vector operator note removed

I am removing my Note 2 on vector operators. Here was the text. Right now I cannot support this stuff because I cannot think of an operator off hand which is a function of x like the (x) below. And this seems to conflict with the notion of kets |x>, the whole thing is too tangential anyway, so out it goes at least until some rainy day when I can make it go. 11.6.11/ Note 2: If V(x) is a contravariant vector field operator in a quantum mechanical (QM) Hilbert Space, and if F=R is a linear transformation, the transformation rule shown above still applies, but there is extra information to consider: '(x') = R (x) = -1(x) x' = R(x) x Here operators in the QM Hilbert Space are indicated by a hat. In this case, the vector field operator (x) is being transformed "forwards" into operator '(x'). The QM Hilbert Space has vectors called kets written |α> and one rotates one of these states foward by doing |α'> = |α>. The "diagonal matrix element" of a vector operator results in a normal vector. Consider these steps: V'(x') = <α' | (x) | α'> = <α | -1 | α> = <α | (-1 ) | α> = <α | ' | α> = <α | R (x) | α> = R <α | (x) | α> = R V This shows that, given the vector operator transformation rule claimed above, then V' = R V R is our same matrix R = F, whereas acts on QM Hilbert Space basis vectors such as <α|-1 = <α'|. Consider an arbitrary Hilbert Space "matrix element" of the above operator equation, <α | '(x') |β> = R <α | (x) |β> = <α |-1(x) |β> = <α' |(x) |β'> Comparing the left end to the right end one sees that the action of transforming an operator forwards is equivalent to the action of transforming the basis vectors backwards. This is analogous to the fact that rotating a vector in 3D space backward and staying in the same frame of reference gives the same vector components as keeping with the original vector in space and rotating the frame of reference forwards. Notice that there are two Hilbert Spaces floating around here: one is RN with vectors x, the other is the QM space with vectors |β>. Another subtle point is that the operator is unitary with respect to the properly normalized infinite-dimensional QM Hilbert Space, meaning -1 = † , whereas the NxN matrix R may be non-unitary with respect to RN. For example, in special relativity a boost matrix has the property R† = RT = R ≠ R-1, as shown in Section 5 (m).