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 ON−1 and the ON−2 corrections are explicitly shown to vanish. Peculiarities of gauge theories in d=1 are discussed in the context of the CPN−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 ϕ4 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 ON−1 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.