Definition Of Metrics Research Articles

A Note on Hyperconvexity in Riemannian Manifolds

Let M denote a connected Riemannian manifold of class C3, with positive definite C2 metric. The curvature tensor then exists, and is continuous.By a classical theorem of J. H. C. Whitehead (1), every point x of M has the property that all sufficiently small spherical neighbourhoods V of x are convex; that is, (i) to every y,z ∈ V there is one and only one geodesic segment yz in M which is the shortest path joining them:f:([0, 1]) → M,f(0) = y, f(1) = z; and (ii) this segment yz lies entirely in V:f([0, 1]) V; (iii) if f is parametrized proportional to arc length, then f(t) is a C2 function of y, t, and z.Let V be a convex set in M; and let y1 y2, Z1, z2 ∈ V.

Schema einer quantenmechanik MIT indefiniter metrik

The problem is examined how to change the axioms of quantum theory in order to introduce an indefinite metric. There are several inequivalent ways to divide the total Hilbert space of indefinite metric into two parts, one of which is of positive definite, the other of negative definite metric (so-called “physical” and “ghost-states”). In view of the arbitrariness in the choice of such decompositions the question is raised whether the completeness of the quantal description only requires the knowledge of the state-vectors and the hermitian operators belonging to the observables, or whether it is necessary to have further knowledge in order to privilege one mode of decomposition of the space into a positive and a negative part. In the paper, the second suggestion is denied. Every commuting system of hermitian operators points to a definite decomposition in a natural way, provided it is “large enough”. We call “decomposing” every system of commuting operators which divides (in a certain sense) the total Hilbert space into a positive and a negative part. Not only does a decomcomposing system induce the decomposition, but there is associated with it a new positive definite metric of the total space. In this way a probabilistic interpretation is valid and the usual axiomatics is applicable. With respect to decomposing systems which induce the same decomposition, all results belonging to a quantum theory with positive definite metric are conserved. But there exists another case: the hermitian operator A may belong simultaneously to two decomposing systems ▪ and ▪ and the decomposition relative to ▪ may not agree with the decomposition relative to ▪. Then, when analysing a quantum state with the help of observables represented by operators belonging to ▪, we may find an expectation value A (1) different from the expectation value A (2) of A with respect to an experiment that determines (in principle) the ▪-observables.

Societies and Academies

Societies and Academies