  The unmodified Heisenberg-Pauli canonical formalism of quantum field theory
applied to a self-interacting scalar boson field is shown to make sense
mathematically in a framework of generalized functions adapted to nonlinear
operations.
  The free-field operators, their commutation relations, and the free-field
Hamiltonian operator are calculated by a straightforward transcription of the
usual formalism expressed in configuration space. This leads to the usual
results, which are essentially independent of the regularization, with the
exception of the zero-point energy which may be set to zero if a particular
regularization is chosen.
  The calculations for the self-interacting field are more difficult,
especially because of the well-known problems due to the unboundness of the
operators and their time-dependent domains. Nevertheless, a proper methodology
is developed and a differentiation on time-dependent domains is defined. The
Heisenberg equations and the interacting-field equation are shown to be
mathematically meaningful as operator-valued nonlinear generalized functions,
which therefore provide an alternative to the usual Bogoliubov-Wightman
interpretation of quantized fields as operator-valued distributions.
  The equation for the time-evolution operator is proved using two different
methods, but no attempt is made to calculate the scattering operator, and the
applications to perturbation theory are left to a subsequent report.
