Teuton wrote:Teuton wrote:

What is the essential physical difference between "classical" fields and quantum fields?

"In very loose terms, the operator valuedness of quantum fields means that to each space-time point (x,t) a field value φ(x,t) is assigned which is an operator. This is the fundamental difference to classical fields because an operator valued quantum field φ(x,t) does not by itself correspond to definite values of a physical quantity like the strength of the electromagnetic field. On this background, Teller has argued in Teller 1995 that the field interpretation of QFT is inappropriate since the alleged fields in QFT are not to be interpreted as physical fields with definite values of some sort which are assigned to space-time points, like in the case of the classical electromagnetic field. Rather, quantum fields are what Teller calls ‘determinables’ (p. 95), as it becomes manifest by the fact that quantum fields are described by mappings from space-time points to operators. Operators are mathematical entities which are defined by how they act on something. They do not represent definite values of quantities but they specify what can be measured, therefore Teller's expression ‘determinables’. (Below it will be discussed why this talk in terms of a field at a point has to be refined using the notion of a smeared field φ(f).)"

(http://plato.stanford.edu/entries/quantum-field-theory/)

"[Operators] do not represent definite values of quantities but they specify what can be measured[.]"

Does this mean that operators represent the range of all possible determinate properties of a determinable quantity that can be possessed by a spacetime point? But how could an actual physical field be a collection of merely possible properties? Is it possible at all for there to be indeterminate physical properties in physical reality? For example, is it possible to have a mass without having a determinate mass?

I don't really speak philosopher language, but maybe the missing ingredient is the state. The field operator φ(x) at the point x can, if you're a philosopher, be thought as a determinable. The specific values of the determinable could be the allowed values <s|φ(x)|s> for each state s, or possibly the set of all expectation values you could construct from the field operator.