Schrödinger quantum mechanics is formulated as an infinite-dimensional Hamiltonian system whose phase space carries an additional structure (uncertainty structure) to account for the probabilistic character of the theory. The algebra of observables is described as an algebra of smooth functions on the quantal phase space, with a product naturally induced by the geometrical structures living on that manifold. The possibility of generalizing Schrödinger mechanics along these lines is discussed.

Making reference to the formalism developed in Part I to formulate Schrödinger quantum mechanics, the properties of Kählerian functions in general, almost Kählerian manifolds, are studied.

We consider the incompressible Euler or Navier-Stokes (NS) equations on a torus T d , in the functional setting of the Sobolev spaces H n Σ0 (T d ) of divergence free, zero mean vector fields on T d , for n ∈ (d/2+1, +∞). We present a general theory of approximate solutions for the Euler/NS Cauchy problem; this allows to infer a lower bound T c on the time of existence of the exact solution u analyzing a posteriori any approximate solution u a , and also to construct a function R n such that u(t) − u a (t) n R n (t) for all t ∈ [0, T c ). Both T c and R n are determined solving suitable "control inequalities", depending on the error of u a ; the fully quantitative implementation of this scheme depends on some previous estimates of ours on the Euler/NS quadratic nonlinearity [15] [16]. To keep in touch with the existing literature on the subject, our results are compared with a setting for approximate Euler/NS solutions proposed in [3]. As a first application of the present framework, we consider the Galerkin approximate solutions of the Euler/NS Cauchy problem, with a specific initial datum considered in [2]: in this case our methods allow, amongst else, to prove global existence for the NS Cauchy problem when the viscosity is above an explicitly given bound.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.Printed in Singapore Local Zeta Regularization and the Scalar Casimir Effect Downloaded from www.worldscientific.com by 54.202.27.70 on 05/12/18. For personal use only. PrefaceZeta regularization is a method to treat the divergent quantities appearing in several areas of mathematics and physics; these are managed by introducing a complex parameter, with the role of a regulator, and defining their "renormalized versions" in terms of the analytic continuation with respect to the regulator. The standard textbook example of this procedure deals with the divergent series +∞ =1(1) which, in the zeta approach, is interpreted as the analytic continuation at s = −1 of the regularized series ζ(s) := +∞ =1 −s . The latter series only converges for s > 1 but the function s → ζ(s) defined in this way, the well-known Riemann zeta function, possesses a unique analytic extension to C \ {1} and ζ(−1) = −1/12; in this sense, the sum (1) "equals" −1/12. In a more pictorial language, one could say that −1/12 is the "renormalized" value of the series (1).The application of zeta regularization to the divergences of quantum field theory was first proposed by Dowker and Critchley [46], Hawking [84] and Wald [146] to renormalize local observables, especially the vacuum expectation value (VEV) of the stress-energy tensor. The ultimate purpose was the semiclassical treatment of quantum effects in general relativity, e.g. using the stress-energy VEV as a source term in Einstein's equations. Due to the attention to vacuum states, the zeta approach was connected from its very beginning to Casimir physics. However, the above mentioned pioneers focused their investigations on the conceptual validity of the method, rather than on its implementation for actual computations in specific configurations. For example, the elegant formula of Dowker-Critchley-Hawking, which gives the VEV of the stress-energy tensor as the functional derivative of the (renormalized) effective field action with respect to the space-time metric is not useful for actual computations in a given geometry, apart from very special cases. This is due to the fact that its use would require computing the effective action for all spacetime metrics close the one under consideration. For this reason, Birrel and Davies have described the application of this formula as "impossibly difficult" (see [18], page 190).Moreover, papers [46,84, 146] deal with a quantized field on the whole spacetime manifold and do not consider the possibility of confining the field to a given space ...