Papers by Juerg Froehlich
Communications in Mathematical Physics, Sep 1, 1989
We study the general mathematical structure of unitary rational conformal field theories in two d... more We study the general mathematical structure of unitary rational conformal field theories in two dimensions, starting from the Euclidean Green functions of the scaling fields. We show that, under certain assumptions, the scaling fields of such theories can be written as sums of products of chiral fields. The chiral fields satisfy an algebra whose structure constants are the matrix elements of Yang-Baxter-or braid-matrices whose properties we analyze. The upshot of our analysis is that two-dimensional conformal field theories of the type considered in this paper appear to be constructible from the representation theory of a pair of chiral algebras.
Journal of Mathematical Physics, Apr 1, 2016
The important problem of how to prepare a quantum mechanical system, S, in a specific initial sta... more The important problem of how to prepare a quantum mechanical system, S, in a specific initial state of interest -e.g., for the purposes of some experiment -is addressed. Three distinct methods of state preparation are described. One of these methods has the attractive feature that it enables one to prepare S in a preassigned initial state with certainty; i.e., the probability of success in preparing S in a given state is unity. This method relies on coupling S to an open quantum-mechanical environment, E, in such a way that the dynamics of S ∨ E pulls the state of S towards an "attractor", which is the desired initial state of S. This method is analyzed in detail.
Nonlinearity, Mar 27, 2007
We study solutions close to solitary waves of the pseudo-relativistic Hartree equation describing... more We study solutions close to solitary waves of the pseudo-relativistic Hartree equation describing boson stars under the influence of an external gravitational field. In particular, we analyze the long-time effective dynamics of such solutions. In essence, we establish a (long-time) stability result for solutions describing boson stars that move under the influence of an external gravitational field.
arXiv (Cornell University), Mar 13, 2017
We give a microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinea... more We give a microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schrödinger equation (NLS) from many-body quantum theory. The starting point of our proof is [10] on the time-independent problem and [15] on the corresponding problem on a finite lattice. An important new obstacle in our analysis is the need to work with a cutoff in the number of particles, which breaks the Gaussian structure of the free quantum field and prevents the use of the Wick theorem. We overcome it by the use of means of complex analytic methods. Our methods apply to the nonlocal NLS with bounded convolution potential. In the periodic setting, we also consider the local NLS, arising from short-range interactions in the many-body setting. To that end, we need the dispersion of the NLS in the form of periodic Strichartz estimates in X s,b spaces.
Annales Henri Poincaré, May 30, 2015
The notion of "closed systems" in Quantum Mechanics is discussed. For this purpose, we study two ... more The notion of "closed systems" in Quantum Mechanics is discussed. For this purpose, we study two models of a quantum-mechanical system P spatially far separated from the "rest of the universe" Q. Under reasonable assumptions on the interaction between P and Q, we show that the system P behaves as a closed system if the initial state of P ∨ Q belongs to a large class of states, including ones exhibiting entanglement between P and Q. We use our results to illustrate the non-deterministic nature of quantum mechanics. Studying a specific example, we show that assigning an initial state and a unitary time evolution to a quantum system is generally not sufficient to predict the results of a measurement with certainty.
Communications in Mathematical Physics, Mar 1, 1985
We prove that, for large disorder or near the band tails, the spectrum of the Anderson tight bind... more We prove that, for large disorder or near the band tails, the spectrum of the Anderson tight binding Hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues. The corresponding eigenfunctions are exponentially well localized. These results hold in arbitrary dimension and with probability one. In one dimension, we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder. Our techniques extend to other physical systems which exhibit localization phenomena, such as infinite systems of coupled harmonic oscillators, or random Schrodinger operators in the continuum.
arXiv (Cornell University), Jan 19, 2022
We prove that the complex Euclidean field theory with local quartic self-interaction in two dimen... more We prove that the complex Euclidean field theory with local quartic self-interaction in two dimensions arises as a limit of an interacting Bose gas at positive temperature, when the density of the gas becomes large and the range of the interaction becomes small. The field theory is supported on distributions of negative regularity, which requires a renormalization by divergent mass and energy counterterms. We obtain convergence of the relative partition function and uniform convergence of the renormalized reduced density matrices. The proof is based on three main ingredients: (a) a quantitative analysis of the infinite-dimensional saddle point argument for the functional integral introduced in [32] using continuity properties of Brownian paths, (b) a Nelson-type estimate for a general nonlocal field theory in two dimensions, and (c) repeated Gaussian integration by parts in field space to obtain uniform control on the renormalized correlation functions. As a byproduct of our proof, in two and three dimensions we also extend the results on the mean-field limit from to unbounded interaction potentials satisfying the optimal integrability conditions proposed by Bourgain .
Communications in Mathematical Physics, Jun 1, 1981
We develop a unified approach, based on Araki's relative entropy concept, to proving absence of s... more We develop a unified approach, based on Araki's relative entropy concept, to proving absence of spontaneous breaking -of continuous, internal symmetries and translation invariance in two-dimensional statisticalmechanical systems. More precisely, we show that, under rather general assumptions on the interactions, all equilibrium states of a two-dimensional system have all the symmetries, compact internal and spatial, of the dynamics, except possibly rotation invariance. (Rotation invariance remains unbroken if connected correlations decay more rapidly than the inverse square distance.) We also prove that two-dimensional systems with a non-compact internal symmetry group, like anharmonic crystals, typically do not have Gibbs states.
Communications in Mathematical Physics, Aug 1, 1990
A variety of problems in quantum physics and classical statistical mechanics, in particular the q... more A variety of problems in quantum physics and classical statistical mechanics, in particular the quantization of topological solitons and the statistical mechanics of defects in ordered media, are described. These problems can be studied within a semi-classical approximation, or with the help of low-temperature expansions, respectively. The calculation of the leading term in such expansions gives rise to variational problems for sections of vector bundles characterized by certain topological constraints. Examples of such problems are the quantization of kinks in the two-dimensional λφ 4 theory and the analysis of Bloch walls in a Landau-Ginzburg model of a threedimensional anisotropic ferromagnet. We state a general existence result for variational problems of this kind and develop regularity and decay estimates for solutions of the Landau-Ginzburg model describing Bloch walls with prescribed boundaries. For certain boundary configurations stability results are established. The relation between the minimizers of the Landau-Ginzburg model in a certain strong-coupling limit and minimal surfaces is pursued in some detail. An open question is whether, asymptotically, the stability of the limit (minimal) surface will imply the stability of the minimizers of the Landau-Ginzburg model. Contents 1. Background from Quantum Field Theory and Statistical Mechanics 431 2. General Existence Results 444 3. Regularity Results for the Landau-Ginzburg Model 447 4. Connections with Minimal Surface Theory 453 5. Uniqueness and Stability of Minimizers 458 ' * x<y (a similar formula holds for s and π); see [6]. Defining s(f) : = JJ s(x, t)f(x, t) dx dt, where / is a suitable testing function, one sees from (1.20), that the "soliton form
Mathematical Physics Analysis and Geometry, Aug 1, 2004
In the context of an idealized model describing an atom coupled to black-body radiation at a suff... more In the context of an idealized model describing an atom coupled to black-body radiation at a sufficiently high positive temperature, we show that the atom will end up being ionized in the limit of large times. Mathematically, this is translated into the statement that the coupled system does not have any time-translation invariant state of positive (asymptotic) temperature, and that the expectation value of an arbitrary finite-dimensional projection in an arbitrary initial state of positive (asymptotic) temperature tends to zero, as time tends to infinity. These results are formulated within the general framework of W *dynamical systems, and the proofs are based on Mourre's theory of positive commutators and a new virial theorem. Results on the socalled standard form of a von Neumann algebra play an important role in our analysis.
Communications in Mathematical Physics, Mar 22, 2011
We consider a heavy quantum particle with an internal degree of freedom moving on the d-dimension... more We consider a heavy quantum particle with an internal degree of freedom moving on the d-dimensional lattice Z d (e.g., a heavy atom with finitely many excited states). The particle is coupled to a thermal bath consisting of free relativistic bosons through an interaction of strength λ linear in creation and annihilation operators. The mass of the quantum particle is assumed to be of order λ -2 , and we assume that the internal degree of freedom is coupled "effectively" to the thermal bath. We prove that the motion of the quantum particle is diffusive in d ≥ 4 and for λ small enough.
Communications in Mathematical Physics, May 1, 1990
Two-dimensional, unitary rational conformal field theory is studied from the point of view of the... more Two-dimensional, unitary rational conformal field theory is studied from the point of view of the representation theory of chiral algebras. Chiral algebras are equipped with a family of co-multiplications which serve to define tensor product representations. Chiral vertices arise as Clebsch-Gordan operators from tensor product representations to irreducible subrepresentations of a chiral algebra. The algebra of chiral vertices is studied and shown to give rise to representations of the braid groups determined by Yang-Baxter (braid) matrices. Chiral fusion is analyzed. It is shown that the braid-and fusion matrices determine invariants of knots and links. Connections between the representation theories of chiral algebras and of quantum groups are sketched. Finally, it is shown how the local fields of a conformal field theory can be reconstructed from the chiral vertices of two chiral algebras.
Annales Henri Poincaré, Oct 1, 2003
The general theory of simple transport processes between quantum mechanical reservoirs is reviewe... more The general theory of simple transport processes between quantum mechanical reservoirs is reviewed and extended. We focus on thermoelectric phenomena, involving exchange of energy and particles. The theory is illustrated on the example of two reservoirs of free fermions coupled through a local interaction. We construct a stationary state and determine energy and particle currents with the help of a convergent perturbation series. We explicitly calculate several interesting quantities to lowest order, such as the entropy production, the resistance, and the heat conductivity. Convergence of the perturbation series allows us to prove that they are strictly positive under suitable smallness and regularity assumptions on the interaction between the reservoirs.
Communications in Mathematical Physics, Jun 1, 1987
Quantization of solitons in terms of Euclidean region functional integrals is developed, and Oste... more Quantization of solitons in terms of Euclidean region functional integrals is developed, and Osterwalder-Schrader reconstruction is extended to theories with topological solitons. The quantization method is applied to several lattice field theories with solitons, and the particle structure in the soliton sectors of such theories is analyzed. A construction of magnetic monopoles in the fourdimensional, compact t/(l)-model, in the QED phase, is indicated as well. </?(*!, * 2 )> " _ , , _ ^ .0.
Communications in Mathematical Physics, Oct 1, 1978
A general method for proving the existence of phase transitions is presented and applied to six n... more A general method for proving the existence of phase transitions is presented and applied to six nearest neighbor models, both classical and quantum mechanical, on the two dimensional square lattice. Included are some two dimensional Heisenberg models. All models are anisotropic in the sense that the groundstate is only finitely degenerate. Using our method which combines a Peierls argument with reflection positivity, i.e. chessboard estimates, and the principle of exponential localization we show that five of them have long range order at sufficiently low temperature. A possible exception is the quantum mechanical, anisotropic Heisenberg ferromagnet for which reflection positivity is not proved, but for which the rest of the proof is valid.
Communications in Mathematical Physics, Oct 1, 1978
We use Ginibre's general formulation of Griffiths' inequalities to derive new correlation inequal... more We use Ginibre's general formulation of Griffiths' inequalities to derive new correlation inequalities for two-component classical and quantum mechanical systems of distinguishable particles interacting via two body potentials of positive type. As a consequence we obtain existence of the thermodynamic limit of the thermodynamic and correlation functions in the grand canonical ensemble at arbitrary temperatures and chemical potentials. For a large class of systems we show that the limiting correlation functions are clustering. (In a subsequent article these results are extended to the correlation functions of two-component quantum mechanical gases with Bose-Einstein statistics). Finally, a general construction of the thermodynamic limit of the pressure for gases which are not H-stable, above collapse temperature, is presented.
Communications in Mathematical Physics, Oct 1, 1976
We estimate the canonical and grand canonical partition function in a finite volume and prove sta... more We estimate the canonical and grand canonical partition function in a finite volume and prove stability and existence of the thermodynamic limit for the pressure of two component classical and quantum systems of particles with charge +ε interacting via two body Yukawa -or Coulomb forces. In the case of Coulomb forces we require neutrality. For the classical system in two dimensions there exists a critical temperature T c at and below which the system collapses. For the classical Yukawa system the correlation functions exist for arbitrary fugacity and the general structure of the pure phases can be analyzed completely.
Communications in Mathematical Physics, Sep 1, 1982
We give a new, elementary proof for the existence of a deconfining transition to a massless (QED)... more We give a new, elementary proof for the existence of a deconfining transition to a massless (QED) phase in the four-dimensional U(l) lattice gauge theory and of an intermediate QED phase, accompanied by dynamical restoration of local U(l) in variance, in the four dimensional Z N models, with N large. Our methods can also be used to prove the existence of a phase transition in the XY model in three or more dimensions, in three-and four-dimensional abelian Higgs models, and in more general models admitting some local, abelian gauge invariance.
Communications in Mathematical Physics, Feb 1, 1982
Ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a so... more Ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion. By using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities. As an application of these inequalities we obtain a simple proof of the mass gap for the λ(φ 4 ) 2 quantum field model. We also establish new upper bounds on critical temperatures.
Communications in Mathematical Physics, Dec 1, 1981
We rigorously establish the existence of a Kosterlitz-Thouless transition in the rotator, the Vil... more We rigorously establish the existence of a Kosterlitz-Thouless transition in the rotator, the Villain, the solid-on-solid, and the 7L n models, for n large enough, and in the Coulomb lattice gas, in two dimensions. Our proof is based on an inductive expansion of the Coulomb gas in the sine-Gordon representation, extending over all possible distance scales, which expresses that gas as a convex superposition of dilute gases of neutral molecules whose activities are small if β is sufficiently large. Such gases are known not to exhibit screening. Abelian spin systems are related to a Coulomb gas by means of a duality transformation.
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Papers by Juerg Froehlich