FREE QUANTUM FIELD THEORY is the theory of quantum fields in which the effect of interactions is entirely neglected. But we had better start by saying what we mean by a field theory. A field theory presupposes a continuum of space-time points. This involves a number of technical considerations that do not concern us here. (The continuum constitutes a differentiable manifold, which can be described by coordinates and to which an affine connection and a metric can be assigned.) The idea of a field enters as the idea that values of physical quantities can be attributed to the space-time points. Specification of the values of all relevant quantities to each space-time point specifies a configuration of the field. And we have a field theory insofar as we have field equations, that is, laws, usually in the form of differential equations, constraining the values of quantities at different space-time points.
I want to focus on the idea that values of various quantities can be attributed to the space-time points. In any specific case we talk about such attribution in much the way we talk about predication of simple properties: One attributes a humdrum property (for example, the color red) to an object (for example, my shirt) by associating predicate
(the word, 'red') with a referring expression ('my shirt'). Similarly, in describing the value of a field quantity (for example, a gravitational potential) at a point, one associates a mathematical entity (for example, a real number) representing the value of the quantity with numerical coordinates representing the point.
What kinds of quantities can be attributed to the space-time points?
We know familiar examples in the form of scalars (a matter density, gravitational potential), vectors (electric and magnetic fields), and higher-order tensors (the stress-energy tensor). When I ask what one means by "the quantum field" I am often told that it is just the asso- ciation of an operator with each of the space-time points. Just as one may characterize a field by associating scalar, a vector, or a higher- order tensor with space-time points, there is no reason why one cannot
further generalize and similarly associate other mathematical entities with the points, for instance, operators (or more generally tensors with operator-valued components).
This specifies a perfectly clear formal sense to the notion of the quan- tum field. I will further examine what this involves in the next chapter, but for now I propose simply to plunge ahead and lay out the well-known connection between the Fock space description of the last chapter and the quantum field in the formal sense just described. I want to do this in order to lay out for the reader the conception of the quantum field as it is generally understood by practitioners and also in order to make clear the connection between this conception and the less orthodox order of presentation given in the last chapter.