arXiv subjects
Publications and source records attributed to S. A. Pol'shin.
We construct a nuclear space $Φ$ as an inductive limit of finite-dimensional subspaces of a Hilbert space $H$ in such a way that $(Φ,H,Φ')$ becomes a rigged Hilbert space, thus simplifying the construction by Bellomonte and Trapani.
Using Grassmann variant of classical mechanics, we construct Lagrangian dynamics of classical spinning particle in (possibly non-abelian) gauge fields. Quantization of this model is briefly discussed.
Let $(M,ω)$ be a symplectic manifold, $\mathcal{D}\subset TM$ a real polarization on $M$ and $\wp$ a leaf of $\mathcal{D}$. We construct a Fedosov-type star-product $\ast_L$ on $M$ such that $C^\infty (\wp)[[h]]$ has a natural structure of left module over the deformed algebra $(C^\infty (M)[[h]], \ast_L)$. This generalizes the results of 0708.2626.
We obtain Bargmann-Michel-Telegdi equations of motion of classical spinning particle using Lagrangian variational principle with Grassmann variables.
We construct a Leibniz bracket on the space $Ω^\bullet (J^k (π))$ of all differential forms over the finite-dimensional jet bundle $J^k (π)$. As an example, we write Maxwell equations with sources in the covariant finite-dimensional hamiltonian form.
In certain neighborhood $U$ of an arbitrary point of a symplectic manifold $M$ we construct a Fedosov-type star-product $\ast_L$ such that for an arbitrary leaf $\wp$ of a given polarization $\mathcal{D}\subset TM$ the algebra $C^\infty (\wp \cap U)[[h]]$ has a natural structure of left module over the deformed algebra $(C^\infty (U)[[h]], \ast_L)$. With certain additional assumptions on $M$, $\ast_L$ becomes a so-called star-product with separation of variables.
We construct the system of generalized coherent states for the quantum Kepler problem corresponds to the homogeneous domain $SU(2,2)/S(U(2)\times U(2))$. We show that the SU(2,2)-equivariant momentum map for this domain yields the momentum map for the classical Kepler problem via appropriate limiting passage. We also show that under this passage the average values of quantum observables in this system of coherent states pass into the functions on classical phase space and $-i/h$ times commutator pass into the Poisson bracket.
We construct a tensionless string model in a four-dimensional space-time ${\bf R}^{2,2}$ with the (2,2) signature and solve the string equations. We find that the signature change radically changes the structure of the holonomy group of the null worldsheet. As a result the introduction of the worldsheet Dirac operator invariant under the holonomy group becomes possible. We show that the such possibility is absent in the tensionless string model in the (3+1)-dimensional Minkowski space.