Appendix A v3

A.1 The first general form and explanation of notation 1 A.2 Special case of the index range: the ε tensor 2 A.3 Antisymmetric Tensor Theorem 3 A.4 A second general form 4 A.5 A third general form 6 A.6 Theorem on raising and lowering P 8 A.7 Ordered Sum Theorem 10 A.8 Sum of permutation signs vanishes 11 A.9 Orthogonality of the Alt and Sym Projection Operators 11 Appendix A. Facts and Theorems involving permutation P A.1 The first general form and explanation of notation Often we encounter objects of the following form Fii...i = ΣP (-1)S(P) RP(i)P(i)...P(i) (A.1.1) We shall now explain this notation. 1. The ΣP is a sum of all k! permutations of the set of integers {1,2,...k}. For example, one permutation is P[1,2,...k] = [2,1,...k], and this permutation swaps the first two integers. We usually think of the first permutation in ΣP as being the identity permutation P[1,2,...k] = [1,2,...k] . Notice that {..} is a set whereas [..] is an ordered sequence. 2. Any two permutations of [1,2,...k] can be linked by a number of pairwise swaps of the integers. For example, if we have P[1,2,...k] = [i1,i2,...ik], one can get from the first integer sequence to the second by doing some number S(P) of pairwise swaps. The integer S(P) is not unique, but whether it is an even or an odd integer is unique, so the factor (-1)S(P) is unique to a particular P (we leave it to the reader to prove this fact.) Example: [1,2,3] → [2,1,3] S(P) = 1 (-1)S(P) = -1 [1,2,3] → [1,3,2] → [2,3,1] → [2,1,3] S(P) = 3 (-1)S(P) = -1 (A.1.2) 3. The indices ir in (A.X.1) can take any values one wants. The ir need not be elements of {1,2,...k}, although the subscript r of ir must be an element of {1,2,...k}. The ir could in theory be Sanskrit symbols, but normally we think of them as real numbers and in particular we usually think of them as being integers in some range 1,2...n. In our applications, usually n = dim(V), the dimension of vector space V, and in general we have k ≠ n. Definitions: A tensor Xii...i is totally symmetric if the tensor is unchanged by the swap of any index pair. An example is Xii...i = xixi ... xi. (A.1.3) A tensor Xii...i is totally antisymmetric if the tensor change sign under the swap of any index pair. An example is εii...i described below. (A.1.4) It is possible for a tensor to be symmetric or antisymmetric on some indices and not others. For example, Xiiii = xixi(yiyi - yiyi) is symmetric on 1↔2, antisymmetric on 3↔4, and has no symmetry at all on 1↔3, 1↔4, 2↔3, 2↔4. For a rank-2 tensor, Xii = xixi is symmetric, while Xii = yiyi - yiyi is antisymmetric, and the adjective "totally" is not meaningful. Example: Here is (A.1.1) for k = 3: Fiii = Riii - Riii + Riii - Riii + Riii - Riii (A.1.5) 1 2 3 4 5 6 where Fiii is seen to be totally antisymmetric. For example Fiii = Riii - Riii + Riii - Riii + Riii - Riii (A.1.6) -6 -5 -4 -3 -2 -1 We shall prove in Section A.3 below that Fii...i in (A.1.1) is totally antisymmetric. A.2 Special case of the index range: the ε tensor In the special case that the indices ir are restricted to have values in the set {1,2,...k}, we can write (A.1.1) using the "ε permutation tensor". The set {ir} does not have to be a permutation of [1,2...k], but the ir have to each be an element in {1,2,...k}. Two ir could be the same, for example. Then (A.1.3) can be written Fiii = εiii F123 ir ϵ {1,2,3} (A.2.1) and more generally for (A.1.1), Fii...i = εii...i F12...k ir ϵ {1,2,..k} (A.2.2) The object ε is usually called "the permutation tensor" even though it is not really a tensor in the strong sense involving transformation. There exists a related object, also written ε, which is the "Levi Civita tensor" and it really is a tensor, see ****. The permutation tensor has very simple properties. If any two indices are swapped, it changes sign, and ε12..k = 1. Thus, εii...i = 0 if any two or more indices are the same. This permutation tensor makes no sense unless the ir are restricted to the range {1,2..k}. The number of indices on ε must match the range of values the indices can take. Interpretation: There is only one rank-k totally antisymmetric tensor which takes the value 1 when indices are 12..k, and that tensor is εii...i . Any other totally antisymmetric rank-k tensor F can be written Fii...i = (constant) εii...i and that constant is therefore F12..k as in (A.2.2). In covariant notation, the permutation "tensor" is the same regardless of whether indices are up or down, so in particular εii...i = εii...i . (A.2.3) Since the ir in (A.1.1) can take arbitrary values, we can certainly choose ir = r to get F12...k = ΣP (-1)S(P) RP(1)P(2)...P(k) (A.2.4) Now applying our restriction ir ϵ {1,2...k}, we claim the above sum can be written in terms of the ε tensor as follows, F12...k = Σii...i εii...i Rii...i ir = 1,2...k (A.2.5) Although this sum has kk nominal terms, many terms vanish because εii....i has two or more indices the same. In fact only k! terms don't vanish since that is the number of non-vanishing values of εii....i : first index can take k values, second then k-1, and so on to get k! . The first term in both sums above is just R12..k . The permutation sign (-1)S(P) is implemented by εii....i . Combining (A.2.2) and (A.2.5) we get Fjj...j = εjj...j F12...k or Fjj...j = εjj...j Σii....i εii....i Rii....i (A.2.6) where jr ϵ {1,2...k} and where all sums are ir = 1,2...k . Warning: Suppose we have jr in the range {1,2...n} where n > k. Then (A.2.6) is invalid because the ε tensor does not make sense. But (A.1.1) is still valid. A.3 Theorem: (A.1.1) is totally antisymmetric The tensor Fii...i ≡ ΣP (-1)S(P) RP(i)P(i)....P(i) is totally antisymmetric. (A.3.1) Note: The sum shown here is defined in (6.5.1) to be [alt(R)]ii...i so this theorem also claims, F = alt(R) = a totally antisymmetric tensor for any R (A.3.2) Proof: We shall show that Fii...i = - Fii...i and then the same argument trivially applies to any pair of indices. Start with, Fii...i = ΣP (-1)S(P) RP(i)P(i)....P(i) . (A.3.3) As noted above, 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 Fii...i = ΣP (-1)S(P) RPP'(i)PP'(i)....PP'(i) . (A.3.4) 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 Fii...i = ΣP (-1)S(PP')-1 RPP'(i)PP'(i)....PP'(i) = (-1) ΣP (-1)S(PP') RPP'(i)PP'(i)....PP'(i) . (A.3.5) 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.3.6) 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 Fii...i = (-1) ΣP (-1)S(P) RP(i)P(i)....P(i) (A.3.7) = - Fii...i QED (A.3.8) Corollary: Tensor F12...k = ΣP (-1)S(P) RP(1)P(2)....P(k) is totally antisymmetric. (A.3.9) This is just (A.3.1) where we select ir = r . A.4 A second general form Consider the following form, Fii...ijj...j = ΣP (-1)S(P) fjP(i) * fjP(i) ... * .fjP(i) (A.4.1) = fji * fji * ... * fji + signed permutations of the second index where * is some operator like or ^ or regular multiplication. Indices ir and jr take arbitrary values. We claim some Facts: Fact: Fii...ijj...j in (A.4.1) totally antisymmetric in the indices ir (regardless of their range). (A.4.2) Proof: For fixed values of the jr, (A.4.1) is just a special case of (A.1.1), Fii...i = ΣP (-1)S(P) RP(i)P(i)...P(i) (A.1.1) which we showed in (A.3.1) to be totally antisymmetric in the ir. Fact: Fii...ijj...j in (A.4.1) totally antisymmetric in the indices jr (regardless of their range), provided that * is a commuting operator. (A.4.3) Proof: We shall show that Fii...ijj...j = - Fii...ijj...j and then the same argument trivially applies to any pair of indices. The proof is very similar to the proof of Section A.3 above. Start with Fii...ijj...j = ΣP (-1)S(P) (fi)P(j) * (fi)P(j)* ...* (fi)P(j) . Define permutation P' exactly as in Section A.3, except here permutations act on the jr subscripts. Then for example j1 = P'(j2) and so on, allowing the above equation to be rewritten as Fii...ijj...j = 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) = (-1) ΣP (-1)S(P) (fi)P(j)* (fi)P(j)...* (fi)P(j) // rearrangement theorem = (-1) ΣP (-1)S(P) (fi)P(j) * (fi)P(j)* ...* (fi)P(j) // * assumed commutative = - Fii...ijj...j QED Fact: The following two expansions are equal: Fii...ijj...j = ΣP (-1)S(P) fjP(i) fjP(i) ... fjP(i) Fii...ijj...j = ΣP (-1)S(P) fP(j)i fP(j)i ...fP(j)i (A.4.4) The first expansion is (A.4.1) where * is taken to be ordinary multiplication. In the second expansion, the P operators have been lowered from the ir indices to the jr indices. Proof: This fact is proven in Section A.6 below. Fact: The expansions shown in (A.4.4) are totally antisymmetric in the ir and in the jr indices. (A.4.5) Proof: For fixed jr the first line in (A.4.4) has the form (A.1.1) and is therefore totally antisymmetric in the ir according to (A.3.1). For fixed ir the second line in (A.4.4) has the form (A.1.1) (in the jr indices) and is therefore totally antisymmetric in the jr according to (A.3.1). An example of form (A.4.4): k!(vj^ vj^ .....^ vj)ii...i = ΣP (-1)S(P) (vP(j))i(vP(j))i ... (vP(j))i (A.4.6) Fact (A.4.5) then tells us that (vj^ vj^ .....^ vj)ii...i is totally antisymmetric in the ir and jr. Fact: The expansions (A.4.4) can be written as a determinant: Fii...ijj...j = det(M) = det( fji) (A.4.7) where Mrs = fji . For example here are det(M) = det(MT) for k = 3, Fiiijjj = det = det (A.4.8) Proof: This is shown in Section A.6 Comment: Suppose fji = (fj)i where fj is a vector and (fj)i is a component of that vector. Suppose these vectors have n components where n > k. Then the second determinant shown above is a minor of matrix MT = [fj, fj, ...fj] whose columns are those vectors. The minor goes the full width of matrix MT, but only includes rows specified by indices i1, i2 ... ik. In the case n = k, the minor is the full det(MT). (A.4.9) The example (A.4.6) can be written in the determinant forms, and the second determinant form can be interpreted as in (A.4.8) where MT = [vj, vj, ..vj] . A.5 A third general form If it happens that the fab objects don't depend on index a, we can write (A.4.1) as Fii...i = ΣP (-1)S(P) fP(i) * fP(i) ... * .fP(i) (A.5.1) where * is a possibly non-commutative operator like or ^ . Fact: Fii...i is totally antisymmetric in the indices ir. (A.5.2) Proof: Expansion (A.5.1) is of the form (A.3.1) which is totally antisymmetric in the ir. Fact: If the indices ir lie in the set {1,2..k}, one can write (A.5.1) as Fii...i = Σii...i εii...i fi * fi ... * fi (A.5.3) Proof: This is an example of going from (A.2.4) to (A.2.5). A special case of (A.5.3) is this: F12...k = ΣP (-1)S(P) fP(1) * fP(2) ... * fP(k) = Σii...i εii...i fi * fi ... * fi ir = 1,2...k (A.5.4) = [ f1 * f2 ... * fk + signed permutations ] where F12...k is still totally antisymmetric under a swap of any two indices. Subtlety. In the first line of (A.5.4), it is clear how the 12...k labels on F appear on the right of the equation since we have just taken i1i2...ik → 1,2...k in (A.5.3). In the second line, this connection is less obvious, so we highlight (red) to bring out this detail F12...k = ΣP (-1)S(P) fP(1) * fP(2) ... * fP(k) = Σii...i εii...i fi * fi ... * .fi ir = 1,2...k (A.5.5) In the second form the labels 12...k match the ir subscripts in factor fi * fi ... * .fi, even though these are dummy summation indices. We know from (A.5.2) that F12...k is fully antisymmetric in all its indices. Here then is how this fact is verified in the second form of (A.5.5): F21...k = Σii...i εii...i fi * fi ... * .fi = Σii...i εii...i fi * fi ... * .fi // swap dummy indices i1 ↔ i2 = Σii...i εii...i fi * fi ... * .fi // summation order is irrelevant = Σii...i ( -εii...i) fi * fi ... * .fi // property of ε tensor = - Σii...i εii...i fi * fi ... * .fi // property of ε tensor = - F12...k (A.5.6) Lowering indices we can restate (A.5.4) as, F12...k = ΣP (-1)S(P) fP(1) * fP(2) ... * fP(k) = Σii...i εii...i fi * fi ... * .fi ir = 1,2...k (A.5.7) Fact: F12...k is totally antisymmetric in its indices. (A.5.8) Proof: This is just a specialization of (A.5.2). Example of form (A.5.7): In (6.1.3) we write ( here * = ), k! (v1^ v2^ .....^ vk) = ΣP (-1)S(P) ( vP(1) vP(2) ..... vP(k)) = Σii...i εii...i ( vi vi ... vi) (6.1.3) From (A.5.8) we conclude that this wedge product of vectors is totally antisymmetric in 12...k. A.6 Theorem on raising and lowering P ΣP (-1)S(P) fjP(i) fjP(i) .... fjP(i) = ΣP (-1)S(P) fP(j)i fP(j)i ... fP(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 fjifji ....fji 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) fjP(i)fjP(i) = fjifji - fjifji second line = ΣP(-1)S(P) fP(j)ifP(j)i = fjifji - fji fji (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) fji fji .... fji = ΣP (-1)S(P) fji fji ... fji . (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 = (fj)i r,s = 1,2...k (A.6.7) so that for example (fj)i = M1P(1) and (fj)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) f1P(i) f2P(i) .... fkP(i) = ΣP(-1)S(P) fP(1)i fP(2i ... fP(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) A.7 Ordered Sum Theorem (ΣP [ΣP(i)<P(i)<...<P(i)]) fii...i = Σi<i<...<i [ΣP fP(i)P(i)...P(i)] . (A.7.1) 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.7.2) 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.7.3) Thus we have proven the Theorem for k = 2, (ΣP [ΣP(i)P(i)]) fii = Σi<i [ΣP fP(i)P(i)] (A.7.4) 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.7.5) 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.7.6) 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.7.7) 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.7.1) A.8 Sum of permutation signs vanishes ΣP (-1)S(P) = 0 (A.8.1) Proof: Suppose in (A.1.1) we take Rii...i = 1 . (A.8.2) Then (A.1.1) says Fii...i ≡ ΣP (-1)S(P) RP(i)P(i)....P(i) = ΣP (-1)S(P). (A.8.3) Theorem (A.1.1) says that Yii...i is totally antisymmetric for any R. But the right side of (A.8.3) does not depend on the indices, so we have for example Fii...i = ΣP (-1)S(P) Fii...i = ΣP (-1)S(P) (A.8.4) Then ΣP (-1)S(P) = Fii...i = - Fii...i = - ΣP (-1)S(P) (A.8.5) and we conclude that ΣP (-1)S(P) = 0. QED Example: Fiii = Riii - Riii + Riii - Riii + Riii - Riii There are three + signs and three - signs. A.9 Orthogonality of the Alt and Sym Projection Operators Alt(Sym(X)) = Sym(Alt(X) = 0 (A.9.1) 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.9.2) [Sym(X)]ii...i ≡ ΣP XP(i)P(i)...P(i) (A.9.3) 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.8.1) 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.8.1) QED