Zeta-nonlocal scalar fields

We consider some nonlocal and nonpolynomial scalar field models originating from p-adic string theory. An infinite number of space-time derivatives is determined by the operator-valued Riemann zeta function through the d’Alembertian □ in its argument. The construction of the corresponding Lagrangians L starts with the exact Lagrangian for the effective field of the p-adic tachyon string, which is generalized by replacing p with an arbitrary natural number n and then summing over all n. We obtain several basic classical properties of these fields. In particular, we study some solutions of the equations of motion and their tachyon spectra. The field theory with Riemann zeta-function dynamics is also interesting in itself. Dedicated to Vasilii Sergeevich Vladimirov on his 85th birthday __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 364–372, December, 2008.     

On nonlinear equations of p-adic strings for scalar tachyon fields

We consider boundary value problems for open and closed p-adic strings for scalar tachyon fields. Estimates for solutions to these problems and possible ways of constructing these solutions are obtained by reducing the problems to linear parabolic equations with nonlinear boundary conditions. We give an application of Gauss-type quadrature formulas to the numerical solution of the boundary value problems, and discuss the possibility of using these methods in multidimensional problems (d = 2). Original Russian Text ? V.S. Vladimirov, 2009, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 265, pp. 254–272.     

Symmetry factors of Feynman diagrams for scalar fields

We calculate the symmetry factors of diagrams for real and complex scalar fields in general form using an analysis of the Wick expansion for Green’s functions. We separate two classes of symmetry factors: factors corresponding to connected diagrams and factors corresponding to vacuum diagrams. The symmetry factors of vacuum diagrams play an important role in constructing the effective action and phase transitions in cosmology. In the complex scalar field theory, diagrams with different topologies can contribute the same, and the inverse symmetry factor for the total contribution is therefore the sum of the inverse symmetry factors.     

Renormalizations in a matrix field theory model in the 1/N expansion

Theoretical and Mathematical Physics -     

Interaction between thermoelastic and scalar oscillation fields

Three-dimensional mathematical problems of the interaction between thermoelastic and scalar oscillation fields in a general physically anisotropic case are studied by the boundary integral equation methods. Uniqueness and existence theorems are proved by the reduction of the original interface problems to equivalent systems of boundary pseudodifferential equations. In the non-resonance case the invertibility of the corresponding matrix pseudodifferential operators in appropriate functional spaces is shown on the basis of the generalized Sommerfeld-Kupradze type thermoradiation conditions for anisotropic bodies. In the resonance case the co-kernels of the pseudodifferential operators are analysed and the efficient conditions of solvability of the original interface problems are established.     

Convergence of a canonical expansion for normal fields

In the present study we consider a normal separable stochastic continuous field, and we prove the convergence of a Karhunen series with probability 1 for all parameter values. This leads in particular, to the nonrandomness of points of the discontinuity and values of the discontinuity. A criterion is presented for the convergence of the canonical expansion in a uniform norm.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 565–572, October, 1973.The author thanks S. A. Molchanov for guidance with the work and Yu. K. Belayev for help and valuable instruction.     

Lax representations for triplets of two-dimensional scalar fields of the chiral type

We consider two-dimensional relativistically invariant systems with a three-dimensional reducible configuration space and a chiral-type Lagrangian that admit higher symmetries given by polynomials in derivatives up to the fifth order. Nine such systems are known: two are Liouville-type systems, and zero-curvature representations for two others have previously been found. We here give zero-curvature representations for the remaining five systems. We show how infinite series of conservation laws can be derived from the established zero-curvature representations. We give the simplest higher symmetries; others can be constructed from the conserved densities using the Hamiltonian operator. We find scalar formulations of the spectral problems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 2, pp. 189–205, August, 2006.