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Re: Fundamental physical properties



On 15:44:04 02/22/97 Joe Darling wrote:
I have a question that has been perplexing me for several years now. The
two physical science texts that I have used since beginning to teach this
course three years ago refer the the four fundamental physical properties,
length, mass, time, and electric charge, and state that all other physical
properties or quantites can be stated as combintations of these four. For
example in Physical Science, 3rd. Ed., by Bill W. tillery, p. 6, it states
that "There are four properties that cannot be described in simpler terms,
and all other properties are combinations of these four." In Physical
Science, by Jerry Schad, p. 10, it states that "It is remarkable that
every physical property you can imagine can in some way be expressed in
terms of one or more of only four fundamental properties: length, time,
mass, and charge. These properties are fundamental in that they cannot be
expressed in simpler terms." I took this as a revelation of some basic
insight into the nature of the Universe. And then I became aware that in
the SI system, there are seven base units. And so for the past few years,
I have been trying to figure out how to express the kelvin, the mole, and
the candela in terms of the meter, kilogram, and second. I have not
researched these units as thoroughly as I might, but I thought I might
be able to get some response as to whether I am headed in the right
direction.

I don't think descriptions such as these by Tillery and Schad above provide
much "insight into the nature of the Universe" since they are quite
misleading and not very true. One cannot quantify the particle properties
"strangeness", "lepton number", "color", or many others as combinations of
these 4 units. Which quantities are chosen as "fundamental physical
properties" (FPPs) and which ones are considered as derived is quite
arbitrary. Even the number of FPPs that one uses is quite arbitary. *The
number of FPPs that one uses is related to the number of dimensioned
universal physical constants one wants to have in one's physical expressions
(i.e. equations of physics)*.

An example of a different choice of FPPs is the U.S. customary system of
engineering units which uses length, force, and time as FFPs. In this system
the mass unit of the slug is a derived quantity, and 1 slug is defined as
that mass which requires a net force of 1 lb to cause it to accelerate it at
1 ft/s^2 (1 slug = 1 lb s^2 / ft).

There is no physical need to have electric current (measured in Amperes) as a
FPP. This is just something that the inventers of the SI system have chosen
to do. The electrical quantities defined in the older, cgs electrostatic,
cgs electromagnetic, Gaussian, and the Lorentz-Heaviside systems do not
introduce a new FPP for electrical quantities. For instance the esu (i.e.
statcoulomb) is made from mechanical units and has the dimension of
(erg cm)^(1/2).

In a description that considers "heat" as a fundamentally different kind of
concept than "work" (sorry Jim G.) one would have separate FPPs for work and
for heat. For instance, if the kcal was a FPP of heat and the Joule was the
derived unit of work then our thermodynamic formulae would have extra
conversion factors in them which related how much work was equivalent (a la
the Joule experiment) to how much heat (i.e. 4184 J/kcal). Such a conversion
factor would appear in any equation that related a work to a heat in the same
equation. This conversion factor would then acquire the status of a
fundamental constant of nature to be experimentally determined via a Joule-
type experiment.

Similarly, since Special Relativity has come on the scene it has become clear
that the speed of light c is such a conversion factor that relates how much
length is equivalent to how much time. (The SI system effectively
ackowledges this in its defintion of the meter.) When converting between
space-time intervals which are measured in meters to those that are measured
in seconds the rate of conversion is that there are 299792458 meters in 1
second. Thus length and time are not both needed as FPPs any more than both
the kcal and the Joule are needed in thermodynamics. Notice that its easier
to do cosmology with the unit of time as the year and the unit of length also
as the (light)year. This makes all velocities dimensionless as they really
are in spacetime. (After all, a velocity is just the hyperbolic tangent of
the rapidity and the rapidity is a dimensionless hyperbolic imaginary angle
characterizing a Lorentz transformation.) Measuring length and time in the
same units has the added advantage that the space-time 4-vector interval then
has all of its components measured in the same units, and all components of
the energy-momentum 4-vector has all of its components measured in the same
units (as they should in any sensible coordinate system). It would be really
cumbersome to do analytic plane geometry where the x-axis is denominated in
km and the y-axis is denominated in microns. We would always have to include
the conversion factor of 10^9 microns/km in our equations every time we
rotated a vector or found its magnitude. We are in just this situation when
we measure time in seconds and space in meters. We keep needing to use the
conversion factor of 299792458 m/s throughout our calculations. Since 1984
the definition of the meter has been 1/299792458 of the distance that EM
radiation travels in a vacuum in 1 second. This effectively defines c to
have a particular numerical value and makes the meter dependent on the
second, so the meter is not strictly an independent FPP any more in the SI
system (even though it's is still used as a base unit). This gives the meter
a status akin to the Ampere in that it is used as a base unit but its
definition refers to other FPPs and it sets the value of a constant of nature
by definition.

Since quantum physics has come on the scene it is now known that the concept
of action is fundamentally a (dimensionless) phase angle. It is the action
difference between different classical paths (trajectories) which measures
how they interfere with each other via the difference in phase angle between
their corresponding complex amplitudes. (Feynman did his Phd thesis on
essentially this.) The conversion factor between a radian of phase and a
joule*second of action is that 1 rad = 1.05457266 x 10^(-34) Js. (In cycles
we have 1 cycle = 6.6260755 x 10^(-34) Js.) Any sensible system of FPPs
would treat action-conjugate pairs of quantities as having a reciprocal
dimensions of each other. Since momentum is a rate of change of action wrt a
change of postion we see that momentum should have an inverse length
dimension (after all, momentum *is* wave number). Also since energy is the
rate of change of action wrt time increments we see that energy *is*
frequency and should be measured in inverse time units. The only reason that
Planck's constant enters the equations of physics is that it is needed as a
conversion factor between action measured in radians or cycles and action
measured in Js. A sensible system of units would have energy and frequency
have the same dimensions and units, and have momentum and wave number
(spatial frequency) have the same dimensions and units. Particle physicists
use a set of units such that c=1 and h_bar=1 and the only FPP is the MeV (or
GeV). Here distances and times are in 1/MeV and energy, mass and momentum
are in MeV, and action, angular momentum, electric charge, strangeness,
hypercharge, color, etc. are all dimensionless.

Since General Relativity has come on the scene it is now apparent that a
fixed amount of the (Ricci) curvature of spacetime is locally produced by
the Stress-Energy-Momentum (2nd rank 4-tensor) of all forms of matter and
radiation present at each spacetime neighborhood. The conversion factor
between one unit of energy density (or stress) having dimensions of
energy/volume and spacetime curvature having dimensions of 1/area is
8*pi*G/c^4 (i.e. curvature = 8*pi*G/c^4 * energy density). A natural choice
of units is to make this conversion factor 1. Making this choice along with
the previous choice above of choosing c=1 and h_bar=1 uniquely fixed all the
fundamental units. This set of units, called Planck units, has no *FPPs
whatsoever* as fundamental quantities that are used to derive all others. In
Planck units all physical quantities are dimensionless.

In short, we can reduce our number of FPPs by 1 for each fundamental constant
of nature that we decide to set to a convenient value (e.g. 1), and we
increase the number of FPPs by one for each extra conversion factor we add to
the equations which allows us to measure related quantities in different
units with different dimensions (e.g. work/heat, time/space, action/angle,
etc.) When such conversion factors are introduced into the theory they
become dimensioned universal constants of nature. Note that we cannot get
rid of all of the universal constants of nature by a jucicious choice of unit
system because there are more constants of nature than there are FPPs to be
eliminated. Things like the intrinsic coupling constants for the fundamental
interactions of nature, the ratios of the masses of the elementary particles,
various group theoretic mixing angles, etc. are all fixed constants in *any*
system of units with any number of FPPs. (It is the hope of the theorists
working on superstring theories of everything that the rest of these
constants will eventually prove to be all calcuable from the theory.
Currently, this is a long way off since those theories have proved to be much
too mathematically intractable extract numerical predictions for these
experimentally measurable constants.)


The candela seems to be a unit of intensity, restricted to the visible
spcetrum, but measured in watts per steradian.

This is true. One candela is the luminous intensity corresponding to
1/683 watts per steradian at the wavelength for which the human eye is most
sensitive (i.e. 555 nm, or 5.40 x 10^14 Hz). If a luminous source has a
different spectral content the number of candelas that it emits is determined
by comparing the source with a monochromatic source at 555 mn with the same
apparent intensity.

The mole seems to be a reflection of having defined the kilogram, and
hence the gram, and then finding that there are Avogadro's number of
carbon-12 atoms in 12 g of carbon-12 atom's. It is a fundamental constant
of nature, but related to the definition of the unit of mass.

I already commented on this statement in a previous post. As I indicated in
that post I don't consider Avagadro's number to be fundamental at all. It's
just a convenient macroscopic measure of quantity for microscopic things.
It's value arbitrarily depends on the ratio of the mass of a C-12 atom to the
mass of the standard kilogram. There is nothing "fundamental" about picking
this ratio. If I was in charge I'd probably choose 10^24 as Avagadro's
number. Then its value would be truely independent of the choice of mass
unit and being a power of 10 it would be closer to the supposed spirit of the
SI system anyway. Actually being a theorist, my natural inclination is to
choose Avagadro's number to be 1. This way we just count the particle
numbers that are actually in our sample which, to my mind, is the most
conceptually simple way to quantify the number of somethings (i.e. just count
them).

But the kelvin has me stumped. The nearest I have come is the internal
energy of an ideal monatomic gas, U=(3/2)nRT, but this does not give a
definition of the kelvin in terms of the meter, kilometer, and second, and
is only true for ideal and monatomic gases. Real gases and molecular
gases contain different amounts of energy even while they are in thermal
equalibrium, i.e.; at the same temperature. Is not the kelvin as
fundamental as the other four units?

The kelvin is a result of considering Boltzmann's constant as a dimensioned
universal constant of nature which appears because someone in the past wanted
to arbitrarily measure thermodynamic temperature in a new unit (i.e. the
kelvin) rather than in the natural unit that temperature comes in, i.e.
energy. Thermodynamic temperature is defined as the partial derivative of
the system's energy wrt its entropy in equilibrium under the condition of no
work allowed to be done while the variation is taken. Thus temperature
naturally has dimensions of energy per entropy. The system's entropy is the
average amount of information needed to specify with certainty the system's
microscopic state given just its macroscopic description. Since the entropy
is fundamentally a dimensionless measure of information it is most natural to
measure it in information units (i.e. bits, bytes, Gbytes., etc.) In the
field of information theory the most convenient unit for entropy is the bit.
But in statistical mechanics the most theoretically simple measure of
thermodynamic entropy is the nat (1 bit = ln(2) nat). The use of the nat
instead of the bit allows one to keep all logarithms to the Natural/Naperian
base e rather than the base 2 as is most convenient in communication and
information theory. By choosing the unit of entropy as the nat (really a
dimensionless quantity like a radian is) then temperature is automatically
measured in joules/nat. Since the nat it dimensionless measuring the
temperature in just joules is sufficient (for the same reason that angular
frequency can be measured in either 1/s or in rad/s). Using this set of
units ends up having the effect of setting Boltamann's constant equal to 1
(just like all the other Planck units do to the other constants of nature).
We can think of the current numerical value of k = 1.380658 x 10 ^(-23) J/K
as being a conversion factor between measuring temperatures in J and
measuring them in K. One kelvin of temperature is 1.380658 x 10^(-23) J in
energy units. If the size of the kelvin were to be changed without changing
any other units then Boltzmann's constant would change accordingly. A
"constant" that so changes cannot rightly be considered a true constant of
nature. It's dimensions gives its away as really being a conversion factor
due to an unfortunate choice of units for temperature and entropy. If we
measured entropy in nats and temperature in J then the Ideal gas law takes on
the appealing form p*V = N*T (where N is just the number of particles in the
volume V). Similarly the internal energy in a classical monatomic ideal gas
is then U=(3/2)*N*T, and the average translational kinetic energy of any
particle in an equilibrium sample of any material made of classical particles
is then u = (3/2)*T.

Again, as in the case of other SI units such as the Ampere, the kelvin unit
is seen to be superfluous. The reason that Boltzmann's constant has the
value k = 1.380658 x 10^(-23)J/K is that the kelvin is defined so that the
triple point of pure water occurs at exactly the value of 273.16 K. This
value is completely arbitrary in that any other value could have been chosen
giving a correspondingly different value for the conversion factor k. The
historical reason that the triple point of water is chosen to have the value
273.16 is that in this case using this scale size for the kelvin there is a
temperature difference of very close to 100 K between the boiling point of
liquid water under 1 atm of pressure and its freezing point under the same
pressure. By choosing the kelvin this way then the Celsius temperature in
deg. C is just the (Kelvin temp.) - 273.15 where 273.15 is then the ice point
temperature in kelvin at 1 atm of pressure.

As can be inferred from this *much* too long post, I am not much of a fan of
SI units. SI units are to scientific measurements what the English language
is to international communications more widely. Both are ubiquitous and it
is their ubiquity that makes them the standard lingua franca of the times
which is understood by nearly everyone else. Neither English nor SI are very
good choices intrinsically. But it's too late now to change scientific
measurements to a more rational system which will probably never catch on, as
it is too late to try to change international communication to Esperanto (or
some other rationalized language) which also hasn't caught on.

David Bowman
dbowman@gtc.georgetown.ky.us