- #1
David Olivier
I have always seen “vector” operators, such as the position operator ##\vec R##, defined as a triplet of three “coordinate” operators; e.g. ##\vec R = (X, Y, Z)##. Each of the latter being a bona fide operator, i.e. a self-adjoint linear mapping on the Hilbert space of states ##\mathcal H##.
(I am putting aside here completely all the subtleties discussed in a recent thread.)
I'd like to be able to express the object ##\vec R## without a reference to a particular coordinate system.
What could such an operator be? A linear mapping ##\mathcal H \to \mathcal E \otimes \mathcal H##, with ##\mathcal E## the vector space of the three-dimensional “physical” space? This would allow us to write ##\vec R | \vec r > = \vec r \otimes | \vec r >##
However, ##\mathcal E## is normally a real vector space, while ##\mathcal H## is a complex one. Perhaps it will do if we first convert ##\mathcal E## into a complex vector space?
Operators are to be self-adjoint. How do we express this condition for such a vector operator?
What about “pseudovectors” such as the angular momentum operator? Would we view them as antisymmetric elements of ##\mathcal E \otimes \mathcal E##?
(I am putting aside here completely all the subtleties discussed in a recent thread.)
I'd like to be able to express the object ##\vec R## without a reference to a particular coordinate system.
What could such an operator be? A linear mapping ##\mathcal H \to \mathcal E \otimes \mathcal H##, with ##\mathcal E## the vector space of the three-dimensional “physical” space? This would allow us to write ##\vec R | \vec r > = \vec r \otimes | \vec r >##
However, ##\mathcal E## is normally a real vector space, while ##\mathcal H## is a complex one. Perhaps it will do if we first convert ##\mathcal E## into a complex vector space?
Operators are to be self-adjoint. How do we express this condition for such a vector operator?
What about “pseudovectors” such as the angular momentum operator? Would we view them as antisymmetric elements of ##\mathcal E \otimes \mathcal E##?