Lessons from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e1176" altimg="si1453.svg"><mml:mrow><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> 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 field theory methods simpler context where many exact results available, but certain subtleties discussed which may be interest active researchers higher dimensional theories as well. Large N introduced zero-dimensional path integral and connection Stirling’s series shown. entire spectrum model, includes familiar l(l+1) eigenvalues rotor special case, found both diagrammatically through large by using Ward identities. already at ON−1 ON−2 corrections explicitly shown vanish. Peculiarities gauge d=1 CPN−1 sigma more general squashed sphere found. precise between linear with ϕ4 interaction discussed. A valid form self-consistent screening approximation (SCSA) applicable models hard constraint presented. point made that least SCSA do worse than simply truncating expansion even for small N. In supersymmetric non-supersymmetric versions naive equations motion relating vacuum expectation values corrected regularization-dependent finite arising from contact terms associated equation constraint.