Appendix C v5

Appendix C v5 PhL 11.22.15 This was installed on 1.26.16, do not edit here. C.1 Theorem One 1 C.2 Theorem Two 4 C.3 Theorem Three 7 C.4 Summary and Generalization 8 Appendix C: Theorems on Pre-Symmetrization The Rearrangement Theorem (A.1.3) is used to prove three other theorems (One, Two and Three) where we have attempted to abstract as much as possible the "permutational nature" of the objects involved by using a generic permutation space with elements |1,2...k>. Then in Section C.4 the theorems are summarized and are generalized to apply to arbitrary tensor products. Finally, the generic theorems are applied to tensors and tensor functions. The reader uninterested in the theorem details would do well to skip right to Section C.4. It is assumed that the reader is familiar with App. A.1 and the first part of App. A.2, C.1 Theorem One Consider the following set of k+k' integers, {1,2....k, k+1,k+2....k+k'} = {1,2....k+k' } . (C.1.1) Partition this list into a low and high group by defining z ≡ 1,2....k Z = k+1,k+2....k+k' (C.1.2) Then {1,2....k+k'} = {z,Z}. (C.1.3) Now let Q be a permutation of the lower integers [1,2...k] = z. There are k! possible permutations, so we know that ΣQ (1) = k! . (C.1.4) We can extend the meaning of Q so it applies to the entire set of integers {1,2....k+k'} merely by stating that this extended Q' does not alter the higher integers. Then Q'(z) = Q(z) = z' = some permutation of the lower integers (C.1.5a) Q'(Z) = Z // since Q has no effect on the higher integers (C.1.5b) Q'(z, Z) = {Q'(z), Q'(Z)} = {Q(z), Z} . *** (C.1.5c) Now imagine we have a function f of the lower integers and a function F of the higher ones, f[z] = f[1,2....k] F[Z] = F[k+1,k+2....k+k'] . (C.1.6) Here are two applications we shall consider later on, f[z] = f[1,2....k] = Tii...i = components of a rank-k tensor f[z] = f[1,2....k] = T(vi,vi, .... vi) = a rank-k tensor function . (C.1.7) Now let P be a general permutation of [1,2....k+k'] = [z,Z]. P(z,Z) = [P(z), P(Z)] . (C.1.8) Notice that QP and PQ are undefined since P and Q operate in different spaces, but Q'P and PQ' are both defined since both permutations Q' and P operate in the space of [1,2....k+k']. Recall now the meaning of S(Q) as the number of swaps required to go from z to Q(z) . This is the same as the number of swaps required to go from [z,Z] to Q'[z,Z] = [Q(z),Z]. Therefore S(Q) = S(Q') (C.1.9) We shall now prove the following theorem : Theorem One ΣP (-1)σS(P) f[P(z)] F[P(Z)] = ΣP (-1)σS(P) f^[P(z)] F[P(Z)] (C.1.10) where f^[z] ≡ (1/k!) ΣQ (-1)σS(Q) f[Q(z)] σ = 1 or 0 The purpose of σ is to state the theorem with and without the (-1)S(P) factor. We shall use the notation f^ in most of this section to apply for both values of σ, but at the end, we shall distinguish these two cases by writing: f^[z] ≡ (1/k!) ΣQ (-1)S(Q) f[Q(z)] // = Alt(f), see (A.2.1) fs[z] ≡ (1/k!) ΣQ f[Q(z)] . // = Sym(f), see (A.3.1) (C.1.11) At the end of this section we will show that the above Theorem One with σ = 1 and σ = 0 is equivalent to the statements: Alt(fF) = Alt(f^F) f^ = Alt(f) σ = 1 Sym(fF) = Sym(fsF) fs = Sym(f) σ = 0 (C.1.12) Proof of Theorem One: Our first task is to process the second line of (C.1.10), f^[z] = (1/k!) ΣQ (-1)σS(Q) f[Q(z)] = (1/k!) ΣQ (-1)σS(Q') f[Q'(z)] . // (C.1.9) and (C.1.5a) (C.1.13) Apply permutation P to the above equation and use (A.2.8) to get P f^[z] = f^[P(z)] = (1/k!) ΣQ (-1)σS(Q') f [PQ'(z)] . (C.1.14) Then, RHS (C.1.10) = ΣP (-1)σS(P) f^[P(z)] F[P(Z)] = ΣP (-1)σS(P) { (1/k!) ΣQ (-1)σS(Q') f[PQ'(z)] } F[P(Z)] // (C.1.14) for f^[P(z)] = (1/k!) ΣQ ΣP (-1)σS(PQ') f[PQ'(z)]} F[P(Z)] // reorder and use (A.1.10) = (1/k!) ΣQ ΣP(-1)σS(PQ') f[PQ'(z)]} F[PQ'(Z)] // Q'(Z) = Z from (C.1.5b) = (1/k!) ΣQ ΣP(-1)σS(P) f[P(z)]} F[P(Z)] // rearrangement theorem (A.1.3) = ΣP(-1)σS(P) f[P(z)]} F[P(Z)] { (1/k!) ΣQ (1)} // reorder = ΣP(-1)σS(P) f[P(z)]} F[P(Z)] { 1 } // ΣQ (1) = k! from (C.1.4) = LHS (C.1.10) QED (C.1.15) Recall now definitions of the generic Alt and Sym operators, [Alt(f)](1,2...k) ≡ (1/k!) ΣP (-1)S(P)f( P(1),P(2)...P(k) ) (A.2.1) (C.1.16) [Sym(f)](1,2...k) ≡ (1/k!) ΣP f( P(1),P(2)...P(k) ) (A.3.1) (C.1.17) Using (C.1.16) and (C.1.17) , the second line of (C.1.10) can be restated f^ = Alt(f) σ = 1 totally antisymmetric fs = Sym(f) σ = 0 totally symmetric . (C.1.18) Recall next the definition of a tensor product in our generic function space, (fg)(1,2,....k+k') ≡ f(1,2...k) g(k+1,k+2....k+k') . (A.2.19) (C.1.19) Then we can write (fF)(1,2,....k+k') = f(1,2..k)F(k+1,k+2...k+k') = f(z) F(Z) (f^F)(1,2,....k+k') = f^(1,2..k)F(k+1,k+2...k+k') = f^(z) F(Z) . (C.1.20) Theorem One (with σ = 1) can then be stated in this manner, ΣP (-1)S(P)(fF)(P(1),P(2),...P(k+k')) = ΣP (-1)S(P)(f^F)(P(1),P(2),...P(k+k')) (C.1.10)σ=1 Add a factor 1/(k+k')! to both sides and use the Alt definition (C.1.15) with k→ k+k' to get, [Alt(fF)](1,2...k+k') = [Alt(f^F)](1,2...k+k') or Alt(fF) = Alt(f^F) f^ = Alt(f) . (C.1.21) Taking σ = 0 in (C.1.10) gives ΣP (fF)(P(1),P(2),...P(k+k')) = ΣP (f^F)(P(1),P(2),...P(k+k')) (C.1.10)σ=0 Use this with the Sym definition (C.1.16) with k→ k+k' to get Sym(fF) = Sym(fsF) fs = Sym(f) (C.1.22) C.2 Theorem Two This section is a copy, paste and edit version of Section C.1. Equations that are the same have italicized equation numbers. Consider the following set of k+k' integers, {1,2....k, k+1,k+2....k+k'} = {1,2....k+k' } (C.1.1) Partition this list into a low and high half by defining z ≡ 1,2....k Z = k+1,k+2....k+k' (C.1.2) Then {1,2....k+k'} = {z,Z} (C.1.3) Now let R be a permutation of the upper integers {k+1,k+2....k+k'} = Z. There are k'! possible permutations, so we know that ΣR (1) = k'! (C.2.4) We can extend the meaning of R so it applies to the entire set of integers {1,2....k+k'} merely by stating that this extended R' does not alter the lower integers. Then R'(Z) = R(Z) = Z'' = some permutation of the upper integers (C.2.5a) R'(z) = z // since R has no effect on the lower integers (C.2.5b) R'(z, Z) = {R'(z), R'(Z)} = {z, R(Z)} *** (C.2.5c) Now imagine we have a function f of the lower integers and a function F of the higher ones, f[z] = f[1,2....k] F[Z] = F[k+1,k+2....k+k'] (C.1.6) We use [] in place of () merely to improve clarity below. Think of the integers as generic labels on the functions. Here are two applications we shall consider later on, F[Z] = F [k+1,k+2....k+k'] = Sii...i = components of a rank-k' tensor F[Z] = F [k+1,k+2....k+k'] = S(vi,vi, .... vi) = a rank-k' tensor function (C.2.7) Now let P be a general permutation of {1,2....k+k'} = {z,Z}. P(z,Z) = {P(z), P(Z)} (C.1.8) Notice that RP and PR are undefined since P and R operate in different spaces, but R'P and PR' are both defined since both permutations R' and P operate in the space of {1,2....k+k'}. Recall now the meaning of S(R) as the number of swaps required to go from Z to R(Z) . This is the same as the number of swaps required to go from {z,Z} to R'{z,Z} = {z,R(Z)}. Therefore S(R) = S(R') (C.2.9) We shall now prove the following theorem : Theorem Two ΣP (-1)σS(P) f[P(z)] F[P(Z)] = ΣP (-1)σS(P) f[P(z)] F^[P(Z)] (C.2.10) where F^[Z] ≡ (1/k'!) ΣR (-1)σS(R) F[R(Z)] σ = 1 or 0 The purpose of σ is to state the theorem with and without the (-1)S(P) factor. We shall use the notation F^ in most of this section to apply for both values of σ, but at the end, we shall distinguish these two cases by writing: F^[z] ≡ (1/k!) ΣQ (-1)S(Q) F[Q(z)] // = Alt(F), see (A.2.1) Fs[z] ≡ (1/k!) ΣQ F[Q(z)] // = Sym(F), see (A.3.1) (C.1.11) At the end of this section we will show that the above Theorem Two with σ = 1 and σ = 0 is equivalent to the statements: Alt(fF) = Alt(fF^) F^ = Alt(F) σ = 1 Sym(fF) = Sym(fFs) Fs = Sym(F) σ = 0 (C.2.12) Proof of Theorem Two: Our first task is to process the second line of (C.2.10), F^[Z] = (1/k'!) ΣR (-1)σS(R) F[R(Z)] = (1/k'!) ΣR (-1)σS(R') F[R'(Z)] / (C.2.9) and (C.2.5a) (C.2.13) Apply permutation P to the above equation and use (A.2.8) to get P F^[Z] = F^[P(Z)] = (1/k'!) ΣR (-1)σS(R') F [PR'(Z)] (C.2.14) Then, RHS (C.2.10) = ΣP (-1)σS(P) f[P(z)] F^[P(Z)] = ΣP (-1)σS(P) f[P(z)]{(1/k'!) ΣR (-1)σS(R') F [PR'(Z)]} // (C.2.14) for F^[P(Z)] = (1/k'!)ΣR ΣP(-1)σS(PR') f[P(z)] F [PR'(Z)] // reorder and (A.1.10) = (1/k'!)ΣR ΣP(-1)σS(PR') f[PR'(z)] F [PR'(Z)] // R'(z) = z from (C.2.5b) = (1/k'!)ΣR ΣP(-1)σS(P) f[P(z)] F [P(Z)] // rearrangement theorem (A.1.3) = ΣP(-1)σS(P) f[P(z)] F [P(Z)] { (1/k'!) ΣR (1) } // reorder = ΣP(-1)σS(P) f[P(z)] F [P(Z)] {1 } // ΣR (1) = k'! from (C.2.4) = LHS (C.2.10) QED (C.2.15) Using (C.1.15) and (C.1.16) , the second line of (C.2.10) can be restated F^ = Alt(F) σ = 1 totally antisymmetric Fs = Sym(F) σ = 0 totally symmetric . (C.2.18) Following the same arguments used the end of Section C.1, one obtains the following equivalent restatement of Theorem Two (just move the subscript from f to F) Alt(fF) = Alt(fF^) F^ = Alt(F) (C.1.21) Sym(fF) = Sym(fFs) Fs = Sym(F) (C.1.22) Alternate Proof of Theorem 2 An alternate proof of Theorem Two is two start with Theorem One and just make these changes z ↔ Z f↔F k↔k' Q→R Here is Theorem One ΣP (-1)σS(P) f[P(z)] F[P(Z)] = ΣP (-1)σS(P) f^[P(z)] F[P(Z)] where f^[z] ≡ (1/k!) ΣQ (-1)σS(Q) f[Q(z)] σ = 1 or 0 (C.1.10) and here is Theorem One with the above changes applied, ΣP (-1)σS(P) F[P(Z)] f[P(z)] = ΣP (-1)σS(P) F^[P(Z)] f[P(z)] where F^[z] ≡ (1/k'!) ΣR (-1)σS(R) F[R(Z)] σ = 1 or 0 . (C.1.10)swap This is the same as Theorem Two which we quote from above, ΣP (-1)σS(P) f[P(z)] F[P(Z)] = ΣP (-1)σS(P) f[P(z)] F^[P(Z)] where F^[Z] ≡ (1/k'!) ΣR (-1)σS(R) F[R(Z)] σ = 1 or 0 (C.2.10) We went ahead with the detailed proof for two reasons. First, the swap proof might not be convincing to the reader. Second, the detailed proof provides steps which are crucial to proving Theorem Three below. C.3 Theorem Three Now both functions have a ^ subscript : ΣP (-1)σS(P) f[P(z)] F[P(Z)] = ΣP (-1)σS(P) f^[P(z)] F^[P(Z)] (C.3.1) where f^[z] ≡ (1/k!) ΣQ (-1)σS(Q) f[Q(z)] F^[Z] ≡ (1/k'!) ΣR (-1)σS(R) F[R(Z)] σ = 1 or 0 The purpose of σ is to state the theorem with and without the (-1)S(P) factor. At the end of this section we will show that the above Theorem Three with σ = 1 and σ = 0 is equivalent to the statements, Alt(fF) = Alt(f^F^) . f^ = Alt(f) F^ = Alt(F) (C.3.5) Sym(fF) = Sym(fsFs) . fs = Sym(f) Fs = Sym(F) (C.3.6) This theorem will involve both R and Q, as well as R' and Q' from earlier sections. Note that R'Q' = Q'R' (C.3.3) because Q' acts only on the lower integers in (1,2...k+k') while R' acts only on the upper integers. Proof of Theorem Three: Recall these results from previous sections, f^[P(z)] = (1/k!) ΣQ (-1)σS(Q') f [PQ'(z)] (C.1.13) F^[P(Z)] = (1/k'!) ΣR (-1)σS(R') F [PR'(Z)] (C.2.13) Then, RHS (C.3.1) = ΣP (-1)σS(P) f^[P(z)] F^[P(Z)] = ΣP (-1)σS(P){ (1/k!) ΣQ (-1)σS(Q') f [PQ'(z)]}{ (1/k'!) ΣR (-1)σS(R') F [PR'(Z)]} = (1/k!)(1/k'!)ΣQΣR ΣP (-1)σS(PR'Q') f [PQ'(z)] F [PR'(Z)] // reorder and (A.1.10) = (1/k!)(1/k'!)ΣQΣR ΣP (-1)σS(PR'Q') f [PQ'R'(z)] F [PR'Q'(Z)] // Q'(Z) = Z from (C.1.5b) // R'(z) = z from (C.2.5b) = (1/k!)(1/k'!)ΣQΣR ΣP (-1)σS(PR'Q') f [PR'Q'(z)] F [PR'Q'(Z)] // (C.3.3) R'Q' = Q'R' = (1/k!)(1/k'!)ΣQΣR ΣP (-1)σS(P[R'Q']) f [P[R'Q'](z)] F [P[R'Q'](Z)] = (1/k!)(1/k'!)ΣQΣR ΣP (-1)σS(P) f [P(z)] F [P(Z)] // rearrangement theorem (A.1.3) = ΣP (-1)σS(P) f [P(z)] F [P(Z)] { (1/k!)ΣQ(1) }{ (1/k'!)ΣR(1) } // reorder = ΣP (-1)σS(P) f [P(z)] F [P(Z)] { 1 }{ 1 } // (C.1.4) and (C.2.4) = LHS (C.3.1) QED (C.3.4) The endgame steps of Section C.1 are identical here with the change F→F^, giving Alt(fF) = Alt(f^F^) . f^ = Alt(f) F^ = Alt(F) (C.3.5) Sym(fF) = Sym(fsFs) . fs = Sym(f) Fs = Sym(F) (C.3.6) C.4 Summary and Generalization Summary of the Three Theorems Theorems One, Two and Three have shown that, in our generic function space, Alt[TS] = Alt[T^S] = Alt[TS^] = Alt[T^S^] where T^ = Alt(T) S^ = Alt(S) (C.4.1) Sym[TS] = Sym[TsS] = Sym[TSs] = Sym[TsSs] where Ts = Sym(T) Ss = Sym(S) . (C.4.2) One can of course rewrite these statements as Alt[TS] = Alt[Alt(T)S] = Alt[TAlt(S)] = Alt[Alt(T)Alt(S)] (C.4.3) Sym[TS] = Sym[Sym(T)S] = Sym[TSym(S)] = Sym[Sym(T)Sym(S)] . (C.4.4) Intuitively these equations are easily interpreted: If one is going to totally antisymmetrize a tensor product, the act of pre-antisymmetrizing one or more of the tensors makes no difference. So adding any ^ subscripts to objects inside an Alt makes no difference. If one is going to totally symmetrize a tensor product, the act of pre-symmetrizing one or more of the tensors makes no difference. So adding any s subscripts to objects inside an Alt makes no difference. Various "theorems" can be generated by "adding hats" to the insides of an Alt expression. Example: Consider. Alt[ABC] = Alt[(AB)C] = Alt[(AB)^C] = Alt[Alt(AB)C] Alt[ABC] = Alt[A(BC)] = Alt[A(BC)^] = Alt[AAlt(BC)] (C.4.5) Therefore Alt[Alt(AB)C] = Alt[ABC] = Alt[AAlt(BC)] (C.4.6) Replacing A,B,C with the obscure names ω,η,θ gives Alt[Alt(ω η) θ] = Alt[ω η θ] = Alt[ω Alt(η θ)] . (C.4.7) This may be compared with Spivak page 80 from which we quote, (C.4.8) Generalization of the three theorems The theorems derived above can be generalized in the following manner. Suppose for example we have a set of integers 1,2,3......(k1+k2+...+kN) = 1,2,3....κN. Instead of partitioning Z into 2 groups Z = (z,z') as done above, we partition the integers into N groups Z = (z1, z2....zN) as follows: κ1 = k1 // "cumulative ranks", as in (7.11.6) κ2 = k1+ k2 κ3 = k1+ k2 + k3 ... κN = k1 + k2 + ... + kN = Σi=1N ki . (7.11.6) Z = (z1, z2....zN) (C.4.9) z1 = 1,2,3...κ1 // the partitions z2 = κ1+1,κ2+2,...κ2 z2 = κ2+1,κ2+2,...κ3 ... zN = κN-1, κN-1 + 1, ...κN . And instead of functions f and F, we have functions f1, f2....fN. Whereas for N = 2 we had 22-1 = 3 theorems, for general N there will be 2N - 1 theorems. If we define L ≡ ΣP (-1)σS(P)f1[P(z1)]f2[P(z2)] ... fN[P(zN)] // Left side of theorems (C.4.10) then here are those theorems: ( exercise for the reader: use induction or brute force ) 1. L = ΣP (-1)σS(P)(f1)^[P(z1)]f2[P(z2)] ... fN[P(zN)] 2. L = ΣP (-1)σS(P)f1[P(z1)](f2)^[P(z2)] ... fN[P(zN)] 3. L = ΣP (-1)σS(P)(f1)^[P(z1)](f2)^[P(z2)] ... fN[P(zN)] 4. L = ΣP(-1)σS(P)f1[P(z1)]f2[P(z2)](f3)^[P(z3)] ...fN[P(zN)] ...... (2N-1). L = ΣP(-1)σS(P)(f1)^[P(z1)](f2)^[P(z2)](f3)^[P(z3)] ...(fN)^[P(zN)] (C.4.11) Translated to Alt/Sym, notation, we then find for the case N = 3 Alt[TSR] = Alt[T^SR] = Alt[TS^R] = Alt[TSR^] = Alt[T^S^R] = Alt[T^SR^] = Alt[TS^R^] = Alt[T^S^R^] (C.4.12) Sym[TSR] = Sym[TsSR] = Sym[TSsR] = Sym[TSRs] = Sym[TsSsR] = Sym[TsSRs] = Sym[TSsRs] = Sym[TsSsRs] (C.4.16) One can write these using X^ = Alt(X) and Xs = Sym(X) to obtain nested equations as we did earlier. In general one can write Alt[(T1)(T2) ...... (TN)] = Alt[(T1)a(T2)a ...... (TN)a] (C.4.17) where each ai can independently be a blank, (Ti), or can be a ^ , (Ti)^. This then is the ultimate statement that arbitrary pre-antisymmetrizing of one or more tensors in a totally antisymmetric product makes no difference. Similarly, Sym[(T1)(T2) ...... (TN)] = Sym[(T1)a(T2)a ...... (TN)a] (C.4.18) where each ai can independently be a blank, (Ti) , or can be an s, (Ti)s. This then is the ultimate statement that arbitrary pre-symmetrizing of one or more tensors in a totally symmetric product makes no difference. Application to Tensors and Tensor Functions All the work done above in Appendix C has been "generic", meaning the various operations are with respect to generic permutation functions like f(1,2...k). The work can be applied to tensors or tensor functions according to these simple translation rules f[1,2....k] = Tii...i = components of a rank-k tensor f[1,2....k] = T(vi,vi, .... vi) = a rank-k tensor function (C.1.7) (C.4.19) For example, consider our result (C.4.1) above that Alt(TS) = Alt(T^S) (C.4.1) (C.4.20) In the generic space this equation means [Alt(TS)](1,2...k+k')= [Alt(T^S)](1,2...k+k') . (C.4.21) Translated from the generic space to the tensor space, one gets [Alt(TS)]ii...i = [Alt(Alt(T)S)]ii...i (C.4.22) where for example [Alt(T)]ii...i = (1/k!) ΣP (-1)S(P) Tii...i (C.4.23) Translated from the generic space to the tensor function space, one gets instead, [Alt(TS)](v1, v2.....vk+k') = [Alt(Alt(T)S)](v1, v2.....vk+k') (C.4.24) where for example [Alt(T)](v1, v2....vk) = (1/k!) ΣP (-1)S(P) T(vP(1), vP(2)....vP(k)) (C.4.25) Here we follow our convention of putting dual-space tensor names into script/italic font.