Related papers: On $\varepsilon$-factorised bases and pure Feynman…
We review some results of calculations, having the property of maximal transcendentality.
The universal method of expansion of integrals is suggested. It allows in particular to derive the threshold expansion of Feynman integrals.
We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality…
The Feynman path integral for the generalized harmonic oscillator is reviewed, and it is shown that the path integral can be used to find a complete set of wave functions for the oscillator. Harmonic oscillators with different…
Here we discuss direct links of the number of fundamental dimensions to the fundamental natural constants using simple arguments of dimensional analysis \corr{based on Maxwell's dimensions length (L), time (T) and mass (M) as well as the…
We evaluate the three-loop five-point pentagon-box-box massless integral family in the dimensional regularization scheme, via canonical differential equation. We use tools from computational algebraic geometry to enable the necessary…
We give a simple proof of perturbative unitarity in gauge theories and quantum gravity using a special gauge that allows us to separate the physical poles of the free propagators, which are quantized by means of the Feynman prescription,…
We consider the weighted eigenvalue problem for a general non-local pseudo-differential operator, depending on a bounded weight function. For such problem, we prove that strict (decreasing) monotonicity of the eigenvalues with respect to…
A sketch of a recent approach to quantum gravity is presented which involves several unconventional aspects. The basic ingredients include: (1) Affine kinematical variables; (2) Affine coherent states; (3) Projection operator approach for…
Using well-known methods we generalize (hyper)virial theorems to case of singular potential. Discussion is carried on for most general second order differential equation, which involves all physically interesting cases, such as…
Hyperuniform particle arrangements are characterized by a local number variance that grows more slowly than the volume of the observation window. We generalize this concept to describe particle systems in which particles carry weights:…
In this paper, we extract natural invariants of a singularity by using the Deligne weight filtration on the cohomology of an exceptional fibre of a resolution, and also on the intersection cohomology of the link. Our primary goal is to…
A Feynman formula is a representation of a solution of an initial (or initial-boundary) value problem for an evolution equation (or, equivalently, a representation of the semigroup resolving the problem) by a limit of $n$-fold iterated…
The method of differential equations in canonical form has proven a powerful tool for solving multiloop Feynman integrals. In this note we test this procedure away from four dimensions. Namely, we consider the simple example of a massless…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…
The role of the measurement process in resolving the gauge ambiguity of the effective gravitational potential is reexamined. The motion of a classical point-like particle in the field of an arbitrary linear source, and in the field of…
It is wellknown that the Feynman kernel for the free particle on the half-line can be expressed as a sum over classical paths if we take the contribution from the reflected path into account. The minus sign for the reflected path needs to…
An approach to evaluation of the smooth Feynman path integrals is developed for the study of quantum fluctuations of particles and fields in Euclidean time-space. The paths are described by sum of Gauss functions and are weighted with…
By introducing the generalized master function of order up to four together with corresponding weight function, we have obtained all quasi-exactly solvable second order differential equations. It is shown that these differntial equations…
Improvement of the classical gravity with the running gravitational coupling obtained from asymptotically safe gravity, is a good way of considering the effects of quantum gravity. This is usually done for metric theories of gravity. Here…