In Search of Physical Solutions in Quantum Field Theory

On the basis of the idea of the fictitious evolution of a system, equations for n-pfs have been expressed in the language of classical mechanics. Imposing ideal reonomic constraints upon possible evolutions, we derive equations in which only negative powers of the coupling constant appear. With the help of non-ideal constraints unbranching equations for n-pfs are derived. Constraints inducing gauge symmetric evolution of the system are considered.     

Integrable Structures in Classical Off-Shell 10D Supersymmetric Yang–Mills Theory

The field equations of supersymmetric Yang–Mills theory in ten dimensions may be formulated as vanishing curvature conditions on light-like rays in superspace. In this article, we investigate the physical content of the modified SO(7) covariant superspace constraints put forward earlier [11]. To this end, group-algebraic methods are developed which allow to derive the set of physical fields and their equations of motion from the superfield expansion of the supercurl, systematically. A set of integrable superspace constraints is identified which drastically reduces the field content of the unconstrained superfield but leaves the spectrum including the original Yang–Mills vector field completely off-shell. A weaker set of constraints gives rise to additional fields obeying first order differential equations. Geometrically, the SO(7) covariant superspace constraints descend from a truncation of Witten's original linear system to particular one-parameter families of light-like rays. Received: 20 April 2000 / Accepted: 10 September 2000     

Geometry of Virasoro constraints in nonperturbative 2-d quantum gravity

The string equation and the Virasoro constraints for arbitrary hermitian multimatrix models are derived using the Lie-Bäcklund symmetries of the generalised KdV equations. From this point of view the origin of the string equation and the meaning of the Virasoro constraints are explained. Some speculation about the appearance of extra constraints, which we suspect to be theW-constraints, is also given.     

Constraints and field equations for ten dimensional super Yang-Mills theory

We give a complete proof of the equivalence between constraint equations and field equations for thed=10,N=1 supersymmetric Yang-Mills theory, a result proposed and partially proved recently by Witten [1]. Our approach explicitly reconstructs the superconnection satisfying the constraints from the on shell component fields. A key ingredient of the method is the choice of a suitable family of gauges, effectively eliminating all gauge dependence on anti-commuting co-ordinates. As a corollary, obtained by dimensional reduction, we also deduce the equivalence of constraints and field equations for thed=4,N=4 theory, as well as ford=6,N=2.Research supported in part by the National Science and Engineering Research Council of Canada and the Israel Ministry of ScienceOn leave of absence from: Department of Pure and Applied Mathematics, Stevens Institute of Technology, Hoboken NJ 07030, USA     

On the problem of constraints in minimally coupled relativistic wave equations for particles of unique mass

We study the problem of a possible change in the number of constraints in linear relativistic wave equations (-iβ _μ ∂ ^μ +m)ψ=0 for particles of unique mass, on introduction of minimal coupling to an external electromagnetic field. Complementing our earlier work in which we obtained conditions for non-loss of constraints in equations characterised by the minimalβ-algebraβ ₀ ⁵ =β ₀ ³ we derive here the conditions for such theories not to generate more constraints than in the free case. The results are illustrated by considering specific equations and a fallacy in certain conclusions of Kobayashi and Shamaly on this problem is pointed out.     

Two-dimensionalR n-gravitation

A system with constraints is considered: a string theory whose Lagrangian is thenth power of the Gauss curvature of a space-time manifold (n∈N,n>1). The problem is solved exactly because after the constraints are utilized we deal with a variational problem with a trivial Lagrangian, i.e., its Euler-Lagrange equations are satisfied identically. One can say that the constraints “swallow” all dynamical degrees of freedom of the field theory. The investigation is a continuation of the 1989 work of Burlankov and Pavlov, who solved the problem of two-dimensionalR ²-gravitation under the gauge γ=1.     

Light scattering by longitudinal acoustic modes in supercooled molecular liquids II: microscopic derivation of the phenomenological equations

The constitutive equations for the orientational dynamics of a liquid formed of linear molecules are derived microscopically. The resulting generalised Langevin equations coincide with the phenomenological approach of Dreyfus et al. [1]. Formally exact expressions are given for the phenomenological coefficients and various constraints are shown to be consequences of this microscopic approach. Received 14 August 2002 Published online 4 February 2003 RID="a" ID="a"e-mail: franosch@hmi.de     

Anomalous Scaling of Structure Functions and Dynamic Constraints on Turbulence Simulations

The connection between anomalous scaling of structure functions (intermittency) and numerical methods for turbulence simulations is discussed. It is argued that the computational work for direct numerical simulations (DNS) of fully developed turbulence increases as Re ⁴, and not as Re ³ expected from Kolmogorov’s theory, where Re is a large-scale Reynolds number. Various relations for the moments of acceleration and velocity derivatives are derived. An infinite set of exact constraints on dynamically consistent subgrid models for Large Eddy Simulations (LES) is derived from the Navier–Stokes equations, and some problems of principle associated with existing LES models are highlighted     

A type of new conserved quantity deduced from Mei symmetry for Nielsen equations in a holonomic system with unilateral constraints

<正>A type of new conserved quantity deduced from Mei symmetry for Nielsen equations in a holonomic system with unilateral constraints is investigated.Nielsen equations and differential equations of motion for the holonomic mechanical system with unilateral constraints are established.The definition and the criterion of Mei symmetry for Nielsen equations in the holonomic systems with unilateral constraints under the infinitesimal transformations of Lie group are also given.The expressions of the structural equation and a type of new conserved quantity of Mei symmetry for Nielsen equations in the holonomic system with unilateral constraints are obtained.An example is given to illustrate the application of the results.     

A new view of the ECSK theory with a Dirac spinor

A formulation of the ECSK (Einstein-Cartan-Sciama-Kibble) theory with a Dirac spinor is given in terms of differential forms with values in exterior vector bundles associated with a fixed principalSL(2, )-bundle over a 4-manifold. In particular, tetrad fields are represented as soldering forms. In this setting, both the scalar curvature (Einstein-Hilbert) action density and the Dirac action density are well-defined polynomial functions of the soldering form and an independentSL(2,)-connection form. Thus, these densities are defined even where the tetrad field is degenerate (e.g. when fluctuations in the gravitational field are large). A careful analysis of the initial-value problem (in terms of an evolving triad field, SU(2)-connection, second-fundamental form and spinor field) reveals a first-order hyperbolic system of 27 evolution equations (not including the 8 evolution equations for the Dirac spinor) and 16 constraints. There are 10 conservation equations (due to local Poincaré invariance) which team up with some of the evolution equations to guarantee that the 16 constraints are preserved under the evolution.     

Geometrization of the physics with teleparallelism. I. The classical interactions

A connection viewed from the perspective of integration has the Bianchi identities as constraints. It is shown that the removal of these constraints admits a natural solution on manifolds endowed with a metric and teleparallelism. In the process, the equations of structure and the Bianchi identities take standard forms of field equations and conservation laws.The Levi-Civita (part of the) connection ends up as the potential for the gravity sector, where the source is geometric and tensorial and contains an explicit gravitational contribution.Nonlinear field equations for the torsion result. In a low-energy approximation (linearity andlow energy-momentumtransfer), the postulate that only charge and velocities contribute to the source transforms these equations into the Maxwell system. Moreover, the affine geodesics become the equations of motion of special relativity with Lorentz force in the same approximation [J. G. Vargas,Found. Phys. 21, 379 (1991)]. The field equations for the torsion must then be viewed as applying to an electromagnetic/strong interaction.A classical unified theory thus arises where the underlying geometry confers their contrasting characters to Maxwell-Lorentz electrodynamics and to an Einstein's-like theory of gravity. The highly compact field equations must, however, be developed in phase-spacetime, since the connection is velocity-dependent, i.e., Finsler-like.Further opportunities for similarities with present-day physics are discussed: (a) teleparallelism allows for the formulation of the torsion sector of the theory as a flat space theory with concomitant point-dependent transformations; (b) spinors should replace Lorentz frames in their role as the subjects to which the connection refers; (c) the Dirac equation consistent with the frame bundle for a velocity-dependent metric with Lorentz signature generates a weak-like interaction in the torsion sector.Work done at the Department of Mathematics and Physics of the Interamerican University of Puerto Rico, San German, Puerto Rico 00683.     

Formulation of Hamiltonian Equations for Fractional Variational Problems

An extension of Riewes fractional Hamiltonian formulation is presented for fractional constrained systems. The conditions of consistency of the set of constraints with equations of motion are investigated. Three examples of fractional constrained systems are analyzed in details.On leave of absence from Institute of Space Sciences, P.O.BOX MG-23, R 76900, Magurele–Bucharest, Romania     

On N-wave type systems and their gauge equivalent

The class of nonlinear evolution equations (NLEE) - gauge equivalent to the N-wave equations related to the simple Lie algebra are derived and analyzed. They are written in terms of (x, t) ∈ satisfying r = rank nonlinear constraints. The corresponding Lax pairs and the time evolution of the scattering data are found. The Zakharov-Shabat dressing method is appropriately modified to construct their soliton solutions. Received 20 October 2001 / Received in final form 30 April 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: gerjikov@inrne.bas.bg     

利用杨辉三角形对称性推导高阶运动微分方程

利用Mathematica数学软件计算函数r=r(q(t),t)各变量之间偏导和高阶导数的关系，发现具有杨辉三角形对称性.结合杨辉三角形的对称性规律和牛顿第二定律推导出了高阶运动微分方程，并讨论了理想约束系统下的高阶运动微分方程. 关键词： 杨辉三角形 牛顿第二定律 高阶运动微分方程 高阶力变率 高阶速度能量 理想约束     

Exactly solvable models through the generalized empty interval method,for multi-species interactions

Multi-species reaction-diffusion systems, with nearest-neighbor interaction on a one-dimensional lattice are considered. Necessary and sufficient constraints on the interaction rates are obtained, that guarantee the closedness of the time evolution equation for E _n(t)'s, the expectation value of the product of certain linear combination of the number operators on n consecutive sites at time t. The constraints are solved for the single-species left-right-symmetric systems. Also, examples of multi-species system for which the evolution equations of E _n(t)'s are closed, are given. Received 25 September 2002 / Received in final form 3 December 2002 Published online 14 February 2003 RID="a" ID="a"e-mail: mamwad@iasbs.ac.ir     

BPS relations from spectral problems and blowup equations

Recently, an exact duality between topological string and the spectral theory of operators constructed from mirror curves to toric Calabi–Yau threefold has been proposed. At the same time, an exact quantization condition for the cluster integrable systems associated with these geometries has been conjectured. The consistency between the two approaches leads to an infinite set of constraints for the refined BPS invariants of the toric Calabi–Yau threefold. We prove these constraints for the \(Y^{{N},m}\) geometries using the K-theoretic blowup equations for SU(N) SYM with generic Chern–Simons invariant m.

     

A complete action for the spinning string

We present an action for the Neveu-Schwarz-Ramond model from which follow both the field equations and the gauge and supergauge constraints. This is done by coupling the free-field action to two-dimensional supergravity in a geometrically clear way. The constraints arise as the supergravity field equations, the supergravity fields playing the role of Lagrange multipliers. The action is invariant under local supersymmetry transformations and, as a consequence, the field equations and the constraints are consistent. The commutator structure of the local supersymmetry algebra is exhibited. It is also shown that there exists a special gauge in which the action, the field equations and the constraints take the free-field from of the usual formulation of the Neveu-Schwarz-Ramond model.     

Review: Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge

We define the rest-frame instant form of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find a quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges. The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.     

Ambient space formulations and statistical mechanics of holonomically constrained Langevin systems

The most classic approach to the dynamics of an n-dimensional mechanical system constrained by d independent holonomic constraints is to pick explicitly a new set of (n − d) curvilinear coordinatesparametrizingthe manifold of configurations satisfying the constraints, and to compute the Lagrangian generating the unconstrained dynamics in these (n − d) configuration coordinates. Starting from this Lagrangian an unconstrained Hamiltonian H(q,p) on 2(n−d) dimensional phase space can then typically be defined in the standard way via a Legendre transform. Furthermore, if the system is in contact with a heat bath, the associated Langevin and Fokker-Planck equations can be introduced. Provided that an appropriate fluctuation-dissipation condition is satisfied, there will be a canonical equilibrium distribution of the Gibbs form exp(−βH) with respect to the flat measure dqdp in these 2(n − d) dimensional curvilinear phase space coordinates. The existence of (n − d) coordinates satisfying the constraints is often guaranteed locally by an implicit function theorem. Nevertheless in many examples these coordinates cannot be constructed in any tractable form, even locally, so that other approaches are of interest. In ambient space formulations the dynamics are defined in the full original n-dimensional configuration space, and associated 2n-dimensional phase space, with some version of Lagrange multipliers introduced so that the 2(n − d) dimensional sub-manifold of phase space implied by the holonomic constraints and their time derivative, is invariant under the dynamics. In this article we review ambient space formulations, and explain that for constrained dynamics there is in fact considerable freedom in how a Hamiltonian form of the dynamics can be constructed. We then discuss and contrast the Langevin and Fokker-Planck equations and their equilibrium distributions for the different forms of ambient space dynamics.