properties of pullback operator REVIEWED

Some properties of the pullback operator PhL 3.28.16 This section has several items: // 5.17.16 (1) show how to prove Sjamaars four theorems listed below. First involves linearity of R. I proved theorem 2 in much more generality. (2) action of linear Dirac operators on tensor products and maybe wedge products as well. This stuff is now incorporated near the end of each of Chapters 5,6,7,8. (3) Perhaps my first proof of the "tensor function derivation of a pullback" Want to show that 1. φ*(aα+bβ) = a(φ*α) + b(φ*β) done 2. φ*(α ^ β) = (φ*α) ^ (φ*β) done and generalized 3. φ*(ψ*α) = (ψ o φ)* α 4. φ*(dα) = d(φ*α) done Maybe just show this for an arbitrary Dirac space operator P, not just for R = φ. And show in the regular space first, not the dual space P | aα+bβ > = obvious Not sure how to do dα in Dirac notation. Maybe <dα| φ* = d ( <α| φ* ) ??? Not sure where I should put these properties, maybe earlier than here in original notation? What do these look like in Dirac? 1. <aα+bβ | φ* = [ <aα| + <bβ| ] φ* = < aα | φ* + <bβ | φ* = a < α | φ* + b <β | φ* or φ*(aα+bβ) = a(φ*α) + b(φ*β) QED here α and β are k-forms in Λk The dual space Λk being a vector space is a linear space, QED. 2. [ < αn1 | ^ < β | ] φ* = use ALT and all that stuff and do more than 2 terms ! 3. (< α | ψ* )φ* = φ*(ψ*α) y = ψ(x ) x = φ(t) y = ψ(φ(t)) = [ψ o φ ](t) = Φ(t) Φ = ψ o φ Φ* = (ψ o φ)* Do in two steps t<α| t<α| φ* = x<β| x<β| ψ* = y<κ| Therefore y<κ | = [x<β | ] ψ* = [t<α | φ*]ψ* = t<α | φ* ψ* or do all at once t<α| Φ* = y<κ| Compare t<α| Φ* = t<α | φ* ψ* Therefore t<α| (ψ o φ)* = t<α | φ* ψ* so (ψ o φ)* = φ* ψ* I really should have a general rule that R [ |T1> ^ |T2>] = R |T1> ^ R|T2> for any vectors in Lk1 ^ Lk2 ????? ******************************** Question 1: How does a general Dirac operator act on a tensor or wedge product of two objects in tensor or wedge space? Here in Chapter 5 is the tensor product of tensors T1T2...TN = ΣI (T1T2....TN)I eI (5.6.13) where eI = ei ei ..... ei = ei ei ..... eκ and (T1T2....TN)I = T1IT2I .... TNI . | T1,T2....TN> = | T1T2....TN> = |T1>|T2>...|TN> = ΣI (T1T2....TN)I |eI> (5.6.13a) Now I would say that P | T1,T2....TN> = P | T1T2....TN> = P|T1>P|T2>...P|TN> = |PT1>|PT2>...|PTN> = ΣI (PT1PT2....PTN)I |eI> = (PT1)I(PT2)I .... (PTN)I |eI> where we have extended the meaning of P to the tensor products space. I don't talk much about this idea. What happens in the corresponding wedge world or the tensor function world? Wedge world is Chapter 7 so start there in non-dual, (T1)^^(T2)^^...^(TN)^ = ΣI (T1IT2I .... TNI) e^I = ΣI (T1T2....TN)I e^I (7.9.d.6) where e^I = ei^ ei .....^ ei = ei^ ei .....^ ei and (T1T2....TN)I = T1IT2I .... TNI . Each tensor has its own rank. Now how does this look in Dirac notation? I never say! |(T1)^> ^ |(T2)^> ^...^ |(TN)^> = ΣI (T1T2....TN)I |e^I> Here each object is a vector in Lki so the product is in LΣki. How is (T1)^^(T2)^^...^(TN)^ related to (T1)^(T2)^...^(TN)^ ?? I know that T1T2...TN = ΣI (T1T2....TN)I eI (T1)^^(T2)^^...^(TN)^ = ΣI (T1T2....TN)I e^I so they seem to have the same coefficient. I know that Fact: (ej ^ ej ^ .... ^ ej) = Alt(ej ej .... ej) e^J = Alt(eJ) (7.3.8) So it should be true that (T1)^^(T2)^^...^(TN)^ = ΣI (T1T2....TN)I e^I = ΣI (T1T2....TN)I AltI(eI) = ΣI QI AltI(eI) But then what do you do? At least the left side is written as a sum of regular tensors in Vk I think. In Dirac we then have |(T1)^> ^ |(T2)^> ^...^ |(TN)^> = ΣI (T1T2....TN)I AltI |eI> The resulting |eI> are in Lκ I think. I would say now that for some operator Q, Q [|(T1)^> ^ |(T2)^> ^...^ |(TN)^>] = ΣI (T1T2....TN)I AltI Q |eI> Q is a linear operator so Q |eI> = ΣJ QIJ |eJ> = linear combination of the basis vectors in Lκ. So then Q [|(T1)^> ^ |(T2)^> ^...^ |(TN)^>] = ΣI (T1T2....TN)I AltI Q |eI> = ΣI (T1T2....TN)I AltI [ΣJ QIJ |eJ>] Now I have these facts Fact: AltI [TJI] = AltJ [TJI] = (1/k!) det(TJI ) . (A.8.30) TJI = TP(J)P(I) where P = any permutation of the index subscripts (A.8.31) ΣI fI = ΣI fP(I) // multiindex notation (A.1.20) the first two valid only if TJI = Tji Tji ..... Tji . But I think that might just apply here at the higher italic index level, so then AltI [ΣJ QIJ |eJ>] = ΣJ AltI [ QIJ |eJ>] If you break this down into individual terms, it might look like this QIJ |eJ> = [ Qij |ej>] [....] ..... [....] = Tji Tji ..... Tji in form. Then I can apply (A.8.30) to get AltI [ΣJ QIJ |eJ>] = ΣJ AltI [ QIJ |eJ>] = ΣJ AltJ [ QIJ |eJ>] Then we have Q [|(T1)^> ^ |(T2)^> ^...^ |(TN)^>] ok = Q [ ΣI (T1T2....TN)I |e^I>] ok = Q [ ΣI (T1T2....TN)I AltI |eI>] ok = ΣI (T1T2....TN)I AltI Q |eI> ok = ΣI (T1T2....TN)I AltI |QeI> ok = ΣI (T1T2....TN)I| (QeI)^> ok Meanwhile [Q |(Tn1)^>] ^ [Q|(Tn2)^>] ^...^ [Q|(TnN)^>] = [ |(QTn1)^>] ^ [|(QTn2)^>] ^...^ [|(QTnN)^>] = ΣI [(QT1)(QT2)....(QTN)]I |e^I> On the other hand Q [|(T1)^> ^ |(T2)^> ^...^ |(TN)^>] = Q [ ΣI (T1T2....TN)I |e^I>] = [ ΣI (T1T2....TN)I Q|e^I>] = [ ΣI (T1T2....TN)I Q|e^I>] = ΣI (T1T2....TN)I AltI [ΣJ QIJ |eJ>] = ΣI (T1T2....TN)I ΣJ AltJ [ QIJ |eJ>] = ΣJ AltJ [ ΣI (T1T2....TN)I QIJ |eJ> ] = ΣJ AltJ [ ΣI (T1IT2I .... TNI) QIJ ......... |eJ> ] = ΣJ AltJ [ ΣI (QIJT1I) .... |eJ> ] = ΣJ AltJ [ ΣI (QT)JIT1I) .... |eJ> ] = ΣJ AltJ [ (QTT1)J ......... |eJ> ] = ΣJ AltJ [ (QTT1)J |eJ>......... ] = ΣJ [(QTT1)J |eJ>] ^ [.....] etc = [Q |(T1)^> ^ Q|(T2)^> ^...^ Q|(TN)^>] ??? Try a simple case just for vectors, Q [ v1 ^ v2 ] = Q [ v1 v2 - v2 v1]/2 = [ Qv1 Qv2 - Qv2 Qv1]/2 = (Qv1) ^ (Qv2) so there should be no transpose. More generally Q [ vi ^ .....] = Q [ vi ..... ] = AltI { Q [ vi ..... ] } = AltI { [ (Qvi) ..... ] } = (Qvi) ^ .... Try product of two general tensors. TS = ΣI (TS)I eI I ≡ I, I' = i1,i2...ik+k', eI ≡ (ei ei .... ei) . (5.6.5) T^^ S^ = ΣI (TS)I e^I I ≡ I, I' = i1,i2...ik+k', e^I ≡ (ei^ ei ....^ ei) . (7.9.a.5) T^^ S^ = Alt(TS) . // see Sec (g) below for this result in Spivak normalization (7.9.a.7) (T1)^^(T2)^^...^(TN)^ = Alt(T1T2...TN) . (7.9.d.10) Plan B. Start with this last result | (T1)^> ^ | (T2)^> ^...^ | (TN)^> = Alt( |T1>|T2>...|TN>) . (7.9.d.10) Now apply operator Q to both sides Q [| (T1)^> ^ | (T2)^> ^...^ | (TN)^> ] = Q [Alt( |T1>|T2>...|TN>) ] = Alt { Q [ |T1> |T2> ... |TN>) ] } = Alt { [ Q|T1> Q|T2> ... Q|TN>) ] } = Alt { [ Q|T1> Q|T2> ... Q|TN>) ] } Stop. Confusion in the main doc right here Alt(TS)J = AltJ(TS)J = ΣI (TS)IAltJ(eI)J // (5.6.5) and (A.5.10) that Alt is linear = ΣI (TS)IAltI(eI)J // use (A.8.26) since (eI)J is totally antisymmetric in I and J = ΣI (TS)I (e^I)J // (7.3.8) = (T^^ S^)J // (7.9.a.5) What is the meaning of: Alt(TS)J ??? I added more detail to A.5.3 which shows that [Alt(TS)]J = AltJ [(TS)J ] = they both mean exactly the same thing Therefore [Alt(TS)]J = AltJ [(TS)J] = ΣI (TS)IAltJ(eI)J // (5.6.5) and (A.5.10) that Alt is linear = ΣI (TS)IAltI(eI)J // use (A.8.26) since (eI)J is totally antisymmetric in I and J = ΣI (TS)I (e^I)J // (7.3.8) = (T^^ S^)J // (7.9.a.5) I suppose it is this Alt { (TS)J } = (1/κ!) ΣP (-1)P (TS)P(J) Now write (TS) = ΣI (TS)I eI (TS)J = ΣI (TS)I (eI)J Then can say Alt { (TS)J } = AltJ{ (TS)J } = AltJ { ΣI (TS)I (eI)J } = ΣI (TS)I AltJ (eI)J = ΣI (TS)I AltI (eI)J = ΣI (TS)I (e^I)J = [ ΣI (TS)I (e^I) ]J = [ (T^^ S^) ]J Then we symbolically abbreviate this result as saying Alt(TS) = T^^ S^ but what this really means is Alt { (TS)J } = (T^^ S^)J Is there something called { Alt(TS) }J ? See Gii...i = [Alt(F)]ii...i = (1/k!) ΣP (-1)S(P) Fii...i (A.5.3) G = Alt(F) . // definition of Alt acting on a tensor OK back up then. The statement T^^ S^ = Alt(TS) MEANS the following [T^^ S^]ii...i = [Alt(TS)]ii...i Gii...i = [Alt(F)]ii...i = (1/k!) ΣP (-1)S(P) Fii...i (A.5.3) G = Alt(F) . // definition of Alt acting on a tensor AltI[ Fii...i] = (1/k!) ΣP (-1)S(P) Fii...i So now I think this: AltI[ Fii...i] = [Alt(F)]ii...i = (1/k!) ΣP (-1)S(P) Fii...i ****** g(1,2...k) = [Alt(f)](1,2...k) ≡ (1/k!) ΣP (-1)S(P)f( P(1),P(2)...P(k) ) (A.2.1) Gii...i = [Alt(F)]ii...i = (1/k!) ΣP (-1)S(P) Fii...i (A.5.3) G = Alt(F) . // definition of Alt acting on a tensor I should maybe add Gii...i = [Alt(F)]ii...i = (1/k!) ΣP (-1)S(P) Fii...i = AltI [ Fii...i ] ********* I have updated (A.5.3) so do the above again ************ First write (7.9.d.10) in Dirac notation, | (T1)^> ^ | (T2)^> ^...^ | (TN)^> = Alt( |T1>|T2>...|TN>) where both sides are vectors in the tensor product space (Vk1 Vk2 .... VkN). Apply an operator Q to both sides: Q [| (T1)^> ^ | (T2)^> ^...^ | (TN)^> ] = Q [Alt( |T1>|T2>...|TN>) ] = Alt [ Q ( |T1> |T2> ... |TN>) ] = Alt [ Q|T1> Q|T2> ... Q|TN>] = Q|T1> ^ Q|T2> ... ^ Q|TN> So this then is the simple rule for applying an operator to an arbitrary wedge product of tensors !!!! This is a theorem I think is very important. Let P = QT. Then transposing we get [<(T1)^| ^ < (T2)^| ^...^ < (TN)^| ] P = <T1|P ^ <T2|P ^ ....^ <TN|P Application 1 Now let P = φ* and write (T1)^ = α1 being a differential form [<α1| ^ < α2| ^...^ < αN| ] φ* = <α1|φ* ^ <α2|φ* ^ ....^ <αN|φ* which then says φ*( α1^ α2 ^ ....^ αN) = φ*(α1) ^ φ*(α2)^ .... ^ φ*(αN) and this is one of "those theorems" generalized! Application 2 (b) Second path: The pullback equation in terms of tensor functions Recall (5.6.17) P [ |T1> |T2>... |TN>] = P |T1> P|T2>... P |TN> (5.6.17) which describes the effect of an operator P on a tensor product of N arbitrary tensors. If the tensors are just vectors and if the operator is R, we apply this to say R| v1,v2...vk> = R [ | v1> | v2> ..... | vk> ] = R | v1> R | v2> ..... R | vk> = = | Rv1> | Rv2> ..... | Rvk> = = | Rv1,Rv2...Rvk> Therefore <αx | φ*| v1,v2...vk> = <αx | R |v1,v2...vk> = <αx | Rv1,Rv2...Rvk> In function notation this is written [φ*(αx)] (v1,v2...vk) = αx(Rv1,Rv2...Rvk) ALL DONE! More: The right side of (10.9.5) takes more work : AltI [ ΣJ RIJ <tuJ | ] = ΣJ AltI [(RIJ) <tuJ | ] = ΣJ AltJ [(RIJ) <tuJ | ] = ΣJ [AltI(RIJ)] <tuJ | = ΣJ [(1/k!) ΣP(-1)P RP(I)J ] <tuJ | // (A.2.1) def of AltI = (1/k!) ΣP(-1)P ( ΣJ RP(I)J <tuJ | ) // reorder = [(1/k!) ΣP(-1)P ( ΣJ RP(I)P(J) ] <tuP(J) | ) // (A.1.20) that ΣJ fJ = ΣJ fP(J) = [(1/k!) ΣP(-1)P ΣJ RIJ ] <tuP(J) | // (A.8.31) that RP(I)P(J) = RIJ = ΣJ RIJ [(1/k!) ΣP(-1)P <tuP(J) | ] // reorder = ΣJ RIJ AltJ <tuJ | // (A.2.1) def of AltJ = ΣJ RIJ <tu^J | where <tu^I| ≡ <tuj | ^ <tuj | ... ^ <tuj| . (10.9.7) ***************8 [ <(T1)^| <(T2)^| ... <(TN)^| ] Q = <(T1)^|Q <(T2)^|Q ... <(TN)^|Q . (8.9.d.14) Using the notation change shown below (10.1.9) we can write this as [ <α1| <α2| ... <αN | ] Q = <α1|Q <α2|Q ... <αN|Q Continuation for Section (8.9.d): In Dirac notation one can write (8.9.d.10) as <(T1)^| ^ < (T2)^| ^...^ < (TN)^| = Alt( <T1| <T2| ... <TN| ) . where both sides of this equation are elements of the tensor product space V*k V*k ... V*k . The action of operator P on a dual tensor product space vector is defined in the obvious manner **** [ <T1| <T2| ... <TN| ] P = <T1|P <T2|P ... <TN|P In other words, the action of P on the larger space is defined in terms of its action on the spaces which make up the tensor product. This result holds as well for the wedge product of N dual tensors, Fact: [<(T1)^| ^ < (T2)^| ^...^ < (TN)^|] P = <(T1)^|P ^ < (T2)^|P ^...^ < (TN)^|P Proof: [<(T1)^| ^ < (T2)^| ^...^ < (TN)^|] P = [ Alt( <(T1)^| <(T2)^| ... <(TN)^| ) ] P = Alt [ ( <(T1)^| <(T2)^| ... <(TN)^| ) P ] = Alt [ ( <(T1)^|P <(T2)^|P ... <(TN)^|P ) ] = <(T1)^|P ^ <(T2)^|P ^ ...^ <(TN)^|P Continuation for Section (7.9.d): In Dirac notation one can write (7.9.d.10) as | (T1)^> ^ | (T2)^> ^ ... ^ | (T2)^> = Alt ( | (T1)^> | (T2)^> ... | (T2)^> ) where both sides of this equation are elements of the tensor product space Vk Vk ... Vk . The action of operator P on a tensor product space vector is defined in the obvious manner **** P [ | (T1)^> | (T2)^> ... | (T2)^> ] = P | (T1)^> P | (T2)^> ... P | (T2)^> In other words, the action of P on the larger space is defined in terms of its action on the spaces which make up the tensor product. This result holds as well for the wedge product of N tensors, Fact: P [ | (T1)^> ^ | (T2)^> ^ ... ^ | (T2)^> ] = P | (T1)^> ^ P | (T2)^> ^ ... ^ P | (T2)^> Proof: P [ | (T1)^> ^ | (T2)^> ^ ... ^ | (T2)^> ] = P [Alt ( | (T1)^> | (T2)^> ... | (T2)^> ) ] = Alt [P ( | (T1)^> | (T2)^> ... | (T2)^> ) ] = Alt [ ( P | (T1)^> P | (T2)^> ... P | (T2)^> ) ] = P | (T1)^> ^ P | (T2)^> ^ ... ^ P | (T2)^>