Correlation functions from a unified variational principle: trial Lie groups

Time-dependent expectation values and correlation functions for many-body quantum systems are evaluated by means of a unified variational principle. It optimizes a generating functional depending on sources associated with the observables of interest $\ldots$Comment: 42 page

Geometry of the Casimir Effect

When the vacuum is partitioned by material boundaries with arbitrary shape, one can define the zero-point energy and the free energy of the electromagnetic waves in it: this can be done, independently of the nature of the boundaries, in the limit that they become perfect conductors, provided their curvature is finite. The first examples we consider are Casimir's original configuration of parallel plates, and the experimental situation of a sphere in front of a plate. For arbitrary geometries, we give an explicit expression for the zero-point energy and the free energy in terms of an integral kernel acting on the boundaries; it can be expanded in a convergent series interpreted as a succession of an even number of scatterings of a wave. The quantum and thermal fluctuations of vacuum then appear as a purely geometric property. The Casimir effect thus defined exists only owing to the electromagnetic nature of the field. It does not exist for thin foils with sharp folds, but Casimir forces between solid wedges are finite. We work out various applications: low temperature, high temperature where wrinkling constraints appear, stability of a plane foil, transfer of energy from one side of a curved boundary to the other, forces between distant conductors, special shapes (parallel plates, sphere, cylinder, honeycomb).Comment: 44 pages, 8 figures; Proceedings of the 15 th SIGRAV Conference on General Relativity and Gravitational Physics, Villa Mondragone, Monte Porzio Catone, Roma, Italy, September 9-12, 200

The entropy-based quantum metric

International audienceThe von Neumann entropy S(ˆD ) generates in the space of quantum density matrices ˆD the Riemannian metric ds2 = −d2S(ˆD ), which is physically founded and which characterises the amount of quantum information lost by mixing ˆD and ˆD + dˆD. A rich geometric structure is thereby implemented in quantum mechanics. It includes a canonical mapping between the spaces of states and of observables, which involves the Legendre transform of S(ˆD). The Kubo scalar product is recovered within the space of observables. Applications are given to equilibrium and non equilibrium quantum statistical mechanics. There the formalism is specialised to the relevant space of observables and to the associated reduced states issued from the maximum entropy criterion, which result from the exact states through an orthogonal projection. Von Neumann's entropy specialises into a relevant entropy. Comparison is made with other metrics. The Riemannian properties of the metric ds2 = −d2S(ˆD) are derived. The curvature arises from the non-Abelian nature of quantum mechanics; its general expression and its explicit form for q-bits are given

Phase transitions and quantum measurements

In a quantum measurement, a coupling $g$ between the system S and the apparatus A triggers the establishment of correlations, which provide statistical information about S. Robust registration requires A to be macroscopic, and a dynamical symmetry breaking of A governed by S allows the absence of any bias. Phase transitions are thus a paradigm for quantum measurement apparatuses, with the order parameter as pointer variable. The coupling $g$ behaves as the source of symmetry breaking. The exact solution of a model where S is a single spin and A a magnetic dot (consisting of $N$ interacting spins and a phonon thermal bath) exhibits the reduction of the state as a relaxation process of the off-diagonal elements of S+A, rapid due to the large size of $N$. The registration of the diagonal elements involves a slower relaxation from the initial paramagnetic state of A to either one of its ferromagnetic states. If $g$ is too weak, the measurement fails due to a ``Buridan's ass'' effect. The probability distribution for the magnetization then develops not one but two narrow peaks at the ferromagnetic values. During its evolution it goes through wide shapes extending between these values.Comment: 12 pages, 2 figure

Lectures on dynamical models for quantum measurements

In textbooks, ideal quantum measurements are described in terms of the tested system only by the collapse postulate and Born's rule. This level of description offers a rather flexible position for the interpretation of quantum mechanics. Here we analyse an ideal measurement as a process of interaction between the tested system S and an apparatus A, so as to derive the properties postulated in textbooks. We thus consider within standard quantum mechanics the measurement of a quantum spin component $\hat s_z$ by an apparatus A, being a magnet coupled to a bath. We first consider the evolution of the density operator of S+A describing a large set of runs of the measurement process. The approach describes the disappearance of the off-diagonal terms ("truncation") of the density matrix as a physical effect due to A, while the registration of the outcome has classical features due to the large size of the pointer variable, the magnetisation. A quantum ambiguity implies that the density matrix at the final time can be decomposed on many bases, not only the one of the measurement. This quantum oddity prevents to connect individual outcomes to measurements, a difficulty known as the "measurement problem". It is shown that it is circumvented by the apparatus as well, since the evolution in a small time interval erases all decompositions, except the one on the measurement basis. Once one can derive the outcome of individual events from quantum theory, the so-called "collapse of the wave function" or the "reduction of the state" appears as the result of a selection of runs among the original large set. Hence nothing more than standard quantum mechanics is needed to explain features of measurements. The employed statistical formulation is advocated for the teaching of quantum theory.Comment: 43 pages, 5 figures. Lectures given in the "Advanced School on Quantum Foundations and Open Quantum Systems", Joao Pessoa, Brazil, summer 2012. To appear in the proceedings and in IJMP

The Quantum Measurement Process: Lessons from an Exactly Solvable Model

The measurement of a spin-\half is modeled by coupling it to an apparatus, that consists of an Ising magnetic dot coupled to a phonon bath. Features of quantum measurements are derived from the dynamical solution of the measurement, regarded as a process of quantum statistical mechanics. Schr\"odinger cat terms involving both the system and the apparatus, die out very quickly, while the registration is a process taking the apparatus from its initially metastable state to one of its stable final states. The occurrence of Born probabilities can be inferred at the macroscopic level, by looking at the pointer alone. Apparent non-unitary behavior of the measurement process is explained by the arisal of small many particle correlations, that characterize relaxation.Comment: 13 pages, discussion of pre-measurement added. World Scientific style. To appear in proceedings "Beyond The Quantum", Th.M. Nieuwenhuizen et al, eds, (World Scientific, 2007

Thomson's formulation of the second law: an exact theorem and limits of its validity

Thomson's formulation of the second law - no work can be extracted from a system coupled to a bath through a cyclic process - is believed to be a fundamental principle of nature. For the equilibrium situation a simple proof is presented, valid for macroscopic sources of work. Thomson's formulation gets limited when the source of work is mesoscopic, i.e. when its number of degrees of freedom is large but finite. Here work-extraction from a single equilibrium thermal bath is possible when its temperature is large enough. This result is illustrated by means of exactly solvable models. Finally we consider the Clausius principle: heat goes from high to low temperature. A theorem and some simple consequences for this statement are pointed out.Comment: 6 pages Latex, uses aip-proceedings style files. Proceedings `Quantum Limits to the Second Law', San Diego, July 200

Hamiltonian structure of thermodynamics with gauge

The state of a thermodynamic system being characterized by its set of extensive variables $q^{i}(i=1,...,n) ,$ we write the associated intensive variables $\gamma_{i},$ the partial derivatives of the entropy $S(q^{1},...,q^{n}) \equiv q_{0},$ in the form $\gamma_{i}=-p_{i}/p_{0}$ where $p_{0}$ behaves as a gauge factor. When regarded as independent, the variables $q^{i},p_{i}(i=0,...,n)$ define a space $\mathbb{T}$ having a canonical symplectic structure where they appear as conjugate. A thermodynamic system is represented by a $n+1$-dimensional gauge-invariant Lagrangian submanifold $\mathbb{M}$ of $\mathbb{T}.$ Any thermodynamic process, even dissipative, taking place on $\mathbb{M}$ is represented by a Hamiltonian trajectory in $\mathbb{T},$ governed by a Hamiltonian function which is zero on $\mathbb{M}.$ A mapping between the equations of state of different systems is likewise represented by a canonical transformation in $\mathbb{T}.$ Moreover a natural Riemannian metric exists for any physical system, with the $q^{i}$'s as contravariant variables, the $p_{i}$'s as covariant ones. Illustrative examples are given.Comment: Proofs corrections latex vali.tex, 1 file, 28 pages [SPhT-T00/099], submitted to Eur. Phys. J.

L’héritage d’Henri Poincaré en physique

Où l’on revient, à la lumière des connaissances actuelles, sur certains apports déterminants de Poincaré : le chaos, l’importance des probabilités – et, en partie, la relativité ; ses positions novatrices comme ses positions conservatrices ; ses apports en physique pratique (TSF, théorie du signal) aussi bien qu’en physique théorique. Poincaré fait partie des savants, assez rares, qui ont beaucoup apporté à la fois à la physique et aux mathématiques