We consider a quantum system of non-interacting fermions at temperature T, in the framework of linear response theory. We show that semiclassical theory is an appropriate framework to describe some of their thermodynamic properties, in particular through asymptotic expansions in $\hbar$ (Planck constant) of the dynamical susceptibilities. We show how the closed orbits of the classical motion in phase space manifest themselves in these expansions, in the regime where T is of the order of $\hbar$.
Consider a free electron gas in a confining potential and a magnetic field in arbitrary dimensions. If this gas is in thermal equilibrium with a reservoir at temperature $T >0$, one can study its orbital magnetic response (omitting the spin). One defines a conveniently ``smeared out'' magnetization $M$, and the corresponding magnetic susceptibility $χ$, which will be analyzed from a semiclassical point of view, namely when $\hbar$ (the Planck constant) is small compared to classical actions characterizing the system. Then various regimes of temperature $T$ are studied where $M$ and $χ$ can be …
The Cauchy problem for the Gross--Pitaevsky equation with quadratic nonlocal nonlinearity is reduced to a similar problem for the correspondent linear equation. The relation between symmetry operators of the linear and nonlinear Gross--Pitaevsky equations is considered.
We consider solutions of the time-dependent Schrödinger equation for a potential localised at the points of a Poisson process. We prove convergence of the phase-space distribution in the annealed Boltzmann-Grad limit to a semiclassical Wigner (or defect) measure and show that it is a solution of the linear Boltzmann equation. Our results hold for a large class of square-integrable initial data associated to Wigner measures, including Langragian states, WKB states and coherent states. This extends important previous work by Eng and Erdős.
The linearization of semiclassical theories of gravity is investigated in a toy model, consisting of a quantum scalar field in interaction with a second classical scalar field which plays the role of a classical background. This toy model mimics also the evolution induced by semiclassical Einstein equations, such as the one which describes the early universe in the cosmological case. The equations governing the dynamics of linear perturbations around simple exact solutions of this toy model are analyzed by constructing the corresponding retarded fundamental solutions, and by discussing the cor…
We introduce a concept of squeezing in collective qutrit systems through a geometrical picture connected to the deformation of the isotropic fluctuations of su(3) operators when evaluated in a coherent state. This kind of squeezing can be generated by Hamiltonians non-linear in the generators of su(3) algebra. A simple model of such a non-linear evolution is analyzed in terms of semiclassical evolution of the SU(3) Wigner function.
In this paper we develop the topics of Quantum Recurrences and of Quantum Fidelity which have attracted great interest in recent years. The return probability is given by the square modulus of the overlap between a given initial wavepacket and the corresponding evolved one; quantum recurrences in time can be observed if this overlap is unity. We provide some conditions under which this is semiclassically achieved taking as initial wavepacket a coherent state located on a closed orbit of the corresponding classical motion. The "quantum fidelity" (or Loschmidt Echo) is the square modulus of the …
A semiclassical approach based on the WKB-Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the kinetic equation under the supposition of weak diffusion. In terms of the approach developed, the local cubic nonlinear term in the original kinetic equation is considered in a nonlocal form. This allows one to transform the nonlinear nonlocal kineti…
In the case of a linear symplectic map A of the 2d-torus, semiclassical measures are A-invariant probability measures associated to sequences of high energy quantum states. Our main result is an explicit lower bound on the entropy of any semiclassical measure of a given quantizable matrix A in Sp(2d,Z). In particular, our result implies that if A has an eigenvalue outside the unit circle, then a semiclassical measure cannot be carried by a closed orbit of A.
We study the linear relaxation Boltzmann equation, a simple semiclassical kinetic model. We provide a resolvent estimate for an associated non-selfadjoint operator as well as an estimate on the return to equilibrium. This is done using a scaling argument and non-semiclassical hypocoercive estimate.
Quantum fields do not satisfy the pointwise energy conditions that are assumed in the original singularity theorems of Penrose and Hawking. Accordingly, semiclassical quantum gravity lies outside their scope. Although a number of singularity theorems have been derived under weakened energy conditions, none is directly derived from quantum field theory. Here, we employ a quantum energy inequality satisfied by the quantized minimally coupled linear scalar field to derive a singularity theorem valid in semiclassical gravity. By considering a toy cosmological model, we show that our result predict…
To study electronic transport through chaotic quantum dots, there are two main theoretical approachs. One involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging. The other treats the transport in the semiclassical approximation and studies correlations among sets of classical trajectories. There are established evaluation procedures within the semiclassical evaluation that, for several linear and non-linear transport moments to which they were applied, have always resulted in the agreement with random matrix predictions. We prove …
A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the 1D nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebr…
In this paper, we establish (1) the classical limit of the Hartree equation leading to the Vlasov equation, (2) the classical limit of the $N$-body linear Schrödinger equation uniformly in N leading to the N-body Liouville equation of classical mechanics and (3) the simultaneous mean-field and classical limit of the N-body linear Schrödinger equation leading to the Vlasov equation. In all these limits, we assume that the gradient of the interaction potential is Lipschitz continuous. All our results are formulated as estimates involving a quantum analogue of the Monge-Kantorovich distance of ex…
We review some recent results obtained in the analysis of two-dimensional quantum field theories by means of semiclassical techniques, which generalize methods introduced during the Seventies by Dashen, Hasllacher and Neveu and by Goldstone and Jackiw. The approach is best suited to deal with quantum field theories characterized by a non-linear interaction potential with different degenerate minima, that generates kink excitations of large mass in the small coupling regime. Under these circumstances, although the results obtained are based on a small coupling assumption, they are nevertheless …
Electronic transport through chaotic quantum dots exhibits universal behaviour which can be understood through the semiclassical approximation. Within the approximation, transport moments reduce to codifying classical correlations between scattering trajectories. These can be represented as ribbon graphs and we develop an algorithmic combinatorial method to generate all such graphs with a given genus. This provides an expansion of the linear transport moments for systems both with and without time reversal symmetry. The computational implementation is then able to progress several orders highe…
We propose a method for deriving Lindblad-like master equations when the environment/reservoir is consigned to a classical description. As a proof of concept, we apply the method to continuous wave (CW) magnetic resonance. We make use of a perturbation scheme we have termed "affine commutation perturbation" (ACP). Unlike traditional perturbation methods, ACP has the advantage of incorporating some effects of the perturbation even at the zeroth-order approximation. Indeed, we concentrate here on the zeroth-order, and show how -- even at this lowest order -- the ACP scheme can still yield non-tr…
The Semiclassical Einstein-Klein-Gordon System: Asymptotic Analysis of Minkowski Spacetime
University of York（约克大学）；Università di Trento（特伦托大学）；GNFM-INdAM（意大利数学国家研究委员会几何与函数论全国小组）；TIFPA-INFN（意大利国家核物理研究所特伦托前沿物理与分析中心）；Università di Genova（热那亚大学）；INFN（意大利国家核物理研究所）；Universität Regensburg（雷根斯堡大学）
The Bochner-Schrödinger operator $H_{p}=\frac 1pΔ^{L^p\otimes E}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold of bounded geometry is studied. For any function $φ\in \mathcal S(\mathbb R)$, we consider the bounded linear operator $φ(H_p)$ in $L^2(X,L^p\otimes E)$ defined by the spectral theorem and describe an asymptotic expansion of its smooth Schwartz kernel in a fixed neighborhood of the diagonal in the semiclassical limit $p\to \infty$. In particular, we prove that the trace of the operator $φ(H_p)$ admits a compl…
