Ch 6 rewrite

New Version of Chapter 6 This was installed at 8 PM Sat 1.12.16, do not edit here! 6. The Tensor Product of k dual vectors : the vector spaces V*k and T(V*) Every equation from Chapter 5 can be converted to an appropriate equation of Chapter 6 using this simple set of translation rules: 1. |X> → <X| and <Y| → |Y>. That is, reverse all Dirac bras and kets. 2. Swap lower and upper indices, indices. eg. ei → ei, Tij → Tij (really: reverse all tilts). 3. |vi> → <αi| // use Greek/script names for functionals; vi → αi 4. Vk → V*k // space goes to dual space 5. < T | v1,v2.....vk > = T( v1,v2.....vk) = a tensor function (a new item) (6.1) In general, translation of a Chapter 5 equation to Chapter 6 is most easily done if the Chapter 5 equation is first stated in Dirac notation. We could end Chapter 6 right here, allowing the reader to apply the above rules, but that seems unsportsmanlike, so we proceed with a partial mimicry of Chapter 5. 6.1 Pure elements, basis elements, and dimension of V*k A generic pure ("decomposable") element of V*k is this tensor product of k functionals, α1 α2 ..... αk . all αi ϵ V* (6.1.1) = <α1| <α2| ... <αk| = <α1,α2, ... αk| . // Dirac notation Since is associative by (2.8.22), one can install parentheses anywhere in (6.1.1) without altering the meaning of the object, for example, α1 (α2 α3) .... αk = α1 α2 a3 .... αk . The basis elements of V*k are λi λi ..... λi = <ei| <ei| ... <ei| = < ei, ei ...ei| . (6.1.2) The subscripts in (6.1.1) and the superscripts in (6.1.2) are labels, not components. The components of these two tensor objects are given by the (2.8.18) outer product form, (α1 α2 ..... αk)jj...j = (α1)j (α2)j .... (αk)j (6.1.3) (ei ei ..... ei)jj...j = (ei)j (ei)j .... (ei)j = δij δij .... δij . = (λi λi ..... λi)jj...j . (6.1.4) If n = dim(V), the total number of such basis elements is nk, so dim(V*k) = nk. (6.1.5) In the full set of dual tensor-product basis elements shown in (5.1.2), two or more of the λi might be the same. This will always be the case if k > n where n ≡ dim(V). For example, for k = 3 and n = 2 one such element would be λ1 λ1 λ2 ≠ 0. In Dirac notation, we can write (6.1.3) and (6.1.4) as <α1,α2, ...αk | ej, ej ...,ej > = <α1| ej><α2| ej>...<αk| ej> = (α1)j (α2)j .... (αk)j (6.1.3a) < ei, ei ...ei | ej, ej ...,ej > = < ei| ej> ... = δij δij .... δij = < λi, λi ...λi | ej, ej ...,ej > . // <eI|eJ> = δIJ (6.1.4a) 6.2 Tensor Expansion for a tensor in V*k ; the ordinary multiindex We apply our translation rules to (5.2.1a) through (5.2.8a) to get this dense translation of Section 5.2 <T| = Σii....i Tii....i < ei, ei .....,ei| . tensor functional (6.2.1) <T| = ΣI TI <eI| <eI| = < ei, ei .....,ei | (6.2.2) T = Σii....i Tii....i (λi λi ..... λi) . (6.2.3) T = ΣI TI λI λI ≡ λi λi ..... λi (6.2.4) T (eiei... ei) = <T| ei, ei .....,ei> = Tii...i = T(ei, ei .....,ei) (6.2.5) T eI = <T|eI> = TI = T(eI) (6.2.6) T (v1v2... vk) = <T| v, v .....,v> = T(v, v .....,v) tensor function (6.2.7) T vZ = <T|vZ> = T(vz) (6.2.8) 6.3 Rules for product of k vectors The tensor product of k vectors is "k-multilinear" meaning it is linear in each of its k factors. This was discussed in (1.1.16) and later in (3.1.4). For example, α1(α2 + α'2)α3.....αk = α1α2α3 .....αk + α1α'2α3 .....αk α1(sα2)α3 ..... αk = s(α1α2α3 .....αk) s = scalar (6.3.1) Here we show linearity in the 2nd factor. All the other factors have similar equations. We impose this k-multilinearity by fiat with the result that: Fact: The space V*k is a vector space. (6.3.2) Proof: Repeat the discussion of Section 5.3 with all vi→ αi, meaning |vi> → <αi| . 6.4 The Tensor Algebra T(V*) T(V*) ≡ V*0 V* V*2 V*3 ....... = Σk=1∞ V*k . (6.4.1) Here V*0 = the space of scalars, V*1 = V the space of dual vectors, V*2 = V**V = the space of rank-2 dual tensors, and so on (tensor = functional). The most general element t of the space T(V*) has the form τ = s ΣiTi λi Σij Tij λiλj Σijk Tijk λiλjλk + ...... s ϵ K (6.4.2) with all coefficients in a field K. Fact: This large space T(V*) is in fact itself a vector space. (6.4.3) The proof is the same as that shown in Section 5.4 with a,b,c,d,e,f replaced by Greek letters. For example k1 α βκ ρση = sum of 4 elements of T(V*) = an element of T(V*) k2 <α| <β,κ| <ρ,σ,η| (6.4.4) For later comparison with the corresponding dual wedge picture, here we have: Object lin comb is Rank(grade) Space s scalar ϵ K 0 V*0 α dual vector 1 V*1 αβ dual rank-2 tensor 2 V*2 αβγ dual rank-3 tensor 3 V*3 αβγδ dual rank-4 tensor 4 V*4 ..... αβγδ.... dual rank-k tensor k V*k ..... arbitrary element of T(V*) dual multivector mixed T(V*) (6.4.6) All objects listed in the left column are tensor functionals, but we just call them tensors above and below. Any linear combination of a set of tensor products of dual k-vectors is a dual rank-k tensor. More generally, a dual rank-k tensor has the form shown in (6.2.3). A dual multivector is any linear combination of dual rank-k tensors for any mixed values of k . The dimensionality of the space T(V*) is as follows, where n = dim(V*), dim[T(V*)] = 1 + n + n2 + n3 + ... = ∞ (6.4.7) 6.5 Comments about Tensor Functions For every rank-k tensor functional <T| = T in V*k there exists a corresponding tensor function: T(v, v .....,v) = <T| v1,v2...vk> // T(vZ) = <T|vZ> T(ei, ei .....,ei) = <T| ei, ei .....,ei> = Tii...i // (6.2.5) (6.5.1) There is a simple one-to-one relationship between the rank-k tensors |T> of Vk and the rank-k tensor functionals <T| of V*k and the rank-k tensor functions T(vZ) of V*k. These functions are manifestly k-multilinear since | v1,v2...vk> = | v1>| v2>...| vk> is k-multilinear. That is to say, each V space in the tensor product Vk = VV ...V is a linear (vector) space. Fact: The vector space V*k is equivalent to the vector space of k-multilinear functions on Vk. (6.5.2) This is the generalization of Fact (4.2.15) from k = 2 to k = k. The point made in Section 5.5 about tensors remaining tensors if their indices are shuffled around is reflected in the space of tensor functions: if T(v,v .....,v) is a rank-k tensor function, then so is the function T(vi, vi .....,vi) where the arguments are any permutation of v,v .....,v . 6.6 The Tensor Product of two or more tensors in T(V*) Were we to write out the full detailed development of Section 5.6, it would begin as follows : Consider two tensor functionals of rank k and k' expanded as in (6.2.3), T = Σii....i Tii....i λi λi ..... λi rank k, T ϵ V*k S = Σjj....j Sjj....j λj λj ..... λj rank k', S ϵ V*k' . (6.6.1) In Dirac notation these equations say T = ΣI TI λI or |T> = ΣI TI <eI | S = ΣI SI λI or |S> = ΣJ SJ <eJ | (6.6.2) and the tensor product of interest is TS = <T| <S| . (6.6.3) The entire development proceeds as shown in Section 5.6 but with the translation rules outlined at the start of Chapter 6, in particular, that all bra-kets are reversed. One then finds for the tensor product of a rank-k tensor functional with a rank-k' one, TS = ΣI,J TISJ λIλJ = ΣI (TS)I λI (TS)I = TISI' (6.6.4) I = {i1, i2, .. ik+k'} λI ≡ λi λi ..... λi I = {i1, i2, .. ik} I' = {ik+1, ik+2, .. ik+k'} . which compare to the non-dual (5.6.5) ( recall that eI = <eI| and λI = <eI| ) TS = ΣI (TS)I eI . (5.6.5) A triple tensor product is then TSR = ΣI,J,K TISJRK λIλJλK = ΣI (TSR)I λI (TSR)I = TISI'RI" (6.6.5) I = {i1, i2, .. ik+k'+k"} λI ≡ λi λi ..... λi I = {i1, i2, .. ik} I' = {ik+1, ik+2, .. ik+k'} I" = {ik+k'+1, ik+k'+2, .. ik+k'+k"} . For an arbitrary set of tensor functionals Ti of rank ki the tensor product is T1T2...TN = ΣI [(T1)I(T2)I .... (TN)I] λI = ΣI (T1T2....TN)I λI (6.6.6) Ti = tensor functional of rank ki , Ii = multiindex range of ir values for tensor Ti I1 = {i1, i2.....ik}, I2 = {ik+1, ik+2.....ik+k}, etc. I = I1 I2 .... IN = {i1, i2......ik+k+...k} λI ≡ λi λi ..... λi (6.6.7) where the resulting tensor has a rank equal to the sum of the ranks of the combined tensors. In Dirac notation, equation (6.6.6) is written, < T1,T2....TN| = < T1T2....TN | = <T1| <T2| ...<TN| = ΣI (T1T2....TN)I <eI| (6.6.8) which is just the Dirac transpose (plus tilt reversal) of the equation (5.6.13a) of Chapter 5. It is understood here that each bra space fits the rank of its tensor, and one could write <Ti| = k<Ti| to make this fact more explicit. Then (6.6.8) would read k<T1| k<T2 | ... k<TN| = ΣI (T1T2....TN)I κ<eI| κN = k1+k2+...+kN . (6.6.9) Consider now the following generic tensor product ket, |v1,v2...vk>k |vk+1,vk+2...vk+k>k .... = |VI>k |VI>k .... |VI>k = |VI> κ = | v1,v2,v3,v4 ................vκ> . (6.6.10) If we close the bra (6.6.9) with this ket, we obtain the simple rule for the tensor product of the corresponding tensor functions, < T1T2....TN | v1,v2 ...vκ> = [ k<T1| k<T2 | ... k<TN| ] [ |VI>k |VI>k .... |VI>k ] = k<T1|VI>k * k<T2|VI>k .... * k<TN|VI>k = T1(vI) T2(vI) ..... TN(vI) (6.6.11) or (T1T2....TN)(v1,v2 ..............vκ) = (T1T2....TN)(vI,vI ...vI) = T1(vI) T2(vI) ..... TN(vI) . (6.6.12) Example: For N = 2 and k1= k and k2 = k' : (TS)(v1,v2....vk, vk+1....vk+k') = T(v1,v2....vk) S(vk+1,vk+2....vk+k') . (6.6.13) Example: For N = 3 and k1= k and k2 = k' and k3 = k" : (TSR)(v1, v2, ... vk+k'+k") T(v1,v2....vk) S(vk+1,vk+2....vk+k')R(vk+k'+1,vk+k'+2....vk+k'+k") . (6.6.14) Here is a direct proof of (6.6.12), independent of Chapter 5, where we make use of the dense multiindex notation. Let, Ti = tensor of rank ki Ii = multiindex range of ir values for tensor Ti I1 = {i1, i2.....ik}, I2 = {ik+1, ik+2.....ik+k}, etc. (6.6.15) Then we have T1T2...TN = ΣII...I (T1)I(T2)I ...(TN)I λIλI...λI (6.6.16) (T1T2...TN)(vI, vI ... vI) = ΣII...I (T1)I(T2)I ...(TN)I (λIλI...λI)(vI, vI ... vI) = ΣII...I (T1)I(T2)I ...(TN)I (vI)I(vI)I ... (vI)I = [ΣI(T1)I(vI)I][ΣI(T2)I(vI)I] ... [ΣI(T2)I(vI)I] = T1(vI) T2(vI) .... TN(vI) . (6.6.17)