Feynman integrals ofp-adic argument in the momentum space. I. Convergence

The regions of convergence of Feynman integrals ofp-adic arguments in the coordinate and momentum spaces are established, and a theorem on the Fourier transformation of Feynman amplitudes is proved.State University, Kazan. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 367–377, March, 1995.     

Feynman integrals with ap-adic argument in momentum space III. Renormalization

The ultraviolet and infrared sequences of poles for massless Feynman integrals having a p-adic argument with an arbitrary number of propagators are found. Within the framework of the MS scheme for the analytic R*-operation, the renormalized amplitudes are proved to be finite. Simple formulas are given for the counter terms (vertex parts) in coordinate and momentum space.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 2, pp. 233–249, February, 1996.     

Operator-valued Feynman integral via conditional Feynman integrals on a function space

Let C ₀ ^r [0; t] denote the analogue of the r-dimensional Wiener space, define X _t : C ^r [0; t] → ?^2r by X _t (x) = (x(0); x(t)). In this paper, we introduce a simple formula for the conditional expectations with the conditioning function X _t . Using this formula, we evaluate the conditional analytic Feynman integral for the functional $$ \Gamma _t \left( x \right) = exp \left\{ {\int_0^t {\theta \left( {s,x\left( s \right)} \right)d\eta \left( s \right)} } \right\}\varphi \left( {x\left( t \right)} \right) x \in C^r \left[ {0,t} \right] $$ , where η is a complex Borel measure on [0, t], and θ(s, ·) and φ are the Fourier-Stieltjes transforms of the complex Borel measures on ? ^r . We then introduce an integral transform as an analytic operator-valued Feynman integral over C ^r [0, t], and evaluate the integral transform for the function Γ _t via the conditional analytic Feynman integral as a kernel.     

Explicit expressions for three-dimensional boundary integrals in linear elasticity

On employing isoparametric, piecewise linear shape functions over a flat triangle, exact formulae are derived for all surface potentials involved in the numerical treatment of three-dimensional singular and hyper-singular boundary integral equations in linear elasticity. These formulae are valid for an arbitrary source point in space and are represented as analytical expressions along the edges of the integration triangle. They can be employed to solve integral equations defined on triangulated surfaces via a collocation method or may be utilized as analytical expressions for the inner integrals in a Galerkin technique. A numerical example involving a unit triangle and a source point located at various distances above it, as well as sample problems solved by a collocation boundary element method for the Lamé equation are included to validate the proposed formulae.     

Transformations of Feynman integrals under nonlinear transformations of the phase space

The Feynman measure is defined as a linear continuous functional on a test-function space (introduced in the paper). The functional is given by means of its Fourier transform. Not only a positive-definite correlation operator but also one without fixed sign is considered (the latter case corresponds to the so-called symplectic, or Hamiltonian, Feynman measure). The Feynman integral is the value of the Feynman measure on a function (in the test-function space). The effect on the Feynman measure of nonlinear transformations of the phase space in the form of shifts along vector fields or along integral curves of vector fields is described. Analogs of the well-known Cameron—Martin, Girsanov—Maruyama, and Ramer formulas in the theory of Gaussian measures are obtained. The results of the paper can be regarded as formulas for a change of variable in Feynman integrals.Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 3–13, July, 1994.     

Feynman path integrals as analysis on path space by time slicing approximation

Using the time slicing approximation, we give a mathematically rigorous definition of Feynman path integrals for a general class of functionals on the path space. As an application, we prove the interchange with Riemann-Stieltjes integrals, the interchange with a limit, the perturbation expansion formula, the semiclassical approximation, and the fundamental theorem of calculus in Feynman path integral.     

Phase space Feynman path integrals with smooth functional derivatives by time slicing approximation

We give two general classes of functionals for which the phase space Feynman path integrals have a mathematically rigorous meaning. More precisely, for any functional belonging to each class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets with respect to the starting point of momentum paths and the endpoint of position paths. Each class is closed under addition, multiplication, translation, real linear transformation and functional differentiation. Therefore, we can produce many functionals which are phase space path integrable. Furthermore, though we need to pay attention for use, the interchange of the order with the integrals with respect to time, the interchange of the order with some limits, the semiclassical approximation of Hamiltonian type, the natural property under translation, the integration by parts with respect to functional differentiation, and the natural property under orthogonal transformation are valid in the phase space path integrals.     

Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations

We give a fairly general class of functionals for which the phase space Feynman path integrals have a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets of the phase space. Our class of functionals is rich because it is closed under addition and multiplication. The interchange of the order with the Riemann integrals, the interchange of the order with a limit and the perturbation expansion formula hold in the phase space path integrals. The use of piecewise bicharacteristic paths naturally leads us to the semiclassical approximation on the phase space.     

Feynman path integrals for the time dependent quartic oscillator

The Schrödinger equation with a time dependence in both a quadratic and a quartic potential is considered. Existence of solutions is shown and a rigorous Feynman path integral representation for the solution is given in terms of well-defined infinite-dimensional oscillatory integrals. To cite this article: S. Albeverio, S. Mazzucchi, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Analytic Feynman integrals of transforms of variation of cylinder type functions over Wiener paths in abstract Wiener space

In this paper, we evaluate various analytic Feynman integrals of first variation, conditional first variation, Fourier-Feynman transform and conditional Fourier-Feynman transform of cylinder type functions defined over Wiener paths in abstract Wiener space. We also derive the analytic Feynman integral of the conditional Fourier-Feynman transform for the product of the cylinder type functions which define the functions in a Banach algebra introduced by Yoo, with n linear factors.     

Singularities of Feynman diagrams in the coordinate space

Theoretical and Mathematical Physics -     

A representation of the Belavkin equation via Feynman path integrals

 The Belavkin equation, describing the continuous measurement of the position of a quantum particle, is studied. A rigorous representation of its solution by means of an infinite dimensional oscillatory integral (Feynman path integral) defined on the complex Cameron-Martin space is given. Received: 7 January 2002 / Revised version: 20 June 2002 / Published online: 19 December 2002 Mathematics Subject Classification (2000): 81, 81S40, 60H15 Key words or phrases: Belavkin equation – Continuous measurement – Quantum theory – Oscillatory integrals – Feynman path integrals