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]/[T2], etc. [Dimensional analysis - Wikipedia]

The Planck Units

Deriving Planck's primary three units of [M], [L] and [T] from the three fundamental physical constants, G0, 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

	G0   - the Newtonian gavitational constant
	ħ      - the "normalized" Planck constant h/(2 π)
	c      - the velocity of light

   In Bridgman's notation

	G0 has units [L3]/[T2 M]
	ħ has units [E][T]  =  [M][L2][T]/[T2]  ( E = mc2)
				=  [M][L2]/[T]

	so ħ has units of "action" [E][T]

	ħ/G0  has units [M][L2]/[T] * [L-3][M][T2]

		                =  [M2][L][T]


	(c ħ/G0)  has units [M2][L][T][L]/[T]

	                	=  [M2][L2]

   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,

	g44    =  -----------------------
	           ( 1 - (G0/c2) (m/r) )

   which relates any mass m to a length ls(m) at the singularity by,

	ls(m) = (G0/c2)  m

   thus relating mass linearly to length, while the Compton wave length
   relates any mass unit to a length unit by

	lc(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"

	lc(m)/ls(m) = ħ/( m c ) * c2/(G0 m)

		= c ħ/( m )   * 1/(G0 m)
		= c ħ/( G0 ) * 1/(m2)

   is dimensionless.  So, when this ratio of these lengths is 1,
   by multiplying by m2 and taking a square root, a specific mass,
   the Planck mass is defined,

	m0  :=  (ħ c/G0)(1/2),

   Now multiply this unit of mass by (G0/c2) 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:

	m0  :=  (ħ c/G0)(1/2)

	l0  :=  (ħ G0/c3)(1/2)

	t0  :=  (ħ G0/c5)(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 the Weak 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

        c    :=  l0 / t0

	ħ    :=  m0 (l0)2 / t0

	G0  :=  (l0)3 / (m0 (t0)2)

   From the above fundamental mechanical Planck units, one can construct
   further mechanical Planck units of:

	E0  =  m0 c2
	     =  (ħ c5 / G0)(1/2)

	p0  =  m0 c
	     =  (ħ c3 / G0)(1/2)

	a0  =  c / t0
	     =  (ħ G0/c7)(-1/2)

	from the 2d law of Newtonian mechanics
	F0  =  m0 a0
	     =  (ħ c/G0)(1/2) (ħ G0/c7)(-1/2)
	     =  c4 / G0

	or, equivalently and consistently, from Newton's gravitational
	force law
	F0  =  G0 (m0 / l0)2
	     =  G0 (ħ c/G0) / (ħ G0/c3)
	     =  c4 / G0

	I0  =  F0 t0
             =  c4 / G0 (ħ G0/c5)(1/2)
             =  ( c8 / G02  ħ G0/c5)(1/2)
             =  ( ħ c3 / G0 )(1/2)
             =  p0

	P0  =  F0 / (l0)2
	     =  c4 / (G0 (l0)2)
	     =  c4 / (ħ (G0)2/c3)
	     =  c7 / (ħ (G0)2)

	V0(n)  =  (l0)n

   energy density of a 3 dimensional space
	D0  =  E0 / (l0)3
	     =  (ħ c5 / G0)(1/2) / (ħ G0/c3)(3/2)
	     =  ( (ħ)(-2) c14 / G04 )(1/2)
	     =  c7 / ( ħ G02 )

   mass density
	M0  =  c5 / ( ħ G02 )

	f0  =  1/t0

   angular frequency
	ω0  =  (2 π)/t0

   angular momentum
	L0  =  m0 c l0
	     =  c m0 l0
	     =  c (ħ c/G0)(1/2) (ħ G0/c3)(1/2)
	     =  c (ħ2 G0/G0)(1/2) (1/c2)(1/2)
	     =  c ħ (1/c)
	     =  ħ

See Also

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

Some Suggestive Relations Among The Three
Fundamental Planck Constants

   Observe that,

	l0 / t0  =  c      <=>      (l0)2 - c2 (t0)2  =  0

	(p0)2 - (E0 / c)2  =  0

	E0  =  m0 c2

	p0  =  m0 c

	l0 p0    =  ħ  (as in Q uncertainties)

	t0 E0    =  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,

	lc(m) ls(m) =  ħ/( m c ) * (G0 m)/c2

	              =  (ħ G0)/c3

	              =  (l0)2

   So, for any mass m, the geometric mean of its lc(m) and ls(m) is
   the Planck length.  The Planck mass finds the simultaneous minimum
   of both lc(m) and ls(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 m0 with the understanding that m0 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.

Extending The Three Fundamental Planck Constants
Of Mechanics To Five

The three physical constants, c, G0 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:

	{pk}  ->  S({pk})  :=  - Σ pk log pk

	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.
	   e be the charge on the electron.

   The Dimensionless fine structure constant can be defined

	   α  :=  e2 / (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

	   T0  =  2 E0 / k
	        =  2.834432 E32 K

   Other secondary units from EMT and Thermodynamics

   An electrostatic potential is written as q/(epsilon r) which has
   units [Q][L-1][Q-2][M][L3][T2] = [Q-1][M][L2][T2], so define

	V0  =  E0 / e

   By simple definition of current,

	i0  =  e / t0

   resistance (from V = I R)
	R0  =  V0/i0  =  E0 t0
	     =  ħ

Remarks On Dimensioned And Dimensionless Constants
Ratios Of Masses, Lengths And Times.

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: and his papers available from his home pages at Queen Mary College 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.)

Interpreting The Planck Constants m0, l0 & t0

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')  =  G0 m m' / r2

G0 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 G0 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 m0, l0, t0, T0. 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 m0, which one might expect to be small, is , in the realm of elementary particles, enormous. It would take about 1019 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.

Numerical Values Of Some Physical Constants

     Symbol     Name                Dimensions            Value
     c        speed of light        [L]/[T]       2.998 E+10 cm/sec

              c = 1/sqrt(epsilon0 mu0)
     ε0  Free space   [Q2]/[M][L3][T2]  1/(36 π) E-8 farad/cm
     μ0       Free space       [M][L]/[Q2]     (4 π) E-9 henry/cm
     ħ    Reduced Planck        [E][T]         1.0546 E-27 gm-cm2/sec
              Constant                             1.381 E-34 cm
     e        Electron charge        [Q]           1.381 E-34 cm
     me      Electron mass          [M]           9.1095 E-28 gm
                                                   6.764 E-56 cm
     re      Electron radius        [L]           2.81794 E-13 cm
              classical (= α(ħ/(me c))
     res     Electron Schwarzschild radius  [L]   1.35264 E-55 cm
              (= 2G0 me/c2)
              22 orders of magnitude *smaller* than the Planck length
     a0      Bohr radius            [L]           0.529177 E-08 cm
              ( = ħ2/(me e2) )
     mp      Proton mass            [M]           1.6726 E-24 gm
                                                   1.2419 E-52 cm
                                                   938 Mev = 0.938 Gev
     α    Fine structure         []            1/137.036
              ( = e2/(4 π ħ c) )
     G0      Gravitational     [L3]/[M][T2]     6.673 E-8 cm3/gm-sec2
     l0      Planck length          [L]           1.616 E-33 cm   
              sqrt(ħ G0/c3)
     t0      Planck time            [T]           5.391 E-44 sec
              sqrt(ħ G0/c5)
     m0      Planck mass            [M]           2.177 E-05 gm
              sqrt(ħ c/G0)
     E0      Planck energy      [M][L2]/[T2]    1.95669 E+16 ergs
              sqrt(ħ c5/G0)

     T0      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
     tH      Heisenberg time        [T]           ? E-24 sec
                                     [L]           ? E-12 cm

     mπ      Pion mass              [M]           0.2240 E-03 erg
                                                   2.29240 E-25 gm

     mK      Kaon mass              [M]           0.7904 E-03 erg
                                                   8.83396 E-25 gm

     mμ      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]/[L2][T][K4]   5.668 E-5 erg/cm2-sec-deg4
               Constant (of Black Body Radiation)
	      ( = (2 π5/15) k4/(h3 c2) )

     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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