtual objects (QED and QGT) or both virtual
and imaginary objects (QGT) is no bar to its success. The
difference between QED and QGT is that QED is abundantly
confirmed by experiment which QGT presently falls short. The
strongest evidence for QGT is the observed action at a distance
of the Newtonian force.
1 Quantum Electrodynamics at http://www.britannica.com/nobel/micro/489_26.html
2 Richard Feynman, QED Princeton University Press pages 5-7.
3 R. Eisberg and R. Resnick, Quantum Physics, John Wiley and Sons 1974 page 690.
4 Dynamic gravity http://en.wikipedia.org/wiki/Generalized_theory_of_gravitation
5 Quantum Gravity at http://www.weburbia.com/pg/qugrav.htm
6 L. Brillouin, Relativity Reexamined, Academic Press 1970 page 34.
Copyright © 2006 by James Constant
By the same author: http://www.coolissues.com/gravitation/sameauthor.htm
. . . . . . . . . . 
>c. This can be seen by recasting the
relativistic energy equation in terms of its equivalent
relativistic mass equation and setting mp=imi
imaginary. While the form of the relativistic energy equation
remains unchanged, the resulting relativistic mass equation is
valid when v>c. A graviton has momentum energy Pc
greater than its total energy E. In the static case,
gravitons have zero total energy and infinite speed. In the
dynamic case, gravitons have finite total energy E<Pc
and v>c but less than infinite speed.