Notes on possible connections between
de Broglie's old wave theory and FCCR:
a bifurcation of Classical time concept into
"Dynamic time" and "Refractory time",
and the reality of de Broglie's "proper frequency".


In his work combining quantum theory and relativity, De Broglie assumed the existence of a cyclic process associated with a massive particle. A question arises immediately: why?

The all too famous, and therefore trivialized, relation

                    E = m_0 c^2,

an interpretive result from special relativity, relates the rest mass m_0 to an equivalent energy E does not tell us, within the mathematical arithmetic equivalence, what physically distinguishes m_0 from E; more bluntly, what distinguishes physical mass from physical energy. This is a question to which there is still no satisfactory answer.

It appears that De Broglie was attempting to give at least some structural answer to the question by saying that somehow mass should be viewed as a spatially localized circulating energy. From this idea or picture comes the frequency of some cyclic process. In more modern times we might call this the picture of a massive particle as a vortex. It is hard to ignore that fact while particles of spin 1/2, and integral multiples are abundant in the pantheon of elementary particles, real encountered particles of spin 0, have more of a theoretic life. The idea of massive particle as quantized vortex seems less intrinsically acceptable as time has passed.

   Possible help from Debroglie's ideas in reconciling the operational
   concept of time as bean counting with both Q & R:
   Within a Q theory (FCCR) this is easy.
   Seeing R in FCCR is related to stating the relativistic invariance of
   bean counting.  This may entail restating and/or modifying the R statement,
   so that bean counting is a thing defined in an inertial frame.
   Bean counting amounts to a quantized invariant, saying for example that
   the value of an analog to
        x^nu x_nu
        p^nu p_nu
   is quantized.  This is exactly the situation on considering the finite
   dimensional IRREPS of sl(2, C), albethey nonunitary.
   Below, consider "time" (that makes the dynamics look good)
   as divided into dynamic and spatial parts.
   Dynamic part expands as usual, the spatial part contracts
   (Lorentz contraction) as a spatial interval in the direction of motion.

        td -> t0 beta^(-1)(v)    The time of motion.
        ts -> t0 beta(v)         The time of stasis or waiting.
        td ts  =  (t0)^2
   and the relativistically invariant quantum t0 is just that.
   Is it necessary (or desirable) to extend conceptually to the (+++-)
   space, and (I_ss I_t) inverted i.e., spatially and temporally inverted
   space (---+)? We can do SR theory with either.
   DeBroglie's wave theory is about reconciling the relativistic frequency
   of a clock associated with a particle having an "internal process"
   associated with mass, with its quantum frequency.
   In the particle's REST frame, define the "frequency" nu_0 by
   associating it with the internal process.
                   h nu_0   =  m_0 c^2
   This defines the frequency nu_0 as that of some internal periodic process,
   that follows the "particle's own clock".  This internal particle clock is
   perfectly ignorant of any observer, and ignorant of the particle velocity.
   As observed by a relatively moving observer, the relativistic kinetic
   energy of the particle can be defined by
                   E(beta)  =  m_0c^2/sqrt(1 - beta^2)
   with beta := v/c.  E(0) = m_0 c^2

   If we divide the first equation by sqrt(1 - beta^2),
                   E(beta) = hnu_0/sqrt(1 - beta^2),
                   T(beta)  =  sqrt(1 - beta^2) T_0
   thus defining a "quantum" energetic frequency of the internal clock
   process as "seen" by a relatively moving observer,
                   nu(beta)  =  nu_0/sqrt(1 - beta^2)
   But, if nu_0 is the frequency of the internal process then,

                   T_0 := 1/nu_0

   is the period of the clock, in the particle's rest frame.
   From the viewpoint of a moving observer, however, this period is longer
   (the clock slows down = "relativistic time dilation")
   and one calculates that the period of the internal process observed 
   by a relatively moving observer is
                   T_1(beta)  =  T_0/sqrt(1 - beta^2)
   if nu_0 is the frequency of the particle's internal
   process as observed in the particle's rest frame, the frequency observed
   by a relatively moving observer, nu_1 = 1/T_1  will be,
                   nu_1(beta)  =  nu_0 sqrt(1 - beta^2)
   It is obvious that nu_1(beta)  not=   nu(beta).
   In fact,
                   nu_1(beta)  <=  nu_0 <=  nu(beta)
                   nu_1(beta) nu(beta)  =  nu_0^2
                   nu_1(beta)/nu(beta)  =  (1 - beta^2)
   with                nu_1(0)  =  nu(0)  =  nu_0
                   lim nu_1(beta)  =  0
   Time quantum(?) lim nu_1^(-1)(beta)  =  infinity
                   lim nu(beta)  =  infinity
   Time quantum(?) lim nu^(-1)(beta)  =  0
   Special Relativity:  Energy <-> Mass    (E  =  m_0 c^2)
              (Connection made with c)
   Quantum theory:      Energy <-> (Time)^(-1) (E  =  h nu)
              (Connection made with h)
   There is apparently a relativistic paradox arising from the
   quantum theoretical association of mass with time.
   Remember that regarding Lorentz Transformations,
        (x, ct)  is a contrvariant 4-vector
        (p, E/c)  is a covariant 4-vector
   Time and energy transform relatively inversly, as is suggested by the
   factors of "c" and "(1/c)" in the fourth component of the 4-vectors.


   It was the apparent irreconcilability of the two frequencies nu(beta)
   and nu_1(beta) that caused DeBroglie subsequently to abandon the picture
   of the internal process associated with nu(beta) and associate it
   with the frequency of a companion wave. (De Broglie - Bohm)
   The problem with these companion waves is that they are then seen to
   travel with a speed greater than c, so the "physical velocity" of the
   particle is then actually equated to the group velocity.
   Here, think in terms of Fourier transforms, power spectra for
   localized wave packets
   and the uncertainty principle.  Think then of the distinction in these
   matters between the standard Fourier transform and the discrete Fourier
   transform.  Dirac delta distributions are replaced with G(n)-null
   Update, August 23, 2005:
   A strong reason for de Broglie's original theory to have been first
   dismissed and then forgotten, yet remembered in quite different terms
   is that experimental results supporting the existence of de Broglie's
   hypothesized internal particle process with "proper frequency" nu_0

			m_0 c²  =  h nu_0

   were fairly lacking.  It appears as if this situation may be a result
   of a genuine lack of trying.  [Gouanère 2005]

   I thank Marcel Gouanère for informing me of the recent experimental work
   involving electron scattering in the neighborhood of 80 MeV by a
   1 micrometer thick Silicon crystal.  As the electron energy is varied,
   there is an 8% dip in the forward scattering within a 0.5% bandwidth
   of a resonance energy of 80.874 MeV that corresponds to the de Broglie
   proper frequency as the frequency of atomic collisions.  This is an
   important result that not only needs replication, but also needs
   replications of variations.

   In a separate addendum [Lochak 2005] to the above cited paper, Georges
   Lochack points out (with references there) that at later date de Broglie
   reconsidered the nature of the expression determining the proper frequency
   nu_0, and suggested that nu_0 be interpreted thermodynamically, as

			k T  =  h nu_0

   seeming to relieve nu_0 of its requirement of involvement with
   relativistic transformations.

   In the discrete situation can one avoid some of DeBroglie's unhappy
   conclusions? There are incompatible notions of time with the two
   frequencies and there are also multiple concepts of time within FCCR.
   [Section XV]
   Associate nu(beta) with the energetic frequency and nu_1(beta) with the
   inverse of a "time quantum", so that when beta -> 1
   Q concept: (Q ties time to Energy) has to do with waves.
           nu(beta) -> infinity
           E(beta) = h nu(beta) -> infinity
   The frequency is that of a wave that is associated to a particle.
   For a photon of mass zero,

           E = h nu

   where nu is the frequency of the electromagnetic wave.  and

           nu lambda  =  c

   To discriminate two spatial points, we need photons with wavelength of
   the order of the separation beteen them:
           (DELTA q)  =  lambda  =  c/nu
                      =  hc/E
   These relations are true regardless of the frame from which they are
   observed since "photons" always move with velocity c.
   Also, from Classical EMT equations,

                     c = 1/sqrt(mu_0 epsilon_0).

   Is there a natural connection with c = (delta q)/(delta t) and
   nu_1(beta) nu(beta)  =  nu_0^2?
   EMT Maxwell Equations in 3+1 form with the electric vectors E and D
   and the magnetic vectors B and H:
           (i)     NABLA X E    =  - (1/c) PARTIAL B/PARTIAL t
           (ii)    NABLA X H    =    (4 pi/c) J + (1/c) PARTIAL D/PARTIAL t
           (iii)   NABLA DOT D  =    (4 pi) rho
           (iv)    NABLA DOT B  =     0
            D  =  epsilon E,                B  =  mu H
            H  =  (1/mu) B
   Permeability and permitivity are inverses of kinds of electric and magnetic
   resistances of space that keep the speed of propagation (ultimately,
   probably in some statistical sense) finite.
   Similarly see h as a diffusion constant for the propagation of state
   functions that causes their spreading in time. (Compare the parabolic
   heat equation to the parabolic Schrödinger equation.)
   If we extend the energy frequency relation to the case of massive particles,
           E(beta) = h nu(beta)  =  m(beta) c^2
           E(beta)  =  m_0 c^2/sqrt(1 - beta^2)
           m(beta)  =  m_0/sqrt(1 - beta^2)
           nu(beta)  =  nu_0/sqrt(1 - beta^2)
   R concept: (R ties time to space) by c, has to do with clocks
              (R ties Energy to mass)
           E(beta)  =  m_0 c^2/sqrt(1 - beta^2)
           nu_1(beta) -> 0
           tau_1(beta) = (nu_1(beta))^(-1) -> infinity
   Discrimination distance of two time points is increased:
   as beta -> 1, time becomes a single point; two or more finitely 
   separated points do not exist; A photon only knows "now".
   nu(beta) and nu_1(beta) are two DIFFERENT
   temporal concepts (moral: waves are not clocks, and conversely!),
   for which there are the interrelations:
                   nu_1(beta)  <=  nu_0 <=  nu(beta)
                   nu_1(beta) nu(beta)  =  nu_0^2
                   nu_1(beta)/nu(beta)  =  (1 - beta^2)
                   nu_1(0)  =  nu(0)  =  nu_0
   The Concepts coalesce exactly for zero relative velocity between frames
   and practically coalesce for low velocities.
   Two Fundamental Concepts of Time contributing to the Classically Observed
   Concept of Time:
   The limiting conditions on the the two frequencies nu(beta) and nu_1(beta)
   might be understood if T_1(beta) = 1/nu_1(beta) is associated with the
   recurrent time or period of the clock, which becomes interminable as beta
   approaches one, while nu(beta) is associated with
   some 'part' of the total transition time between clock states.
   Total transition time might be divided into a refractory time and a time
   of dynamical transition during which the clock pointer actually moves,
   necessarily with some velocity.
   From the viewpoint of time dilation, what does it mean during a time where
   nothing happens (the refractory time) for the time to dilate?
   As the finite "clock slows down" with increasing beta, the clock period can
   become infinite and one of the subdivisions of a transition increment
   can become infinite also in order to keep the number of pointer positions
   invariant, while the other subdivision of an increment goes to zero.
   Then associate T(beta) = 1/nu(beta) with the time interval that goes
   to zero.
   Should the refractory time dilate and dynamical time go to zero,
   or, should the refractory time go to zero and the dynamical time dilate?

   It would seem that if the clock slows down that dead time (refractory time)
   should dominate the behaviour, and it should probably become longer (dilate)
   while the dynamic time going to zero would actually cooperate with the net
   effect of slowing down the clock.
   Examine the dynamical time taking into account that a clock part moves with
   some velocity u <= c and with an adiabatic acceleration.
   Is the direction of u relative to v an important complication?
   Assume both velocities are colinear, then if u is the clock part velocity
   in the clock's rest frame and v is the velocity of the clock, the apparent
   observed velocity of the part is  [Bergmann 1942], p. 43,
                   v + u
           w  =  -----------
                 1 + (vu/c_0)
                        (1 - v/c)(1 - u/c)
              = c [ 1 - ------------------ ]
                            1 + (vu/c_0)
   In addition to the three frequencies above nu_0 nu_1(beta), and nu(beta)
   there is also the Planck frequency nu_p (determining the ultimate absolute 
   time quantum by its inverse?  It would seem so.)
           nu_p  =  [c^5/(h-bar G_0)]^(1/2)
           nu_x  =  2 pi c h/(G_0) m_0)
           [nu_0 nu_x]^(1/2)  =  nu_p
           [nu_1(beta) (nu_x sqrt(1 - beta^2))]^(1/2)  =  nu_p
           [nu(beta) (nu_x/ sqrt(1 - beta^2)]^(1/2)    =  nu_p
   Zitterbewegung averages over lightlike transitions.
           c  =  (delta q)/(delta t)
           v  =  k (delta q)/(delta t) > c
   cannot happen, but
           v  =  (delta q)/(k (delta t)) < c

   This follows immediately if (delta t) is understood as the fundamental
   quantum of the refractory time; that is the minimal time quantum that
   must elapse before a state transition can take place.
   For lightlike transitions, the elapsed refractory time is ALWAYS minimal;
   there is only the compulsory waiting time, and no more; a transition
   takes place every time refractory quantum.  To each clock eigenstate


   attach a Hilbert space with a number operator N_k(n).
   Define the refractory operators R_k(n) = N_k(n) + 1.
   Every clock state has an attached Hilbert space and an attached
   refractory state so each clock state is associated with an expected
   waiting time w_k expressed in refractory units for a transition
   to some other state.
           w_k  =  <rho(k)|R_k(n)|rho(k)>/<rho(k)|rho(k)>
   where |rho(k)> is the refractory state in the Hilbert
   space Hilb_k(n) attached to |c_k>.
   We have actually a direct product of Hilbert spaces
           (Clocktime H) X (Refractory H)
   Then we represent "process" as convex combinations of lightlike process.
   NB It is not quite so clear that they will be timelike. See  [Section XII].
   Anything seems representable by convex combinations of G(n)-null vectors.
   Remember that they do span the Hilbert space!
   FCCR oscillator (clock) exhibits an angular frequency omega_T(n)
   determined by the requirement that time be measured in integral multiples
   of a fundamental time unit tau_0.  Then,
           omega_T(n)  =  2 pi/(n tau_0)
   This is the frequency associated to the process-clock period
           T_n  =  n tau_0
           omega_T(n) T_n  =  2 pi
   If we consider higher frequencies than omega_T(n), then
   the clock period must be smaller.  There is a limit to this, and seemingly
   the smallest clock period must then be (2 tau_0).  (A single state clock
   is not a clock since it cannot measure a time difference; it is simply in
   one state and that is all there is ever to say about it.)
   For n fixed, consider clock frequencies and periods for k > 2
           T(n, k)  =  k tau_0
           omega_T(n, k)  =  2 pi/(k tau_0)
           omega_T(n, k) T(n, k)  =  2 pi
   For smaller k, (smaller T(n, k)), omega_T(n, k) must be larger.  Clock
   frequencies are not necessarily associated with clock energies.  Let's
   say that we have a set of clock pointer positions.  Assume that the
   transit time for the clock is the least it can be, tau_0.
   The clock period then depends on the number of positions available.
   Let that be k.  We can distribute that same amount of energy over any
   number of clock states, so the number of clock states available is so far
   independent of the total clock energy.  Choosing the least transit time
   seems associated to a maximal energy.

   Do we properly speak of transit time or do we properly speak
   of relative transition probabilities (amplitudes)?
   FCCR oscillator also exhibits an angular frequency omega_H
   which is associated with the oscillator Hamiltonian, hence the energy
   spectrum of the oscillator, in particular with a minimum energy.
           E_k  =  h-bar omega_H (k + 1/2)
   Define an energetic angular frequency spectrum by
           omega_H(n, k)  :=  omega_H (k + 1/2)
           omega_H(0)  =  omega_H/2 < omega_H
   and for k > 0
           omega_H(k)  >  omega_H(0)
   The two fundamental frequencies are related by
           omega_H  =  n omega_T(n)
   and the two frequency spectra have the functional forms
           omega_T(n, k)  =  (2 pi/tau_0)(1/k)
                          =  omega_0 (1/k)

           omega_H(n, k)  =  (2 pi/tau_0)(k + 1/2)
                          =  omega_0 (k + 1/2)
   (For integral k, these two spectra are not immediately reconcilable.)
   For k, j > 1,
           omega_T(n, k) < omega_0  <  omega_H(n, j)
           omega_H  =  2 pi/tau_0  =  omega_0
           omega_T(n)/omega_T(n, k)  =  k/n
   Correlate the companion wave velocities of de Broglie to the FCCR
   "spacelike" or "statelike" process.  In the fundamental quadratic form
   G(n), it appears that we have a c-like quantity in sqrt(n-1) or 1/sqrt(n-1).
   DeBroglie's abandonment of the concept of "internal process" associated with
   a particle may have been premature.
   It is clear that the tau_0 appearing in omega_H and omega_T
   should not be taken as a fundamental time unit if the oscillator is supposed
   to have a mass.  Rather use the relation
                   h-bar  omega_H  =  m_0 c^2
                   tau_0  =  h/(m_0 c^2)
                   omega_H  =  (m_0 c^2)/h-bar 
                   omega_T  =  (m_0 c^2)/(n h-bar )

   In a time operator context, tau_0 should be associated with
           (DELTA t(n))^2  :=  < (<t(n)> - t(n))^2 >

   an uncertainty in t(n) [Standard deviation in a probability distribution].


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Created: November 1997
Last Updated: May 28, 2000
Last Updated: August 23, 2005