Eigenvalues of a quantum field

Question

In the book 'Quantum field theory for the Gifted Amateur", the following is stated, cf. 9.3:

"A quantum field $\hat{\phi}(x)$ takes a position in spacetime and returns an operator whose eigenvalues can be a scalar, a vector (the $W^{\pm}$ and $Z^0$ particles are described by vector fields), a spinor (the object that describes a spin-1 particle such as an electron), or a tensor."

My question: is this statement correct, or should the word "eigenvalue" be replaced with "eigenvector"? If I naively think of a quantum field as being a function valued in some Hilbert space, then it seems to me eigenvalue of the operator $\hat{\phi}(x)$ should be a scalar quantity.

Answer 1

I believe the statement is correct, although one can argue that it expands the definition of "eigenvalue" beyond what one can be used to. The quantum field has more than one component in case of spinors, vectors, or tensors, so components of spinors etc. are eigenvalues of the components of the quantum fields. As for eigenvectors of quantum field, remember that they are vectors in the Fock space.