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The constitutive map and some of its ramifications
ANNALS
OF PHYSICS:
71, 497-518 (1972)
The Constitutive
Map and Some of Its Ramifications E. J. POST
523 Altenkirchen,
Im Vogelsang
4, West-Germany
Received May 20, 1971
The constitutive map is introduced to replace the familiar relation D = CE and B = pn. Its objective is a rigorous functional separation between field- and constitutive equations. The possibility to implement this functional separation is shown to depend critically on the introduction of electric charge as an independent space-time invariant unit, rather than as a unit derived from mass, length, and time. The linear map is shown to be invariant under the conformal group, while the field equations remain invariant under general space-time diffeomorphisms. Algebraic and differential comitants of the linear map are discussed, their physical roles are identified. The impedance of a medium emerges as a space-time invariant. The new concepts are then applied to prove the invariance and constancy of the fine structure “constant” and the ratio ee/fi.
I. INTRODUCTION In the following we are concerned with an attempt at reorganizing a well-known and well-established part of physical theory. The objective is one of opening up possible perspectives that cannot be easily obtained in any other fashion. The general subject matter is electromagnetic theory. The specific aspect under scrutiny is the need for a recasting of the conventional constitutive relations in a form that is more susceptible to a deductive treatment. Major attention will be given to the constitutive properties of the free-space medium. The new insights resulting from such a reorganizational move are varied. Most of them are qualitative in nature, which should not be surprising. Nontrivial and fundamental, quantitative, experimental results normally precede theories; they are frequently given an exhaustive description before a theory reachesfuller maturity. The result is that the incentive for a further development of the theory drops below threshold. Yet, qualitative features in theory and observation can also be most significant and useful. They assist in classifying and ordering of already existing results of a quantitative nature. Important highlights of the present investigation
are: an inertial
frame definition
basedon the single question: what transformation group preservescentral symmetry and spatial isotropy? In addition a further delineation is obtained of the distinction between physical frames and the family of permissible coordinate neighborhoods 497 Copyright All rights
0 1972 by Academic Press, Inc. of reproduction in any form reserved.
498
POST
associated with such physical frames. The criterion is a group criterion which is independent of the choice of coordinate neighborhood. More specifically, the physical role of the conformal group in electromagnetic theory can be identified with greater precision on the basis of its decomposition in subgroups. The Lorentz subgroup is found to characterize the inertial frame situation, whereas the Abelian subgroup of inversions-translations-inversions identifies a family of physically equivalent situations in isotropic space. The latter conclusion completely bears out the original motivation of Cunningham [I] and Bateman [2] to extend the technique of equivalent mirror charges in electrostatics by studying conformal transformations. In contrast to previous discussions, the coqformalgroup can now be identiJied as characterizing a purely constitutive situation. The traditional lack of clarity concerning the physical role of the conformal group can thus be relegated to the traditional absence of a clear-cut distinction between physical frames and the large family of coordinate references that can be associated with that single frame. Sometimes, but not always, one may be able to single out in these families a subgroup representing in a meaningful manner a group of physically equivalent situations. The Abelian subgroup in the conformal group is an example. The mentioned distinctions can be cast into a consistent mathematical framework by following an early lead by Cartan [3], elaborated by van Dantzig [4] and the present author [5]. To accommodate all conceivable situations it is now necessary to consider the complete family of space-time diffeomorphisms, i.e., all linear and nonlinear space-time transformations that are continuous, differentiable and nonsingular. The essence of Cartan’s [3] idea is that the basic laws of electrodynamics permit a complete functional separation between medium independent law statements and the constitutive specification of the medium. To include free-space the medium independent law statements should be independent of the space-time metric structure. The geometric structure of space-time is subsequently covered by a scrupulously deductive development of the constitutive relations. The major emphasis accordingly focuses on the so-called “constitutive map”; a new word meant to accommodate the familiar and old relations B = EE, B = $7 in a physically more “transparent” garb. A number of additional benefits then ensues apart from those already mentioned. Here are a few of them. The constitutive map has a general algebraic invariant that can be identified as an impedance. The fine structure constant cx can accordingly be identified as a ratio of two generally invariant impedances. The observed constancy of 01throughout the universe now leads to better guided considerations concerning the cosmological behavior of the invariant impedances of which oi is the ratio. An examination of the constitutive map also reveals that the condition of net charge neutrality of a medium leads very naturally to a study of its differential
CONSTITUTIVE
MAP
499
comitants. Not all differential comitants of the linear constitutive map are presently known. In the present context this means that the n.a.s. condition to be imposed on the constitutive map to securenet charge neutrality is not available. Yet a more complete mathematical information about this subject seemsessential to answer the question whether or not electric charge has a structurally determining influence on space-time geometry, apart from the massthat carries the charge.
II. ISOLATING
THE PURE FIELD RELATIONS
The phenomenological description of electromagnetism is characterized by a separation into two functionally distinct domains. First, there is the concern with properties that all systemsshare; secondly, there is the concern with a systematic characterization of specific systems.This policy of approaching the subject matter has led, in a quite natural manner, to the introduction of two setsof basic relations. They are known as the electromagnetic field equations and the constitutive equations of the medium. The early historic developmentsomehowfailed to accept the free-space medium as worthy of a complete constitutive treatment. When at the turn of the century the Lorentz symmetry of this “unmedium” was uncovered, convention and tradition were already so well established that adoption of free-space as a full-fledged member of the family of electromagnetic media was not quite possible. As a result of this historic development, Lorentz invariance is today commonly presented as a symmetry of the free-spacefield equations rather than as a symmetry of the free-space constitutive equations. For the purpose of the present investigation, it is necessaryto abandon these historic conventions. The following program permits a rendering of the field equations characterized by a maximal independence of preconceived or prejudicial notions concerning the electromagnetic media to which they apply. Hence preconceived notions of geometric space-time structure will have to be removed from the field equations if we are to include free-space in the family of electromagnetic media. The field equations are thus erected on the austere assumption that space-time is a differentiable manifold. The fundamental physical facts may then be introduced in the following logical order.* 1. The Faraday induction law and the absence of magnetic monopoles are locally expressed by the vanishing exterior derivative of the differential 2-form F of electrified E and magnetic induction B: dF = 0.
(1)
1 The differential-form method has been selected to present the basic facts in a concise yet complete manner. Additional explanations for this decision are provided in an appendix.
500
POST
Equation (1) expresses in a space-time invariant manner the “local” conservation of flux or equivalently the local absence of magnetic monopoles. The local conservation statement (1) can be extended into a global (universally valid) conservation statement, if the integrals of F over all possible 2-dimensional closed domains vanish. One then concludes from de Rham’s theorem2 on the existence of a globally defined l-form A with the property dA = F.
(2)
Relation (1) is implied by relation (2), because dd = 0. The inverse statement that (1) implies (2) does not hold in a global manner unless the conditions for de Rham’s theorem are met. 2. The conservation of net charge, which reflects the well-observed fact that charge creation always occurs in pairs of opposite polarity, is expressed by a vanishing exterior derivative of the differential 3-form C of charge and current density dC = 0. (3) Equation (3) expresses in a space-time invariant manner the local conservation of charge or equivalently the local absence of a net charge creation mechanism. The local conservation statement (3) can be extended into a global (universally valid) conservation statement, if the integrals of C over all possible 3-dimensional closed domains vanish. One then concludes from de Rham’s theorem on the existence of a globally defined 2-form G with the property dG = C.
(4)
Relation (3) is implied by relation (4), because dd = 0. The inverse statement that (3) implies (4) does not hold in a global manner unless the conditions for de Rham’s theorem are met. The relation (1) is a medium-independent, space-time invariant rendering of the Maxwell equations curl E = -B
and
div B = 0.
The relation (4) is a medium independent, space-time invariant, expression that so far only resembles the Maxwell equations curl H = D + C
and
div D = p.
(14 rendering of an (44
a See Ref. [6, Chap. III, p. 391. A very interesting and physically digestable discussion of the de Rham theorems appear in two French papers, Math. Congress (1932), p. 195, Enseignement Math. 35 (1936), 213, and a German paper, Sem. Hamburg Un. 12, p. 313. See also the appendix of this article.
CONSTlTUTlVE
501
MAP
The relations (4) and also (4a) become expressionsof the Biot-Savart and Amp&e force laws only after one de$nesthe medium in which they are supposedto apply. The fields, or rather the forms, A and G, share the property of a gauge undeterminedness;each is defined module an exact form.
III. THE LINEAR CONSTITUTIVE
MAP x
Equation (4) (in principle also 4a) was introduced as a mere mathematical consequenceof the global validity of the law of net charge conservation. Through de Rham’s theorem, one thus carries Maxwell’s argument for the displacement current to its logical conclusion. Simultaneously, however, one modifies through this deed the physical interpretation of the fields D and R (i.e., G); they have been reduced to the role of potential fields with a nonunique gauge. The relations (4) acquire a more specific physical meaning in the senseof the Biot-Savart and Ampere force laws after the introduction of a constitutive relation which in fact should also fix the gauge of G. In analogy with the well-known relations D = l .i? and B = pi?, one may assumethat the coefficients of G are linearly related to the coefficients of F
The tensorial quantity x obeys the obvious symmetry relations
becausethe coefficients of G and Fare both antisymmetric. The relations (5) do not presupposea specific symmetry of the medium, neither is it assumedthat the medium is homogeneous; x may depend on the coordinates and on the time. Furthermore, the algebraic nature of (5) indicates that local and instantaneous interaction has been assumed,which meansthat the coefficients of G at the space-time point P are solely determined by the values of the coefficients F at that samepoint P. These assumptionsare well justified if all properties of a material medium can be interpreted as local statistical averages of free-space properties. The relations (5) may thus be thought of as primarily applicable to free-space, although at times it may be convenient to consider their direct applicability to a material medium in a macroscopic sense. The relations (5) may be called the constitutive map. They map an exact 2-form F = dA into a general 2-form G. The 2-form G is called closed if dG = 0, which means at that point electric charges are absent. For a general map x, however,
POST
502
one may not expect that dG = 0. Hence the nature of x determines whether C = 0 or C # 0. The linear constitutive map x can be reexpressedin an equivalent form by means of the totally antisymmetric Levi-Civita unit tensor @for four dimensions.
The equivalent form obeys the symmetry relations (independent of the medium) X AVoK=
_
X
vim =
_
x
AYKO .
W
If a Lagrangian function 2 exists and one defines the G field according to Mie: G = 39/S’, there is now the additional symmetry (6b) The form (6) of course, relates to a corresponding modification of G
Finally, where free-spaceis not known to be birefringent, an additional restriction may be placed on x in matter-free domains. It can be shown that x is not birefringent if it is constructed as the alternating product of a symmetric tensor of valence 2. Drawing on the existing knowledge about a metric space-time with metric tensor g,, one may consider the form AWJK =
X
y,
1 g ~l/z(glogvK
-
gAy7)
(8)
in which g is the determinant of g,, , Y, is an invariant scalar to be determined later.3 In the sequel primarily the unstructured forms (5) and (6) will be used to circumvent unnecessary prejudicial notions concerning a space-time geometric structure based on a metric. The expression (8) is related to the so-called Hodge-star operator [17] defined for a nonorientable space-time manifold. The reader should be alerted to the fact that it is common in the literature on differential geometry to restrict the discussions to orientable manifolds. Space and space-time are not orientable by virtue of the existence of enantiomorphic structures in space; e.g., crystals, molecules, and elementary particles that lack a center of symmetry. 3 A discussion is found in Ref. [5, Chap. 9, formula (9.4)].
CONSTITUTIVE
IV. THE TRANSFORMATION
503
MAP
PROPERTIES OF THE LINEAR
MAP
x
The map x is, of course, tensorial by virtue of the tensorial properties of the coefficients of F and G. Yet its transformation deservesspecial attention becausex in the form (5) is not an ordinary tensor but a pseudo or W tensor.4 It transforms with the sign factor d/j d / in which d is the Jacobian and j d 1its absolute value A -A’“’ A:‘A$4~~&1”,, x. O’K’= IAl
.
Note that the transformationalfeature introduced in (9) is most essential.It is not to be considered as a show of overconscientiousnessor mathematical pedantry. One can easily convince onself why this sign factor A// A 1is indispensable. An inspection of the equations (la) and (4a) showsthat the electric and magnetic fields trade places. Hence to secure the always present relation between D and E and B, H in a centrally symmetric medium, it is necessarythat the coefficients j$,, , etc., are preserved under a spatial inversion. In other words, D and E are of same “parity” (E, D, polar vectors) while B and Hare also of same“parity” but opposite to D and E (B, H axial vectors). Let the spatial inversion be given by the Jacobian matrix 1 A’=
0
0
i 0 O-l 0-1 0 ,o
0
0 0i f
(10)
O-l
An application of (10) to (9) showsx!.‘,, cannot be preserved unlessthe sign factor A/l A / is included in the transformation (9). The W-transformation properties of the mixed tensor 2 has rather important consequencesfor the differential 2-forms F and G. The result is sufficiently significant to be stated in the form of a theorem: Thefundamental 2-forms F and G of electromagnetic theory have oppositeparity.5 It follows that either For G should transform as a W tensor. One retains the best 4 Wtensorsor pseudotensorsare denoted by a “tilda”. The W stands for the nameof H. Weyl who was one of the first to introduce this distinction explicitly. Earlier usage, however, occurred in crystal physics by W. Voigt. See Schouten [7]. In the French literature the names “pair” and “impair” are used. See Ref. [6, pp. 19 and 221. 5 Differential forms of opposite parity are discussed by de Rhan [6, pp. 19 and 221. Note that the term opposite parity is unique because the coefficients of F and G are necessarily covariant by nature. By contrast the distinction: tensor, W tensor is not unique as will become clear in the sequel.
504
POST
conformity with existing conventions if G is taken to be the W form. Its coefficient thus transforms according to the rule C&,, = (O/l d I) A;#A;&
while F retains the standard transformation
,
(11)
rule of a covariant tensor
Fl’u’ = A;,A;,FA, .
(12)
The distinction between ordinary quantities and W quantities is not a standard feature in physical theory except in crystal physics or elementary particle physics. Tradition has it that these distinctions in field theory are at most retained in a somewhat ad-hoc manner to ensure that the vectors E, D have polar spatial character while H, B have axial spatial character. Moreover, it is common to convert the W-tensor e and the W-mixed tensor 2 into tensor densities that transform with the absolute value 1 d I. The totally antisymmetric unit tensor of Levi-Civita is itself a W-tensor density to possess the property of absolute invariance under “improper” transformations [see (lo), Ll = -11. The equivalent density versions of G and 2 are defined by ($ji”’ = gpO~&
)
(13)
and (14) Their transformation
laws are 6 h’v’ = 1d 1-l AfA;‘(liAY,
(15)
A’“‘O’K’ = I d 1-l &&fA,“‘xA”“K. X
(16)
and
V. ALGEBRAIC
COMITANTS
OF THE MAP
x
Algebraic invariants and comitants of the map x can be obtained by standard tensorial methods. An obvious invariant of the mixed W form (9) is the pseudo scalar f1 = 2;;“‘. (17) The same pseudo scalar can, of course, be obtained from (16) by the process of alternation, i.e., summing over all even transpositions of [XYUK] with a plus sign
CONSTITUTIVE
and all odd transpositions brackets. X It follows
505
MAP
with a minus sign. The operation is denoted by square [A’Y’U’K’I
=
, d
Af&‘A$A~;Xr”YoHI*
1-l
(18)
that A = A~;A:‘AfA:]
Hence, with the appropriate
.
normalization
and from (17) or (18) and (19) it follows
that
A 21’ = jdi
j&
is a pseudo scalar or W scalar. An algebraic comitant of the constitutive map is obtained by contraction one index 2;” = g;;v” = &rj,v. The constitutive
09)
(20)
over (21)
map x has also a true scalar invariant x2
=
-..OK-..A” XA” XOK
(22)
or equivalently with the help of the totally antisymmetric Levi-Civita unit tensor one can write x2 also in terms of the tensor density representation of x
The invariant x2 is different from zero in any medium. The W-scalar g1 , of course, vanishes in any centrally symmetric medium. Yet for a long time it seemed impossible to show experimentally that f1 could exist in a medium without central symmetry [5]. In fact quantum theoretical explanations have been suggested why a so-called “directive effect” between electric and magnetic dipole moments in a material medium cannot exist. However, recently Hou and Bloembergen [8] have shown that the effect indeed exists in nickel sulfite crystals at low temperature. Previous quantum mechanical discussions gave a negative result because of a hidden central symmetry in the Hamiltonians that were used for the discussion. Hence the moral of this story: the Neumann principle of crystalphysics rules supreme in classical as well in (nonclassical) quantum physics. In a microscopic sense the W-scalar g1 can only exist in particles that do not have
506
P0S-r
central symmetry. Free-space has a vanishing f1 . The latter fact can be easily established from the metric representation (8) of the constitutive map x. Alternation over hUK clearly leads to zero because the metric tensor g,, is symmetric.
VI.
DIFFERENTIAL
In a charge-free medium closed differential 2-form F requirement would lead to In a charge-free medium,
COMITANTS
OF THE
MAP x
the constitutive map x is required to map the (exact) into a closed differential 2-form G. An exception to this the presence of net electric charge. the Maxwell Eqs. (1) and (4) are source-free dF = 0,
(23)
dG = 0.
(24)
The 2-form F is always closed, in fact it is exact; the latter property only refers to its global behavior, not considered at this moment. In a charge-free medium the 2-form G is closed, because C = 0. The form G is definitely not exact. It follows that x maps a global property into a local property. Hence the map x is local in nature. The local conditions of charge absence may thus be formulated in the following abstract mathematical way if we realize that the condition should hold for all F. The constitutive map x in a charge-free domain of space-time gives a local automorphism of all closed 2-forms. Question: What are the n.a.s. conditions to be imposed on x so that it preserves the local integrability of all closed 2-forms? This problems has been solved for a linear map of l-forms by Nijenhuis [9]. No solution is available for 2-forms. One may expect, however, that the integrability preserving condition to be imposed on x, if it exists, should be a differential comitant of x only. Hence it should be tensorial, which means the condition is independent of arbitrary coordinate changes because of its homogeneous transformation. It is possible, however, to obtain at least a necessary condition for the preserving of charge absence. Consider the map in the form (14) @ji”y= &x~v~~FoK. Now suppose that the invariant
(25)
21 (see 17a) is not zero: j?? =
XWKI
#
0.
(26)
Subtract this part from x and call the remaining part x0 x
=
x0
+
x1*
(27)
CONSTITUTIVE
507
MAP
Substitution of (27) in (25) gives, after taking the divergence (exterior of G) of (25),
derivative
Now assume that the remaining part x0 preserves charge absence. The relation (28) thus reduces to a”w” = @“X[A”~“lF,, . (30) The differentiation
The alternation
operating on x and F individually
over hvo~ and the summation
finally gives
over WK means that only
(32) contributes in the first sum. Hence the first term in (31) vanishes because of (1); the expression (32) is the equivalent of the exterior derivative of F and vanishes. The divergence on the left (dG = C) gives the charge and current density QA. Hence we obtain the following interesting relation from (31):
For arbitrary
F, OA vanishes if or
a,g, = 0.
(34)
The condition is not sufficient because we assumed that x0 preserves charge absence. Taking the divergence of (33) one easily finds that (33) meets the requirement of charge conservation. In the previous section, it was found that the free-space map (8) has a vanishing & . It follows that (8) also meets the condition (34). At this point it is of interest to consider the situation that the map x does not preserve charge absence through a nonvanishing gradient of g1 . The expression
a”?, f 0 is a W-four vector. Equation (33) can be converted in three-dimensional
aof,= 00
and
(35) vector form if we write
a,& = a I = 1,2,3.
(36)
508
POST
The expression (33) is now p=G*B,
(37)
c = UJ + 0 A E.
It appears that 6 is of the nature of a Hall vector. The fourth component cr,,cannot exist in a centrally symmetric medium. The spatial part G vanishesin an isotropic surrounding.
From a microscopic point of view one can make a statement concerning a sufficient requirement for electric charge to occur. The first equation (37) shows that G # 0 demands absence of isotropy. A condition that is faithfully met by most elementary charges. Yet with the absence of complete knowledge about the differential comitants of x no more conclusive statements can be made. VII.
THE SPATIALLY
ISOTROPIC AND CENTRALLY
SYMMETRIC
MAP
x
Locally two of the most conspicuous properties of space without matter are probably its isotropy and central symmetry. Let us examine how these properties affect the general linear map x, conveniently depicted by (38) for the tensorial representations (5) and (6) both. I.
.OK
-+
Xnv 4
-D
01 23 31 12
01 R
02 03
B
E
02
03
23
31
--El1
--El2
-El3
Fll
$2
-621
-E22
-E23
721
722
-E31
-E32
-E33
/ 731
732
-x”“p”
12 Y13
4 01
(38)
551
.i;21
931
x11
Xl2
xl3
23
712
f22
732
x21
x22
x23
31
553
?23
733
x31
x32
x33
12
B
The matrix xZk is the inverse of the permeability matrix, cllc is the permittivity matrix and Flk is the matrix that describes the rare but existing directive effect between magnetic and electric polarization. Yet the absence of polarization mechanisms in free-space may not be taken as a tacit admission that ylle always vanishes identically in matter-free space. The matrix j& , which is a 3-dimensional W-tensor is nonzero in acceleratedframes [lo] even infree-space. For material media, Flk # 0 if they are in uniform motion (Fizeau effect). There are 21 nonzero independent elements in (38). They reduce to two nonzero
CONSTITUTIVE
509
MAP
independent elements if the conditions of isotropy and central symmetry are imposed. The Wtensor rllc vanishes in a central-symmetric medium, because it relates polar and axial vectors. The isotropy condition further reduces the permittivity tensor clJCand the inverse permeability tensor xllE to diagonal form. Hence f& = 0
and
Elk
=
E,,6,k
;
XZk
=
CLo%r
(39)
.
At this point the symbols E,, and p0 can be chosen in accord with a reasonably well-established custom to characterize free-space. The matrix (38) now takes the reduced form * +- DKDT E B X i&P 01 02 03 J 23 31 12 J
-D
23 31 12
-CO
0 0 0 0 0
I-
Y”
0 -E.
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
p;l 0 0
0 pi1 0
0 0 p;l
0
0
01 02 -D 03
(40)
B
01 02 03
-
L
23 31 12
R
The W-scalar g1 clearly vanishes for a medium with the reduced x (40). The invariant xZ is nonzero: x2 = -12EdP.o -
(41)
The ratio E~/~.L~ is known to determine the reflection and transmissionproperties at an interface if its value changesgoing from one sideto the other. The sameholds for a continuous change of ~~~~~in the medium. One may then speak of a light d@ision in the medium. -The ratio ~~~~~is commonly known in the form Z = ~/cL~/c~as the impedance of the medium.s The quantity xZ is an invariant under general coordinate transformations in space-time, regardlesswhether the medium is material or free-space. A reexamination of expression (8) shows after a comparison with (40) that the scalar invariant Y, in (8) is simply the inverse free-space impedance, because &” = {EOPO 9 -1, -1, -l}
and
gllz _ I g,b vz = (Eopo)-1’2.
B In engineering applications one normally defines the ratio 1E/H 1 as the impedance of the medium. For isotropic media at rest, this leads to 2 = l/pO/q,. In birefringent and in moving media, it is not possible to correlate the two definitions in a simple manner. 595/71/2-14
510
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One thus arrives at the theorem: Thefree-space impedanceZ, = l/t~,,lq, is a general space-time invariant.
If one furthermore accepts the reasonable assumption that free-space, without the presence of matter, does not exhibit the property of diffusing light, one may add the additional conclusion: The free-space impedanceZ = (~,,/E~)~‘~is a universal space-time constant.
The space-time scalar property is clearly a prerequisite for being a space-time constant.
VIII.
THE LOCAL
SYMMETRIES OF THE FREE-SPACE MAP x
The initially assumed symmetries of isotropy and central symmetry generate for the free-space case a larger symmetry group under which the form (40) is preserved. This conclusion is evident if one examines the expression (8) where x is constructed from the metric tensor. The metric tensor gAV= {q,,u,, , - 1, - 1, - 1) remains invariant under the Lorentz group, hence the Lorentz group is also an invariance group of the free-space map x defined by (40).
The statement is only physically meaningful for free-space because a translating material medium that is central-symmetric and isotropic in a rest-frame looses these properties after translation (Fizeau effect). While no direct proof of construction (8) was presented in the previous section, an independent verification of the Lorentz invariance of (40) may be useful. The calculation is somewhat involved and will be presented in an abbreviated form. We select for this purpose the transformation formula (16). There is a total of six nonvanishing independent elements. However, if one includes the index transpositions that follow from the index symmetries (6a) and (6b), one finds that the following index combinations have to be included in the summation of the transformation (16). The & per column denotes with what sign each element should enter the summation. 0101 0202 0303 2323 3131 1212 +
1001 2002 3003 3223 1331 2112 -
0110 0220 0330 2332 3113 1221 -
1010 2020 3030 3232 13 13 2121 +
(42)
CONSTITUTIVE
511
MAP
We know already that the spatial rotation group R3 is a symmetry group of the free-space map x. Hence, rather than trying a general linear transformation in all four variables, one may consider a transformation in t and x. The transformation Jacobian is then represented by
(43)
A = ) Ai’ 1 = ad ~ bc. Now it is a question of writing out the transformation (16) mation matrix (43). One finds that all nondiagonal elements identically after the transformation (43). The invariance of the of (40) then yields the following conditions for the elements a, The invariance of X
0101
(ad
+
-
) ad -
=
bc /
,
’
ad--c==
*I,
(-a2Eo + b2/po) = -eo,
X2323 +Ll= X 3131-
bc)'
with the transforof x still vanish diagonal elements b, c and d of (43).
(-c%
(441
&l, + d2ipo) = 1/po .
Note that the result d = fl critically depends on the presence of I d 1 rather than A in the transformation formula (16). The minus sign refers to the “improper” transformations. The independent restriction on a, b, c and d as given by (44) are thus dZ - 1 = c~/E,~~, a2 ~ 1 = b2/eopo,
(444
ad - bc = fl. It is easily seen that the special Lorentz Hence
group L,s satisfies the conditions
(44a). (45)
with y = (1 - u~E~,~~)-~/~. The product copLoequals the inverse square of the free-space light velocity and the parameter v denotes the mutual velocity of the translating frames.
512
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We know already that R, is a symmetry of (40), hence the full Lorentz group L, x R, = L is a symmetry of (40). The full Lorentz group may be extended with space-time translations xA + aA+ xA’ expressing local space-time homogeneity. The four-vector aA is assumed to be a constant; hence the transformation (16) is trivially
obeyed. It follows that
The free-space map x given by (40) is invariant under the Lorentz-PoincarP group.
The Lorentz-Poincare group is, however, not yet the complete local symmetry group of the form (40). Consider the transformation
(46)
when u is a constant. It is now evident from the transformation (16) that not only the free-space map x given by (40) is unchanged under the “scaling” (46), but any linear map x permits the scaling (46). Namely each coefficient Ai’ in (16) acquires a factor u, hence the product of four of such factors gives a u4, while the inverse determinant 1d 1-l gives a factor G-~. We have thus established the theorem. The free-space map x given by (40) exhibits a local symmetry characterized by the direct product qf the Lorentz-Poincare’ group and the Abelian scaling (or calibration) group (46).
IX.
THE CONFORMAL
INVARIANCE
OF THE FREE-SPACE
MAP
Although x is essentially local in nature, for special purposes it may be of interest to consider a nonlocal extension of the Lorentz-Poincare invariance. The LorentzPoincare group has 10 parameters and the Abelian scaling has 1 parameter. Hence the local symmetry of the free-space x is characterized by a 1 l-parameter group. An inspection of the form x as given by (40) immediately suggests that its invariance would be preserved under a 6-dimensional orthogonal group if x for that purpose is considered as a symmetric tensor of valence 2 in 6-dimension. The B-dimensional orthogonal group has 6 x 5/2 = 15 parameters. Its transformation elements consist of bilinear expressions of the transformation elements of 4dimensional space-time. A direct proof from (16) and (40) that a transformation group in space-time exists isomorphic with the 6-dimensional orthogonal group is somewhat involved. Moreover, the problem has been discussed at some length in the mathematical literature [l 11.
CONSTITUTIVE
MAP
513
The proof of conformal invariance of the free space x can be much simplified if we now use immediately the construction (8). One obtains the conformal group as an extension of the Lorentz-Poincare group and the scaling group (46). The extension is an Abelian group consisting of an inversion-a translation followed by another inversion [13]. Symbolically, j = x/x2, z=y+iG, w = 2/z2,
inversion; translation inversion.
(4 parameters);
For the relation between X and W one finds (47) To prove the invariance of the free-space x under the transformation (47), it is again sufficient that x is invariant under a single inversion. We first prove that the metric tensor transforms according to the rule ifA” = hA” 7 EAU= x-4P9 I& = x161gA”I,
(48)
y” = x”lg,,xAx” = XKX-2.
(49)
under the inversion
The Jacobian transformation
matrix of (49) is -aYA = x-2(S,A - 2xAx,x-7,
a32
if we write for simplicity x, = gy,xo.
(51)
The new metric thus becomes
= g~~x-ys~A - 2x”x,x-2) = x-4gAu,
x-2(sKv - 2X”X,X3
(52)
which immediately establishes all relations (48). The application of (48) to (8) confirms the invariance of the free-space map under the inversions (49). The invariance under translation is trivially obeyed. Hence (8) is invariant under the
514
POST
Abelian group defined by (47). Having established already that (8) is invariant under scalings and the full Lorentz-Poincare group, one can now state the theorem: The free-space constitutive conformal group.
map [as given by (8) or (40)] is invariant
under the
For an arbitrary medium, say, (38), there is always an orthogonal transformation in 6-space that brings (38) on diagonal form with in general six distinct eigen values. This principal axis transformation has a conform image in 4-space. Hence one obtains the remarkable theorem: There is always a conformal constitutive map.
tran?formation
that diagonalizes the general linear
X. THE FINE STRUCTURE “CONSTANT” The major spectral features of a hydrogen-like atom are given by energy states that are proportional to the rest energy of the electron m,,c2,the square of the fine structure constant CLand some functionf(n) of the principal quantum number n; the values of which give the discrete energy state E, - a2m,c2f(n).
(53)
The structure of the functionf(n) depends only to a minor extent on important physical constants. Its major structural components are the charge number Z of the nucleus and a minor correction due to the small ratio of electron and nuclear mass. The formula (53) is an adequate tool to identify the sources of the spectra that are obtained from distant stellar objects. Thesespectral observations have the common feature of a redshifting of the lines. This shift can be attributed to a number of causes.The commonly accepted external causesare Doppler shift and gravitational redshifts. In addition to the external causes,one may assumethat the physical constants 01,m, and c2 in formula (53) vary with time and position. Although the Doppler shift has played a major role in the interpretation of stellar spectra, the possibility of space-time changes of the major physical “constants” has also been subject to speculative considerations [14]. The degreesof freedom for speculation are much restricted if the fine structure of the spectrum can be observed in greater detail. The formula (53) should then be replaced by a formula with the additional quantum numberj E,,j -
a2m,,c2F(n,j, CX”).
(5%
A comparison of (53) and (53a) showsthat the fine-structure permits an independent determination of 01.
CONSTITUTIVE
MAP
515
Recent observations [ 151 on spectra of quasi-stellar objects made an independent observation of 01possible, with the spectacular result that it equals within experimental error the cydetermined in the laboratory. One may conclude from this observation that 01is truly a universal constant of nature. A prerequisite for a quantity to be a universal constant is that it is a spacetime scalar under arbitrary transformations. The conventional rendering of the fine structure constant 01 = e2/Ac does not make its invariant scalar properties explicit. There is the very strong physical suggestion that 01,e2 and #i are space-time invariant scalars under general transformations. Yet, the light velocity c is only a Lorentz invariant. How can we remove this inconsistency? The conventional rendering of the fine structure constant results from using a derived electrical unit of charge e in 01.The preceding discussion, as explained in Section II, strongly depends on the introduction of electric charge as an independent and separate unit. It will now be shown that this minor change of adopting an independent electrical charge is the key to a generally invariant rendering of 01.
Following a well-established procedure, one replaces the derived unit e2 by an independent unit e2 requiring the unit adjustment factor l/473, : (54) E,, is the parameter that occurs in the free-space constitutive map x given by (40). The light velocity c expressed in the free-space parameters E” and pLois c = (Eopp=. Using (54) and (55), the conventional
(55)
iy. = e2/Ac becomes (56)
The transformational inconsistency has now been removed. The fine structure constant as given by (56) is the ratio of two generally invariant impedances:
They are zl = 4&c
9
(57)
and Z,,, = 4di/e2 = 2hle2.
(58)
The general invariance of the free-space impedance was established in Section VII, formula (41). The general invariance of Z, is provided by ample theoretical evidence that e and fi each are generally invariant space-time quantities.
516
POST
At the end of Section VII, it was established that 2, is not only a general spacetime invariant, there is ample evidence that Z,, is also a universal constant. It follows that the ratio Z, = 2file2 should be a universal constant of nature. This result would appear as a foregone conclusion if it were not for the many speculations in the course of time that 01,e and fi might change in time. These proposed changes are much restricted unless one wants to drop the constancy of Z,, which means that light can be scattered and diffused without the presence of matter. There is, for the time being, little evidence to justify such a radical assumption. An additional comment is indicated concerning the primary role of the quantized elementary units of action (h/2) and charge (e). The notion of an elementary quantum stands and falls with the identical magnitudes to be associated with individual quanta. A time and position dependence of e and fi would then become a contradiction in terms unless one admits to only a local validity of the quantum concept. By contrast, the integrals of the differential 3-form of current and charge density (phys. dim. [e]) and of action (phys. dim. [fi]) are determined by a specification of their domain of integration. Their values become multiples of e and fi/2 if one accepts the global validity of the quantum concept. Hence the magnitude and possible change of the individual quanta e and fi become a cosmological matter if one accepts the global validity of the quantum concept. Note. During the preparation of this manuscript, two remarkable notes by A. Wyler [18] came to my attention. In a sense subsuming the invariance and constancy of (Yas discussed in this paper, Wyler obtains a numerical value for cy which agrees in 6 decimal places with the best observed values. The physical argument is shrouded in abstruse group considerations. Hopefully a more elaborate discussion will be forthcoming.
APPENDIX
A differential form rendering of the Maxwell equations was introduced by Cartan [3] almost half a century ago. Recently it has been revived through the work of Misner and Wheeler [16], and Flanders [17]. The work of de Rham on the topology of multiple integrals may be accepted as a major motivation for the cited revival. The de Rham theorems are discussed by Flanders [17, p. 681 as well as by Misner and Wheeler [16]. The differential-form calculus consists of an exterior algebra and a calculus of exterior derivatives. It is a calculus geared to the treatment of totally antisymmetric quantities that emerge from a study of Pfaffian integrability theory.
CONSTITUTlVE
MAP
517
The differential-form calculus as such may be seen as an important subdiscipline of general tensor calculus, which it transcends by far in elegance, because of its conspicuous and beautiful algebraic structure. The homogeneity in algebraic structure, as compared to tensor analysis, permits an extremely concise notation to adequately account for this structure.The relegation of structural detail to a solid algebraic foundation permits greater emphasis on fundamental conceptual problems such as topological features related to properties in the “large” and properties in the “small”. Subtle distinctions now emerged. The property of a differential form of having a vanishing exterior derivative led to the concepts of closed forms (local) and exact forms (global); the latter is a property in the “large”. The old Stokes and Gauss integral laws received new attention from a point of view of topology. In fact they spawned new integral theorems such as de Rham’s theorems, which, gross0 mode may be considered as appropriate extensions of Cauchy’s famous residue theorem in complex analysis. How do these new developments tie in with Maxwell theory and why should we use them? First of all, the basic electromagnetic laws are all expressable as differential forms. Then the new mathematical developments open up the possibility of discussing the much neglected properties in the “large”. The distinction between “local” and “global” properties for the first time permits more precise statements about conservation laws in the “small” and in the “large”. The conservation of flux and net charge are typical examples of global laws that hold in the large. By contrast conservation of energy and momentum is a typical example of a law of only local significance. These are some reasons that made it tempting to use the differential-form formalism in Section II of this paper: to make global aspects more explicit even if one cannot use the formalism throughout. Finally, an important philosophical distinction should be mentioned between the work of Misner and Wheeler and the presentation given in Section II of this paper. Misner and Wheeler interpret their field sources as a microscopic multiple connectedness of the ordering manifold of space-time in which these fields are defined: the worm hole concept of electricity. Epistemologically it is better if the cohomology of fields is given a primary physical role rather than the homology of chains over which these fields are being integrated. The philosophy is simply that we take cognizance of space-time through what exists and happens in space-time. The a priori existence of worm holes in space-time is not well compatible with a rigorous pursuit of the idea that magnetic monopoles do not exist. The latter assumption is a cornerstone of the development in Section II.
518
POST ACKNOWLEDGMENTS
A long time ago, Francis J. Zucker, then at the Air Force Cambridge Research Laboratories, asked me whether the widest invariance group of the constitutive tensor might have a true image as a transformation group in space-time. My initial response was one of scepticism in the sense that the result might be of mere mathematical interest without much further physical perspective. More recently the same question arose in a discussion with Professor La&o Tisza (MIT). The present paper is an attempt at a further inquiry into the subject. The result may be viewed as a new angle to an old problem. I thank the above-mentioned gentlemen for their insistence to look into the matter, Professor Tisza for references and discussions of the relevant work of Felix Klein and W. Blaschke, and Professor Kenneth Johnson for a discussion of the properties of the conformal group. Any remaining inadequacies are the sole responsibility of the author.
REFERENCES 1. E. CUNNLNGHAM, Proc. London Math. Sot. 8 (lPlO), 77. 2. H. BATEMAN, Proc. London Math. Sot. 8 (lPlO), 223. 3. E. CARTAN, Ann. Econ. Norm. 41 (1924), 1. 4. D. VAN DANTZIG, Proc. Cambridge Phil. Sot. 30 (1934), 421. D. VAN DANTZIG, Proc. Amsterdam Acad. 37 (1934), 521, 526, 644, 825. 5. E. J. POST, “Formal Structure of Electromagnetics,” North-Holland, Amsterdam, 1962. 6. G. DE RHAM, “Varietes Differentiables,” Hermann, Paris, 1955. 7. J. A. SCHOUTEN, “Ricci Calculus,” 2d ed., Springer, Berlin, 1954. J. A. SCHOUTEN, “Tensor Analysis for Physicists,” Oxford Univ. Press, London, 1951. 8. S. L. How AND N. BLOEMBERGEN, Phys. Rev. 138 (1965), A1218. 9. A. NIJENHUIS, Proc. Amsterdam Acad. 54 (1967), 200. 10. E. J. POST, Rev. Mod. Phys. 39 (1967), 475. 11. F. KLEIN, “Hohere Geometrie,” 246-258, Springer, Berlin, 1926. 12. W. BLASCHKE, “Differential Geometric,” Vol. III, Springer, Berlin, 1925. 13. J. WESS, Nuovo Cimento 18 (1960), 1086. 14. G. GAMOW, Phys. Rev. Lett. 19 (1967), 913. 15. J. N. BAHCALL AND M. SCHMIDT, Phys. Rev. Lett. 19 (1967), 1294. 16. C. MISNER AND J. A. WHEELER, “Geometrodynamics” (J. A. Wheeler, Ed.), Academic Press, New York, 1962. 17. H. FLANDERS, “Differential Forms,” Academic Press, New York, 1963. 18. A. WALER, C. R. Acad. Sci. Paris 269 (1969), 743; A. WYLER, C. R. Acad. Sci. Paris 271 (1971), 186.
OF PHYSICS:
71, 497-518 (1972)
The Constitutive
Map and Some of Its Ramifications E. J. POST
523 Altenkirchen,
Im Vogelsang
4, West-Germany
Received May 20, 1971
The constitutive map is introduced to replace the familiar relation D = CE and B = pn. Its objective is a rigorous functional separation between field- and constitutive equations. The possibility to implement this functional separation is shown to depend critically on the introduction of electric charge as an independent space-time invariant unit, rather than as a unit derived from mass, length, and time. The linear map is shown to be invariant under the conformal group, while the field equations remain invariant under general space-time diffeomorphisms. Algebraic and differential comitants of the linear map are discussed, their physical roles are identified. The impedance of a medium emerges as a space-time invariant. The new concepts are then applied to prove the invariance and constancy of the fine structure “constant” and the ratio ee/fi.
I. INTRODUCTION In the following we are concerned with an attempt at reorganizing a well-known and well-established part of physical theory. The objective is one of opening up possible perspectives that cannot be easily obtained in any other fashion. The general subject matter is electromagnetic theory. The specific aspect under scrutiny is the need for a recasting of the conventional constitutive relations in a form that is more susceptible to a deductive treatment. Major attention will be given to the constitutive properties of the free-space medium. The new insights resulting from such a reorganizational move are varied. Most of them are qualitative in nature, which should not be surprising. Nontrivial and fundamental, quantitative, experimental results normally precede theories; they are frequently given an exhaustive description before a theory reachesfuller maturity. The result is that the incentive for a further development of the theory drops below threshold. Yet, qualitative features in theory and observation can also be most significant and useful. They assist in classifying and ordering of already existing results of a quantitative nature. Important highlights of the present investigation
are: an inertial
frame definition
basedon the single question: what transformation group preservescentral symmetry and spatial isotropy? In addition a further delineation is obtained of the distinction between physical frames and the family of permissible coordinate neighborhoods 497 Copyright All rights
0 1972 by Academic Press, Inc. of reproduction in any form reserved.
498
POST
associated with such physical frames. The criterion is a group criterion which is independent of the choice of coordinate neighborhood. More specifically, the physical role of the conformal group in electromagnetic theory can be identified with greater precision on the basis of its decomposition in subgroups. The Lorentz subgroup is found to characterize the inertial frame situation, whereas the Abelian subgroup of inversions-translations-inversions identifies a family of physically equivalent situations in isotropic space. The latter conclusion completely bears out the original motivation of Cunningham [I] and Bateman [2] to extend the technique of equivalent mirror charges in electrostatics by studying conformal transformations. In contrast to previous discussions, the coqformalgroup can now be identiJied as characterizing a purely constitutive situation. The traditional lack of clarity concerning the physical role of the conformal group can thus be relegated to the traditional absence of a clear-cut distinction between physical frames and the large family of coordinate references that can be associated with that single frame. Sometimes, but not always, one may be able to single out in these families a subgroup representing in a meaningful manner a group of physically equivalent situations. The Abelian subgroup in the conformal group is an example. The mentioned distinctions can be cast into a consistent mathematical framework by following an early lead by Cartan [3], elaborated by van Dantzig [4] and the present author [5]. To accommodate all conceivable situations it is now necessary to consider the complete family of space-time diffeomorphisms, i.e., all linear and nonlinear space-time transformations that are continuous, differentiable and nonsingular. The essence of Cartan’s [3] idea is that the basic laws of electrodynamics permit a complete functional separation between medium independent law statements and the constitutive specification of the medium. To include free-space the medium independent law statements should be independent of the space-time metric structure. The geometric structure of space-time is subsequently covered by a scrupulously deductive development of the constitutive relations. The major emphasis accordingly focuses on the so-called “constitutive map”; a new word meant to accommodate the familiar and old relations B = EE, B = $7 in a physically more “transparent” garb. A number of additional benefits then ensues apart from those already mentioned. Here are a few of them. The constitutive map has a general algebraic invariant that can be identified as an impedance. The fine structure constant cx can accordingly be identified as a ratio of two generally invariant impedances. The observed constancy of 01throughout the universe now leads to better guided considerations concerning the cosmological behavior of the invariant impedances of which oi is the ratio. An examination of the constitutive map also reveals that the condition of net charge neutrality of a medium leads very naturally to a study of its differential
CONSTITUTIVE
MAP
499
comitants. Not all differential comitants of the linear constitutive map are presently known. In the present context this means that the n.a.s. condition to be imposed on the constitutive map to securenet charge neutrality is not available. Yet a more complete mathematical information about this subject seemsessential to answer the question whether or not electric charge has a structurally determining influence on space-time geometry, apart from the massthat carries the charge.
II. ISOLATING
THE PURE FIELD RELATIONS
The phenomenological description of electromagnetism is characterized by a separation into two functionally distinct domains. First, there is the concern with properties that all systemsshare; secondly, there is the concern with a systematic characterization of specific systems.This policy of approaching the subject matter has led, in a quite natural manner, to the introduction of two setsof basic relations. They are known as the electromagnetic field equations and the constitutive equations of the medium. The early historic developmentsomehowfailed to accept the free-space medium as worthy of a complete constitutive treatment. When at the turn of the century the Lorentz symmetry of this “unmedium” was uncovered, convention and tradition were already so well established that adoption of free-space as a full-fledged member of the family of electromagnetic media was not quite possible. As a result of this historic development, Lorentz invariance is today commonly presented as a symmetry of the free-spacefield equations rather than as a symmetry of the free-space constitutive equations. For the purpose of the present investigation, it is necessaryto abandon these historic conventions. The following program permits a rendering of the field equations characterized by a maximal independence of preconceived or prejudicial notions concerning the electromagnetic media to which they apply. Hence preconceived notions of geometric space-time structure will have to be removed from the field equations if we are to include free-space in the family of electromagnetic media. The field equations are thus erected on the austere assumption that space-time is a differentiable manifold. The fundamental physical facts may then be introduced in the following logical order.* 1. The Faraday induction law and the absence of magnetic monopoles are locally expressed by the vanishing exterior derivative of the differential 2-form F of electrified E and magnetic induction B: dF = 0.
(1)
1 The differential-form method has been selected to present the basic facts in a concise yet complete manner. Additional explanations for this decision are provided in an appendix.
500
POST
Equation (1) expresses in a space-time invariant manner the “local” conservation of flux or equivalently the local absence of magnetic monopoles. The local conservation statement (1) can be extended into a global (universally valid) conservation statement, if the integrals of F over all possible 2-dimensional closed domains vanish. One then concludes from de Rham’s theorem2 on the existence of a globally defined l-form A with the property dA = F.
(2)
Relation (1) is implied by relation (2), because dd = 0. The inverse statement that (1) implies (2) does not hold in a global manner unless the conditions for de Rham’s theorem are met. 2. The conservation of net charge, which reflects the well-observed fact that charge creation always occurs in pairs of opposite polarity, is expressed by a vanishing exterior derivative of the differential 3-form C of charge and current density dC = 0. (3) Equation (3) expresses in a space-time invariant manner the local conservation of charge or equivalently the local absence of a net charge creation mechanism. The local conservation statement (3) can be extended into a global (universally valid) conservation statement, if the integrals of C over all possible 3-dimensional closed domains vanish. One then concludes from de Rham’s theorem on the existence of a globally defined 2-form G with the property dG = C.
(4)
Relation (3) is implied by relation (4), because dd = 0. The inverse statement that (3) implies (4) does not hold in a global manner unless the conditions for de Rham’s theorem are met. The relation (1) is a medium-independent, space-time invariant rendering of the Maxwell equations curl E = -B
and
div B = 0.
The relation (4) is a medium independent, space-time invariant, expression that so far only resembles the Maxwell equations curl H = D + C
and
div D = p.
(14 rendering of an (44
a See Ref. [6, Chap. III, p. 391. A very interesting and physically digestable discussion of the de Rham theorems appear in two French papers, Math. Congress (1932), p. 195, Enseignement Math. 35 (1936), 213, and a German paper, Sem. Hamburg Un. 12, p. 313. See also the appendix of this article.
CONSTlTUTlVE
501
MAP
The relations (4) and also (4a) become expressionsof the Biot-Savart and Amp&e force laws only after one de$nesthe medium in which they are supposedto apply. The fields, or rather the forms, A and G, share the property of a gauge undeterminedness;each is defined module an exact form.
III. THE LINEAR CONSTITUTIVE
MAP x
Equation (4) (in principle also 4a) was introduced as a mere mathematical consequenceof the global validity of the law of net charge conservation. Through de Rham’s theorem, one thus carries Maxwell’s argument for the displacement current to its logical conclusion. Simultaneously, however, one modifies through this deed the physical interpretation of the fields D and R (i.e., G); they have been reduced to the role of potential fields with a nonunique gauge. The relations (4) acquire a more specific physical meaning in the senseof the Biot-Savart and Ampere force laws after the introduction of a constitutive relation which in fact should also fix the gauge of G. In analogy with the well-known relations D = l .i? and B = pi?, one may assumethat the coefficients of G are linearly related to the coefficients of F
The tensorial quantity x obeys the obvious symmetry relations
becausethe coefficients of G and Fare both antisymmetric. The relations (5) do not presupposea specific symmetry of the medium, neither is it assumedthat the medium is homogeneous; x may depend on the coordinates and on the time. Furthermore, the algebraic nature of (5) indicates that local and instantaneous interaction has been assumed,which meansthat the coefficients of G at the space-time point P are solely determined by the values of the coefficients F at that samepoint P. These assumptionsare well justified if all properties of a material medium can be interpreted as local statistical averages of free-space properties. The relations (5) may thus be thought of as primarily applicable to free-space, although at times it may be convenient to consider their direct applicability to a material medium in a macroscopic sense. The relations (5) may be called the constitutive map. They map an exact 2-form F = dA into a general 2-form G. The 2-form G is called closed if dG = 0, which means at that point electric charges are absent. For a general map x, however,
POST
502
one may not expect that dG = 0. Hence the nature of x determines whether C = 0 or C # 0. The linear constitutive map x can be reexpressedin an equivalent form by means of the totally antisymmetric Levi-Civita unit tensor @for four dimensions.
The equivalent form obeys the symmetry relations (independent of the medium) X AVoK=
_
X
vim =
_
x
AYKO .
W
If a Lagrangian function 2 exists and one defines the G field according to Mie: G = 39/S’, there is now the additional symmetry (6b) The form (6) of course, relates to a corresponding modification of G
Finally, where free-spaceis not known to be birefringent, an additional restriction may be placed on x in matter-free domains. It can be shown that x is not birefringent if it is constructed as the alternating product of a symmetric tensor of valence 2. Drawing on the existing knowledge about a metric space-time with metric tensor g,, one may consider the form AWJK =
X
y,
1 g ~l/z(glogvK
-
gAy7)
(8)
in which g is the determinant of g,, , Y, is an invariant scalar to be determined later.3 In the sequel primarily the unstructured forms (5) and (6) will be used to circumvent unnecessary prejudicial notions concerning a space-time geometric structure based on a metric. The expression (8) is related to the so-called Hodge-star operator [17] defined for a nonorientable space-time manifold. The reader should be alerted to the fact that it is common in the literature on differential geometry to restrict the discussions to orientable manifolds. Space and space-time are not orientable by virtue of the existence of enantiomorphic structures in space; e.g., crystals, molecules, and elementary particles that lack a center of symmetry. 3 A discussion is found in Ref. [5, Chap. 9, formula (9.4)].
CONSTITUTIVE
IV. THE TRANSFORMATION
503
MAP
PROPERTIES OF THE LINEAR
MAP
x
The map x is, of course, tensorial by virtue of the tensorial properties of the coefficients of F and G. Yet its transformation deservesspecial attention becausex in the form (5) is not an ordinary tensor but a pseudo or W tensor.4 It transforms with the sign factor d/j d / in which d is the Jacobian and j d 1its absolute value A -A’“’ A:‘A$4~~&1”,, x. O’K’= IAl
.
Note that the transformationalfeature introduced in (9) is most essential.It is not to be considered as a show of overconscientiousnessor mathematical pedantry. One can easily convince onself why this sign factor A// A 1is indispensable. An inspection of the equations (la) and (4a) showsthat the electric and magnetic fields trade places. Hence to secure the always present relation between D and E and B, H in a centrally symmetric medium, it is necessarythat the coefficients j$,, , etc., are preserved under a spatial inversion. In other words, D and E are of same “parity” (E, D, polar vectors) while B and Hare also of same“parity” but opposite to D and E (B, H axial vectors). Let the spatial inversion be given by the Jacobian matrix 1 A’=
0
0
i 0 O-l 0-1 0 ,o
0
0 0i f
(10)
O-l
An application of (10) to (9) showsx!.‘,, cannot be preserved unlessthe sign factor A/l A / is included in the transformation (9). The W-transformation properties of the mixed tensor 2 has rather important consequencesfor the differential 2-forms F and G. The result is sufficiently significant to be stated in the form of a theorem: Thefundamental 2-forms F and G of electromagnetic theory have oppositeparity.5 It follows that either For G should transform as a W tensor. One retains the best 4 Wtensorsor pseudotensorsare denoted by a “tilda”. The W stands for the nameof H. Weyl who was one of the first to introduce this distinction explicitly. Earlier usage, however, occurred in crystal physics by W. Voigt. See Schouten [7]. In the French literature the names “pair” and “impair” are used. See Ref. [6, pp. 19 and 221. 5 Differential forms of opposite parity are discussed by de Rhan [6, pp. 19 and 221. Note that the term opposite parity is unique because the coefficients of F and G are necessarily covariant by nature. By contrast the distinction: tensor, W tensor is not unique as will become clear in the sequel.
504
POST
conformity with existing conventions if G is taken to be the W form. Its coefficient thus transforms according to the rule C&,, = (O/l d I) A;#A;&
while F retains the standard transformation
,
(11)
rule of a covariant tensor
Fl’u’ = A;,A;,FA, .
(12)
The distinction between ordinary quantities and W quantities is not a standard feature in physical theory except in crystal physics or elementary particle physics. Tradition has it that these distinctions in field theory are at most retained in a somewhat ad-hoc manner to ensure that the vectors E, D have polar spatial character while H, B have axial spatial character. Moreover, it is common to convert the W-tensor e and the W-mixed tensor 2 into tensor densities that transform with the absolute value 1 d I. The totally antisymmetric unit tensor of Levi-Civita is itself a W-tensor density to possess the property of absolute invariance under “improper” transformations [see (lo), Ll = -11. The equivalent density versions of G and 2 are defined by ($ji”’ = gpO~&
)
(13)
and (14) Their transformation
laws are 6 h’v’ = 1d 1-l AfA;‘(liAY,
(15)
A’“‘O’K’ = I d 1-l &&fA,“‘xA”“K. X
(16)
and
V. ALGEBRAIC
COMITANTS
OF THE MAP
x
Algebraic invariants and comitants of the map x can be obtained by standard tensorial methods. An obvious invariant of the mixed W form (9) is the pseudo scalar f1 = 2;;“‘. (17) The same pseudo scalar can, of course, be obtained from (16) by the process of alternation, i.e., summing over all even transpositions of [XYUK] with a plus sign
CONSTITUTIVE
and all odd transpositions brackets. X It follows
505
MAP
with a minus sign. The operation is denoted by square [A’Y’U’K’I
=
, d
Af&‘A$A~;Xr”YoHI*
1-l
(18)
that A = A~;A:‘AfA:]
Hence, with the appropriate
.
normalization
and from (17) or (18) and (19) it follows
that
A 21’ = jdi
j&
is a pseudo scalar or W scalar. An algebraic comitant of the constitutive map is obtained by contraction one index 2;” = g;;v” = &rj,v. The constitutive
09)
(20)
over (21)
map x has also a true scalar invariant x2
=
-..OK-..A” XA” XOK
(22)
or equivalently with the help of the totally antisymmetric Levi-Civita unit tensor one can write x2 also in terms of the tensor density representation of x
The invariant x2 is different from zero in any medium. The W-scalar g1 , of course, vanishes in any centrally symmetric medium. Yet for a long time it seemed impossible to show experimentally that f1 could exist in a medium without central symmetry [5]. In fact quantum theoretical explanations have been suggested why a so-called “directive effect” between electric and magnetic dipole moments in a material medium cannot exist. However, recently Hou and Bloembergen [8] have shown that the effect indeed exists in nickel sulfite crystals at low temperature. Previous quantum mechanical discussions gave a negative result because of a hidden central symmetry in the Hamiltonians that were used for the discussion. Hence the moral of this story: the Neumann principle of crystalphysics rules supreme in classical as well in (nonclassical) quantum physics. In a microscopic sense the W-scalar g1 can only exist in particles that do not have
506
P0S-r
central symmetry. Free-space has a vanishing f1 . The latter fact can be easily established from the metric representation (8) of the constitutive map x. Alternation over hUK clearly leads to zero because the metric tensor g,, is symmetric.
VI.
DIFFERENTIAL
In a charge-free medium closed differential 2-form F requirement would lead to In a charge-free medium,
COMITANTS
OF THE
MAP x
the constitutive map x is required to map the (exact) into a closed differential 2-form G. An exception to this the presence of net electric charge. the Maxwell Eqs. (1) and (4) are source-free dF = 0,
(23)
dG = 0.
(24)
The 2-form F is always closed, in fact it is exact; the latter property only refers to its global behavior, not considered at this moment. In a charge-free medium the 2-form G is closed, because C = 0. The form G is definitely not exact. It follows that x maps a global property into a local property. Hence the map x is local in nature. The local conditions of charge absence may thus be formulated in the following abstract mathematical way if we realize that the condition should hold for all F. The constitutive map x in a charge-free domain of space-time gives a local automorphism of all closed 2-forms. Question: What are the n.a.s. conditions to be imposed on x so that it preserves the local integrability of all closed 2-forms? This problems has been solved for a linear map of l-forms by Nijenhuis [9]. No solution is available for 2-forms. One may expect, however, that the integrability preserving condition to be imposed on x, if it exists, should be a differential comitant of x only. Hence it should be tensorial, which means the condition is independent of arbitrary coordinate changes because of its homogeneous transformation. It is possible, however, to obtain at least a necessary condition for the preserving of charge absence. Consider the map in the form (14) @ji”y= &x~v~~FoK. Now suppose that the invariant
(25)
21 (see 17a) is not zero: j?? =
XWKI
#
0.
(26)
Subtract this part from x and call the remaining part x0 x
=
x0
+
x1*
(27)
CONSTITUTIVE
507
MAP
Substitution of (27) in (25) gives, after taking the divergence (exterior of G) of (25),
derivative
Now assume that the remaining part x0 preserves charge absence. The relation (28) thus reduces to a”w” = @“X[A”~“lF,, . (30) The differentiation
The alternation
operating on x and F individually
over hvo~ and the summation
finally gives
over WK means that only
(32) contributes in the first sum. Hence the first term in (31) vanishes because of (1); the expression (32) is the equivalent of the exterior derivative of F and vanishes. The divergence on the left (dG = C) gives the charge and current density QA. Hence we obtain the following interesting relation from (31):
For arbitrary
F, OA vanishes if or
a,g, = 0.
(34)
The condition is not sufficient because we assumed that x0 preserves charge absence. Taking the divergence of (33) one easily finds that (33) meets the requirement of charge conservation. In the previous section, it was found that the free-space map (8) has a vanishing & . It follows that (8) also meets the condition (34). At this point it is of interest to consider the situation that the map x does not preserve charge absence through a nonvanishing gradient of g1 . The expression
a”?, f 0 is a W-four vector. Equation (33) can be converted in three-dimensional
aof,= 00
and
(35) vector form if we write
a,& = a I = 1,2,3.
(36)
508
POST
The expression (33) is now p=G*B,
(37)
c = UJ + 0 A E.
It appears that 6 is of the nature of a Hall vector. The fourth component cr,,cannot exist in a centrally symmetric medium. The spatial part G vanishesin an isotropic surrounding.
From a microscopic point of view one can make a statement concerning a sufficient requirement for electric charge to occur. The first equation (37) shows that G # 0 demands absence of isotropy. A condition that is faithfully met by most elementary charges. Yet with the absence of complete knowledge about the differential comitants of x no more conclusive statements can be made. VII.
THE SPATIALLY
ISOTROPIC AND CENTRALLY
SYMMETRIC
MAP
x
Locally two of the most conspicuous properties of space without matter are probably its isotropy and central symmetry. Let us examine how these properties affect the general linear map x, conveniently depicted by (38) for the tensorial representations (5) and (6) both. I.
.OK
-+
Xnv 4
-D
01 23 31 12
01 R
02 03
B
E
02
03
23
31
--El1
--El2
-El3
Fll
$2
-621
-E22
-E23
721
722
-E31
-E32
-E33
/ 731
732
-x”“p”
12 Y13
4 01
(38)
551
.i;21
931
x11
Xl2
xl3
23
712
f22
732
x21
x22
x23
31
553
?23
733
x31
x32
x33
12
B
The matrix xZk is the inverse of the permeability matrix, cllc is the permittivity matrix and Flk is the matrix that describes the rare but existing directive effect between magnetic and electric polarization. Yet the absence of polarization mechanisms in free-space may not be taken as a tacit admission that ylle always vanishes identically in matter-free space. The matrix j& , which is a 3-dimensional W-tensor is nonzero in acceleratedframes [lo] even infree-space. For material media, Flk # 0 if they are in uniform motion (Fizeau effect). There are 21 nonzero independent elements in (38). They reduce to two nonzero
CONSTITUTIVE
509
MAP
independent elements if the conditions of isotropy and central symmetry are imposed. The Wtensor rllc vanishes in a central-symmetric medium, because it relates polar and axial vectors. The isotropy condition further reduces the permittivity tensor clJCand the inverse permeability tensor xllE to diagonal form. Hence f& = 0
and
Elk
=
E,,6,k
;
XZk
=
CLo%r
(39)
.
At this point the symbols E,, and p0 can be chosen in accord with a reasonably well-established custom to characterize free-space. The matrix (38) now takes the reduced form * +- DKDT E B X i&P 01 02 03 J 23 31 12 J
-D
23 31 12
-CO
0 0 0 0 0
I-
Y”
0 -E.
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
p;l 0 0
0 pi1 0
0 0 p;l
0
0
01 02 -D 03
(40)
B
01 02 03
-
L
23 31 12
R
The W-scalar g1 clearly vanishes for a medium with the reduced x (40). The invariant xZ is nonzero: x2 = -12EdP.o -
(41)
The ratio E~/~.L~ is known to determine the reflection and transmissionproperties at an interface if its value changesgoing from one sideto the other. The sameholds for a continuous change of ~~~~~in the medium. One may then speak of a light d@ision in the medium. -The ratio ~~~~~is commonly known in the form Z = ~/cL~/c~as the impedance of the medium.s The quantity xZ is an invariant under general coordinate transformations in space-time, regardlesswhether the medium is material or free-space. A reexamination of expression (8) shows after a comparison with (40) that the scalar invariant Y, in (8) is simply the inverse free-space impedance, because &” = {EOPO 9 -1, -1, -l}
and
gllz _ I g,b vz = (Eopo)-1’2.
B In engineering applications one normally defines the ratio 1E/H 1 as the impedance of the medium. For isotropic media at rest, this leads to 2 = l/pO/q,. In birefringent and in moving media, it is not possible to correlate the two definitions in a simple manner. 595/71/2-14
510
POST
One thus arrives at the theorem: Thefree-space impedanceZ, = l/t~,,lq, is a general space-time invariant.
If one furthermore accepts the reasonable assumption that free-space, without the presence of matter, does not exhibit the property of diffusing light, one may add the additional conclusion: The free-space impedanceZ = (~,,/E~)~‘~is a universal space-time constant.
The space-time scalar property is clearly a prerequisite for being a space-time constant.
VIII.
THE LOCAL
SYMMETRIES OF THE FREE-SPACE MAP x
The initially assumed symmetries of isotropy and central symmetry generate for the free-space case a larger symmetry group under which the form (40) is preserved. This conclusion is evident if one examines the expression (8) where x is constructed from the metric tensor. The metric tensor gAV= {q,,u,, , - 1, - 1, - 1) remains invariant under the Lorentz group, hence the Lorentz group is also an invariance group of the free-space map x defined by (40).
The statement is only physically meaningful for free-space because a translating material medium that is central-symmetric and isotropic in a rest-frame looses these properties after translation (Fizeau effect). While no direct proof of construction (8) was presented in the previous section, an independent verification of the Lorentz invariance of (40) may be useful. The calculation is somewhat involved and will be presented in an abbreviated form. We select for this purpose the transformation formula (16). There is a total of six nonvanishing independent elements. However, if one includes the index transpositions that follow from the index symmetries (6a) and (6b), one finds that the following index combinations have to be included in the summation of the transformation (16). The & per column denotes with what sign each element should enter the summation. 0101 0202 0303 2323 3131 1212 +
1001 2002 3003 3223 1331 2112 -
0110 0220 0330 2332 3113 1221 -
1010 2020 3030 3232 13 13 2121 +
(42)
CONSTITUTIVE
511
MAP
We know already that the spatial rotation group R3 is a symmetry group of the free-space map x. Hence, rather than trying a general linear transformation in all four variables, one may consider a transformation in t and x. The transformation Jacobian is then represented by
(43)
A = ) Ai’ 1 = ad ~ bc. Now it is a question of writing out the transformation (16) mation matrix (43). One finds that all nondiagonal elements identically after the transformation (43). The invariance of the of (40) then yields the following conditions for the elements a, The invariance of X
0101
(ad
+
-
) ad -
=
bc /
,
’
ad--c==
*I,
(-a2Eo + b2/po) = -eo,
X2323 +Ll= X 3131-
bc)'
with the transforof x still vanish diagonal elements b, c and d of (43).
(-c%
(441
&l, + d2ipo) = 1/po .
Note that the result d = fl critically depends on the presence of I d 1 rather than A in the transformation formula (16). The minus sign refers to the “improper” transformations. The independent restriction on a, b, c and d as given by (44) are thus dZ - 1 = c~/E,~~, a2 ~ 1 = b2/eopo,
(444
ad - bc = fl. It is easily seen that the special Lorentz Hence
group L,s satisfies the conditions
(44a). (45)
with y = (1 - u~E~,~~)-~/~. The product copLoequals the inverse square of the free-space light velocity and the parameter v denotes the mutual velocity of the translating frames.
512
POST
We know already that R, is a symmetry of (40), hence the full Lorentz group L, x R, = L is a symmetry of (40). The full Lorentz group may be extended with space-time translations xA + aA+ xA’ expressing local space-time homogeneity. The four-vector aA is assumed to be a constant; hence the transformation (16) is trivially
obeyed. It follows that
The free-space map x given by (40) is invariant under the Lorentz-PoincarP group.
The Lorentz-Poincare group is, however, not yet the complete local symmetry group of the form (40). Consider the transformation
(46)
when u is a constant. It is now evident from the transformation (16) that not only the free-space map x given by (40) is unchanged under the “scaling” (46), but any linear map x permits the scaling (46). Namely each coefficient Ai’ in (16) acquires a factor u, hence the product of four of such factors gives a u4, while the inverse determinant 1d 1-l gives a factor G-~. We have thus established the theorem. The free-space map x given by (40) exhibits a local symmetry characterized by the direct product qf the Lorentz-Poincare’ group and the Abelian scaling (or calibration) group (46).
IX.
THE CONFORMAL
INVARIANCE
OF THE FREE-SPACE
MAP
Although x is essentially local in nature, for special purposes it may be of interest to consider a nonlocal extension of the Lorentz-Poincare invariance. The LorentzPoincare group has 10 parameters and the Abelian scaling has 1 parameter. Hence the local symmetry of the free-space x is characterized by a 1 l-parameter group. An inspection of the form x as given by (40) immediately suggests that its invariance would be preserved under a 6-dimensional orthogonal group if x for that purpose is considered as a symmetric tensor of valence 2 in 6-dimension. The B-dimensional orthogonal group has 6 x 5/2 = 15 parameters. Its transformation elements consist of bilinear expressions of the transformation elements of 4dimensional space-time. A direct proof from (16) and (40) that a transformation group in space-time exists isomorphic with the 6-dimensional orthogonal group is somewhat involved. Moreover, the problem has been discussed at some length in the mathematical literature [l 11.
CONSTITUTIVE
MAP
513
The proof of conformal invariance of the free space x can be much simplified if we now use immediately the construction (8). One obtains the conformal group as an extension of the Lorentz-Poincare group and the scaling group (46). The extension is an Abelian group consisting of an inversion-a translation followed by another inversion [13]. Symbolically, j = x/x2, z=y+iG, w = 2/z2,
inversion; translation inversion.
(4 parameters);
For the relation between X and W one finds (47) To prove the invariance of the free-space x under the transformation (47), it is again sufficient that x is invariant under a single inversion. We first prove that the metric tensor transforms according to the rule ifA” = hA” 7 EAU= x-4P9 I& = x161gA”I,
(48)
y” = x”lg,,xAx” = XKX-2.
(49)
under the inversion
The Jacobian transformation
matrix of (49) is -aYA = x-2(S,A - 2xAx,x-7,
a32
if we write for simplicity x, = gy,xo.
(51)
The new metric thus becomes
= g~~x-ys~A - 2x”x,x-2) = x-4gAu,
x-2(sKv - 2X”X,X3
(52)
which immediately establishes all relations (48). The application of (48) to (8) confirms the invariance of the free-space map under the inversions (49). The invariance under translation is trivially obeyed. Hence (8) is invariant under the
514
POST
Abelian group defined by (47). Having established already that (8) is invariant under scalings and the full Lorentz-Poincare group, one can now state the theorem: The free-space constitutive conformal group.
map [as given by (8) or (40)] is invariant
under the
For an arbitrary medium, say, (38), there is always an orthogonal transformation in 6-space that brings (38) on diagonal form with in general six distinct eigen values. This principal axis transformation has a conform image in 4-space. Hence one obtains the remarkable theorem: There is always a conformal constitutive map.
tran?formation
that diagonalizes the general linear
X. THE FINE STRUCTURE “CONSTANT” The major spectral features of a hydrogen-like atom are given by energy states that are proportional to the rest energy of the electron m,,c2,the square of the fine structure constant CLand some functionf(n) of the principal quantum number n; the values of which give the discrete energy state E, - a2m,c2f(n).
(53)
The structure of the functionf(n) depends only to a minor extent on important physical constants. Its major structural components are the charge number Z of the nucleus and a minor correction due to the small ratio of electron and nuclear mass. The formula (53) is an adequate tool to identify the sources of the spectra that are obtained from distant stellar objects. Thesespectral observations have the common feature of a redshifting of the lines. This shift can be attributed to a number of causes.The commonly accepted external causesare Doppler shift and gravitational redshifts. In addition to the external causes,one may assumethat the physical constants 01,m, and c2 in formula (53) vary with time and position. Although the Doppler shift has played a major role in the interpretation of stellar spectra, the possibility of space-time changes of the major physical “constants” has also been subject to speculative considerations [14]. The degreesof freedom for speculation are much restricted if the fine structure of the spectrum can be observed in greater detail. The formula (53) should then be replaced by a formula with the additional quantum numberj E,,j -
a2m,,c2F(n,j, CX”).
(5%
A comparison of (53) and (53a) showsthat the fine-structure permits an independent determination of 01.
CONSTITUTIVE
MAP
515
Recent observations [ 151 on spectra of quasi-stellar objects made an independent observation of 01possible, with the spectacular result that it equals within experimental error the cydetermined in the laboratory. One may conclude from this observation that 01is truly a universal constant of nature. A prerequisite for a quantity to be a universal constant is that it is a spacetime scalar under arbitrary transformations. The conventional rendering of the fine structure constant 01 = e2/Ac does not make its invariant scalar properties explicit. There is the very strong physical suggestion that 01,e2 and #i are space-time invariant scalars under general transformations. Yet, the light velocity c is only a Lorentz invariant. How can we remove this inconsistency? The conventional rendering of the fine structure constant results from using a derived electrical unit of charge e in 01.The preceding discussion, as explained in Section II, strongly depends on the introduction of electric charge as an independent and separate unit. It will now be shown that this minor change of adopting an independent electrical charge is the key to a generally invariant rendering of 01.
Following a well-established procedure, one replaces the derived unit e2 by an independent unit e2 requiring the unit adjustment factor l/473, : (54) E,, is the parameter that occurs in the free-space constitutive map x given by (40). The light velocity c expressed in the free-space parameters E” and pLois c = (Eopp=. Using (54) and (55), the conventional
(55)
iy. = e2/Ac becomes (56)
The transformational inconsistency has now been removed. The fine structure constant as given by (56) is the ratio of two generally invariant impedances:
They are zl = 4&c
9
(57)
and Z,,, = 4di/e2 = 2hle2.
(58)
The general invariance of the free-space impedance was established in Section VII, formula (41). The general invariance of Z, is provided by ample theoretical evidence that e and fi each are generally invariant space-time quantities.
516
POST
At the end of Section VII, it was established that 2, is not only a general spacetime invariant, there is ample evidence that Z,, is also a universal constant. It follows that the ratio Z, = 2file2 should be a universal constant of nature. This result would appear as a foregone conclusion if it were not for the many speculations in the course of time that 01,e and fi might change in time. These proposed changes are much restricted unless one wants to drop the constancy of Z,, which means that light can be scattered and diffused without the presence of matter. There is, for the time being, little evidence to justify such a radical assumption. An additional comment is indicated concerning the primary role of the quantized elementary units of action (h/2) and charge (e). The notion of an elementary quantum stands and falls with the identical magnitudes to be associated with individual quanta. A time and position dependence of e and fi would then become a contradiction in terms unless one admits to only a local validity of the quantum concept. By contrast, the integrals of the differential 3-form of current and charge density (phys. dim. [e]) and of action (phys. dim. [fi]) are determined by a specification of their domain of integration. Their values become multiples of e and fi/2 if one accepts the global validity of the quantum concept. Hence the magnitude and possible change of the individual quanta e and fi become a cosmological matter if one accepts the global validity of the quantum concept. Note. During the preparation of this manuscript, two remarkable notes by A. Wyler [18] came to my attention. In a sense subsuming the invariance and constancy of (Yas discussed in this paper, Wyler obtains a numerical value for cy which agrees in 6 decimal places with the best observed values. The physical argument is shrouded in abstruse group considerations. Hopefully a more elaborate discussion will be forthcoming.
APPENDIX
A differential form rendering of the Maxwell equations was introduced by Cartan [3] almost half a century ago. Recently it has been revived through the work of Misner and Wheeler [16], and Flanders [17]. The work of de Rham on the topology of multiple integrals may be accepted as a major motivation for the cited revival. The de Rham theorems are discussed by Flanders [17, p. 681 as well as by Misner and Wheeler [16]. The differential-form calculus consists of an exterior algebra and a calculus of exterior derivatives. It is a calculus geared to the treatment of totally antisymmetric quantities that emerge from a study of Pfaffian integrability theory.
CONSTITUTlVE
MAP
517
The differential-form calculus as such may be seen as an important subdiscipline of general tensor calculus, which it transcends by far in elegance, because of its conspicuous and beautiful algebraic structure. The homogeneity in algebraic structure, as compared to tensor analysis, permits an extremely concise notation to adequately account for this structure.The relegation of structural detail to a solid algebraic foundation permits greater emphasis on fundamental conceptual problems such as topological features related to properties in the “large” and properties in the “small”. Subtle distinctions now emerged. The property of a differential form of having a vanishing exterior derivative led to the concepts of closed forms (local) and exact forms (global); the latter is a property in the “large”. The old Stokes and Gauss integral laws received new attention from a point of view of topology. In fact they spawned new integral theorems such as de Rham’s theorems, which, gross0 mode may be considered as appropriate extensions of Cauchy’s famous residue theorem in complex analysis. How do these new developments tie in with Maxwell theory and why should we use them? First of all, the basic electromagnetic laws are all expressable as differential forms. Then the new mathematical developments open up the possibility of discussing the much neglected properties in the “large”. The distinction between “local” and “global” properties for the first time permits more precise statements about conservation laws in the “small” and in the “large”. The conservation of flux and net charge are typical examples of global laws that hold in the large. By contrast conservation of energy and momentum is a typical example of a law of only local significance. These are some reasons that made it tempting to use the differential-form formalism in Section II of this paper: to make global aspects more explicit even if one cannot use the formalism throughout. Finally, an important philosophical distinction should be mentioned between the work of Misner and Wheeler and the presentation given in Section II of this paper. Misner and Wheeler interpret their field sources as a microscopic multiple connectedness of the ordering manifold of space-time in which these fields are defined: the worm hole concept of electricity. Epistemologically it is better if the cohomology of fields is given a primary physical role rather than the homology of chains over which these fields are being integrated. The philosophy is simply that we take cognizance of space-time through what exists and happens in space-time. The a priori existence of worm holes in space-time is not well compatible with a rigorous pursuit of the idea that magnetic monopoles do not exist. The latter assumption is a cornerstone of the development in Section II.
518
POST ACKNOWLEDGMENTS
A long time ago, Francis J. Zucker, then at the Air Force Cambridge Research Laboratories, asked me whether the widest invariance group of the constitutive tensor might have a true image as a transformation group in space-time. My initial response was one of scepticism in the sense that the result might be of mere mathematical interest without much further physical perspective. More recently the same question arose in a discussion with Professor La&o Tisza (MIT). The present paper is an attempt at a further inquiry into the subject. The result may be viewed as a new angle to an old problem. I thank the above-mentioned gentlemen for their insistence to look into the matter, Professor Tisza for references and discussions of the relevant work of Felix Klein and W. Blaschke, and Professor Kenneth Johnson for a discussion of the properties of the conformal group. Any remaining inadequacies are the sole responsibility of the author.
REFERENCES 1. E. CUNNLNGHAM, Proc. London Math. Sot. 8 (lPlO), 77. 2. H. BATEMAN, Proc. London Math. Sot. 8 (lPlO), 223. 3. E. CARTAN, Ann. Econ. Norm. 41 (1924), 1. 4. D. VAN DANTZIG, Proc. Cambridge Phil. Sot. 30 (1934), 421. D. VAN DANTZIG, Proc. Amsterdam Acad. 37 (1934), 521, 526, 644, 825. 5. E. J. POST, “Formal Structure of Electromagnetics,” North-Holland, Amsterdam, 1962. 6. G. DE RHAM, “Varietes Differentiables,” Hermann, Paris, 1955. 7. J. A. SCHOUTEN, “Ricci Calculus,” 2d ed., Springer, Berlin, 1954. J. A. SCHOUTEN, “Tensor Analysis for Physicists,” Oxford Univ. Press, London, 1951. 8. S. L. How AND N. BLOEMBERGEN, Phys. Rev. 138 (1965), A1218. 9. A. NIJENHUIS, Proc. Amsterdam Acad. 54 (1967), 200. 10. E. J. POST, Rev. Mod. Phys. 39 (1967), 475. 11. F. KLEIN, “Hohere Geometrie,” 246-258, Springer, Berlin, 1926. 12. W. BLASCHKE, “Differential Geometric,” Vol. III, Springer, Berlin, 1925. 13. J. WESS, Nuovo Cimento 18 (1960), 1086. 14. G. GAMOW, Phys. Rev. Lett. 19 (1967), 913. 15. J. N. BAHCALL AND M. SCHMIDT, Phys. Rev. Lett. 19 (1967), 1294. 16. C. MISNER AND J. A. WHEELER, “Geometrodynamics” (J. A. Wheeler, Ed.), Academic Press, New York, 1962. 17. H. FLANDERS, “Differential Forms,” Academic Press, New York, 1963. 18. A. WALER, C. R. Acad. Sci. Paris 269 (1969), 743; A. WYLER, C. R. Acad. Sci. Paris 271 (1971), 186.