  Finite-size critical systems defined on a parallel plate geometry of finite
extent along one single ($z$) direction with Dirichlet and Neumann boundary
conditions at $z=0,L$ are analyzed in momentum space. We introduce a modified
representation for the discrete eigenfunctions in a renormalized one-particle
irreducible vertex part ($1PI$) scalar field-theoretic framework using either
massless or massive fields. The appearance of multiplicities in the Feynman
rules to construct diagrams due to this choice of representation of the basis
functions is discussed along with the modified normalization conditions. For
nonvanishing external quasi-momenta, Dirichlet and Neumann boundary conditions
are shown to be unified within a single formalism. We examine the dimensional
crossover regimes for these and show a correspondence with those from
antiperiodic and periodic boundary conditions. It is demonstrated that
finite-size effects for Dirichlet and Neumann boundary conditions do not
require surface fields necessarily but are implemented nontrivially from the
Feynman rules involving only bulk terms in the Lagrangian. As an application,
the critical exponents $\eta$ and $\nu$ are evaluated at least up to two-loop
level through diagrammatic means. We show that the critical indices are the
same as those from the bulk (infinite) system irrespective of the boundary
conditions.
