1. Random Flight and FCCR
2. Standard Dichotomic Random Flight
3. Polychotomic Random Flight
4. Probability Proportional to Inverse Distance
5. Probability Proportional to Inverse Square Distance
6. The G(n)-cone of Hilb(n)
7. 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  αk |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 αk.

   If the αk determine a real convex 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
   λ0 sqrt(n).
   where τ0 is the time quantum of each displacement, λ0 is the
   displacement quantum, and n is the number of displacements, so that the
   velocity of each displacement is c := λ0/τ0.

   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 ps associated with a step of length s,
   so

           Σ  ps  =  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  ps s²
                        s

   The random flight distribution is given by

     W(n, {m})  =  (n {m}) Π ps(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 ms!
                            s

   and

           Σ  ms  =  n
           s

   Normalization of the multinomial probability distribution holds without
   any further requirement:

           Σ  W(n, {m})  =  1
          {m}

   If ps = 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  ms)
                               s

                =  (n {m}) pn

                  | (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 ms, 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 ms << 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 ms! + n log p

   The distribution will peak when all of the ms 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  ms  =  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 ) Sn (S/n)(S/2) pn 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 ps = 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  ps 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 ps = 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 )