LETTERE AL NUOVO CIMENTO
VOL. 2, X. 6
7 Agosto 1971
Black-Body Radiation and Generalized Theory of Physical
Dimensions (*)
F~. I~ECAMI (**)
Center ]or Particle Theory, P h y s i c s D e p a r t m e n t
University o] Texas at A u s t i n - Austin,, 7'ex.
(ricevuto il 14 Giugm, 1971)
1.
-
Introduction.
Since s o m e d e c a d e s , m a n y p h y s i c i s t s w o n d e r e d a b o u t t h e so-called R a y l e i g h - J e a n s
f o r m u l a for classical b l a c k - b o d y r a d i a t i o n d e n s i t y (v --- f r e q u e n c y ) :
(1)
Q:oc v~ .
In fact, eq. (l) m a y s c a r c e l y a p p e a r as d e r i v e d i n a correct w a y f r o m classical p h y s i c s .
For some a u t h o r s , it looks like a relation violating t h e e n e r g y c o n s e r v a t i o n p r i n ciple (~). A t least, it is clearly e x p e c t e d to b r i n g to t h e w e l l - k n o w n (~u l t r a v i o l e t
c a t a s t r o p h e ~). In a n y case, cq. (1) o u g h t to r e f e r to an e q u i l i b r i u m d i s t r i b u t i o n ; on
t h e c o n t r a r y , it yields an instability c o n d i t i o n (e.g., it does n o t alh)w defining a t e m p e r a t u r e , for t h e black b o d y ) w h i c h c a n n o t evolve i n t o an e q u i l i b r i u m s t a t e w i t h i n
a finite t i m e . This s e e m s to be i n c o m p a t i b l e w i t h classical t h e r m o d y n a m i c s a n d
e l e c t r o d y n a m i c s (2).
Evell if P l a n c k ' s solution m a d e m a n y people f o r g e t t h e p r e v i o u s d o u b t s , in r c c e n t
t i m e s s o m e a u t h o r s (~-6) t o o k again t h e p r o b l e m i n t o c o n s i d e r a t i o n . I n s t e a d of eq. (1),
t h e y t r i e d to get a relation c o n s i s t e n t w i t h t h e r e m a i n i n g classical t h e o r y , even if n o t
well r e p r o d u c i n g t h e e x p e r i m e n t a l d a t a . F o r i n s t a n c e , one of t h e first p o i n t s q u e s t i o n e d
was t h e use of t h e e n e r g y e q u i p a r t i t i o n a s s u n l p t i o n w h e n t h e (, b l a c k - b o d y b o x ~) is
(*) Work partially supported by the Consiglio Nazionalc dcllc Rieerche and by AEC/AT(40-1)3992.
('*) On leave from the lstituto di Fisica Teorica dcll'Univcrsith, Catania.
(t) Cf. A. SO3E~IERFELD: Thermodynamics and Statistical Mechanics, edited by F. BoPP and J..'~IEIX-N'ER
(New York, 1956), p. 145.
(2) See also, e.g., P. BOCCIIIERI and A. LOINGER: Left. Nuovo Cimenfo, l, 709 (1971).
(s) •. GADIOLI and E. ]~EC)--~I~: private communication (Milan, 1965).
(4) T . H . B O Y E R : Phys. Rev., 182, 1374 (1969).
(~) E. C. G. SUI)ARSIIA.~
~ and J. 3IEHRA: Int. Journ. Theor. Phys., 3, 245 (1970).
(e) 1). BOCCmER~ and A. LOI.~'(~ER: LeH. Nuovo Cimento, 4, 310 11970).
297
298
~. RECAMI
made tend to infinity (according to the common procedure for getting a continuum
radiation spectrum); i.e. the applicability of the Boltzmann distribution to an infinite
series of classes of objects ((( oscillators ~ with frequency v).
Many details of the classical treatment are actually defective and worth of
re-analysis.
But, here, our aim is in trying to clarify the whole subject on the basis of the
rather general considerations which come out from the generalized theory of physical
dimensions (7).
2. - The generalized theory of physical dimensions.
It is well known t h a t the physical laws are relations between physical quantities
and not merely between pure numbers. The (~homogeneity requirement,) asks that
any physical law be written in the form
(2)
~v(p~, p~ . . . . . P . ) = o .
The P ' s are the various dimensionless products which can be built up by (rational powers
of) the m physical quantities Qx, Q2 . . . . . Qm, which have been recognized to affect the
phenomenon under examination.
In mechanics no difficulties were involved, since the constants appearing in the
mechanical laws are dimensionless. Let us recall, e.g., the laws for the oscillation period
of a pendulum, whose form may be foreseen on the only basis of requirement (2).
But, in dealing with the whole classical physics, there are difficulties due to the
presence of (( universal constants ~>, whose physical dimensions are a priori unknown.
This fact prevented the use of the homogeneity requirement outside the field of
mechanics.
Let us now choose the usual (three) mechanical fundamental quantities L, T, M,
and call A the fourth (fundamental) quantity introduced for the field under examination. Of course, a generic (( dimensional expression ,~ for the universal constants such as
(3)
[const] = [L x T ~ M s A ~] ,
with 2, r, #, a arbitrary exponents, would allow writing any arbitrary (( physical ,) law
(chaos).
On the contrary, we know that only one self-consistent group of laws holds for
every field, in correspondence to the physical interpretation we want associate with
those phenomena. The real laws can be actually derived by dimensional considerations,
provided that we limit the generality of eq. (3) by the introduction of linear relations
between the exponents (8). For instance, in the Maxwell electrodynamics, the universal
constants can be recognized (7) to obey the relation
(4)
[C] = [L a T ~ ~ / 2 + v Q - 2 ( , l + v ) ] ,
(7) P. STRANEO: IVuovo Cimento, 17, 183, 506 (1940); Rendiconti del S e m i n a r i o M a t e m a t i c o Fisico di
.Vlilano, Vol. 14 (Milano, 1940). I n a following p a p e r t h e t r a n s l a t i o n - - f r o m I t a l i a n - - w i l l be i n s e r t e d of
t h e m a i n p a r t s of t h e w o r k s b y STRANEO.
(s) P, STRANEO: A t t i Regia Accademia delle Scienze di Torino, Vol. 52 (Torino, 1918).
BLACK-BODY
RADIATION
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299
where t h e f o u r t h f u n d a m e n t a l q u a n t i t y i n t r o d u c e d here is t h e electric charge ( A - Q).
This is enough for writing, e.g., the force ] between two p o i n t charges e~, e.~ at a distance r
in the forrn
(5)
1 ele 2
[el~
where ~ is t h e dielectric constant. S y m m e t r y reasons, then, suggest t h e n u m e r i c a l
function ~0 to be identically c o n s t a n t : ~0~ 1. Thus, one finds i m m e d i a t e l y Coulonlb's
law. Of course, t h e p u r e l y numerical factor e n t e r i n g in eq. (5) depends on the measure
u n i t s adopted.
More generally, all t h e e l e c t r o d y n a m i c a l theories, compatible w i t h t h e energy
conservation principles, can be deduced (7) by means of mere dimensional evaluations,
n a m e l y b y assuming
(6)
[C] = [L ~ T ~ M<~+~)/nQ-2(~+r>/n]
as t h e generic expression of the possible dimensional constants for the new laws (n being
a rational number). F o r n = 0 we get t h e <(plane electrology ~; for n = 1 t h e usual
e l c c t r o d y n a m i c s ; for n > 1 infinite new electrodynamics, unphysical but still satisfying
t h e e n e r g y conservation principles (and all h y p o t h e t i c a l l y possible).
Analogously, as new geonletries were i n t r o d u c e d besides the Euclidean one, so we
m a y consistently consider more general kinematics and d y n a m i c s (as t h e relativistic
ones).
Also for the g r a v i t a t i o n theory, if we accept for simplicity t h e degeneration coming
from the coincidence of inertial and g r a v i t a t i o n a l masses, we m a y write directly
[G] = [L 3 T -2 M -1]
and
(7)
mira2
/ ml ~
r'
where t h e experience c o n f i r m e d - - t i l l n o w - - t h e assumption ~ ~ 1, suggested by obvious
s y m m e t r y reasons.
B u t let us now confine ourselves to thermodynamics, since we are i n t e r e s t e d in the
black-body problem.
L e t us n a m e l y consider all the h y p o t h e t i c a l (classical) t h e r m o d y n a m i c s , c o m p a t ible w i t h the conservation principles and such t h a t can account also for a radiation
(with a certain velocity).
Such theories will contain (s.~) universal constants of the t y p e (*) (0 ~ t e m p e r a t u r e )
(8)
[C] = [L~T~M~O~/~-I']
(n # O),
i.e.
(9)
(0) P. STItA~EO" .~tti Regia Accademia delle Scienze di Torino, Vol. 60 (Torino, 1924).
(*) Within this framework, very fundamental laws may be derived, still independent of any partieuJar
optical ,> hypothesis about the <,radiation *.
300
E. RECAMI
w h e r e e is t h e r a d i a t i o n velocity, k t h e B o l t z m a n n c o n s t a n t , a n d w h e r e t h e c o n s t a n t X
is a (~n e w ~ u n i v e r s a l c o n s t a n t (s.9)
(10)
[Z] = [Z2 Tn-2 M ] .
3. - Classical thermodynamics.
Corpuscular and ondulatory radiations.
W e w a n t n o w specify t h e v a l u e of n e n t e r i n g in r e l a t i o n (8). T h e big p o i n t is to recognize t h a t in classical t h e r m o d y n a m i c s t w o di]]erent specifications a p p e a r ( w h i c h
we shall call ~(o p t i c a l t>).
T h e first one (s) c o r r e s p o n d s to n = 1 (wave theory, or o n d u l a t o r y t h e r m o d y n a m i c s ) ,
a n d t h e s e c o n d one (s) to n = 2 (corpuscle theory, or c o r p u s c u l a r t h e r m o d y n a m i c s ) .
I n p a r t i c u l a r , t h e generic u n i v e r s a l c o n s t a n t X b e c o m e s
(11)
x=h
for n =
1 (waves),
(12)
z=i
for n = 2 ( c o r p u s c l e s ) ,
w h e r e h is e s s e n t i a l l y P l a n c k ' s c o n s t a n t (i.e. it is d i m e n s i o n a l l y a n (~a c t i o n t>) a n d j
is a n e w m H v e r s a l c o n s t a n t (10) ( w h i c h is d i m e n s i o n a l l y a (, m o m e n t u m of i n e r t i a ~)).
Therefore, the wave thermodynamics can be derived from the generalized physicald i m e n s i o n t h e o r y , b y m a k i n g recourse to c o n s t a n t s of t h e t y p e
(13)
[C~] = [L ~ T ~ M . 0~+~-~] ~ [c~-~, k-~-~ +, h~+~] ;
a n d t h e c o r p u s c u l a r t h e r m o d y n a m i c s b y c o n s t a n t s of t h e t y p e
(14)
[Co] = [L~ T ~ Mg0(~+T),2-~] = [c~-2~k-(~+,)/2+l~ ~'(x+~/z] .
L e t us r e p e a t t h a t t h e c o n s t a n t c is a velocity, k a n e n t r o p y , h a n a c t i o n a n d j a mom e n t u m of i n e r t i a . As we c a n see, P l a n c k ' s c o s t a n t h a n d t h e n e w (10) (( S t r a n e o ' s cons t a n t )>j were a l r e a d y i m p l i c i t l y c o n t a i n e d in classical physics. C o n c e r n i n g t h e c o n s t a n t
of P l a n c k , e.g., t h i s fact was r e a l i z e d long t i m e ago (1) a n d in m o r e r e c e n t t i m e s (11)
(PLANCK h i m s e l f m e t t h e c o n s t a n t h also in t h e classical f r a m e w o r k (1,12)).
L e t us first d e r i v e a n e x p r e s s i o n for t h e (( i n t e g r a l d e n s i t y t> (energy) of a r a d i a t i o n of w h a t s o e v e r n a t u r e . F r o m t h e d i m e n s i o n t h e o r y (7) we get ( w i t h iV a p u r e n u m b e r )
(15)
U = N [ L -1 T -2 MO-(3+n)/n] 90 (3+n)ln ,
w h e r e t h e coefficient of 0 (3+n)ln is a d i m e n s i o n a l c o n s t a n t . F o r n = 1, we h a v e t h e
S t e f a n ' s law for t h e w a v e r a d i a t i o n s a n d for n = 2 t h e a n a l o g o u s law for t h e corpuscular radiations, U=const.0~.
(to) See also L. DE BROGLIE: Une nouveUe conception de la lumi~re (Paris, 1934); and Nouvelles recherches
sur la tnmi~re (Paris, 1936).
(11) See, e.g., B. LIEBOWITZ: NUOVO Cimento, 63A, 1235 (1969). See also, e.g., R. K. VAm~CA:Phys. Rev.
Left., 26, 417 (1971).
('~) See, e.g., J. MEHRA and H. RECHENBERG: preprint CPT-78 (Austin, Tex.).
BLACK-BODY
RADIATION
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301
As regards the (specific) density % one gets, in the u = 1 case (*),
(16)
{
u~= A'v3"r
[A] ~ [ L - ~ T 2 M ] ,
,
[B]
------I T - 1 0 - 1 ] .
This is t h e well-known W i e n ' s law, holding for any wave radiation. The W i c n ' s law
expresses t h e best specification of u~ which can be classically reached on t h e basis of
t h e r m o d y n a m i c s and of t h e e l e c t r o m a g n e t i c n a t u r e of radiation. W e shall r e t u r n on
this p o i n t in t h e following.
4. - More a b o u t classical t h e r m o d y n a m i c s .
I n order to clarify how t h e (( t h e o r y of dimensions ~ works in the case of interest,
let us add w h a t follows. W e h a v e seen t h a t the u s u a l t h e r m o d y n a m i c a l theories (with
o n d u l a t o r y or corpuscular radiation) are particular, (( irreducible ~ cases of a t h e o r y
containing universal constants of the t y p e (8)-(9)
(18)
[C] = [L~T~M~O (~+~)/n-I~] ~ [c~-21'k-(~§
(~+v)/~] .
H o w e v e r , the constants k and X do not d e p e n d on t h e p a r t i c u l a r n a t u r e of the bodies
considered and are actually universal constants. L e t us explicitly state here t h a t k
assures t h a t t h e energy conservation principles ( = t h e r m o d y n a m i c s ) are respected,
and X assures the presence of a radiation w i t h generic n a t u r e (the d e t e r m i n a t i o n s of 7.
speci]ying t h e notion of radiation, i.e. specifying t h e (( optics ~>).
On t h e c o n t r a r y , the v e l o c i t y c depends on the m e d i u m in which t h a t radiation
propagates and, therefore, it is not strictly a (, universal )) constant, in a n y case. Thus,
t h e m o s t general laws, v a l i d / o r all the matter, can be o b t a i n e d b y eliminating t h e propagation v e l o c i t y e in eq. (18)--i.e. imposing 2 - - 2 / ~ = 0 - - a n d
putting c among the
physical quantities which m a y c v c n t u a l l y enter in the law expressions. I n fact, one
will h a v e now
(19)
[C] -- [L 2~ T ~ M/' 0(2t~+~)/~-I~] = [k -(2~+~)/"+~ Z(~+~)/~] ,
and, following the usual procedure, one could directly get b o t h t h e so-called Kirchhoff's
laws (~'~).
(*) A t this point, if we w a n t e d to a s s u m e t h a t our laws m a y m e a n i n g f u l l y d e p e n d also on a * freq u e n c y 9 ~, even in t h e g e n e r a l case (**) in w h i c h n is n o t specified (i.e. for e v e r y a r b i t r a r y * optics ~),
one m i g h t write a generalized W i e n ' s law for any r a d i a t i o n w h a t s o e v e r
(17)
uv = A.vn+~. r
[A] ~- [L-1Tn+IM],
[B] =- [T-IO -']~]
.
E q u a t i o n (17) should yield the best d e t e r m i n a t i o n of u v w h i c h m a y be classically p u r s u e d w i t h i n t h e
field of all t h e possible * optical theories * (without a d d i n g /urther hypotheses),
(**) I n such a context, one could notice t h a t - - e v e n w i t h i n classical p h y s i c s - - t h e possibility of associating
a w a v e l e n g t h to a n y kind of r a d i a t i o n (also of c o r p u s c u l a r t y p e ) m i g h t h a v e b e e n conceived a n d p e r h a p s
guessed f r o m dimensional considerations.
302
E.
RECAMI
L e t n s c o n s i d e r t h e (, first, l a w ,~, in t h e case of o n d u l a t o r y r a d i a t i o n s ( n = 1).
I f u~ is t h e r a d i a t i o n specific e n e r g y , f r o m t h e t h e o r y of p h y s i c a l d i m e n s i o n s o n e e a s i l y
g e t s t h e first K i r e h h o f f ' s l a w
(20)
u,,c 3 -
b . v 3.
[h v J - - F ( v ' O ) '
w h i c h , of c o u r s e , h o l d s for a n y m e d i u m w h a t s o e v e r (*).
G o i n g f a r t h e r , w e m a y a t l a s t c o n s i d e r t h e p a r t of t h e r m o d y n a m i c s t o t a l l y independent of a n y n o t i o n of r a d i a t i o n . I n t h i s case, we g e t s i m p l y o l d t h e r m o d y n a m i c s
o f m a t e r i a l b o d i e s ( w i t h o u t h e a t r a d i a t i o n ! , e.g.). I n f a c t , f r o m u n i v e r s a l c o n s t a n t s
of the type
(22)
[C] = [ L 21' T -2/~ ]'[#0 -~'] = [M'] ,
w e o b t a i n (7) t h e law of p e r f e c t g a s e s
PV=
toRT,
R bcing an entropy.
5. - T h e b l a c k b o d y .
W e h a v e seen t h a t classical ( o n d u l a t o r y ) p h y s i c s c a n n o t go b o y ( r o d t h e W i e n ' s law.
W e k n o w , also f r o m t h e d e v e l o p m e n t of q u a n t u m p h y s i c s , t h a t it is p o s s i b l e to go
f a r t h e r b y p r o p e r l y (~c o n n e c t i n g ~> t h e o n d u l a t o r y a n d c o r p u s c u l a r c o n c e p t i o n s .
T h e classical b l a c k - b o d y r a d i a t i o n d e n s i t y q u e s t i o n (i.e. t h e q u e s t i o n of t h e e n e r g y
l ) a r t i t i o n in a s y s t e m of b o t h c o r p u s c u l a r m a t t e r a n d w a v e r a d i a t i o n ) is, in f a c t , t h e
classical p r o b l e m in w h i c h o n e o u g h t c o n t e m p o r a n e o u s l y to use b o t h t h e o n d u l a t o r y
a n d c o r p u s c u l a r i n t e r p r e t a t i o n s . B u t w e s a w t h a t , i n classical physics, t h e s e t w o t h e o r i e s
a r c ( , i r r e d u c i b l e ~>, i n c o m p a t i b l e , a n d it is i m p o s s i b l e t o c o m p o s e s u c h a d i c h o t o m y
( w i t h o u t f u r t h e r , nonclassical h y t ) o t h e s e s ).
T h e classical dichotomy b e t w e e n t h e (, w a v e ,) a.nd (, c o r p u s c l e ~ t h e o r i e s is s t r i k i n g l y
f o r w a r d e d b y t h e f a c t t h a t n t a k e s t h e v a l u e n = 1 in t h e first t h e o r y a n d n = : 2
in t h e s e c o n d o n e (**). L e t us r e w r i t e
(23)
[Z,,] -- [ L2 T ' - 2 M ] .
(*) If we wanted again to assume that our laws may generally depend also on a * frequency * v, for
any whatsover value of n, one might ~Tite the generalized Kirchhoff's laws. For instance, hmtead of
eq. (20), we would have (~)
which states that for any radiation wbatsover, in thermodynamical equilibrium, the quantity uvc*
equals always the same universal function Fly, 0), in any medium whatsoever. Analogously, for the
second law ,. The generalized Kirchhoff's laws should be the most general laws of classical physics,
holding for every medium and for an)" radiation whatever (provided that the conservative principles
are preserved).
(~ By the way, considering values of n such tllat 1 < ~ < 2, we might build * i~dermediate ~)theories,
self-consistent but unphyslcal.
BLACK-BODY RADIATION AND GENERALIZED
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303
It is necessary t o l i n k the theories based respectively on the universal constants (13)
and (14).
But clearly, in the general expression of the universal constants, we cannot introduce both
Zl- h
and
Ze- J
without any constraint, otherwise our laws would be quite arbitrary. We would actually
have no laws (chaos).
From the dimensional point of view (7), Planck's idea was the following. The
.quantity (( action ~ appears in a certain sense to regulate the (~corpuscular thermodynamics ~, for the very fact that it does not enter into eq. (14), i.e. it remains constant
in the related phenomena. One may enlarge the corpuscular-physics framework by
introducing variations of the ((action)~ h. Of course, not arbitrary variations (for
avoiding the chaos), but conditional variations {h} (namely, multiple of the Planck's
constant). I n other words, ]~LANCKdeveloped from the corpuscular t h e r m o d y n a m i e s - - a
(quantum) theory founded on universal constants of the type
c = c(c, k, j, {h}).
Within the classical theories, any such composition between constants of the
types (14) and (13) cannot obviously be achieved. All the authors who tried to do that
could do nothing but equating to zero the exponent of Z in eq. (18). Such a procedure,
independently o/ the mathematical details, forwards (7) inevitably the Rayleigh-Jeans
formula.
This Rayleigh-Jeans formula was always reached just following implicitly that way.
B u t that trivial procedure is wrong. In fact, it yields a thermodynamics in accord with
the conservation principles, but in which no (( optics )) at all is specified, and no type
of radiation is described (even if a velocity c enters in it, somewhat inconsistently).
Since no radiation can be accounted for, in this ~(theory ~) the very concept of the
usual ((frequency)) v becomes ineaningless. Therefore, the Rayleigh-Jeans relation
comes out from a hybrid context, and cannot be regarded as correctly derived by classical
physics (2) (or as belonging to the classical theories). When Z is lacking, with universal
constants of the type
(24)
[C]
[L~'T-~JI"O -f'] ~ [c~-2"k ~]
one would even fail in building a formula (of Stefan's kind) for the ((integral density ~)!
Constants (24) would not allow writing down any ((integral density ~) depending only
on the temperature.
6.
-
Conclusions.
Thus, independently of the mathematicai approaches and tricks, we cannot derive
within classical theories any distribution law for the (specific) density % as a function
of v and 0 (besides of the universal constants).
hi particular, all the receut papers (3-5) aimed at obtaining a Planck-type formula in
classical physics got some results only because they implicitly introduced ]urther
((( quantum ~)) hypotheses. It is easy to recognize in each work (a-5) the point or the
points in which the q u a n t u m hypothesis was assumed, more or less consciously. Ill
304
E. RECAM I
t h e B o y e r ' s w o r k (4), for i n s t a n c e , t h e c o n c e p t of (~z e r o - p o i n t r a d i a t i o n ~)--by no m e a n s
c l a s s i c a l - - h a s b e e n e x t e n s i v e l y e m p h a s i z e d a n d used.
I f w e a d m i t , i n d e e d , s o m e q u a n t u m h y p o t h e s e s , w e m a y r e a c h ((Planck-type)~
f o r m u l a e in a v a r i e t y of w a y s : see, e.g., ref. (3-5.1a).
F o r i n s t a n c e , if we a d m i t t h e b l a c k - b o d y r a d i a t i o n to be o r i g i n a t e d or c o n s t i t u t e d
b y e l e m e n t a r y oscillators (3), w h i c h , b e f o r e escaping, r e a c h a s u i t a b l e e n e r g e t i c a l equil i b r i u m inside t h e (( black b o d y ~), w e m i g h t e v e n h a v e for u, a M a x w e l l i a n d i s t r i b u t i o n (*) (i.e. a d i s t r i b u t i o n a t least free of u l t r a v i o l e t c a t a s t r o p h e a n d (( similar ~ to t h e
P l a n c k ' s one).
As is clear f r o m all w h a t p r e c e d e s , our s t a r t i n g p h i l o s o p h y is t h a t p h y s i c a l dim e n s i o n s ( a l t h o u g h w e are s o m e w h a t free in choosing (~) t h e f u n d a m e n t a l q u a n t i t i e s i
are n o t a t all a r b i t r a r y c o n v e n t i o n s , b u t on t h e c o n t r a r y e x p r e s s i n t r i n s i c c h a r a c t e r i s t i c s
of t h e v a r i o u s p h y s i c a l q u a n t i t i e s (7). A n o p p o s i t e c o n v i c t i o n is still d i f f u s e d , - - n o t w i t h s t a n d i n g t h e r e c e n t w a r n i n g s of t h e I n t e r n a t i o n a l E l e c t r o t e c h n i c C o m m i s s i o n ,
A j a a n d Brussels, 1935, a n d of t h e I X Conf6rence G~n~rale des P o i d s a n d Mesures, 1 9 4 8 , - - o w i n g t o t h e k n o w n abuses (v) of t h e l a s t - c e n t u r y p h y s i c i s t s , a n d to t h e
s u b s e q u e n t use of p h y s i c a l l y i n c o n s i s t e n t m e a s u r e u n i t s y s t e m s (as t h e CGSes, CGSem,
or Gauss ones).
T h e e x p l i c i t r e q u i r e m e n t of d i m e n s i o n a l h o m o g e n e i t y w o u l d be v e r y p r o b a b l y useful
also in m o d e r n p h y s i c s (1~). I n p a r t i c u l a r , a f t e r t h a t HEISENBERG c l a i m e d for t h e
i n t r o d u c t i o n of a ~ f u n d a m e n t a l l e n g t h ~, t h e n e e d of a greater n u m b e r of u n i v e r s a l
c o n s t a n t s h a s b e e n s t r e s s e d b y m a n y a u t h o r s , also r e c e n t l y (~), in a c c o r d w i t h t h e
g e n e r a l i z e d t h e o r y of p h y s i c a l d i m e n s i o n s .
***
T h e a u t h o r is v e r y i n d e b t e d to Dr. M. BALDO, Dr. G. DE MARCO, P r o f . E. GADIOLI,
P r o f . P . STRANEO for m a n y s t i m u l a t i n g , helpful discussions ; a n d t h a n k s Profs. A. AGODI
L. LANZ, A. LOINGER, J . MEHRA, R . POTENZA, S . RECHENBERG, E. C. G. SUDARSHAN
for t h e i r i n t e r e s t in t h i s w o r k . Besides, h e is g r a t e f u l to P r o f s . F. W . DE WETTE a n d
E. C. G. SUDARSHAN for t h e h o s p i t a l i t y e x t e n d e d to h i m a t t h e D e p a r t m e n t of P h y s ics, U n i v e r s i t y of Texas, in A u s t i n .
(~3) D. LEITER: Nuovo Cimento, 63 A, 1087 (1969).
(*) We consider the roughest model for a black body. Let our * cavity * contain elementary oscillators
(each one behaving as a whole), which reach their stationary condition by mutual exchange of collision
energy, such as the molecules of a perfect gas. Then, at stationarity, each elementary oscillator will
dissipate as much energy as it receives, and it will not suffer damped oscillations. At last, let us suppose
that an average oscillation width a may be defined for every frequency v. In a first approximation,
we may even assume a to be independent of v.
Thus, as for the molecules of a perfect gas we get the Maxwell distribution of velocities, so for our
elementary oscillators we shall get a Maxwellian distribution of energies. The black-body radiation distribution is then assumed to be ~ sample of the very oscillator distribution for the various frequencies.
We immediately reach the following equation for the differential distribution of the number of oscillators:
(25)
dP(~) = 4~a3(2zra/kT)~ ~' exp [-- 2u~ma~/kT] d~.
This equation, dimensionally correct, is in accord with the Wien's law. In it T is the absolute temperature, k the Boitzmann constant and m the electron (e.g.) mass.
(,4) See, e.g., J. RAYSKI: Acta Phys. Polon., A37, 269 (1970).
(x~) See, e.g., YA. B. ZEL'DOVICH: SOY. Phys. Usp., 11, 381 (1968); ]~. A. Ir
preprint
UCRL-19893 (submitted to Phys. Rev.) and UCRL-20627; M. CREUTZ and R. JAFFE: preprint SLACPUB-791 (submitted to Phys. Rev.); E. G ~ : private communication.