Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userReflection positivity on real intervals
91
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
In the present paper we continue our investigations of the representation theoretic side of reflection positivity by studying positive definite functions psi on the additive group (R,+) satisfying a suitably defined KMS condition. These functions take values in the space Bil(V) of bilinear forms on a real vector space V. As in quantum statistical mechanics, the KMS condition is defined in terms of an analytic continuation of psi to the strip { z in C: 0 leq Im z leq b} with a coupling condition psi (ib + t) = oline{psi (t)} on the boundary. Our first main result consists of a characterization of these functions in terms of modular objects (Delta, J) (J an antilinear involution and Delta > 0 selfadjoint with JDelta J = Delta^{-1}) and an integral representation. Our second main result is the existence of a Bil(V)-valued positive definite function f on the group R_tau = R rtimes {id_R,tau} with tau(t) = -t satisfying f(t,tau) = psi(it) for t in R. We thus obtain a 2b-periodic unitary one-parameter group on the GNS space H_f for which the one-parameter group on the GNS space H_psi is obtained by Osterwalder--Schrader quantization. Finally, we show that the building blocks of these representations arise from bundle-valued Sobolev spaces corresponding to the kernels 1/(lambda^2 - (d^2)/(dt^2}) on the circle R/bZ of length b.
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.
Recently the behavior of operator monotone functions on unbounded intervals with respect to the relation of strictly positivity has been investigated. In this paper we deeply study such behavior not only for operator monotone functions but also for operator convex functions on bounded intervals. More precisely, we prove that if $f$ is a nonlinear operator convex function on a bounded interval $(a,b)$ and $A, B$ are bounded linear operators acting on a Hilbert space with spectra in $(a,b)$ and $A-B$ is invertible, then $sf(A)+(1-s)f(B)>f(sA+(1-s)B)$. A short proof for a similar known result concerning a nonconstant operator monotone function on $[0,infty)$ is presented. Another purpose is to find a lower bound for $f(A)-f(B)$, where $f$ is a nonconstant operator monotone function, by using a key lemma. We also give an estimation of the Furuta inequality, which is an excellent extension of the Lowner--Heinz inequality.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
