# Re: Constant 'c'?

David Bowman (dbowman@tiger.gtc.georgetown.ky.us)
Wed, 05 Nov 1997 11:05:02 EST

[Warning! More physics.]

In commenting on Moorad's last post George Murphy writes:

>> For the same reason, Planck's E = h f and de Broglie's p =
>> h/ lambda are fundamental features of nature (or our understanding of nature
>> if you like) and not mere conversion factors. These findings of Planck and
>> de Broglie indicate the fundamental particle/wave duality of nature (or our
>> understanding of nature if you like).
> You point out here something that I've puzzled about. Of the 3
>basic constants c, G, & h (which can be combined to give "natural"
>Planck units for length, time, & mass), the first 2 can be seen as
>conversion factors: From special relativity, c converts space units to
>time units, & from general relativity G converts inertial mass units to
>gravitational mass units. But h can't be seen as such a conversion
>factor: Quantum theory (at least in versions I know) doesn't say that
>energy & frequency are really the same thing in different units. Any
>suggestions?
> George Murphy

Actually, all the dimensioned universal constants of nature, including h, can
be thought of as unit conversion factors. The path integral formulation of
QM (first worked out by Feynman) demonstrates that the real meaning of action
is that action represents the phase of the complex amplitude for a virtual
process which requires the requisite amount of action. The quantum
interference between two alternative classical paths for a process is
determined by the phase shift between the amplitudes for those two paths.
This phase shift is the difference in the classical action for those two
paths denominated in units of Planck's constant. Thus QM teaches us that
action is really the phase of a complex amplitude, and one radian of phase
corresponds to h_bar = 1.0546 x 10^(-34) J*s of action, whereas one cycle of
phase corresponds to an action of h = 6.6261 x 10^(-34) J*s. The integrand
of the Feynman path integral for the overall amplitude for a process is
exp(i*S/h_bar) where S is the classical action for each path to be integrated
over and i = sqrt(-1).

Since one can think of the momentum of a system as being the rate of change
of the system's action w.r.t the system's position we see that momentum has
the natural dimension of action/distance, and since action is really a
dimensionless angle (in radians) we see that momentum comes in dimensions of
radians/distance or a wave number. Thus the DeBroglie relation p = h_bar*k =
= h/[lambda] *is* just a statement of the conversion of units between
momentum measured in mass*length/time units to momentum measured in wave
number units. Similarly, the energy of the system can be described as the
(negative of) the rate of change of the system's action w.r.t the time. Thus
energy naturally comes in phase/time or angular frequency units, and the
Planck equation E = h_bar*[omega] = h*f is again just a relation describing a
change in units between energy measured in mass*(length/time)^2 units to
energy measured in (angular) frequency units.

Similarly, in the realm of thermal physics it can be shown that Boltzmann's
constant k = 1.38063 x 10^(-23) J/K is just a unit conversion constant which
converts temperature measured in kelvins to temperature measured in energy
units such as joules. A factor of k appears in front of the stat. mech.
expressions for the entropy of a system since in that case it merely
converts between entropy measured in J/K and the entropy measured in 'nat's.
Here 1 bit = ln(2) nat and 1 nat = log_2(e) bit. The real meaning of the
entropy is that it is the amount of information necessary to exactly
determine the precise microscopic state of a thermodynamic system given only
the system's macroscopic description. Since the entropy really measures an
information it most naturally comes in information units such as bits,
bytes, gbytes, (or nats). The presence of the factor k in stat mech is an
historical artifact of a prior definition of the kelvin temperature scale
based in the requirement that the triple point of pure water is supposed to
have a temperature of exactly 273.16 K rather than measuring temperature in
the natural energy units joules/nat that temperature would naturally come in
if entropy were measured in the natural nat units. The *definition* of the
thermodynamic temperature of a thermodynamic system is that it is the partial
derivative T = dE/dS where the variation is taken quasistatically and no
work is done for the variation. From the definition we see that temperature
naturally has energy/entropy or energy/information units.

David Bowman
dbowman@gtc.georgetown.ky.us