On converse to Koopman's Lemma

Koopman's Lemma states that if a flow T_t is measure preserving for a measure μ on a constant energy surface Ω, then the flow generates a one parameter family of unitary operators U_t on L² (Ω, μ). We show here a converse, namely that under certain (physically motivated) conditions a unitary operator family U_t can be made to generate a corresponding underlying family T_t of point transformations. This result comes out of questions of independent interest in the study of relationships between reversibility and irreversibility, and has application to the foundations of statistical mechanics. In particular, it establishes the principle often used intuitively in chemistry that a forward moving (e.g., Markov) process that loses information cannot be reversed. In a different setting, it provides the answer to a question in the representation theory of isometries on L^p spaces a Banach-Lamperti theorem). These results also allow an interesting reformulation of Ornstein's isomorphism theorem on Bernoulli systems.     

Binary collision expansion and partial green's functions of Kadanoff-Baym

A new formula for the binary collision expansion of the unitary operator U (t₂, t₁) is proposed. The formula is applied to the expansion of the partial Green's functions of Kadanoff-Baym in powers of the correct binary scattering amplitude. It is shown that certain linked diagrams of left-multidentate structure should be taken into account. The duration of the binary collision is seen to play an important role in the rigorous formulation. Upon neglecting this duration, a useful approximation is found for the analysis of correlations on a macroscopic time scale.     

Higher order field equations II; Limit properties of green's functions

In a previous paper wave propagation was studied according to a sixth-order partial differential equation involving a complex mass M. The corresponding Yang-Feldman integral equations (indicated as SM-YF-equations), were formulated using modified Green's functions G^M_R(x) and G^M_A(x), which then incorporate the partial differential equation together with certain boundary conditions. In this paper certain limit properties of these modified Green's functions are derived: (a) It is shown that for |M| → ∞ the Green's functions G^M_R(x) and G^M_A(x) approach the Green's functions Δ_R(x) and Δ_A(x) of the corresponding KG-equation (Klein-Gordon equation). (b) It is further shown that the asymptotic behaviour of G^M_A(x) and G^M_A(x) is the same as of Δ_R(x) and Δ_A(x) - and also the same as for D_R(x) and D_A(x) for t→ ± ∞, where D_R and D_A are the Green n's functions for the KG-equation with mass zero. It is essential to take limits in the sense of distribution theory in both cases (a) and (b). The property (b) indicates that the wave propagation properties of the SM-YF-equations, the KG-equation with finite mass and the KG-equation with mass zero are closely related in an asymptotic sense.     

Wick's theorem and Fokker-Planck dynamics

The correlation and response functions are studied through a Wick's theorem for a Fokker-Planck system in a non-stationary situation. Non gaussian correlations are considered. The effect of the stochastic force is a dressing of the free correlation function.     

A slice theorem for the space of solutions of Einstein's equations

A slice for the action of a group G on a manifold X at a point x ? X is, roughly speaking, a submanifold S_x which is transverse to the orbits of G near x. Ebin and Palais proved the existence of a slice for the diffeomorphism group of a compact manifold acting on the space of all Riemannian metrics. We prove a slice theorem for the group $\text{D}$ of diffeomorphisms of spacetime acting on the space $\text{E}$ of spatially compact, globally hyperbolic solutions of Einstein's equations. New difficulties beyond those encountered by Ebin and Palais arise because of the Lorentz signature of the spacetime metrics in $\text{E}$ and because $\text{E}$ is not a smooth manifold- it is known to have conical singularities at each spacetime metric with symmetries. These difficulties are overcome through the use of the dynamic formulation of general relativity as an infinite dimensional Hamiltonian system (ADM formalism) and through the use of constant mean curvature foliations of the spacetimes in $\text{E}$. (We devote considerable space to a review and extension of some special properties of constant mean curvature surfaces and foliations that we need.) The conical singularity structure of $\text{E}$, the sympletic aspects of the ADM formalism, and the uniqueness of constant mean curvature foliations play key roles in the proof of the slice theorem for the action of $\text{D}$ on $\text{E}$. As a consequence of this slice theorem, we find that the space $\text{D}$ = $\text{E}$/$\text{D}$ of gravitational degrees of freedom is a stratified manifold with each stratum being a sympletic manifold. The spaces for homogeneous cosmologies of particular Bianchi types give rise to special finite dimensional symplectic strata in this space $\text{G}$. Our results should extend to such coupled field theories as the Einstein-Yang-Mills equations, since the Yang-Mills system in a given background spacetime admits a slice theorem for the action of the gauge transformation group on the space of Yang-Mills solutions, since there is a satisfactory Hamiltonian treatment of the Einstein-Yang-Mills system, and since the singularity structure of the solution set is known.     

K1-Dominance of the Ke3 form factor and Sirlin's relation

We discuss the precise momentum dependence of the K_e3 form factor f₊(t) by studying some of the recent experimental results on K⁰_L → π^± e^? ν decays. The parametrization of f₊(t) based on the assumption of K¹-dominance can considerably improve the agreement of Sirlin's relation with existing data.     

Local versus global conservation and Noether's theorem

A distinction between local and global conservation resolves a long standing dispute between Hilbert and Klein, which inadvertently has precipitated an inadequate use of Noether's theorem in contemporary work.     

On the physical basis for Urbach's rule

The low-energy tail of the E 6 a exciton in GeS obeys Urbach's rule, with Urbach parameters σ₀ = 1.45 ± 0.05, and $\text{h}\text{?}\text{ω}{}_{\mathrm{p}}\text{= 8.7 ± 0.6 meV}$. The energy ?ω_p corresponds to a previously measured rigid-layer vibrational mode which has no associated electric field. This finding is inconsistent with Dow and Redfield's unified theory of Urbach and exponential absorption edges. Our results are consistent with Sumi and Toyazawa's theory of Urbach edges, and with Fivaz and Mooser's model for electron-phonon interactions in layered compounds.     

Weyl's association,Wigner's function and affine geometry

According to Weyl one may associate a function with a dynamical operator; these functions depend on the parameters p and q and can be displayed in a p, q manifold, the W space. In the classical limit the W space becomes the phase space parametrised by the canonical variables. The function associated in this manner with the density operator is Wigner's function. It turns out that if Wigner's function is a delta function it cannot represent the density operator of a physically realisable state unless the argument of the delta-function is linear in the parameters a and q. In all other cases Wigner's function associated with a physically realisable state has a finite width, proportional to $\text{h}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$. Consequently straightness (linear combination of p and q) has a fundamental significance in the W space. Since this property is preserved under linear inhomogeneous transformations the W space will have a geometry generated by these transformations, the affine geometry of Euler, Moebius and Blaschke. In the present note we show how this comes about, how it simplifies the semiclassical approximations of Wigner's function, and makes one understand how in the classical limit this geometry is lost, allowing to be replaced by the geometry of canonical transformations.     

Noether's theorem and invariants of certain nonlinear systems

The invariants of certain nonlinear systems (having the form of inhomogeneous time-dependent harmonic oscillators) proposed by Ray and Reid are derived from Noether's theorem.     

Extending Kraitchman's equations

Kraitchman's method for the determination of the substitution (or r_s) coordinates of an atom in a rotating molecule (asymmetric rotor) from the two sets of three principal moments of inertia of the common isotope of the molecule and the species where the particular atom has been isotopically substituted, has been extended to include the complete coordinate transformation between the principal axis systems of both molecular species. No additional information is required. Several applications are discussed, including a method which helps solve the problem of “small coordinates” by following an appropriate “substitution sequence.”     

Generalized Noether's theorem. I. theory

This article is a theoretical investigation of generalized Noether's theorem, which, though unconcerned with considerations such as coordinate transformations, symmetry, and invariance, is the basic mechanism of conventional Noether's theorem, its extensions, and its inverse. The generalized theorem is a completely new approach to the subject—formally, conceptually, and practically. It is an association, for a set of field equations, of field variations with conserved currents. The theorem is stated from two points of view and analyzed with regard to its interpretation and its formal and conceptual relation to conventional Noether's theorem and extensions, transformation groups, and Hamilton's principle. The inverse theorem is also treated. The role of coordinate transformations in conventional Noether's theorem is analyzed.     

Applications of photoelectron spectroscopy to molecular propertles: XVIII. The vertlcal ionization potentials of diphosphene as calculated by perturbation corrections to koopmans' theorem

The vertlcal ionization potentials (VIP's) of dimethyl diphosphene have been computed using perturbation corrections to Koopmans' theorem. These corrections allow an estimate of the effects of polarization and correlation, and show for this molecule an important contribution of the specific correlation of the ion for the ²A_g and ²B_u states. The results predict for the first three VIP's the ordering ²A_g, ²A_u, ²B_u, in agreement with our previous assignment.     

Devil's attractors and chaos of a driven impact oscillator

The motion of a driven elastic impact oscillator: $\text{x}\text{?}\text{+ 0.4}\text{x}\text{?}\text{+ x =}\text{cos}\text{(ωt)}$, x > 0 and $\text{x}\text{?}\text{(t}{}^{\mathrm{+}}\text{) = -}\text{x}\text{?}\text{(t}{}^{\mathrm{?}}\text{)}$ at x = 0, is studied for ω ≈ 2?4. The oscillator exhibits Feigenbaum's bifurcations (computed δ ≈ 4.70), the Feigenbaum and intermittent transitions to chaos, crises in chaos and a strong hysteresis region for ω ≈ 3.18–3.20 where the impact/period ratios of a group of attractors show the Devil's staircase behaviour with locking values between $\text{3}\text{5}$ and $\text{3}\text{4}$.     

Application of the Green's function method to lattice vibrations in thin films

The Green's function method is used to investigate the lattice vibrations in a simple cubic thin film with nearest neighbour interactions. Appropriate Green's functions for the study of the surface waves for (001) and (011) orientations of the surface are determined. For a thin film with two parallel (001) free surfaces the atomic force tensors at the surface are modified in order to satisfy the rotational invariance condition. The surface waves are determined and a detailed discussion is made for the long wave-length limit k_y = 0 and k_x ? 1. For a given k_x there is a critical thickness below which only an antisymmetrical solution can exist. For the (011) orientation only the case of a single free surface is discussed.     

A computation of Adler's β sum rule II

We improve our previous calculation of Adler's β sum rule by including the Born contributions to the πN states. The result is that the sum rule is only balanced if the weak axial nucleon form factor g_A(q²) and the strong pion-nucleon form factor have roughly the same q² dependance.