Felix Alexandrovich Berezin and his work

This is a survey of Berezin's work focused on three topics: representation theory, general concept of quantization, and supermathematics.Comment: LaTeX, 27 page

Dynamics of a thin shell in the Reissner-Nordstrom metric

We describe the dynamics of a thin spherically symmetric gravitating shell in the Reissner-Nordstrom metric of the electrically charged black hole. The energy-momentum tensor of electrically neutral shell is modelled by the perfect fluid with a polytropic equation of state. The motion of a shell is described fully analytically in the particular case of the dust equation of state. We construct the Carter-Penrose diagrams for the global geometry of the eternal black hole, which illustrate all possible types of solutions for moving shell. It is shown that for some specific range of initial parameters there are possible the stable oscillating motion of the shell transferring it consecutively in infinite series of internal universes. We demonstrate also that this oscillating type of motion is possible for an arbitrary polytropic equation of state on the shell.Comment: 17 pages, 7 figure

SUSY structures, representations and Peter-Weyl theorem for S1∣1S^{1|1}

The real compact supergroup $S^{1|1}$ is analized from different perspectives and its representation theory is studied. We prove it is the only (up to isomorphism) supergroup, which is a real form of $({\mathbf C}^{1|1})^\times$ with reduced Lie group $S^1$, and a link with SUSY structures on ${\mathbf C}^{1|1}$ is established. We describe a large family of complex semisimple representations of $S^{1|1}$ and we show that any $S^{1|1}$-representation whose weights are all nonzero is a direct sum of members of our family. We also compute the matrix elements of the members of this family and we give a proof of the Peter-Weyl theorem for $S^{1|1}$

N-particle sector of quantum field theory as a quantum open system

We give an exposition of a technique, based on the Zwanzig projection formalism, to construct the evolution equation for the reduced density matrix corresponding to the n-particle sector of a field theory. We consider the case of a scalar field with a $g \phi^3$ interaction as an example and construct the master equation at the lowest non-zero order in perturbation theory.Comment: 12 pages, Late

Killing spinors are Killing vector fields in Riemannian Supergeometry

A supermanifold M is canonically associated to any pseudo Riemannian spin manifold (M_0,g_0). Extending the metric g_0 to a field g of bilinear forms g(p) on T_p M, p\in M_0, the pseudo Riemannian supergeometry of (M,g) is formulated as G-structure on M, where G is a supergroup with even part G_0\cong Spin(k,l); (k,l) the signature of (M_0,g_0). Killing vector fields on (M,g) are, by definition, infinitesimal automorphisms of this G-structure. For every spinor field s there exists a corresponding odd vector field X_s on M. Our main result is that X_s is a Killing vector field on (M,g) if and only if s is a twistor spinor. In particular, any Killing spinor s defines a Killing vector field X_s.Comment: 14 pages, latex, one typo correcte

Path integral representations in noncommutative quantum mechanics and noncommutative version of Berezin-Marinov action

It is known that actions of field theories on a noncommutative space-time can be written as some modified (we call them $\theta$-modified) classical actions already on the commutative space-time (introducing a star product). Then the quantization of such modified actions reproduces both space-time noncommutativity and usual quantum mechanical features of the corresponding field theory. The $\theta$-modification for arbitrary finite-dimensional nonrelativistic system was proposed by Deriglazov (2003). In the present article, we discuss the problem of constructing $\theta$-modified actions for relativistic QM. We construct such actions for relativistic spinless and spinning particles. The key idea is to extract $\theta$-modified actions of the relativistic particles from path integral representations of the corresponding noncommtative field theory propagators. We consider Klein-Gordon and Dirac equations for the causal propagators in such theories. Then we construct for the propagators path-integral representations. Effective actions in such representations we treat as $\theta$-modified actions of the relativistic particles. To confirm the interpretation, we quantize canonically these actions. Thus, we obtain the Klein-Gordon and Dirac equations in the noncommutative field theories. The $\theta$-modified action of the relativistic spinning particle is just a generalization of the Berezin-Marinov pseudoclassical action for the noncommutative case

Canonical and D-transformations in Theories with Constraints

A class class of transformations in a super phase space (we call them D-transformations) is described, which play in theories with second-class constraints the role of ordinary canonical transformations in theories without constraints.Comment: 16 pages, LaTe

Boundary sources in the Doran - Lobo - Crawford spacetime

We take a null hypersurface (the causal horizon) generated by a congruence of null geodesics as the boundary of the Doran-Lobo-Crawford spacetime, to be the place where the Brown-York quasilocal energy is located. The components of the outer and inner stress tensors are computed and shown to depend on time and on the impact parameter $b$ of the test particle trajectory. The surface energy density $\sigma$ on the boundary is given by the same expression as that obtained previously for the energy stored on a Rindler horizon.Comment: 4 pages, title changed, no figures, minor text change

On the Coherent State Path Integral for Linear Systems

We present a computation of the coherent state path integral for a generic linear system using ``functional methods'' (as opposed to discrete time approaches). The Gaussian phase space path integral is formally given by a determinant built from a first-order differential operator with coherent state boundary conditions. We show how this determinant can be expressed in terms of the symplectic transformation generated by the (in general, time-dependent) quadratic Hamiltonian for the system. We briefly discuss the conditions under which the coherent state path integral for a linear system actually exists. A necessary -- but not sufficient -- condition for existence of the path integral is that the symplectic transformation generated by the Hamiltonian is (unitarily) implementable on the Fock space for the system.Comment: 15 pages, plain Te

On SUSY curves

In this note we give a summary of some elementary results in the theory of super Riemann surfaces (SUSY curves)