Tony's Home

```Are "Other" Quantum Theories
Really Different
from Many - Worlds ? ```

### Prigogine - Nonequilibrium Thermodynamics

```

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

```

## The 15-dimensional Conformal Group Spin(4,2)

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

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

The Conformal Group Spin(4,2) = SU(2,2) is used in the D4-D5-E6 physics model to describe Gravity and the Higgs Mechanism, and also shows relationships between Special Relativity and Quantum Theory.

Barut and Raczka, in their book Theory of Group Representations and Applications (World 1986), describe the 15 Lie algebra basis elements of the Conformal Group Spin(4,2) = SU(2,2), which are 3 Spatial Rotations, 3 Lorentz Boosts, 4 Spacetime Translations, 1 Scale Transformation, and 4 Special Conformal Transformations.

In Chapter 13 (particularly section 13.4) Barut and Raczka show that wave equations for massive particles are formally invariant under the Conformal Group if the mass is transformed by the 1 Scale Transformation and the 4 Special Conformal Transformations of the Conformal Group.

Barut and Raczka also show in Chapter 13 that in the Non-Relativistic limit the generators of the 15-dimensional Conformal Group contain the 10-dimensional projective Galilei Group, and that the Schrodinger Operator is invariant under a 12-dimensional Schrodinger Lie Algebra that is made up of the 10-dimensional Galilei Lie Algebra plus 1 modified Scale Transformation and 1 modified Special Conformal Transformation. Then the 12 Schrodinger Lie Algebra generators 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 subgroup the 3-dimensional group SU(1,1) = Spin(2,1) = SL(2,R) generated by Ho, D, and Ko. In particular, the Lie Algebra SU(1,1) solves the 3-dimensional quantum oscillator with Hamiltonian H = (p^2 / 2m) + L q^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 Lie Algebra SU(2) and that, if the remaining 9 generators from P, J, and M are identified with a U(3) Lie Algebra, then the 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 Raczka consider the Relativistic case in which the 15-dimensional Conformal Group Spin(4,2) is considered as an extension of the 6-dimensional Lorentz Group Spin(3,1) = SL(2,C). In this case, the Scale Transformation plus a Special Conformal Transformation plus Time Translation 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). They then show that one representation of this SU(1,1) subgroup gives the Klein-Gordon equation, and another representation gives the second order Dirac equation for the Coulomb potential.

Paul J. Freitas uses 4-Dim Euclidean Spatial Space to derive Special Relativity from Quantum Theory in his paper Connections between Special Relativity, Charge Conservation, and Quantum Mechanics dated 25 March 1998. Freitas describes particles as existing in a 4-dimensional Euclidean Spatial Space, as opposed to a 3-dimensional Euclidean Spatial Space in 4-dimensional Spacetime. As he says, "... It is possible to add one dimension to our three dimensions of space in such a way that we can treat simple objects with and without rest mass exactly the same way. This new four-space has the nice property of being Euclidean, and yields all of the usual relativistic properties through a few simple, familiar postulates. ... simple particles of matter can be described as existing in a Euclidean four-space as described above, with the proper time of the particle being related to the fourth, non-obvious position component. In such a space, all simple forms of matter are constantly moving at the speed of light. ... [distance in the fourth spatial dimension is] what is commonly referred to as the spacetime interval [dw^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2] ... By considering a few well-understood physical processes, we shall see that the momentum of a particle in the fourth direction may correspond to its charge, which means that charge conservation is just a form of momentum conservation. ... In fact, we can start with four-dimensional quantum mechanics and derive the results of special relativity very naturally, with even fewer assumptions. ...".

The 4-dimensional Euclidean Space of Freitas seems to be related to the 4-dimensional Euclidean subspace of the vector space of the Conformal Group Spin(4,2) = SU(2,2) that is used by Barut and Raczka to describe Quantum Theory and Spacetime Symmetry, and by me in the D4-D5-E6 physics model to describe Gravity and the Higgs Mechanism.

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 to D4 / (D3 x U(1)) where D3 is the 15-dimensional Conformal Group whose compact version is Spin(6), and to D3 / (D2 x U(1)) where D2 is the 6-dimensional Lorentz Group whose compact version is Spin(4). Electromagnetism, Gravity, and the ZPF all have in common the symmetry of the 15-dimensional D3 Conformal Group whose compact version is Spin(6), as can be seen by the following structures with D3 Conformal Group symmetry:

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

## Click Here to see Segal's Conformal Theory and GraviPhotons.

```

Another probabilistic approach to Quantum Theory
is ```

## Edward Nelson's Quantum Fluctuations

```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 the frequency-cubed power spectrum Lorentz invariant, it is possible to describe realistic physics, including even massive particles, using only light-cone particles. Background fields, especially, can be so described. 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 to a 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 are defined) are both automatically massless free fields. ... We may think of the total interacting field as composed of pieces in which fields propagate for a while along null straight lines as massless free fields, but scatter repeatedly at points in this interior region. The novel feature that arises in our approach is the propagation entirely along null lines between scatterings. ... [We briefly indicate] a corresponding treatment of the Maxwell-Dirac system. ... It would be interesting also to develop our procedure (with or without twistors) into a full description of quantum electrodynamics. ... The diagrams arising here are in many ways analogous to Feynman diagrams. But there is the unusual feature that here we are concerned only with null space-time separations, even for massive fields. ..." Iwo Bialynicki-Birula used a similar approach to formulate Dirac and Weyl equations on a cubic lattice as Quantum Cellular Automata. His Quantum Cellular Automaton is fundmentally a Feynman Checkerboard, in which mass is the amplitude to change light-cone direction, and motion of all particles is a sequence 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 Conservative Markovian Diffusions on the configuration space M of unordered N-tuples of spatial coordinates in R3 for N indistinguishable particles "... fall into two sharply different classes according to the statistics obeyed. ...". "... In the first case (Bose-Einstein statistics) it is a symmetric function ... and in the second case (Fermi-Dirac statistics) it is an antisymmetric function ... a superposition of a symmetric and an antisymmetric wave function does not 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 Hidden Variables, printed in the book Quantum Concepts in Space and Time (Penrose and Isham, eds., Oxford 1986) describes Nelson's derivation of quantum mechanics as a Brownian motion process, discussing the wave function

PSI = sqrt(rho) exp( i S / hbar )

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

v = (1/m) divS

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

Vquantum = (hbar^2 / 2m) div^2 (sqrt(rho) / rho )

... In stochastic mechanics, the term Vquantum is derived ... from the general theory of Brownian motion ... by specifying that the Brownian 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 delicately correlated that an exactly conserved energy of the form (2) may be defined in terms of their probability distribution [and which obey the other condidtions] will behave as if each member of the ensemble is coupled to the probability distribution of the whole ensemble ...".

In some respects, Dirac anticipated some of the fundamental ideas of Nelson's Stochastic Quantum Theory. In 1951-1954, Dirac advocated the reality and utility of the aether, as shown in this quote from pages 202-203 of Dirac: A Scientific Biography, by Helge Kragh (Cambridge 1990): "... "Let us imagine the aether to be in a state for which all values of the velocity of any bit of aether, less than the velocity of light, are equally probable. ... In this way the existence of an aether can be brought into complete harmony with the principle of relativity." Dirac identified the ether velocity with the stream velocity of his classical electron theory ... it was the velocity with which small charges would flow if they were introduced. ... in the spring of 1953, Dirac proposed that absolute time be reconsidered. ... The ether, absolute simultaneity, and absolute time "... can be incorporated into a Lorentz invariant theory with the help of quantum mechanics ..." ... he was unable to work out a satisfactory quantum theory with absolute time and had to rest content with the conclusion that "one can try to build up a more elaborate 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 Concepts in Space and Time (Penrose and Isham, eds), at page 156) a diffusion constant that "... is inversely proportional to the inertial mass of the particle, with the constant of proportionality being a universal constant hbar: v = hbar / m ...". Compare this with Dirac's 1951 suggestion that the electromagnetism U(1) gauge-fixing condition should be A A = k^2 where (see page 199 in Kragh's book I am omitting some sub and superscript mus and nus): "... In order to get agreement with the Lorentz equation, the constant k was indentified with m/e The four-velocity v of a stream of electrons ws found to be related to A by v = (1/k) A ..." which gives for Dirac's theory v = e / m.

Carlos Rodriguez in section 8 of physics/9808010 describes a generalization of the Markov property that permits derivation of the Schrodinger equation for Clifford Algebra valued conditional measures, such as might be used to construct the D4-D5-E6 physics model.

With respect to the Schrodinger equation, Rodriguez cites quant-ph/9804012 by Ariel Caticha, whose abstract says: "Quantum theory is formulated as the only consistent way to manipulate probability amplitudes. The crucial ingredient is a consistency constraint: if there are two different ways to compute an amplitude the two answers must agree. This constraint is expressed in the form of functional equations the solution of which leads to the usual sum and product rules for amplitudes. A consequence is that the Schrodinger equation must be linear: non-linear variants of quantum mechanics are inconsistent. The physical interpretation of the theory is given in terms of a single natural rule. This rule, which does not itself involve probabilities, is used to obtain a proof of Born's statistical postulate. Thus, consistency leads to indeterminism." Caticha also says that "... the fact that Born's postulate is actually a theorem was independently discovered long ago by Gleason, by Finkelstein, by Hartle and by Graham. ...".

Guido Bacciagaluppi in quant-ph/9811040 says: "In de Broglie and Bohm's pilot-wave theory, as is well known, it is possible to consider alternative particle dynamics while still preserving the | PSI |^2 distribution. I present the analogous result for Nelson's stochastic theory, thus characterising the most general diffusion processes that preserve the quantum equilibrium distribution, and discuss the analogy with the construction of the dynamics for Bell's beable theories. I briefly comment on the problem of convergence to | PSI |^2 and on possible experimental constraints on the alternative dynamics."

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

### quantum mechanics is not a complete theory, but rather is an emergent phenomenon arising from the statistical mechanics of matrix models that have a global unitary invariance ...

The exposition of the text is based on dynamical variables that are matrices in complex Hilbert space, but many of the ideas carry over to a statistical dynamics of matrix models in real or quaternionic Hilbert space ...

our idea is to start from a classical dynamics in which the dynamical variables are non-commutative matrices or operators. (We will use the terms matrix and operator interchangeably throughout this book, and do not commit ourselves as to whether they are finite NxN dimensional, or infinite dimensional as obtained in the limit N --> oo.) Despite the non-commutativity, a sensible Lagrangian and Hamiltonian dynamics is obtained by forming the Lagrangian and Hamiltonian as traces of polynomials in the dynamical variables, and repeatedly using cyclic permutation under the trace ... We further assume that the Lagrangian and Hamiltonian are constructed without use of non-dynamical matrix coefficients, so that there is an invariance under simultaneous, identical unitary transformations of all the dynamical variables, that is, there is a global unitary invariance. We assume that the complicated dynamical equations resulting from this system rapidly reach statistical equilibrium, and then show that with suitable approximations, the statistical thermodynamics of the canonical ensemble for this system takes the form of quantum field theory. ... The requirements for the underlying trace dynamics to yield quantum theory at the level of thermodynamics are stringent, and include both the generation of a mass hierarchy and the existence of boson-fermion balance. ... From the equilibrium statistical mechanics of trace dynamics, the rules of quantum mechanics emerge as an approximate thermodynamic description of the behavior of low energy phenomena. "Low energy" here means small relative to the natural energy scale implicit in the canonical ensemble for trace dynamics, which we identify with the Planck scale, and by "equilibrium" we mean local equilibrium, permitting spatial variations associated with dynamics on the low energy scale. Brownian motion corrections to the thermodynamics of trace dynamics then lead to fluctuation corrections to quantum mechanics which take the form of stochastic modifications of the Schrodinger equation, that can account in a mathematically precise way for state vector reduction with Born rule probabilities. ...

there is a conserved operator with the dimensions of action, which we call Cbar, which is equal to the sum of bosonic commutators [p,q] minus the corresponding sum of fermionic anticommutators {p,q}, and which is the conserved matrix-valued Noether 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 natural measure for matrix phase space, and show that this measure obeys a generalized Liouville theorem. ... the generic conserved quantities H, N, and Cbar appear multiplied by Lagrange multipliers that represent generalized "temperatures." ... we specialized the ensemble to 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 will play the role of i ... hbar will play the role of reduced Planck's constant ...

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

.. the Copenhagen interpretation ... state[s] by fiat that the unitary state vector evolution of quantum mechanics does not apply to measurement situations. One then adds to the unitary evolution postulate a second postulate, that of state vector reduction, which states that after a measurement one sees a unit normalized state corresponding to the measurement outcome ... with a probability given by the Born rule ...

In the "many-worlds" interpretation introduced by Everett ... there is no state vector reduction, but only Schrodinger evolution of the entire universe. ... to describe N successive quantum measurements requires consideration of an N-fold tensor product wave function. The mathematical framework can be enlarged to create a sample space by considering the space of all possible such tensor products, and defining a suitable measure on this space. ... This procedure ... is the basis for arguments obtaining the Born rule as the probability for the occurrence of a particular outcome, that is, the probability of finding oneself on a particular branch of the universal 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 in configuration space is assumed to obey the Born rule ... at some initial time, the Bohmian equations then imply that this continues to be true at all subsequent times. Arguments have been given that the Bohmian initial time probability postulate follows from considerations of "typicality" of initial configurations. ...

Trace dynamics: the classical Lagrangian and Hamiltonian dynamics of matrix models ... The fundamental idea is to set up an analog of classical dynamics in which the phase space variables are non-commutative, and the basic tool that allows one to accomplish this is cyclic invariance under a trace. ... Quantum mechanical behavior will be seen to emerge only when ... we study the statistical mechanics of the classical matrix dynamics formulated here. ...

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

The matrix model for M theory ... theta is a 16-component fermionic spinor ... with the transpose T ... so that thetaT is simply the 16-component row spinor corresponding to the 16-component column 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 the canonical ensemble, and of the equipartition of Cbar, is the emergence of relativistic quantum field theory as the low energy effective approximation to a relativistic trace dynamics ... the equipartition theorem can be viewed as a Ward identity application in classical statistical mechanics ...

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

... our Ward identity derivation ... contains a source of violation of local causality, which is the way the operator Cbar enters ... Since ... the boson and fermion contributions to Cbar largely cancel ... it can have large fluctuations over the operator phase space. Correspondingly, in any finite subsystem of the universe described by the canonical ensemble, Cbar has large fluctuations over the ensemble and hence as a function of time. These fluctuations give rise to corrections to the emergent quantum mechanics that we derived by replacing Cbar by its ensemble average. ... these fluctuations do not affect the unit normalization of states, but add stochastic terms to the effective Schrodinger equation that describes the time development of a state. In order to preserve state normalization, this Schrodinger equation in the generic case must be nonlinear in the state, which introduces violation of local causality, since changes in the wave function at one spatial point are instantaneously communicated, via the noise terms, to all spatial points. ... the fluctuations in Cbar .. play the role of a Brownian "noise" which drives state vector reduction, in such a way as to be precisely consistent with Born rule probabilities. ... the average over the noise of the density matrix obeys a linear evolution equation ...

Brownian motion corrections to emergent quantum mechanics can provide the mechanism responsible both for the reduction of the state vector, and for the emergence of the Born and ... (in the case of degeneracies) ... Luders probability rules. ... with suitable assumptions, one can derive the standard stochastic Schrodinger equation for objective state vector reduction. Depending on the details of the model, the stochastic driving terms in this equation can couple to the total energy, to a local density ... , or to both ... when the stochastic drivign terms involve a set of mutually commuting operators, this equation leads to state vector reduction with Born rule probabilities. ...

there is a plausible route leading from the underlying trace dynamics to CSL ... continuous spontaneous localization ... 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 the correlation function ... no particular choice of functional form is needed ... the results are independent of the value of the correlation length rC, provided that ... rC ... lies between microscopic and macroscopic dimensions. The value rC = 10^(-5) cm is typically assumed in the CSL literature. ... Ghirardi, Pearle, and Rimini ... assume a correlation length rC = 10^(-5) cm, and propose the value ... [of the] stochasticity parameter ... gamma = 10^(-30) cm^3 s^(-1) GeV^(-2) ... any instrument pointer displacement involving at least 10^13 nucleons gives a reduction time ... 10^7 s^(-1) .... [which is] less than typical experimental measurement times. ...

the underlying dynamics is not unitary, with the unitary dynamics of quantum field theory emerging ... as a thermodynamic approximation;  this suggests an amelioration, in the underlying dynamics, of the infinities of quantum field theory, provides a basis for understanding the nonlocal "paradoxes" of quantum theory, and may ... play a role in establishing the large-scale uniformity of the universe. ...

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

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

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

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

## Gravity from Vacuum Zero Point Fluctuation

is a fundamental ingredient of

### Sidharth's Compton Radius Vortex model of the Electron,

and Paul Davies notes that the ZPF may provide a physical basis for Mach's Principle.

Gravity from Vacuum Zero Point Fluctuation 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 of the Zero Point Fluctuation (ZPF) of the Vacuum (in the form of virtual particles and particle-antiparticle pairs) is zero.

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

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

( hbar / 2 pi^2 ) INT k^3 dk

where k is the wave number of the ZPF and INT denotes integral.

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

L(R) = A hbar INT k^3 dk + B hbar R INT k dk + terms of order 2 or higher in R

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

The first term is just the Standard Model ZPF part of the Lagrangian 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 gravitational part of the Hilbert action Lagrangian

( 1 / 16 pi G ) INT R (-g)^(1/2) dx = ( - c^3 R / 16 pi G ) INT dx

so that

B hbar R INT k dk = ( - c^3 R / 16 pi G ) and G = c^3 / ( - c^3 / 16 pi B hbar INT k dk )

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

k(cut-off) = ( c^3 / hbar G )^(1/2) = 1 / Planck Length = 1 / 1.6x10^(-33) cm.

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

G = 8 pi T - LAMBDA g

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

Subsequently, Haisch and Rueda showed in physics/9802030 and physics/9802031 that inertia can be due to interaction with the heat bath of the ZPF (which ZPF heat bath may be similar to the Background Field of Nelson). They noted that inertial mass is related to gravitational mass by the equivalence principle, and they feel that "... all matter at the level of quarks and electrons is driven to oscillate (zitterbewegung in the terminology of Schroedinger) by the ZPF. But every oscillating charge will generate its own ... fields. Thus any particle will experience the ZPF as modified ever so slightly by the fields of adjacent particles ... and that is gravitation! It is a kind of long-range van der Waals force. ...", but they do not claim to have shown that Sakharov's conjecture is true in detail.

Petkov in physics/9805028 says that "... One of the consequences of general relativity is that the velocity of electromagnetic signals (or simply the velocity of light) in the vicinity of massive objects is anisotropic; it is believed that this anisotropy is caused by the spacetime curvature. ... Due to the anisotropy of the speed of light the electric field of an electron on the Earth's surface is distorted which gives rise to a self-force originating from the interaction of the electron's charge with its distorted electric field. This self-force tries to force the electron to move down wards and coincides with what is traditionally called a gravitational force. The electric self-force is proportional to the gravitational acceleration g and the coefficient of proportionality is the mass "attached" to the electron's electric field which proves to be equal to the electron's mass. In such a way the electron's passive gravitational mass turns out to be purely electromagnetic in origin. ... "

Petkov also says "... At this stage it appears that quantum mechanical treatment of the electromagnetic mass is not possible since quantum mechanics does not offer a model for the quantum object. ...", but that may be too pessimistic a statement. A geometric model for the quantum object can be given by a geometric model for the Zero Point Fluctuations of the Vacuum, which from the Many-Worlds point of view is MacroSpace and from Bohm's point of view is the Super Implicate Order.

It also may be true that Petkov is incorrect in identifying Electromagnetic ZPF with Gravity, as that does not include the other two Standard Model forces, the Color Force and the Weak Force, nor does it include the Higgs mechanism. Perhaps Petkov's statement of identification should have been a weaker, but still interesting, statement of similarity, based on the common Conformal Group symmetry of:

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

Further,

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

in the same way that

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

` `

and that, according to pages 1192-1193 of Misner, Thorne, and Wheeler (Gravitation, Freeman 1973), the magnitude DR of ZPF flucturations in the curvature of space in an Electon Compton Radius Vortex are

DR of the order of about Lp / (Rc)^3

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

DR are of the order of about 10^(-33) / (10^(-11))^3 = unity

so that, as Sidharth says,

"... In other words the entire curvature of the [Electron Compton Radius Vortex] ... can be thought to have been created by these fluctuations alone ...".

Since Sidharth invokes "... the DeBrogiie-Bohm Hydrodynamical Formulation [of Quantum Theory] to picture an [Electron] as a ... Vortex ...",

## Mark J. Hadley has a Quantum Theory based on Topology.

```
A useful aspect of ```

## the density matrix approach

```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 ```

## Hiroomi Umezawa in his Thermo Field Dynamics

```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.

```

## Ilya Prigogine

```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 both
very brief times ... and very long times.  However, in spite of
a great number of experimental studies, no deviations from
exponential behavior have yet been detected.
...The precise exponential behaviour observed thus far shows
the 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 UPDATE
The American Institute of Physics Bulletin of Physics News
Number 327 June 25, 1997
by 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 will
characteristically undergo a process (quantum tunneling)
whereby the number of nuclei remaining in an initial state
after 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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