Appendix A v2

Appendix A: Supporting Theorems 1 A.1 Antisymmetric Tensor Theorem 1 A.2 Antisymmetric Function Theorem 2 A.3 Ordered Sum Theorem 3 A.4 Sum of permutation signs vanishes 4 A.5 Orthogonality of the Alt and Sym Projection Operators 5 A.6 Theorem on raising and lowering P 5 Appendix A: Supporting Theorems A.1 Antisymmetric Tensor Theorem The tensor Yii...i ≡ ΣP (-1)S(P) RP(i)P(i)....P(i) is totally antisymmetric. (A.1.1) Note: The sum shown here is defined in ** to be [alt(R)]ii...i so this theorem also claims, Y = alt(R) = a totally antisymmetric tensor for any R (A.1.2) Proof: We shall show that Yii...i = - Yii...i and then the same argument trivially applies to any pair of indices. Start with, Yii...i = ΣP (-1)S(P) RP(i)P(i)....P(i) . (A.1.3) In this sum, P represents a permutation of the integers {1,2...k}, and P(ir) ≡ iP(r) so that P really acts on the subscript r. S(P) is the number of swaps involved in permutation P. For example, if we define the specific permutation P' by P'{1,2...k} = {2,1...k}, then S(P') = 1. Since in this case i2 = P'(ii), we can write i1 = P(i2) = PP'(i1). Similarly, i2 = P(i1) = PP'(i2). For any other subscript like i3 we have i3 = P(i3) = PP'(i3). The above sum can then be written Yii...i = ΣP (-1)S(P) RPP'(i)PP'(i)....PP'(i) . (A.1.4) Since S(PP') = S(P) + S(P') for any P and P', and since S(P') = 1, we know S(P) = S(PP') - 1, so then Yii...i = ΣP (-1)S(PP')-1 RPP'(i)PP'(i)....PP'(i) = (-1) ΣP (-1)S(PP') RPP'(i)PP'(i)....PP'(i) . (A.1.5) Finally we can use the rearrangement theorem of group theory applied to the permutation group (see PL ***). This theorem says that, ΣP f(QP) = ΣP f(PQ) = ΣP f(P) (A.1.6) where Q is any fixed permutation. The first two sums are just reorderings of the third sum and so equal the third sum. Selecting Q = P' we have ΣP f(PP') = ΣP f(P) so then Yii...i = (-1) ΣP (-1)S(P) RP(i)P(i)....P(i) (A.1.7) = - Yii...i QED (A.1.8) Corollary: Tensor Y12...k = ΣP (-1)S(P) RP(1)P(2)....P(k) is totally antisymmetric. (A.1.9) Proof: This is an application of Theorem 1 where we take in = n. The theorem and corollary are fairly obvious when one writes Yii...i = ΣP (-1)S(P) RP(i)P(i)....P(i) = Rii...i + signed permutations (A.1.10) or Y12...k = ΣP (-1)S(P) RP(1)P(2)....P(k) = R12...k + signed permutations (A.1.11) Examples: Yiii = Riii - Riii + Riii - Riii + Riii - Riii Y123 = R123 - R132 + R312 - R321 + R231 - R213 (A.1.12) ________________________________________________________________________________________________ A.2 Antisymmetric Function Theorem The function F(vi,vi.....vi) ≡ ΣP (-1)S(P) (fi)P(j)(fi)P(j)...(fi)P(j) is totally antisymmetric in its arguments. superseded (A.2.1) We shall show that F(vi,vi.....vi) = - F(vi,vi.....vi) and then the same argument trivially applies to any pair of indices. The proof is very similar to the proof of Theorem A.1 above. Start with F(vi,vi.....vi) = ΣP (-1)S(P) (fi)P(j)(fi)P(j)...(fi)P(j) . (A.2.2) Define permutation P' exactly as in Section A.2, except here permutations act on the j indices. Then for example j1 = P'(j2) and so on, allowing the above equation to be rewritten as F(vi,vi.....vi) = ΣP (-1)S(P) (fi)PP'(j)(fi)PP'(j)...(fi)PP'(j) = (-1) ΣP (-1)S(PP') (fi)PP'(j)(fi)PP'(j)...(fi)PP'(j) (A.2.3) = (-1) ΣP (-1)S(P) (fi)P(j)(fi)P(j)...(fi)P(j) // rearrangement theorem (A.2.4) = - F(vi,vi.....vi) QED (A.2.5) _______________________________________________________________________________________________ A.3 Ordered Sum Theorem (ΣP [ΣP(i)<P(i)<...<P(i)]) fii...i = Σi<i<...<i [ΣP fP(i)P(i)...P(i)] . (A.3.1) Rather than present a formal proof, we look at the two simplest cases and the general case is then obvious. k = 2: First, consider Q ≡ Σi≠i fii = [Σi<i + Σi>i] fii = [Σi<i + Σi<i] fii= (ΣP[ΣP(i)<P(i)]) fii . (A.3.2) On the other hand, Q = [Σi<i + Σi<i] fii = Σi<i fii + Σi<i fii = Σi<i fii + Σi<i fii //dummy swap i1↔i2 in 2nd term = Σi<i [ fii + fii] = Σi<i [ΣP fP(i)P(i)] . (A.3.3) Thus we have proven the Theorem for k = 2, (ΣP [ΣP(i)P(i)]) fii = Σi<i [ΣP fP(i)P(i)] (A.3.4) k = 3: First, consider Q ≡ Σi≠i≠i fiii = (Σi<i<i + Σi<i<i + Σi<i<i + Σi<i<i + Σi<i<i + Σi<i<i) fiii = (ΣP [ΣP(i)<P(i)<P(i)]) fiii . (A.3.5) On the other hand we can rename the summation indices in all but the first sum to get Q = (Σi<i<i + Σi<i<i + Σi<i<i + Σi<i<i + Σi<i<i + Σi<i<i) fiii as is 2↔3 1↔3 = Σi<i<i fiii + Σi<i<i fiii + Σi<i<i fiii + 3 more sums = Σi<i<i [ fiii + fiii + fiii + 3 more terms ] = Σi<i<i [ΣP fP(i)P(i)P(i)]. (A.3.6) Thus we have proven the Theorem for k =3, (ΣP [ΣP(i)<P(i)<P(i)]) fiii = Σi<i<i [ΣP fP(i)P(i)P(i)] . (A.3.7) The argument for a k-fold sum proceeds in the same manner, and we end up with (ΣP [ΣP(i)<P(i)<...<P(i)]) fii...i = Σi<i<...<i [ΣP fP(i)P(i)...P(i)] . (A.3.1) ________________________________________________________________________________________________ A.4 Sum of permutation signs vanishes ΣP (-1)S(P) = 0 (A.4.1) Proof: Suppose in (A.1.1) we take Rii...i = 1 . (A.4.2) Then (A.1.1) says Yii...i ≡ ΣP (-1)S(P) RP(i)P(i)....P(i) = ΣP (-1)S(P). (A.4.3) Theorem (A.1.1) says that Yii...i is totally antisymmetric for any R. But the right side of (A.4.3) does not depend on the indices, so we have for example Yii...i = ΣP (-1)S(P) Yii...i = ΣP (-1)S(P) (A.4.4) Then ΣP (-1)S(P) = Yii...i = - Yii...i = - ΣP (-1)S(P) (A.4.5) and we conclude that ΣP (-1)S(P) = 0. QED Example: Yiii = Riii - Riii + Riii - Riii + Riii - Riii There are three + signs and three - signs. ________________________________________________________________________________________________ A.5 Orthogonality of the Alt and Sym Projection Operators Alt(Sym(X)) = Sym(Alt(X) = 0 (A.5.1) The Alt and Sym operators are defined in ** and ** as [Alt(X)]ii...i ≡ ΣP (-1)S(P) XP(i)P(i)...P(i) (A.5.2) [Sym(X)]ii...i ≡ ΣP XP(i)P(i)...P(i) (A.5.3) Proof that Alt(Sym(X)) = 0 Alt(Sym(X)) = Alt( ΣP XP(i)P(i)...P(i)) = ΣP [Alt(XP(i)P(i)...P(i))] = ΣP [ ΣP' (-1)S(P')XP(P'i)P(P'i)...P(P'i)] = ( )2 ΣP'(-1)S(P'){ ΣP [XP(P'i)P(P'i)...P(P'i)] } = ( )2 [ΣP'(-1)S(P')]{ ΣP [XP(i)P(i)...P(i)] } // rearrangement theorem (A.1.6) = 0 // because [ΣP'(-1)S(P')] = 0 by (A.3.1) QED Sym(Alt(T)) = Sym (ΣP(-1)S(P)XP(i)P(i)...P(i))= ΣP(-1)S(P)[Sym(XP(i)P(i)...P(i))] = ΣP(-1)S(P) { ΣP' XP(P'i)P(P'i)...P(P'i)} = ( )2 [ΣP(-1)S(P)]{ ΣP' [XP'(i)P'(i)...P'(i)] } // rearrangement theorem (A.1.6) = 0 // because [ΣP(-1)S(P)] = 0 by (A.3.1) QED _______________________________________________________________________________________________ A.6 Theorem on raising and lowering P ΣP (-1)S(P) (vj)P(i) (vj)P(i) .... (vj)P(i) = ΣP (-1)S(P) (vP(j))i (vP(j))i ... (vP(j))i (A.6.1) The theorem claims one can move the P operator from the ir indices to the jr labels without changing the sum. It is the index subscripts that P permutes, for example, P(i1) ≡ iP(1), and the P are permutations of {1,2..k}. The actual indices ir and labels jr can take arbitrary values. For either expansion the P = 1 term in is (vj)i(vj)i ....(vj)i so the claim seems at least feasible. It is easy to make an arm-waving argument that (A.6.1) is obvious by just "writing out a few terms", but we nevertheless attempt a precise proof. Example: For k = 2 one has first line = ΣP (-1)S(P) (vj)P(i)(vj)P(i) = (vj)i(vj)i - (vj)i(vj)i second line = ΣP(-1)S(P) (vP(j))i(vP(j))i = (vj)i(vj)i - (vj)i (vj)i (A.6.2) so the theorem is valid for k = 2. Lemma: The determinant of a covariant (lower indices) k x k matrix Mab can be written det(Mab) = Σii...i εii...iM1iM2i ...Mki = ΣP (-1)S(P) M1P(1)M2P(2) ...MkP(k) = [ M11M22....Mkk + signed permutations of the second index ] . (A.6.3) Using the second form, and the fact that det(M) = det(MT) one knows that ΣP (-1)S(P) M1P(1)M2P(2) ... MkP(k) = ΣP (-1)S(P) MP(1)1MP(2)2 ... MP(k)k . (A.6.4) For the "mixed rank 2 tensor" Mab the above equation is trivially rewritten, ΣP (-1)S(P) M1P(1)M2P(2) ... MkP(k) = ΣP (-1)S(P) MP(1)1MP(2)2 ... MP(k)k (A.6.5) where either side is det(Mab). This last equation is our "Lemma". Proof of Theorem (A.6.1). We first restate (A.6.1) in its more precise form, ΣP (-1)S(P) (vj)i (vj)i .... (vj)i = ΣP (-1)S(P) (vj)i (vj)i ... (vj)i . (A.6.6) showing how P in fact acts only on subscripts of the indices ir and the labels jr . Define a k x k matrix M to be Mrs = (vj)i r,s = 1,2...k (A.6.7) so that for example (vj)i = M1P(1) and (vj)i = MP(1)1 . (A.6.8) Then (A.6.6) becomes ΣP (-1)S(P) M1P(1) M2P(2) .... MkP(k) = ΣP (-1)S(P) MP(1)1 MP(2)2 ... MP(k)k . (A.6.9) But this is precisely the claim of Lemma (A.6.5). Therefore (A.6.6) is proved, and so (A.6.1) is proved. Corollary 1: Theorem (A.6.1) is valid for any values of {j1,j2...jk}, and so in particular it is valid if we select {j1,j2...jk} = {1,2,..k}. Then, ΣP (-1)S(P) (v1)P(i) (v2)P(i) .... (vk)P(i) = ΣP(-1)S(P) (vP(1))i (vP(2)i ... (vP(k))i (A.6.10) Corollary 2: Application to Kronecker deltas, ΣP (-1)S(P) δjP(i) δjP(i)...δjP(i) = ΣP (-1)S(P) δP(j)i δP(j)i... δP(j)i (A.6.11) Although this is not quite a special case of (A.6.1), the proof is identical if one replaces (A.6.7) by Mrs = δji . (A.6.12) _______________________________________________________________________________________________