Quantum and Classical Ergodicity of Spinning Particles

We propose to study the L2-norm distance between classical and quantum phase space distributions, where for the latter we choose the Wigner function, as a global phase space indicator of quantum–classical correspondence. For example, this quantity should be the key to understanding the correspondence between quantum and classical Loschmidt echoes. We concentrate on fully chaotic systems with compact (finite) classical phase space. By means of numerical simulations and heuristic arguments we find that the quantum–classical fidelity stays at one up to the Ehrenfest-type time scale, which is proportional to the logarithm of effective Planck constant, and decays exponentially with a maximal classical Lyapunov exponent, after that time.

The structure of the global “quantum phase space” is analyzed for the harmonic oscillator perturbed by a monochromatic wave in the limit when the perturbation amplitude is small. Usually, the phenomenon of quantum resonance was studied in nondegenerate [G. M. Zaslavsky, Chaos in Dynamic Systems (Harwood Academic, Chur, 1985)] and degenerate [Demikhovskii, Kamenev, and Luna-Acosta, Phys. Rev. E 52, 3351 (1995)] classically chaotic systems only in the particular regions of the classical phase space, such as the center of the resonance or near the separatrix. The system under consideration is degenerate, and even an infinitely small perturbation generates in the classical phase space an infinite number of the resonant cells which are arranged in the pattern with the axial symmetry of the order 2μ (where μ is the resonance number). We show analytically that the Husimi functions of all Floquet states (the quantum phase space) have the same symmetry as the classical phase space. This correspondence is demonstrated numerically for the Husimi functions of the Floquet states corresponding to the motion near the elliptic stable points (centers of the classical resonance cells). The derived results are valid in the resonance approximation when the perturbation amplitude is small enough, and the stochastic layers in the classical phase space are exponentially thin. The developed approach can be used for studying a global symmetry of more complicated quantum systems with chaotic behavior.

We now come to the mathematical foundation of quantum mechanics on phase space. In the two previous chapters, motivation for and some of the desired mathematical structure of quantum mechanics on phase space has emerged. In this chapter, we give a rigorous mathematical analysis of the origin, the variety, and the structure of quantum mechanics on phase space. To review, the discovered components of the structure of quantum mechanics on phase space are: (i) A symmetry group; (ii) A classical phase space, Γ; (iii) Representations of the symmetry group by unitary operators acting on the Hilbert space L2(Γ); (iv) Localization operators (P.V.M.’s) on these representations; (v) Harmonic decompositions of these representations into irreducible sub-representations; (vi) Intertwining operators between these irreducible sub-representations and the standard irreducible representations [as commonly used in physics and obtained by the Mackey-Frobenius method of induced representations.] The intertwined images of the (P.V.M.) localization operators become P.O.V.M.’s. (vii) Informational completeness criteria for these P.O.V.M.’s.

We look at the high-lying eigenstates (from the 10,001st to the 13,000th) in the Robnik billiard (defined as a quadratic conformal map of the unit disk) with the shape parameter $\lambda=0.15$. All the 3,000 eigenstates have been numerically calculated and examined in the configuration space and in the phase space which - in comparison with the classical phase space - enabled a clear cut classification of energy levels into regular and irregular. This is the first successful separation of energy levels based on purely dynamical rather than special geometrical symmetry properties. We calculate the fractional measure of regular levels as $\rho_1=0.365\pm 0.01$ which is in remarkable agreement with the classical estimate $\rho_1=0.360\pm 0.001$. This finding confirms the Percival's (1973) classification scheme, the assumption in Berry-Robnik (1984) theory and the rigorous result by Lazutkin (1981,1991). The regular levels obey the Poissonian statistics quite well whereas the irregular sequence exhibits the fractional power law level repulsion and globally Brody-like statistics with $\beta = 0.286\pm0.001$. This is due to the strong localization of irregular eigenstates in the classically chaotic regions. Therefore in the entire spectrum we see that the Berry-Robnik regime is not yet fully established so that the level spacing distribution is correctly captured by the Berry-Robnik-Brody distribution (Prosen and Robnik 1994).

Quantum processes on phase space

We consider the Laplacian with a delta potential (a “point scatterer”) on an irrational torus, where the square of the side ratio is diophantine . The eigenfunctions fall into two classes—“old” eigenfunctions (75%) of the Laplacian which vanish at the support of the delta potential, and therefore are not affected, and “new” eigenfunctions (25%) which are affected, and as a result feature a logarithmic singularity at the location of the delta potential. Within a full density subsequence of the new eigenfunctions we determine all semiclassical measures in the weak coupling regime and show that they are localized along 4 wave vectors in momentum space — we therefore prove the existence of so-called “superscars” as predicted by Bogomolny and Schmit. This result contrasts the phase space equidistribution which is observed for a full density subset of the new eigenfunctions of a point scatterer on a rational torus. Further, in the strong coupling limit we show that a weaker form of localization holds for an essentially full density subsequence of the new eigenvalues; in particular quantum ergodicity does not hold. We also explain how our results can be modified for rectangles with Dirichlet boundary conditions with a point scatterer in the interior. In this case our results extend previous work of Keating, Marklof and Winn who proved the existence of localized semiclassical measures under a clustering condition on the spectrum of the Laplacian.

A new formulation of phase space of classical mechanics is proposed based on a known four-dimensional formulation of the space-time. With space-time represented by a four-dimensional manifold S (with affine structure and stratified by Euclidean three dimensional spaces) the phase space \(\bar \Gamma \) for a single particle is defined as the cotangent bundle of S. Thus \(\bar \Gamma \) is an eight dimensional manifold and a vector bundle with symplectic structure and tensor fields transferred from S. Some remarks on the constants of motion are made. In the general case of a system with n degrees of freedom \(\bar \Gamma \) is a manifold with 2n+2 dimensions.

Geometrical semantics for linear logic (multiplicative fragment)

We examine the relationship between the mathematical structures of classical mechanics, quantum mechanics, and special relativity, with a view toward building a consistent framework for all three. The usual idea of ’’canonical quantization,’’ with its emphasis on the transition from functions over classical phase space to operators, appears to be inconsistent with relativistic covariance. On the other hand, the spectral condition of relativistic quantum mechanics, which is merely a covariant statement that the energy is nonnegative, naturally leads to the construction of a covariant extension of classical phase space. By giving up the idea of sharp ’’localizability’’ in space—known to be at odds with covariance— and adopting instead a notion of ’’soft’’ localizability in phase space, a consistent theory of relativistic quantum mechanics is seen to emerge whose structure naturally incorporates the classical sympletic geometry as well. Furthermore, the new theory deals directly and covariantly with extended particles rather than point particles and is free of the various inconsistencies known to plague the usual theory. The new notion of ’’microlocality’’ in phase space leads to a new form of gauge freedom which is similar to the usual one but simpler and more powerful, using methods of complex analysis. The phase-space version of Yang–Mills theory is worked out.

A wave dynamics of fields φ(p, q; t) ∈ L2(Γ) over the phase space Γ(p, q) of a classical system 𝒮 is derived from the Liouville theorem. We define the energy contained in a given field φ(p, q; t). We show that for a special class of fields, selected on physical grounds, the energy spectrum is given by a time-independent Schrödinger equation. This allows us to associate with 𝒮 an ordinary quantum system Q such that the values of the quantized energy coincide for the fields in the phase space of 𝒮 and for Q. Then we make use of Wiener's stochastic integral based on the theory of Brownian motion to derive probabilities which are the same as those one would obtain through Born's statistical postulate of quantum theory. From this it follows that we can regard normalized fields φ(p, q; t) as ``probability amplitudes'' leading to a probability density function ρ(p, q; t) = φφ* in the sense of Gibbs' statistical mechanics. Our work therefore appears as a bridge between a statistical theory (in the sense of Gibbs) of a mechanical system 𝒮 and the usual quantum theory of the related quantum system Q.

At a fixed point in spacetime (say, x0), gravitational phase space consists of the space of symmetric matrices {Fab} (corresponding to the canonical momentum πab(x0)) and of symmetric matrices {Gab} (corresponding to the canonical metric gab(x0)), where 1 ⩽ a, b ⩽ n, and, crucially, the matrix {Gab} is necessarily positive definite, i.e. ∑uaGabub > 0 whenever ∑(ua)2 > 0. In an alternative quantization procedure, known as metrical quantization, the first and most important ingredient is the specification of a suitable metric on classical phase space. Our choice of phase-space metrics, guided by a recent study of affine quantum gravity, leads to gravitational phase-space geometries which possess constant scalar curvature and may be regarded as higher-dimensional analogues of the Poincaré plane, which applies when n = 1. This result is important because phase spaces endowed with such symmetry lead naturally, via the procedures of metrical quantization, to acceptable Hilbert spaces of high dimension.

Within the scope of simple quantum mechanics we present a semiclassical theory which is exact. While the semiclassical theory of canonical phase space path integrals is now well established [1, 2] we examine here the case where the classical phase space is the two-sphere. After summarizing some relevant features of a classical spin, we briefly discuss the localization of classical phase space integrals and then present an extension for a quantum spin. The semiclassical propagator is employed to solve the Jaynes-Cummings model.

On the similarities between the Wigner distribution function in classical and quantum optics

On classical phase spaces admitting just one complex-differentiable structure, there is no indeterminacy in the choice of the creation operators that create quanta out of a given vacuum. In these cases the notion of a quantum is universal, i.e. independent of the observer on classical phase space. Such is the case in all standard applications of quantum mechanics. However, recent developments suggest that the notion of a quantum may not be universal. Transformations between observers that do not agree on the notion of an elementary quantum are called dualities. Classical phase spaces admitting more than one complex-differentiable structure thus provide a natural framework to study dualities in quantum mechanics. As an example we quantise a classical mechanics whose phase space is a torus and prove explicitly that it exhibits dualities.

The condition for distinguishability of a countably infinite number of pure states by a single measurement is given. Distinguishability is to be understood as the possibility of an unambiguous measurement. For a finite number of states, it is known that the necessary and sufficient condition of distinguishability is that the states are linearly independent. For an infinite number of states, several natural classes of distinguishability can be defined. We give a necessary and sufficient condition for a system of pure states to be distinguishable. It turns out that each level of distinguishability naturally corresponds to one of the generalizations of linear independence to families of infinite vectors. As an important example, we apply the general theory to von Neumann’s lattice, a subsystem of coherent states which corresponds to a lattice in the classical phase space. We prove that the condition for distinguishability is that the area of the fundamental region of the lattice is greater than the Planck constant, and also find subtle behavior on the threshold. These facts reveal the measurement theoretical meaning of the Planck constant and give a justification for the interpretation that it is the smallest unit of area in the phase space. The cases of uncountably many states and of mixed states are also discussed.