A singleFEYNMAN CHECKERBOARDconfiguration is one of all possible ways for an electron to go from an initial point to a destination point. Choice of a particular single configuration is aRANDOMquantum choice.

Feynman's Relativistic Chessboard as an Ising Model, by H. A. Gersch (Int. J. Theor. Phys. 20 (1981) 491), shows that the (1+1)-dimensional Feynman Checkerboard, which describes the (1+1)-dimensional Dirac equation, is equivalent to the 1-dimensional Ising model and

is related to **Bernoulli Schemes, Julia Sets, **and
**Chebyshev Polynomials**.

The Feynman Checkerboard for an electron in 1+1 dimensional spacetime can be represented as a 1-dimensional Cellular Automaton evolving in 1 time dimension. The initial state is a point in the t=0 initial time, the initial location of the electron in 1-dimensional space. Each possible state at time t=N is a point locating the electron on the spatial line at t=N. All possible paths from the initial state at t=0 to the state at t=N are summed (the Quantum Sum Over Histories) to get the amplitude for the electron to go from the initial state at t=0 to each of the possible states at t=N. The Cellular Automaton evolves to time t=N+1 by considering all possible paths that go from t=0 to t=N+1. In terms of amplitudes (rather than probabilities), the paths to t=N+1 are just the extensions of the paths to t=N. Here is a rough diagram of one such possible path to one such possible point, up to time t=N+1 : t=N+1 --------------*---------------------------- \ t=N ----------------*-------------------------- / t=N-1 --------------*---------------------------- \ .................................................. .................................................. .................................................. / t=3 -------------------*----------------------- \ t=2 ---------------------*--------------------- / t=1 -------------------*----------------------- / t=0 -----------------*------------------------- Each Initial State of the Feynman Checkerboard, or point on the space-like 1-dimensional line at t=0, can be represented on the real line interval [0,4] by a binary semi-sequence of +/- signs in the form of 2 +/- sqrt(2 +/- sqrt(2 +/- sqrt(2 +/- ... ))) Such sequences of length M are the zeroes of the Chebyshev polynomials of degree 2^M.

Consider the unit circle S1 and the map T_L: S1 -> S1 defined on C, the complex plane, by T_L(x) = (x - L)^2, where L is a real number in [0,2]. If L = 0, then the unit circle S1 invariant under T_L and, for any n, under n iterations denoted by T_L^n. S1 and its interior is the subset of C that remains bounded under the iterated map T_0^n as n becomes arbitrarily large. Denote by K_L the subset of C that remains bounded under the iterated map T_L^n as n becomes large. Call the boundary of K_L the Julia set for L, denoted by B_L. For the complex plane, the intersection of K_L with B_L is just K_L and the Julia set B_L is defined for the complex plane. Julia sets for the complex plane have been studied by Barnsley, Geronimo, and Harrington (1983), whose treatment is followed herein. Some Julia sets are shown here:

Clearly, the Julia sets B_L range from B_0 = Unit Circle S1 through a number of complicated shapes to B_2 = [0,4] on the real axis. For alI L in (0,2], B_L can be represented as the set of all complex numbers of the form

where sqrt is defined by using sqrt(-1) = +/- i. **The sequence
of sqrt signs +/- 1 defines equivalence classes of points in B_L for
all L in (0,2]**. Those equivalence classes, along with
continuity, define a map W from B_0, through B_L for L in
[0,2], to B_2. Consider the finite sequence of length n. The
corresponding **points in B_2 = [0,4] are the zeroes of the
Chebyshev polynomials of degree 2^n**. The intervals containing
those zeroes form nth order Borel sets for B_2. The corresponding
**Borel measure is the singular measure concentrated at the zeroes
of the Chebyshev polynomials of degree 2 taking the value 2^(-n) at
each zero**. Note that T_2^n maps each nth order Borel set densely
onto the whole set B_2 = [0,4]. T_2 acts as a Bernoulli shift
operator for the Chebyshev measure system on B_2 and, **as n becomes
large, the Chebyshev measure goes to the arccosine measure**. The
**Bernoulli scheme is isomorphic to the scheme defined for Lebesgue
measure and the usual Borel sets on the unit interval**. Use the
map W: B_0 --> B_2. Define Zn: B_0 = S1 --> B_2 = [0,4]
as the composition of W and the inverse of T_2^n. Note that B_2 =
[0,4] has a Chebyshev measure structure. **The nth order
Borel sets and Chebyshev measure on the B_2 induce nth order Borel
sets and Chebyshev measure on B_2. T_2 then acts as a Bernoulli Shift
on B_2, and T_2^n maps each nth order Borel set densely onto the
whole set B2.** Therefore, **the inverse image under Z_n of each
Borel set in B_2 is dense in B_0 = S1**.

Define M_n as the lattice constructed from B_2 by identifying the
nth order Borel sets of B_2, each with its Chebyshev measure, as
vertices of the lattice. **M_n is a 1-dimensional lattice with
structure of the Chebyshev measures on B2. The fineness of the
lattice M_n Is determined by the order n of the Borel sets**. As
the nth order Chebyshev measure is a singular measure concentrated at
the zeroes of the Chebyshev polynomials of degree 2^n, **the lattice
M_n has a natural singular measure that converges** as n becomes
large, or as lattice spacing becomes small, **to the Chebyshev
measure that is isomorphic as a Bernoulli scheme to Lebesgue measure
on the closed interval [0,4]**.

Define a map X_n: S1 --> M_n by composing Z_n with the defining map from BB_2 to M_n. Then the inverse image under X_n of any vertex in the lattice M_n is dense in S1.

M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, in their paper On the invariant sets of a family of quadratic maps (Georgia Tech Math preprint December 1981)(Communications in Mathematical Physics 88 (1983) 479-501), say:

"... we consider a family of invariant sets which are associated with a one-parameter family of quadratic maps from the complex plane into itself. The parameter is L, which may be real or complex, and the mapping T_L: C -> C is defined byT_L z = ( L - z )^2 , z in C. For L =/= 0, T_L is related to the mapping T~_L z = 1 - L z^2

which has been extensively studied in the context of iterated maps ... The relationship is T~_L = A^(-1) T_L A where A is the affine transformation

A z = L - L z T~_L is usually viewed as a mapping from the real interval [-1,1] into itself, and then L lies in the interval [0,2]. This is equivalent to looking at T_L as a mapping from the interval [0,2L] into itself, which agains corresponds to L in [0,2]. ... the critical values of L, at which occur such phenomena as the first appearance of k-cycles and the onset of ergodic behavior, are the same for both mappings ... The invariant set which we study will be denoted by B ... With the notation T_L^0 z = z and T_L^(n+1) z = T_L (T_L^n z ) ... B_L can be defined to be those points in the complex plane where the sequence {T_L^n z} is not normal. ... However, we will ... begin with the set B~_L of formal objects { L +/- (L +/- (L +/- ... ad infinitum where all sequences of plus and minus signs are included. ... The calculated set appears to become the unit circle ... as L approaches zero, whilst for L = 2 it lies thickly on the real interval [0,4]. For 0 < L < 2 more complicated figures were obtained ... For L > 2 it was found to lie upon the real axis and to consist of distinct points. ... The set B_L ... associated ... with B~_L ... provides insight into ... such phenomena as the cascades of bifurcations, the Feigenbaum numbers, and the onset of (nontrivial) ergodic properties. ...

... One reason why we first became interested in B_L was because it arose for L = 3 in the context of the Diophantine Moment Problem (D.M.P.) ...[which]... appeared in an attempt to predict the critical indices for Ising model lattice gases ... The ...[D.M.P.]... was completely solved for L < 4 and largely solved for L = 4 (leading, incidentally, to a novel resolution for a one-dimensional Ising model). For oo > L > 6 it was shown that there exists a transformation upon the generating function .. which yields a new generating function .. with L~ < 6. ... the support of the measure for a fixed point of the transformation was associated with B~_3. ...

.. we examine the case 2

<L < oo, for which ... B_L lies entirely on the real axis. ... the operation of T_L on B_L ...[is]... equivalent to that of the right-shift operator upon the set OMEGA of half-infinite Bernoulli sequences ...[there exists]... a distance function which is natural to B_L, yielding ... that ... for 2 < L< oo, B_L is a Cantor set with Lebesgue measure ... zero. For 2<L < 00, B_L is compact and perfect. ... This measure is singular with respect to Lebesgue measure and has no purely atomic component. ... the action of T_L upon B_L has entropy equal to ln 2. ... the approximating measures are related to a set of monic polynomials, orthogonal with respect to the invariant measure. ... the polynomials are none other than the Tchebycheff polynomials when L = 2, and ... they generalize the latter in a nontrivial way when L > 2. ... they are related to the Tchebycheff polynomials { T_n(x) = cos(n cos^(-1)x) }_n=0^oo according toLim(L->2+) Pn(x) = 2 T_n((1/2)x - 1) ... the Tchebycheff polynomials are the only family of 2_F_1 hypergeometric polynomials which are mapped into themselves under the change of variable x -> (x-2)^2. The polynomials { P_n(x) }_n=-1^oo are also of interest because their measure is totally singular (it has no absolutely continuous part and no discrete jumps). ...

... we consider the cases -(1/4)

<L<2 and | L | < (1/4) with L in C. ...... for -(1/4)

<L<2 , B is realized as the boundary of the image under a conformal mapping F_L of the exterior of the unit circle, with the propertyT_L o F_L = F_L o T_0 ... which relates the action of T_L on B_L to that of T_0 on B_0, the unit circle. ...[Another]... type of approximation to B_L is provided by a second sequence of analytic functions {f_n*(z)}, which are associated with an increasing set of boundaries, and which ... approach B_L from the "inside." ...

... for | L | < (1/4) with L in C ... we first restrict attention to the case 0

<L<0.2 ...[for which]... F_L(z) is defined for all z in C such that | z = 1 ...[then]... we extend the results of the case L ...[in]... { L in C | | L | < (1/4) } by showing that F_L(z) is both defined and analytic in L for L in ...[that set]...".

According to the Encyclopedic Dictionary of Mathematics, Second Edition (MIT Press 1993)(section 136D,G):

"... A generalized Bernoulli shift is always ergodic. A Markov shift is ergodic if and only if the corresponding Markov process is irreducible ... D. Anosov considered a class of flows and diffeomorphisms satisfying a condition that characterizes unstable motions such as geodesic flows on a manifold of negative curvature. ... Sinai constructed for transitive Anosov diffeomorphisms (and for Anosov flows) a special partition of the underlying manifold M called a Markov partition ... they enable one to represent such diffeomorphisms as Markov shifts. Starting with a measure invariant fo rsuch a Markov shift having the maximal entropy ... Sinai constructed a Gibbs measure .. which ... gave rise to a natural invariant measure mu for the corresponding diffeomorphism. ... the diffeomorphism phi considered as an automorphism of the measure space ... was shown ... by R. Azencott ...[to be]... Bernoulli with respect to mu ... ".

Christian Beck, in his book Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized Fields (World Scientific 2002)(summary at hep-th/0207081), says (images in the following book excerpts taken from similar passages/figures in his earlier paper hep-th/0105152):

"... The Tchebyscheff polynomials T_N(F) = cos(N arccos F). The first few polynomials are given by

- T_1(F) = F ,
- T_2(F) = 2 F^2 - 1 ,
- T_3(F) = 4 F^3 - 3 F ,
- T_4(F) = 8 F^4 - 8 F^2 + 1 ,
- T_5(F) = 16 F^5 - 20 F^3 + 5 F ,
- T_6(F) = 32 F^6 - 48 F^4 + 18 F^2 -1
- ...
- ...
The most remarkable and distinguished feature of Tchebyscheff maps T_N with N

>2 is that these maps are semi-conjugated to a Bernoulli shift with N symbols ...... For any map conjugated to a Bernoulli shift, rescaled sums of the iterates satisfy a functional central limit theorem guaranteeing the convergence to the Wiener preocess in the limit ... Since the derivative of the Weiner process is Gaussian white noise, in the limit the deterministic chaotic ...[iterates]... may ... serve as a source of Gaussian white noise for any field theory quantized according to the Parisi-Wu approach ... of stochastic quantization ...

[ My comment here is that I prefer the Nelson approach to stochastic quantization. ]

... All Tchebyscheff maps with N > 2 are ergodic and mixing, with ... natural invariant probability density ... independent of N. ... the non-vanishing correlations of hte Tchebyscheff maps correspond to tuples (n_1, ... , n_r) that are the solutions of the diophantic equationsSUM_i=1^r sigma_i N^(n_i) = 0 , sigma_i = +/-1. ... other maps ... have more non-vanishing higher-order correlations than the Tchebyscheff maps. ... Tshebyscheff maps are distinguished by a minimum skeleton of higher-order correlations. In that sense they are closest to uncorrelated Gaussian white noise, as close as possible for a smooth deterministic system. This makes them an attractive model for a deterministic dynamics that generates 'noise' at the smallest quantum mechanical scales. ...

... Note that the first-order correction to the Gaussian behaviour is of order sqrt(g) for N = 2 and of order g for N

>3. For N>4, up to second order in sqrt(g), only trivial trees contribute, and as a result of this the leading order perturbative expression ... is the same as generated by independent discrete random variables ... In other words, only N = 2 and N = 3 yield nontrivial behaviour in leading order of chaotic quantization. ...... The principal idea of this book is to assume that the noise fields necessary for the quantization of standard model fields have dynamical origin. We will now spatially couple the chaotic noise. This leads to chaotic noise objects, which, ... we will call 'chaotic strings'. ... . ... From a mathematical point of view, chaotic strings are 1-dimensional coupled map lattices of diffusively coupled Tchebyscheff maps. ... Tchebyscheff maps satisfy a Central Limit Theorem which guarantees the convergence to the Wiener process (and hence to ordinary path integrals) if sums of iterates are looked at from large scales. Moreover, Tschebyscheff maps have least higher-order correlations among all systems conjugated to a Bernoulli shift, and are in that sense closest to Gaussian white noise, though being completely deterministic.

Now let us discuss possible ways of spatially coupling the chaotic noise. ... physically it is most reasonable that the coupling should result from a Laplacian coupling ... This leads to coupled map lattices of the diffusive coupling form. ... we will mainly consider the 1-dimensional case. ... One obtains a 'chaotic string' defined by

F_n+1^i = = (1 - a)T_N(F_n^i ) + s (a/2)(T_N^b ( F_n^(i-1)) + T_N^b ( F_n^(i+1))

... F_N^i is a discrete chaotic noise field taking continuous values on the interval [-1,1]. The initial values F_0^i are randomly distributed in this interval. i is a 1-dimensional spatial lattice coordinate an n a discrete time coordinate (in our case identified with the fictitious time of the Parisi-Wu approach). T_N denotes the N-th order Tchebyscheff polynomial. ... The variable a is a coupling constant taking values in the interval [0,1]. ... Since a determines the strength of the Laplacian coupling, a^-1 can be regarded as a kind of metric in the 1-dimensional string space indexed by i. s is a sign variable taking on the values ±1. The choice s = +1 is called 'diffusive coupling', but for symmetry reasons it also makes sense sense to study the choice s = -1, which we call 'anti-diffusive coupling'. The integer b distinguishes between the forward and backward coupling form, b = 1 corresponds to forward coupling (T_N^1(F) := T_N( F)), b = 0 to backward coupling (T_N^0(F) := F). We consider periodic boundary conditions and large lattices ...

... restricting ourselves to N = 2 and N = 3, in total 6 different chaotic string theories arise, characterized by (N,b,s) = (2,1,+1), (2,0,+1), (2,1,-1), (2,0,-1), and (N,b) = (3,1), (3,0). ... we will denote these ... theories as 2A, 2B, 2A-, 2B-, 3A, 3B, respectively. ...

...[There are]... two types of vacuum energies ... V_+/-(a) ... the self energy ... and W_+/-(a) ... the interaction energy ... the evoluton in space i is governed by the postential W ... and the corresponding expectation of the energy density is +/- W(a). We should thus impose the condition of constraint that W(a) should vanish for physically observable states. ... the evolution in fictitious time n is governed by the self-interacting potential V ... This potential generates a shift of informaiton, since the Tchebyxcheff maps T_N are conjugated to a Bernoulli shift of N symbols. Hence V(a) can be regarded as the expectation of a kind of information potential or entropy function, which, motivated by theromodynamics, should be extremized for physically observable states. Note that the action of V and W alternates in the n and i direction. Both types of vacuum energies describe different relevant observables ... and are of equal importance. ...

... A ... physical interpretation would be that at each time step n a fermion-antifermion pair f_1, fbar_2 is spontaneously created in cell i by the field energy of the self-interacting potential. ... The interact with particles in neighbored cells by exchange of a ... gauge boson B_2, then they annihilate into another boson B_1 and the next ... creation of a particle-antiparticle pair ... takes place. This can be symbolically described by the Feynman graph ...

... the graph could ... be called a 'Feynman web' ... in this interpretation a is a ... standard model coupling constant, since it describes the strength of momentum exchange of neighored particles. At the same time, a can be regarded as an inverse metric in the 1-dimensional ... space, since it determines the strength of the Laplacian coupling. ... we will present ... numerical evidence that minima of the vacuum energy ... are observed for certain ... couplings a_i, and these ... couplings ... coincide with running standard model couplings,

[ My comment here is that, particularly in his context of
quantum theory of diffusively coupled Chebyshev maps, Christian
Beck's use of the Renormalization Group
Equations might be inappropriate because he does not seem to
consider the possibilities mentioned by Alexei Morozov and Antti J.
Niemi in their
paper, Can Renormalization Group Flow End in a Big Mess?,
hep-th/0304178, where they say: "... The field theoretical
renormalization group equations have many common features with the
equations of dynamical systems. In particular, the manner how
Callan-Symanzik equation ensures the independence of a theory from
its subtraction point is reminiscent of self-similarity in autonomous
flows towards attractors. Motivated by such analogies we propose that
**besides isolated fixed points, the couplings in a renormalizable
field theory may also flow towards more general, even fractal
attractors. This could lead to Big Mess scenarios** in applications
to multiphase systems, from spin-glasses and neural networks to
fundamental ... theory. We argue that ... such chaotic flows ... pose
no obvious contradictions with the known properties of effective
actions, the existence of dissipative Lyapunov functions, and even
the strong version of the c-theorem. We also explain the difficulties
encountered when constructing effective actions with chaotic
renormalization group flows and observe that they have many common
virtues with realistic field theory effective actions. We conclude
that if chaotic renormalization group flows are to be excluded,
conceptually novel no-go theorems must be developed. ... in the
classical Yang-Mills theory chaotic behaviour has already been well
established ... Consequently such chaotic behaviour will not be
considered here. Obviously, a chaotic RG flow also necessitates the
consideration of field (string) theories with at least three
couplings. In the present article we shall be interested in the
possibility of chaotic RG flows in the IR limits of quantum field and
string theories. ... we consider limit cycles from the point of view
of RG flows, and inspect vorticity as a RG scheme independent tool
for describing multicoupling flows. ... we explain how to construct
model effective actions from the beta-function flows. In particular,
we explain how the construction fails in case of chaotic flows and
suggests this parallels the problems encountered in constructing
actual field theory effective actions. This also explains why it is
very hard to construct actual field theory models with chaotic RG
flow. ...". ]

the energy (or temperature) being given byE = (1/2) N (m_B_1 + m_f_1 + m_f_2 ). Here N is the index of he Tchebyscheff map ... and m_B_1 . m_f_1 , m_f_2 denote the masses of the particle involved in the Feynman web interpretation. ... the ... spectrum appears to reproduce the masses and coupling constants of the known quarks, leptons, and gauge bosons of the standard model ...

... formally ...[the equation for E]... reminds us of the energy levels ... of a quantum mechanical harmonic oscillator, with low-energy levels ... given by the masses of the standard model particles. ... The boson B_2 is virtual and does not contribute to the energy scale. The factor N can be understood as a multiplicity factor counting the number of degrees of freedom. ...

... One period of the self energy of the 2A/B string in the scaling region ...

... It is ... possible to provide a fully symmetric attribution of the 11 minima to the 12 known fermions ...... if one assumes that the minimum b_1 is degenerated, i.e. that it describes two different fermionic particles with the same mass modulo 2. ... Since all minima are only defined modulo 4, we have identified b_0 := b_11 / 4 . ...

... The above scheme ... fix[ex] the neutrino masses modulo 2 as ...

... or equivalently ...

... If ...[the currently accepted]... experimental estimate is correct with a precisionof a factor 2, then ...[the equations of the model]... impl[y] n_3 = 92 ... n_2 = 96 ...[and]... n_1 = 104. This means that ...

...

[ To me, the large number of adjustable factors such as 2 and powers of 2, and the use of experimental input, as well the fact that the V(a) vs. a curve appears to me to have a lot of bumps and flat parts that seem to allow flexibility in choosing where to place the points b_i, indicate that the Christopher Beck model is NOT as clearly predictive as his writings suggest. In my opinion, his model does NOT clearly predict PRECISE values of particle masses, etc., but DOES indicate that there exists a Quantum Structure that is roughly consistent with the (in my opinion much more precise and clearly defined) numerical values calculated in the D4-D5-E6-E7-E8 VoDou Physics Model. I take his results as a confirmation that the D4-D5-E6-E7-E8 VoDou Physics Model calculations are substantially consistent with a Chebyshev-Bernoulli-Feynman Sum-Over-Histories Quantum Theory. ]

... Within the above scheme all up and down members of the same family are described by neighbored minima, and the up member always has a larger self energy than the down member ...

[ This last statement seems to me to be inconsistent with the next quoted statement of Christian Beck. ]

... Light fermion masses as obtained from the minima of the self energy of the 2A/B string ...[are]......

[ I note that here Christian Beck says that the up quark mass is about 5 MeV while the down quark mass is about 9 MeV. I also note that Christian Beck's results produce quark current masses, unlike the D4-D5-E6-E7-E8 VoDou Physics Model calculations which produce quark constituent masses. However, the two sets of masses are interconvertible, and I do agree with Christian Beck that calculations in his model are likely to produce current masses, which are more naturally related to dynamical processes than the stable configurations that are more natural for the D4-D5-E6-E7-E8 VoDou Physics Model constituent quark mass calculations. ]

... Heavy fermions masses as obtained from the minima of the self energy of the 2A/B string ...[are]...... The t quark is put in parenthesis since m_t is so large that the corresponding Yukawa coupling falls out of the scaling region. ...

... The self energy of the 2B string is ...

... The most pronounced minimum ...[is]... a'_3^(2B) = 0.2235(2) ... However, most interesting for us is another minimum, namely a'_2^(2B) = 0.03440(2). This can be interpreted in terms of Yukawa interaction of the top quark. ...... The Yukawa coupling of any fermion f is proportional to the square of its mass. It is given by

alpha_Yu^f = (1/4) alpha_2 ( m_f / m_W )^2 ...

[ I am not sure that formula can be applied accurately to the T-quark, which is a lot heavier than the other quarks ( even Christian Beck puts the T-quark "...in parenthesis since m_t is so large that the corresponding Yukawa coupling falls out of the scaling region ...". ). For example, Chris Quigg says in hep-ph/0001145: "... the electron's Yukawa term in the electroweak Lagrangian is ....Y_e ... the electron mass is m_e = Y_e v / sqrt(2) ...[where v is the Higgs Vacuum Expectation Value]... Similar expressions obtain for the quark Yukawa couplings. ... we do not know how to calculate the fermion Yukawa couplings Y_f ... The values of the Yukawa couplings are vastly different for different fermions: for the top quark, Y_t = 1, for the electron Y_e = 3 x 10^(-6), and if the neutrinos have Dirac masses, presumably Y_nu = 10^(_10) ... I [Chris Quigg] am quoting the values of the Yukawa couplings at a low scale typical of the masses themselves. ...".

Any uncertainty in the T-quark Yukawa coupling formula would affect Christain Beck's T-quark mass calculations, described immediately below. ]

... Our interpretation of the minimum a'_2^(2B) is ...a'_2^(2B) = alpha_Yu^t (m_H + 2 m_t) = = (1/4) alpha_2(m_H + 2m_t) (m_t / m_W)^2

... From our earlier consideration we know that alpha_2(m_Z + 2m_b) = 0.03369(1). Transferring the running alpha_2 to the higher energy scale E = m_H + 2 m_t we obtain for m_H = 154 Gev [which is similar to the D4-D5-E6-E7-E8 VoDou Physics Model calculated value of about 146 GeV for the Higgs mass]

alpha_2(E) = 0.03284(1) ... we thus get m_t = 164.5(2) GeV ... The corresponding pole mass M_t ...[is]... M_t = 174.4(3) GeV. ...

[ I am not convinced by Christian Beck's T-quark mass calculations. From the graph it appears to me that the curve near a'_2^(2B) is pretty flat between about 0.025 and about 0.0375, so that the value of 0.03440(2) that he uses for a'_2^(2B) of 0.03440(2) could just as well have been chosen to be at the left side of the flat region (where the slope of the curve changes dramatically on the way down from left to right), which value of about 0.025 would give by his method of calculation a T-quark mass value of 2 sqrt( 0.025 / 0.03284 ) 80.35 GeV = 140.2 GeV, which is within 10% the 130 GeV Truth Quark Mass of the D4-D5-E6-E7-E8 VoDou Physics Model constituent quark mass calculations. Perhaps Christian Beck's 164.5 to 174.4 GeV mass calculation might shed light on the second peak that appears in that energy region in Fermilab data.

Similar calculations for the peak at a'_3^(2B) = 0.2235(2) gives mass

Using D4-D5-E6-E7-E8 VoDou Physics Model constituent quark mass calculations, the 421.5 GeV value is on the order of the mass of 3 Truth Quarks (red, green, blue) = 3 x 130 = 390 GeV, plus 3 Beauty Quarks (red, green, blue) = 3 x 5.6 = 16.8 GeV, for a total of about 406.8 GeV. Such a Third-Generation 6-quark configuration would be color neutral and would have electric charge +1. Perhaps such a Third-Generation-Deuteron marks then end of conventional hadron formation, which might account for the apparent importance of the minimum peak at a'_3^(2B) = 0.2235(2).

Further, uncertainty in the Yukawa coupling formula used by Christian Beck could have a substantial effect on the validity of his interpretations. ]

... Interaction energy of the 3A string ... for the low-coupling region a in [0, 0.018] ...[is]...

... the zero a_2^(3A) appears to approximately coincide with the fine structure constant alpha_el = 1 / 137. Arbitrary couplings a (in the basin of attraction) will evolve to this stable fixed point under the flow generated by ... adot = a^2 G W(a) + noise ... where [adot is the first derivative of a and] G is a positive constant and W(a) is the interaction energy ... The fictitious time t of the Parisi-Wu approach has dimension energy^(-2), hence if W(a) is ... dimensionless ... then the constant G should have the dimension energy^2 ... Alternatively, we could regard ...[the flow equation]... as an equation describing the renormalization flow under energy scale transformations. ... To construct a ... Feynman web ... choose B_1 to be any massless boson and B_2 = y (photon) , f_1 = e- , fbar_2 = e+ (electrons and positrons). The relevant energy scale ... is ... E = (3/2) ( m_y + 2 m_e) = 3 m_e , ... Hence ... a_2^(3A) = alpha_el(3 m_e) . ... let us estimate the running electromagnetic coupling at this energy scale ... we get alpha_el(3m_e) = 0.007303 ...[ = 1 / 136.93 ]... , to be compared with a_2^(3A) = 0.0073038(17) ...[ = 1 / 136.915 ] ...

[ I note here that 1 / 1367.9 is similar to the D4-D5-E6-E7-E8 VoDou Physics Model calculated value of 1/137.03608 for the electromagnetic fine structure constant, so Chebyshev Quantum Model of Christian Beck is here consistent with the D4-D5-E6-E7-E8 VoDou Physics Model. ]

... There are ... further stable zeros ...[some of which]... cannot be identified with standard model couplings in a straightforward way. ...".

[ I note here that the multiple stable zeros indicate a flexibility in the model of Christian Beck that indicates that it is NOT as constrained and predictive as the D4-D5-E6-E7-E8 VoDou Physics Model.

My impressions of Christian Beck's further calculations are similar, such as for example his use of interaction energy of 3B strings to calculate the Weinberg angle to be

(compared to the D4-D5-E6-E7-E8 VoDou Physics Model calculated value of 0.235)

again finding at least one stable zero for which he does not find a straightforward standard model interpretation.

All in all, as I said above, I think that the work of Christian Beck DOES indicate that there exists a Quantum Structure that is roughly consistent with the (in my opinion much more precise and clearly defined) numerical values calculated in the D4-D5-E6-E7-E8 VoDou Physics Model. I take his results as a confirmation that the D4-D5-E6-E7-E8 VoDou Physics Model calculations are substantially consistent with a Chebyshev-Bernoulli-Feynman Sum-Over-Histories Quantum Theory.

The work of Christian Beck is deals with (1+1)-dimensional spacetime, and so with Complex Plane structure.

Simlarly to the way that the binary sequences can represent the spatial part of the 1+1-dim Feynman Checkerboard as points on a line segment, the spatial part of the 1+1-dim Feynman Checkerboard can also be represented as a 1-dimensional sphere S1. This is analogous to the S7 space of the RP1 x S7 spacetime of the D4-D5-E6-E7-E8 VoDou Physics model, which is discussed below as an Octonion Julia Set. Intervals containing those zeroes form Mth order Borel sets for [0,4]. The corresponding Borel measure is the singular measure concentrated at the zeroes with value 2^(-M) at each zero. As M goes to infinity, the Chebyshev measure goes to the arccosine measure. TheBernoulli Schemeof the right shift operator acting on such semi-infinite sequences of +/- signs. is isomorphic to the Bernoulli Scheme for Lebesgue measure and the usual Borel sets on the unit interval, as shown by Barnsley, Geronimo, and Harrington in Comm. Math. Phys. 88 (1983) 479-501. Consider such a binary semi-sequence of length KM. Then, a Bernoulli shift of length PN for P less than K takes a sequence whose initial M elements correspond to one Initial State to a sequence whose initial M elements correspond to another Initial State, described by the Pth set of M elements in the original sequence. Consider such a sequence of length KM with K = 2^M such that all subsequences of length M beginning with 0, M, 2M, ... are distinct. Such a sequence contains All Possible Initial States for the Feynman Checkerboard at t = 0. Therefore: a Bernoulli Scheme of shift operator of length M acts on such sequences of length KM which describe the Ensemble of all Possible Initial States. By extending the process to include all possible states at each time up to time t=N, you get that a compound Bernoulli Scheme of a shift operator of length KM and a second shift operator of length M acts on such sequences of length NKM which describe the Ensemble of All Sums Over Histories of all paths from all initial states to all final points, or, equivalently, of All Ensembles of All Possible Initial States. Generalize the binary semi-sequence of +/- signs in the form of 2 +/- sqrt(2 +/- sqrt(2 +/- sqrt(2 +/- ... ))) to the binary semi-sequence of +/- signs in the form of L +/- sqrt(L +/- sqrt(L +/- sqrt(L +/- ... ))) where L is in the real interval (0,2], and extend the result to L=0 by continuity. Then, as discussed by Barnsley, Geronimo, and Harrington in Comm. Math. Phys. 88 (1983) 479-501, sequences of +/- signs of length N correspond to points on theJulia SetJL of N-th order iterations of F(z,L), where z is a complex number and F(z,L) = (z - L)^2, which form a Bernoulli Scheme as described above. Such a Julia Set JL is defined as the boundary of the region in z-space for which all iterations of the map F remain bounded (for that value of L). For F(z,0), the the Julia Set J0 is the unit circle S1, the limit of the sequence representations as L approaches 0. In that way, the spatial part of the 1+1-dim Feynman Checkerboard can be represented by the binary sequences as a 1-dimensional sphere S1. This is analogous to the S7 space of the RP1 x S7 spacetime of the D4-D5-E6-E7-E8 VoDou Physics model, which is discussed below as an Octonion Julia Set.

If L is greater than 2, the Julia set JL is a set of points on the real axis, is perfect (all its points are limit points and it contains all its limit points), is compact, non-denumerable. and contains no intervals, is of Lebesgue measure zero, and can be regarded as a dust of disconnected points. If L is allowed to take values in the entire complex plane, the region in L-space for which all iterations of the map F remain bounded (for z = 0) is the Mandelbrot Set. An equivalent definition is the set of values of L for which the Julia set JL is connected, and not a dust. Here is a web page about the Mandelbrot Set images. Here is a Fractal FAQ. Note that instead of the map F(z,L) = (z - L)^2 others often use the closely related map G(z,L) = z^2 + L. For F(z,L) = (z - L)^2 the Mandelbrot set look like this:

However, the structure can be generalized to higher dimensions. The conventional Feynman Checkerboard has 1 space dimension. The Quantum Cellular Automaton of Iwo Bialynicki-Birula is fundamentally a Feynman Checkerboard with 3 space dimensions, and is similar to the Penrose-Rindler light-cone formulation of massive fields and a frequency-cubed power spectrum Background Field. In the D4-D5-E6-E7-E8 VoDou Physics model, the generalized HyperDiamond Feynman Checkerboard has 3 physical space dimensions plus 4 internal symmetry space-like dimensions, for a total of 7 space dimensions. If z in F(z,L) is quaternion or octonion instead of complex, you get similar figures in each 2-dimensional subspace containing the real axis.

Generalization to the S1 x S7 (1+7)-dimensional spacetime of the D4-D5-E6-E7-E8 VoDou Physics Model prior to dimensional reduction can be described by using 8 separate Chebyshev maps:

- one mapping from the timelike S1 related to the Octonion real axis onto a lattice (whose fineness is determined by the order n of the Chebyshev iterations) in the real line segment [0,4], which real line segment is the timelike segment of a spacetime volume element [0,4] x [0,4]^7 with signature -+++++++; and
- 7 mapping each from the 7 basis S1 circles of S7 generated by the 7 imaginary Octonions {i,j,k,E,I,J,K} onto a lattice (whose fineness is determined by the order n of the Chebyshev iterations) in the real line segment [0,4], which 7 real line segments form the 7-dimensional cube [0,4]^7 of a spacetime volume element [0,4] x [0,4]^7 with signature -+++++++.

The vertices thus constructed on the 8 axes {1,i,j,k,E,I,J,K} of the 8-dimensional spacetime volume element 8-dimensional lattice should be completed throughout the 8-dimensional spacetime volume element so that the full 8-dimensional spacetime lattice is made up of 8-dimensional E8 HyperDiamond Lattices with 8-dimensional Feynman Checkerboard Structure .

The structure is based on sequences and limits of the form

and the fact that the square roots of -1 include i, j, k, E, I, J, K and their negatives.

The role played by the 7 distinct E8 lattices is in representation of the 7 charged first-generation fermions (electron, 3 up quarks, 3 down quarks) in the Generalized Feynman Checkerboard.

After Dimensional Reduction to 4-dimensional Physical SpaceTime, forms a 4-dimensional Generalized Feynman Checkerboard.

From this Octonion Chebyshev Bernoulli Shift point of view, only 3 {i,j,E} of the 7 {i,j,k,E,I,J,K} imaginary octonions are algebraically independent, so physical spacetime should have 3 spatial dimensions.

After Dimensional Reduction, physical spacetime has S1 x S3 (1+3)-dimensional Quaternionic Structure, which can be described using 4 separate Chebyshev maps:

- one mapping from the timelike S1 related to the Quaternion real axis onto a lattice (whose fineness is determined by the order n of the Chebyshev iterations) in the real line segment [0,4], which real line segment is the timelike segment of a spacetime volume element [0,4] x [0,4]^3 with signature -+++; and
- 3 mapping each from the 3 basis S1 circles of S3 generated by the 3 imaginary Quaternions {i,j,k} onto a lattice (whose fineness is determined by the order n of the Chebyshev iterations) in the real line segment [0,4], which 3 real line segments form the 3-dimensional cube [0,4]^3 of a spacetime volume element [0,4] x [0,4]^3 with signature -+++.

The vertices thus constructed on the 4 axes {1,i,j,k} of the 4-dimensional physical spacetime volume element 4-dimensional lattice should be completed throughout the 4-dimensional spacetime volume element so that the full 8-dimensional spacetime lattice is a 4-dimensional Physical SpaceTime which forms a 4-dimensional Generalized Feynman Checkerboard.

The structure is based on sequences and limits of the form

and the fact that the square roots of -1 include i, j, k and their negatives. ]

As to the general properties of

an interesting web page describing Bernoulli Numbers is maintained by S. C. Woon, who also wrote a paper Analytic Continuation of Bernoulli Numbers, a New Formula for the Riemann Zeta Function, and the Phenomenon of Scattering of Zeroes.

can be seen as coming from binary decision trees, similar to the binary decision trees that produce Markov Processes and the binary decision trees that produce the Surreal Numbers:

Bernoulli Numbers are also related to Homotopy Theory.

A single ISING MODEL STATE configuration is one of all possible spin configuration states. Choice of a particular single configuration is aRANDOMstatistical choice.

Feynman's Relativistic Chessboard as an Ising Model, by H. A. Gersch (Int. J. Theor. Phys. 20 (1981) 491), shows that the (1+1)-dimensional Feynman Checkerboard, which describes the (1+1)-dimensional Dirac equation, is equivalent to the 1-dimensional Ising Model. There are interesting relationships among Ising Models and Cellular Automata and Spin Networks. Generalized Ising Models based on E8 lattices and octonions might accurately represent the HyperDiamond Feynman Checkerboard picture of the D4-D5-E6-E7-E8 VoDou Physics model.

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