Showing papers in "Communications in Mathematical Physics in 1984"
TL;DR: In this article, a non-abelian generalization of the usual formulas for bosonization of fermions in 1+1 dimensions is presented, which is equivalent to a local bose theory which manifestly possesses all the symmetries of the fermi theory.
Abstract: A non-abelian generalization of the usual formulas for bosonization of fermions in 1+1 dimensions is presented. Any fermi theory in 1+1 dimensions is equivalent to a local bose theory which manifestly possesses all the symmetries of the fermi theory.
2,231 citations
TL;DR: In this article, the authors prove that the maximum norm of the vorticity of a solution of the Euler equation is a function of the smoothness of the solution, and that if a solution is initially smooth and loses its regularity at some later time, then the maximum vortivity necessarily grows without bound as the critical time approaches.
Abstract: The authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3-D Euler equations. In other words, if a solution of the Euler equations is initially smooth and loses its regularity at some later time, then the maximum vorticity necessarily grows without bound as the critical time approaches; equivalently, if the vorticity remains bounded, a smooth solution persists.
1,595 citations
TL;DR: Using only the Boson canonical commutation relations and the Riemann-Lebesgue integral, this article constructed a simple theory of stochastic integrals and differentials with respect to the basic field operator processes.
Abstract: Using only the Boson canonical commutation relations and the Riemann-Lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes. This leads to a noncommutative Ito product formula, a realisation of the classical Poisson process in Fock space which gives a noncommutative central limit theorem, the construction of solutions of certain noncommutative stochastic differential equations, and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps. The classical Ito product formula for stochastic differentials with respect to Brownian motion and the Poisson process is a special case.
1,298 citations
TL;DR: The sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages and it is suggested that such undecidability is common in these and other dynamical systems.
Abstract: Self-organizing behaviour in cellular automata is discussed as a computational process. Formal language theory is used to extend dynamical systems theory descriptions of cellular automata. The sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages. Many examples are given. The sizes of the minimal grammars for these languages provide measures of the complexities of the sets. This complexity is usually found to be non-decreasing with time. The limit sets generated by some classes of cellular automata correspond to regular languages. For other classes of cellular automata they appear to correspond to more complicated languages. Many properties of these sets are then formally non-computable. It is suggested that such undecidability is common in these and other dynamical systems.
579 citations
TL;DR: Algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata, and the complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities.
Abstract: Cellular automata are discrete dynamical systems, of simple construction but complex and varied behaviour. Algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata. The complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities. The systems are usually irreversible, and are found to evolve through transients to attractors consisting of cycles sometimes containing a large number of configurations.
429 citations
TL;DR: The behavior of the outgoing light rays in the gravitational collapse of an inhomogeneous spherically symmetric dust cloud is analyzed in this paper, where it is shown that, for an open subset of initial density distributions, the first singular event, which occurs at the center of symmetry, is the vertex of an infinity of future null geodesic cones which intersect future null infinity.
Abstract: The behaviour of the outgoing light rays in the gravitational collapse of an inhomogeneous spherically symmetric dust cloud is analyzed. It is shown that, for an open subset of initial density distributions, the first singular event, which occurs at the center of symmetry, is the vertex of an infinity of future null geodesic cones which intersect future null infinity. The frequency of the corresponding light rays is infinitely redshifted.
398 citations
TL;DR: In this paper, an explicit recurrence relation for the conformal block functions is presented, which permits one to evaluate the X-expansion of these functions order-by-order and appropriate for numerical calculations.
Abstract: An explicit recurrence relation for the conformal block functions is presented. This relation permits one to evaluate theX-expansion of these functions order-by-order and appropriate for numerical calculations.
382 citations
TL;DR: In this paper, the Lipschitz-Killing curvatures of smooth Riemannian manifolds for piecewise flat spaces have been studied in the special case of scalar curvature.
Abstract: We consider analogs of the Lipschitz-Killing curvatures of smooth Riemannian manifolds for piecewise flat spaces. In the special case of scalar curvature, the definition is due to T. Regge; considerations in this spirit date back to J. Steiner. We show that if a piecewise flat space approximates a smooth space in a suitable sense, then the corresponding curvatures are close in the sense of measures.
361 citations
TL;DR: In this article, an explicit parametrisation of the solutions to the Bogomolny equation over ℝ3 is presented. But the parametrization is not explicit.
Abstract: Solutions of Nahm's system of ordinary differential equations are produced by variational methods. This leads to an explicit parametrisation of the solutions to the Bogomolny equation over ℝ3.
295 citations
TL;DR: In this article, it was shown that the Yang-Mills instantons can be described in terms of certain holomorphic bundles on the projective plane, using explicit matrix descriptions arising from monads and an analysis of the corresponding groups of symmetries.
Abstract: We show that the Yang-Mills instantons can be described in terms of certain holomorphic bundles on the projective plane. The proof uses explicit matrix descriptions arising from monads and an analysis of the corresponding groups of symmetries.
273 citations
TL;DR: In this article, the inverse scattering method was used for the calculation of correlation functions in completely integrable quantum models with the R-matrix of XXX-type, including the Bose-gas and the Heisenberg XXX-model.
Abstract: The inverse scattering method approach is developed for calculation of correlation functions in completely integrable quantum models with theR-matrix of XXX-type. These models include the one-dimensional Bose-gas and the Heisenberg XXX-model. The algebraic questions of the problem are considered.
TL;DR: In this article, the existence of maximal surfaces in asymptotically flat spacetime satisfying an interior condition is proved using a priori estimates which can also be applied to prescribed mean curvature surfaces in cosmological spacetimes and the Dirichlet problem.
Abstract: We prove the existence of maximal surfaces in asymptotically flat spacetime satisfying an interior condition. This uses a priori estimates which can also be applied to prescribed mean curvature surfaces in cosmological spacetimes and the Dirichlet problem.
TL;DR: In this article, a new approach to the Pirogov-Sinai theory of phase transitions is developed, not employing the contour models with a parameter, and the completeness of the phase diagram is proven.
Abstract: A new approach to the Pirogov-Sinai theory of phase transitions is developed, not employing the contour models with a parameter. The completeness of the phase diagram is proven.
TL;DR: In this paper, it was shown that the Boltzmann equation can be solved globally in time under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough.
Abstract: Solutions of the Boltzmann equation are proved to exist, globally in time, under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough. It is also proved, as might be expected in this case, that the density of the gas at each point in space goes to zero as time goes to infinity.
TL;DR: In this paper, it was shown that Yang-Mills instantons in four dimensions can naturally be identified with the instantons of a two-dimensional theory with values in the loop group.
Abstract: It is shown that Yang-Mills instantons in four dimensions can naturally be identified with the instantons of a two-dimensional theory with values in the loop group.
TL;DR: In this paper, it was proved that the action minimization problem has a solution, u 0, of finite action and that this solution also minimizes the action within the class {v is a solution and v has finite action, u0}.
Abstract: The system of equations studied in this paper is— Δui = gi(u) on Rd, d≧ 2, with u:Rd-→]Rn and gi(u)=∂G/∂i. Associated with this system is the action, S(u) = f {1/2|2 — G(u)}. Under appropriate conditions on G (which differ for d = 2 and d≧ 3) it is proved that the system has a solution, u 0, of finite action and that this solution also minimizes the action within the class {v is a solution, v has finite action, u0}.
TL;DR: In this paper, it was shown that QCD with a sufficient number of fermions of zero bare mass has physical massless particles, and that result also follows from triangle anomalies, so only their method is novel.
Abstract: We show that QCD with a sufficient number of fermions of zero bare mass has physical massless particles. That result also follows from triangle anomalies, so only our method is novel. Our method involves proving special cases of recently conjectured paramagnetic inequalities for fermions. The proofs are simple applications of the Atiyah-Patodi-Singer theorem on spectral flow.
TL;DR: In this article, a random walk on the one-dimensional semi-lattice was considered, and the authors proved that the moving particle walks mainly occur in a finite neighbourhood of a point depending only on time and a realization of the random environment.
Abstract: We consider a random walk on the one-dimensional semi-lattice ℤ={0, 1, 2,...}. We prove that the moving particle walks mainly in a finite neighbourhood of a point depending only on time and a realization of the random environment. The size of this neighbourhood is estimated. The limit parameters of the walks are also determined.
TL;DR: In this article, the use of the family's index theorem in the study of gravitational anomalies is discussed and the geometrical framework required to apply the index theorem is presented and the relation to gravitational anomalies are discussed.
Abstract: We discuss the use of the family's index theorem in the study of gravitational anomalies. The geometrical framework required to apply the family's index theorem is presented and the relation to gravitational anomalies is discussed. We show how physics necessitates the introduction of the notion oflocal cohomology which is distinct from the ordinary topological cohomology. The recent results of Alvarez-Gaume and Witten are derived by using the family's index theorem.
TL;DR: In this article, a principle of local definiteness is introduced which is needed beyond equations of motion and commutation relations to fix the theory uniquely, and it also allows us to formulate local stability.
Abstract: We discuss quantum fields on Riemannian space-time. A principle of local definiteness is introduced which is needed beyond equations of motion and commutation relations to fix the theory uniquely. It also allows us to formulate local stability. In application to a region with a time-like Killing vector field and horizons it yields the value of the Hawking temperature. The concept of vacuum and particles in a non-stationary metric is treated in the example of the Robertson-Walker metric and some remarks on detectors in non-inertial motion are added.
TL;DR: The Lagrangian based theory of the gravitational field and its sources at the arbitrary background space-time is developed in this paper, where the equations of motion and the energy-momentum tensor of the gravity field are derived by applying the variational principle.
Abstract: The Lagrangian based theory of the gravitational field and its sources at the arbitrary background space-time is developed The equations of motion and the energy-momentum tensor of the gravitational field are derived by applying the variational principle The gauge symmetries of the theory and the associated conservation laws are investigated Some properties of the energymomentum tensor of the gravitational field are described in detail and the examples of its application are given The desire to have the total energymomentum tensor as a source for the linear part of the gravitational field leads to the universal coupling of gravity with other fields (as well as to the self-interaction) and finally to the Einstein theory
TL;DR: In this article, the Callan-Symanzikγ-and β-functions are calculated analytically for Q.E.D. in the limit of a large number of leptons up to terms of order 1/NαπF inclusive, and closed analytic expressions for the coefficients of these terms in their series expansion in powers of K ≡αNαNF/π.
Abstract: The Callan-Symanzikγ- andβ-functions are calculated analytically for Q.E.D. in the limit of a large number of leptons (N
F→∞) up to terms of order 1/N
F inclusive. We give closed analytic expressions for the coefficients of these terms in their series expansion in powers ofK ≡αN
F/π. We have been able to sum these series and to obtain some striking results.
TL;DR: In this article, an Ito product formula is proved for stochastic integrals against Fermion Brownian motion, which can be used to construct unitary processes satisfying the Boson theory.
Abstract: An Ito product formula is proved for stochastic integrals against Fermion Brownian motion, and used to construct unitary processes satisfying stochastic differential equations. As in the corresponding Boson theory [10, 11] these give rise to stochastic dilations of completely positive semigroups.
TL;DR: In this article, the Navier-Stokes flow of an incompressible fluid in a domain Θ is shown to be valid in dimensions ≧ 3 and not 2, and in the limit Θ → ∞.
Abstract: Ruelle has found upper bounds to the magnitude and to the number of non-negative characteristic exponents for the Navier-Stokes flow of an incompressible fluid in a domain Θ. The latter is particularly important because it yields an upper bound to the Hausdorff dimension of attracting sets. However, Ruelle's bound on the number has three deficiences: (i) it relies on some unproved conjectures about certain constants; (ii) it is valid only in dimensions ≧ 3 and not 2; (iii) it is valid only in the limit Θ → ∞. In this paper these deficiences are remedied and, in addition, the final constants in the inequality are improved.
TL;DR: In this paper, the spectral radius of the Perron-Frobenius operator for piecewise expanding transformations is derived and the speed of convergence to equilibrium in such one-dimensional systems is analyzed.
Abstract: We determine the essential spectral radius of the Perron-Frobenius-operator for piecewise expanding transformations considered as an operator on the space of functions of bounded variation and relate the speed of convergence to equilibrium in such one-dimensional systems to the greatest eigenvalues of generalized Perron-Frobenius-operators of the transformations (operators which yield singular invariant measures).
TL;DR: In this article, the Hopf map admits a sourceless, topologically non-trivial gauge field, which is shown to be Spin (9) symmetric and leads to a new generalized duality condition.
Abstract: We will show that the Hopf map\(S^{15} \xrightarrow{{S^7 }}S^8\) admits a sourceless, topologically non-trivial gauge field This result will be cast in the form of a solution to eight dimensional Euclidean Yang-Mills field equations with topological chargeQ=1 This solution is Spin (9) symmetric and leads to a new generalized duality conditionF∧F=±(F∧F)*
TL;DR: In this paper, the dressing method of Zakharov, Mikhailov, and Shabat is shown to be equivalent to a Backlund transformation for an associated, linearly extended system.
Abstract: This work deals with Backlund transformations for the principal SL(n, ℂ) sigma model together with all reduced models with values in Riemannian symmetric spaces. First, the dressing method of Zakharov, Mikhailov, and Shabat is shown, for the case of a meromorphic dressing matrix, to be equivalent to a Backlund transformation for an associated, linearly extended system. Comparison of this multi-Backlund transformation with the composition of ordinary ones leads to a new proof of the permutability theorem. A new method of solution for such multi-Backlund transformations (MBT) is developed, by the introduction of a “soliton correlation matrix” which satisfies a Riccati system equivalent to the MBT. Using the geometric structure of this system, a linearization is achieved, leading to a nonlinear superposition formula expressing the solution explicitly in terms of solutions of a single Backlund transformation through purely linear algebraic relations. A systematic study of all reductions of the system by involutive automorphisms is made, thereby defining the multi-Backlund transformations and their solution for all Riemannian symmetric spaces.
TL;DR: In this article, the authors studied the oscillatory critical amplitudes of the q-state Potts model on a diamond hierarchical lattice and compared the magnitude of the oscillations with geometrical characteristics of the Julia set of zeros of the partition function.
Abstract: We study the oscillatory critical amplitudes of theq-states Potts model on a diamond hierarchical lattice. We consider an example of the generic case (finite critical index), as well as the degenerate case (essential singularity). In both cases, we compare the magnitude of the oscillations with geometrical characteristics of the Julia set of zeroes of the partition function.
TL;DR: In this paper, a viscous incompressible fluid enclosed in a bounded region of ℝ 2 or 3 was considered and subjected to time dependent forces, and bound state estimates for the Schrodinger operator were obtained for the characteristic exponents, entropy (Kolmogorov-Sinai invariant), and Hausdorff dimension of attracting sets.
Abstract: We consider a viscous incompressible fluid enclosed in a bounded region of ℝ2 or ℝ3, and subjected to time dependent forces. Using bound state estimates for the Schrodinger operator, we obtain rigorous bounds for the characteristic exponents, entropy (Kolmogorov-Sinai invariant), and Hausdorff dimension of attracting sets. Our methods are of potential use for more general time evolutions described by nonlinear partial differential equations.
TL;DR: The absence of the analytic continuation for the free energy near the point of the first order phase transition in the d-dimensional Ising model is proved in this paper, where it is shown that thermodynamic functions in the metastable phase do not have certain values and can be derived only with an uncertainty δ.
Abstract: The absence of the analytic continuation for the free energy near the point of the first order phase transition in thed-dimensional Ising model is proved. It is shown that thermodynamic functions in the metastable phase do not have certain values and can be derived only with an uncertaintyδ. The asymptotic expansion near the point of the phase transition yields the values of thermodynamic functions with the same uncertainty.