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Limits with physical constants and scaling (Contractions)

   Limits of the phase operator and its canonical commutator

   In the limit as n->infinity, for k, j = 0, 1, ..., n-1, the variable

          omega  :=   pi (k-j)/n                            (17.1)

   becomes a continuous variable with range - pi <= omega <=  + pi.
   In the limit as n->infinity, for m = 0, 1, ..., n-1, the variable

          s :=  m / sqrt(n)                                            (17.1)

   becomes a continuous variable with range 0 <= s < infinity.
   The limit of the unitarity expression for the Fourier transform
   UPSILON(n) defined by  (7.21):

        (1/n) SIGMA exp[ i2 pi(k/ sqrt(n) - j/ sqrt(n))(m/ sqrt(n)) ]  ->

        (1/(2 pi)) INTEGRAL exp(i 2 pi(x - y)s)  d(2 pi s)             (17.2)

   which defines a Heisenberg delta distribution.

      INTEGRAL exp(i 2 pi(x - y)s) d(2 pi s)  =  delta_+(x - y)     (17.3)

   The unitarity of UPSILON(n) is metaphorically cut in half.
   Considering also the limit of the phase operator F(n) given by

           F(n)  =  UPSILON(n) N(n) UPSILON!(n) 

        <n, k|F(n)|n, j>  =
   SIGMA (m/sqrt(n)) exp[ i 2 pi(k/sqrt(n) - j/sqrt(n))(m/sqrt(n)) ] (1/sqrt(n))

   we have with alpha = lim k/sqrt(n), beta = lim j/sqrt(n), s = lim m/sqrt(n)

        <n, k|F(n)|n, j>  ->  INTEGRAL s exp(i 2 pi(alpha - beta)s) ds


        delta_+(2 pi beta)  =  (1/2 pi)delta_+(beta)                    (17.5)


        (PARTIAL/PARTIAL beta) delta_+(2 pi beta)  =

      (i 2 pi) INTEGRAL s exp(i 2 pi beta s) ds                      (17.6)

   The distribution delta_+( beta ) has the property

     beta (PARTIAL/PARTIAL beta)delta_+(beta)   =  - delta_+(beta)   (17.7)

   So in the limit

                                i     delta_+(alpha - beta)
     <alpha|F(n)|beta>  =  ---------- ---------------------
                           (2 pi)^(2)   (alpha - beta)

                 =  ---------- delta^(1+)(alpha - beta)              (17.8)
                    (2 pi)^(2)


     <alpha|[N, F]|beta>  =  ---------- delta_+(alpha - beta)
                             (2 pi)^(2)

   where delta^(1+)(alpha - beta) is a first derivative, and formally,

     N   :=  lim (N(n) / sqrt(n))                                    (17.10)

   with a continuous spectrum bounded below.  This limit then
   is not that of the QM harmonic oscillator.

A Discrete Analog of the Derivative

In QM, when the position operator is in diagonal form and treated as a "multiplication operator", the momentum operator is represented as a derivative operator. We say this as simple hand waving, and do not bother here about the messy details of distributions in defining what it means for a square integrable SCRIPT-L^(2) function to be differentiable. In fact, it is not so odd that P(n), or Q(n) can behave like a derivative since mappings A -> [A, M] are derivations in matrix algebras and obey non-commutative versions of all the Leibnitz formulas. [Putnam 1951] and [Appendix B].

The Fourier transform has been used to extend the notion of derivative [Dold 1975], and can also be used here in its discrete incarnation to define a notion of discrete derivative.

In the |q(n, k)> in which Q(n) is diagonal, Q(n) acts like a multiplication operator. Cf. [Section IX].

   Let psi(q) := <psi|q>, (q = q(n, m)) in Sp( Q(n) )
   be a function defined on the spectrum Q(n).

     (Q(n) psi)(q) = <psi|Q(n)|q>
            = q <psi|q>
            = q psi(q)                                               (17.11)


     (P(n) psi)(q) = <psi|P(n)|q>
            = SIGMA_(jk) <psi|q(n, j)><q(n, j)|P(n)|q(n, k)><q(n, k)|q>
            = SIGMA_j  <psi|q(n, j)><q(n, j)|P(n)|q>

                                       exp(i pi (j-m))
            = -i SIGMA psi( q(n, j) ) -----------------              (17.12)
                (q(n,j) not= q)       q(n, j) - q(n, m)

which has the formal appearance of a Hilbert integral transform of a complex valued function made discrete, or a discrete version of an integral kernel whose limit is a Dirac delta function.

Limits of FCCR approaching QM

In the following, a limiting process for FCCR is examined that has as a limit, the typical conditions for QM with the absence of bound states: both space and time are effectively continuous paramaters, energy is not discretely quantized, a fundamental velocity is bounded, and there exists a constant with the physical dimensions of the gravitational constant, which is bounded and not zero.

The limiting procedure is not unlike that of group contraction, but is distinct in the following way. Group contraction is set up by scaling and therefore parametrizing the group structure constants of a Lie group in such a way that their defining properties are maintained for any value of the parameter(s), then tying the parameter values to representations. The contraction is then the structural limit obtained by passing through a sequence of representations. Here, however, the structural limit is that of a one parameter sequence of Lie groups in their defining representations, where each Lie group in the sequence contains as a subgroup, the preceding group. While this is all stated in terms of Lie groups, the machinery actually used are the associated Lie algebras. In [Section VI], it is shown that Q(n) and P(n) generate the Lie algebra su(n) by commutation. The limits discussed here are only in terms of the fundamental commutation relation (2.8).

   Introduce physically dimensioned, nonzero, absolute
   and effective constants (functions of n) of:

        length:     l_0, l(n)
        momentum:   p_0, p(n)
        time:       tau_0, tau(n)
        action:     h-bar,   h-bar(n)

   with, as usual

        h-bar := h/(2 pi)

   For FCCR
        [Q(n), P(n)]  =  i G(n)

   dress the fundamental operators with multiplicative
   scalar functions of the dimension, and of appropriate physical

     Q(n)  ->  ~Q(n)  =  (l(n)/(sqrt(2))) Q(n)
     P(n)  ->  ~P(n)  =  (p(n)/(sqrt(2))) P(n)                    (17.13)
     G(n)  ->  ~G(n)  =  (h-bar(n))G(n)

   and assume that for the fundamental dimensioned constants

     l_0 p_0  =  h-bar                                               (17.14)

   Then FCCR implies for the effective constants that

     l(n) p(n)  =  h-bar(n)                                          (17.15)

   If FCCR has as a limit, CCR, when n->infinity

     lim h-bar(n)  =  h-bar                                          (17.16)
     lim tau(n)  =  0

   Without loss of generality let

     l(n)    :=  l_0 alpha(n)
     p(n)    :=  p_0/beta(n)                                         (17.17)
     tau(n)  :=  tau_0 gamma(n)

   where alpha(n), beta(n) and gamma(n) are then defined as
   dimensionless functions of n.

      h-bar(n)  =  h-bar (alpha(n)/beta(n))
      lim (alpha(n)/beta(n))  =  1
      lim gamma(n)  =  0

If one assumes that a single l(n) transition requires at least one tau(n) transition, an effective maximal velocity c(n) can be defined

      c(n)  :=  DELTA(~q)(n)/DELTA(~t)(n)                            (17.20)
             =  sqrt(pi/(2n)) (l_0 alpha(n))/(tau_0 gamma(n))

   where DELTA(~q)(n) is the spectral spacing of ~Q(n).
   The gravitational coupling constant G_0 of the Einstein
   equations which couples stress-energy-momentum to
   geometrical curvature has the physical units (Bridgeman-dimensions):

        [L^(3)] [M^(-1)] [T^(-2)]  =  c^(4) [L] [E^(-1)]

   Introduce first, an effective energy quantum

     e_0(n)  :=  epsilon(n) h-bar(n)/tau(n)                          (17.21)

where now epsilon(n) is defined as a dimensionless function of n which allows the freedom to express the energy scaling and structure that might be appropriate for some given system in the QM limit.

   Now express an effective gravitational constant as

     g_0(n)  :=  c^4(n) (DELTA(~q)(n)/e_0(n))                        (17.22)
              =  c^5(n) tau(n)/e_0(n)


The dynamical Ansatz which determines a time operator from a given Hamiltonian is given in Theorem 8.20. Let |E_k> be the eigenbasis of the Hamiltonian H(n). Construct the unitary cyclic matrix on this eigenbasis, so

     C(n) |E_k>  =  |E_(k-1)>

where the index of eigenvectors is understood mod n. C(n) is a unitary operator and may therefore be expressed as

     C(n)  =  exp( i omega_H(n) t_H(n) )

where omega_H(n) is a frequency constant attached to the Hamiltonian, and t_H(n) is the time operator for the Hamiltonian H(n). We assume that omega_H(n) is real and that t_H(n) is Hermitean; this consistent with C(n) being cyclic and so also then unitary. But we also know from the structure of C(n), that

     C(n)  =  exp( i ((2 pi)/n) N_t)(n) )

   where N_t(n) is the number operator in the eigenbasis
   of t_H(n).  Therefore

     ((2 pi)/n) N_t(n)  =  omega_H(n) t_H(n)                         (17.23)

   Then since N_t(n) has an additive spectrum, so does t_H(n) and

     omega_H(n)  =  ((2 pi))/(n tau_0 gamma(n))                      (17.24)

   Desiderata for n=2 and for the limit of unbounded n:

        lim tau(n)  =  0
        lim Sup Sp( t(n) )  =  infinity
             implies for large n
                  O( gamma(n) )  =  n^(delta(t))
                  0 < delta(t) < 1

        tau(2)  =  tau_0
                  gamma(n)  =  (n-1)^(-delta(t))

     Therefore take

               gamma(n)  :=  (n-1)^(-delta(t))                       (17.25)
               0 < delta(t) < 1


     Sp( t_H(n) )  ->  [0, infinity]


     omega_H(n)  =  ((2 pi)/(n tau_0))) (n-1)^(delta(t))
                 =  ((2 pi)/tau_0)) n^((delta(t)-1)), for large n,

   and then

     lim omega_H(n)  =  0                                            (17.26)

   for all values of delta(t).

   Recalling that

     C(n) = exp( i omega_H(n) t_H(n) )

   The limit

     lim C(n)  =  SHA_H(infinity)  =  SHA

is a one sided shift operator and is not cyclic; the exponential representation cannot exist in the limit: [Section II]. Therefore, one should *not* *even* *expect* that omega_H(n) t_H(n) will have a proper limit. The phase in the t(n)-eigenbasis has a limit of a continuous phase operator with eigenvalues defined on the unit circle in C. But the t(n)-eigenbasis has no physically acceptable existence in the limit since UPSILON_H(n) the transformation connecting the H(n)-eigenbasis with the t(n)-eigenbasis has no physically acceptable limit.

     <n, k|UPSILON_H(n)|n, j>  =   (1/sqrt(n)) exp( i((2 pi)/n)kj )


Substituting the expression (17.25) for gamma(n) into the previous expression (17.20) for c(n),

  c(n)  =  (l_0/tau_0) sqrt(pi/2) alpha(n) n^(-1/2) (n-1)^(delta(t)) (17.27)

   It is a relativistic desideratum that

        lim c(n)  =  c  < infinity

   This can be arranged by taking

     alpha(n)  :=  n^(1/2) (n-1)^(-delta(t))                         (17.28) 

   In this case

     c(n) = c = (l_0/tau_0) sqrt(pi/2)                               (17.29)

   which is independent of n.
   Also take

     beta(n)  =  alpha(n)                                            (17.30)


     h-bar(n)  =  h-bar                                              (17.31)

which is also independent of n. Incorporating these results into the effective energy quantum and the effective gravitational constant:

  e_0(n)  =  (h-bar/tau_0) epsilon(n) (n-1)^(delta(t))               (17.32)

  g_0(n)  =  (c^5 tau_0^2/h-bar) (n-1)^(-2delta(t)) epsilon^(-1)(n)  (17.33)

   Minimal physical desideratum of the energy quantum:
   In the limit, one should have,

        lim e_0(n)  < infinity

   Note that

     If we take
          epsilon(n)  =  (n-1)^(-delta(t))

     lim (n-1)^(delta(t)) epsilon(n)  =  1

     otherwise take
          epsilon(n)  =  (n-1)^(-(delta(t)+delta(e)))
          0 < delta(e)

          lim (n-1)^(delta(t)) epsilon(n)  =  1

     Therefore, generally, for a bounded limit, take

          epsilon(n)  =  (n-1)^(-(delta(t)+delta(e)))                (17.34)

          0 < delta(e)
     so that
          epsilon(n)  ->  1, when delta(e) = 0
          epsilon(n)  ->  0, when delta(e) > 0

   Now consider the limit of 

     g_0(n)  =  (c^5 tau_0^2 /h-bar) (n-1)^(-2 delta(t)) epsilon^(-1)(n)
             =  (c^5 tau_0^2 /h-bar) (n-1)^((delta(e)-delta(t)))

   There are three cases:

         Infinite gravitational coupling (physically unacceptable)
     i)  delta(e) > delta(t), g_0(n)  -> infinity

         Finite gravitational coupling (independent of n)
    ii)  delta(e) = delta(t), g_0(n)  =  (c^5 tau_0^2 /h-bar)        (17.35)

         Complete gravitational decoupling
   iii)  delta(e) < delta(t), g_0(n)  ->  0

   where the only restrictions within these three cases on
   the delta(e) and delta(t) are

           0 <= delta(e)
           0 < delta(t) < 1

   Eliminating unphysical case (i), if delta(e) -> 0,
   then for case (iii)

           e_0(n) -> e_0 not= 0,
           g_0(n) -> 0

   and for case (ii) (the case of physical choice)

           delta(e)  =  delta(t)
           0 < delta(t) < 1
           e_0(n) -> 0
           g_0(n) = (c^5 tau_0^2 / h-bar)

   Asymptotic Spectral Radii of operators for large n are:

           rho( Q(n) )  =  tau (pi/2n) (n-1)/2
           rho( P(n) )  =  tau (pi/2n) (n-1)/2
           rho( G(n) )  =  (n-1)

           rho( ~Q(n) )  =  (l(n)/tau 2)tau(pi/2n) (n-1)/2
           rho( ~P(n) )  =  (p(n)/tau 2)tau(pi/2n) (n-1)/2
           rho( ~G(n) )  =  h-bar (n-1)

           rho( t_H(n) )  =  tau_0 (n-1)^(1-delta(t))
           0 < delta(t) < 1

   For all allowed values of delta(t),

           lim rho( ~Q(n) )  =  infinity
           lim rho( ~P(n) )  =  infinity
           lim rho( ~t(n) )  =  infinity

   We have now proved the following

   Theorem 17.1:

   By Introducing nonzero, absolute and effective
   dimensioned constants of:

        length:     l_0, l(n)
        momentum:   p_0, p(n)
        time:       tau_0, tau(n)
        action:     h-bar,   h-bar(n)


        l_0 p_0  =  h-bar

   for FCCR

        [Q(n), P(n)]  =  i G(n)

   and defining the dressed operators

        ~Q(n), ~P(n)

   as above, so that

        [~Q(n), ~P(n)]  =  i h-bar G(n),

   a limit with no bound states, of FCCR can be defined with

        FCCR  ->  CCR

when a "Hamiltonian" H(n) is defined and the associated phase-time operator t_H(n) is defined by the Ansatz as above and in Theorem 8.20 and beyond. The limits of the operator spectral spacings are:

        DELTA(~q(n))    ->  0
        DELTA(~p(n))    ->  0
        DELTA(t_H)(n)   ->  0

        DELTA(~q(n)) DELTA(~p(n))  =  (pi/(2n)) h-bar
        e_0(n) tau(n)  =  epsilon(n) h-bar

and, therefore that ~Q(n), ~P(n), t_H(n) all have as a limit operators with unbounded, dense spectra that are linearly extendible to continuous spectra. Moreover, in this limit, the constructed constants, c, h-bar, g_0 all have finite nonzero values consistent with their standard physical interrelationships. This limiting behavior can be achieved by choosing the undetermined dimensionless scaling functions to be contrained by:

     alpha(n)  =  alpha(n)  =  (n-1)^(-delta(t)+1/2)
     gamma(n)  =  (n-1)^(-delta(t))
     epsilon(n)  =  (n-1)^(-2 delta(t))
     0 < delta(t) < 1

   Relation of our assumed physical constants to the Planck units

   The fundamental and Planck units are given by:

Symbol   Name              Abstract Units    cgs approximate value    

c        Speed of Light         [LT^(-1)]       2.998E+10 cm/sec
h-bar    Reduced Planck         [ET]            1.0546E-27 gm-cm^2/sec
G_0      Gravitation            [M^(-1)L^3T^(-2)]   6.673E-8 cm^(3)/gm-sec^(2)
l_P      Planck length          [L]             1.616E-33 cm   
          sqrt(h-bar G_0)/c^3)
tau_P    Planck time            [T]             5.391E-44 sec
          sqrt(h-bar G_0)/c^5)
mu_P     Planck mass            [M]             2.177E-05 gm
          sqrt(h-bar c/G_0))
e_P      Planck energy          [ML^(2)T^(-2)]  1.95669E+16 ergs
          sqrt(h-bar c^5/G_0))

p_P      Planck momentum        [MLT^(-1)]      6.52389E+4 gm-cm/sec
          sqrt(h-bar c^3/G_0))

   which are related by

          e_P tau_P  =  h-bar
          l_P p_P    =  h-bar

          l_P        =  c tau_P
          e_P        =  mu_P c^2

   The standard gravitational constant may then be expressed

          G_0  =  (tau_P^2 c^5 )/h-bar

From this it appears that we have specific identifications within the Planck system since our c(n), g_0(n) and h-bar(n) are actually independent of the dimension n, we identify them with the fundamental constants upon which the Planck system is founded, then:

        tau_0  =  tau_P
        l_0  =  l_P
        g_0  =  G_0

Our effective quantum of energy, hence also effective mass quantum has a zero limit, and so quantities of these units are allowed to be continuously selected in the limit. Since for these quanta

        e_0(n)  =  m_0(n) c^2

we need only be concerned with the behavior of one of them and the behavior of the other necessarily follows. First remember that the exponent delta(t) has *not* yet been determined, but only constrained by

        0 < delta(t) < 1.

and so the other effective constants as functions of n have not yet been determined. Consider the energy quantum. Given the relations so far established, write the expression for e_0(n):

        e_0(n)  =  h-bar epsilon(n)/tau_0(n)
               =  (h-bar/tau_0) (n-1)^(-delta(t))

   If for some n finite

        e_0(n)  =  e_P
        (h-bar/tau_0) (n-1)^(-delta(t))  =   sqrt(h-bar c^5/G_0)

        (h-bar/tau_0^2) (n-1)^(-2 delta(t))  =  (c^5/G_0)
        (n-1)^(-2 delta(t))  =  (c^5/G_0)/(h-bar/tau_0^2)
        (n-1)^(2 delta(t))   =  (h-bar/tau_0^2)/(c^5/G_0)
        (n-1)^(2 delta(t))   =  (e_P/tau_0)
        (n-1)^(2 delta(t))   =  1

   For n > 2, this equation can only hold if the constraint
   on delta(t) is violated.  Therefore,

        e_0(n)  not=  e_P

   for any n > 2.  If n = 2, the equation is identically
   satisfied and there is no further constraint on delta(t),
   we have

        e_0(2)  =  e_P

   consequently also then

        m_0(2)  =  e_P/c^2
        m_0(2)  =  mu_P

   Generally for n > 2

        e_0(n)  =  (h-bar/tau_0) (n-1)^(-delta(t))
        m_0(n)  =  (h-bar/(tau_0 c^2)) (n-1)^(-delta(t))

   are energy and mass quanta.  But still delta(t) is not determined.
   The quanta e_0(n) and m_0(n) converge to zero
   as n->infinity more slowly, the smaller delta(t).

Limit of the Harmonic Oscillator

Since the entire notion of FCCR started with a variant of the simple harmonic oscillator of QM, it would be rather damning of this mathematical venture should one not address and show explicitly the limiting process which arrives back at the standard quantum mechanical oscillator. Using the above limit as a model for procedure, begin again almost from the beginning. The simple oscillator of QM is a classical system, and so the the maximal velocity c(n) should have an unbounded limit.

Let ....

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Created: August 1997
Last Updated: May 28, 2000
Last Updated: February 12, 2004