Appendix D rewrite

The use of abs value |J| in tensor density PhL 2.25.12 This is a full Appendix D rewrite! Opening comments: Weinberg does not use the absolute value in his book. I have run into a "σ problem" when I use absolute values all weights I have seen so far are integers. the only place I ever make use of tensor densities is my Section 8 which is now under rewrite! So what would my Appendix D look like if I dispensed with the absolute value in the definition of a tensor density? Let's copy and paste it right here, then edit and see what happens. This is a very LONG 22 page appendix, so this will take a while to do. I will convert each section to blue after it is processed. Log of editing. (1) First trouble point is that my Weight Changing Theorem fails. Where do I use this theorem? One place is in section (h) where I am trying to show that C = curl B transforms as a regular vector. Maybe this conclusion is wrong if parity is included! Basically I am now trying to correctly incorporate parity into this appendix. I use the weight changing theorem 3 more times toward the end of App D. So, when parity is included, is the definition of a tensor density as in Weinberg and me correct? T' abcde = J-W Raa' Rbb' Rcc' Rdd' Ree' Ta'b'c'd'e' Or should there be a factor σW present, in which case I am back to the old appendix D! Digression: I am suddenly suspicious of the σ factor which I put in this equation (An)i = σ (-1)n-1 εiabc..x (e1)a(e2)b.... (eN)x // en missing For example, if N = 3 we would have A3 = σ e1 x e2 σ = sign of det(S) Why should the direction of this N-piped face area be a function of S which after all is the transformation from x space to x'-space, whereas here I am just doing everything in x-space which does not know about S. Ok, we have to now push down another level on "the stack". Go resolve that in a different doc, then when that is stabilized, return here and continue. Resolution: (see "should sigma be in the dA form.. ") Yes, x-space stays the same, but under parity the N-piped spanned by the en inverts itself, so that all outfacing faces have a change in vector direction. The front of a cube becomes the back. If we want the vector definition to be outfacing, the sign has to change. Since the sign of e1 x e2 does not change when each of these en changes sign, the σ brings in that required sign change. I am happy again. Comment: I used to have the full edited Appendix D sitting right here, all converted to blue as I edited it and fixed things up. The main edit was getting rid of the Weight Changing theorem which, though valid, was not necessary since it is just a special case of the usual weight addition rule. Another major change was to repair the C = curl B result, and I showed that this is a vector of weight -1, a change in my thinking on this subject. I repaired the curl derivation Section 12 and all went well there. Another major change was going from |J| to J in all the tensor density stuff! I did a full spelling check right here before my cut and paste. All done, we have a brand new Appendix D installed.