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

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) (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, 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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Last Updated: August 23, 2005