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Register a new userSpherical harmonics and rigged Hilbert spaces
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Added by
Mariano A. del Olmo Prof.
Publication date
2018
fields
Physics
and research's language is
English
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This paper is devoted to study discrete and continuous bases for spaces supporting representations of SO(3) and SO(3,2) where the spherical harmonics are involved. We show how discrete and continuous bases coexist on appropriate choices of rigged Hilbert spaces. We prove the continuity of relevant operators and the operators in the algebras spanned by them using appropriate topologies on our spaces. Finally, we discuss the properties of the functionals that form the continuous basis.
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Spherical Harmonics, $Y_ell^m(theta,phi)$, are derived and presented (in a Table) for half-odd-integer values of $ell$ and $m$. These functions are eigenfunctions of $L^2$ and $L_z$ written as differential operators in the spherical-polar angles, $theta$ and $phi$. The Fermion Spherical Harmonics are a new, scalar and angular-coordinate-dependent representation of fermion spin angular momentum. They have $4pi$ symmetry in the angle $phi$, and hence are not single-valued functions on the Euclidean unit sphere; they are double-valued functions on the sphere, or alternatively are interpreted as having a double-sphere as their domain.
The introduction of operator states and of observables in various fields of quantum physics has raised questions about the mathematical structures of the corresponding spaces. In the framework of third quantization it had been conjectured that we deal with Hilbert spaces although the mathematical background was not entirely clear, particularly, when dealing with bosonic operators. This in turn caused some doubts about the correct way to combine bosonic and fermionic operators or, in other words, regular and Grassmann variables. In this paper we present a formal answer to the problems on a simple and very general basis. We illustrate the resulting construction by revisiting the Bargmann transform and finding the known connection between L^2(R) and the Bargmann-Hilbert space. We then use the formalism to give an explicit formulation for Fock spaces involving both fermions and bosons thus solving the problem at the origin of our considerations.
A notion of resolvent set for an operator acting in a rigged Hilbert space $D subset Hsubset D^times$ is proposed. This set depends on a family of intermediate locally convex spaces living between $D$ and $D^times$, called interspaces. Some properties of the resolvent set and of the corresponding multivalued resolvent function are derived and some examples are discussed.
The thermal equilibrium distribution over quantum-mechanical wave functions is a so-called Gaussian adjusted projected (GAP) measure, $GAP(rho_beta)$, for a thermal density operator $rho_beta$ at inverse temperature $beta$. More generally, $GAP(rho)$ is a probability measure on the unit sphere in Hilbert space for any density operator $rho$ (i.e., a positive operator with trace 1). In this note, we collect the mathematical details concerning the rigorous definition of $GAP(rho)$ in infinite-dimensional separable Hilbert spaces. Its existence and uniqueness follows from Prohorovs theorem on the existence and uniqueness of Gaussian measures in Hilbert spaces with given mean and covariance. We also give an alternative existence proof. Finally, we give a proof that $GAP(rho)$ depends continuously on $rho$ in the sense that convergence of $rho$ in the trace norm implies weak convergence of $GAP(rho)$.
We build a family of explicit one-forms on $S^3$ which are shown to form a complete set of eigenmodes for the Laplace-de Rahm operator.
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