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PHYSICS (578 journals)

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Journal Cover Reviews in Mathematical Physics
  [SJR: 1.092]   [H-I: 36]   [0 followers]  Follow
   Hybrid Journal Hybrid journal (It can contain Open Access articles)
   ISSN (Print) 0129-055X - ISSN (Online) 1793-6659
   Published by World Scientific Homepage  [118 journals]
  • Covariant KSGNS construction and quantum instruments
    • Abstract: Reviews in Mathematical Physics, Ahead of Print.
      We study completely positive (CP) [math]-sesquilinear-form-valued maps on a unital [math]-algebra [math], where the sesquilinear forms operate on a module over a [math]-algebra [math]. We also study the cases when either one or both of the algebras are von Neumann algebras. Moreover, we assume that the CP maps are covariant with respect to actions of a symmetry group. This allows us to view these maps as generalizations of covariant quantum instruments. We determine minimal covariant dilations (KSGNS constructions) for covariant CP maps to find necessary and sufficient conditions for a CP map to be extreme in convex subsets of normalized covariant CP maps. As a special case, we study covariant quantum observables and instruments whose value space is a transitive space of a unimodular type-I group. Finally, we discuss the case of instruments that are covariant with respect to a square-integrable representation.
      Citation: Reviews in Mathematical Physics
      PubDate: 2017-06-13T09:19:49Z
      DOI: 10.1142/S0129055X17500209
  • Spectral analysis of a model for quantum friction
    • Abstract: Reviews in Mathematical Physics, Ahead of Print.
      An otherwise free classical particle moving through an extended spatially homogeneous medium with which it may exchange energy and momentum will undergo a frictional drag force in the direction opposite to its velocity with a magnitude which is typically proportional to a power of its speed. We study here the quantum equivalent of a classical Hamiltonian model for this friction phenomenon that was proposed in [11]. More precisely, we study the spectral properties of the quantum Hamiltonian and compare the quantum and classical situations. Under suitable conditions on the infrared behavior of the model, we prove that the Hamiltonian at fixed total momentum has no ground state except when the total momentum vanishes, and that its spectrum is otherwise absolutely continuous.
      Citation: Reviews in Mathematical Physics
      PubDate: 2017-05-24T09:41:46Z
      DOI: 10.1142/S0129055X17500192
  • Local disorder, topological ground state degeneracy and entanglement
           entropy, and discrete anyons
    • Abstract: Reviews in Mathematical Physics, Ahead of Print.
      In this comprehensive study of Kitaev’s abelian models defined on a graph embedded on a closed orientable surface, we provide complete proofs of the topological ground state degeneracy, the absence of local order parameters, compute the entanglement entropy exactly and characterize the elementary anyonic excitations. The homology and cohomology groups of the cell complex play a central role and allow for a rigorous understanding of the relations between the above characterizations of topological order.
      Citation: Reviews in Mathematical Physics
      PubDate: 2017-05-24T09:41:45Z
      DOI: 10.1142/S0129055X17500180
  • General construction of reproducing kernels on a quaternionic Hilbert
    • Abstract: Reviews in Mathematical Physics, Ahead of Print.
      A general theory of reproducing kernels and reproducing kernel Hilbert spaces on a right quaternionic Hilbert space is presented. Positive operator-valued measures and their connection to a class of generalized quaternionic coherent states are examined. A Naimark type extension theorem associated with the positive operator-valued measures is proved in a right quaternionic Hilbert space. As illustrative examples, real, complex and quaternionic reproducing kernels and reproducing kernel Hilbert spaces arising from Hermite and Laguerre polynomials are presented. In particular, in the Laguerre case, the Naimark type extension theorem on the associated quaternionic Hilbert space is indicated.
      Citation: Reviews in Mathematical Physics
      PubDate: 2017-05-02T07:44:27Z
      DOI: 10.1142/S0129055X17500179
  • Brownian motion and finite approximations of quantum systems over local
    • Abstract: Reviews in Mathematical Physics, Ahead of Print.
      We give a stochastic proof of the finite approximability of a class of Schrödinger operators over a local field, thereby completing a program of establishing in a non-Archimedean setting corresponding results and methods from the Archimedean (real) setting. A key ingredient of our proof is to show that Brownian motion over a local field can be obtained as a limit of random walks over finite grids. Also, we prove a Feynman–Kac formula for the finite systems, and show that the propagator at the finite level converges to the propagator at the infinite level.
      Citation: Reviews in Mathematical Physics
      PubDate: 2017-04-28T08:48:12Z
      DOI: 10.1142/S0129055X17500167
  • Topological field theories on manifolds with Wu structures
    • Abstract: Reviews in Mathematical Physics, Ahead of Print.
      We construct invertible field theories generalizing abelian prequantum spin Chern–Simons theory to manifolds of dimension [math] endowed with a Wu structure of degree [math]. After analyzing the anomalies of a certain discrete symmetry, we gauge it, producing topological field theories whose path integral reduces to a finite sum, akin to Dijkgraaf–Witten theories. We take a general point of view where the Chern–Simons gauge group and its couplings are encoded in a local system of integral lattices. The Lagrangian of these theories has to be interpreted as a class in a generalized cohomology theory in order to obtain a gauge invariant action. We develop a computationally friendly cochain model for this generalized cohomology and use it in a detailed study of the properties of the Wu Chern–Simons action. In the 3-dimensional spin case, the latter provides a definition of the “fermionic correction” introduced recently in the literature on fermionic symmetry protected topological phases. In order to construct the state space of the gauged theories, we develop an analogue of geometric quantization for finite abelian groups endowed with a skew-symmetric pairing. The physical motivation for this work comes from the fact that in the [math] case, the gauged 7-dimensional topological field theories constructed here are essentially the anomaly field theories of the 6-dimensional conformal field theories with (2, 0) supersymmetry, as will be discussed elsewhere.
      Citation: Reviews in Mathematical Physics
      PubDate: 2017-04-13T01:50:46Z
      DOI: 10.1142/S0129055X17500155
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