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Register a new userKMS conditions, standard real subspaces and reflection positivity on the circle group
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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.
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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.
A new characterization of the singular packing subspaces of general bounded self-adjoint operators is presented, which is used to show that the set of operators whose spectral measures have upper packing dimension equal to one is a $G_delta$ (in suitable metric spaces). As an application, it is proven that, generically (in space of continuous sampling functions), spectral measures of the limit-periodic Schrodinger operators have upper packing dimensions equal to one. Consequently, in a generic set, these operators are quasiballistic.
An analytic definition of a $mathbb{Z}_2$-valued spectral flow for paths of real skew-adjoint Fredholm operators is given. It counts the parity of the number of changes in the orientation of the eigenfunctions at eigenvalue crossings through $0$ along the path. The $mathbb{Z}_2$-valued spectral flow is shown to satisfy a concatenation property and homotopy invariance, and it provides an isomorphism on the fundamental group of the real skew-adjoint Fredholm operators. Moreover, it is connected to a $mathbb{Z}_2$-index pairing for suitable paths. Applications concern the zero energy bound states at defects in a Majorana chain and a spectral flow interpretation for the $mathbb{Z}_2$-polarization in these models.
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 this paper we study the convex cone of not necessarily smooth measures satisfying the classical KMS condition within the context of Poisson geometry. We discuss the general properties of KMS measures and its relation with the underlying Poisson geometry in analogy to Weinsteins seminal work in the smooth case. Moreover, by generalizing results from the symplectic case, we focus on the case of $b$-Poisson manifolds, where we provide a complete characterization of the convex cone of KMS measures.
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