Ch 4 Dirac Sections

This is the Title PhL 1.11.15 Reviewed on 5.16.16, can discard, Dirac sections for Ch 4 Note that page numbering is turned on in this template and view is 125%, located in phil/roaming/microsoft/templates size about 219K. ei = |ei> e'i = |e'i> bases for V and W eie'j = |ei> |e' > basis for the tensor product space VW vw = |v> |w> a pure "vector" in the tensor product space VW (4.1.1a) T = |T> = Σij Tij |ei> |e'j> rank-2 tensor in VW (a b)ij = <ei| <ej| |a> |b> = <ei|a> <ej|b> = aibj outer product a b = <a | b> dot product |v> |w> = Σijviwj |ei> |e'j> expansion of pure vector on basis vectors (4.1.6a) T ≡ |T> = Σij Tij |ei> |e'j> expansion of a rank-2 tensor (4.1.7a) <a| <b| |c> |d> = <a|c><b|d> = Σiaici Σjbjdj = Σijaibjcidj dot product in V2 (4.1.12a) ********************** Section 4.3 (a) |a> ^ |b> = ( |a> |b> - |b> |a> )/2 (4.3.1a) |a> ^ |b> = - |b> ^ |a> (4.3.2a) |a> ^ |a> = 0 (4.3.3a) T^ = |T^> = Σij Tij |ei> ^ |ej> (4.3.5a) T^ab = < ea| <eb | |T^> = < ea| <eb | Σij Tij |ei> ^ |ej> = Σij Tij <ea| <eb | [ |ei> |ej> - |ej> |ei> ]/2 = Σij Tij [ <ea |ei> <eb |ej> - <ea |ej> <eb |ei> ]/2 = Σij Tij [ δaiδbj- δajδbi]/2 = (Tab - Tba)/2 = - T^ba (4.3.7a) Section 4.3 (b) T^ = |T^> = Σi<j Aij |ei> ^ |ej> (4.3.10a) Section 4.3 (c) |a> ^ |b> = Σij aibj |ei> ^ |ej> = Σi<j det |ei> ^ |ej> (4.3.12a) |a> ^ |b> ^ |c> = det(a,b,c) |e1> ^ |e2>^ |e3> n = 3 (4.3.15a) Section 4.3 (d) (a ^ b)rs = <er| <es | |a> ^ |b> = <er| <es | ( |a> |b> - |b> |a> )/2 = [ <er| <es | |a> |b> - <er| <es | |b> |a> ]/2 = [ <er |a><es |b> - <er |b><es |a> ]/2 = (arbs - asbr)/2 (4.3.27a) Section 4.3 (e) (ei^ej) (cd) = [(eic)(ejd) - (ejc)(eid)]/2 = [cidj - cjdi]/2 . <ei| ^ <ej| |c> |d> = (1/2)[<ei| <ej| - <ej| <ei| ] |c> |d> = (1/2)[ <ei| c><ej|d> - <ej| c><ei|d> ] = (1/2) [ cidj - cjdi] (4.3.33a) **************** Section 4.4 (a) <α| ^ <β| = (<α| <β| - <α| ^ <β| )/2 (4.4.1)D <α| ^ <β| = - <β| ^ <α| (4.4.2)D <α| ^ <α| = 0 (4.4.3)D T^ = <T^| = Σij Tij <ei| ^ <ej| // λi = <ei| (4.4.5)D T^ = <T^| = Σijαiβj <ei| ^ <ej| = ( Σiαi<ei| )( Σjβi<ej| ) = <α| ^ <β| (4.4.6)D Section 4.4 (b) T^ = <T^| = Σi<j Aij <ei| ^ <ej| (4.4.10)D Section 4.4 (c) α ^ β = <α| ^ <β| = Σi<j det <ei| ^ <ej| (4.4.12)D Section 4.4 (d) (λi^ λj)(vr,vs) = <ei| ^ <ej| | vr> | vs> = (1/2) (<ei| <ej| - <ej| <ei| ) | vr> | vs> = (1/2) ( <ei|vr><ej| vs> - <ei|vs><ej| vr>) = (1/2) [ (vr)i(vs)j - (vr)j(vs)i] (4.4.20b)D The vector spaces Λ2 and Λ2f Looking at (4.4.20), (4.4.23) and (4.4.29), one sees that (λi^ λj)(vr,vs), (α ^ β)(vr,vs) and T^(vr,vs) are all antisymmetric bilinear functions of the two arguments vr, vs ϵ V . In Section 4.3 we declare that the rank-2 tensor T^ = |T^> is antisymmetric (alternating) if T^ab = - T^ba . In similar fashion, we declare that the rank-2 tensor functional T^ = <T^| is antisymmetric (alternating) if T^(v,v') = - T^(v',v). That is to say, saying that the functional is alternating means that the corresponding tensor function is alternating. Section A.10 writes <T^| = <Alt T | in an attempt to declare the asymmetry of <T^| without reference to the corresponding tensor function, "it is just notation". We can regard both T^ = <T^| and T^(v,v') = <T^|v,v'> as representations of the same antisymmetric rank-2 tensor functional <T^| in V* ^ V* = Λ2. The object T^ is an asymmetric bilinear rank-2 tensor functional, whereas T^(v,v') is an asymmetric bilinear rank-2 tensor function (a Spivak 2-tensor). There is a 1-to-1 correspondence between T^ and T^(v,v') . We shall say T^ ϵ Λ2 while T^(v,v') ϵ Λ2f (f = function), and the two spaces are isomorphic. Fact: The vector space Λ2 is equivalent to the vector space Λ2f of asymmetric bilinear functions on V2. (4.4.34) This may be compared to our earlier statement for the larger space V*2 = V*V* , Fact: The vector space V*2 is equivalent to the vector space V*2f of bilinear functions on V2. (4.2.15)