Proof of the polar decomposition of the Wiener measure
In the paper, we give the proof of the polar decomposition of the Wiener measure according to the orbits of the group of diffeomorphisms.
math-ph↗
arXiv subjects
E. T. Shavgulidze
Publications and source records attributed to E. T. Shavgulidze.
Correlation functions in the Schwarzian theory
A mathematically correct approach to study theories with infinite-dimensional groups of symmetries is presented. It is based on quasi-invariant measures on the groups. In this paper, the properties of the measure on the group of diffeomorphisms are used to evaluate the functional integrals in the Schwarzian theory. As an important example of the application of the new technique, we explicitly evaluate the correlation functions in the Schwarzian theory.
hep-th↗
The Exact Solution of the Schwarzian Theory
The explicit evaluation of the partition function in the Schwarzian theory is presented.
hep-th↗
Peculiar spaces of relativistic quantum fields
We study how quantum field theory models are modified under the reparametrizations of the space-time coordinates and some simultaneous transformations of the field function. The transformations that turn the action of the massive field in the Minkowski space-time into the action of the massless field in some curved space are presented.
hep-th↗
Masses of quantum particles as a result of diffeomorphisms
We present the explicit form of the diffeomorphisms of the time variable that transform the measure for quantum free massless particle into the measure for quantum free massive particle.
math-ph↗
A quantum mechanical model of "dark matter"
The role of singular solutions in some simple quantum mechanical models is studied. The space of the states of two-dimensional quantum harmonic oscillator is shown to be separated into sets of states with different properties.
math-ph↗
A class of exactly solvable quantum models of scalar gravity
Exactly solvable quantum theory of a singular at the origin scalar field with the self-interaction of Liouville type is proposed. The mean value of the scale factor in the FLRW metric as a function of conformal time is evaluated explicitly.
hep-th↗
Paths with singularities in functional integrals of quantum field theory
In the toy model ($ ϕ^{4}$-interacting quantum field theory in one-dimensional "Euclidean" space-time) we prove that the functional integrals of the free field theory evaluated over the space of continuous functions are equal to the functional integrals of the interacting field theory evaluated over a set of spaces containing the spaces of discontinuous functions.
hep-th↗
Quantum restoration of broken symmetries
A certain non-linear non-local substitution is shown to transform the action of the self-interacting quantum field to the free one. The functional integrals in both theories are equal to each other. However, the integrations are performed over different functional spaces. The classical action and the classical limit of the corresponding quantum theory turn out to be different. And the symmetry originally broken in the classical theory is restored in the classical limit of the quantum theory.
math-ph↗
On the local limit of quantum field theories defined on the loop space
The local limit of a quantum field theory on the loop space is studied. It is proved that the invariance of the theory with respect to the group of diffeomorphisms leads to Feynman diagrams convergence in the local limit.
hep-th↗
About amenability of subgroups of the group of diffeomorphisms of the interval [0,1]
Averaging linear functional on the space continuous functions of the group of diffeomorphisms of interval is found. Amenability of several discrete subgroups of the group of diffeomorphisms $\Diff^3([0,1])$ of interval is prove. In particular, a solution of the problem of amenability of the Thompson's group $F$ is given.
math.GR↗
About amenability of subgroups of the group of diffeomorphisms of the interval
Averaging linear functional on the space continuous functions of the group of diffeomorphisms of interval is found. Amenability of several discrete subgroups of the group of diffeomorphisms $\Diff^3([0,1])$ of interval is prove. In particular, a solution of the problem of amenability of the Thompson's group $F$ is given.
math.GR↗
Renormalizable quantum field theory as a limit of a quantum field model on the loop space
A nonlocal generalization of quantum field theory in which momentum space is the space of continuous maps of a circle into $\mathbf{R}^4$ is proposed. Functional integrals in this theory are proved to exist. Renormalized quantum field model is obtained as a local limit of the proposed theory.
hep-th↗