VIRTUAL BLACK HOLES AND PROTON DECAY
The standard model of particle physics is a spectacular success. But this
virtue also presents a problem. Science advances by scouting out the
limitations of exiting theories, and the experimental success of the standard model
have made finding these limitations very difficult. But we know that the
standard model can not be the ultimate theory of nature. The standard model for all
its success fails to incorporate gravity, and our best theory of gravity,
Einstein’s General Theory of Relativity fails to incorporate Quantum theory.
An interesting question is whether Quantum theory or General Relativity is
more fundamental or if there is a theory which transcends both. . The various
approaches being used in an attempt to advance beyond the standard model have
different answers to this question. However, an advance “along the way” is
to unify the strong and electroweak force.
One method to effect this unification is, based on the symmetry structure of
the standard model, to postulate a more unified group structure. Given the
symmetry of the standard model:
SU (3) _c X SU (2) _L X U (1) _Y there are many possibilities.
These possibilities include SU (5), SO (8), SO (9), Sp (8), and F_4 based
on selecting the most compact groups. The most parsimonious choice, based on
the relationship of charge conjugation to the double-singlet structure of
fermions is SU (5).
If SU(5) is a good symmetry of nature then the quark and lepton sectors are
coupled by a vector field and the proton is not a stable composite particle.
However, the predicted decay of the proton has not been observed in several
careful experiments. This has prompted efforts to modify SU(5) by expanding it
to incorporate Super Symmetry , a symmetry between fermions and bosons , and
by postulating a less compact unified Symmetry SO (10) which predicts longer
mean lifetimes for proton decay.
I had originally thought I had found an interesting objection to unifying
the quark and Lepton sector with a gauge field, but in reality the objection
disappears if one postulates that the lepton and quark sector are time reversed
from each other in terms of energy flow and energy sign. This seems a
reasonable assumption based on charge structure, especially B-L charge.
However, there may be another mechanism in nature which can result in proton
decay. Based on the work of John Wheeler, we have reason to believe that
near the Planck scale we have ever present quantum fluctuations of all the
existing quantum fields sufficient to create virtual black holes.
This is due to the fact that energy couples to space time curvature which
means that near the Planck scale we have violent fluctuations of the geometry of
space time. Evidence for this prediction is in the Lamb Shift due to EM
field fluctuations which gives us an additional effective electric field
potential given by;
E(x) = Integral E (w)*exp [-i*w*t] dw
This adds to the static field of the nucleus affecting the energy level of
the orbiting electron.
We can write the ground state probability amplitude as a function of
position
Psi(x) = {(m*w/pi*hbar) ^ (1/4)}* exp [- (m*w/2hbar)*x^2]
This cause the particle to be displaced from its classical position by
Delta(x) = approx sqrt [hbar/m*w]
The electromagnetic field can be treated as an infinite collection of these
field oscillations over w. So that
Psi( E) = N *exp [ - { E_1 +E_2 ….. } ]
One can rewrite this in terms of the magnetic field configuration of the
ground state (Wheeler (1962)
Psi ( B(x,y,z) = N* exp [ - double
integral(B(x_1)*{B(x_2))/(16*pi*^3*hbar*c*r_12^2)}d^3X_1 d^3x_2
This gives approximately the fluctuations of the magnetic field in a region
of dimension L
Delta (B) = sqrt [hbar*c]/ L^2
Because space time geometry is coupled to the energy fluctuations of the
various quantum fields we get a related fluctuation in the metric coefficients
defining the local geometry of space time.
Delta (g) = L_plk/L Based on the breakdown of space time at the Planck
scale. So that we get
Delta(R) = Delta (g) /L^2 = L_plk/L^3
This is the fluctuation of the curvature of space-time in a region that is
defined by length L.
We find that due to quantum fluctuations that something like gravitational
collapse is taking place everywhere in space and all the time: that in effect
gravitational collapse is perpetually being done and undone. Therefore we get
fluctuations of sufficient magnitude to create virtual black holes.
The existence of virtual black holes has interesting consequences for the
proton and the neutron.
We will model the nucleon as a hollow sphere of dimension 1E-15M. Within
this sphere one should expect the existence of numerous virtual black holes
based on the prediction offered by Wheeler. The consequence of a single quark
falling into a virtual black hole is nil bases on the no hair theorem and the
conservation of mass energy as well as the conservation of gauges quantum
charges. At the Planck scale we should expect that B_L charge is gauged therefore
conserved. (This charge is conserved in SU (5) theory also.)
So nucleon decay events require at least two quarks to fall into a virtual
black hole at the same time.
The means decay time for this event is equal to
t_mean = t_c* / (P (2q)*P (bh))
Where t_c is the Compton time for the nucleon, P (2q) is the probability of
two quarks being within one Planck distance and P (bh) is the probability of
the existence of black at this point.
P(2q)= V_pl/V(nuc)_c = m_nuc^3 / M_plk^3
Where V are the volumes of the Planck distance and the Compton wavelength
of the nucleon.
P( bh) = R_plk/ R(nuc)_c = m_nuc/ m_plk
Giving
t_mean= {hbar/2&M+nuc*c^2}* {( m_plk/m_nuc)^4 }
Which is on the order of 1E44 years.
We get the following decay events
U_1+U_2 + BH = dbar_3+ ebar
Therefore
Proton to Pion (0) + positron
U_1 + d_2 + BH = dbar_3+ nubar
Therefore
Proton to Pion (+) + antineutrino
Neutron to Pion (0) + antineutrino
d_1 +d_1 +BH = Ubar_3 + nubar
Therefore
Neutron to Pion (0) +antineutrino
So if Wheeler is correct about the existence of virtual black hole , even
absent Grand Unification, protons are not eternal.
Bob Zannelli