Download QUANTUM ELECTRODYNAMICS AND GRAVITATION

Journal of Theoretics Vol.3-5
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QUANTUM ELECTRODYNAMICS AND GRAVITATION
Richard M. Blaber, MA, PhD. <[email protected]>
Abstract: This paper demonstrates a simple mathematical relationship between the
fine-structure constant (which determines the strength of the electromagnetic
interaction), the electron-proton mass ratio, and the gravitational constant. These
together determine the relative strength of the gravitational interaction. The
conclusion is drawn that the relationship described here underpins any valid unified
field theory, and the paper ends by speculating about the possibility offered of an
‘electro-gravitic’ propulsion system.
Introduction
Quantum electrodynamics (QED) is the quantum theory of light and its
interaction with matter. It is what is termed a ‘relativistic quantum field theory’ indeed, it is the prototype of all such theories, and the foundation of the ‘standard
model’ of particle physics. As is well known, QED argues that electromagnetism is
the result of the exchange of either virtual or observable photons between charged
particles, the emission and absorption of which modify these particles’ momenta in
specific ways, which can be calculated using Feynman diagrams.
The probability that an electron, for example, will emit a virtual photon in a
given time period is determined by
α = e2/4πε0Åc = 0.00729735246323 ,
(1)
where α is the fine-structure constant, e the electronic charge, ε0 the electric constant,
Å Dirac’s constant, and c the speed of light in a vacuum. This is, in effect, the ratio
of two characteristic time periods,
e2/4πε0mec3 and Å/mec2 ,
(2)
and says that, whereas the odds against a virtual photon emission (or re-absorption) in
the former (about 9.4E-24 s) are ~137:1, those in the latter case (1.288E-21 s) are
obviously a good deal better, because the length of time involved is ~137 times
longer.
The problem is that in QED, whereas the electron’s charge is thought of as
α1/2 - which is the probability amplitude for an electron-photon interaction, and also
the electronic charge in the Planck system of units (which treats Å and c as if they
were both equal to 1, and ε0 as if it equaled 1/4π) - neither the charge nor the mass of
the electron or any other other charged particle can actually be calculated in QED they have to be assumed. This is because they are seen as the ‘end-products’, as it
were, of an infinite series of Feynman diagrams - the charge and mass of the electron
are infinite, until they have been ‘renormalized’ back to finite reality by what
Feynman himself termed a ‘shell game’ and ‘a dippy process…hocus-pocus’i.
Instead, QED has to imagine an ideal electron (or proton, or whatever), with an ideal
mass and charge, which interacts in only the simplest possible way, and which is used
to model the real electron for purposes of calculation. These ideal values are not the
same as the ‘bare’ mass and charge, which are both infinite. The ideal charge, j, is
~0.1, and corresponds to an ideal interaction, or ‘coupling’, probability, j2.
The Relationship Between α and j2
It is reasonable to assume that, if there is any sort of relationship between the
actual coupling probability, α, and its ideal or theoretical counterpart, j2, then the
mediator of that relationship should itself be a probability, and this is indeed the case.
Let us begin by stating the relationship in its most general form:
j2 = |(ααg)1/2| = 0.009999996694668 .
(3)
In other words, j2 is the geometric mean of two other probabilities - α
Q1/137, and αg Q 1/73. This is obviously not good enough, however. We need to
know what the derivation of αg is, and what we find is that
αg = 4πgeme/mp = Ge/4πε0κÅc2 = 0.01370359104934 .
(4)
Here, ge is the absolute value of the electron’s g-factor (its magnetic moment in Bohr
magnetons, µe, divided by its spin, s = 1/2), me and mp are the masses of the electron
and proton respectively, G is the gravitational constant, and κ is a constant, given by
κ = vre/eme = 0.7397655255432 m2s-1kg-1C-1 ,
(5)
where re is the classical radius of the electron, and v is a ‘speed’ (if that is the right
word) of 3.831440356112 10-35 ms-1, or about 2.37 Planck lengths/s! At that
speed, an object would take about 830,000 trillion years (over 55 million times the
present age of the universe!) to traverse a single nanometer of space. Continental
drift would seem remarkably rapid in comparison. There is obviously no way any
such ‘motion’ could ever be detected. On the other hand,
V = (Ge/4πε0κÅ)1/2 = 3.509440613913E7 ms-1 ,
(6)
or ~11.706% of c, and that is clearly detectable. We see how αg1/2.c manifests itself
physically by the relationship:
(½mpV2)/mec2 = 4πµe ,
(7)
but this is a special case, whereas αc is the normal speed for an electron in the first
orbit (ground state) of a Bohr atom, and the most probable speed for an electron in the
first orbital of the wave-mechanical model of the atom.
Implications
We have seen how we can derive the QED ‘ideal’ coupling probability j2 from
α and αg. It is a corollary of this that we can relate the fine-structure constant to the
gravitational constant and the electron-proton mass ratio by means of:
α = 4πegeκmec/Gmp .
(8)
In dimensionless Planck units, this becomes:
α1/2 = 4πκgeme/mp .
(9)
Now, the strength of the gravitational force relative to the electromagnetic is given
by:
Gmemp/αÅc = 4.406758405743E-40 .
(10)
Substituting the expression for α1/2 above, and again using Planck units, this becomes:
mp3/me(4πκge)2 .
(11)
mp/me = 4παge/j4 ,
(12)
Bearing in mind that:
we can turn this into:
asmp2/2geκ2j4 = asα(memp)2/2ge(revj2)2 ,
(13)
where as is Schwinger’s approximation to the electron g-factor anomaly, α/2π. Given
that the gravitational force between an electron and a proton re apart is proportional to
memp/re2 Q 1.06E-82 (and is equal to it in Planck units, re being ~1.74E20 Planck
lengths), this implies that the expression
2gev2j4/asαmemp
(14)
yields the electromagnetic force between the two particles. Given that v is
approximately 1.28E-43 of a Planck length per Planck time unit, me is ~4.18E-23 of a
Planck mass and mp is ~7.68E-20 of a Planck mass, this gives ~2.4E-43, which is
~2.275E39 times greater than the gravitational force, a figure in broad agreement with
eq. (10) above, as one would expect.
Conclusion
As can be seen from eq. (14), the part played by the masses of the electron and
proton in the electromagnetic interaction is substantial - and startling, at least to this
researcher. What (13) and (14) taken together say is that if the masses of the electron
and proton were zero, the gravitational force between them at an equivalent distance
to re would be zero (as would that at any non-zero distance), but the electromagnetic
force would be infinite. The smaller the mass, the larger the electromagnetic force.
This is probably why there are no charged particles less massive than the electron. Of
course, eq. (14) only applies when the electron is involved. The equation for the
electromagnetic force between a muon and a proton would have to take account not
only of the muon’s mass, but of the different value of its g-factor. It would also need
a different value for v - but this would render the equation invalid, as the value of the
constant κ (~6.23 in Planck units) depends on v having the magnitude given above,
and this in turn co-determines the value of αg, and thus, indirectly, of α. Likewise,
given the part played by mp in determining the value of α, the equation would be
rendered invalid without its presence.
In spite of this restriction on its validity, however, the relationship outlined
here between α, j, ge, the masses of the electron and proton, the gravitational constant
and the relative strengths of the gravitational and electromagnetic interactions clearly
underpins any valid unified field theory - to the extent that no such theory which
contradicted the relationship could be validii. It goes without saying, however, that it
is not a unified field theory, and no such claim is made for it. What it does do, though
- and here the prospect is tantalizing - is offer the possibility of what I can only call an
‘electro-gravitic’ propulsion system.
If we recall eqs. (6) and (7), we can see that
EK = ½mpV2 = Gemp/8πε0κÅ = 4πµemec2 ,
(15)
where EK is the kinetic energy of a proton traveling with a velocity, V, which as we
have seen is ~11.706% of c. Of course, this equation is non-relativistic; to reconcile it
with relativity, we need to introduce the correctional term (1 - V2/c2)-1/2, but as this is
only ~1.0069, it is safe to ignore it for our purposes here. What this implies is the
possibility of an electro-gravitic force, given by
Ge/4πε0κcd2 ,
(16)
specifically between a proton and an electron, where d is the distance separating
them. If this is taken to be the Bohr radius, a0 Q 5.292E-11 m, then this ‘electrogravitic force’ would be ~1.547E-7 N. By mpaa0 = ½mpv2 this corresponds to a
proton velocity, v, of ~9.89E4 ms-1, and an acceleration from rest, a, of ~9.25E19 ms2
. (Obviously this is only sustained for a very short time!) If we substitute re for a0,
then we get an ‘electro-gravitic energy’ of ~1.537E-13 J and a force of 54.56 N,
corresponding to a velocity of ~1.356E7 ms-1, which is ~0.386 V. To achieve V by
this means the proton’s separation from the electron, on this account, would have to
be ~0.149 of re.
This is clearly not possible in practice; and as it obeys an inverse square law,
the ‘electro-gravitic force’ falls off rapidly with distance. Nevertheless, the
possibility remains that some way may be found to harness this force, assuming it
exists, and make use of it - perhaps in a propulsion system for unmanned (and
therefore relatively lightweight) spacecraft. The author of this paper, for one, would
be gratified if such a prospect could ultimately be realized.
References:
i
Feynman, R.P. (1985), ‘QED. The Strange Theory of Light and Matter,’
Harmondsworth, Middlesex, England: Penguin Books, p.128.
ii
Greene, B. (1999), ‘The Elegant Universe. Superstrings, Hidden Dimensions, and
the Quest for the Ultimate Theory,’ London: Jonathan Cape, pp.130, 283.
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