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##.(adsbygoogle = window.adsbygoogle || []).push({});

(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##?

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I “Vector” operators

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads - “Vector” operators | Date |
---|---|

I Operators and vectors in infinite dimensional vector spaces | Mar 2, 2018 |

I Inner products and adjoint operators | Apr 25, 2016 |

How Fourier components of vector potential becomes operators | Sep 30, 2015 |

Vector Operator | May 24, 2015 |

Linear operator and linear vector space? | Dec 23, 2014 |

**Physics Forums - The Fusion of Science and Community**