  In this work, we use tools from non-standard analysis to introduce
infinite-dimensional quantum systems and quantum fields within the framework of
Categorical Quantum Mechanics. We define a dagger compact category *Hilb
suitable for the algebraic manipulation of unbounded operators, Dirac deltas
and plane-waves. We cover in detail the construction of quantum systems for
particles in boxes with periodic boundary conditions, particles on cubic
lattices, and particles in real space. Not quite satisfied with this, we show
how certain non-separable Hilbert spaces can also be modelled in our
non-standard framework, and we explicitly treat the cases of quantum fields on
cubic lattices and quantum fields in real space.
