Logic of infinite quantum systems

Limits of sequences of finite-dimensional (AF)C ^*-algebras, such as the CAR algebra for the ideal Fermi gas, are a standard mathematical tool to describe quantum statistical systems arising as thermodynamic limits of finite spin systems. Only in the infinite-volume limit one can, for instance, describe phase transitions as singularities in the thermodynamic potentials, and handle the proliferation of physically inequivalent Hilbert space representations of a system with infinitely many degrees of freedom. As is well known, commutative AFC ^*-algebras correspond to countable Boolean algebras, i.e., algebras of propositions in the classical two-valued calculus. We investigate thenoncommutative logic properties of general AFC ^*-algebras, and their corresponding systems. We stress the interplay between Gödel incompleteness and quotient structures in the light of the nature does not have ideals program, stating that there are no quotient structures in physics. We interpret AFC ^*-algebras as algebras of the infinite-valued calculus of Lukasiewicz, i.e., algebras of propositions in Ulam's twenty questions game with lies.     

On Quasi-static processes in linear response theory

It is shown that the difference between adiabatic and isothermal susceptibilities for homogeneous external forces is accounted for by aδ-singularity at zero frequency in the spectral function of linear response theory.     

Open quantum systems with time-dependent Hamiltonians and their linear response

We give a rigorous (Hamiltonian) treatment of a quantum system weakly coupled to an infinite free reservoir and subject to an external time-dependent driving potential varying on the scale of dissipation. The linear response of the system initially in thermal equilibrium is determined and compared with the usual expressions of linear response theory.Work partly supported by NSF grant no. MPS 75-11864-A04.On leave of absence from Fachbereich Physik der Universität München. Work supported by a DFG fellowship.     

W *-dynamics of infinite quantum systems

We prove that the dynamics of an infinite quantum system, as formulated in the Schrödinger representation within the framework constructed in an earlier work [1], corresponds to ^*-automorphisms of the W ^*-algebra dual to the space of its physical states.     

The dynamics of infinite open quantum systems

We establish the existence of the thermodynamic limit of quantum dynamical semigroups describing the irreversible dynamics of a class of mean-field quantum systems coupled to collective or individual reservoirs.     

On linear response theory and area preserving mappings

In this report the linear response theory of Kubo and the verification of the Kubo formula for electrical conduction by experiments and numerical calculations is reviewed. A survey of area reserving mappings is given, emphasizing the significance of these mappings for mechanics and statistical mechanics. The response of area preserving mappings with different statistical properties is calculated. The relation between the response, the statistical properties of the mappings and the fundamental assumptions of linear response theory is discussed.     

Notes on infinite layer quantum Hall systems

We study the fractional quantum Hall effect in three-dimensional systems consisting of infinitely many stacked two-dimensional electron gases placed in transverse magnetic fields. This limit introduces new features into the bulk physics such as quasiparticles with non-trivial internal structure, irrational braiding phases, and the necessity of a boundary hierarchy construction for interlayer correlated states. The bulk states host a family of surface phases obtained by hybridizing the edge states in each layer. We analyze the surface conduction in these phases by means of sum rule and renormalization group arguments and by explicit computations at weak tunneling in the presence of disorder. We find that in cases where the interlayer electron tunneling is not relevant in the clean limit, the surface phases are chiral semi-metals that conduct only in the presence of disorder or at finite temperature. We show that this class of problems which are naturally formulated as interacting bosonic theories can be fermionized by a general technique that could prove useful in the solution of such “one and a half” dimensional problems.     

On van Kampen's objections against linear response theory

A discussion is given of the derivation of the Kubo-Green formulas. A reformulation of the derivation is presented, which leads to an exact result, involving field-dependent transport coefficients. Kubo's result is obtained as the first-order term in a resolvent expansion. For the general Liouville operator case, in accord with van Kampen's objections, it cannot be argued that the other terms are negligible. However, for a large class of systems, this can be justified. This is shown for systems where dissipation is due to weak interactions, amenable to the Van Hove limit, and having sufficiently short relaxation times.     

On the quantum theory of superradiance in two-and three-level macroscopic systems

Modified quantum kinetic equations for the superradiance in two-and three-level macroscopic systems that interact with electromagnetic fields are derived based on the method for elimination of the boson variables. It is demonstrated that the theoretical predictions are in good agreement with the experimental data.     

Correlation functionals of infinite volume quantum spin systems

The existence and analyticity of the correlation functionals of a quantum lattice in the infinite volume limit is proved. The result is valid at sufficiently high temperatures and for a large class of interactions. Our method estimates the kernelK for a set of Kirkwood-Salzburg equations. While a naive estimate would indicate that K =, we take into account cancellations between different contributions toK in order to show that for sufficiently high temperatures K <1, and this estimate is independent of the volume of the system.Supported in part by the Air Force Office of Scientific Research.     

Strong and weak thermalization of infinite nonintegrable quantum systems

When a nonintegrable system evolves out of equilibrium for a long time, local observables are in general expected to attain stationary expectation values, independent of the details of the initial state. But the thermalization of a closed quantum system is not yet well understood. Here we show that it presents indeed a much richer phenomenology than its classical counterpart. Using a new numerical technique, we identify two distinct regimes, strong and weak, occurring for different initial states. Strong thermalization, intrinsically quantum, happens when instantaneous local expectation values converge to the thermal ones. Weak thermalization, well known in classical systems, shows convergence to thermal values only after time averaging. Remarkably, we find a third group of states showing no thermalization, neither strong nor weak, to the time scales one can reliably simulate.     

On the theory of quantum groups

By using the results of S. L. Woronowicz, we show that for the twisted version of the classical compact matrix groups, the Hopf algebraA _h of representative elements is isomorphic as a co-algebra to the Hopf algebraA _O of representative functions on the classical group. As a consequence,A _h can be identified withA _O as a co-algebra but with an associative product, called the star-product, which is a deformation of the original commutative product ofA _O . Furthermore, the construction of this star product from the original product is connected to the Fourier transformation in a manner which is similar to the construction of quantum mechanics from classical mechanics on phase space. In fact, we shall describe the analog of the Weyl correspondence.     

Entropy production and its positivity in nonlinear response theory of quantum dynamical systems

A formulation ofentropy production is given with the aid of relative entropy in the nonlinear response theory of a quantum dynamical system. It allows a natural interpretation of the quantity in terms of familiar thermodynamic notions, such as force and response current conjugate to it, without sacrificing the full nonlinearity in the perturbing force. For the understanding ofdissipativity aspositive entropy production, the stationarity of states and coarse graining of time scale turn out to be essential, which are implemented by some time averaging procedures involvingalmost periodic external forces. Finally, it is shown that the obtained result reduces, in the linear response regime, to the power dissipation appearing in the well-known fluctuation-dissipation relation.     

About the exactness of the linear response theory

For quantum lattice systems with finite range potentials and integrable space clustering, we prove the linearity of the response theory when dealing with fluctuation observables.     

Semiclassical results in the linear response theory

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 ? (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 ?.     

On the asymptotic behaviour of infinite gradient systems

We consider gradient systems of infinitely many particles in one-dimensional space interacting via a positive invariant pair potential with a hard core. The main assumption is that is strictly convex within the rangeR of (whereR is a fixed number ). Under some technical conditions we prove the following theorems: Let the initial distribution be given by a translation invariant point process onR ¹. Then there exists only one extreme equilibrium state with a given intensityI() satisfyingI()R ^–1, and all ergodic initial distributions with an intensityI()R ^–1 converge weakly ast to the extreme equilibrium state with the same intensity.     

Zero-mass infinite spin representations of the Poincaré group and quantum field theory

It is shown that a local quantized field with a manifestly covariant transformation law under the Poincaré group cannot have nonvanishing matrix elements between the vacuum and an irreducible subspace of zero mass and infinite spin.