Quenching Evolution in a Quantum-Classical Hybrid System

The adiabatic theorem, an important theory in quantum mechanics, tells that a quantum system subjected to gradually changing external conditions remains to the same instantaneous eigenstate of its Hamiltonian as it initially in. In this paper, we study the quench evolution that is another extreme circumstance where the external conditions vary rapidly such that the quantum system can not follow the change and remains in its initial state (or wavefunction). We examine the matter-wave pressure and derive the requirement for such an evolution. The study is conducted by considering a quantum particle in an infinitely deep potential, the potential width Q is assumed to be change rapidly. We show that the total energy of the quantum subsystem decreases as Q increases, and this rapidly change exerts a force on the wall which plays the role of boundary of the potential. For Q < Q₀ (Q₀ is the initial width of the potential), the force is repulsive, and for Q > Q₀, the force is positive. The condition for the quenching evolution evolution is given via a spin-\( \frac{1}{2} \) in a rotating magnetic field.

     

Quantum mechanical version of classical Liouville theorem

In terms of the coherent state evolution in phase space, we present a quantum mechanical version of the classical Liouville theorem. The evolution of coherent state from |z〉to |sz-rz^*〉angle corresponds to the motion from a point z(q,p) to another point sz-rz^* with |s|²-|r|²=1. The evolution is governed by the so-called Fresnel operator U(s,r) recently proposed in quantum optics theory, which classically corresponds to the matrix optics law and the optical Fresnel transformation and obeys the group product rules. In another word, we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space, which seems to be a combination of quantum statistics and quantum optics.     

Incompressible Flows of an Ideal Fluid¶with Unbounded Vorticity

In this paper we study solutions to the Euler equations of an ideal incompressible fluid in R ⁿ singular at the origin with a finite symmetry group. For an “admissible” class of finite groups we prove a local existence and uniqueness theorem. In even dimensions this theorem covers some symmetric flows with essentially unbounded vorticity. In arbitrary dimension (including n=3) we construct local in time solutions with vorticity that behaves, e.g., like a function of homogeneous degree zero near the origin. The symmetry condition provides necessary additional cancellations and is preserved by the evolution due to uniqueness. Received: 31 March 1999 / Accepted: 10 July 2000     

Birkhoff’s theorem in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">T</Emphasis>) gravity

Generalized from the so-called teleparallel gravity, which is exactly equivalent to general relativity, f(T) gravity has been proposed as an alternative gravity model to account for the dark energy phenomena. In this letter we prove that the external vacuum gravitational field for a spherically symmetric distribution of source matter in the f(T) gravity framework must be static. The conclusion is independent of the radial distribution and spherically symmetric motion of the source matter, that is, whether it is in motion or static. As a consequence, the Birkhoff’s theorem is valid in the general nonsingular f(T) theory at the un-perturbative level. We also discuss its application in the de Sitter spacetime evolution phase as preferred by present dark energy observations.     

On G-Trace and G-Index in Deformation Quantization

For a compact group G of symplectomorphisms we define a G-trace on the algebra of quantum observables by postulating its properties. We give an explicit construction of such a trace and prove a G-index theorem similar to the Atiyah–Segal–Singer equivariant index theorem for elliptic operators.     

Beliefs and stochastic modelling of interest rate scenario risk

We present a framework that allows for a systematic assessment of risk given a specific model and belief on the market. Within this framework the time evolution of risk is modeled in a twofold way. On the one hand, risk is modeled by the time discrete and nonlinear garch(1,1) process, which allows for a (time-)local understanding of its level, together with a short term forecast. On the other hand, via a diffusion approximation, the time evolution of the probability density of risk is modeled by a Fokker-Planck equation. Then, as a final step, using Bayes theorem, beliefs are conditioned on the stationary probability density function as obtained from the Fokker-Planck equation. We believe this to be a highly rigorous framework to integrate subjective judgments of future market behavior and underlying models. In order to demonstrate the approach, we apply it to risk assessment of empirical interest rate scenario methodologies, i.e. the application of Principal Component Analysis to the the dynamics of bonds. Received 1st August 2000     

On quantization of free fields in stationary space-times

In Section 1 we analyse the structure of the infinite-dimensional Hamiltonian system described by the Klein-Gordon equation (free real scalar field) in stationary space-times with closed space sections; we give an existence and uniqueness theorem for the Lichnerowicz distribution kernelG ₁ together with its proper Fourier expansion, and we construct the Hilbert spaces of frequency-part solutions defined by means ofG ₁.In Section 2 an analysis, a theorem and a construction similar to the above are formulated for thefree real field spin 1, massm>0, in one kind of static space-times.In this letter, only results are given. For detailed proofs and further results, see reference [9], [10] and [11].     

超短脉冲激光在DBASVP分子中传播时的双光子面积演化和光限幅效应

以一维不对称π共轭分子体系（DBASVP分子）为介质,在双光子共振条件下,从双光子面积定理和严格数值求解Maxwell-Bloch方程两方面出发,分别研究超短脉冲激光在该有机分子介质中的传播过程,从而探讨双光子面积的演化规律,并分析双光子面积定理的适用性.提出了一种数值模拟分子介质光限幅特性的理论方法.分子的电子结构和电偶极矩是基于密度泛函理论利用从头计算方法得到的.研究结果表明,基于慢变幅和慢变相近似以及单模场条件下的双光子面积定理不能很好地描述超短脉冲的双光子面积在该分子介质中的演化规律.基于双光子吸收的分子光限幅特性与分子介质的厚度有关. 关键词： 双光子吸收 光限幅效应 双光子面积定理 超短脉冲激光     

Spectral Extension of the Quantum Group Cotangent Bundle

The structure of a cotangent bundle is investigated for quantum linear groups GL _q (n) and SL _q (n). Using a q-version of the Cayley-Hamilton theorem we construct an extension of the algebra of differential operators on SL _q (n) (otherwise called the Heisenberg double) by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators—the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive SL _q (n) type dynamical R-matrices in a surprisingly simple way. Second, we calculate an evolution operator for the model of the q-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. The relation between the two operators is given by a modular functional equation for the Riemann theta function.     

Solution of the Crow-Kimura and Eigen Models for Alphabets of Arbitrary Size by Schwinger Spin Coherent States

To represent the evolution of nucleic acid and protein sequence, we express the parallel and Eigen models for molecular evolution in terms of a functional integral representation with an h-letter alphabet, lifting the two-state, purine/pyrimidine assumption often made in quasi-species theory. For arbitrary h and a general mutation scheme, we obtain the solution of this model in terms of a maximum principle. Euler’s theorem for homogeneous functions is used to derive this ‘thermodynamic’ formulation of evolution. The general result for the parallel model reduces to known results for the purine/pyrimidine h=2 alphabet and the nucleic acid h=4 alphabet for the Kimura 3 ST mutation scheme. Examples are presented for the h=4 and h=20 cases. We also derive the maximum principle for the Eigen model for general h. The general result for the Eigen model reduces to a known result for h=2. Examples are presented for the nucleic acid h=4 and the amino acid h=20 alphabet. An error catastrophe phase transition occurs in these models, and the order of the phase transition changes from second to first order for smooth fitness functions when the alphabet size is increased beyond two letters to the generic case. As examples, we analyze the general analytic solution for sharp peak, linear, quadratic, and quartic fitness functions.     

Entanglement Entropy Evolution under Double‐trace Deformation

In this paper, we study the bulk entanglement entropy evolution in conical BTZ black bole background using the heat kernel method. This is motivated by exploring the new examples where the quantum correction of the entanglement entropy gives the leading contribution. We find that in the large black hole limit the bulk entanglement entropy decreases under the double‐trace deformation which is consistent with the holographic c theorem and in the small black hole limit the bulk entanglement entropy increases under the deformation. We also discuss the minimal area correction.     

Micro-local approach to the Hadamard condition in quantum field theory on curved space-time

For the two-point distribution of a quasi-free Klein-Gordon neutral scalar quantum field on an arbitrary four dimensional globally hyperbolic curved space-time we prove the equivalence of (1) the global Hadamard condition, (2) the property that the Feynman propagator is a distinguished parametrix in the sense of Duistermaat and Hörmander, and (3) a new property referred to as the wave front set spectral condition (WFSSC), because it is reminiscent of the spectral condition in axiomatic quantum field theory on Minkowski space. Results in micro-local analysis such as the propagation of singularities theorem and the uniqueness up toC of distinguished parametrices are employed in the proof. We include a review of Kay and Wald's rigorous definition of the global Hadamard condition and the theory of distinguished parametrices, specializing to the case of the Klein-Gordon operator on a globally hyperbolic space-time. As an alternative to a recent computation of the wave front set of a globally Hadamard two-point distribution on a globally hyperbolic curved space-time, given elsewhere by Köhler (to correct an incomplete computation in [32]), we present a version of this computation that does not use a deformation argument such as that used in Fulling, Narcowich and Wald and is independent of the Cauchy evolution argument of Fulling, Sweeny and Wald (both of which are relied upon in Köhler's proof). This leads to a simple micro-local proof of the preservation of Hadamard form under Cauchy evolution (first shown by Fulling, Sweeny and Wald) relying only on the propagation of singularities theorem. In another paper [33], the equivalence theorem is used to prove a conjecture by Kay that a locally Hadamard quasi-free Klein-Gordon state on any globally hyperbolic curved space-time must be globally Hadamard.To my parents     

Wave equation for dark coherence in three-level media

We report the derivation of a wave equation for coherence in "dark state" two-photon-resonance spectroscopy. One of its consequences is a dark state area theorem. The dark area theorem is a single ordinary differential equation which is globally equivalent, in a way we describe, to the full set of five coupled nonlinear partial differential equations that govern space-time evolution of two-pulse coherence in a lambda medium. The predictions of the dark area theorem are open to test via laser spectroscopy in dilute vapors and inhomogeneously broadened solids.     

Some results on the quantum dynamics of a particle in a Markovian potential

We consider the quantum dynamics of a particle in a time dependent potentialV(t), assuming it to be a Markovian random function of time. We derive a formula for the density matrix at timet averaged over the realisations of the potential. We then obtain a kind of RAGE theorem for the time evolution of compact observables, and some information on the phase space behaviour of the system.     

Uniqueness theorems in ideal magnetofluid dynamics

Summary If based on the hyperbolic PDE general theory, the proof of a uniqueness theorem for the ideal magnetofluid-dynamic evolution problem presupposes a number of laborious verifications, as well as a good acquaintance with that theory. This paper provides a direct, ?ad hoc? alternative proof, which is immediately accessible to the nonspecialist and also removes a significant, but unnecessary, limitation tied to the usual theory.     

A Gallavotti–Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics

We extend the work of Kurchan on the Gallavotti–Cohen fluctuation theorem, which yields a symmetry property of the large deviation function, to general Markov processes. These include jump processes describing the evolution of stochastic lattice gases driven in the bulk or through particle reservoirs, general diffusive processes in physical and/or velocity space, as well as Hamiltonian systems with stochastic boundary conditions. For dynamics satisfying local detailed balance we establish a link between the average of the action functional in the fluctuation theorem and the macroscopic entropy production. This gives, in the linear regime, an alternative derivation of the Green–Kubo formula and the Onsager reciprocity relations. In the nonlinear regime consequences of the new symmetry are harder to come by and the large deviation functional difficult to compute. For the asymmetric simple exclusion process the latter is determined explicitly using the Bethe ansatz in the limit of large N.     

Solution of the local mass problem in general relativity

The local mass problem is solved. That is, in suitable function spaces, it is shown that for any vacuum space-time near flat space, its massm is strictly positive. The relationship to other work in the field and some discussion of the global problem is given. Our proof is, in effect, a version of critical point analysis in infinite dimensions, but detailedL ^p and Sobolev-type estimates are needed for the precise proof, as well as careful attention to the coordinate invariance group. For the latter, we prove a suitable slice theorem based on the use of harmonic coordinates.Partially supported by the University of Toronto, Université de Paris VI and NSF Grant MPS-75-05576     

Uniqueness and symmetry breaking in S-matrix theory

Assuming that the physical world is a solution of theS matrix equations, nonlinear functional analysis enables its uniqueness to be tested experimentally. As a first step, we develop such tests within the limits of partial wave dispersion relations,with crossing symmetry included. They are closely related to Levinson's theorem. We show that they give conditions for the validity of the bootstrap hypothesis, of the dynamical generation of symmetries, and of Dashen-Frautschi perturbation theory. They do not appear to be satisfied experimentally.     

Planar clusters and perimeter bounds

We provide upper and lower bounds on the least-perimeter way to enclose and separate n regions of equal area in the plane (theorem 3.1). Along the way, inside the hexagonal honeycomb, we provide minimizers for each n (theorem 2.7).     

Asymptotic behavior of nucleon electromagnetic form factors in time-like region

We study the asymptotic behavior of the ratio of Pauli and Dirac electromagnetic nucleon form factors, F₂/F₁, in time-like region, for different parametrizations built for the space-like region. We investigate how fast the ratio F₂/F₁ approaches the asymptotic limits according to the Phragmèn-Lindel?f theorem. We show that the QCD-inspired logarithmic behavior of this ratio results in very far asymptotics, experimentally unachievable. This is also confirmed by the normal component of the nucleon polarization, P_y, in e⁺ + e^-↦N + ˉ (in collisions of unpolarized leptons), which is a very interesting observable, with respect to this theorem. Finally we observe that the 1/Q parametrization of F₂/F₁ contradicts this theorem.