Theoretical Physics Group

The Theoretical Physics Group works on two main topics: Deformation Quantization and Complex methods in General Relativity.

At present we quite well understand physical phenomena at micro scale and at the large scale separately. The first are described by quantum mechanics the other by general theory of relativity. They use different mathematical language. The most successful mathematical language of quantum mechanics are linear operators acting in a Hilbert space or rigged Hilbert space (Gelfand triple). General relativity, in contrary, defines the geometry of spacetime itself. A lot of problems have been solved within a respective frameworks, nevertheless one should be aware of the areas of weaknesses of each of them. Quantization procedure of a system can be performed in Cartesian coordinates and in general can not be applied to systems on curved space. On the other hand there are difficulties in canonical quantization of general relativity. Physicists are still far from reconciling these two theories.

The logical structure of quantum mechanics consists of observables and states. In deformation quantization the noncommutative associative multiplication of operators (observables) is considered as a formal associative deformation of the multiplication of the ‘symbols of these operators’ i.e. formal complex power series. According to Kontsevich theorem this construction is universal in a sense that it is possible not only for R^{2n} but for any Poisson manifold. The representation of quantum states is a Wigner function. It reproduces the expectation values for physical observables in phase-space picture of quantum theory.

Complex methods in general theory of relativity was first proposed as a possible path to a theory of quantum gravity and as well as a way to solve real Einstein equation of Lorentzian signature. It results in invention of twistor theory which is especially natural for solving the equations of motion of massless fields of arbitrary spin. Complex methods in general relativity have attracted a great interest on its own. Null tetrad formalism, twistor analysis and heavenly and hyperheavenly spaces play an important role in physics and mathematics, let us remind SDYM integrable models, string theory, Walker spaces and Walker-Osserman spaces as a few examples. Hyperheavenly spaces are complex spacetimes with algebraically degenerate self-dual or anti-self-dual part of the Weyl tensor satisfying the vacuum Einstein equations with cosmological constant. The transparent advantage of hyperheavenly spaces is the reduction of Einstein equations to one, nonlinear differential equation of the second order, i.e. hyperheavenly equation. It opens a way to find new real vacuum solutions of Einstein field equations of Lorentzian (physical + - - - ) signature. This reaserch programme (Plebanski programme) is as follows: solve the hyperheavenly equations and then find Lorentzian slices of respective complex spacetimes. Unfortunately, obtaining the real slices appeared to be more difficult that anyone has ever suspected. Believing that symmetry of the complex spacetime simplifies the problem, Killing vectors (defining symmetries) are studied.

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