Are "Other" Quantum Theories Really Different fromMany-Worlds?

What are the relationships among:Feynman Checkerboards; Ising Models;Cellular Automata, and Wei Qi?

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is the maximal symmetry group of both **Maxwell's Equations**and the **Hydrogen Atom**,

as well as, as shown in hep-th/9907009by Liu, Ma, and Hou, the **canonical Dirac Lagrangian formassive fermions**.

**The ****Conformal****Group Spin(4,2) = SU(2,2) is used in the ****D4-D5-E6physics model**** to describe ****Gravityand the Higgs Mechanism****, and also shows relationshipsbetween Special Relativity and Quantum Theory. **

**Barut and Raczka**, in their book Theory of GroupRepresentations and Applications (World 1986), describe the 15 Liealgebra basis elements of the Conformal Group Spin(4,2) = SU(2,2),which are 3 Spatial Rotations, 3 Lorentz Boosts, 4 SpacetimeTranslations, 1 Scale Transformation, and 4 Special ConformalTransformations.

In Chapter 13 (particularly section 13.4) Barut and Raczka showthat wave equations for massive particles are formally invariantunder the Conformal Group if the mass is transformed by the 1 ScaleTransformation and the 4 Special Conformal Transformations of theConformal Group.

Barut and Raczka also show in Chapter 13 that in theNon-Relativistic limit the generators of the 15-dimensional ConformalGroup contain the 10-dimensional projective Galilei Group, and thatthe Schrodinger Operator is invariant under a 12-dimensionalSchrodinger Lie Algebra that is made up of the 10-dimensional GalileiLie Algebra plus 1 modified Scale Transformation and 1 modifiedSpecial Conformal Transformation. Then the 12 Schrodinger Lie Algebragenerators are represented in momentum space by

- Ho = (1 / 2m)
**p**^2 **P**=**p****J**=**q**x**p****M**= - t**p**+ m**q**- D = (t / m)
**p**^2 + (1 / 2) (**p**.**q**+**q**.**p**) - Ko = - (t^2 / 2m)
**p**^2 + (1 / 2) t (**p**.**q**+**q**.**p**) - (m / 2)**q**^2

and the 12-dimensional Schrodinger Group contains as a subgroupthe 3-dimensional group SU(1,1) = Spin(2,1) = SL(2,R) generated byHo, D, and Ko. In particular, the Lie Algebra SU(1,1) solves the3-dimensional quantum oscillator with Hamiltonian H = (p^2 / 2m) + Lq^2 and also solves the free particle with Hamiltonian H = p^2 /2m.

Note that SU(1,1) is a non-compact version of the Weak Force LieAlgebra SU(2) and that, if the remaining 9 generators from **P**,**J**, and **M** are identified with a U(3) Lie Algebra, thenthe Schrodinger Lie Algebra is a version of SU(2)xU(3) =SU(2)xU(1)xSU(3) of the Standard Model.

In Chapter 21 (particularly section 21.3.D) Barut and Raczkaconsider the Relativistic case in which the 15-dimensional ConformalGroup Spin(4,2) is considered as an extension of the 6-dimensionalLorentz Group Spin(3,1) = SL(2,C). In this case, the ScaleTransformation plus a Special Conformal Transformation plus TimeTranslation make up a 3-dimensional subgroup of the Conformal Group,that is, the 3-dimensional group SU(1,1) = Spin(2,1) = SL(2,R). Theythen show that one representation of this SU(1,1) subgroup gives theKlein-Gordon equation, and another representation gives the secondorder Dirac equation for the Coulomb potential.

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**Paul J. Freitas** uses 4-Dim Euclidean Spatial Space toderive Special Relativity from Quantum Theory in his paper Connectionsbetween Special Relativity, Charge Conservation, and QuantumMechanics dated 25 March 1998. Freitas describes particles asexisting in a 4-dimensional Euclidean Spatial Space, as opposed to a3-dimensional Euclidean Spatial Space in 4-dimensional Spacetime. Ashe says, "... It is possible to add one dimension to our threedimensions of space in such a way that we can treat simple objectswith and without rest mass exactly the same way. This new four-spacehas the nice property of being Euclidean, and yields all of the usualrelativistic properties through a few simple, familiar postulates.... simple particles of matter can be described as existing in aEuclidean four-space as described above, with the proper time of theparticle being related to the fourth, non-obvious position component.In such a space, all simple forms of matter are constantly moving atthe speed of light. ... [distance in the fourth spatial dimensionis] what is commonly referred to as the spacetime interval[dw^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2] ... By considering afew well-understood physical processes, we shall see that themomentum of a particle in the fourth direction may correspond to itscharge, which means that charge conservation is just a form ofmomentum conservation. ... In fact, we can start withfour-dimensional quantum mechanics and derive the results of specialrelativity very naturally, with even fewer assumptions. ...".

The 4-dimensional Euclidean Space of Freitas seems to be relatedto the 4-dimensional Euclidean subspace of the vector space of theConformal Group Spin(4,2) = SU(2,2) that is used by Barut and Raczkato describe Quantum Theory and Spacetime Symmetry, and by me in theD4-D5-E6 physics model to describeGravity and the Higgs Mechanism.

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**The ****D4-D5-E6 model****coset spaces ****E6**** /(****D5**** x U(1)) and****D5**** / (****D4****x U(1)) are Conformal Spaces. ** You can continue the chain toD4 / (D3x U(1)) where D3 is the 15-dimensional Conformal Group whose compactversion is Spin(6), and to D3 /(D2 x U(1)) where D2 is the6-dimensional Lorentz Group whose compact version is Spin(4).Electromagnetism, Gravity, and theZPF all have in common the symmetry of the 15-dimensional D3Conformal Group whose compact version is Spin(6), as can be seen bythe following structures with D3 Conformal Group symmetry:

- Maxwell's equations of Electromagnetism
- Gravity derived from the Conformal Group using the MacDowell-Mansouri mechanism
- the Quantum Theoretical Hydrogen atom
- the Lie Sphere geometry of SpaceTime Correlations in the Many-Worlds picture

Further, the 12-dimensional Standard Model Lie AlgebraU(1)xSU(2)xSU(3) may be related to the D3 Conformal Group Lie Algebrain the same way that the 12-dimensionalSchrodinger Lie Algebra is related to the D3 Conformal Group LieAlgebra.

Another probabilistic approach to Quantum Theory is

in which the Schrodinger equation is derived from the variational principle using Markovian gaussian diffusion processes acting through a Background Field. The Background Field, according to Robert Bass, has its source in the Coulomb forces from all the leptons and quarks of our universe. Today, most of them are electrons and protons of hydrogen atoms. The Background Field can be represented from the Newton-Coulomb differential equation for a hydrogen atom, with the RHS being, not zero, but hW where W = N+ - N- is a White Noise source of zero mean and non-zero variance whose power spectrum is Lorentz invariant frequency-cubed with a Planck energy cut-off, and h is Planck's constant.

Although only for light-cone (massless) particles is thefrequency-cubed power spectrum Lorentz invariant, it is possible todescribe realistic physics, including even massive particles, usingonly light-cone particles. Background fields, especially, can be sodescribed. Penrose and Rindler (Spinors and Space-time, vol. 1,section 5.12, Cambridge 1984, reprinted with corrections1986) state:"... in classical electrodynamics when applying the Lorentz force toa point charge [s]ome concept of background field is required... since the full field diverges ... at the charge itself ...[T]he advanced and retarded background fields (where they aredefined) are both automatically massless free fields. ... We maythink of the total interacting field as composed of pieces in whichfields propagate for a while along null straight lines as masslessfree fields, but scatter repeatedly at points in this interiorregion. The novel feature that arises in our approach is thepropagation entirely along null lines between scatterings. ...[We briefly indicate] a corresponding treatment of theMaxwell-Dirac system. ... It would be interesting also to develop ourprocedure (with or without twistors) into a full description ofquantum electrodynamics. ... The diagrams arising here are in manyways analogous to Feynman diagrams. But there is the unusual featurethat here we are concerned only with null space-time separations,even for massive fields. ..." IwoBialynicki-Birula used a similar approach to formulate Dirac andWeyl equations on a cubic lattice as Quantum Cellular Automata. HisQuantum Cellular Automaton isfundmentally a Feynman Checkerboard,in which mass is the amplitude tochange light-cone direction, and motion of all particles is asequence of light-cone motions.

The mean position of the electron in the atom is Newton+Coulomb, but hW causes the elecron to follow a Zitterbewegung path described by probability distribution RHO, which can be factored RHO = PSI* PSI where PSI is a complex function that must evolve by Schroedinger's Equation.Puthofff has shown that an electron in an atom emits radiation due to its orbital motion, but that the energy loss by radiation is matched by the energy it absorbs from the Background Field if and only if the electron path follows a Bohr orbit. The Background Field contains dipole radiation of every frequency and every direction, so that there are plane waves simultaneously touching each and every particle in the universe. Each particle does not know what the other is doing, but, as Nelson says, the Background Field knows. The diffusion processes of Nelson may be related to the diffusion calculations of fundmental physical constants, such as force strengths, in the theory under development by Michael Gibbs which in turn is related to the D4-D5-E6 physics model.

As Nelson says in his book Quantum Fluctuations (Princeton 1985,pp. 101-102), his ConservativeMarkovian Diffusions on the configuration space M of unorderedN-tuples of spatial coordinates in R3 for N indistinguishableparticles "... fall into two sharply different classes according tothe statistics obeyed. ...". "... In the first case (Bose-Einsteinstatistics) it is a symmetric function ... and in the second case(Fermi-Dirac statistics) it is an antisymmetric function ... asuperposition of a symmetric and an antisymmetric wave function doesnot define a diffusion on M. ..."

The Background Field may be related to the Zero Point Fluctuation - Gravity idea of Sakharov.and the PSI-field of Bohm, Philip Pearle, in his review of Nelson's book Quantum Fluctuations (Princeton 1985), says (Nature (6 Feb 1986)): "...Nelson shows that, in this theory, ... [there is] ...non-local behaviour, and ... writes: '... a theory that violates locality is untenable'. ... It is interesting that upon encountering this difficulty, David Bohm, the author of a similar theory (differing from Nelson's in that the equation of motion is deterministic), has embraced it, delighting in the principle that everything is connected to everything else."

Lee Smolin, in his paper Stochastic Mechanics and HiddenVariables, printed in the book Quantum Concepts in Space and Time(Penrose and Isham, eds., Oxford 1986) describes Nelson's derivationof quantum mechanics as a Brownian motion process, discussing thewave function

in which "... Schrodinger's equation ... decomposes into aconservation equation with the current velocity defined as [notethat v = (1/2)(b + b*), where b and b* are motions for forward andbackward time steps, is distinct from the osmotic velocity u =(1/2)(b - b*)]

and the dynamical equation [which] ... has the form of aHamilton-Jacobi equation for the motion of a particle in a potentialV plus an additional term

... In stochastic mechanics, the term Vquantum is derived ... fromthe general theory of Brownian motion ... by specifying that theBrownian motion processes satisfy three additional conditions ...

- (1) The current velocity is irrotational. Thus, there exists a function S(x,t) such that m v = div S.
- (2) In spite of the fact that the particle is subject to random alterations in its motion there exists a conserved energy, defined ... as E = INT d^3x rho((x,t) [ (1/2) m v^2 + (1/2) m u^2 + V(x).
- (3) The diffusion constant VC is inversely proportional to the inertial mass of the particle, with the constant of proportionality being a universal constant hbar: VC = hbar / m.

... an ensemble of Brownian processes which are so delicatelycorrelated that an exactly conserved energy of the form (2) may bedefined in terms of their probability distribution [and whichobey the other condidtions] will behave as if each member of theensemble is coupled to the probability distribution of the wholeensemble ...".

In some respects, Dirac anticipated some of the fundamental ideasof Nelson's Stochastic Quantum Theory. In 1951-1954, Dirac advocatedthe reality and utility of the aether, as shown in this quote frompages 202-203 of Dirac: A Scientific Biography, by Helge Kragh(Cambridge 1990): "... "Let us imagine **the aether to be in a statefor which all values of the velocity of any bit of aether, less thanthe velocity of light, are equally probable**. ... In this way theexistence of an aether can be brought into complete harmony with theprinciple of relativity." Dirac identified the ether velocity withthe stream velocity of his classical electron theory ... it was thevelocity with which small charges would flow if they were introduced.... in the spring of 1953, Dirac proposed that absolute time bereconsidered. ... The ether, absolute simultaneity, and absolute time"... can be incorporated into a Lorentz invariant theory with thehelp of quantum mechanics ..." ... he was unable to work out asatisfactory quantum theory with absolute time and had to restcontent with the conclusion that "one can try to build up a moreelaborate theory with absolute time involving electron spins ...".Recall that **Nelson's non-local stochastic quantum mechanics**(which I think can be formulated consistently with Bohm theory)**involves** (see the paper by Smolin in the book Quantum Conceptsin Space and Time (Penrose and Isham, eds), at page 156) **adiffusion constant that "... is inversely proportional to theinertial mass of the particle, with the constant of proportionalitybeing a universal constant hbar: v = hbar / m** ...". Compare thiswith **Dirac's 1951 suggestion** that the electromagnetism U(1)gauge-fixing condition should be A A = k^2 where (see page 199 inKragh's book I am omitting some sub and superscript mus and nus):"... In order to get agreement with the Lorentz equation, theconstant k was indentified with m/e The four-velocity v of a streamof electrons ws found to be related to A by v = (1/k) A ..." **whichgives for Dirac's theory v = e / m**.

Carlos Rodriguezin section 8 of physics/9808010describes a generalization of the Markov property that permitsderivation of the Schrodinger equation for CliffordAlgebra valued conditional measures, such as might be used toconstruct the D4-D5-E6 physicsmodel.

With respect to the Schrodinger equation, Rodriguez citesquant-ph/9804012by Ariel Caticha, whose abstract says: "Quantum theory is formulatedas the only consistent way to manipulate probability amplitudes. Thecrucial ingredient is a consistency constraint: if there are twodifferent ways to compute an amplitude the two answers must agree.This constraint is expressed in the form of functional equations thesolution of which leads to the usual sum and product rules foramplitudes. A consequence is that the Schrodinger equation must belinear: non-linear variants of quantum mechanics are inconsistent.The physical interpretation of the theory is given in terms of asingle natural rule. This rule, which does not itself involveprobabilities, is used to obtain a proof of Born's statisticalpostulate. Thus, consistency leads to indeterminism." Caticha alsosays that "... the fact that Born's postulate is actually a theoremwas independently discovered long ago by Gleason, by Finkelstein,by Hartle and by Graham. ...".

GuidoBacciagaluppi in quant-ph/9811040 says: "In de Broglie and Bohm'spilot-wave theory, as is well known, it is possible to consideralternative particle dynamics while still preserving the | PSI |^2distribution. I present the analogous result for Nelson's stochastictheory, thus characterising the most general diffusion processes thatpreserve the quantum equilibrium distribution, and discuss theanalogy with the construction of the dynamics for Bell's beabletheories. I briefly comment on the problem of convergence to | PSI|^2 and on possible experimental constraints on the alternativedynamics."

In his book** ****QuantumTheory as an Emergent Phenomenon** (Cambridge 2004), **Stephen L.Adler** says: "...

The exposition of the text is based on dynamical variables thatare matrices in complex Hilbert space, but many of the ideas carryover to a statistical dynamics of matrix models in real orquaternionic Hilbert space ...

our idea is to start from a classical dynamics in which thedynamical variables are non-commutative matrices or operators. (Wewill use the terms matrix and operator interchangeably throughoutthis book, and do not commit ourselves as to whether they are finiteNxN dimensional, or infinite dimensional as obtained in the limit N--> oo.) Despite the non-commutativity, a sensible Lagrangian andHamiltonian dynamics is obtained by forming the Lagrangian andHamiltonian as traces of polynomials in the dynamical variables, andrepeatedly using cyclic permutation under the trace ... We furtherassume that the Lagrangian and Hamiltonian are constructed withoutuse of non-dynamical matrix coefficients, so that there is aninvariance under simultaneous, identical unitary transformations ofall the dynamical variables, that is, there is a global unitaryinvariance. We assume that the complicated dynamical equationsresulting from this system rapidly reach statistical equilibrium, andthen show that with suitable approximations, the statisticalthermodynamics of the canonical ensemble for this system takes theform of quantum field theory. ... The requirements for the underlyingtrace dynamics to yield quantum theory at the level of thermodynamicsare stringent, and include both the generation of a mass hierarchyand the existence of boson-fermion balance. ... From the equilibriumstatistical mechanics of trace dynamics, the rules of quantummechanics emerge as an approximate thermodynamic description of thebehavior of low energy phenomena. "Low energy" here means smallrelative to the natural energy scale implicit in the canonicalensemble for trace dynamics, which we identify with the Planck scale,and by "equilibrium" we mean local equilibrium, permitting spatialvariations associated with dynamics on the low energy scale. Brownianmotion corrections to the thermodynamics of trace dynamics then leadto fluctuation corrections to quantum mechanics which take the formof stochastic modifications of the Schrodinger equation, that canaccount in a mathematically precise way for state vector reductionwith Born rule probabilities. ...

there is a conserved operator with the dimensions of action, whichwe call Cbar, which is equal to the sum of bosonic commutators[p,q] minus the corresponding sum of fermionicanticommutators {p,q}, and which is the conserved matrix-valuedNoether charge corresponding to the assumed global unitary invariance...

we treat the classical dynamics of matrix models as fundamental... To introduce statistical methods, we first define a naturalmeasure for matrix phase space, and show that this measure obeys ageneralized Liouville theorem. ... the generic conserved quantitiesH, N, and Cbar appear multiplied by Lagrange multipliers thatrepresent generalized "temperatures." ... we specialized the ensembleto one that has maximal symmetry consistent with the ensemble average<Cbar>AV being non-zero, which we show implies that<Cbar>AV can be written as ieff hbar ... The matrix ieff willplay the role of i ... hbar will play the role of reduced Planck'sconstant ...

Smolin considers classical matrix models, with an explicitstochastic stochastic noise along the lines of that used by Nelson... giving rise to the quantum behavior ... elements of theirapproaches that will ultimately be seen to share common ground withours ...

.. the Copenhagen interpretation ... state[s] by fiat thatthe unitary state vector evolution of quantum mechanics does notapply to measurement situations. One then adds to the unitaryevolution postulate a second postulate, that of state vectorreduction, which states that after a measurement one sees a unitnormalized state corresponding to the measurement outcome ... with aprobability given by the Born rule ...

In the "many-worlds" interpretation introduced by Everett ...there is no state vector reduction, but only Schrodinger evolution ofthe entire universe. ... to describe N successive quantummeasurements requires consideration of an N-fold tensor product wavefunction. The mathematical framework can be enlarged to create asample space by considering the space of all possible such tensorproducts, and defining a suitable measure on this space. ... Thisprocedure ... is the basis for arguments obtaining the Born rule asthe probability for the occurrence of a particular outcome, that is,the probability of finding oneself on a particular branch of theuniversal wave function. ...

In Bohmian mechanics ... in addition to the Schrodinger equation... one enlarges the mathematical framework by introducing hidden"particles" moving in configuration space ... If the probability inconfiguration space is assumed to obey the Born rule ... at someinitial time, the Bohmian equations then imply that this continues tobe true at all subsequent times. Arguments have been given that theBohmian initial time probability postulate follows fromconsiderations of "typicality" of initial configurations. ...

Trace dynamics: the classical Lagrangian and Hamiltoniandynamics of matrix models ... The fundamental idea is to set up ananalog of classical dynamics in which the phase space variables arenon-commutative, and the basic tool that allows one to accomplishthis is cyclic invariance under a trace. ... Quantum mechanicalbehavior will be seen to emerge only when ... we study thestatistical mechanics of the classical matrix dynamics formulatedhere. ...

... In general ... matrix dynamics ... is not unitary ... Thre is,however, a special case ... in which the trace dynamics and theunitary Heisenberg picture evolutions coincide. ... consider ...Weyl-ordered Hamiltonians, in which the bosonic operators are alltotally symmetrized with respect to one another and to the fermionicoperators, and in which the fermionic operators are totallyantisymmetrized with respect to one another. ...

The matrix model for M theory ... theta is a 16-componentfermionic spinor ... with the transpose T ... so that thetaT issimply the 16-component row spinor corresponding to the 16-componentcolumn spinor theta ... the gammai are a set of nine 16x16 matrices,which are related to ... the Dirac matrices of spin(8)...

... the combined effect of a decoupling of the effective,ensemble averaged, dynamics from the non-covariant ... term in thecanonical ensemble, and of the equipartition of Cbar, is theemergence of relativistic quantum field theory as the low energyeffective approximation to a relativistic trace dynamics ... theequipartition theorem can be viewed as a Ward identity application inclassical statistical mechanics ...

... the emergence of quantum field dynamics from trace dynamicsevades the Kochen-Specker theorem and Bell inequality argumentsagainst a "hidden variable" completion of quantum mechanics ...

... our Ward identity derivation ... contains a source ofviolation of local causality, which is the way the operator Cbarenters ... Since ... the boson and fermion contributions to Cbarlargely cancel ... it can have large fluctuations over the operatorphase space. Correspondingly, in any finite subsystem of the universedescribed by the canonical ensemble, Cbar has large fluctuations overthe ensemble and hence as a function of time. These fluctuations giverise to corrections to the emergent quantum mechanics that we derivedby replacing Cbar by its ensemble average. ... these fluctuations donot affect the unit normalization of states, but add stochastic termsto the effective Schrodinger equation that describes the timedevelopment of a state. In order to preserve state normalization,this Schrodinger equation in the generic case must be nonlinear inthe state, which introduces violation of local causality, sincechanges in the wave function at one spatial point are instantaneouslycommunicated, via the noise terms, to all spatial points. ... thefluctuations in Cbar .. play the role of a Brownian "noise" whichdrives state vector reduction, in such a way as to be preciselyconsistent with Born rule probabilities. ... the average over thenoise of the density matrix obeys a linear evolution equation ...

Brownian motion corrections to emergent quantum mechanics canprovide the mechanism responsible both for the reduction of the statevector, and for the emergence of the Born and ... (in the case ofdegeneracies) ... Luders probability rules. ... with suitableassumptions, one can derive the standard stochastic Schrodingerequation for objective state vector reduction. Depending on thedetails of the model, the stochastic driving terms in this equationcan couple to the total energy, to a local density ... , or to both... when the stochastic drivign terms involve a set of mutuallycommuting operators, this equation leads to state vector reductionwith Born rule probabilities. ...

there is a plausible route leading from theunderlying trace dynamics to CSL ... continuous spontaneouslocalization ... reduction with mass proportional couplings ...which is the phenomenonlogically favored form of the CSL model ...although the CSL literature often assumes a Gaussian form for thecorrelation function ... no particular choice of functional form isneeded ... the results are independent of the value of thecorrelation length rC, provided that ... rC ... lies betweenmicroscopic and macroscopic dimensions. The value rC = 10^(-5) cm istypically assumed in the CSL literature. ... Ghirardi, Pearle, andRimini ... assume a correlation length rC = 10^(-5) cm, and proposethe value ... [of the] stochasticity parameter ... gamma =10^(-30) cm^3 s^(-1) GeV^(-2) ... any instrument pointer displacementinvolving at least 10^13 nucleons gives a reduction time ... 10^7s^(-1) .... [which is] less than typical experimentalmeasurement times. ...

the underlying dynamics is not unitary, with the unitary dynamicsof quantum field theory emerging ... as a thermodynamicapproximation; this suggests an amelioration, in the underlyingdynamics, of the infinities of quantum field theory, provides a basisfor understanding the nonlocal "paradoxes" of quantum theory, and may... play a role in establishing the large-scale uniformity of theuniverse. ...

our framework leads to not one but two copies of quantum fieldtheory, corresponding to the eigenvalues +/-i of ieff; we have notattempted to assign a physical role to the second copy, nor to theadditional "off-diagonal" degrees of freedom corresponding to theparts of the underlying matrices that anticommute with ieff. ... inreal Hilbert space ... the second copy of quantum field theory isabsent. The reason is that in real Hilbert space ieff cannot bediagonalized, and so itself plays the role of the imaginary unit ofthe emergent complex quantum theory.

We stress that we have not identified a candidate for the specificmatrix model that realizes our assumptions ... there may be only one,which could then provide the underlying unified theory of physicalphenomena that is the goal of current researches in high-energyphysics and cosmology. ...

It is possible that the underlying dynamics may be discrete, andthis could naturally be implemented within our framework of basing anunderlying dynamics on trace class matrices. ...

the ideas of this book suggest, one should seek a common originfor both gravitation and quantum field theory at the deeper level ofphysical phenomena from which quantum field theory emerges. ...".

Gravity from Vacuum Zero PointFluctuation was a conjecture of **Sakharov** in the 1960s.According to Misner, Thorne, and Wheeler (Gravitation, Freeman 1973),Sakharov noted that, since in Flat Spacetime, the total density ofthe Zero Point Fluctuation (ZPF) ofthe Vacuum (in the form of virtual particles andparticle-antiparticle pairs) is zero.

In the D4-D5-E6 physics model, the zero value is not only due torenormalization (the argument of Sakharov) but may be due toits ultraviolet finiteness asa unified theory of Gravity and the Standard Model.

Sakharov roughly estimates Standard Model part of the Zero PointFluctuation energy density as

where k is the wave number of the ZPF and INT denotesintegral.

Then, Sakharov looked at the Lagrangian L(0) for the StandardModel particles and fields (expressed in terms of wave number k) inflat spacetime and its value L(R) in spacetime with Gravity expressedas curvature R

where A and B are coefficients, of order unity, in power seriesexpansion in R.

The first term is just the Standard Model ZPF part of theLagrangian L(R), so that if you ignore terms of order 2 or higher,the second term should correspond to the Gravitational part of L(R),and in fact the second part is of the same form as the gravitationalpart of the Hilbert action Lagrangian

so that

and, since B is of order unity, the effective cutoff of the wavenumber k should be about

Interpreted in terms of the D4-D5-E6 physicsmodel, since the ZPF for Gravity + the Standard Model is zero due toultraviolet finiteness, the ZPF for the Standard Model is equal inmagnitude to the ZPF for Gravity, and the Einstein curvature tensor Gof Gravity can be written in terms of the stress-enregy tensor T ofthe particles and fields of the Standard Model and a cosmologicalconstant LAMBDA (which, despite the name "constant", can be avariable - Overduin and Cooperstock, in astro-ph/9805260,have described some other cosmological models with variablecosmological constant) as

Sakharov's conjecture in the 1960s was based only on in terms ofwave number k, and not in terms of physical particles and fields.When Puthoff attempted to work out Sakahrov's conjecture in terms ofElectromagnetism (in a formalism of stochastic electrodynamics butnot in full Quantum ElectroDynamics), difficulties wereencountered.

Subsequently, Haischand Ruedashowed in physics/9802030and physics/9802031that inertia can be due to interaction with the heat bath of the ZPF(which ZPF heat bath may be similar to theBackground Field of Nelson). They noted that inertial mass isrelated to gravitational mass by the equivalence principle, andtheyfeel that "... all matter at the level of quarks and electrons isdriven to oscillate (zitterbewegung in the terminology ofSchroedinger) by the ZPF. But every oscillating charge will generateits own ... fields. Thus any particle will experience the ZPF asmodified ever so slightly by the fields of adjacent particles ... andthat is gravitation! It is a kind of long-range van der Waals force....", but they do not claim to have shown that Sakharov's conjectureis true in detail.

Petkov in physics/9805028says that "... One of the consequences of general relativity is thatthe velocity of electromagnetic signals (or simply the velocity oflight) in the vicinity of massive objects is anisotropic; it isbelieved that this anisotropy is caused by the spacetime curvature.... Due to the anisotropy of the speed of light the electric field ofan electron on the Earth's surface is distorted which gives rise to aself-force originating from the interaction of the electron's chargewith its distorted electric field. This self-force tries to force theelectron to move down wards and coincides with what is traditionallycalled a gravitational force. The electric self-force is proportionalto the gravitational acceleration g and the coefficient ofproportionality is the mass "attached" to the electron's electricfield which proves to be equal to the electron's mass. In such a waythe electron's passive gravitational mass turns out to be purelyelectromagnetic in origin. ... "

Petkov alsosays "... At this stage it appears that quantum mechanicaltreatment of the electromagnetic mass is not possible since quantummechanics does not offer a model for the quantum object. ...", butthat may be too pessimistic a statement. A geometric model for thequantum object can be given by a geometric model for the Zero PointFluctuations of the Vacuum, which from the Many-Worldspoint of view is MacroSpaceand from Bohm's point of view is the Super ImplicateOrder.

It also may be true that Petkov is incorrect in identifyingElectromagnetic ZPF with Gravity, as that does not include the othertwo Standard Model forces, the Color Force and the Weak Force, nordoes it include the Higgs mechanism. Perhaps Petkov's statement ofidentification should have been a weaker, but still interesting,statement of similarity, based on the common Conformal Group symmetryof:

- Maxwell's equations of Electromagnetism
- Gravity derived from the Conformal Group using the MacDowell-Mansouri mechanism
- the Quantum Theoretical Hydrogen atom
- the Lie Sphere geometry of SpaceTime Correlations in the Many-Worlds picture

Electromagnetism, Gravity, and the ZPF all have in common thesymmetry of the Conformal Group whose compact version is Spin(6).

Further,

the 12-dimensional Standard Model Lie Algebra U(1)xSU(2)xSU(3) maybe related to the Conformal Group Lie Algebra

in the same way that

the 12-dimensional Schrodinger LieAlgebra is related to the Conformal Group Lie Algebra.

B. G. Sidharth notes that ZPF may describethe electron within its Compton Radius Region,

and that, according to pages 1192-1193 of Misner, Thorne, andWheeler (Gravitation, Freeman 1973), the magnitude DR of ZPFflucturations in the curvature of space in an Electon Compton RadiusVortex are

where Lp is the Planck Length (about 1.6 x 10^(-33) cm), and Rc isthe Electron Compton Radius (about 3.86 x 10^(-11) cm). If you lookonly at orders of magnitude, you see that the fluctuations

so that, asSidharth says,

Since Sidharthinvokes "... the DeBrogiie-BohmHydrodynamical Formulation [of Quantum Theory] to picture an[Electron] as a ... Vortex ...",

A useful aspect of

to Quantum Theory is that it can be formulated in terms of the Liouville equation. As Wilkie and Brumer have shown, quantum Liouville spectral projection operators and spectral densities, and hence classical dynamics, are shown to approach their classical analogs in the limit as h approaches 0. The correspondence is shown to occur via the elimination of essential singularities. Brumer acknowledges discussions with Joe Ford, whose interest in the Liouville equation and Quantum Chaos continued throughout his life.

The Liouville equation and density matrices are used by

described in Advanced Field Theory (AIP 1993). Umezawa uses a superoperator formalism in which the density matrix is treated as a vector. If the density matrix is regarded as a vector, each of its components might be regarded as a state, which state might itself be represented as a vector. Such a "self-similar reflexive structure" is shared by the Octonions. As Onar Aam has noted, the 7 imaginary octonions are in 1-1 correspondence with the 7 E8 lattices, and each E8 lattice is an 8-dimensional lattice of integral octonions. These characteristics of octonions are the basis for construction of the D4-D5-E6 physics model.

also uses the Liouville equation and density matrices in his book, The End of Certainty (Simon & Schuster (The Free Press) 1997). I disagree with Prigogine with respect to one point: Prigogine does not like to use Hilbert space in his Quantum Theory. He says: "As long as we remain in Hilbert space,there are deviations from the exponential [ decay ] for bothvery brief times ... and very long times. However, in spite ofa great number of experimental studies, no deviations fromexponential behavior have yet been detected....The precise exponential behaviour observed thus far showsthe inadequacy of Hilbert space description..." The non-exponential decay that Prigogine describes is known as the "watched pot" theorem (a watched pot does not boil,and a frequently observed quantum system does not decay). Since Prigogine's book was published, the watched pot phenomenon HAS been experimentally observed, according to Physics News Update, at the University of Texas: PHYSICS NEWS UPDATEThe American Institute of Physics Bulletin of Physics NewsNumber 327 June 25, 1997by Phillip F. Schewe and Ben Stein"...NON-EXPONENTIAL DECAY of a quantum system has been observed for the first time.Many unstable systems such as a mass of radioactive nuclei willcharacteristically undergo a process (quantum tunneling)whereby the number of nuclei remaining in an initial stateafter a time t will be proportional to e raised to a power of -at,where a is a constant.Quantum mechanics does not expressly forbid non-exponential decay,and physicists at the University of Texas have now devised a scheme---sodium atoms trapped in a web of laser light---in which the rate of atoms escaping(under the auspices of quantum tunneling) is non-exponential,at least over intervals on the order of 10 microseconds.(Nature, 5 June 1997.)..." However, Prigogine's use of the Liouville approach to Quantum Theory with density matrices to try to understand the Arrow of Time in terms of chaos in the corresponding classical limit is interesting to me. In particular, Prigogine describes the Arrow of Time in terms of Bernoulli Schemes.

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