The math about the way the Higgs forms when 8-dim spacetime isreduced to 4-dim spacetime can be found on my web site at
The math proof was not done by me, but by Meinhard Mayer who usedProposition 11.4 of chapter II of the book Foundations ofDifferential Geometry, vol. 1, John Wiley 1963, by Kobayashi andNomizu.
My effort was to give Mayer's math proof a physical interpretationas part of my physics model. I think that Mayer is now a professoremeritus of physics and math at U.C. Irvine.
The geometric volumes related to the physics groups that I use,along with some combinatoric rules, produce the force strengths andconstituent masses that are seen in physics. My two most recentpapers about that can be found at
As long as my physical representation of math objects are usedconsistently with the structure of the math objects,
that is all that is needed for physics model building.Conventional physicists often do the same thing. For example, AFAIKnobody has given any "math proof" justifying the use of path integralquantization of the Standard Model Lagrangian, but conventionalphysicists do use that all the time.
However, there is one part of my model where I am using a mathconjecture that I have not proven, which is that a lot of real 8-dimClifford algebras when put together form a generalization of theHyperfinite II1 von Neumann factor. See
I would like to prove that conjecture, or see somebody else proveit, but I have not yet had the time to do it.
As to whether there are "... any new prediction behind your theory...", the answer is yes. Recently I have put two papers on my website at
They describe my model of the Tquark - Higgs - Triviality system,and they predict that the LHC will see three states of the Higgsboson:
Since the LHC should be able to see any Higgs state below about250 GeV soon (within a couple of years or so) after it begins to getresults in 2008, my masses of three Higgs states are predictions thatshould be tested then. There have been some very inconclusive resultsfrom Fermilab that encourage me, which I have discussed at
In my model you have (in the fundamental first generation) 8fermion particles and 8 fermion antiparticles. In the binomialexpansion used in my 8-dim Clifford algebra you have
1 8 28 56 70 56 28 8 1
So, maybe only in the special dimensions of my model (8 reduced to4 of spacetime) can you formulate bosons as fermion-antifermionpairs.
As to how "... 8 of the  2-grade bivectors [of the8-dim real Clifford algebra] AFTER DIMENSIONAL REDUCTION TO 4-DPHYSICAL SPACETIME corrspond to the 8 generators of color for SU(3) ?...", the process is set out in some detail on my web site at
and from some other equivalent points of view at
Further, this text from e-mail messages I sent to Garrett Lisiback in September 2005 might be useful:
"...There are several levels at which you can look atthe Spin(8) group associated with Cl(8).(Here I am using Euclidean signature being sloppy for simplicity.)...The Lie algebra of Spin(8) is generated by the Weyl symmetriesof the root vector polytope of Spin(8), which is the 24-cellin 4-dim Euclidean space.So, you can describe Spin(8) in three ways:As a Lie group.As its Lie algebra.As its root vector polytope.What I agree cannot be done is to factor the Lie group Spin(8)into the Cartesian product of Lie groups U(4) x SU(3) x SU(2) x U(1)....However, what I contend CAN be done,is to do the decomposition at the level of the root vector polytope,...the way I decompose the 24-cell plus 4 Cartan dimensions for 28-dim Spin(8)into12-vertex cuboctahedron plus 4 Cartan dimensions for 16-dim U(4)and8-vertex cube for SU(3)andline with 4 vertices for U(2) = SU(2) x U(1)is an unambiguous procedure ...However,it requires thinking of the Gauge Groupsnot as Lie groups,and not even as Lie algebras generating them,butas the root vector polytopes that generate the Lie algebras.Such a way of thinking is clearly defined mathematically,as in texts that describe how to construct Lie algebras fromthe root vector polytopes. For example, see Chapter 21 (especiallysection 21.3) of the book Representation Theory by Fulton and Harris(Springer 1991) which does that using the Dynkin diagrams that areassociated with the root vector polytopes and therefore the Lie algebras....However,such a way of thinking is not common for mathematicians,much less for physicists.Even though it is not a common way of thinking, it does work,in the sense of producing a realistic physics model,andit works much better than the more well-known way of avoidingthe Coleman-Mandula theorem by using Lie superalgebras insteadof Lie algebras....In other words,my work began in the early 1980s, and was motivated by supergravity.I wanted to make a better and more radical departure fromLie group/algebra than the supergravity departure to Lie superalgebra.I read a paper by Saul-Paul Sirag about physics and the Weyl group,andI realized that my departure could be going down tothe Weyl group / root vector level, so I began to work on it,and (even though my then-advisor David Finkelstein was unenthusiasticabout it) my work on the Weyl group / root vector stuff led me tomy present realistic model....AFTER you use the root vector process for the decomposition,THEN AND ONLY THEN do you construct Lie algebra/group structuresfrom the decomposed root vector polytopes,and then you have the conventionalMacDowell-Mansouri gravity from the U(2,2)andthe Standard Model from the SU(3)xSU(2)xU(1),each in their own sandbox(in the computer system sense of the term "sandbox") ...".
and whether "... there ... Is ... a way to associate to each oneof the 256 CA a matrix and stablish a kind isomorfism between Cl(8)and the 1-D CA ? ...".
Effectively, Wolfram does that in his book by assigning to eachsuch automaton a unique number between 0 and 255, which can bewritten in 8-digit binary form, because the resulting 256 binarynumbers can be assigned to grades in the 256-element Clifford algebraaccording to how many 1s they contain. If the number of 1s had beenused, the result would be the Hodge dual of Cl(8), which is alsoisomorphic to Cl(8).
As to whether I think that all "... the ... CA ... giving...[the same]... [graphic] results ... [should beseen as equivalent because they] ... give the same information...",
No, because even if they give the same graphic results, theirrules are really fundamentally different, which is why they areassigned different 8-digit binary numbers. However, having the samegraph does indicate that their physical properties may be similar.For example, 11000000 corresponds to the U(1) photon and 10100000corresponds to the SU(2) neutral weak boson, and they both have thesame graphic picture of just one point and then blank, which forconvenience I will call a "blank" graph, and the SU(2) neutral weakboson acts very much like a photon except that at the low energieswhere we do experiments the Higgs mechanism gives mass to the SU(2)neutral weak boson, so it acts physically somewhat like a "heavyphoton".
As to "... why ... [I] ... consider those 8 2-gradebivectors as the generators of color force SU(3) and not others ofthe 2-grade bivectors ? ...", I did not do a nice math proof (I thinkthat one could be done, but I have not taken the time to do it yet),but just intuitively looked at the pictures. The breakdown of the 28that I wanted was:
and the other 28 - 12 - 4 = 12 should correspond to
(You can see that my choices fit those criteria.)
Note that the 8 graphs for SU(3) look different from the other16+1+3 = 20 in that they have more "volume", which I think is relatedto the fact that in my physics model the SU(3) acts globally on theinternal symmetry space while the SU(2) and U(1) act internally onthe internal symmetry space and the U(2,2) of gravity acts internallyon the physical space.
Segrob Siul sent a nice list of problems, which I see as:
Here are some comments on them:
but I do not have any more formal "proof" of why that assignment should be chosen (other than the fact that it works in the sense that it produces realistic particle masses, force strengths, etc).
As to "... where to place in time the initial condition ... How[my] model reconciles with the Big Bang theory ? ...", asSegrob Siul says, in my model "... the Universe starts from the voidsby introducing a binary choice ...". Roughly, what happens is thatbefore any space or time or gauge bosons or fermions exist:
The12-step program above built our universe out of just one pointout of the huge El Aleph, so that El Aleph is really big and reallycomprehensive, like the Vedic unified Krishna.
PS - As to why I use Clifford algebra,it is because other math structures seem too limited.For instance:Lie algebras are included in the Clifford algebra bivectors,and so are not as comprehensive;Exterior (Grassmann) algebras don't have spinors;etc.Maybe that is not a formal justification, and I should justsay that after trial and error, the Clifford algebras work inthe sense that the physics model constructed with them givesthe "right answers" when compared with experimental results (whichI regard as the voice of g-d).PPS - You might also ask why I use Lagrangian structure,and I can only say that to me Lagrangians seem natural(there are physicists who disagree and don't like Lagrangians,but here, for once, I am on the side of the conventional physicists),andLagrangians are a very effective way to describe the physicsthat we see in experiments.
As to using the isomorphism between CAs and Cl(8) to construct an"... operation between CAs so that computer experiments can beperformed and reproduce the results from ...[my]... model(locally, because the isomorphism is with Cl(8) ) ...", that would benice, and it need not be only local because tensor products of Cl(8)make up larger-than-local neighborhoods and tensor products of the CAoperation should be definable and workable.
On a large scale, what I would like to see would be to use suchcomputer experiments to describe basins of attraction etc to see howour future (or possible futures) might look, by considering ourfutures to be described by quantum game theory played out among thepossibilities of Many-Worlds quantum theory. For a very muchoversimplified example see http://www.valdostamuseum.org/hamsmith/ManyFates.html#example
Such basins of attraction have been described for individual CAs(see for example the book "The Global Dynamics of Cellular Automata"by Wuensche and Lesser (Addison-Wesley 1992)) and it would be fun tosee that for lots of CAs acting as Clifford algebra elements.
On a smaller scale, maybe such an operation would allow computerCA experiments that would make difficult calculations (such as QCDSU(3) color force calculations) easier. As to whether that approachto QCD might work, when I look at 2D successes with respect to fluiddynamics (see the book "Lattice Gas Methods for Partial DifferentialEquations" ed. by Gary Doolen (Addison-Wesley 1990)) I getoptimistic, but when I note that Wolfram even as late as his New Kindof Science book seems to have made no substantial progress withrespect to QCD calculations, I get pessimistic. However, maybeWolfram never thought about combining CAs with Clifford algebras andusing the resulting operation.
As to "... whether Clifford algebra could be applied to cellularautomata in Planck scale geometry, whether e.g. each Planck volumecould be considered a discrete cell in a cellular automaton. ...",that seems to me to be related to the work of Paola Zizzi on theuniverse as a big quantum computer. In checking the web, I saw athttp://www.quantumbionet.org/eng/index.php?pagina=60a description of her current work, and an article The "Poetry of aLogical Truth" by her at http://www.quantumbionet.org/eng/index.php?pagina=84
As to Paula Zizzi saying "and corresponds to a superposed state of10^9 quantum registers", which is much smaller than the number oftubulins in the human brain, she and I discussed that back in 2000and we agreed, as she said, that "... As far as the number oftubulins is concerned ... the total number of them in our brain is10^18 ... the selected quantum register, which is the n=10^9,contains 10^18 qubits...".
Here is why I have been using 10^18 tubulins per human brain: Thehuman brain contains about 10^11 neurons; and there are 10^7 TubulinDimers per neuron. As references, see the Osaka paper QUANTUMCOMPUTING IN MICROTUBULES - AN INTRA-NEURAL CORRELATE OFCONSCIOUSNESS? by Stuart Hameroff in which he mentions: "... thehuman brain's 10^11 neurons ..." and the Orch OR paperhttp://www.quantumconsciousness.org/penrose-hameroff/orchOR.htmlOrchestrated Objective Reduction of Quantum Coherence in BrainMicrotubules: The "Orch OR" Model for Consciousness by StuartHameroff and Roger Penrose in which they say: "... Yu and Baas (1994)measured about 10^7 tubulins per neuron. ...". Their Yu and Baas(1994) reference is "... Yu, W., and Baas, P.W. (1994) Changes inmicrotubule number and length during axon differentiation. J.Neuroscience 14(5):2818-2829....".
The Osaka paper was on the web some years ago, and I downloadedits text back then, and my quote from it is from that text download.I don't know exactly where to find it on the web nowadays. Areference for the number of neurons that is on the web now isPHYSICAL REVIEW E, VOLUME 65, 061901(Received 2 May 2000; revisedmanuscript received 7 August 2001; published 10 June 2002) Quantumcomputation in brain microtubules: Decoherence and biologicalfeasibility by S. Hagan, S. R. Hameroff2 and J. A. Tuszynski athttp://www.quantumconsciousness.org/pdfs/decoherence.pdfwhere they mention "... about 10^8 neurons approximately 0.1%-1% ofthe entire brain) ...".
The Samsonovich-Hameroff et al ideas about patterns of excitedtubulins in a microtubule do remind me of the cellular automatapatterns that may be isomorphic to the 256 elements of the Cl(8)Clifford algebra. Where my ideas may differ from Samsonovich-Hameroffet al may be in their ACT (acusto-conformational transformation)mechanism by which MTs communicate with each other. My idea is thatthe communication is by resonant gravity connection. It is based onPenrose's use of gravity as an Orch OR mechanism and on generalizingto gravitational gravitons Carver Mead's description of resonancecausing atomic emission of electromagnetic photons - see http://www.valdostamuseum.org/hamsmith/QuanConResonance.html#resonance
Samsonovich-Hameroff et al describe two ways for ACT to work:
I think that 1 is too slow in propagating throughout the brain. 2is basically the picture that I have, but I use resonant gravity asthe basic underlying force connecting all the MTs, and although theydo use Penrose's idea of gravity for Orch OR collapse, I think thatthey may not use gravity to maintain coherent superpositions of MTsthroughout the brain.
As to why I think that 1 - "... two neighboring coupled MTs (ortwo parts of the same MT) ..." may be too slow to be a way for ACT tocommunicate among MTs to maintain superposition coherence throughoutthe brain, Samsonovich-Hameroff et al say about that mechanism: "...global coherent oscillations are initiated spontaneously by thermalfluctuations and amplified by energy release due to conformationalmotions stimulated by these oscillations ...". I think that thethermal fluctuations have a time-scale, and their generation ofoscillations introduces another time-delay, and the conformationalmotions introduce yet another time-delay, and all those time factorsare much slower than graviton speed-of-light connections, and arewhen added up slower than the ambient thermal fluctuations that canmake a superposition decohere in the way used by Tegmark in hiscriticism of quantum consciousness, so it seems to me that mechanism1 is doomed to failure as a means of maintaining superpositioncoherence.
In my picture, each tubulin emits and absorbs gravitons (at speedof light) from every other tubulin involved in the superposition, sothe gravitons keep all the tubulins in step to be in coherentsuperposition by exchanging gravitons much faster than the time-scaleof the thermal fluctuations that Tegmark tries to use fordecoherence. Since the tubulins interact much faster than the thermalfluctuations, they can easily evade any decohering effects related tothe thermal fluctuations.
An analogy occurred to me. Consider the USA stealth aircraft F117.Before its development, fixed-wing aircraft were generallyaerodynamically stable in that, left alone, they tended to continueon a predictable path. However, the F117 was aerodynamically unstablein that, left alone, small fluctuations of turbulence woulddestabilize the aircraft, and no human pilot could react fast enoughto correct those instabilities, so with only a human pilot it wouldnot continue on a predictable path (and would likely crash). Due toits instabilities, it was known to its test pilots as the "Wobblin'Goblin". Only with the development of automated computer controlsystems that reacted much faster than the time scale of turbulentfluctuations could such an aircraft be useful to an air force. Sincesuch reaction time was far faster than any human reaction time, sometest pilots quit because there was no way they themselves could be in"control" of the aircraft. At the risk of belaboring the obvious:
turbulent fluctuations that = thermal fluctuations that Tegmark used todestabilize the F117 argue for decoherenceslow human reflexes (slower = slow processes (not much faster thanthan the turbulent fluctuations) thermal fluctuations) fail to stopfail to stabilize the F117 Tegmark-type decoherencefast computer control = fast processes (carried by speed-of-lightsystems (much faster than gravitons) allow maintenance ofthe turbulent fluctuations) coherence of MT state superpositionscan and do stabilize the F117
In one of his papers known as a "water paper" (I downloaded ityears ago, but the link I used then seems to be invalid now) StuartHameroff says: "... Herbert Frohlich, an early contributor to theunderstanding of superconductivity, also predicted quantum coherencein living cells (based on earlier work by Oliver Penrose and LarsOnsager ... Frohlich ... theorized that sets of protein dipoles in acommon electromagnetic field (e.g. proteins within a polarizedmembrane, subunits within an electret polymer like microtubules)undergo coherent conformational excitations if energy is supplied.Frohlich postulated that biochemical and thermal energy from thesurrounding "heat bath" provides such energy. Cooperative, organizedprocesses leading to coherent excitations emerged, according toFrohlich, because of structural coherenceof hydrophobic dipoles in acommon voltage gradient. ...".
That would be using electromagnetic photon processes to maintainthe coherence, although they may use gravity for Orch OR decoherence.I prefer to use gravity for both things.
As to the ideas described above as Frohlich's, I see a problemwith his source of energy for a driven non-equlibrium system: How canthe Frohlich energy source (surrounding heat bath) produce acoherence that is stable against a decohering influence (the samesurrounding heat bath) that is just as strong as the energy source ?Note that this situation is VERY different from the sun and plantlife which was mentioned by Penrose in Emperor's New Mind. At page320, Penrose says "... All this is made possible by the fact that thesun is a hot-spot in the sky! The sky is in a state of temperatureimbalance: one small region of it, namely that occupied by the sun,is at a very much higher temperature than the rest. ... The earthgets energy from that hot-spot in a low entropy form ... and itre-radiates to the cold regions in a high-entropy form ...". Unlikethe sun in the sky, Frohlich's surrounding heat bath source is at theSAME temperature as the rest of the brain (sky). For the brain towork like the sun and sky, you will have to find a part of the brain(sun) that is a lot hotter than most of the brain (sky). That shouldbe easy to find experimentally, and in fact I seriously doubt that itexists, because it should be so easy to find that it would havealready been found if it existed. So, I don't like the Frohlichelectromagnetic coherence mechanism for maintaining brain-widecoherent states of MTs. However, I should day that electromagneticprocesses are useful and may play some other roles in brainfunction.
I found some work of Nanopoulos et al (as to whether it isoriginal or application of ideas of others, I do not know) to beinteresting, particularly application of "... error- correctingmathematical code known as the K2(13, 2^6, 5) code. ..." to MTinformation.
For maintaining a coherent superposition it is indeed 'the more,the better' because if they are linked together (as in Carver Mead'sbook Collective Electrodynamics) anything trying to decohere thesuperposition must be strong enough to do all of them, not just oneof them. Philip Anderson calls that collective phenomenon a QuantumProtectorate - see http://www.valdostamuseum.org/hamsmith/QuanCon2.html#quantumprotectorate
On the other hand, Penrose Orch OR collapse-decoherence is basedon the time-energy uncertainty principle h = T E which gives a time T= h / E at which consciousness-collapse-decoherence takes place. Thedetails of the actual calculation in my model are too long for e-mailbut can be found at http://www.valdostamuseum.org/hamsmith/QuanCon.html#orchorand all the material on that page. Roughly, the resulting decoherencetime T_N for N tubulins is T_N = N^(-5/3) 10^26 sec For example, for4 x 10^15 tubulins (far from all 10^18 in the brain, but still alarge number) the time is about 100 milliseconds which is roughly theEEG alpha frequency and for 10^17 tubulins the time is about half amillisecond.
So, the more tubulins you have the more protected they are by theQuantum Protectorate, but the faster they collapse by Penrose Orch ORso the less time you have think your thought.
There is in my model a third relevant process, which is collapsedue to the quantum fluctuations of the universe at large which I callGRW decoherence. My view of GRW itself is described at http://www.valdostamuseum.org/hamsmith/GRW.htmland the relationship between GRW and Orch OR decoherence is shown athttp://www.valdostamuseum.org/hamsmith/QuanCon.html#timegraphThe equation for GRW decoherence time is roughly T_N = ( 1 / N ) x 3x 10^14 sec which in the chart on the link immediately above iscompared with the Orch OR decoherence time of about T_N = ( 1 /N^(5/3)) x 10^26 sec
There is still a fourth relevant process that limits the size of asingle brain (i.e., limits the number of tubulins N), which is basedon Paola Zizzi's quantum decoherence model, which I like and soinclude in my model. It gives the upper limit of about N = 2^64 orroughly 10^19. For details see http://www.valdostamuseum.org/hamsmith/QuanCon3.html#quinflation
So, there are four relevant processes in my model:
For less than about 10^15 Tubulins or so, GRW decoheres thesuperposition BEFORE the Orch OR decoherence takes place, so you musthave AT LEAST 10^15 Tubulins in order to have consciousness based onPenrose-type Orch OR.
Given the human brain size limit of about 10^18 tubulins, thefastest that humans can think would be about 10^(-5) seconds.
As the Zizzi upper limit is about N = 10^19, the human brain hasevolved to be almost as smart as a single brain can get. How couldhumanity get smarter? Maybe by cooperating more and fighting less.Maybe by having multiple brains (dolphins have 2). Maybe as in themovie Matrix or the Star Trek Borg by being forced involuntarily tocooperate. Of course, there is always the possibility that humanitymight just stay the same, and be superseded by something else.
As to relevant experimental tests, some might be:
As to what I think of the Hameroff-Penrose article, I like partsof it (after all my model is based on some of their ideas) but I dodiffer a bit on some points and as to some calculated numbers. Forexample, Segrob Siul says that "They calculate 2 x 10^11 tubulins insuperposition will reach threshold in 25 msec (40 Hz)" while mycalculations (using 10^6 tubulins per neuron, or 10% of all tubulins)give:
Time Number of Number of T_N Tubulins Neurons 10^(-5) sec 10^18 10^12 5 x 10^(-4) sec (2 kHz) 10^17 10^11 25 x 10^(-3) sec (40 Hz) 10^16 10^10100 x 10^(-3) sec (EEG alpha) 4 x 10^15 4 x 10^9500 x 10^(-3) sec (Radin/Bierman) 1.5 x 10^15 1.5 x 10^9
I don't worry about numerical differences even of 1 or 2 orders ofmagnitude (factors of 10 or 100) because exact details of processesare not well known, and to me if a lot of calculations are all moreor less consistent in that range it indicates to me that the overallmodel is OK, and it is worth the effort to make more refined versionsof the model.
However, I have not done much more work in this area since 2002,because the first paper that the Cornell arXiv rejected when I wasblacklisted was a quantum consciousness paper, so in a futile attemptto get off the blacklist, I started writing some more clearly (to me)plain vanilla physics papers caculating things like neutrino massesand mixing angles and the conformal gravity model that explains thePioneer effect and the WMAP observations of the ratio Dark Energy :Dark Matter : Ordinary Matter. I even thought that if I wrote out mymodel in terms of string theory they might let me off the blacklist,but they did not (my string model did not have conventional 1-1supersymmetry and had a physical interpretation of strings asworld-lines of point particles, and they probably disliked that).Anyhow, I realized around 2004 that writing goodplain-vanilla-physics stuff would not help, but I have gotten offinto writing that kind of stuff (lately some Tquark and Higgs stuffthat might be seen at the LHC, and stuff about tapping into DarkEnergy with Josephson Junction arrays), and I have not yet gottenaround to doing more work on refining quantum consciousness models,constructing a generalized II1 Hyperfinite von Neumann factor, etc.There is not enough time for me, working by myself, to do all thesethings.
As to "... why both Penrose OR and GRW are needed. ... ? ", GRWand Zizzi are not needed in a bare-bones Penrose-Hameroff model toshow that human consciousness is based on quantum gravity oftubulins. I include them because I think in the context of my overallClifford algebra generalized II1 Hyperfinite von Neumann factorphysics model it is natural for them to exist, and so I should takethem into account. When I do take them into account:
wish I had time to work all such stuff out in detail, but probablythe best I can do is a little bit about a few things in the shorttime that probably remains of my lifetime.
My model is based on Penrose-Hameroff because of two original andvery useful ideas of theirs: coding the information using tubulins inmicrotubules as 2-state quantum systems; and using quantum gravityfor Orch OR decoherence. I like and use some of their related ideas,such as information theory codes (I would use quantum informationtheory) related to patterns of tubulin states in the cylindricalmicrotubules. I also like the things that I added to their model:
So, the things that I like about my model that Penrose-Hameroffdoes not have are the graviton quantum protectorate and lower andupper bounds for Orch OR brain size, with the human brain fitting inOK.
The things I added do not contradict basic Penrose-Hameroff, theyjust add more stuff to it that seems to work.
Penrose's book "The Emperor's New Mind" (ENM), is indeed veryinteresting. However, Penrose wrote ENM around 1990, years before thedevelopment of quantum computing, quantum information theory, andquantum game theory, which began to be developed around 1995. In hisolder books (Emperor's New Mind and Shadows of the Mind) did mentionquantum computing, but only on the level of early ideas ofpossibility-in-principle due to Deutsch and Feynman and he did notthen seem to take into account the major advances beginning around1995 such as Cerf and Adami quant-ph/9512022 etc. For example, inEmperor's New Mind, Penrose said about quantum computing "... So farthese results are a little disappointing ... but these are early daysyet. ...".
Papers like that of Cerf and Adami at quant-ph/9512022 show thatquantum information theory is very much like particle physics quantumtheory, and since my model has a Clifford algebra basis for particlephysics and for quantum consciousness, quantum computing seems to beto be effectively what Penrose is looking for (but had not beendeveloped when he wrote ENM).
Penrose's latest book, "The Road to Reality" (2006), does mentionquantum computers, saying that they "... would make use of thevastness of the sizes of the kinds of space of wavefunctions ..." andalso mentions quantum information theory, which he calls"quanglement", and about which he says "... the precise role ofquanglement in ... the circumstances under which R takes over from U... is not yet very clear, to my [Penrose's] mind. ... A morepromising connection is with some of the ideas of twistor theory, andthese will be examined briefly in section 33.2 ...". In section 33.2,Penrose discusses twistors, and says "... it turns out that theconformal group has an important place in twistor theory ... We shallsee more explicitly what the role of this group is in the next twosections ...". Those "next two sections" 33.3 and 33.4 say "... a15-dimensional symmetry group - the conformal group - ... is ...Apart from the two-to-one nature of the correspondence arising from... reversibility of the generator directions, O(2,4) ... Theshortest ... way to describe a ... twistor is to say that it is a ...half spinor ... for O(2,4) ...". Since my model gets gravity fromthat conformal group, it is fundamentally a twistor theory, and sincemy model is also fundamentally a Clifford algebra connected toquantum information theory, I think that my model is a concreteexample of what Penrose needs to complete his program.
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