KMS condition in a Schrödinger picture of the dynamics

Given the C¹-algebra $\text{A}$ of observables, the KMS condition is formulated in terms of the time evolution α¹_t of a set $\text{S}$₀ ? $\text{S}$($\text{A}$) of “physical” states subject to certain natural conditions. α¹_t need not be defined by an automorphism group of $\text{A}$. It is shown that, for a KMS state ω, α¹_t induces a 1-automorphism α^ω_t of the von Neumann algebra π_ω($\text{A}$)″ generated by the representation π_ω belonging to ω.     

The strong topology on the algebra of polynomials

It is shown that under a quite general condition on the operator T (unbounded, symmetric) and on the domain $\text{D}$ for the representation x → T of the algebra $\text{P}$(x) on $\text{D}$ in $\text{P}$(T) the strongest locally convex topology τ_∞ coincides with the strong topology σ^$\text{D}$.     

Quantum logics and chemical kinetics

A statistical theory of chemical kinetics is presented based on the quantum logical concept of chemical observables. The apparatus of Boolean algebra $\text{B}$ is applied for the construction of appropriate composition polynomials referring to any stipulated arrangement of the atomic constituents. A physically motivated probability measure μ(F) is introduced on the field $\text{B}$ of chemical observables, which considers the occurrence of the yes response of a given F ? $\text{B}$. The equations for the time evolution of the species density operators and the master equations for the corresponding number densities are derived. The general treatment is applied to a superposition of elementary substitution reactions (AB)_α + C ? (AC)_β + B. The expressions for the reaction rate coefficients are established.     

On invariant states and the commutant of a group of quasi-free automorphisms of the car algebra

We study primary states of the CAR algebra which are left invariant under quasi-free automorphisms α_U corresponding to unitaries U of a von Neumann algebra $\text{M}$ on the one-particle Hilbert space, and show that they are quasi-free states ?_A corresponding to self-adjoint operators A in $\text{M}$′ with 0 ? A ? 1, under the assumption that $\text{M}$ does not contain any finite type Ifactor direct summands. Next we study automorphisms of the CAR algebra which commute with α_U for U in a von Neumann algebra $\text{M}$ and show that they are quasi-free automorphisms α_U with U in $\text{M}$′ under the same assumption on $\text{M}$ as above. Finally by using the latter result we obtain a generalization of a theorem of Hugenholtz and Kadison [3].     

The uniform and the strong topology on realizations of the algebra of polynomials

It is shown that under quite general assumptions on the operators A₁,…,A_n (unbounded, symmetric) and on the domain $\text{D}$ on the realization $\text{P}\text{(A}{}_1\text{,…,A}{}_{\mathrm{n}}\text{)}$ of the algebra of polynomials $\text{P}\text{(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{)}$, the strongest locally convex topology τ_st coincides with the uniform topology τ_$\text{D}$ as well as with the strong operator topology τ_s. In the case n = 2 some conditions are given, under which these general assumptions are fulfilled.     

States on the current algebra

We introduce the field algebra Σ$\text{D}$(M;ⁿ?ⁿ$\text{g}$) associated with the current algebra $\text{D}$_r(M;$\text{g}$) for the Lie algebra $\text{g}$ over physical space M. The Heisenberg magnet model is generalized to this continuum. It is shown that the Hamiltonian can be given meaning as implementing a derivation of the field algebra in certain representations.We introduce new representations of the current algebra. For example, if G = SU(2), a representation in L²(R³)?³ is [σ(?)F]^j = ε^jkl?_kψ_l for (?_k) = ? in $\text{D}$_r(M;$\text{g}$)(ψ_l = F. This has cyclic subrepresentations with prime parts.     

Stereoelectronic properties of photosynthetic and related systems: Ab initio configuration interaction calculations on the ground and lower excited singlet and triplet states of magnesium chlorin and chlorin

Ab initio configuration interaction wavefunctions and energies are reported for the ground state and many low-lying singlet and triplet states of magnesium chlorin and chlorin, and are employed in an analysis of the electronic absorption spectra of these systems.In chlorin, the calculated visible spectrum consists of two ${}^1\text{(π, π}{}^1\text{)}$ states, the lower energy, y-polarized state exhibiting moderate absorption intensity in contrast to the very weak absorption of the higher energy x-polarized state. The configurational composition of both states is well described by the four-orbital model. Five ${}^1\text{(π, π}{}^1\text{)}$ states are responsible for the Soret band envelope. A moderately intense y-state lies under the low energy edge of the band envelope, while two x-polarized states of moderate and strong intensity, respectively, are responsible for the band maximum. The final two ${}^1\text{(π, π}{}^1\text{)}$ states lie at the high energy edge of the Soret band and introduce a measure of asymmetry into the band envelope. Two ${}^1\text{(n, π}{}^1\text{)}$ states of very weak oscillator strength are also found in this region of the spectrum. All the Soret states are of complex configurational composition, and several of the higher lying states contain contributions from doubly excited configurations.The calculated visible spectrum of magnesium chlorin also consists of two ${}^1\text{(π, π}{}^1\text{)}$ states, with the weakly absorbing x-polarized state lying approximately 200 cm^?1 lower in energy than the moderately intense y-polarized state. The configurational composition of both states is well described by the four-orbital model. Four ${}^1\text{(π, π}{}^1\text{)}$ states constitute the bulk of the intensity in the Soret band envelope. In distinction to chlorin, the moderately intense ${}^1\text{(π, π}{}^1\text{)}$ state at the low energy edge of the band envelope is x-polarized. Two intense ${}^1\text{(π, π}{}^1\text{)}$ states of y- and x-polarization, respectively, constitute the band maximum region, and a single x-polarized state of moderately strong intensity can be assigned to the high energy shoulder of the band envelope. Two other weakly absorbing ${}^1\text{(π, π}{}^1\text{)}$ states are also found in this region, along with another weakly absorbing state of mixed in-plane and out-of-plane polarization. No clearly defined ${}^1\text{(n, π}{}^1\text{)}$ states are observed. As was the case for chlorin, all the Soret states are of complex configurational composition, and some of the higher energy states contain significant contributions from doubly excited configurations.Chlorin and magnesium chlorin both possess three ${}^3\text{(π, π}{}^1\text{)}$ states which lie below S₁ and a single ${}^3\text{(π, π}{}^1\text{)}$ which lies slightly above S₂. All four of the low-lying ${}^3\text{(π, π}{}^1\text{)}$ states in each molecule are well described by the four-orbital model, with T₁ being essentially a single configuration in each case. The remainder of the ${}^3\text{(π, π}{}^1\text{)}$ states are clustered in the same energetic region as the comparable ${}^1\text{(π, π}{}^1\text{)}$ Soret states, with comparably complex configurational compositions.Dipole moments and charge distributions for low-lying singlet and triplet states are also reported, and are used to rationalize chemical reactivity characteristics.     

Quarks and reggeons

We propose a new algebra which will hopefully contribute to the joint understanding of electromagnetic and weak structure functions and of Regge couplings. It involves an infinite number of form factors of arbitrary complex spin J (with Re(J?1) > 0) and definite signature. The algebra holds exactly in a quark-gluon model when quantized canonically on equal light fronts. If all form factors have k = 0 and J = 1, the algebra closes and becomes U(6) × U(6), with the connecting subgroups being associated with pure quarks and pure antiquark currents, respectively. This doubling of the group leads to exact exchange degeneracy if baryons and mesons consist of qqq and q$\text{q}$ wave functions only.To some lowest approximation, Regge couplings can be calculated by representing U(6) × U(6) on properly mixed quark wave functions. With the mixing parameters fixed by the chiral subgroup, these couplings are determined up to an overall normalization.As an illustration, the coupling of the trajectories ?, ω; A₂, f; A₁, D; Z, Z_sg; π, η; B, H trajectories are estimated for the meson resonances ?, ω; π, η; A₂, f; A₁, D; B, H; A₀, σ. The trajectories π, η; B, H are not in the algebra initially but can be inferred from an extension of PCAC to the bilocal currents (“PCBC”).     

Impact parameter analysis of KN and πN scattering in the intermediate energy region

We have evaluated the s-channel-helicity partial-wave amplitudes for $\text{K}\text{N and πN}$ scattering as functions of the impact parameter using partial-wave data in the energy range P_L≈1.0–2.0 GeV/c. We find that the $\text{K}\text{N}$ background and resonance amplitudes exhibit features consistent with the dual absorptive picture for pomeron and f + ω and A₂ + ? exchanges. Comparison of the πN low-energy amplitude with the partial-wave decomposition of a quantitative Regge model gives evidence for local duality between the s-channel resonance and t- and u-channel Regge exchanges.     

Decomposable operators application to K.M.S. weights in a decomposable von Neumann algebra

Let $\text{U}\text{=}\text{?}\text{⊕}\text{U}\text{(ξ) dμ(ξ)}$be a decomposable von Neumann algebra, ? a faithful strictly semi-finite weight on $\text{U}$, σ^?_! the modular automorphism group associated with ?; if $\text{U}$ admits a cyclic and separating vector (or if π_?($\text{U}$)′ is countably decomposable, (π_?, H_?) being the representation of $\text{U}$, given by ?), then we get $\text{?=}\text{?}\text{⊕}\text{?ξ dμ(ξ)}$ (the ?ξ being faithful strictly semi-finite weights) and $\text{σ}{}^{\mathrm{?}}{}_{\mathrm{!}}\text{=}\text{?}\text{?}\text{σ}{}^{\mathrm{?}}{}_{\mathrm{!}}\text{dμ(ξ)}$ (σ^?_! is the modular automorphism group in $\text{U}$(ξ), associated with ?).These results are obtained by developing the functional calculus for decomposable operators. More precisely, let $\text{Δ=}\text{?}\text{⊕}\text{Δ(ξ)dμ(ξ)}$ be a Hermitian (bounded) decomposable operators, and f a complex essentially bounded function on R; then $\text{f(Δ) =}\text{?}\text{?}\text{f(Δ(ξ))dμ(ξ)}$.Extensions of these results are given for nonessentially bounded measurable fields of bounded operators and for unbounded self-adjoint operators commuting with diagonalizable operators.     

High pT π0 production from αα and αp collisions at the CERN ISR

The invariant cross sections for π⁰ meson production in alpha—alpha and alpha—proton collisions at the ISR were meas- ured up to transverse momenta of $\text{7}\text{GeV}\text{c}$ and $\text{8}\text{GeV}\text{c}$, respectively. These measurements are compared with π⁰ production in pp collisions at the same values of $\text{s}\text{/}$nucleon, and the variation of the nuclear A-dependence with p_T is determined.     

On the stochastic theory of a bistable chemical reaction

The stochastic kinetics of a bistable chemical reaction is studied in the birth and death formalism. An elementary perturbation technique allows to estimate the first two nontrivial eigenvalues μ₁ and μ₂ of the evolution matrix and the corresponding engenvectors. This shows that once a quasitationary state is established in a time of order $\text{т}{}_2\text{=∣μ}{}_2\text{∣}{}^{-1}$, the final evolution only changes the probability weight of each stable state, with the relaxation time $\text{т}{}_2\text{=∣μ}{}_1\text{∣}{}^{-1}\text{?т}{}_2$ (Kramer's time). More accurate estimation of μ₂ and μ₁ are proposed and compared with exact numerical results.     

Polarization observables in π-d scattering

Exact calculations of vector and tensor polarizations are presented for the reaction $\text{π +}\text{d}\text{→ π +}\text{d}$ at T_π = 142 MeV. They are aimed at meeting the apparent demand raised by upcoming pertinent experiments. The effects of (i) deuteron D state, (ii) participating π-N partial waves and (iii) coupling of different π-d orbital angular momenta are found to be very important.     

Affine extensions of supersymmetry: The finite case

We examine the graded Poincaré (GP) Lie algebra of supersymmetry with a view to constructing possible affine extensions of the algebra, i.e. extensions of the GP algebra which contain as a subalgebra the Lie algebra $\text{ga}\text{(4,}\text{R}\text{)}$. We restrict our attention in this paper to an examination of the finite extensions. We demonstrate explicitly that if we adjoin only a symmetric tensor generator to the GP algebra, then such a generator cannot generate all the deformations, in particular the shear, of the general affine group GA(4, $\text{R}$). Similarly, we show that adjoining the supersymmetry generator to $\text{ga}\text{(4,}\text{R}\text{)}$ cannot lead to closure of the resulting algebra, even in the trivial case. We further demonstrate that the GLA $\text{ga}\text{(}\text{4}\text{4}\text{,}\text{R}\text{)}$ does not contain the Lie algebra $\text{ga}\text{(4,}\text{R}\text{)}$ represented over the entire superspace upon which $\text{ga}\text{(}\text{4}\text{4}\text{,}\text{R}\text{)}$ is defined.     

An effective method of investigation of positive maps on the set of positive definite operators

Let $\text{A}$₁ be the algebra of linear operators on the n-dimensional Hilbert space $\text{H}$₁, and let $\text{A}$₂ be the algebra of linear operators of the m-dimensional Hilbert space $\text{H}$₂. Let $\text{L}$($\text{A}$₁, $\text{A}$₂) denote the complex space of linear maps from $\text{A}$₁ to $\text{A}$₂. By a positive map we mean an element of the space $\text{L}$($\text{A}$₁, $\text{A}$₂) (superoperator with respect to $\text{H}$₁) which maps positive definite operators in $\text{A}$₁ into positive definite operators in $\text{A}$₂. The aim of this paper is to present an effective method which allows to verify whether a given superoperator Λ∈$\text{L}$($\text{A}$₁, $\text{A}$₂) is a positive map. Besides that necessary and sufficient conditions for the positive definiteness of even-degree forms in many variables are given.     

Etude comparative des structures des alumines β stoechiometrique et non stoechiometrique: Transition de phase dans l'alumine β stoechiometrique (II)

The usual preparation methods of β alumina lead to a non stoichiometric compound $\text{(β“N.S.”)}$ of formula 11Al₂O₃?(1 + x) B₂O with x ≈-0.3; a metastable phase with a composition close to stoichiometry $\text{(β“S”}\text{and}\text{x ? 0)}$ can however also be obtained. X-Ray diffuse scattering studies of this stoichiometric form of silver β alumina reveals a sharp order—disorder phase transition at about 307 K. The low temperature ordered state of the silver ions is found to correspond to a 3D hexagonal superstructure with the lattice constants $\text{a}\text{3}\text{, a}\text{3}\text{, c}$. Above the transition temperature 3D short range order is observed up to about 315 K, where a cross over occurs towards a higher temperature 2D short range state, similar to that previously observed at low temperature in $\text{β}{}_{\mathrm{<mtext>Ag</mtext>}}\text{“N.S.”}$. Above 500 K the conducting silver ions are found to be in a 2D quasi liquid state. A similar type of order—disorder phase transition seems to occur in stoichiometric sodium β alumina at lower temperature. It is concluded that the very particular behaviour namely the absence of phase transition in the usual forms of β alumina is a direct consequence of non stoichiometry.     

Determination of the one-pion exchange potential from pn and pp charge exchange scattering

The πNN vertex function is determined from $\text{d}\text{σ}\text{d}\text{t}$ for pn → np and $\text{p}\text{p →}\text{n}\text{n}$ at 8 GeV/c in the interval 0 < ? t < 0.1 GeV². A “regularor mass” of 3.5m_π=488 MeV is found, corresponding to an “extension” of 0.40 fm of the πNN vertex. The resulting OPE potential is discussed.     

Correlations between two large transverse momentum πO mesons at the CERN ISR

The correlations between two π^O's of high transverse momentum were studied at the CERN ISR at a centre-of-mass energy $\text{s}\text{= 52.7}\text{GeV}$. Results are presented for two π^O's detected with an azimuthal separation near 180° and also near 0°. Angular and p_⊥ correlations are discussed, as well as invariant mass plots. The observations are compared with some theoretical models of high p_⊥ phenomena. The most striking feature of the data is a strong correlation in the transverse momentum of the two π^O's.     

Pion-nuclear total cross sections near the 32, sol32 resonance

The π⁺ total nuclear cross sections for ⁹Be, ¹²C, ¹⁶O, and ²⁷Al have been measured in the energy range 114 to 210 MeV, while π^? data have been taken at 153 MeV. In the center-of-mass coordinate system the π-nucleus total cross sections reach their maximum value near the π-nucleon $\text{(}\text{3}\text{2}\text{,}\text{3}\text{2}\text{)}$ resonance (T_π^C.M.= 1.52 MeV) where they scale as $\text{A}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$. The measured total cross sections exhibit considerable broadening with respect to the π-nucleon resonance.     

The transitional process in the Cu3Au alloy from the disordered state to the ordered state

The ordering process in the Cu₃Au alloy from the disordered state is examined experimentally. The time evolution of the cluster size is observed by measuring the X-ray superstructure line width at several temperatures. It is found that the cluster size grows very slowly in the early stage, and then increases more rapidly in obedience to the $\text{t}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ law which is apparently explained by the theory of Kawasaki et al.