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4
ëø öø
incorporate nth order derivatives of whose choice is of no consequence for the distinction of
ìø ÷ø
n
íø øø
actually differing U-fields. Thus the number of expansion coeffi cients relevant for the enumeration of
4
ëø öø
the U-fields is decreased by . By the enumeration method we obtain for the number of free nth
ìø ÷ø
n
íø øø
order coefficients
îø 444 4 ùø
ëø öø ëø öø ëø öø ëø öø
z =
ïø16 n÷ø + 64ìø n -1÷ø - 4ìø n +1÷ø - ìø
ìø
n÷øúø
íø øø íø øø íø øø íø øø
ðøûø
îø 44 ùø
ëø öø ëø öø
- (23)
ïø16 n - 2÷ø + 64ìø n -1÷øúø
ìø
íø øø íø øø
ðøûø
îø 44 ùø
ëø öø ëø öø
+
ïø4 n - 3÷ø + n - 2÷øúø
ìø ìø
íø øø íø øø
ðøûø
The first bracket represents the total number of relevant nth order coefficients which characterize the
-U-field, the second the reduction of this number due to the existence of the field equat ions, and the
third bracket gives the correction to this reduction on account of the identities (18) and (19).
Computing the asymptotic value for large n we find
4
ëø öø z1
z (23a)
ìø
n÷ø n
íø øø
where
z1 = 42
The field equations of the non-symmetric field are thus considerably weaker than those of the pure
gravitational field (z1 = 12).
The influence -invariance on the strength the system equations. One may be tempted to
bring about transposition invariance of the theory by starting from the transposition invariant
expression
96
1
ik
= Rik + Rik
2
(instead of introducing the U as field variables). Of course, the resulting theory will be different
from the one expounded above. It can be shown that for this no -invariance exists. Here, too, we
obtain field equations of the type (16a), (16b), which are transposition invariant (with resp ect to and
“). Between them, there exist, however, only the four "Bianchi identities." If one applies the method
of enumeration to this system, then, in the formula corresponding to (23), the fourth term in the first
bracket and the second term in the th ird bracket are missing. One obtains
z1 = 48
» » » » » »
.
Comparison with the previous system of field equations. This is given by
“s s = 0 Rik = 0
i
s s
gik ,l - gsk“il - gis“lk = 0 R i k,l + Rk l ,i + R l i , k = 0
1
where Rik is defined by (4a) as a function of the “ (and where Ri k =+ Ri k Rki ,
2
1
R i k =- Ri k Rk i ).
2
This system is entirely equivalent to the new system (16a), (16b) since it has been derived from
the same integral by variation. It is transposition invariant with respect to the gi k and “l k . The
i
difference, however, lies in the following. The integral to be varied is itself not trans position
invariant, nor is the system of equations that is at first obtained by its variation; it is, howe ver,
invariant with respect to the -transformations (5). In order to obtain transposition invariance
here, one has to use an artifice. One formally introduces four new field variables, , which after
i
variation are so chosen that the equations “s s = 0 are satisfied.* Thus the equations obtained by
i
variation with respect to the “ are brought into the indicated transposition invariant form. But the
Rik-equations still contain the auxiliary variables . One can, however, eliminate them, which
i
leads to a decomposition of these equations in the manner stated above. The equations obtained
are then also transposition invariant (with respect to the and “).
Postulating the equations “s s = 0 involves a normalization of the “-field, which removes the
i
-invariance off the system of equations. As a result, not all equivalent representations of a
“ -field appear as solutions of this system. What takes place here, is comparable to the procedure
of adjoining to the field equations of pure g ravitation arbitrary additional equations which restrict
the choice of coordinates. In our case, moreover, the system of equations becomes un -
l
*
By s et t i ng “l k* =“l k +
i i i k
97
necessarily complicated. These difficulties are avoided in the new representation by starting
from a variational principle that is transposition invariant with respect to the and U, and by
using throughout the and U as field variables.
The divergence law and the conservation law of momentum and energy
If the field equations are satisfied and if, moreover, the variation is a transformation variation,
then, in (14), not only Sik and vanish, but also , so that the field equations imply the
ik
equations
ik s
Uik ,s = 0
i
where Uisk is given by (10c). This divergence law holds for any choice of the vector . The
i
simplest special choice, i.e. independent of the x, leads to the four equations
s ik
a" Uisk ,t ,s = 0
t,s
These can be interpreted and applied as the equations o f conservation of momentum and
energy. It should be noted that such conservation equations are never uniquely determined by the
system of field equations. It is interesting that according to the equations
s ik
a" Uisk ,t
t
1 2 3 4
the density of the energy current , , as well as the energy density vanish for a
4 4 4 4
field that is independent of x4. From this one can conclude that according to this theory a
stationary field free from singularities can never represent a mass different from zero.
The derivation as well as the form of the conservation laws become much more complicated if
the former formulation of the field equations is used.
GENERAL REMARKS
A. In my opinion the theory presented here is the l ogically simplest relativistic field theory
which is at all possible. But this does not mean that nature might not obey a more complex field
theory.
More complex field theories have frequently been pro posed. They may be classified according to
the following characteristic features:
(a) Increase of the number of dimensions of the con tinuum. In this case one must explain why
the continuum is apparently restricted to four dimensions.
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