  Various topics related to the $O(N)$ model in one spacetime dimension (i.e.
ordinary quantum mechanics) are considered. The focus is on a pedagogical
presentation of quantum field theory methods in a simpler context where many
exact results are available, but certain subtleties are discussed which may be
of interest to active researchers in higher dimensional field theories as well.
  Large $N$ methods are introduced in the context of the zero-dimensional path
integral and the connection to Stirling's series is shown. The entire spectrum
of the $O(N)$ model, which includes the familiar $l(l+1)$ eigenvalues of the
quantum rotor as a special case, is found both diagrammatically through large
$N$ methods and by using Ward identities. The large $N$ methods are already
exact at subleading order and the $\mathcal{O}\!\left(N^{-2}\right)$
corrections are explicitly shown to vanish. Peculiarities of gauge theories in
$d=1$ are discussed in the context of the $CP^{N-1}$ sigma model, and the
spectrum of a more general squashed sphere sigma model is found. The precise
connection between the $O(N)$ model and the linear sigma model with a $\phi^4$
interaction is discussed. A valid form of the self-consistent screening
approximation (SCSA) applicable to $O(N)$ models with a hard constraint is
presented. The point is made that at least in $d=1$ the SCSA may do worse than
simply truncating the large $N$ expansion to subleading order even for small
$N$. In both the supersymmetric and non-supersymmetric versions of the $O(N)$
model, naive equations of motion relating vacuum expectation values are shown
to be corrected by regularization-dependent finite corrections arising from
contact terms associated to the equation of constraint.
