variants, with a single velocity replaced with a

spectrum of velocities, and associated probabilities

related to FCCR.

- Random Flight and FCCR
- Standard Dichotomic Random Flight
- Polychotomic Random Flight
- Probability Proportional to Inverse Distance
- Probability Proportional to Inverse Square Distance
- The G(n)-cone of Hilb(n)
- The Emergence of Stochastic Relativity

In what way is an element of n-dimensional Hilbert space for FCCR associated with a path in the Zitterbewegung picture of space-time paths presented by Feynman for the construction of a propagation kernal? Let |y> = Σ α_{k}|q(n, k)> k Presumably each |q(n, k)> represents a lightlike path segment (or process) of spatial displacement q(n, k). These displacements have sign and |y> is a linear combination of these displacements with complex weighting α_{k}. If the α_{k}determine arealconvex combination of the |q(n, k)>, then <y| G(n) |y> < 0, and said to be time-like or within the light cone. In the random standard flight problem in one spatial dimension, a waiting time of (n τ_0) gives an expected displacement from the origin of λ_{0}sqrt(n). where τ_{0}is the time quantum of each displacement, λ_{0}is the displacement quantum, and n is the number of displacements, so that the velocity of each displacement is c := λ_{0}/τ_{0}. In FCCR, n is the dimension of the Hilbert space. The spectral radius of Q(n) (Then also of P(n)) is asymptotically given by its maximal eigenvalue: π (n-1) h(n, k_max) = ------------ -> (π/4) sqrt( n ) 2 sqrt(n) 2 which increases as sqrt( n ). |y> that are real convex combinations of the eigenvectors of Q(n) have not only <y| G(n) |y> < 0, but also <y| Q(n) |y> < h(n, k_max). In FCCR then, n the dimension of the Hilbert space can be taken proportional to a waiting time. Restriction of linear combinations to convex combinations seems to restrict expected q values to those less than or equal to the expected value for random flight. The expected random flight displacement then assumes the value of a maximal displacement. Note, however, that for a random flight problem associated to FCCR, that the displacements are not uniform. It would appear that there are more complications to the problem; however, the displacements approximate additivity rather quickly with increasing n. The general situation for arbitrary |y> is not that at every time quantum the particle suffers a displacement of +1 or -1 with equal probability, but rather that at every time quantum the "particle" suffers a displacement of q(n, k) with relative probability |<q(n, k)|y>|². With this picture, and the alternatives of displacement, the significance of the linear combinations of the Hilbert space becomes clearer as the expression of interferring alternatives of process. With this as the general picture, Feynman's construction of the relativistic kernal can be seen as the case n=2. This indicates a connection with the Dirac equation and therefore with spin-1/2. Then also, higher n could be associated with higher spin phenomena. "Spin is a statistical effect, where QM is already a thermodynamic limit." But also see Classical Physics & Geometry Redux. A true QM limit then should probably involve replications of finite FCCR; but then the limit will probably not be QM, but QFT.

A particle is allowed to take a single positive step or a single negative step at each time increment. The "local velocity" is a constant. Assume simple step additivity and probablistic independence of each step The Variables n: "waiting time" m: displacement waited for m+: the number of positive steps in a path m-: the number of negative steps in a path p+: probability of a positive step p-: probability of a negative step The dichotomic random flight distribution is binomial W(n, m) = (n m+) (p+)^{n}(p-)^{(n-m+)}= (n m+) (p+ p-)^{n}(p-)^{(- m+)}with the binomial coefficient (n m+) = n!/((n - m+)! m+!) So normalization of the distribution holds when p+ + p- = 1: n Σ W(n, m) = 1 m=0 With n fixed, the expected value of m, n E(m) := Σ m W(n, m) = n p+ = n (1 - p-) m=0 p+ = E(m)/n p- = 1 - E(m)/n the variance, Var(m) := E( (m - E(m))² ) = n p+ p- p+ p- = Var(m)/n = (E(m)/n)(1 - E(m)/n) So, with simple substitutions, W(n, m) = (n m+) ( Var(m)/n )^{n}(1 - E(m)/n)^{(-(n+m)/2)}where m+ = (n+m)/2 m- = (n-m)/2 so m+ + m- = n m+² + m-² = m² + n² m+ - m- = m m+² - m-² = n m n² - m² = 4 m+ m- ln W(n, m) = ln (n m+) + n ln ( Var(m)/n ) - ((n+m)/2) ln (1 - E(m)/n) Using Stirling's asymptotic approximation, and --- The distribution is specialized to Bernoullian if p+ = p- =: p, then p = 1/2, and W(n, m) = (n m+) 2^{(-n)}= (n m-) 2^{(-n)}For large n, and m << n, Cf. [Wilks 1962] p. 257 "DeMoivre-Laplace theorem" W(n, m) -> (2/pn)^{(1/2)}exp( -(m²)/2n ) = (2/n^{(1/2)}) exp( -(m²)/2n ) Similarly for the FCCR(2) clock of two pointer positions, without memory: The Variables n: "waiting time" m: clock displacement waited for m+: the number of clock steps in a path m0: the number of static steps in a path p+: probability of a positive step p0: probability of no step, static

Now, the particle is allowed to take steps of length s in the set {-r, -r+1, ..., r-1, r} at each time increment. The local velocity is statistical but bounded by r. Its values are, as the the step length values additive. For r even, s has 2r possible values; for r odd s has 2r+1 possible values. Odd allows for the possibility that no actual step takes place. Generally, there is a probability p_{s}associated with a step of length s, so Σ p_{s}= 1 s Then assuming that waiting time is measured in integers and that displacements are also measured in integers, the expectation value of the square of the local velocity is: E( v² ) = Σ p_{s}s² s The random flight distribution is given by W(n, {m}) = (n {m}) Π p_{s}^{(m^s)}s where {m} := (m_(-r), m_(-r+1), ..., m_(r-1), m_r) and (n {m}) is a multinomial coefficient: (n {m}) := n! / Π m_{s}! s and Σ m_{s}= n s Normalization of the multinomial probability distribution holds without any further requirement: Σ W(n, {m}) = 1 {m} If p_{s}= p for all s, (Homogeneous Polychotomic Random Flight) so then p = 1/(2r) for r even, p = 1/(2r+1) for r odd The random flight distribution simplifies to W(n, {m}) = (n {m}) p^( Σ m_{s}) s = (n {m}) p^{n}| (n {m}) (2r)^{-n}for r even = | | (n {m}) (2r+1)^{-n}for r odd The expected value of the square of the local velocity is | r | Σ 1/(2r) s² for r even | s=1 E( v² ) = | | r | Σ 1/(2r+1) s² for r odd | s=0 or | (r+1)(2r+1)/12 for r even E( v² ) = | | r(r+1)/6 for r odd So the root mean square local velocity is | sqrt[(r+1)(2r+1)/12] for even v_rms(r) = | | sqrt[r(r+1)/6] for r odd | sqrt[(r+1)/2] sqrt[r/3 + 1/6] for even = | | sqrt[(r+1)/2] sqrt[r/3] for r odd Which, asymptotically grows linearly with r. An interesting distinction for r odd and even: | 1/sqrt(6) for even lim v_rms(r)/r = | r->infinity | 0 for r odd A bound on r hence on s would be implied by a physical bound on the space in which the particle resides. Such a physical bound then also bounds the allowed values of each of the m_{s}, therefore also their sum, hence n. A bound on r might also be considered as a bound on available energy, (uncertainty of energy?) with the idea of some kind of binding potential for particles in space: space or spacetime is slightly sticky. A new sticky position removes sticky constraints of an old sticky position. In this respect consider a local SHO model with finite, cutoff for the range of attraction. Consider the asymptotic form of the homogeneous random flight distribution for large n and m_{s}<< n, using Stirling's asymptotic formula: log n! = (n + (1/2)) log n - n + (1/2)log( 2 π ) log W(n, {m}) = log (n {m}) + n log p = (n + (1/2)) log n - n + (1/2)log( 2 π ) - Σ log m_{s}! + n log p The distribution will peak when all of the m_{s}are approximately equal; call this value just m, then, log W(n, {m}) = (n + (1/2)) log n - n + (1/2)log( 2 π ) - s_max log m! + n log p But we knew that Σ m_{s}= n so in the peak approximation where S is the maximal value of s, S m = n Therefore log W(n, {m}) = (n + (1/2)) log n - n + (1/2)log( 2 π ) - S log (n/S)! + n log p Now S << n and is fixed, so using Stirling's formula again, log W(n, {m}) = (n + (1/2)) log n - n + (1/2)log( 2 π ) - S (n/S + (1/2)) log (n/S) + (n/S) - (1/2)log( 2 π ) + n log p = (n + (1/2)) log n - n - S (n/S + (1/2)) log (n/S) + (n/S) + n log p Collecting terms and exponentiating we have the asymptotic formula, W(n, {m}) = sqrt( n ) S^{n}(S/n)^{(S/2)}p^{n}exp( -n(1 - (1/S)) ) For r even or odd, in the homogeneous case we always have S = 1/p, and m = np p = m/n Therefore, W(n, {m}) = sqrt( n ) (pn)^{(-1/2p)}exp( -n(1 - p) ) W(n, m) = sqrt( n ) (m)^{(-n/2m)}exp( -(n - m) )

What if the probability of displacement is inversely proportional to the magnitude of displacement? If p_{s}= A/|s| for all s Σ 1/|s| = 1/A s For n = 2r and 2r+1 r 2 Σ 1/k = 1/A k=1 so asymptotically for large n A = 1/[2(C + log n)] A Σ s/|s| = 0 E( v² ) = Σ p_{s}s² = s A Σ s²/|s| = A (n-1)/2 if n is odd A n/2 if n is even which approximate each other for large n. Applying L'Hospital's rule, the rms velocity is unbounded with r. Asymptotically [Jolley 1961] #70 n Σ 1/k -> C + log n + 1/2n - 1/[12n(n+1)] - 1/[12n(n+1)(n+2)] ... k=1 where C is the Euler-Mascheroni constant C = 0.577216

Now try the probabilities of displacement being proportional to the inverse square of the displacements. If p_{s}= A/s² for all s Σ 1/s² = 1/A s For n = 2r and 2r+1 r 2 Σ 1/k² = 1/A k=1 so asymptotically for large n A = 1/[2(π / 6)] = 3 / π E( v² ) = Σ p s² = Σ (A/s²) s² s s = Σ A = n A = 3n/π s v_rms(n) = sqrt( 3n/π ) which increases as sqrt(n), as do the spectral radii of Q(n) and P(n). QM which in a limit of FCCR is not a relativistic theory and has a fundamentally infinite propagation velocity. If then, FCCR(n) correctly replaces CCR as the fundamental kinematical restriction of quantum theory, and the existence of a finite fundamental velocity of propagation is required, the indication is that taking the n->∞ limit would be in error. A stab at a lower bound for an appropriate value of n for n would be; sqrt( 3(n+1)/π ) - sqrt( 3n/π ) ≤ [precision by which c is known] Rather amazingly, the CODATA value of the velocity of light in a vacuum, 299 792 458 meters/sec is there claimed to be "exact"! Perhaps the US government has redefined "exact". Nevertheless, if we take this to be true (or true enough), or that this is as good as it gets, and that relative to decimal representations the error or uncertainty is anything less precise than this, then the error is of the order of 10^-11 m/s, and so, sqrt( 3/π ) ( sqrt(n+1) - sqrt(n) ) ≤ 10^-11 For very large n, using a Taylor expansion, sqrt(n+1) = sqrt(n) + (1/2)(1/sqrt(n)) So, sqrt( 3/π ) ( (1/2)(1/sqrt(n)) ) ≤ 10^-11 and squaring, and inverting, n ≥ (3/(4π)) 10^22 This is a Hilbert space dimension quite large enough that the physics, cosmologically calculated using it will be well approximated by the usual continuum model involving differential equations. It also informs the derivation of local Newtonian time from fundamental quantum principles.

In special relativity, the vertex angle θ of the light cone, defined by x² = c² t² is related to the velocity of light by tan( θ/2 ) = 1/c where θ is the vertex angle of the cone. In FCCR, the vertex angle θ(n) of the G(n)-cone in Hilb(n) is given by tan( θ(n) / 2 ) = 1/sqrt(n-1) := 1/c(n) The cone opens up in the limit and the "timelike" basis vector |n, n-1> disappears if the limit is QM. While the limit of FCCR(n) can be QM, it need not be. See [Section III] on limits in FCCR, and especially [Section XVII] more limits in FCCR. Numerically, and approximately, of course, sqrt( 3/π ) = 0.977205 If we relate the two cone angles, by c(n) = sqrt( (3/π) (n-1)/n ) v_rms(n) = sqrt( n-1 ) sqrt( (3/π) (n-1)/n ) → sqrt( 3/π ) So, asymptotically, c(n) = sqrt( (3/π) ) v_rms(n) = sqrt( n-1 )

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