1. Sufficient conditions for uniqueness In the quantisation of constrained systems it can happen that one obtains a representation of an algebra of quantum operators on a vector space without a preferred inner product. Since an inner product is necessary for the probabilistic interpretation of quantum theory, some way needs to be found of introducing an appropriate inner product on this vector ...

A Hilbert Space is an inner product space that is complete. Let H be a Hilbert space and for f, g ∈ H, let 〈f, g〉 be the inner product and ‖f‖ = √ 〈f, f〉. (1.1) We will consider Hilbert spaces over C. Most arguments go through for Hilbert spaces over R, and the arguments are simpler. We assume that the reader is familiar with some of the basic facts about the inner product. Let us prove a few t...

The concepts of 2−inner products and 2−inner product spaces have been intensively studied by many authors in the last three decades. A systematic presentation of the recent results related to the theory of 2−inner product spaces as well as an extensive list of the related references can be found in the book [5]. We recall here the basic definitions and the elementary properties of 2−inner produ...

Throughout, we work in the Euclidean vector space V = R, the space of column vectors with n real entries. As inner product, we will only use the dot product v ·w = vw and corresponding Euclidean norm ‖v ‖ = √v · v . Two vectors v,w ∈ V are called orthogonal if their inner product vanishes: v ·w = 0. In the case of vectors in Euclidean space, orthogonality under the dot product means that they m...

In this article, we formalize in Mizar [5] the definition of embedding of lattice and its properties. We formally define an inner product on an embedded module. We also formalize properties of Gram matrix. We formally prove that an inverse of Gram matrix for a rational lattice exists. Lattice of Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorit...

It is shown that the support of an irreducible weight module over the SchrödingerVirasoro Lie algebra with an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite-dimensional. As a side-product, it is obtained that every simple weight module over the Schrödinger-Virasoro Lie algebra with a nontrivial finite-dime...

We study the covariant free bosonic string field theory and explore its locality (causality) properties. We find covariant string fields which are strictly local and covariant, but act on an unconstrained Hilbert space with an indefinite inner product. From these we also define observable fields which act on the physical Hilbert space with an definite inner product. These are shown to be approx...

Let G0 denote a compact semisimple Lie algebra and U a finite dimensional real G0 module. The vector space N0 = U ⊕ G0 admits a canonical 2-step nilpotent Lie algebra structure with [N0,N0] = G0 and an inner product 〈, 〉, unique up to scaling, for which the elements of G0 are skew symmetric derivations of N0. Let N0 denote the corresponding simply connected 2-step nilpotent Lie group with Lie a...