  The transition from a classical field theory to a quantum field theory is characterized by the occurrence of operator-valued quantum fields \(\hat{\phi}(\mathbf{x},t)\), and corresponding conjugate fields, for both of which certain canonical commutation relations hold. Thus there is an obvious formal analogy between classical and quantum fields: in both cases field values are attached to space-time points, where these values are specified by real numbers in the case of classical fields and operators in the case of quantum fields. That is, the mapping \(\mathbf{x} \mapsto \hat{\phi}(\mathbf{x},t)\) in QFT is analogous to the classical mapping \(\mathbf{x} \mapsto \phi(\mathbf{x},t)\).  Due to this formal analogy it appears to be beyond any doubt that QFT is a field theory. 