Surveys in Differential Geometry

Volume 20 (2015)

Euclidean-signature semi-classical methods for quantum cosmology

Pages: 277 – 319



Vincent Moncrief (Department of Physics, Yale University, New Haven, Connecticut, U.S.A.)


We show how certain microlocal analysis methods, already well-developed for the study of conventional Schrödinger eigenvalue problems, can be extended to apply to the (mini-superspace) Wheeler–DeWitt equation for the quantized Bianchi type IX (or ‘Mixmaster’) cosmological model. We use the methods to construct smooth, globally defined expansions, for both ‘ground’ and ‘excited state’ wave functions, on the Mixmaster mini-superspace. We then review an expansive, ongoing program to further broaden the scope of such microlocal methods to encompass a class of interacting, bosonic quantum field theories and conclude with a discussion of the feasibility of applying this ‘Euclidean-signature semi-classical’ quantization program to the Einstein equations themselves—in the general, non-symmetric case—by exploiting certain established geometric results such as the positive action theorem.


cosmological solutions, quantized cosmological model, mixmaster, Einstein equations, microlocal analysis

2010 Mathematics Subject Classification

35A01, 83C05, 83C45, 83F05

Published 7 July 2015