Open Journal of Microphysics, 2011, 1, 13-18
doi:10.4236/ojm.2011.11002 Published Online May 2011 (http://www.SciRP.org/journal/ojm)
Octonionic Gravity Formation, Its Connections to
Micro Physics
Andrew Beckwith
Department of Physics, Chongqing University, Chongqing, China
Received March 14, 2011; revised May 1, 2011; accepted May 11, 2011
Abstract
We ask if Octonionic quantum gravity is a relevant consideration near the Planck scale. Furthermore, we
examine whether gravitational waves would be generated during the initial phase, 0

, of the universe when
triggered by changes in spacetime geometry; i.e. what role would an increase in degrees of freedom have in
setting the conditions during 0

, so that the result of these conditions can be observed and analyzed by a
gravitational detector. The micro physics interaction is due to the formation of a pre Planckian to Planckian
space time transition in spatial dimensions at and near the Planck dimensional values, i.e. 10–33 centimeters in
spatial dimensions. This transition would be abrupt and arising in micro physics regimes of space time.
Keywords: High-Frequency Gravitational Waves (HFGW), Symmetry, Causal Discontinuity
1. Introduction
The Planck epoch has remained mysterious, and may be
invisible to all other kinds of detectors, but the universe’s
gravity wave background radiation likely contains the
imprint of even the very earliest events. Changes in the
geometry of space-time near the Planck scale could be
revealed or studied in this manner. We discuss how to
obtain insights into 0

, initially, while looking at the
geometric considerations determining space and time
development which would create relevant space-time
geometry phase changes during the early universe. The
formation of Planckian space time is in what is called
Octonionic quantum gravity [1] Each such phase change
leading to Octonionic Quantum gravity should produce
gravitational waves. The geometry change alluded to is
effectively initially on an infinitesimal scale, which is
why Micro physics is referred to at all. (‘quantum me-
chanics’) and de facto quantum gravity, at the start of
Planckian space time. This Planckian space time would
mark the beginning of inflation. We give conditions for
detection of [2,3] 0

if, for example, one can isolate an
appropriate first-order perturbative electromagnetic power
flux, [2-4] in scenarios where the graviton has a
(1)
uv
T
vanishingly small – but non-zero – rest mass [2]. Now
for how to get non zero graviton rest mass. This is where
the micro physics regime becomes so important.
The Micro Physics Problem. How to Solve the Corre-
spondence Problem, I. E. (Gravitons in 4D with Slight
Mass (5 Degrees of Freedom) Versus Gravitons in 4 D
with NO Mass (2 Degrees of Freedom).
This formation of a graviton, must be done before the
formation of QM, in space time.
The correspondence problem, is that a Graviton with
slight rest mass in 4 D will have five degrees of freedom
wheras a Graviton in 4 D, with no mass has two degrees
of freedom. I.e. there is no way about this, and that if
gravitons as brought up by Beckwith [2] have a small
rest mass, which contributes to DE speed up of accelera-
tion of the universe a billion years ago, gravitons cannot
be synthesized during the time when QM becomes
dominant. This document gives a regime of space time
before QM, which embeds quantum mechanics. If a 4D
graviton with rest mass not equal to zero is formed, it
will be in the template of space time before to up to 10–44
seconds. I.e. as stated , so as to have GW/Gravitons with
rest mass not equal to zero forming [2], The document
mentions a phase alteration, which we claim is due to
this alteration from classical embedding of quantum me-
chanics, to its full expression in the Planckian space time
regime. This micro physics leads to [2-4] 0

What we propose is the following evolution in the mi-
cro physics regime, up to 10–44 seconds
1) That the degrees of freedom increase, with an in-

A. BECKWITH 14
crease in temperature, during a transition to a Rindler
Geometry flat space regime of space time. As given
in Equation (16), with increasing temperature, more
degrees of freedom unfold from a topological transi-
tion. Degrees of freedom likely approach a maxima
as temperature does, but this is a subject needing ex-
perimental exploration and verification.
2) That for low but non zero initial temperature, the so
called cold universe model, in pre space time in the
pre Planckian regime, one has initially a huge degree
of generated entropy. At the same time, we have
about 2 degrees of freedom, with complex geometry
in each geometrical slot, geoinfometric instantiation,
or “infometron” of space time, which large quantities
of stored entropy enveloped in the ‘crevices’ between
infometrons, or lattice points.
3) Low degrees of freedom for low temperature corre-
sponds to a complex geometry storing large amounts
of total entropy in a complex geometric structure, and
that later the entropy is released, with a break down
of this complex geometric structure, i.e. equivalent to
having many lattices, highly ordered, with low de-
grees of freedom per ‘lattice’, to many degrees of
freedom (DOF) as space time ‘lattices’ are broken,
releasing entropy. The analogy is not perfect, but ap-
proximates what would happen as one goes from
complex curved space geometry with many ‘crevices’
for storage of entropy, which are released, “appar-
ently” leading to a lot of entropy,.
Further elaboration of what is being brought up is tied
in with mutually unbiased basis (MUB), [5] .The values
of 0

are set by the difference between Renyi entropy
[6], and a particle count version of entropy, i.e. Sn.
Are predictions also possible regarding signal strength of
evolutionary artifacts of early universe HFGW? Again,
yes. What we are talking about is the break down, due to
thermal heat flux of an initial mutually unbiased basis set
for a very complex initial geometry, and a reconstitution
of space time geometry in flat Euclidian space time re-
gime. The topological transition is due to a change in
basis / geometry from the regime of Renyi entropy [6]to
entropy in a particle count version of entropy, i.e. Sn.
The choice of a Gaussian mapping, with two variable
inputs, as given by Equation (16) below is done as a
simplest case model. We will model inputs into the ini-
tial value of as high energy fluctuations, and see if
they contribute to examination of the formation of non
commutative geometry in the beginning/just before the
inflationary era.

1.1. First, Thermal Input into the New Universe.
In Terms of Vacuum Energy
We will briefly allude to temperature drivers which may
say something about how thermal energy will be intro-
duced into the onset of a universe. This will be the
‘thermal driver’ for the increase in degrees of freedom.
Begin first with looking at different value of the cosmo-
logical vacuum energy parameters, in four and five di-
mensions [7]

5dim1 1cT



 (1)
in contrast with the more traditional four-dimensional
version of the same, minus the minus sign of the brane
world theory version. as given by Park [8]
4dim 2
cT



 (2)
Right after the gravitons are released, one still sees a
drop-off of temperature contributions to the cosmological
constant .Then for time values1
P
tt   , and
integer n [7]
1
01  
4dim
5dim
1
1n



 (3)
In terms of its import the following has been suggested
in the initial phases of the big bang, with large vacuum
energy

 and


0, 01at at

 , the following
relation, which violates (signal) causality, is obtained for
small fluctuation


P
at l


If we examine

5dim 2
cT


We assume in this that we have, a discontinuity in the
pre Planckian regime, for scale factors [7].




1( )1
at tval ue
at







 


 (4)
Furthermore, in the transition for 0
P
tt the fol-
lowing increase in degrees of freedom is driven by ther-
mal energy from a prior universe starting with [9]
0
1
2
thermalB temperature
EkT T

 

 
 (5)
The assumption is that there is an initial fixed entropy
arising, with N as a nucleated structure in short time
interval as temperature ar-
rives.

19
0,10
temperature
TG 
eV
If the inputs into the inflaton

, as given by a random
influx of thermal energy from temperature, we will see
the particle count on the right hand side of Equation (13)

2
22
22
1612161
24π4π4π
Planck
STk M
 



















 

 







 (6)

A. BECKWITH15
above a random creation of
P
article Count
n
e degrees of
. The way to
introduce the expansion of thfreedom from
3 is to definearly zero to having N(T) ~ 10ne the classical
and quantum regimes of gravity as to minimize the point
of the bifurcation diagram affected by quantum processes
[9]. Dynamical systems modeling is employed right ‘af-
ter’ evolution through the ‘quantum dot’ regime. The
diagram, would look like an application of the Gauss
mapping of [9].
2
expxx
1ii






 (7)
The inputs of change of iterated st
hand side of Equation (14) may indeed s
de
eps on the right
how increase in
grees of freedom. Change of temperature, as given,
over a short distance, is [5,6]

52
B
temp
N
kT
dist di
  

netelectricfield
qE
st 
change in degree of freedom


(8)
We would regard this as being the regime in which we
see a thermal increase in temperature, up to the Planckian
ph
Temperatures as Given in Table 1 Leads to
n
To d[1] as to how to look at the way we
may have , if temperatures increase, as stated in Table 1
ysics regime. If so, then we can next look at what is
the feeding in mechanism from the end of a universe, or
universes, and inputs into Equation (7), Equation (8).
1.2. Formal Proof that Increase in Thermal
Approaching Quantum Mechanics, As a
End Run about the Correspondence
Problem.
o this we look at
above, from a low point to a higher one, for there to be a
flattening of space time and the end of non communative
geometry. This non communative geometry due to rising
temperatures signifies conditions for the emergence of
Equation (23) to become [1]
,,
j
i
Te
xpi ij
mp


 
  (9)
In order to get conditions for Equation
lowing can be referred to about non comm
om
smological in 5 and 4 dimensions [7], with
phasis upon the Michysics initial regime.
(24) the fol-
utative ge-
etry [1]
Table 1. Co
ro pem

0P tt
(PLANCK TIME ~ 10–44 S CONDS).
Time0
E
P
tt Time 0
P
tt

 Time
P
tt
5
 undefined,
T 
32
10T
almost
,0
jiji Temp
xx i


 
 (10)
The essentials step is to imetric rl
tensor is proportional to the square of 1 over the Parks
representation of the “Planck constnt”, which has a
temperature dependence built in it [1].
say the ant symea
a

2
22
4
~~ 10
ji NCDimT
T






 
 (11)
When Equation (26) goes to zero, leading to Equation
(10) going to zero, we submit that then Equation (10) is
recovering quantum/Octonian gravity. The Equation (24)
is linkable to initial violations of Lorentz invaria
submit that the entire argument of Equation (22) to
Eq
f
Emergent Structure Cosmology
We
and wave
gene given
by Led in Chong-
quing the following representation of amplitude, i.e. as
nce. We
uation (25), as given by Equation (11) with rising
temperature is a way to understand the removal of non
Euclidian space to approach Euclidian flat space. We shall
next examine how this increasing temperature may lead to
an explosion of the degrees of freedom present.
1.3. Understanding how Phase Shift in
Gravitational Waves may be Affected by the
Transition to and from a Causal
Discontinuity, and Different Models o
will outline research initiated by . Beckwith and Li,
Yang Nang, gives us details of gravitational
ration by early universe conditions. In [3] as
i, and Yang, 2009, Beckwith [2] outlin
by reading [3] the following case for amplitude
A
AA


 (12)
Furthermore, the first order perturbative (E and M)
terms of an E&M field may have its components written
as [3]
(1)
0 20
(1)
1
F
iF

(13)
Secondly, there is a way to represent the” number” of
transverse first order perturbative photon flux density as
given by (in an earth bound high frequency GW detector).
[3]

(1)
0
Re
2
r
e
c
n


 (14)
  
(1)*
0 1
exp y
x
e
i
iiF yx 



 
 





 (15)
Here the quantity
y
x
e
i
yx 








K
4dim
 
5


 ,
extremy
large
4dim
 el
32 12
10 10
K
T K
54
 ,
dim

T much smaller
than 10 TK12
represents the
z component of the magnetic field of a Gaussian b
be used in an EM cavity to detect GW. We introduce the
eam to

A. BECKWITH 16
quy factor of the dete
to
antity Q, the qualitctor cavity set up
observe GW, and
A

, the experimental GW amplitude.
In the simplest case, (0)
ˆ
y
B is a static magnetic field.
Then the (1) (1)
0 20 1
F
iF

will lead to [2,3]
(1) (0)
0 10
π
ˆ
2snexp
yg
nz
FiABQit
b
i




rmula

 

 



 (16)
The fo1
aB temperature
kT
2
therm l
E


is a feed
into
g

provided that we pick time Planc
and set Equation (16) with
tk time,
g
g

  by setting up the
1
2
thermalB temp
T
erature
Ek

ds, for relic 

[9]. In other wor
raranGW/gviton production, a topological tsformation and
interrelationship between

 and
1
2
thermalB temperature
EkT 


for increases in (topologi-
cal) degrees of freedom, as a change in geometry, i.e.
before quantum gravity. Passing gravitons through to a
new univis not the same thinerse g though as a pre
Planckian geometry, f or Octonian gravity conditions
arise in early Planckian space time. This is a different
perspective than what is normally used in analyzing what
happens in a transition between initial Planck time ~
44
10 seconds, and cosmological evolution up to 30 10
seconds We will specify how to locate massive gravitons,
via an experimental set up which may enable obtaining
data for supporting a value [2,3] for 0

. The next dis-
cussion is on research done by Dr. F. Li, et al., 2003, [4],
2. Re Casting the Problem of GW/Graviton
in a Detector for “Massive” Gravitons
We now turn to the problem of detection. The following
/
-
sea time
gy
discussion is based upon with the work of Dr. Li, Dr
Beckwith, and other Institute of theoretical physics re
rchers in Chongquing University [2-4]. It is now
to consider what happens if one is looking for traces of
gravitons which may have a small rest mass in four di-
mensions. What Li et al. have shown in 2003 [2-4] which
Beckwith commented upon and made an extension in [2]
is to obtain a way to present what is called here in the
literature [3,4]a first order perturbative electromagnetic
power flux, i.e. what was called
(1)
uv
Tin terms of a non
zero four dimensional graviton rest mass, in a detector ,
in the presence of uniform magnetic field, when exam-
ining the following situation, i.e. [2-4] what if we have
curved space time with say an ener momentum tensor
of the electro magnetic fields in GW fields as given by
0
11
4
uvv v
TFFgFF





 


(17)
Li et al. [3,4] state that (0) (1)
vv v
F
FF



, with
(1) (0)
vv
F
F

 will lead to
(0) (1)
uvuv uv uv
TTTT
(2)


The 1st term to the right hand side of Equation (18) is
thf the back groun
tro magnetic 2 term to the right hand side
of Equation (18) is the first order perturbation of an elec-
tro magnetic field due to the presen
w
(18)
e energy – momentum tensor od elec-
field, and the nd
ce of gravitational
aves. The above Equation (17) and Equation (18) will
eventually lead to a curved space version of the Maxwell
equations as [3]


0
1v
v
g
ggF J
x
g

 



 

 (19)
as well as

,0 v
F

 (20)
Eventually, with GW affecting the above
ve a way to isolate If one looks at if a
sional graviton with a very small rest mass
included [2-4] we can write[2] an
[3,4]
two equa-
tions, we ha
four dimen
(1)
uv
T.
extension of Li et al.
.


0effecti ve
vgggFJJ
x
g


1v
 

 

 (21) 
where for 0



but very small [3,4]

,v
F



 (22)
The claim which A. Beckwith made [2] is that
4effectivecountDGravition
Jnm

 (23)
As stated by Beckwith, in [2],
grams, while
Be
tenut an appropriate
as ass, and u
of
65
410
DGravition
m

 
of
count n is the number
gravitons which may be in the detector sample. What
ckwith, Li, and Chonquing university researchers in-
d to do is to try to isolate o
suming a non zero graviton res
(1)
uv
T
t msing Equa-
tion (21), Equation (22) and Equation (23). From there,
the energy density order contributions of,
(1)
uv
T i.e.
(1)
00
T
can be isolated, and reviewed in order to obtain traces


, which can be used to interpret Equation (14). I.e. use
F
 

and make a linkage of sorts with
(1)
00
T. The term
(1)
00
T isolated out from
(1)
uv
T present day dhe ata. Tpoint
va
be to cosen
of diffe
Working with a unified The value picked for
here that the detected GW would help constrain and
lidate Equation (14). If this is done, the next step will
me up with a protocol as far as making se out
rent GW measurement protocols.
2
0GW
h.

A. BECKWITH17
m
2.1. Wavelength, Sensitivity a
] treatment of both wavelength,
early
serve,
suchle of
earlyation of GW at the
generaleasurements was
26
0~10 GW
h  (24)
Next, after we tabulate results with this measurement
standard, we will commence to note the difference and
the variances from using 26
0~10
GW
h  as a unified
measurement which will be in the different models dis-
cussed right afterwards
nd other such
Constructions from Maggiore, with our
Adaptations and Comments
We will next give several of our basic considerations as
to early universe geometry which we think are appropri-
ate as to Maggiore’s [10
strain, and GW  among other things. As far as
universe geometry and what we may be able to ob
considerations are make or break as to the ro
universe geometry and the gener
start of the universe. To begin with, we will look at
Maggiore’s [10] GW  formulation, his ideas of strain.
The idea will be to look at how the ten to the tenth
stretch out of generated wave length may tie in with
early universe models. We will from there proceed to
look at, and speculate how the presented conclusions
factor in with information exchange between different
universes.
We begin with the following tables, Table 1 and Ta-
ble 2. The idea will be to, if one has 00.51 0.14h, as
a degree of measurement uncertainty begin as to under-
stand what may be affecting an expansion of the wave
lengths of pre Planckian GW/gravitons which are then
increased up to ten orders of magnitude. What we have
stated below in Table 2 will have major consequences as
far as not only information flow from a prior to present
universe, but also fine tuning the degree of GW variance
What we are expecting, as given to us by L. Crowell,
is that initial waves, synthesized in the initial part of the
Planckian regime would have about 14
~10
GW meters


for 22 ~10
GW
f
Hertz which would turn into
1
~10
GW meters

, for 9
~10
GW
f
Hertz, and sensitivity
Table 2. Managing GW generation from pre planckian
physics and its immediate aftermath.
33
2.82 10
C
h  12 ~10
GW
f
Hertz 4 ~10
GW meters


31
2.82 10
C
h  10 ~10
GW
f
Hertz 2 ~10
GW meters


29
2.82 10
C

 h8 ~10
GW
f
Hertz 0 ~10
GW meters

27
2.82 10
C
h  6 ~10
GW
f
Hertz 2 ~10
GW meters

25
2.82 10
C
h  4 ~10
GW
f
Hertz 1 ~10
GW kilo meter

23 2
2.82 10
C
h  ~10
GW
f
Hertz 3 ~10
GW kilometer

30
2.82 0
C
h. This is assuming that 2
0
h of 1


6GW
~10

, using Magg
. It is por
discuss the differen
iorie’s [10] analy
tant to note
t models th
2
0GW
h
i all of this, th
at the
tical
at when
26
~
GW
ex-
pression imn
we 0
h10


is drastically al-
n (26
eality, on
, the fi
) should
ly the
pe being seri-
ons of
rst col-
is th
tered
be
2nd
e first mesurem
. C
h which is m
also noted t be
and 3rd columns in
ly off and very
vita
n
aent metr
entioned i
an upper b
Table 1
ic which
Equatio
ound. In r
above esca
quark
n
o
ous different., since the interacti
grational waves/gravitons with – gluon plas-
mas and even neutrinos would serve to deform by at least
an order of magnitude C
h. So for Table 1
umn is meant to be an upper bound which, even if using
Equatio(40.c) may be off by an order of magnitude.
More seriously, the number of gravitons per unit volume
of phase space as estimated, is heavily dependent upon
26
0~10
GW
h

.
The particle per phase state count will be given as, if
26
0~10
GW
h

 [10]
4
37
2
0
10 1000
~3.6
fGW
H
z
nh f

 


(25)
Secondly we have that a detector strain for device
physics is given by [10]

21 1
2.82 10
C
H
z
h
f



(26)

These values of strain, the numerical count, and also
of
f
n
ssibl
give a bit count and entropy which
poe limits as to how much information is trans
fe
will lead to
rred. Note that the beginning of relic inflation GW

16
~10 10
f
meters n
 /graviton unitphase 
9
space
for ~10
GW
f
Hertz
in pre inflationary physics of
This is to have, say a starting point
22
10~
GW
f
Hertz when
14
~10
,
GW meters i.e. a change of ders of
magnitude in about 25 10

13
~10 or

seconds, or less.
3. 1st Part of Conclusion. Can We Justify/
Isolate out an Appropriate if one ha
on Zero Gra
It is difficult. It tandi
strucat the mathe-
matical self organized criticality structure is akin to a
fla-
Wh itting
(1)
uv
T s
Nviton Rest Mass?
depends upon undersng what is
meant by emergent ture, we assert th
definition as to how Dp branes arise at the start of in
tion. at is the emergent structure perm
,ik ik
pdx




to hold? [1] What is
ticality structure leading to forming an
the self organized
cri appropriate
(1)
uv
T if one has non zero graviton rest mass? Answering
such questions will permit us to understand how to link

A. BECKWITH 18
finding
(1)
uv
T in a GW detector, its full analytical link-
age to

in Equation (16). The following construction
is used to elucid
GW detector. This construction below is to be used to
inve
/gra
ity
c
space timrresponds, as brought up in the
sy
chni-
cal assistance to the author as to understanding the phase
s due to early universe cos-
gnificant encouragement to
[2] A. W. Beckwith, “Identifying a Kaluza- Klein treatment
permitting a de-celeration parameter Q(Z)
e to standard DE “,Journal of Cosmology,
104008
018-1026, 2008
n Brane
Use the Cosmological
ate how a EM Gaussian sense beam can
(1)
perhaps be used to eventually help in isolating uv
T in a
stigate ‘massive gravitons’/and also the initial struc-
ture of self organized criticality, in the aftermath of
gravitonvitational wave generation.
3.1. 2nd Part of Conclusion: In Terms of the
Planckian Evolution and a Pre Planckian
Space Time
We wish to summarize what we have presented in an
orderly fashion. Doing so is a way of stating that Analog,
(Pre Ocononian) reality is the driving force behind the
evolution of inflationary physics
1) Pre Octonian gravity physics (analog regime of
ity) features a break down of the Octonian grav
ommutation relationships when one has curve
e. This co
real-
Volume 13, 2011,
http://journalofcosmology/BeckwithGraviton.pdf
[3] F. Li, and N. Yang, “Phase and Polarization State of High
d Fr
Jacobi iterated mapping for the evolution of degrees
of freedom to a build up of temperature for an in-
crease in degrees of freedom from 2 to over 1000. Per
unit volume of space time. The peak regime of where
the degrees of freedom maximize is where the Octo-
nian regime holds.
2) Analog physics, prior to the build up of temperature
can be represented by Equation (7) and Equation (8).
The first of these mappings is an ergotic mapping, a
perfect mixing regime from many universes into our
own present universe. This mapping requires a de-
terministic quantum limit as similar to what t’Hooft
included in his embedding of Quantum physics in a
larger, non linear theory [11].
This will help to localize the regime of space time
build up potentially giving experimental details to the
formation of Equation (14) and Equation (15) above. The
claim is that the physical evolution described by Equa-
tion (7) and Equation (8) would arise, as a dynamic evo-
lutionary process in the Micro physics regime leading to
an embedding of quantum physics, in a larger, non linear
theory, which would lead to massive gravitons being
nthesized whole scale. This whole scale synthesis of
gravitons with a small rest mass occurs in initial micro
physics background as indicated above, a result we hope
to confirm via accurate measurements of the phase.
4. Acknowledgements
This work is supported in part by National Nature Sci-
ence Foundation of China grant No. 11075224 Dr Fan-
gyu Li of Chongquing University gave extensive te
change in gravitational wave
mology. Stuart Allen gave si
the author as to finalizing this document for publication.
5. References
[1] L. Crowell, Quantum Fluctuations of Space-time, in
World Scientific Series in Contemporary Chemical Phys-
ics, Volume 25, Singapore, Republic of Singapore, 2005.
of a Graviton
as an alternativ
equency Gravitational waves”, Chin Phys. Lett. Vol
236, No 5(2009), 050402, pp 1-4
[4] F. Li,. M.,Tang, D. Shi, “Electromagnetic response of a
Gaussian beam to high frequency relic gravitational waves
in quintessential inflationary models”, PRD 67,
(2003), pp1-17
[5] S. Chaturvedi, “Mutually Unbiased Bases”, Pramana
Journal of Physics, Vol. 52, No. 2, pp 345-350
[6] John C. Baez, “Renyi Entropy and Free Energy”,
http://arxiv.org/abs/1102.2098
[7] A. W. Beckwith, arXiv:0804.0196 [physics.gen-ph], AIP
Conf.Proc. 969:1
[8] D. K., Park, H. Kim, H., and S. Tamarayan, “Nonvanish-
ing Cosmological Constant of Flat Universe i
world Senarios,” Phys. Lett. B 535, 5-10 (2002).
[9] A. W. Beckwith,” How to
Schwinger Principle for Energy Entropy, and “atoms of
Space-Time” to Create a Thermodynamic Space-Time
and Multiverse”; accepted for the DICE 2010 proceed-
ings; http://vixra.org/abs/1101.0024, 2010
[10] M. Maggiorie,. “Gravitational Wave Experiments and
Early Universe Cosmology”, Physics Reports 331 (2000)
pp. 283-367
[11] G. t’ Hooft, Beyond the Quantum, edited by Th. M. Nieu-
wenhuizen et al. Singapore, World Press Scientific, 2006;
http://arxiv.org/PS_cache/quant-ph/pdf/0604/0604008v2.
pdf.