Narrow Results

A careful study of the classical/quantum connection with the aid of coherent states offers new insights into various technical problems. This analysis includes both canonical as well as closely related affine quantization procedures. The new tools are applied to several examples including: (1) A quantum formulation that is invariant under arbitrary classical canonical transformations of coordinates; (2) A toy model that for all positive energy solutions has singularities which are removed at the classical level when the correct quantum corrections are applied; (3) A fairly simple model field theory with non-trivial classical behavior that, when conventionally quantized, becomes trivial, but nevertheless finds a proper solution using the enhanced procedures; (4) A model of scalar field theories with non-trivial classical behavior that, when conventionally quantized, becomes trivial, but nevertheless finds a proper solution using the enhanced procedures; (5) A viable formulation of the kinematics of quantum gravity that respects the strict positivity of the spatial metric in both its classical and quantum versions; and (6) A proposal for a non-trivial quantization of that is ripe for study by Monte Carlo computational methods. All of these examples use fairly general arguments that can be understood by a broad audience.

Increasing tensor powers of the $k\times k$ matrices $M_k(ℂ)$ are known to give rise to a continuous bundle of $C^{\ast }$-algebras over $I=\{0\}\cup 1/\mathbb{N}\subset [0,1]$ with fibers $A_{1/N}=M_k{(ℂ)}^{\otimes N}$ and $A_0=C(X_k)$, where $X_k=S(M_k(ℂ))$, the state space of $M_k(ℂ)$, which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of $X_k$ à la Rieffel, defined by perfectly natural quantization maps $Q_{1/N}:{Ã}_0\to A_{1/N}$ (where ${Ã}_0$ is an equally natural dense Poisson subalgebra of $A_0$).

We apply this quantization formalism to the Curie–Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its ${\mathbb{Z}}_2$ symmetry is spontaneously broken in the thermodynamic limit $N\to \infty$. If this limit is taken with respect to the macroscopic observables of the model (as opposed to the quasi-local observables), it yields a classical theory with phase space $X_2\cong B^3$ (i.e. the unit three-ball in ${ℝ}^3$). Our quantization map then enables us to take the classical limit of the sequence of (unique) algebraic vector states induced by the ground state eigenvectors ${\mathrm{\Psi }}_N^{(0)}$ of this model as $N\to \infty$, in which the sequence converges to a probability measure $\mu$ on the associated classical phase space $X_2$. This measure is a symmetric convex sum of two Dirac measures related by the underlying ${\mathbb{Z}}_2$-symmetry of the model, and as such the classical limit exhibits spontaneous symmetry breaking, too. Our proof of convergence is heavily based on Perelomov-style coherent spin states and at some stage it relies on (quite strong) numerical evidence. Hence the proof is not completely analytic, but somewhat hybrid.

A continuous bundle of $C^{\ast }$-algebras provides a rigorous framework to study the thermodynamic limit of quantum theories. If the bundle admits the additional structure of a strict deformation quantization (in the sense of Rieffel) one is allowed to study the classical limit of the quantum system, i.e. a mathematical formalism that examines the convergence of algebraic quantum states to probability measures on phase space (typically a Poisson or symplectic manifold). In this manner, we first prove the existence of the classical limit of Gibbs states illustrated with a class of Schrödinger operators in the regime where Planck’s constant $\hslash$ appearing in front of the Laplacian approaches zero. We additionally show that the ensuing limit corresponds to the unique probability measure satisfying the so-called classical or static KMS-condition. Subsequently, we conduct a similar study on the free energy of mean-field quantum spin systems in the regime of large particles, and discuss the existence of the classical limit of the relevant Gibbs states. Finally, a short section is devoted to single site quantum spin systems in the large spin limit.

We showed that there is a complete analogue of a representation of the quantum plane where |q| = 1, with the classical ax+b group. We showed that the Fourier transform of the representation of on has a limit (in the dual corepresentation) toward the Mellin transform of the unitary representation of the ax+b group, and furthermore the intertwiners of the tensor products representation has a limit toward the intertwiners of the Mellin transform of the classical ax+b representation. We also wrote explicitly the multiplicative unitary defining the quantum ax+b semigroup and showed that it defines the corepresentation that is dual to the representation of above, and also correspond precisely to the classical family of unitary representation of the ax+b group.

We prove that, for a smooth two-body potential, the quantum mean-field approximation to the nonlinear Schrödinger equation of the Hartree type is stable at the classical limit h → 0, yielding the classical Vlasov equation.

In this paper we have investigated the classical limit in Bohmian quantum cosmology. It is observed that in the quantum regime where the quantum potential is greater than the classical one, one has an expansion in terms of negative powers of the Planck constant. But in the classical limit there are regions having positive powers of the Planck constant, and regions having negative powers and also regions having both. The conclusion is that the Bohmian classical limit cannot be obtained by letting the Planck constant goes to zero.

Classical and quantum mechanical descriptions of motion are fundamentally different. The Universality of Free Fall (UFF) is a distinguishing feature of the classical motion (which has been verified with astonishing precision), while quantum theory tell us only about probabilities and uncertainties thus breaking the UFF. There are strong reasons to believe that the classical description must emerge, under plausible hypothesis, from quantum mechanics. In this paper, we show that the UFF is an emergent phenomenon: the coarse-grained quantum distribution for high-energy levels leads to the classical distribution as the lowest order plus quantum corrections. We estimate the size of these corrections on the Eötvös parameter and discuss the physical implications.

The algebraic properties of a strict deformation quantization are analyzed on the classical phase space ${ℝ}^{2n}$. The corresponding quantization maps enable us to take the limit for $\hslash \to 0$ of a suitable sequence of algebraic vector states induced by $\hslash$-dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on ${ℝ}^{2n}$, defining a classical algebraic state. The observables are here represented in terms of a Berezin quantization map which associates classical observables (functions on the phase space) to quantum observables (elements of $C^{\ast }$ algebras) parametrized by $\hslash$. The existence of this classical limit is in particular proved for ground states of a wide class of Schrödinger operators, where the classical limiting state is obtained in terms of a Haar integral. The support of the classical state (a probability measure on the phase space) is included in certain orbits in ${ℝ}^{2n}$ depending on the symmetry of the potential. In addition, since this $C^{\ast }$-algebraic approach allows for both quantum and classical theories, it is highly suitable to study the theoretical concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by switching off $\hslash$. To this end, a detailed mathematical description is outlined and it is shown how this algebraic approach sheds new light on spontaneous symmetry breaking in several physical models.

We discuss a generalization of the Ehrenfest theorem to the recently proposed precanonical quantization of vielbein gravity which proceeds from a space-time symmetric generalization of the Hamiltonian formalism to field theory. Classical Einstein-Palatini equations are derived as the equations of expectation values of precanonical quantum operators. This is preceded by a consideration of an interacting scalar field theory in curved space-time, which shows how the classical field equations emerge from the results of precanonical quantization. It also allows us to identify the connection term in the covariant generalization of the precanonical Schrödinger equation with the spin connection.