Functional integral representations of partition function without limiting procedure. Techniques of calculation of moments

A definition of the Feynman path integral which does not rest on a limiting procedure based on time-slicing has been given by DeWitt-Morette. We present in this paper a discussion of real Gaussian measures and formulate expressions for the quantum statistical partition function directly in terms of measures of integration on the topological vector space ø₀ of continuous functions defined on the time intervalT = (t _a ,t _b ), such thatx(t _a ,t _b )=0 for allx ? ø₀. We give a definition of a measure for the space ø₀ equivalent to the path integral based on the Uhlenbeck-Ornstein probability distribution. We give expressions for the partition function using the Wiener-Feynman measure and the Uhlenbeck-Ornstein measure. As an exercise in the use of the new techniques, we present calculations of moments of potential functions. The techniques will enable one to solve in a rigorous manner practical problems in quantum statistical mechanics.     

Ambiguities in theη-nucleon scattering length determination

The positive sign of theη-nucleon scattering lengthb _ηN was predicted in [1] assuming the resonance mechanism for theπ ^?p → ηn reaction. We demonstrate that

1. the assumption about thet-channel mechanism of the reaction leads tob _ηN < 0 and
2. the experimental data on theη production cross section are equally compatible with both resonance andt-channel mechanisms.

     

On the universality classes of the Henon map

Within an appropriate renormalization framework, we discuss a¹(b) and δ associated with the bifurcation road to chaos of a Hénon-like map generalized as follows: (x_t+1, y_t+1) = (1?a|x_t|^z + y_t, ?bx_t); (b?0, z?1). For fixed z, we obtained (i) only two universality classes, namely the conservative (b = 1) and non-conservative (b≠1) ones and (ii) $\text{a}{}^1\text{(}\text{1}\text{b}\text{) =}\text{a}{}^1\text{(b)}\text{b}{}^{\mathrm{z}}$. For b = 1, δ(z) presents a minimum, and diverges for z → 1 and z → ∞ (this contrasts with the b≠1 case).     

Probing the sign-changeable interaction between dark energy and dark matter with current observations

We consider the models of vacuum energy interacting with cold dark matter in this study, in which the coupling can change sigh during the cosmological evolution. We parameterize the running coupling b by the form b(a) = b_0 a + b_e(1-a), where at the earlytime the coupling is given by a constant b_e and today the coupling is described by another constant b_0. We explore six specific models with(i) Q = b(a)H_0ρ_0,(ii) Q = b(a)H_0ρ_(de),(iii) Q = b(a)H_0ρ_c,(iv) Q = b(a)Hρ_0,(v) Q = b(a)Hρ_(de), and(vi) Q = b(a)Hρ_c.The current observational data sets we use to constrain the models include the JLA compilation of type Ia supernova data, the Planck 2015 distance priors data of cosmic microwave background observation, the baryon acoustic oscillations measurements,and the Hubble constant direct measurement. We find that, for all the models, we have b_0 0 and b_e 0 at around the 1σ level,and b_0 and b_e are in extremely strong anti-correlation. Our results show that the coupling changes sign during the evolution at about the 1σ level, i.e., the energy transfer is from dark matter to dark energy when dark matter dominates the universe and the energy transfer is from dark energy to dark matter when dark energy dominates the universe.     

The phase shifts leading to the broadening and shift of spectral lines

The classical theory of collisional broadening and shift parameters (β, δ) of an isolated spectral line was used to obtain simple analytical formulas for calculating both β and δ. These formulas were obtained on the assumption that the short range interaction is effective only in the broadening while the long range is effective in the shift of the spectral line. These parameters β and δ depend on the limiting phase shifts responsible for broadening η_b and shift ηδ. It was found that the values of η_b and η_δ are not equal to each other as was proposed by Weisskopf η_b=η_δ=1. The maximum and average values of η_b (η_b max, η_b av) and η_δ (η_δ max, η_δ av) were obtained by numerical evaluation, using different inverse power potentials. By introducing these parameters into the approximated formulas for β and δ using Van der Waals and Lennard-Jones potential, it was found that the results of calculations for (β and δ) with different atomic transitions perturbed by different inert gases are in close agreement with earlier results. Those results, obtained earlier, were based on the Lindholm-Foley theory especially with the average values of η_b [η_b av=0.6057] and the maximum values of η_δ [η_δ max=1.57625]. The impact parameters ρ_b and ρ_δ leading to the broadening and shift of the spectral line were also obtained for different interactions. It was found that the end parameter for the broadening ρ_b is not equal to the starting parameter for the shift ρ_δ.     

On a factor associated with the unordered phase of λ-model on a cayley tree

In this paper we consider nearest-neighbours models, where the spin takes values in the set Φ = {η₁, η₁, …, η_q} and is assigned to the vertices of the Cayley tree Γ^k. The Hamiltonian is defined by some given λ-function. We find a condition for the function λ to determine a type of von Neumann algebra generated by the GNS construction associated with the unordered phase of the λ-model. We give also some physical applications of the obtained result.     

γγ→π0 transition and asymptotics of γγ and γγ transitions of unflavored pseudoscalar mesons

For the spacelike momenta k of the virtual photon γ, the π⁰(p)γ(k)γ(k^′) transition form factor is considered in the coupled Schwinger–Dyson and Bethe-Salpeter approach in conjunction with the generalized impulse approximation using the dressed quark–photon–quark vertices of the Ball–Chiu and Curtis-Pennington type. These form factors are compared with the ones predicted by the vector meson dominance, operator product expansion, QCD sum rules, and the perturbative QCD for the large spacelike transferred momenta k. The most important qualitative feature of the asymptotic behavior, namely the 1/k² dependence, is in our approach obtained in the model-independent way. Again model-independently, our approach reproduces also the Adler–Bell–Jackiw anomaly result for the limit of both photons being real. For the case of one highly virtual photon, we find in the closed form the asymptotic expression which is then easily generalized both to the case of other unflavored pseudoscalar mesons P⁰=π⁰,η₈,η₀,η_c,η_b, and to the case of arbitrary virtuality of the other photon.     

Some estimates for magnetic fluctuations in a turbulent medium

A new integral relationship between the fluctuations b(r, t) of a magnetic field and its mean B ₀(r, t) is derived for the steady-state magnetic field in a turbulent medium. This formula provides the estimate 〈b?curlb〉=?B ₀?curlB ₀. Simultaneously, the coefficient of amplification of the mean magnetic field α effect) is obtained: α=(η+β)B ₀? curlB ₀/B ₀ ² . The formula for α allows for a decrease in this coefficient owing to the back action of the magnetic field on the turbulent velocity field. It is shown that the Zel’dovich’s estimate 〈 b ²〉?β/η B ₀ ² for two-dimensional turbulence holds for magnetic fields at the instant the fluctuations 〈a ²〉 of the vector potential, rather than 〈b ²〉, reach a maximum. Here, η and β are the ohmic (molecular) and turbulent diffusion coefficients, respectively. This estimate is refined with allowance made for the fact that the condition for diffusion approximation itself relates the β, b, and B ₀ quantities to each other.     

A simple Fourier photopolarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices

The classic and simplest polarimetric scheme of examining a “sample” by placing it between a pair of linear polarizers and observing the intensity of the transmitted light can be transformed into a powerful photopolarimeter if the two polarizers are synchronously rotated at different speeds and the transmitted flux is linearly detected and its periodic waveform Fourier analyzed. In particular, if the angular speed of rotation of one polarizer is ω and that of the other is 3ω, the detected signal has the waveform, i=a₀ + Σ⁴_n=1a_n cos nω_ft + b_n sin nω_ft, where ω_f=2ω is the fundamental frequency. The nine Fourier amplitudes a₀ and (a_n, b_n), n=1,2,3,4, to be derived by performing a discrete Fourier transform (DFT) of the signal i, determine all nine elements of the 3×3 submatrix M_3×3 obtained by deleting the fourth row and fourth column of the Mueller matrix M. If the sample is nondepolarizing, the absolute values of all the four elements of the equivalent Jones matrix $\text{J}\text{=(J}{}_{\mathrm{ij}}\text{)}$, i,j=1,2 and their angle differences θ_ij?θ₂₂ (where θ_ij=argJ_ij) can be determined.     

Interpreting data from polarized proton beams

The reactions pp → pp and pp → Δ⁺⁺n with polarized beam and/or polarized target are currently under investigation at the Argonne ZGS. We discuss how to interpret various measured quantities in terms of amplitudes whose behavior is familiar (as functions of s, t). For pp total cross sections and elastic scattering, Argonne measurements will yield Im ?₂ (s,t = 0) and the rather complicated combination $\text{2|?}{}_5\text{|}{}^2\text{+}\text{Re}\text{(?}{}_1\text{?}{}_2{}^1\text{? ?}{}_3\text{?}{}_4{}^1\text{)}$, where ?_i (i = 1, … 5) are conventional s-channel helicity amplitudes. The forward direction (t = 0) is of special interest. We find that for both pp → pp and pp → Δ⁺⁺n, polarized beam — polarized target experiments plus the rather general (testable) assumption that amplitudes with the same s-channel helicity flip quantum numbers are proportional, are sufficient to fully determine all non-vanishing amplitudes at t = 0. Numerical estimates of some observables, based on calculations in a specific model, are also given.     

Some t-channel aspects of a geometrical Pomeron

We discuss the geometrical scaling in the case of total cross-sections behaving at high energies like (lns)^η and show that i) the case ?1?η?0 is inconsistent with t-channel unitarity; ii) the structure of the t-channel partial wave amplitudes is $\text{A(t,j)∝}\text{1}\text{(?t)}{}^{\mathrm{1+r}}\text{H(k≡}\text{(j?1)}\text{(?t)}{}^{\mathrm{r}}\text{)}$ with r=η^? and iii) the t-dependence of the j-plane singularities is given for η > 0 by $\text{α(t)=1+a(?t)}{}^{\mathrm{<mtext>1</mtext><mtext>\eta </mtext>}}$ with a being const.     

Crystal and magnetic structures of Y2CrS4

Chromium(II) sulfide, Y₂CrS₄, prepared by a solid-state reaction of Y₂S₃ and CrS, showed an antiferromagnetic transition at 65 K. The neutron diffraction patterns at 10 and 90 K were both well refined with the space group Pca2₁. At 90 K, cell parameters were a=12.5518(13) Å, b=7.5245(8) Å, and c=12.4918(13) Å. At 10 K, magnetic peaks were observed, which could be indexed on the same unit cell. Magnetic moments of chromium ions were parallel to the b-axis and antiferromagnetically ordered in each set of the 4a sites.     

Lower bounds on the slope parameter

A rigorous lower bound on the slope parameter γ(s, t) = d ln A(s, t)/dt is derived for 0 < t < t₀ where A(s, t) is the absorptive part of the elastic scattering amplitude and t₀ is related to the right extreme of the Lehmann-Martin ellipse. When A(s, t) has high-energy behavior like s^α(t)ln^η(t)s…, this lower bound on α(s, t) is used to obtain lower bounds on α′(t) for 0 < t < t₀, which saturate for ‘parabolic trajectories’. We also obtain a lower bound on γ(s, t) for t < 0 which can be used to find the nearforward region in which γ(s, t) cannot vanish.     

Theory of the froissaron exchange

The results of the multi-pomeron exchange theory considered earlier, are shown to depend crucially on the threshold behaviour of the pomeron contribution at t = 4μ_π², under the condition α(0) ? 1 = Δ > α′(0) 4μ_π². The t-channel partial wave f(t) of the multi-pomeron exchange is calculated. In the limit ξ · Δ > 1, where ξ = ln s/4μ², it corresponds to the scattering on a black disk of expanding radius b₀ ≈ a · ξ where a = s/2μ_π ? α′2μ_π. Due to the threshold singularity influence, it does not violate the t-channel unitarity condition. At a sufficiently small value of the froissaron coupling constant g₀₀ ～ Δ³/a², the theory is shown to be simultaneously s-unitary.     

Multiport beam-splitter controlled transformations with postselection of superposition single-photon states

Two components of superposition single-photon state are entering into two different ports of multiport interferometer. Interference effects in which the inputs into the interferometer are controlling the outputs are studied. In particular we study a multiport beam-splitter (BS) configuration, in which eight input operators a₀, a₁, a₂…a₇ are mixed by 12 BS's to produce the output operators b₀, b₁, b₂…b₇. Multiport beam-splitter transformations of superposition single-photon states are analyzed with a possible use of postselection effects.     

Conversion ratio to F-H center pairs from self-trapped excitons in alkali chloride crystals

Abstract

The time delayed double excitation spectroscopy has been utilized to determine the conversion ratio to F-H center pairs from self-trapped excitons(STE_L) at the lowest state (1s[sgrave]_g;a_1g). The final conversion ratios, η_F/(η_F+η_X), were 0.86, 0.49 and 0.20 for NaCl, KCl and RbCl at 14K, respectively. The conversion efficiency (η =η_F+η_X) from STE_L to F-H center pairs(η_F) and to unknown states(η_X) were 0.25, 0.90 and 0.76 for the hole excitation to π_g, while 0.03, 0.01 and 0.01 for the electron excitation to b_1u, b_2u or b_3u, in NaCl, KCl and RbCl, respectively.     

New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation

We analyze the time evolution of mixed state ρ₀ in a dissipative channel, characteristic of a decay constant κ, by virtue of the elegant properties of entangled state representation 〈η|. We find that the matrix element of the mixed state ρ(t) at time t in 〈η| representation is proportional to that of the initial ρ₀ in the decayed entangled state 〈ηe^-κt| representation, accompanying with a Gaussian damping factor . Thus we have a new insight about the nature of the dissipative process.     

Application of K-matrix CMSR'S to single charged pion photoproduction

The background terms, required in addition to Regge poles with some definite properties, in order to describe the high energy behavior of the t-channel single pion photoproduction amplitudes F_t^(?)(v, t), i = 1, …, 4, are investigated in the framework of a reggeized K-matrix model. To this end continuous moment sum rules (CMSR's) are applied, not to the full amplitudes, but just to the two-particle K-matrix amplitudes assumed to show pure Regge pole asymptotic behavior. The K-matrix amplitudes are defined by taking into account in the Heitler equation the elastic (πN) intermediate state alone. Such a definition corresponds to a weak-cut model with only final-state interactions included. It turns out that strong background contributions are still present on the high energy side of the K-matrix CMSR's. The Regge cut contributions generated within our formalism are found to be too weak to account for the background terms in the full amplitudes. This is presumably to a great extent due to the neglect of the (?N) diffraction-dissociation intermediate state in the definition of the K-matrix amplitudes.