In Bridgman's Dimensional Analysis
[Bridgman 1931]
a notation was used for "physical dimensions" that is independent
of any particular system, e.g., cgs or mks of units. In the physics
of mechanics, any physical quantity is expressible in terms of units
of mass [M], length [L] and time [T], so that, for example, a velocity
has units of [L]/[T], acceleration has units of [L]/[T^{2}], etc.
[Dimensional analysis - Wikipedia]

Deriving Planck's primary three units of [M], [L] and [T] from
the three fundamental physical constants, G_{0}, ħ and c, is at
first nothing more than a bit of prestidigitation involving
dimensional analysis, the purpose of which is to establish a
set of units appropriate to a discussion of fundamental physics.
Or - is it?
I'll save that question for later.
A derivation first.

In [Planck 1899] fundamental units of mass, length and time, and therefore all mechanical quantities, are expressed in terms of the physical constants

G_{0}- the Newtonian gavitational constant ħ - the "normalized" Planck constant h/(2 π) c - the velocity of light In Bridgman's notation G_{0}has units [L^{3}]/[T^{2}M] ħ has units [E][T] = [M][L^{2}][T]/[T^{2}] ( E = mc^{2}) = [M][L^{2}]/[T] so ħ has units of "action" [E][T] ħ/G_{0}has units [M][L^{2}]/[T] * [L^{-3}][M][T^{2}] = [M^{2}][L][T] So, (c ħ/G_{0}) has units [M^{2}][L][T][L]/[T] = [M^{2}][L^{2}] The Schwarzschild radius appearing in the spheristatic solution of the Einstein equations as the radius of the singular surface determined by the gravitational potential given by the time-time component of the covariant, symmetric, 2nd rank, gravimetric tensor, 1 g_{44}= ----------------------- ( 1 - (G_{0}/c^{2}) (m/r) ) which relates any mass m to a length l_{s}(m) at the singularity by, l_{s}(m) = (G_{0}/c^{2}) m thus relating mass linearly to length, while the Compton wave length relates any mass unit to a length unit by l_{c}(m) = ħ/(mc) These functions of m, simply use the fundamental units in order to define equivalences of mass in other types of physical units. NB: As in special relativity, the mass parameter m appears as a constant. In the Schwarzschild solution of general relativity as a constant of integration of the differential equations that are the Einstein equations reduced by the imposition of the condition of spheristasis. Then, the ratio of the two mass "constants" l_{c}(m)/l_{s}(m) = ħ/( m c ) * c^{2}/(G_{0}m) = c ħ/( m ) * 1/(G_{0}m) = c ħ/( G_{0}) * 1/(m^{2}) is dimensionless. So, when this ratio of these lengths is 1, by multiplying by m^{2}and taking a square root, a specific mass, the Planck mass is defined, m_{0}:= (ħ c/G_{0})^{(1/2)}, Now multiply this unit of mass by (G_{0}/c^{2}) as above to get a unit of length. Then multiply the unit of length by (1/c) to get units of time. Et, voilá! - the fundamental [M], [L], [T] Planck constants are readily constructed: m_{0}:= (ħ c/G_{0})^{(1/2)}l_{0}:= (ħ G_{0}/c^{3})^{(1/2)}t_{0}:= (ħ G_{0}/c^{5})^{(1/2)}If you missed or didn't take into account the Schwarzschild radius and the Compton wavelength, you may never have gotten there simply. They *aren't* necessary to the derivation, merely helpful, and rather suggestive in the way of interpreting the mathematics of the physics. It should be noted that theWeak Principle of Equivalence, where inertial mass is taken to be equivalent to gravitational mass is being assumed. Since we know the dimensions of the three original constants, these three relations can be inverted to reconstruct the original three constants: c := l_{0}/ t_{0}ħ := m_{0}(l_{0})^{2}/ t_{0}G_{0}:= (l_{0})^{3}/ (m_{0}(t_{0})^{2}) From the above fundamental mechanical Planck units, one can construct further mechanical Planck units of: energy E_{0}= m_{0}c^{2}= (ħ c^{5}/ G_{0})^{(1/2)}momentum p_{0}= m_{0}c = (ħ c^{3}/ G_{0})^{(1/2)}acceleration a_{0}= c / t_{0}= (ħ G_{0}/c^{7})^{(-1/2)}force from the 2d law of Newtonian mechanics F_{0}= m_{0}a_{0}= (ħ c/G_{0})^{(1/2)}(ħ G_{0}/c^{7})^{(-1/2)}= c^{4}/ G_{0}or, equivalently and consistently, from Newton's gravitational force law F_{0}= G_{0}(m_{0}/ l_{0})^{2}= G_{0}(ħ c/G_{0}) / (ħ G_{0}/c^{3}) = c^{4}/ G_{0}impulse I_{0}= F_{0}t_{0}= c^{4}/ G_{0}(ħ G_{0}/c^{5})^{(1/2)}= ( c^{8}/ G_{0}^{2}ħ G_{0}/c^{5})^{(1/2)}= ( ħ c^{3}/ G_{0})^{(1/2)}= p_{0}pressure P_{0}= F_{0}/ (l_{0})^{2}= c^{4}/ (G_{0}(l_{0})^{2}) = c^{4}/ (ħ (G_{0})^{2}/c^{3}) = c^{7}/ (ħ (G_{0})^{2}) n-volume V_{0}(n) = (l_{0})^{n}energy density of a 3 dimensional space D_{0}= E_{0}/ (l_{0})^{3}= (ħ c^{5}/ G_{0})^{(1/2)}/ (ħ G_{0}/c^{3})^{(3/2)}= ( (ħ)^{(-2)}c_{14}/ G_{0}^{4})^{(1/2)}= c^{7}/ ( ħ G_{0}^{2}) mass density M_{0}= c^{5}/ ( ħ G_{0}^{2}) frequency f_{0}= 1/t_{0}angular frequency ω_{0}= (2 π)/t_{0}angular momentum L_{0}= m_{0}c l_{0}= c m_{0}l_{0}= c (ħ c/G_{0})^{(1/2)}(ħ G_{0}/c^{3})^{(1/2)}= c (ħ^{2}G_{0}/G_{0})^{(1/2)}(1/c^{2})^{(1/2)}= c ħ (1/c) = ħ

- Planck units - Wikipedia
- [physics/0110060] Trialogue on the number of fundamental constants
- Frank Wilczek - Scaling Mount Planck I: A View From The Bottom
- Planck's original paper on Natural Units
- Interpreting the Planck Mass

Observe that, l_{0}/ t_{0}= c <=> (l_{0})^{2}- c^{2}(t_{0})^{2}= 0 (p_{0})^{2}- (E_{0}/ c)^{2}= 0 E_{0}= m_{0}c^{2}p_{0}= m_{0}c l_{0}p_{0}= ħ (as in Q uncertainties) t_{0}E_{0}= ħ (as in E uncertainties) If, in the arrival at the Planck mass above, instead of taking the ratio of the Compton and Schwarzschild lengths of a given mass, we took the product, l_{c}(m) l_{s}(m) = ħ/( m c ) * (G_{0}m)/c^{2}= (ħ G_{0})/c^{3}= (l_{0})^{2}So, for any mass m, the geometric mean of its l_{c}(m) and l_{s}(m) is the Planck length. The Planck mass finds the simultaneous minimum of both l_{c}(m) and l_{s}(m) subject to this constraint. One can obviously construct similar Compton and Schwarzschild, times, energies, velocities, and then actions, etc., associated to any given m, which have the same form of algebraic relationships.

In the above four relationships, the first seems simply to redefine the speed of light. In terms of dimensional analysis, that's all it is; however, in the context of the physical theory of relativity, c is understood as a bounding velocity for massive particles, and the required velocity for massless particles.

The second, interestingly enough, is the relativistic expression for
a particle of rest mass zero, telling that one must be careful in
interpreting m_{0} with the understanding that m_{0} represents, from
a relativistic standpoint, an *equivalent* mass corresponding to
a given energy.
The exact nature of this equivalence is not specified; it has
yet to be.

The remaining two equalities can be recognized as those of minimal uncertainty products (in quantum theory) of canonically conjugate pairs of variables.

The mechanical Planck units can be extended to electomagnetic theory, and further to thermodynamics (statistical mechanics), addressed in the following.

The three physical constants, c, G_{0} and ħ are extracted from theories of
relativity, gravitation and quantum mechanics.
While electromagnetic theory is relativistically covariant, there
is also the matter of electric charge and the observation that observed
particles exhibit charges that are integral multiples of the charge
of the electron, which should be included as a fundamental physical constant.

Further, there is also the essential area of statistical mechanics where mechanics being statisticalized gives connection with and explanation for thermodynamic properties of matter, the key concept there being entropy and its definition through the Boltzmann equation

S := k log p the epitaph on Boltzman's tombstone. Aside: Entropy, or Information (negentropy) is a little more Complicated than that, and is generally defined as a functional of a probability distribution: {p_{k}} -> S({p_{k}}) := - Σ p_{k}log p_{k}k The powerful idea of Boltzman, which was actually rejected during his lifetime, understood by many to be the precipitating factor, and possibly the reason for his suicide, was that a then unaccepted atomic theory provided for an explanation of thermodynamics by treating N-body problems with N exceedingly large as a statistical system, thus launching the physical theory of statistical mechanics. Now, extend the idea of Planck to electromagnetic theory. Let e be the charge on the electron. The Dimensionless fine structure constant can be defined α := e^{2}/ (4 π ħ c) is dimensionless in units, and is a measure of the coupling strength of the EM field to its charge. The charge on the electron was chosen as standard since charge in sustainable particles appears always to be integral multiples of e. Introduce the Boltzmann constant of statistical mechanics k The Boltzmann constant with units [E]/[K] to extend to thermodynamics, to define a Planck temperature T_{0}= 2 E_{0}/ k = 2.834432 E32 KOther secondary units from EMT and ThermodynamicsAn electrostatic potential is written as q/(epsilon r) which has units [Q][L^{-1}][Q^{-2}][M][L^{3}][T^{2}] = [Q-1][M][L^{2}][T^{2}], so define potential V_{0}= E_{0}/ e By simple definition of current, current i_{0}= e / t_{0}resistance (from V = I R) R_{0}= V_{0}/i_{0}= E_{0}t_{0}= ħ

Dr. John Baez has remarked and argued that dimensioned constants of physics are ultimately not near so important at the dimensionless constants. Prototypically and alone in the world of dimensionaless physical constants is the fine structure constant, a ratio of forces, presumably from two distinct origins. Dr. Baez points specifically to the ratios of masses of elemetary particles as being of essential physical importance; at the same time, however, he almost dismisses the importance of dimensioned physical constants.

While I respect his more than considerable work, intelligence and pedegogic prowess, I disagree with Baez on this matter, particularly in the great importance of mass ratios. It's not a scientific disagreement; it's more of a disagreement regarding what we each believe as a matter of experience and thought to be fundamental. I would not be surprised to find out that the masses of fundamental particles are not fundamental things, and that even their ratios have actually varied in cosmological history.

If dimensioned constants are of no real theoretical interest, then the model of spacetime physics is scale invariant in the context of its customary continua, and there is no essential change "in the way things look" when probing increasingly small spatial distances and time intervals, and then also that the Planck length and time are merely funny fictions and have no fundamental theoretic significance. General arguments from quantum field theory and various specific attempts at constructing a theory of quantum gravity say this continuum model is essentially untenable, and that the untenability is directly connected with Planck units. At the Planck level, things like coordinates cease to have any meaning. This essential Planck level quantization is a fundamental idea in Loop Quantum Gravity as pursued by Lee Smolin, which owes its genesis to a formulation of QCD due to Kenneth Wilson, in terms of loops. It almost goes without saying that an essential quantization of space and time should eliminate the "ultraviolet" divergences of field theories, and that an associated overall finiteness of space and time should eliminate the "infrared" divergences.

Loops conjure a concept of closed strings, and as it turns out, FCCR(n) is also about quantized closed strings of size n.

By the way, Baez, with his omnivorousness, perception and Feynmanesque skills at making truly arcane mathematics and physics reasonable and reasonably understandable constantly reminds me through his own excitement just how utterly fascinating and compelling the subject of of mathematical and theoretical physics really is. Do follow his This Week's Finds in Mathematical Physics; there will be epiphanies even if you don't understand everything.

Fine structure as coupling constant:

Its value being significantly less than 1 provides for the convergence
of perturbative series in quantum electrodynamics and quantum
electrodynamics.
In this sense, the numerical value is of enormous importance.
However, all the quantities that measure the strength of an interaction
are not dimensionless.

The speed of light is said to be "merely" a convenient device for
equating or transforming between units [L] and [T].
That merely, however, misses a point: we do not measure times
with rulers, nor distances with clocks, and rulers are not clocks.
It is burdensomely difficult to deal with a positivistic physics that measures
lengths with rulers, but less difficult to deal with measurements
of times by clocks which can *in abstracto* be seen to have
well defined constraints imposed by general considerations of quantum
and relativistic principles.
See, e.g.,
[Wigner 1957].
The existence of c implies more than the obvious algebra.

Exactly dimensioned constants are essential for the construction and statement of physical laws. The meaning or interpretation of the fine structure constant is not something that enters at any fundamental theoretical level - yet. That is to say that theory has not predicted its value. It would have been interesting bit of numerology if α was merely the inverse of the prime number 137; but, it is not.

Regarding the exciting idea of a theory of the fine structure constant, see prof. James G. Gilson's rather compelling theory of the fine structure constant, indeed of coupling constants generally: http://www.btinternet.com/~ugah174/ and his papers available from his home pages at Queen Mary College http://www.maths.qmul.ac.uk/~jgg/. I am hoping to have more to say about this soon. It is worth noting that while almost all previous purported explanations of the fine structure constant have been little more than mystical numerology that might have amused Pythagoras, Gilson's argument is soundly routed in physics, and indicates more than what amounts to a single immutable constant. Because of that, the theory actually has predictions and is actually disprovable, unlike the previous explanations. Moerover, the physical reasoning behind it gives suggestions as to how the theory might be modified if that is even found necessary.

It is not so much the value of dimensioned constants that is important, since arbitrarily convenient measures of any physical dimension can always be chosen. Rather it is that they exist - and as constants. The structure of special relativity depends not on the value of c, but that c exists as a finite fundamental velocity of massless physical entities, and that structure, unlike Newtonian physics recognizes that massless physical entities exist. Unlike Newtonian theory, relativistic theory predicts that massless particles exist, and this is in agreement with an empirical reality.

Given an understanding of, and agreement to the complexity and essential violence of of quantum field theoretic "vacuum", the idea that the observed mass spectrum of elementary particles is not truly elementary, but dynamical in nature is a common and not unreasonable hypothesis of particle physics, which is to say that the observed spectrum is not something easily explainable by any cheap and simple simple algebraic tricks.

Are the constants really constants? Or, do they just appear to be constant as a result our highly limited experiences relative to cosmological expanses?

Dirac, considered this question in [Dirac 1939] which I actually have, someplace.

That things are what they appear to be is always too easy an assumption. It is as dangerous as are the absolute idealisms of Plato. These are tools, powers of thought that like any other powers can be used for good or ill. Only on Tuesdays between 3:00 PM and 4:00 PM am I Platonist. It works for me.

One should not be too quick to assume that the fundamental constants of physics are indeed constants. "Variables don't, and constants aren't." (Parameters are another matter.)

The physical constants can be considered independent of each other, but not necessarily independent of the age of the universe. If we notice the origin of Newton's gravitational constant in his law of universal gravitation,

F(m, m') = G_{0}m m' / r^{2}

G_{0} is a constant that relates the *gravitational* expression of
force to the *inertial* expression of force in the second law of
Newton's Principia (F=ma). If G_{0} were zero, it would indicate an
completely general absence of gravitational forces.

In contrast, the constants c and ħ, express limitations on the kinematics of mechanics. Removing the limitations from theory involves some limit where ħ -> 0 and c -> infinity. Were c -> 0, this would involve a decoupling of space and time in such a way that only the velocity v = 0 for massive particles would be possible: a frozen existence. If ħ -> infinity, no classical existence would be possible, and all existence would be a quantum fluctuation within the tolerance of uncertainty relations.

So too then the constants fundamental m_{0}, l_{0}, t_{0}, T_{0}.
The Planck length and time are so small as to be beyond the reach
of current experimental possibility.
The first surprise is that the mass m_{0}, which one might expect to
be small, is , in the realm of elementary particles, enormous.
It would take about 10^{19} nucleons to make one Planck mass, or
as someone (I forget who) remarked, the Planck mass is about the
order of the mass of a flea. That sounds like it should have been
Richard Feynman.

Interpreting the Planck Mass I address in a separate essay. The Planck energy, momentum and temperatures are equally enormous.

Symbol Name Dimensions Value c speed of light [L]/[T] 2.998 E+10 cm/sec c = 1/sqrt(epsilon_{0}mu_{0}) ε_{0}Free space [Q^{2}]/[M][L^{3}][T^{2}] 1/(36 π) E-8 farad/cm permittivity μ_{0}Free space [M][L]/[Q^{2}] (4 π) E-9 henry/cm permeability ħ Reduced Planck [E][T] 1.0546 E-27 gm-cm^{2}/sec Constant 1.381 E-34 cm e Electron charge [Q] 1.381 E-34 cm m_{e}Electron mass [M] 9.1095 E-28 gm 6.764 E-56 cm r_{e}Electron radius [L] 2.81794 E-13 cm classical (= α(ħ/(m_{e}c)) r_{es}Electron Schwarzschild radius [L] 1.35264 E-55 cm (= 2G_{0}m_{e}/c^{2}) 22 orders of magnitude *smaller* than the Planck length a_{0}Bohr radius [L] 0.529177 E-08 cm ( = ħ^{2}/(m_{e}e^{2}) ) m_{p}Proton mass [M] 1.6726 E-24 gm 1.2419 E-52 cm 938 Mev = 0.938 Gev α Fine structure [] 1/137.036 ( = e^{2}/(4 π ħ c) ) G_{0}Gravitational [L^{3}]/[M][T^{2}] 6.673 E-8 cm^{3}/gm-sec^{2}Constant l_{0}Planck length [L] 1.616 E-33 cm sqrt(ħ G_{0}/c^{3}) t_{0}Planck time [T] 5.391 E-44 sec sqrt(ħ G_{0}/c^{5}) m_{0}Planck mass [M] 2.177 E-05 gm sqrt(ħ c/G_{0}) E_{0}Planck energy [M][L^{2}]/[T^{2}] 1.95669 E+16 ergs sqrt(ħ c^{5}/G_{0}) T_{0}Planck Temperature 2.834432 E32 K H Hubble [L^{-1}] 1/(1.7 E+28 cm) Hubble constant 15 km/sec/million light years Cf Weinberg First 3 minutes p. 23 t_{H}Heisenberg time [T] ? E-24 sec [L] ? E-12 cm m_{π}Pion mass [M] 0.2240 E-03 erg 2.29240 E-25 gm m_{K}Kaon mass [M] 0.7904 E-03 erg 8.83396 E-25 gm m_{μ}Muon mass [M] 0.1696E-03 erg 1.88695E-25 gm k Boltzmann's Constant [E/K] 1.380658 E-16 erg/K sigma Stefan-Boltzmann [E]/[L^{2}][T][K^{4}] 5.668 E-5 erg/cm^{2}-sec-deg^{4}Constant (of Black Body Radiation) ( = (2 π^{5}/15) k^{4}/(h^{3}c^{2}) ) Age of the universe < 20 billion years Age of the earth 4.6 billion years -------------------------------------------------------------------- The Schwarzschild radius for the Planck mass is twice the Planck length. The Schwarzschild diameter for the Planck mass is the Planck length. Weinberg highest 1.5 E12 K, first frame E11 K [Weinberg 1977] Planck Temperature = 2 * Planck Energy / Boltzmann's const = 2 * 1.95669 /1.380658 10+32 K = 2 * 1.417216 E32 K = 2.834432 E32 K

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