By analogy with the Wiener measure on the Euclidean plane that is invariant under the group of rotations and quasi-invariant under the group of diffeomorphisms, we construct the path integrals measure that is invariant under the Lorentz group and quasi-invariant under the group of diffeomorphisms. The correspondence between the paths in the future cone of the Minkowskian plane and the paths in the coverings of the Euclidean plane is established.

The action A of Quadratic Gravity in FLRW metric is invariant under the group of diffeomorphisms of the time coordinate and can be written in terms of the only dynamical variable g(tau). We construct perturbation theory for calculating path integrals of the form integral F(g)exp{-A(g)}dg, and find the averaged value of the scale factor in the first nontrivial perturbative order.

In the paper, we give the proof of the polar decomposition of the Wiener measure according to the orbits of the group of diffeomorphisms.

We consider a model of 2D gravity with the action quadratic in curvature and represent path integrals as integrals over the SL(2,R) invariant Gaussian functional measure. We reduce these path integrals to the products of Wiener path integrals and calculate the correlation function of the metric in the first perturbative order.

In memory of Andrei Alekseevich Slavnov, Aref’eva I.Ya., Belokurov V.V., Boos E.E., Bykov D.V., Volovich I.V., Kazakov D.I., Kozlov V.V., Libanov M.V., Matveev V.A., Rubakov V.A., Treshchev D.V., Trubnikov G.V.

A bstract Using the invariance of Quadratic Gravity in FLRW metric under the group of diffeomorphisms of the time coordinate, we rewrite the action A of the theory in terms of the invariant dynamical variable g ( τ ) . We propose to consider the path integrals ∫ F ( g ) exp {− A } dg as the integrals over the functional measure μ ( g ) = exp {− A 2 } dg, where A 2 is the part of the action A quadratic in R. The rest part of the action in the exponent stands in the integrand as the “interaction” term. We prove the measure μ ( g ) to be equivalent to the Wiener measure, and, as an example, calculate the averaged scale factor in the first nontrivial perturbative order.

theoretical physicist, director of the JINR Laboratory of Theoretical Physics, doctor of physical and mathematical sciences, professor, and corresponding member of the Russian Academy of Sciences (RAS), Dmitrii Igorevich Kazakov. Dmitrii Igorevich was born in Moscow. His father, Igor' Efimovich Kazakov, was doctor of technical sciences, professor, major general, and aviation engineer. He was a prominent scientist in the field of automatic control systems and was head of the Chair of Guided Aircraft Missiles at the N E Zhukovskii Air Force Engineering Academy. In 1968, D I Kazakov entered the Physics Department of Lomonosov Moscow State University (MSU). He studied in the Chair of Quantum Statistics and Field Theory, headed by academician N N Bogoliubov. Upon graduating from MSU in 1974, D I Kazakov began working at the Laboratory of Theoretical Physics of the Joint Institute for Nuclear Research (JINR, town of Dubna, Moscow region) as a trainee-researcher. There, in 1974, he successfully defended his candidate thesis, ``Renormalizations in dynamic symmetry theories,'' under the guidance of correspondingmember of the USSRAS, DV Shirkov (academician of RAS since 1994). Dmitrii Igorevich's further scientific biography is connected to the Bogoliubov Laboratory of Theoretical Physics (LTP) of JINR, where he progressed from trainee-researcher and junior research fellow to director of the laboratory. At LTP, he became leader of theoretical studies in high energy physics, and in 1988, defended his doctoral thesis, ``Finite supersymmetric models in quantum field theory.'' Since 2004, he has conducted the largest thematic group at LTP, Fundamental Interactions of Fields and Particles. In 2016, Dmitrii Igorevich was appointed to the post of head of the Scientific Department of the Theory of Fundamental Interactions, and, in 2017, after elections at the Scientific Council of JINR, he became director of the Bogoliubov Laboratory of Theoretical Physics. In 2004±2016, D I Kazakov also headed the Laboratory of Fundamental Interactions at the Institute of Theoretical and Experimental Physics inMoscow. In 2016, he was elected corresponding member of the Russian Academy of Sciences. In the Russian Academy of Sciences, he has held the post of deputy academician secretary of the Division of Physical Sciences. D I Kazakov is a prominent specialist in elementary particle physics, quantum field theory, and supersymmetry. He has repeatedly been invited by many leading international scientific centers for joint work. In the early 1990s, he worked over a year at the University of Southampton (Great Britain). He was an invited professor at the University of Karlsruhe (Germany) and at KEKÐthe High Energy Accelerator Research Organization (Japan). He has published more than 200 papers, of which the most well known are devoted to renormalizations in theories with broken supersymmetry and to a phenomenological analysis of supersymmetric extensions of the Standard ModelÐ the modern elementary particle theory. D I Kazakov successfully developed effective methods of calculation and summation of higher-order quantum corrections both in the Standard Model and in other field-theoretic models appearing in high-energy physics and in condensed matter physics. His original ideas made it possible to perform record calculations of critical indices in scalar models and to systematically find renormalization group functions in broken supersymmetry models. Dmitrii Igorevich developed a large class of supersymmetric theories without ultraviolet divergences and suggested a new approach to renormalization of models in spaces with extra dimensions. He developed an effective method of renormalizations in theories with spontaneous supersymmetry breaking. Of great importance were D I Kazakov's results concerning the mass spectrum of superpartners and Higgs bosons in the supersymmetric StandardModel and the idea of experimentally searching for them, suggested on the basis of Uspekhi Fizicheskikh Nauk 191 (12) 1403 ± 1404 (2021) Translated by M V Tsaplina PERSONALIA PACS number: 01.60.+q

We derive the general rules of functional integration in the theories of Schwarzian type, thus completing the elaboration of Schwarzian functional integrals calculus initiated in [1, 2]. Our approach is mathematically rigorous and does not contain any unproved conjectures. It is based on the analysis of the properties of the measures on the groups of diffeomorphisms, and does not appeal for the experience from other physical models. Its great merit consists in reducing a problem of functional integration to that of the only functional integral (24) that is calculated explicitly with the result written in the form of the ordinary integral. We evaluate two-point and four-point correlation functions defined as functional integrals over the groups Diff(+)(1) (R) and Diff(+)(1)(S-1), and discuss the difference between the results in the two cases.

n explicit form of the functional measure on the factor space Diff_ + ^1(S^1)/ . -0emSL(2,𝐑) is obtained that makes Schwarzian functional integrals calculus more simple and transparent.

It is shown that nonlinear nonlocal substitutions in functional integrals lead to the need of integration over functional spaces that include functions with singularities. This makes it possible to formulate a quantum theory in cases where singularities are essential, e.g., in quantum cosmology. The proper accounting of singularities in functional integrals gives an additional unexpected effect, which we call “quantum restoration of broken symmetry.”

We find an explicit form of the polar decomposition of the Wiener measure and obtain an equation relating functional integrals in conformai quantum mechanics to functional integrals in the Schwarzian theory. Using this relation, we evaluate some nontrivial functional integrals in the Schwarzian theory and also find the fundamental solution of the Schrödinger equation in imaginary time in the model of conformal quantum mechanics.

We derive the general rules of functional integration in the theories of the Schwarzian type, and evaluate explicitly the functional integrals assigning correlation functions in the SYK model.

A regular approach to evaluate the functional integrals over the quasi-invariant measure on the group of diffeomorphisms is presented. As an important example of the application of this technique, we explicitly evaluate the correlation functions in the Schwarzian theory.

A polar decomposition of the Wiener measure based on its quasi-invariance under the group of diffeomorphisms is proposed. As a result, functional integrals in the Schwarzian theory can be written as the Fourier transform of the functional integrals in the quantum oscillator model with the Calogero potential.

We give some examples of the problems where non-Hilbert spaces could be relevant.

A review of the work of the authors is presented, in which corollaries of the quasi-invariance of functional integrals on the Wiener measure with respect to the action of a group of diffeomorphisms are studied, and the behavior of functional integrals with nonlinear nonlocal change of variables of integration is investigated as well. Using these substitutions, the functional integrals over discontinuous paths can be determined. The simplest models of the (Euclidean) quantum field theory are offered, in which the presence of hidden internal symmetries or the allowance for discontinuous paths in functional integrals leads to a number of paradoxical properties contradicting the conventional view.

The explicit evaluation of the partition function in the Schwarzian theory is presented.

We study the quantum theory of the singular scalar field $φ$ minimally coupled to gravity. In our approach, the scalar field is treated as a true quantum variable, while the scale factor $a(t)$ is supposed to be classical. We evaluate the quantum average of the self-interaction potential $V(φ)$ and use it in the modified differential equation for the Hubble parameter.

The standard lattice perturbation theory leads to the asymptotic series because of the incorrect interchange of the summation and integration. However, changing the initial approximation of the perturbation theory, one can generate the convergent series. We study the lattice $\phi^4$-model and compare the operator $\langle\phi_n^2\rangle$ calculated using the convergent series and obtained by Monte Carlo simulations.

Памяти Дмитрия Васильевича Ширкова, Белокуров В.В., Воронов В.В., Казаков Д.И., Матвеев В.А., Мешков И.Н., Оганесян Ю.Ц., Рубаков В.А., Скринский А.Н., Славнов А.А., Трубников Г.В., Трутнев Ю.А., Фортов В.Е.