App E g rewrite

This doc should be deleted in a few days after 4.11.12 Rewrite of Appendix E (g) (g) Operators and Matrices for Rank-2 tensors Operator concept. As discussed in Section 5 (i), x-space and x'-space of Picture A are both N-dimensional real Hilbert Spaces with scalar product indicated by the large dot , and one can regard V as a vector in either space. Expressed as a "vector" in x-space one can write, as done above with generic basis bi , V = Σi αi bi Moreover, one can regard a rank-2 tensor A as an "operator" in this Hilbert space, A = Σij αij bibjT Application of (bT)n on the left and bm on the right, and then a double use of (bT)nbi = bn bi = δni gives [A(b)]nm = (bT)n A bm Here, one regards A as an operator in the x Hilbert space, whereas [A(b)]nm is a "matrix" which is associated with the operator A in the particular bn basis. The idea of A as operator has an abstract meaning distinct from the matrix Aij. In the above equation the symbol A is this abstract operator and (bT)n A bm has a meaning distinct from our interpretation of it in terms of the matrix combination of three objects. In the matrix interpretation, one writes (bT)n A bm = [(bT)n]i Aij [bm]j = [bT]i Aij [bm]j and only then does A become a "matrix". This matrix happens to be the contravariant Aij matrix because we happened to select the un basis to write the components like [bm]j = uj bm. Bra-ket Notation. For the author of this document, the bra-ket notation commonly used in quantum mechanics (Paul Dirac 1939) provides a clean way to look at a rank-2 tensor A as an operator. It is true that in quantum mechanics one usually deals with infinite dimensional Hilbert spaces and complex numbers, but the formalism applies just as well to real Hilbert spaces with finite dimensions. In bra-ket notation one writes bi → |bi>, biT→ <bi| , so that the above equations become |V> = Σi αi |bi> <bj|bi> = δji orthogonality of the basis A = Σij αij | bi> <bj| 1 = Σi | bi><bi| completeness of the basis [A(b)]nm = <bn | A | bm > <U | V> = U V = scalar product In this notation, the N |bi> are a set of basis vectors which span an N-dimensional real Hilbert Space, while <bi| span the so-called adjoint (or transpose in our case) Hilbert Space. One then refers to αnm as the "matrix element of the operator A in the bi basis ". In general, |bi> and |bi> are different vectors. In this notation, based on what was presented earlier, one can write, Anm = <un | A | um > = the x-space components of tensor A (basis un) raise/lower with g A'nm = <en | A | em > = the x'-space components of tensor A (basis en) raise/lower with g' [A(b)]nm = <bn | A | bm > = the matrix of A in the bn basis raise/lower with w In the first of these three lines, one can raise and lower indices with gab and gab on both sides of the equation. On the second line this can be done with g'ab and g'ab. It was shown in Section 6 (b) that bn = wnmbk and conversely bn = wnmbk where wnm is the metric tensor g' one would get for some underlying transformation Fb which causes bn to be its tangent base vectors. Thus, on the third line above we can raise and lower indices on each side with wab and wab where wnm = bn bm . Notice in the last three equations that the operator A between the vertical bars is the exact same operator in each case. The matrices are different not because the operator has changed, but because the basis vectors are different. [A(b)]nm are the components of a rank-2 tensor in only two cases -- those shown in the first pair of equations above. In the first case Anm are components of a tensor in x-space, and in the second case the A'nm are components of a tensor in x'-space. In a consistent notation one might write Anm = [A(u)]nm and A'nm = [A(e)]nm . Bases are related by a transformation. Consider again, [A(b)]nm = <bn | A | bm > = (bn)T A bm = [bn]i Aij [bm]j = <bn|ui><ui|A|uj><uj|bm> We lower index m on both sides (using wij as noted above) and reverse the j tilt to get [A(b)]nm = <bn | A | bm > = (bn)T A bm = [bn]i Aij [bm]j = <bn|ui><ui|A|uj><uj|bm> One could then define the following tensor-like object, Bni ≡ [bn]i . The first index on B is raised and lowered by w, while the second is raised and lowered by g, so this object is a bit like R and S in its non-tensor nature. Lowering n and raising i then gives Bni = [bn]i = (BT)in where we use the notion of the transpose of a tilted matrix described in Section 7 (i) item 8. One then has [A(b)]nm = Bni Aij(BT)jm Since all the matrices are tilted the same way and summed indices are contractions, this is one of the "legal" Standard Notation matrix forms and we then write, A(b) = BABT or more precisely [A(b) = BA BT ]SN,dt where SN,dt means Standard Notation, down-tilt, as described in Section 7 (i) item 7. The matrix equation A(b) = BABT shows that the [A(b)]nm are related to the Aij by a "congruence transformation" with a matrix Bni = [bn]i whose rows are the basis vectors bm . When bm = um , matrix B is the identity matrix, and when bm = em one has Bni = [en]i = Rni, so that B = R in this case. It was shown in Section 7 (i) that in standard notation R is real orthogonal, so in fact one has for the bm = em basis, A(e) = BABT = R A RT = R A R-1 = R A S Specifically in this case, [A(e)]nm = RniAijSjm = RniRmjAij = A'nm .