Lessons from $O(N)$ models in one dimension

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.