3.2.1. Transformation of a Superconservation Law
A discussion by DeWitt [15, page 113] shows that a super-conservation law is an expression of the form
(11)(-1)iV,ii=0,
where Vi is a supervector density of weight +1, and that the transformation law for a super vector density is
(12)Vi¯=∂x∂x¯Vjjxi¯,
where ∂x/∂x¯ is the Jacobian superdeterminant of xi¯j. (note: DeWitt uses the symbol μXi instead of our Vi, and the symbol J-1 instead of our ∂x/∂x¯.).

If we differentiate (12) with respect to xi¯ and multiply by (-1)i, we get
(13)(-1)iV,i¯i¯=(-1)i+j+ij(∂x∂x¯Vjxji¯),i¯,
where we have used jxi¯=(-1)j(j+i) xji¯, which corresponds to DeWitt's equation (1.7.25). Continuing, we have
(14)(-1)iV,i¯i¯=(-1)i+j+ij+i(i+j)(∂x∂x¯Vj),i¯xji¯+(-1)i+j+ij∂x∂x¯Vjxji¯,i¯=(-1)i(∂x∂x¯Vi),i+(-1)i+j+ij∂x∂x¯Vjxj,ki¯xi¯k=∂x∂x¯ (-1)iV,ii+(∂x∂x¯),jVj+(-1)j+k+jk∂x∂x¯Vjxi¯kxj,ki¯,
and since (∂x/∂x¯),j=-(-)k(∂x/∂x¯)xi¯kxj,ki¯, we get
(15)(-1)iV,i¯i¯=∂x∂x¯(-1)iV,ii-(-)k∂x∂x¯xi¯kxk,ji¯Vj+(-1)j+k+jk ∂x∂x¯Vjxi¯kxj,ki¯=∂x∂x¯(-1)iV,ii-(-)j+k∂x∂x¯Vjxi¯kxk,ji¯+(-1)j+k+jk∂x∂x¯Vjxi¯kxj,ki¯.
Thus, we have
(16)(-1)iV,i¯i¯=∂x∂x¯(-1)i V,ii-(-1)j+k ∂x∂x¯Vjxi¯k[xk,ji¯-(-1)jkxj,ki¯].
We see from (16) that we have both (-1)iV,ii=0 and (-1)i V,i¯i¯=0 (i.e., a super-conservation law is an invariant statement) if and only if (-1)j+k(∂x/∂x¯)Vjxi¯k[xk,ji¯-(-1)jkxj,ki¯]=0. Since Vj is arbitrary and ∂x/∂x¯≠0, we must have (-1)j+kxi¯k [xk,ji¯-(-1)jkxj,ki¯]=0. The index j is not summed, so we divide by (-1)j to obtain (10). We therefore call transformations that satisfy (10) “super-conservative.”

3.2.2. Proof That the Super-Conservative Transformations Form a Group Which Contains the Superdiffeomorphisms as a Proper Subgroup
We begin by noting that the identity transformation xi¯=xi is a super-conservative transformation that is, it satisfies (10). Next, we consider the result of following a transformation from xi to xi¯ by a transformation from xi¯ to xi^. Upon differentiating xki^=xr¯i^xkr¯ with respect to xj, we get
(17)xk,ji^=(-1)j(k+r)xr¯,ji^xkr¯+xr¯i^xk,jr¯=(-1)jk+jrxr¯,s¯i^xjs¯xkr¯+xr¯i^xk,jr¯=(-1)ki+kr+ri+rxkr¯xr¯,s¯i^xjs¯+xr¯i^xk,jr¯,
and we multiply by (-1)kxi^k to get
(18)(-1)kxi^kxk,ji^=(-1)ki+kr+ri+r+kxi^kxkr¯xr¯,s¯i^xjs¯+(-1)kxi^kxr¯i^xk,jr¯=(-1)rxkr¯xi^kxr¯,s¯i^xjs¯+(-1)kxr¯kxk,jr¯.
Thus,
(19)(-1)kxi^kxk,ji^ =(-1)rxi^r¯xr¯,s¯i^xjs¯+(-1)kxr¯kxk,jr¯.

We interchange j and k in the middle line of (17) to obtain
(20)xj,ki^=(-1)jk+krxr¯,s¯i^xks¯xjr¯+xr¯,i^xj,kr¯=(-1)jk+r+rs+ri+kr+kixkr¯xs¯,r¯i^xjs¯+xr¯i^xj,kr¯,
and we multiply by (-1)k+jk xki^ to get
(21)(-1)k+jkxi^kxj,ki^=(-1)k+r+rs+ri+kr+kixi^kxkr¯ xs¯,r¯i^xjs¯+(-1)k+jkxi^kxr¯i^xj,kr¯=(-1)r+rsxkr¯xi^kxs¯,r¯i^xjs¯+(-1)k+jkxi^kxr¯i^xj,kr¯,
so
(22)(-1)k+jkxi^kxj,ki^ =(-1)r+rsxi^r¯xs¯,r¯i^xjs¯+(-1)k+jkxr¯kxj,kr¯.
Upon subtracting (22) from (19), we obtain
(23)(-1)kxi^k[xk,ji^-(-1)jkxj,ki^] =(-1)rxi^r¯[xr¯,s¯i^-(-1)rsxs¯,r¯i^]xjs¯ +(-1)kxr¯k[xk,jr¯-(-1)jk xj,kr¯].
Equation (23) shows that if the quantities (-1)rxi^r¯[xr¯,s¯i^-(-1)rsxs¯,r¯i^] and (-1)kxr¯k[xk,jr¯-(-1)jkxj,kr¯] both vanish, then it follows that the quantity (-1)kxi^k[xk,ji^-(-1)jkxj,ki^] vanishes. Thus, if the transformations from xi to xi¯ and from xi¯ to xi^ are super-conservative, then the product transformation from xi to xi^ is super-conservative. If we let xi^=xi, we see from (23) that the inverse of a super-conservative transformation is super-conservative. As in Section 2.2.2, the relation xki^=xr¯i^xkr¯ guarantees that the associative law is satisfied. This completes the proof that the super-conservative transformations form a group.

We note that if (9) is satisfied, then (10) is satisfied; that is, the super-conservation group contains the super-diffeomorphisms as a subgroup. Thus, to show that it contains the super-diffeomorphisms as a proper subgroup, we need only to exhibit transformation coefficients which satisfy (10), but do not satisfy (9). Such transformation coefficients are the same as those in (7). This is clear, because the quantities in (7) are all ordinary numbers. Therefore, as DeWitt notes, they are in the even part of our supermanifold. Hence, in this case, the exponents j and k in (9) and (10) are even numbers. Thus, for the transformation coefficients in (7), (9), and (10) reduce to (1) and (2). We have shown previously in Section 2.2.2 that the transformation coefficients in (7) do not satisfy (1), but satisfy (2). q.e.d.