The classical limit ofn-vector spin models

It is proved that the free energy of a system ofn-dimensional spins with Kac type potential is equal, in the infinite range zero strength limit, to the free energy of the corresponding Curie-Weiss system in which every spin interacts equally with every other spin.     

On the classical limit of relativistic quantum theories with arbitrary spin

Using the general theory of classical limit developed by the author, we show the existence of classical limit for positive energy representations of the Poincaré group $\text{B}$ of arbitrary spin. The resulting classical phase space is an orbit of $\text{B}$ in the dual of its Lie algebra corresponding to given mass and spin.     

The classical limit of quantum partition functions

We extend Lieb's limit theorem [which asserts that SO(3) quantum spins approachS ² classical spins asL] to general compact Lie groups. We also discuss the classical limit for various continuum systems. To control the compact group case, we discuss coherent states built up from a maximal weight vector in an irreducible representation and we prove that every bounded operator is an integral of projections onto coherent vectors (i.e. every operator has diagonal form).Supported by USNSF Grant MCS-78-01885     

The classical limit for quantum dynamical semigroups

We describe a class of single-particle quantum-mechanical dynamical semigroups which, in the classical limit, give rise to Markov semigroups on phase space.     

Emergence of chaos in quantum systems far from the classical limit

The dynamical status of isolated quantum systems is unclear as conventional measures fail to detect chaos in such systems. However, when quantum systems are subjected to observation--as all experimental systems must be--their dynamics is no longer linear and, in the appropriate limit(s), the evolution of expectation values, conditioned on the observations, closely approaches the behavior of classical trajectories. Here we show, by analyzing a specific example, that microscopic continuously observed quantum systems, even far from any classical limit, can have a positive Lyapunov exponent, and thus be truly chaotic.     

Low temperature phase diagrams for quantum perturbations of classical spin systems

We consider a quantum spin system with Hamiltonian $$H = H^{(0)} + \lambda V,$$ whereH ⁽⁰⁾ is diagonal in a basis ∣s〉=? _x ∣s _x 〉 which may be labeled by the configurationss={s_x} of a suitable classical spin system on ? ^d , $$H^{(0)} |s\rangle = H^{(0)} (s)|s\rangle .$$ We assume thatH ⁽⁰⁾(s) is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitation, whileV is a finite range or exponentially decaying quantum perturbation. Mapping thed dimensional quantum system onto aclassical contour system on ad+1 dimensional lattice, we use standard Pirogov-Sinai theory to show that the low temperature phase diagram of the quantum spin system is a small perturbation of the zero temperature phase diagram of the classical HamiltonianH ⁽⁰⁾, provided λ is sufficiently small. Our method can be applied to bosonic systems without substantial change. The extension to fermionic systems will be discussed in a subsequent paper.     

The classical limit for quantum mechanical correlation functions

For quantum systems of finitely many particles as well as for boson quantum field theories, the classical limit of the expectation values of products of Weyl operators, translated in time by the quantum mechanical Hamiltonian and taken in coherent states centered inx- andp-space around? ^?1/2 (coordinates of a point in classical phase space) are shown to become the exponentials of coordinate functions of the classical orbit in phase space. In the same sense,? ^?1/2 [(quantum operator) (t) — (classical function) (t)] converges to the solution of the linear quantum mechanical system, which is obtained by linearizing the non-linear Heisenberg equations of motion around the classical orbit.     

Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems

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.Research supported in part by the U.S. National Science Foundation under grant MPS 75-11864A Sloan Foundation Fellow     

The van der Waals limit for classical systems

For a -dimensional system of particles with the two-body potentialq(r)+ ^v K(r) and density , it is proved under fairly weak conditions onq andK that the canonical pressure (, ) and chemical potential (, ) tend to definite limits when 0. The limiting functions are absolutely continuous and are given in terms of the derivative of the limiting free energy density which was found in Part I.     

Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems

We consider quantum unbounded spin systems (lattice boson systems) in -dimensional lattice space Z. Under appropriate conditions on the interactions we prove that in a region of high temperatures the Gibbs state is unique, is translationally invariant, and has clustering properties. The main methods we use are the Wiener integral representation, the cluster expansions for zero boundary conditions and for general Gibbs state, and explicitly -dependent probability estimates. For one-dimensional systems we show the uniqueness of Gibbs states for any value of temperature by using the method of perturbed states. We also consider classical unbounded spin systems. We derive necessary estimates so that all of the results for the quantum systems hold for the classical systems by straightforward applications of the methods used in the quantum case.     

Activated scaling of classical and quantum spin glasses

A model for thermally activated dynamics in disordered systems shows that the linear and nonlinear susceptibility follows a generic exponential form with a "critical rounding," chi(1) proportional to chi(3) proportional to [T ln(t/tau(0)')/K](gamma/b phi) exp - [Tt(g)(phi b)ln(t/tau(0)'/K)](nu/b) (T=temperature, t=time, K=barrier constant, t(g) = 1 - T(SG)/T, and T(SG) = transition temperature; gamma>0 for chi(3) and <0 for chi(1)). This model, also valid in the presence of resonant tunneling states at energies K(0) < K [provided that K is replaced by K(0)+2T ln (1/Gamma(0)), where Gamma(0)(2) proportional, variant tunnel splitting of a spin S=1], is potentially applicable to a wide variety of systems opening the way for the study of thermally activated quantum phase transitions. The famous spin-glass system LiHo(x)Y(1-x) seems to follow this model.     

The limit diffusion mechanism of relaxation for spin systems

The diffusion limit theorem for stochastic differential equations is applied to analyse the dynamical evolutions of spin systems. Bloch equations are derived and the stability of asymptotic evolutions is proved. The theory is applied to nuclear magnetic relaxation of two spins.     

The finite group velocity of quantum spin systems

It is shown that if Φ is a finite range interaction of a quantum spin system,τ _t ^Φ the associated group of time translations, τ _x the group of space translations, andA, B local observables, then $$\mathop {\lim }\limits_{\begin{array}{*{20}c} {|t| \to \infty } \\ {|x| > \upsilon |t|} \\ \end{array} } ||[\tau _t^\Phi \tau _x (A),B]||e^{\mu (\upsilon )t} = 0$$ wheneverv is sufficiently large (v>V_Φ) where μ(v)>0. The physical content of the statement is that information can propagate in the system only with a finite group velocity.     

Differential equations of spin dynamics and asymptotic expansion of quantum expectation values near the classical limit. II

The method developed in Part I is used to obtain asymptotic expansions of the functions that represent spin operators and of the quantum expectation values near the classical limit J→∞. We calculate with accuracy to the first quantum correction the correlation functions responsible for the statistical properties of cooperative spontaneous emission (super-radiance) of a gas of twolevel molecules. It is shown that the light emitted by a finite number of molecules does not have coherence of higher orders. The statistics of the field of quantum fluctuations is discussed.     

Dual symmetry in classical and quantum spin theory

Apropos the continuing discussion of the observability of the magnetic charge (the Dirac monopole [1]), there is interest in investigating the dual symmetry of the spin equations of motion in classical and quantum theory. It is known that in dual symmetry theory the presence of a magnetic charge is a simple consequence of the dual invariance of the equations of electrodynamics [2], The matter of the observability of a magnetic charge is not tackled there, and becomes only experimentally verifiable [2, p. 64].Translated from Izvestiya Vysshikh Uchebnkh, Zavedenii, Fizika, No. 7, pp. 56–59, July, 1932.     

Differential equations of spin dynamics and asymptotic expansion of quantum mean values near the classical limit. III

A study is made of the asymptotic behavior of the functional analogs of the coefficients of vector addition (Clebsch-Gordan coefficients) introduced by means of atomic coherent states for large values of the angular momenta. The cases J₁ J₂, J₃ 1 and J₁, J₃ J₂, where J₁ and J₂ are the original angular momenta and J₃ is the resultant, are investigated. It is shown that the investigated functions are transformed for large J into narrow distributions, which makes it possible to expand integrals containing such functions in asymptotic series in powers of J^–1. The quantum rule for adding angular momenta, formulated in functional language, goes over into the classical one in the limit J The possibility of using these relations to describe molecules with rotational degrees of freedom is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 142–148, April, 1977.     

Differential equations of spin dynamics and asymptotic expansion of quantum expectation values near the classical limit. I

A large class of functional representations for spin operators in the spaces of the phase functions of a two-dimensional oscillator and functions on the unit sphere is constructed by means of generating functionals. The group-theoretical meaning of the formalism and its connection with the method of coherent states are discussed. Rules are found for rewriting the operator equations of motion in the form of differential equations for the functions that represent the operators.     

Vortices in quantum spin systems

We examine spin vortices in ferromagnetic quantum Heisenberg models with planar anisotropy on two-dimensional lattices. The symmetry properties and the time evolution of vortices built up from spin-coherent states are studied in detail. Although these states show a dispersion typical for wave packets, important features of classical vortices are conserved. Moreover, the results on symmetry properties provide a construction scheme for vortex-like excitations from exact eigenstates, which have a well-controlled time evolution. Our approach works for arbitrary spin length both on triangular and square lattices. Received 2 October 1998     

Renarks on the classical limit of quantum field theories

Recently, there has been an increasing interest in computing quantum mechanical corrections to solutions of classical field equations. In this note, we want to proceed in the opposite way and we summarize theorems about the classical limit of relativistic quantum field models. These results are a byproduct of the so called constructive approach to quantum field theory.After a section on generalities, we discuss in Section 2 the situation where no phase transitions occur in the limith0 and in Section 3 we reformulate one result in the case where such a transition occurs (Glimmet al. [7]). We discuss the validity of the loop expansion. It seems however that the tools to show the rigorous validity of soliton calculations are not yet prepared.     

The random-walk representation of classical spin systems and correlation inequalities

We use the random-walk representation to prove the first few of a new family of correlation inequalities for ferromagnetic ?⁴ lattice models. These inequalities state that the finite partial sums of the propagator-resummed perturbation expansion for the 4-point function form an alternating set of rigorous upper and lower bounds for the exact 4-point function. Generalizations to 2n-point functions are also given. A simple construction of the continuum ? _d ⁴ quantum field theory (d<4), based on these inequalities, is described in a companion paper.