The Schrödinger Equation, Path Integration and Applications

The Schrödinger equation is fundamental in quantum mechanics as it makes it possible to determine the wave function from energies and to use this function in the mean calculation of variables, for example, as the most likely position of a group of one or more massive particles. In this paper, we present a survey on some theories involving the Schrödinger equation and the Feynman path integral. We also consider a Feynman–Kac-type formula, as introduced by Patrick Muldowney, with the Henstock integral in the description of the expectation of random walks of a particle. It is well known that the non-absolute integral defined by R. Henstock “fixes” the defects of the Feynman integral. Possible applications where the potential in the Schrödinger equation can be highly oscillating, discontinuous or delayed are mentioned in the end of the paper.

Exactly solvable single-trace four point correlators in χCFT4

In this paper we study a wide class of planar single-trace four point correlators in the chiral conformal field theory (χCFT4) arising as a double scaling limit of the γ-deformed $$ \mathcal{N} $$ N = 4 SYM theory. In the planar (t’Hooft) limit, each of such correlators is described by a single Feynman integral having the bulk topology of a square lattice “fishnet” and/or of an honeycomb lattice of Yukawa vertices. The computation of this class of Feynmann integrals at any loop is achieved by means of an exactly-solvable spin chain magnet with SO(1, 5) symmetry. In this paper we explain in detail the solution of the magnet model as presented in our recent letter and we obtain a general formula for the representation of the Feynman integrals over the spectrum of the separated variables of the magnet, for any number of scalar and fermionic fields in the corresponding correlator. For the particular choice of scalar fields only, our formula reproduces the conjecture of B. Basso and L. Dixon for the fishnet integrals.

State-dependent graviton noise in the equation of geodesic deviation

We consider an equation of the geodesic deviation appearing in the problem of gravitational wave detection in an environment of gravitons. We investigate a state-dependent graviton noise (as discussed in a recent paper by Parikh,Wilczek and Zahariade) from the point of view of the Feynman integral and stochastic differential equations. The evolution of the density matrix and the transition probability in an environment of gravitons is obtained. We express the time evolution by a solution of a stochastic geodesic deviation equation with a noise dependent on the quantum state of the gravitational field.

A Partial Derivative Approach to the Change of Scale Formula for the Function Space Integral

We investigate the partial derivative approach to the change of scale formula for the functon space integral and we investigate the vector calculus approach to the directional derivative on the function space and prove relationships among the Wiener integral and the Feynman integral about the directional derivative of a Fourier transform.

A Change of Scale Formula for Wiener Integrals about the First Variation on the Product Abstract Wiener Space

We shall prove the existence of the Wiener integral and the analytic Wiener and Feynman integral and we obtain those relationships and later, we prove the change of scale formula for the Wiener integral about the first variation of a function defined on the product abstract Wiener space. Later, we obtain those relationships in the Fresnel class as it’s corollaries.

Partial Derivative Approach to the Change of Scale Formula for the Function Space Integral

We investigate the behavior of the partial derivative approach to the change of scale formula and prove relationships among the analytic Wiener integral and the analytic Feynman integral of the partial derivative for the function space integral.

Analytic Feynman Integral and a Change of Scale Formula for Wiener Integrals of an Unbounded Cylinder Function

We investigate the behavior of the unbounded cylinder function F x = ∫ 0 T α 1 t d x t 2 k ⋅ ∫ 0 T α 2 t d x t 2 k ⋅ ⋯ ⋅ ∫ 0 T α n t d x t 2 k , k = 1,2 , … whose analytic Wiener integral and analytic Feynman integral exist, we prove some relationships among the analytic Wiener integral, the analytic Feynman integral, and the Wiener integral, and we prove a change of scale formula for the Wiener integral about the unbounded function on the Wiener space C 0 0 , T .

Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform

We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of F(x)=exp{∫0Tθ(t,x(t))dt} successfully exist under the certain condition, where θ(t,u)=∫Rexp{iuv}dσt(v) is a Fourier–Stieltjes transform of a complex Borel measure σt∈M(R) and M(R) is a set of complex Borel measures defined on R. We will find this condition. Moreover, we prove that the change of scale formula for Wiener integrals about the first variation of F(x) sucessfully holds on the Wiener space.

Hypergeometric expression for a three-loop vacuum integral

Using the corresponding Mellin–Barnes representation, we derive holonomic hypergeometric system of linear partial differential equations (PDEs) satisfied by Feynman integral of a three-loop vacuum with five propagators. Through the multidimensional residue theorem in dimensional regularization, the scalar integral can be written as the summation of multiple hypergeometric functions, whose convergent regions can be obtained by the Horn’s convergent theory. The numerical continuation of the scalar integral from convergent regions to whole kinematic regions can be accomplished with the finite element methods, when the system of PDEs can be treated as the stationary conditions of a functional under the restrictions.