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\section{\usemenu{slacpub7205::context::slacpub72050012}{Quantum Gravity}}\label{section::slacpub72050012}
The initial intent of Eddington, and following him of Bastin and
Kilmister, was to achieve the reconciliation of quantum mechanics with
general relativity. I emphasize here that they were aware of the
problem, and thought they had a research program which might solve it,
long before the buzzwords ``Grand Unified Theory'', ``Theory of
Everything'', ``Final Theory'' or ``Ultimate Dynamical Theory''
\cite{90,67} became popular. In a sense they achieved the
first major step with the publication of Ted Bastin's paper in
1966\cite{7}. According to John Amson, that paper came about
after many attempts to understand Fredrick's
breakthrough\cite{79} had led John to the discovery of
discriminate closure\cite{2} which gave more mathematical coherence to the
scheme, and also convinced Clive and Fredrick that it was time to
publish. In the event, the four authors could not agree on a text in
time to meet a deadline, and authorized Bastin to go ahead with his
version as sole author. This paper really does unify electromagnetism
with gravitation (which was also Einstein's long sought and unachieved
goal) in the sense that both coupling constants are derived
from a common theory. Ted also correctly identified $(256)^4$ with the
weak interactions, but missed the $\sqrt{2}$ needed to connect
it numerically with the Fermi constant because he was unfamiliar with
the difference between 3vertices (Yukawatype couplings) and
4vertices (Fermitype couplings) in the quantum theory of fields.
As we all know, this paper was met by resounding silence.
I am optimistic, in spite of my past failures, that
the bitstring theory now has enough points of contact with
more conventional approaches to fundamental physics to get us all
into court.
Since the critical problem for many physicists is how we deal with
``quantum gravity'', I start there. For weak gravitational fields
it makes sense to start in a flat
spacetime\cite{95}. Then it can be shown
that spin 2 gravitons of zero mass lead to the Einstein field
equations, but as Meisner, Thorne and Wheeler note\cite{52},
the ``Resulting theory eradicates original flat geometry from all
equations, showing it to be unobservable.'' Consequently they feel
that this approach says nothing about ``... the greatest single crisis
of physics to emerge from these equations: complete gravitational
collapse.'' My qualitative answer is that this crisis arises from using a continuum
theory at short distance where only a quantum theory makes sense.
I now try to make the alternative presented here plausible.
My first step is to establish the existence of quantum gravitational effects
for {\it neutral} particles, namely neutrons. That neutrons are
gravitating objects in the classical sense was proved at Brookhaven
soon after the physicists there learned how to extract epithermal neutrons from their
high flux reactor and send them down an evacuated pipe a quarter of a
mile long. The neutrons fell (within experimental error) by just the amount
that Galileo would have predicted. That they are quantum mechanical
objects was proved by Overhauser\cite{77} by cutting a single silicon crystal 10
centimeters long into three connected planes and using critical
reflection, both {\it calculable} from measured $nSi$ cross sections
and demonstrable {\it experimentally}, to form in effect
a twoslit apparatus for neutrons with the positions known to atomic
precision over a distance of ten centimeters. Then the shift in the interference
pattern between the case when two beams were both horizontal to
the case when they were in the vertical plane with one higher than the
other for part of its path was proved to be precisely that predicted
by nonrelativistic quantum mechanics using the Newtonian
gravitational potential in the Schroedinger equation. It was this
brilliant experiment which convinced me that quantum mechanics is a
general theory and not just a peculiarity in the behavior of
electrically charged particles at short distance. In my opinion,
Overhauser deserves the Nobel prize for this work, which opened
up the study of the foundations of quantum mechanics to high
precision experimental investigation. In the hands of Rausch\cite{82}
and others this technique has led to many tests of the model of
the neutron as a quantum mechanical particle acting coherently
with a precisely known mass and magnetic dipole.
Having established that neutral particles react gravitationally to
the Newtonian gravitational potential $V_N(m_1,m_2;r) =G_N{m_1m_2\over
r}$ as expected, it makes sense to extend our relativistic bitstring
model for the Coulomb potential $V_C(m_1,m_2;r) ={Z_1Z_2e^2\over r}$
to the gravitational case.
Here $Z_1,Z_2$ are the electric charges expressed in units of the electronic
charge $e$. If we are guided by Bastin's remark
quoted above to the effect that the basic quantization is the
quantization of mass, the analogy suggests that there is a (currently
unknown) unit of mass, which we will call $\Delta m$. Then, to complete
the analogy with the Coulomb case, we can replace $\alpha _C=e^2/\hbar
c\approx 1/137$ with a much smaller constant $\alpha_N=G_N{\Delta m^2/\hbar
c}$. If we also define $N_i=m_i/\Delta m$ for any particle $i$ with
{\it gravitational mass} $m_i$, the quantized version of the Coulomb and
Newtonian interactions become formally equivalent, differing only by two
dimensionless constants and two {\it quantum numbers}, independent of the
units of charge or mass:
\begin{equation}
V_C(Z_1,Z_2;r) = Z_1Z_2\alpha_C{\hbar c\over r}; \ \ \ \
V_N(N_1,N_2;r) = N_1N_2\alpha_N{\hbar c\over r}
\end{equation}
We can now apply the Dyson argument to gravitation with more precision.
We have seen that renormalized QED extended to enough precision to generate
$N_e=137$ electronpositron pairs, becomes unstable because of
(statistically rare) clumping of clusters with enough electrostatic
energy to form another pair. We interpret the fact that this disaster
does not occur to the formation of a pion with mass $m_{\pi} \approx 2\times
137 m_e$. In the neutral particle case, if one assumes CPT invariance,
one can still distinguish fermions from antifermions by their
spin even if they have no other quantum numbers. Hence, independent
of whether or how neutral fermions and antifermions interact, we can
expect gravitational clumping to occur for each type separately.
Recall Dyson's remark that the system is dilute enough so that
the nonrelativistic potential can be used reliably to estimate
the interaction energy of the clump. We know experimentally
that $\Delta m$ is much smaller than the electron mass so that
the critical radius of the clump is $\hbar/\Delta m c >> \hbar/m_ec$,
so Dyson's comment still applies..
In contrast to the electromagnetic case where the cutoff massenergy
of the pion requires us to go outside QED for the physics,
in the gravitational case we have a cutoff energy ready to hand,
namely the Planck mass $M_{Pk} \equiv [{\hbar c/G_N}]^{{1\over 2}}$.
If we assemble a Planck's mass worth of neutral particles of mass
$\Delta m$ at rest within their own Compton wavelength, i.e.
$N_G\Delta m=M_{Pk}$, and nothing
else intervenes, they will fall together until they are all within
a distance of $\hbar/M_{Pk}c$. At that point they will have a
gravitostatic energy $N_G G_N\Delta m M_{Pk}/[\hbar/c\Delta
M]=M_{Pk}c^2$, which is just sufficient
to contain the kinetic energy they acquired in reaching this
concentration. Inserting the definition of the Planck Mass
into this gravitational energy equation we find that it is
algebraically equivalent to the boundary condition with which
we started: $N_G\Delta m=M_{Pk}$.
They will form a black hole with the Planck radius.
We conclude that $N_G=M_{Pk}/\Delta m$ neutral, gravitating
objects of mass $\Delta m$ at rest within their own Compton wavelength will
collapse to a black hole with the Planck radius, a quantum version of
the disaster that Wheeler is concerned about.
If there are, in fact,
neutral fermions with no other properties than their mass, they would form such
such black holes and might serve as a model for the dark matter
which we know to be at least ten times as prevalent in the universe as
ordinary matter. It then becomes a question in bigbang cosmology
whether or not they contribute the needed effects to correlate
additional observations. We defer that question to another occasion.
If we assemble enough particles of the
types we know about to add up to a Planck mass, they
can start collapsing and will radiate much of their energy on the
way down to higher concentrations. If attractions are
balanced by repulsions they could end up close enough together to form
a black hole. However, as Hawking showed, small black holes
interact with the ``vacuum'' outside the
event horizon and radiate electromagnetically with the consequence
that they are ``white hot''
and soon evaporate; the calculation was extended to rotating, charged
black holes by Zurek and Thorne\cite{98}.
At the quantum scale a new possibility enters, namely that
a quantum number may be possessed by the system which {\it cannot}
be radiated away by emitting a single particle with that quantum number
while conserving energy, momentum and spin. The obvious candidates
for such conserved quantum numbers are baryon number, charge and lepton
number, suggesting that the lightest baryon (the proton), the
lightest charged lepton (the electron) and the lightest neutral lepton
(the electrontype neutrino) are {\it gravitationally stabilized}
black holes with spin ${1\over 2}\hbar$. This idea did not make it
into the mainstream literature\cite{63}. We use it freely
in what follows. The problem then is to explain why $(M_{Pk}/m_p)^2
\approx 2^{127}$, why $m_p/m_e \approx 1836$ and why $m_{\nu_e}/m_e
\leq 5\times 10^{5} m_e$.
Before we leave gravitation, however, we need to show within bitstring
physics that the graviton has spin 2. We know from our discussion of
the handydandy formula that we can account for spin ${1\over 2}$
electrons, positrons and protons and their interactions with the
appropriate spin 1 photons. We have shown elsewhere\cite{65}
that we can construct the quantum numbers of the standard model of
quarks and leptons. In particular, this will include the electron,
muon and tau neutrinos and their antiparticles. Extending this
approach to a string
of length 10 we can have 6 spin ${1\over 2}$ fermions and 2 spin 1
bosons. On another occasion I will show how these construct 5 gravitons
and 5 antigravitons represented in terms of strings of length 10,
and go on from that to make a model for dark matter that can be expected
to be approximately 12.7 times as prevalent in the universe as electrons
and nucleons.
It remains to show that we can meet the three classical
tests of general relativity, a problem met on another occasion
\cite{71}. Briefly, any relativistic theory gives the
solar red shift (Test 1), the factor of 2 compared to special relativity
in the bending of light by the sun comes from the spin 1 of the photon
(Test 2), and the factor 6 compared to special relativity for the precession
of the perihelion of Mercury\cite{13} from the spin 2
of the graviton (Test 3). The calculation by Sommerfeld on which the third
argument partly depends comes from simply replacing the factors
$N_i=m_i/\Delta m$ in the Coulomb potential by $E_i/\Delta m$,
where $E_i$ includes the changing velocity of the orbiting particle
in elliptical orbits, and hence is natural in our theory.
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