Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userReflection positivity on spheres
66
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this article we specialize a construction of a reflection positive Hilbert space due to Dimock and Jaffe--Ritter to the sphere $mathbb{S}^n$. We determine the resulting Osterwalder--Schrader Hilbert space, a construction that can be viewed as the step from euclidean to relativistic quantum field theory. We show that this process gives rise to an irreducible unitary spherical representation of the orthochronous Lorentz group $G^c = mathrm{O}_{1,n}(mathbb{R})^{uparrow}$ and that the representations thus obtained are the irreducible unitary spherical representations of this group. A key tool is a certain complex domain $Xi$, known as the crown of the hyperboloid, containing a half-sphere $mathbb{S}^n_+$ and the hyperboloid $mathbb{H}^n$ as totally real submanifolds. This domain provides a bridge between those two manifolds when we study unitary representations of $G^c$ in spaces of holomorphic functions on $Xi$. We connect this analysis with the boundary components which are the de Sitter space and the Lorentz cone of future pointing light like vectors.
rate research
Read More
We study functions f : (a,b) ---> R on open intervals in R with respect to various kinds of positive and negative definiteness conditions. We say that f is positive definite if the kernel f((x + y)/2) is positive definite. We call f negative definite if, for every h > 0, the function e^{-hf} is positive definite. Our first main result is a Levy--Khintchine formula (an integral representation) for negative definite functions on arbitrary intervals. For (a,b) = (0,infty) it generalizes classical results by Bernstein and Horn. On a symmetric interval (-a,a), we call f reflection positive if it is positive definite and, in addition, the kernel f((x - y)/2) is positive definite. We likewise define reflection negative functions and obtain a Levy--Khintchine formula for reflection negative functions on all of R. Finally, we obtain a characterization of germs of reflection negative functions on 0-neighborhoods in R.
We prove that the only entrywise transforms of rectangular matrices which preserve total positivity or total non-negativity are either constant or linear. This follows from an extended classification of preservers of these two properties for matrices of fixed dimension. We also prove that the same assertions hold upon working only with symmetric matrices; for total-positivity preservers our proofs proceed through solving two totally positive completion problems.
We specify the structure of completely positive operators and quantum Markov semigroup generators that are symmetric with respect to a family of inner products, also providing new information on the order strucure an extreme points in some previously studied cases.
Entrywise functions preserving the cone of positive semidefinite matrices have been studied by many authors, most notably by Schoenberg [Duke Math. J. 9, 1942] and Rudin [Duke Math. J. 26, 1959]. Following their work, it is well-known that entrywise functions preserving Loewner positivity in all dimensions are precisely the absolutely monotonic functions. However, there are strong theoretical and practical motivations to study functions preserving positivity in a fixed dimension $n$. Such characterizations for a fixed value of $n$ are difficult to obtain, and in fact are only known in the $2 times 2$ case. In this paper, using a novel and intuitive approach, we study entrywise functions preserving positivity on distinguished submanifolds inside the cone obtained by imposing rank constraints. These rank constraints are prevalent in applications, and provide a natural way to relax the elusive original problem of preserving positivity in a fixed dimension. In our main result, we characterize entrywise functions mapping $n times n$ positive semidefinite matrices of rank at most $l$ into positive semidefinite matrices of rank at most $k$ for $1 leq l leq n$ and $1 leq k < n$. We also demonstrate how an important necessary condition for preserving positivity by Horn and Loewner [Trans. Amer. Math. Soc. 136, 1969] can be significantly generalized by adding rank constraints. Finally, our techniques allow us to obtain an elementary proof of the classical characterization of functions preserving positivity in all dimensions obtained by Schoenberg and Rudin.
We characterize positivity preserving, translation invariant, linear operators in $L^p(mathbb{R}^n)^m$, $p in [1,infty)$, $m,n in mathbb{N}$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
