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Heisenberg's relations in discrete N-dimensional parameterized metric vector spaces

Abstract:

This paper shows that Heisenberg's relations are deducible from the structure of parameterized metric vector spaces of arbitrary dimension, by means of some new ideas, not entirely found in the current and vast literature about this subject. In order to allow this task to be done, some new concepts are put forward, as the real inward product of two vectors, described within vector spaces defined over the complex field, which permit further construction of real scalar products and Euclidian norms. Also, the definition of triads in parameterized metric vector spaces, which are constructs formed by three linearly independent vectors: a parameterized vector, the triad generator, and a pair of vectors orthogonal to the generator, the triad companions, one is simply made of the generator multiplied by the parameter and the other by using the first derivative with respect the parameter over the generator. Triads appear as forming a crucial building block for deduction of Heisenberg's relations. Building Heisenberg's relations is based on setting up the Gram matrix of a vector pair, which provides a new definition of the cosine of the angle subtended by two vectors in complex metric spaces, as well as an ancillary redefinition of the Schwarz inequality. It must be noted the fact that covariance of conjugated pairs of quantum mechanical variables appears to be a constant in the present scheme and others, a property which seems to become equivalent to the Heisenberg's relations, where in the current descriptions only variances or uncertainties are involved.