Transport in two dimensions. I the self-diffusion coefficient

The Mori formalism is used to study generalized transport coefficients in two dimensions. All finite multilinear products of the single particle density and momentum density comprise the set of the variables in the calculation of the self-diffusion coefficient. A self-consistent equation, which is a non-linear integral equation, is obtained for the leading asymptotic behavior of the generalized self-diffusion coefficient. An asymptotic solution is presented which for small wavevector (k) and frequency (s) behaves like $\text{In}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(s + k}{}^2\text{D)}{}^{\mathrm{?1}}$, where D has the dimensions of a diffusion coefficient. The mean square and mean fourth displacements of a tagged particle are also calculated. The long time behavior of the momentum correlation function exhibits a tail of the form $\text{[t}\text{In}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(t)]}{}^{\mathrm{?1}}$ whose coefficient is dependent of the intermolecular potential.     

Investigation of the phase of the electromagnetic pion form factor in the asymptotic region

The phase of the electromagnetic pion form factor G(t) in the asymptotic region (t > 1 (GeV)²) is investigated by the dispersion sum-rule method. Using experimental data for G(t) at t < 1 (GeV)² we find that the mean weighted value of the asymptotic phase does not show strong deviations from the asymptotic value of the phase coming from an extrapolation of the data from the ?-resonance region with the help of the Gounaris-Sakurai formula. The contribution of the hypothetical ?-like resonances to the dispersion sum rules is roughly estimated. These estimates and the results of the calculation of the dispersion sum rules show that the existence of strong ?-like elastic resonances is forbidden in the 1–1.7 GeV $\text{t}$ range at least.     

D-state effects in (d, t) and (d, 3He) reactions

The influence of the deuteron and triton [³He] D-state on the form factor required in the DWBA analyses of the ($\text{d}$, t) [($\text{d}$, ³He)] reactions has been studied. The wave functions of Phillips are used for three-nucleon systems because they are designed to yield the correct asymptotic form for the (d, t) and (d, ³He) overlaps. The zero-range parameters D₀ and D₂ which enter into the calculation of the cross section and the tensor analysing power in the DWBA with the local energy approximation are calculated and found to yield values consistent with experimental data.     

Cascading through the 1P1 (1s)(2p) state of fluorine

The ¹P₁ (1s)(2p) state of He-like fluorine produced in a foil-excited fluorine beam is shown to decay with a $\text{t}{}^{\mathrm{<mtext>-3</mtext><mtext>2</mtext>}}$ time function. This behavior is identical to that recently reported for oxygen beams and thus demonstrates that there is no strong Z dependence for this time behavior.     

Correlation functions of the transverse Ising chain at the critical field for large temporal and spatial separations

We study the behavior of 〈σ₀^x(t)σ_n^x(0)〉 and 〈σ₀^y(t)σ_n^y(0)〉 for the transverse Ising chain at the critical magnetic field at T = 0. Explicit results are obtained for the three distinct regions where t → ∞ and n → ∞with $\text{0 ?}\text{n}\text{t}\text{<1}$, $\text{1 <}\text{n}\text{t}$, or $\text{t = n + n}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{(}\text{z}\text{2}\text{)}$ where z is fixed of order one. In this latter region the general Painlevé V solution is shown to reduce to a Painlevé II function. We use our results to discuss the general problem of long-time behavior of Toda equations with slowly decaying initial values.     

Spin diffusion in Heisenberg Ferromagnets in the high temperature limit

A study is made of the time dependence of the memory function given by the mode-mode theory for the Heisenberg Ferromagnet in the high temperature limit. For very long times this function is found to go as $\text{t}{}^{\mathrm{<mtext>?5</mtext><mtext>2</mtext>}}$ exp (-Dq²t/2). This leads to the conclusion that spin diffusion is not predicted in the asymptotic time region, in contrast to what has been stated before.     

Solving the linearized Φ4 equation in external driving fields

The Φ⁴ equation, linearized about a kink at rest, is solved for external driving fields F(t) = Σ_i=n^∞f_itⁱ. The x-dependent part of the solution $\text{u}\text{?}\text{(x, t)}$ is proportional to the translation mode. This implies that also for large times the kink's dynamics is adiabatic, if the external field is sufficiently weak and $\text{|}\text{u}\text{?}\text{|? z1}$|ν| ? 1.     

A light scattering investigation of the incommensurate transition in BaMnF4

The results of a Brillouin scattering investigation of BaMnF₄ near its incommensurate structural phase transition at T_i=247 K are reported. We have observed an anomalous dynamic central peak in the polarized (bb) spectrum, with $\text{q}\text{6}\text{c}$. This feature is visible only very near T_i, and possesses a width which is strongly wavevector dependent. The indicated behavior is Λ=D_tq², with D_t=0.14 ± 0.02 cm²sec^-1. It is identified as the result of a coupling between the LA phonon and the soft mode, where the latter contains a relaxing self energy due to a linear coupling to a diffusive process, possibly entropy fluctuations. The narrow central peak is not visible for $\text{q}\text{6}\text{a}$. In addition, we report an anomalous dispersion (～4%), in the LA mode propagating along $\text{c}$ (V_cc) near T_i.     

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.     

Low momentum transfer theorem for two photon exchange in lepton hadron scattering

The two photon exchange contribution to lepton-hadron scattering is considered. Under the assumptions of Lorentz covariance, gauge invariance, unitarity and analyticity, we prove a low momentum transfer theorem for the relevant amplitudes. Fixed energy dispersion relations tell us that their nonanalytic part, in the neighbourhood of t = 0, is given by the contribution of the two photon cut in the t-channel. The t-channel absorptive parts are obtained from unitarity. Their calculation has as input the two amplitudes corresponding to Compton scattering on the hadron with a pole contribution, and the continuum controlled at low t by the electromagnetic polarizabilities. By means of the dispersion integral, one proves the expansion $\text{k}{}_1\text{(s)+k}{}_2\text{(s)}\text{?t}\text{+k}{}_3\text{(s)t}\text{log}\text{(?t)+O(t)}$ for the continuum contribution, where k₁(s) is the only unknown. Explicit expressions are obtained for the pole contribution as M → ∞, where M is the hadron mass, and for the continuum when (?t) <Λ and (?t) < 4m², where m is the muon mass and Λ is a characteristic parameter of the hadron structure.     

Reducible scale invariance at short distances

It is proven that, using reducible scale invariance at short distances, conformal symmetry implies canonical (Bjorken) scaling, provided diagonal dimensions of dilatation multiplets occuring in the operator product expansion of two electromagnetic currents have the canonical value l_n = 2 + n. If the electromagnetic current itself belongs to such multiplets then the hadron production cross section in e⁺e^? annihilation falls off faster than $\text{1}\text{s}$ at asymptotic energy.     

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.     

On a least upper bound Tc of superconductors with paramagnetic impurities

We have obtained a least upper bound, $\text{k}{}_{\mathrm{B}}\text{T}{}_{\mathrm{c}}\text{? c(μ}{}^1\text{, t)A}$, on the critical temperature T_c of an isotropic superconductor with paramagnetic impurities described by the scattering matrix t for fixed values of μ¹. We have also obtained the corresponding optimal spectrum $\text{α}{}^2\text{F(m) = Aδ[ω?d(μ}{}^1\text{, A]}$. The numerical results for the functions $\text{c(μ}{}^1\text{, t)}\text{and}\text{d(μ}{}^1\text{, t)}$ are presented for $\text{α}{}^1\text{= 0.1}$ and 0.16 in the form of universal curves representing $\text{c(μ}{}^1\text{, t)}\text{and}\text{d(μ}{}^1\text{, t)}$ as functions of the reduced impurity concentration $\text{t}\text{= t/A}$. We have also established an upper limit to the reduced critical concentration $\text{t}{}_{\mathrm{crit}}$ for an arbitrary shape of α²F(ω)1.     

Note on the harmonic oscillator calculation of two-nucleon transfer form factors

In a recent publication, Iano and Pinkston showed that the zero-range form factor for two-nucleon transfer reactions obtained through their shell model calculation was well approximated in the asymptotic region by the one calculated using the standard well-depth procedure. We wish to show that such an agreement is merely due to the restricted space which they have used. It is found that by including a larger working basis, the form factor may increase in the asymptotic region by as much as 50 % over the one obtained by the well-depth procedure; this in turn will bring the theoretical cross section to within $\text{2}\text{3}$ of the experimental one for the reaction ⁴⁰Ca(t, p)⁴²Ca.     

Asymptotic behavior of the pion electromagnetic form factors in Bethe-Salpeter model

Asymptotic behavior of the pion form factors has been discussed treating pion as a bound state of the elementary nucleon and antinucleon system. The matrix-element of the electromagnetic current is written in terms of the Bethe-Salpeter amplitude for the nucleon-antinucleon bound system. Using suitable approximations the Bethe-Salpeter equation for nucleon-antinucleon bound state with nucleons as Dirac spinors has been solved and the corresponding off-shell pion form factors are determined. The form factors are strongly interaction dependent. For coupling $\text{g}{}^2\text{16π}{}^2\text{=}\text{15}\text{4}$ the form factors vanish asymptotically like t^?1 and for $\text{14}\text{4}\text{<}\text{g}{}^2\text{16π}{}^2\text{<}\text{15}\text{4}$ the form factors still vanish but less rapidly than t^?1.     

Asymptotic planarity: An S-matrix basis for the Okubo-Zweig-Iizuka rule

A mechanism is exhibited that monotonically depresses the cylinder component of the topological expansion with increasing t, and it is conjectured that all non-planar S-matrix components diminish as t increases, exchange degeneracy and the Okubo-Zweig-Iizuka rule becoming more accurately satisfied. Such asymptotic planarity is compared to the field-theoretical concept of asymptotic freedom. The characteristics low-t cylinder “quenching interval” is found to be the inverse of the mean value over a two-reggeon loop, of $\text{1}\text{2}\text{π}{}^2\text{(α′)}{}^2\text{(t}{}_1\text{? t}{}_2\text{)}{}^2\text{/(?t),}\text{where}\text{t}{}_1\text{and}\text{t}{}_2$ are the squared masses of the loop reggeons and α′ is the trajectory slope. For leading trajectories the low-t cylinder quenching interval is predicted by this formula to be roughly 0.5 GeV²-consistent with the observed pomeron intercept and slope, with the p-ω and f-A₂ mass differences and with the (φ,ω) deviation from ideal mixing. As t grows negatively over a corresponding interval, it is predicted that the pomeron will become nearly a pure SU(3) singlet. If the pion mass helps to set the scale for reggeon loops coupled to unnatural-parity trajectories, the cylinder quenching interval will be larger, explaining the large (η, η′) deviation from ideal mixing as well as the large π-η mass difference. Even when the small-t cylinder quenching is rapid (“precocious planarity”) the large-t approach to the planar limit turns out to be gentle. A by-product of this study is an explanation of the approximate reality and linearity of trajectories at large t.     

Supersymmetry and the division algebras

We show how spinors in space-times of dimension D = t + s, where t is the time dimension, are associated for s - t = 1, 2, 4, 8 (and if t = 0, 1, 2) with the number systems (division algebras), $\text{|R, C, H, O}$. For t = 1 and s - t = 1, 2, 4 this association is “realized” by the sequence of Lorentz groups S1(2,$\text{|R}$), S1(2;$\text{|C}$), S1(2;$\text{|H}$) for D = 3, 4, 6 respectively. We discuss how octonions may be related to D = 10. For D = 6 we give details of S1(2; $\text{|H}$) spinors and construct supersymmetric models with them. These results explain various “empirical” observations in the literature relating quaternions and supersymmetry.     

Coupled channel study on the S-wave ππ → KK interaction

We perform a dispersion relation type calculation of the I = 0 S-wave amplitude g₀⁺ (t) for the process $\text{ππ →}\text{K}\text{K}$, in the region from t = 4μ² to t = (1100 MeV)². Crossing is imposed by generalizing the newly developed hyperbolic partial-wave relations to our reaction and by imposing them on our amplitude using experimental Kπ phase-shift information as input. Analyticity and unitarity is imposed by generalizing the formalism of BFP [1] for parametrizing partial-wave amplitudes. This readily allows us to impose the experimental $\text{ππ →}\text{K}\text{K}$ cross section. Finally the low energy behaviour is constrained to lie within certain limits recently deduced from fixed-tKπ dispersion relation studies [2] and including the current algebra prediction as a special case. We are able to resolve a previous controversy regarding the sign of Im g₀⁽⁺⁾ (t).