Are operators some kind of tensors?

Question

My question is may seem vague to some, but for me is a concept I am now learning for the first time (Tensors). I'm now reading from "APPLICATIONS OF CLASSICAL PHYSICS" by Roger D. Blandford and Kip S. Thorne. There in Ch.1 they introduce a the notion of a tensor being a "thing" that accepts vectors as inputs and gives out numbers or more specifically a real value.

So the detailed version of my question is, if someone can think of Tensors as operators acting on multiple vectors at once. Or are vectors a special kind of operators? I know that scalars and vectors can be thought as operators. But scalars and vectors can also be thought as tensors. Does the concept of tensors incorporate operators or the other way around? Or am I losing the point entirely

Answer 1

The term "operator" can have different meanings depending on the context. In general, an operator is any kind of mapping which maps elements of one space to elements of another (or the same) space.

In theoretical physics, however, the term "operator" is mostly used in the context of functional analysis: There, an operator is a map which takes functions as arguments. Typical examples are differentiation or integration operators.

A tensor is a multilinear map which takes vectors (and/or covectors) as arguments and maps them to scalars (such maps are called multilinear functionals). So it is an operator in the broader sense but not in the functional analysis sense.

However, in some areas of physics, there are objects which are both tensors and operators (in the functional analysis sense) at the same time. For example, the field strength tensor $F$ (with components $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ where $A$ is the vector potential) is obviously a tensor. In classical electrodynamics, that's about it. But in quantum electrodynamics $F$ becomes an operator as well, since the vector potential $A$ becomes a quantum field, that is, an operator which can act on quantum mechanical states.