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...

We establish a new outer bound for the capacity region of product broadcast channels. This outer bound matches Marton’s inner bound for a variety of classes of product broadcast channels whose capacity regions were previously unknown. These classes include product of reversely semi-deterministic and product of reversely more-capable channels. A significant consequence of this new outer bound is...

We study estimation of (semi-)inner products between two nonparametric probability distributions, given IID samples from each distribution. These products include relatively well-studied classical L and Sobolev inner products, as well as those induced by translation-invariant reproducing kernels, for which we believe our results are the first. We first propose estimators for these quantities, a...

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...

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...

we define a new function-valued inner product on l2(g), called ?-bracket product, where g is a locally compact abelian group and ? is a topological isomorphism on g. we investigate the notion of ?-orthogonality, bessel's inequality and ?-orthonormal bases with respect to this inner product on l2(g).

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...