# What is the value of a quantum field?

As far as I'm aware (please correct me if I'm wrong) quantum fields are simply operators, constructed from a linear combination of creation and annihilation operators, which are defined at every point in space. These then act on the vacuum or existing states. We have one of these fields for every type of particle we know. E.g. we have one electron field for all electrons. So what does it mean to say that a quantum field is real or complex valued. What do we have that takes a real or complex value? Is it the operator itself or the eigenvalue given back after it acts on a state? Similarly when we have fermion fields that are Grassmann valued what is it that we get that takes the form of a Grassmann number?

The original reason I considered this is that I read boson fields take real or complex values whilst fermionic fields take Grassmann variables as their values. But I was confused by what these values actually tell us.

When physicists say that a quantum field $\phi(x)$ is real-valued, they are usually referring to Feynman's path integral formulation of quantum field theory, which is equivalent to Schwinger's operator formulation.
The values of a field $\phi(x)$ in the path integral formulations are numbers. E.g.:
• If the numbers are real, we say that the field $\phi(x)$ is real-valued. (Such a field $\phi(x)$ typically corresponds to a Hermitian field operator $\hat{\phi}(x)$ in the operator formalism.)
• If the numbers are complex, we say that the field $\phi(x)$ is complex-valued.
• If the numbers are Grassmann-odd, we say that the field $\phi(x)$ is Grassmann-odd. (The numbers in this case are so-called supernumbers. See also this Phys.SE post.)