Quantum supergroups VI: roots of 1

Letters in Mathematical Physics - A quantum covering group is an algebra with parameters q and $$\pi $$ subject to $$\pi ^2=1$$, and it admits an integral form; it specializes to the usual quantum...     

Quantum Spin Systems at Positive Temperature

We develop a novel approach to phase transitions in quantum spin models based on a relation to their classical counterparts. Explicitly, we show that whenever chessboard estimates can be used to prove a phase transition in the classical model, the corresponding quantum model will have a similar phase transition, provided the inverse temperature β and the magnitude of the quantum spins satisfy . From the quantum system we require that it is reflection positive and that it has a meaningful classical limit; the core technical estimate may be described as an extension of the Berezin-Lieb inequalities down to the level of matrix elements. The general theory is applied to prove phase transitions in various quantum spin systems with . The most notable examples are the quantum orbital-compass model on and the quantum 120-degree model on which are shown to exhibit symmetry breaking at low-temperatures despite the infinite degeneracy of their (classical) ground state.     

The notion of observable and the moment problem for $$*$$∗-algebras and their GNS representations

Letters in Mathematical Physics - We address some usually overlooked issues concerning the use of $$*$$-algebras in quantum theory and their physical interpretation. If $${\mathfrak {A}}$$ is a...     

Stochastization of Long Living Spin-Cyclotron Excitations in a Spin-Unpolarised Quantum Hall System

JETP Letters - We address the kinetics of long-lived excitations at zero temperature in an electronic quantum Hall system with filling factor $$\nu = 2$$ . The initial coherent state of...     

Born’s rule for arbitrary Cauchy surfaces

Letters in Mathematical Physics - Suppose that particle detectors are placed along a Cauchy surface $$\Sigma $$ in Minkowski space-time, and consider a quantum theory with fixed or variable number...     

Perfect state transfer on weighted graphs of the Johnson scheme

Letters in Mathematical Physics - We characterize quantum perfect state transfer on real-weighted graphs of the Johnson scheme $${\mathcal {J}}(n,k)$$ , which represent spin networks with...     

A Characterization of Right Coideals of Quotient Type and its Application to Classification of Poisson Boundaries

Let be a co-amenable compact quantum group. We show that a right coideal of is of quotient type if and only if it is the range of a conditional expectation preserving the Haar state and is globally invariant under the left action of the dual discrete quantum group. We apply this result to the theory of Poisson boundaries introduced by Izumi for discrete quantum groups and generalize a work of Izumi-Neshveyev-Tuset on SU _q (N) for co-amenable compact quantum groups with the commutative fusion rules. More precisely, we prove that the Poisson integral is an isomorphism between the Poisson boundary and the right coideal of quotient type by a maximal quantum subgroup of Kac type. In particular, the Poisson boundary and the quantum flag manifold are isomorphic for any q-deformed classical compact Lie group.     

Non-relativistic Limit of a Dirac Polaron in Relativistic Quantum Electrodynamics

A quantum system of a Dirac particle interacting with the quantum radiation field is considered in the case where no external potentials exist. Then the total momentum of the system is conserved and the total Hamiltonian is unitarily equivalent to the direct integral of a family of self-adjoint operators acting in the Hilbert space , where is the Hilbert space of the quantum radiation field. The fiber operator is called the Hamiltonian of the Dirac polaron with total momentum . The main result of this paper is concerned with the non-relativistic (scaling) limit of . It is proven that the non-relativistic limit of yields a self-adjoint extension of a Hamiltonian of a polaron with spin 1/2 in non-relativistic quantum electrodynamics.     

$${f(\mathcal R)}$$ quantum cosmology

We have quantized a flat cosmological model in the context of the metric models, using the causal Bohmian quantum theory. The equations are solved and then we have obtained how the quantum corrections influence the classical equations.     

Local Asymptotic Normality in Quantum Statistics

The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state ϕ _u of an algebra of canonical commutation relations. The convergence holds for all “local parameters” such that parametrizes a neighborhood of a fixed point . In order to prove the result we define weak and strong convergence of quantum statistical experiments which extend to the asymptotic framework the notion of quantum sufficiency introduces by Petz. Along the way we introduce the concept of canonical state of a statistical experiment, and investigate the relation between the two notions of convergence. For the reader’s convenience and completeness we review the relevant results of the classical as well as the quantum theory. Dedicated to Slava Belavkin on the occasion of his 60th anniversary     

Imperfect Instantaneous Energy Conservation in Wheeler–Feynman Absorber Theory

Foundations of Physics - In standard quantum mechanics (QM), a state vector $$| \psi \rangle $$ may belong to infinitely many different orthogonal bases, as soon as the dimension N of the Hilbert...     

Fast synthesize ZnO quantum dots via ultrasonic method

Green emission ZnO quantum dots were synthesized by an ultrasonic sol–gel method. The ZnO quantum dots were synthesized in various ultrasonic temperature and time. Photoluminescence properties of these ZnO quantum dots were measured. Time-resolved photoluminescence decay spectra were also taken to discover the change of defects amount during the reaction. Both ultrasonic temperature and time could affect the type and amount of defects in ZnO quantum dots. Total defects of ZnO quantum dots decreased with the increasing of ultrasonic temperature and time. The dangling bonds defects disappeared faster than the optical defects. Types of optical defects first changed from oxygen interstitial defects to oxygen vacancy and zinc interstitial defects. Then transformed back to oxygen interstitial defects again. The sizes of ZnO quantum dots would be controlled by both ultrasonic temperature and time as well. That is, with the increasing of ultrasonic temperature and time, the sizes of ZnO quantum dots first decreased then increased. Moreover, concentrated raw materials solution brought larger sizes and more optical defects of ZnO quantum dots.     

$${{{v}}_{2}}$$ -Dependences of the Rotational Contributions to the Effective Dipole Moment of the H2O Molecule and Their Effect on the Broadening and Shift of Lines Induced by the Pressure of Buffer Gases

Optics and Spectroscopy - The dependences of the rotational contributions to the effective dipole moment of the H2O molecule on vibrational quantum number $${{{v}}_{2}}$$ , which corresponds to the...     

Critical Temperature of Heisenberg Models on Regular Trees,via Random Loops

Journal of Statistical Physics - We estimate the critical temperature of a family of quantum spin systems on regular trees of large degree. The systems include the spin- $$\frac{1}{2}$$ XXZ model...     

Casimir and Surface Tension Forces on a Single Interacting Bose–Einstein Condensate in Canonical Ensemble

The forces on a single Bose–Einstein condensate confined between two parallel plates consist of two components, namely, surface tension force and Casimir force. In canonical ensemble, these forces are quite different from the one in grand canonical ensemble. In small region with distance $$\ell $$ between two parallel plates, using double parabola approximation, we find that surface tension force decreases as $${{\ell }^{{ - 3}}}$$, whereas the Casimir force, in one-loop approximation of the quantum field, is proportional to $${{\ell }^{{ - 13/2}}}$$. The total force is also considered and its veer is found.     

Determination of Stoichiometry,Concentration of OH Groups,and Point Defects in Lithium Niobate Crystals from Their IR Absorption Spectra

Optics and Spectroscopy - The concentration of OH impurity groups, the Li/Nb ratio, the concentration of Nb $$_{{{\text{Li}}}}^{{4 + }}$$ and V $$_{{{\text{Li}}}}^{ - }$$ point defects in...     

Quasi-Linear Quantum Field Theories for Maps to Groups and Their Quotients

I describe a functional integral for maps from to a Lie group or its quotient which has a simple renormalization that leads to a quantum field theory for maps from into the Lie group or its quotient whose Hamiltonian is the time translation generator for a unitary action of the n+1 dimensional Poincaré group on the quantum Hilbert space. I also explain how the renormalization provides a functional integral for maps from a Riemann surface to a compact Lie group or its quotient that exhibits many conformal field theoretic properties.Support in part by a grant from the National Science Foundation     

Orthocomplementation and Compound Systems

In their 1936 founding paper on quantum logic, Birkhoff and von Neumann postulated that the lattice describing the experimental propositions concerning a quantum system is orthocomplemented. We prove that this postulate fails for the lattice _sep describing a compound system consisting of so called separated quantum systems. By separated we mean two systems prepared in different “rooms” of the lab, and before any interaction takes place. In that case, the state of the compound system is necessarily a product state. As a consequence, Dirac’s superposition principle fails, and therefore _sep cannot satisfy all Piron’s axioms. In previous works, assuming that _sep is orthocomplemented, it was argued that _sep is not orthomodular and fails to have the covering property. Here we prove that _sep cannot admit an orthocomplementation. Moreover, we propose a natural model for _sep which has the covering property. PACS: 03.65.Ta, 03.65.Ca