Hilbert space

We examine how best to associate quantum states of a single particle to modes of a narrowly collimated beam of classical radiation modeled in the paraxial approximation, both for scalar particles and for photons. Our analysis stresses the importance of the relationship between the inner product used to define orthogonal modes of the paraxial beam, on the one hand, and the inner product underlying the statistical interpretation of the quantum theory, on the other. While several candidates for such an association have been proposed in the literature, we argue that one of them is uniquely well suited to the task. Specifically, the mapping from beam modes to ”henochromatic” fields on spacetime is unique within a large class of similar mappings in that it is unitary in a mathematically precise sense. We also show that the single-particle quantum states associated to the orthogonal modes of a classical beam in the henochromatic approach are not only orthogonal, but also complete in the quantum Hilbert space.

The purpose of this thesis is to provide complete
proofs for several results on integral-valued polynomials,
which are used in Serre's proof of Hilbert's Theorem
found in the theory of characteristic polynomials. These
results, however elementary, are not found in the
literature.
The proof of Hilbert's Theorem is also given.

In this work, we discuss the conceptual framework of quantum mechanics in the Hilbert space formalism from a group representation point of view. After a brief review of the main results of the theory of groups and their representations, we describe mathematical models of the subject, and show the applications of this theory for getting numerical answers to problems in elementary particle physics.

This thesis discusses propagators and uses them to
construct a solution to the following initial value problem
under some special conditions: {du(t)/dt = A(t)u(t) u(s) = φ where A(t) is a linear operator in a Banach space X for
every t ε R, φ ε X, and s ε R.

We analyze some topics of the theory of integration in
infinite dimensional vector spaces. In the first five
chapters we deal mainly with I. E. Segal's approach to
integration in Hilbert spaces and prove the existence of
the canonical normal distribution in a Hilbert space;
we also give a simplified proof of Segal's version of the
Plancherel Theorem. In the last chapter we discuss a
theorem due to A. M. Gleason relevant to separable Hilbert
spaces. We deduce that the behavior of measures on finite
dimensional subspaces determines the behavior of the
measures in a separable Hilbert space.