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".

A PRELUDE

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
or
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.
Then

td ts  =  (t0)^2

and the relativistically invariant quantum t0 is just that.

Aside:
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)

and

nu_1(beta) nu(beta)  =  nu_0^2

and

nu_1(beta)/nu(beta)  =  (1 - beta^2)

with                nu_1(0)  =  nu(0)  =  nu_0

lim nu_1(beta)  =  0
beta->1

Time quantum(?) lim nu_1^(-1)(beta)  =  infinity
beta->1

lim nu(beta)  =  infinity
beta->1

Time quantum(?) lim nu^(-1)(beta)  =  0
beta->1

Special Relativity:  Energy <-> Mass    (E  =  m_0 c^2)

Quantum theory:      Energy <-> (Time)^(-1) (E  =  h nu)

There is apparently a relativistic paradox arising from the
quantum theoretical association of mass with time.
Remember that regarding Lorentz Transformations,
if
(x, ct)  is a contrvariant 4-vector
then
(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
vectors.

---------------------------------------------------------------------------
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.

Aside:
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,
then
E(beta) = h nu(beta)  =  m(beta) c^2
where

E(beta)  =  m_0 c^2/sqrt(1 - beta^2)
m(beta)  =  m_0/sqrt(1 - beta^2)
so
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.

Problem:
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)
If
nu_x  =  2 pi c h/(G_0) m_0)
then
[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)
Apparently
v  =  k (delta q)/(delta t) > c
cannot happen, but
v  =  (delta q)/(k (delta t)) < c
can.

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

|c_k>

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
Then
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)
so
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)

Then
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)
so
omega_H  =  2 pi/tau_0  =  omega_0
and
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
so
tau_0  =  h/(m_0 c^2)
so,
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].
--------------------------------------------------------------------------

[UNFINISHED]
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