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Quantum mechanics as an infinite‐dimensional Hamiltonian system with uncertainty structure: Part II
1.R. Cirelli, A. Manià, and L. Pizzocchero, J. Math. Phys. 31, 2891 (1990).
2.In Ref. 1, we were prevalently interested in the case because of applications to quantum mechanics where In the present framework, which is a bit more abstract, we shall also admit the possibility that ν is negative. The case in which both and product become pointwise product, is of a particular nature; for example, for points (d) and (e) in Proposition 2.1. are not equivalent to (a), (b), and (c).
3.The theory of connections in vector bundles is well known in the finite‐dimensional case. A suitable formulation is possible also in the Banachic case (or in the Hilbertian case, for Riemannian connections), in which all the results needed here can be generalized from finite dimensions.
4.S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Wiley, New York, 1969), Vols. I and II.
5.In order to avoid misunderstandings, we explain in which sense the identity (2.7) is the covariant derivative of the identity (2.6). Equation (2.6) tells us that where A and B are the sections of such that and for every one form α and every vector field Z. From it follows for every vector field Y; but, for every pair Z, Y of vector fields and every one‐form α, and are just the left and the right side of (2.7). Similar manipulations will be frequently used in this paper, without any further comments.
6.R. S. Hamilton, “The inverse function theorem of Nash and Moser,” Bull. Am. Math. Soc. 7, 65 (1982). The theory of infinite‐dimensional connections proposed in this reference is suitable for a more general framework than the one in which we work here (indeed, it is formulated in the context of Fréchet manifolds and Fréchet vector bundles). We quote this paper only to give an indication on how to define the curvature tensor.
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