Statistical mechanics of probabilistic cellular automata

We investigate the behavior of discrete-time probabilistic cellular automata (PCA), which are Markov processes on spin configurations on ad-dimensional lattice, from a rigorous statistical mechanics point of view. In particular, we exploit, whenever possible, the correspondence between stationary measures on the space-time histories of PCAs on ^d and translation-invariant Gibbs states for a related Hamiltonian on ( ^d+1). This leads to a simple large-deviation formula for the space-time histories of the PCA and a proof that in a high-temperature regime the stationary states of the PCA are Gibbsian. We also obtain results about entropy, fluctuations, and correlation inequalities, and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high-noise regime. We discuss phase transitions in the low-noise (or low-temperature) regime and review Toom's proof of nonergodicity of a certain class of PCAs.     

Solutions of Polyakov's loop equations and their contribution to loop averages

The solutions of the classical equations for the Wilson loop presented by Polyakov are studied forU(1) andSU(2) gauge theories on a lattice. All the solutions are found ford=2 space-time dimensions while a particular set is obtained ford-3,4. A possible application of this result to the computation of quantum loop averages in the weak coupling region is discussed: ind=2 a saddle point approximation is shown to be very accurate and simple in terms of loop variables.     

The27Al(n,d)26Mg reaction at 22 MeV

Deuteron spectra from the reaction²⁷Al(n, d)²⁶Mg has been measured at reaction angles 0°→60°, with a spectrometer comprising three multiwire proportional counters followed by a curved plastic scintillator. Spectra were obtained with an energy resolution of ～0.7 MeV and an angle resolution of ～5°. Angular distributions for transitions to the ground state and excited states at 1.81, 2.93 and 4.3 MeV in²⁶Mg were obtained. Distorted waves analysis of the angular distributions yield spectroscopic factors of 0.29(8), 0.9(3), 0.18(7) and 1.9(6) for the ground and excited states, respectively. The results are compared with previous (d,³He) and (n, d) measurements, and with theoretical predictions.     

On phase transitions in Schlögl's second model

We study Schlögl's second model, characterized by chemical reactions $$\begin{array}{*{20}c} {2X\underset{{k_2 }}{\overset{{k_1 }}{\longleftrightarrow}}3X,} & {X\underset{{k_4 }}{\overset{{k_3 }}{\longleftrightarrow}}0,} \\ \end{array} $$ ind-dimensional space. The reactions are assumed to be local; local fluctuations are fully taken into account, and particle transport occurs via diffusion. In contrast to previous investigations, we find no phase transition whenk ₄≠0 andd<4. Fork ₄=0,k ₃≠0, and 1≦d<4, we find a second-order phase transition which is in the same universality class as the transition in Schlögl's first model. Only ford≧4 we do find the first-order transition found also by previous authors. These claims are supported by extensive Monte Carlo calculations for various realizations of this process on discrete space-time lattices.     

Is there a c-theorem in four dimensions?

The difficulties of extending Zamolodchikov's c-theorem to dimensions d ≠ 2 are discussed. It is shown that, for d even, the one-point function of the trace of the stress tensor on the sphere, S^d, when suitably regularized, defines a c-function, which, at least to one loop order, is decreasing along RG trajectories and is stationary at RG fixed points, where it is proportional to the usual conformal anomaly. It is shown that the existence of such a c-function, if it satisfies these properties to all orders, is consistent with the expected behavior of QCD in four dimensions.     

Ramond-Neveu-Schwarz string with new boundary conditions

For the Ramond-Neveu-Schwarz string in D space-time dimensions we seek boundary conditions which preserve Poincaré invariance in d dimensions, d<D. We obtain twisted closed and twisted open strings preserving Gervais-Sakita supersymmetry. Covariant BRST quantization yields D=10. For some boundary conditions, partition functions exhibit space-time supersymmetry.     

Proton pick-up from nuclei in the middle of the 1p shell

The (d,³He) reaction has been used to excite proton hole states in⁸Li,⁹Be and¹⁰Be. Angular distributions have been measured and have been analyzed in terms of the DWBA to get spectroscopic factors for the considered transitions. Excitation energies and transition strengths are compared with the results of Cohen and Kurath's intermediate coupling shell model calculations, where the two models of the effective interaction produce different results especially for transitions to final states in mass 8 and 10 nuclei. The experimental results are clearly in favour of the (6–16) 2 BME interaction. A positive parity state in¹⁰Be predicted by the calculations has been looked for and found at 9.60 MeV.     

On the bound states of the massless free field in two space-time dimensions

We point out the peculiar kinematics of zero-mass fields in two space-time dimensions by showing that the n-fold tensor product of one-particle representations of the Poincaré group for n ? 2 contains a countable infinity of discrete, zero-mass representations. We relate this result to the well-known “bound” states in every charge sector of the fermion Fock space.     

Uniqueness of Gibbs states for generalP(ϕ)2-weak coupling models by cluster expansion

We consider quantum fields with weak coupling in two space-time dimensions. We prove that the set of their ultraregular Gibbs states consists of only one point and this point is an extremal Gibbs state.     

Critical dynamics of the sine-Gordon model ind=2−ε dimensions

The renormalization scheme of Amit, Goldschmidt and Grinstein is extended tod=2?ε dimensions. The exponent ν of the correlation lengthv ^?1=2ε+O(ε²) is in agreement with the result of Kosterlitz for the Coulomb gas. The exponent η of the correlation function of the sine-Gordon field is η=ε+O(ε²). The scaling form of the dynamical structure factorS(q,ω) of the dynamic sine-Gordon ModelA is studied ind=2?ε dimensions. The dynamic exponentz is found to bez=2+(b?1)ε+O(ε²) for ε≧0. The constantb is given by the integral $$b = \int\limits_0^\infty {dss^{ - 2} \exp } \left( { - 2\int\limits_s^\infty {dxx^{ - 1} e^{ - x} } } \right) = 2,371544...$$      

Markov and stability properties of equilibrium states for nearest-neighbor interactions

Consider models on the lattice ^d with finite spin space per lattice point and nearest-neighbor interaction. Under the condition that the transfer matrix is invertible we use a transfer-matrix formalism to show that each Gibbs state is determined by its restriction to any pair of adjacent (hyper)planes. Thus we prove that (also in multiphase regions) translationally invariant states have a global Markov property. The transfer-matrix formalism permits us to view the correlation functions of a classicald-dimensional system as obtained by a linear functional on a noncommutative (quantum) system in (d – 1)-dimensions. More precisely, for reflection positive classical states and an invertible transfer matrix the linear functional is a state. For such states there is a decomposition theory available implying statements on the ergodic decompositions of the classical state ind dimensions. In this way we show stability properties of _ev ^d -ergodic states and the absence of certain types of breaking of translational invariance.     

Generalizations of CPn−1 with non-abelian gauge symmetry in 2 + ϵ dimensions

This paper considers non-linear σ models having global U(n) and local U(k) symmetries (1 ? k < n) in space-time dimensions d > 2. The special case k = 1 is the CP^n?1 model. The renormalizability to all orders of these models (in the presence of gauge invariant sources) is demonstrated in 2 + ? dimensions. A second order phase transition is shown to occur at a coupling strength of order ?, analogous to that in the O(n) model. Certain critical exponents associated with this transition are evaluated at two loop order. We are able to compare the two phases only in the limit n → ∞, k fixed.     

Supersymmetric gauge anomalies in 2 and 4 dimensions from generalized descent-equations

Via a supersymmetric generalization of the descent-equations we derive in Wess-Zumino gauge explicit expressions for chiral anomalies inN=1 supersymmetric Yang-Mills theory for space-time dimensions 2 and 4.     

The renormalization group and the ϵ expansion

The modern formulation of the renormalization group is explained for both critical phenomena in classical statistical mechanics and quantum field theory. The expansion in ? = 4?d is explained [d is the dimension of space (statistical mechanics) or space-time (quantum field theory)]. The emphasis is on principles, not particular applications. Sections 1–8 provide a self-contained introduction at a fairly elementary level to the statistical mechanical theory. No background is required except for some prior experience with diagrams. In particular, a diagrammatic approximation to an exact renormalization group equation is presented in sections 4 and 5; sections 6–8 include the approximate renormalization group recursion formula and the Feyman graph method for calculating exponents. Sections 10–13 go deeper into renormalization group theory (section 9 presents a calculation of anomalous dimensions). The equivalence of quantum field theory and classical statistical mechanics near the critical point is established in section 10; sections 11–13 concern problems common to both subjects. Specific field theoretic references assume some background in quantum field theory. An exact renormalization group equation is presented in section 11; sections 12 and 13 concern fundamental topological questions.     

The structure ofT=1 levels inA=14 nuclei

States in¹⁴C, populated via the¹¹B(α, p) reaction at 14 MeV bombarding energy, were investigated with the Doppler Shift Attenuation Method (DSAM). The analysis ofpγ coincidence spectra taken in a Ge(Li) detector at 0? and 120? with the particle detector near 0? with respect to the beam direction yielded the following lifetimes and lifetime limits for the states at 6.09, 6.59, 7.01 and 7.34 MeV, respectively, <20 fs, <1,200 fs, <7 fs and 160±60 fs. Shell model calculations using the MSDI and an empirical interaction fitted to nuclear states in 1p and 2s 1d shell nuclei, are shown to account very well for the experimental levels andγ-transition rates of 5 different multipolarities. The structure of the (J ^*,T)=(2⁺, 1) levels is discussed in the light of the experimentally observed T_z-dependence of the 2⁺, 1→0⁺, 1E2 matrix elements.     

Supersymmetries and their representations

We determine all manifest supersymmetries in more than 1 + 1 dimensions, including those with conformal or de Sitter space-time symmetry. For the supersymmetries in flat space we determine the structure of all representations and give formulae for an effective computation. In particular we show that at least for masses m² = 0, 1, 2 the states of the spinning string form supersymmetry multiplets.     

Alpha-particle unstable states in12C

Alpha particle unstable states in¹²C have been investigated using the three reactions¹⁴N(d, α′)¹²C,¹⁴N(d, α′α)⁸Be and¹²C(α, α′)¹²C. Excitation cross sections and angular distributions have been measured for the reactions¹⁴N(d, α′) at 52 MeV and¹²C(α, α′) at 90 MeV. For a few angle pairs the α-decay of excited states in¹²C have been observed in a¹⁴N(d, α′α) correlation measurement. The reactions selectively excite onlyT=0 states. A previously undetected level with a large α-decay width (Γ=1.2 MeV) has been observed at 15.62 MeV excitation. This level shows up clearly in both reactions and is further distinguished from the nearby 14.08 state in the correlation measurement because of the distinctly different energy distributions of the decay products. On the basis of a particle angular distributions the 15.62 MeV level was assigned spin and parity 4⁺ and the level at 14.08 was assigned 3^?. The latter differs from the value suggest by earlier work. Comparison with DWBA calculations indicates that angular distributions of all other prominent levels are in agreement with their earlier assignments. Two levels at 19.20 and 20.30 MeV (both Γ?0.4 MeV) and three further levels at 21.81, 22.7 and 24.24 MeV also decay predominantly by α-emission.