The God has created a man in order that he creates that the God fails to do

Wednesday, 23 May 2012

My lectures on mathematical physics

G.Sardanashvily, Five lectures on the jet manifold methods in field theory, hep-th/ 9411089

G.Sardanashvily, Ten lectures on jet manifolds in classical and quantum field theory, math-ph/ 0203040

G.Sardanashvily, Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians, arXiv: 0908.1886v1

G.Sardanashvily, Lectures on supergeometry, arXiv: 0910.0092v1

G.Sardanashvily, Lectures on differential geometry of modules and rings, arXiv: 0910.1515v1

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced mechanics. Mathematical introduction, arXiv: 0911.0411

Saturday, 12 May 2012

Lagrangian BRST field theory (from my Scientific biography)

BRST theory emerged in the framework of the quantum theory of gauge fields, where the timing of a degeneration of the Yang - Mills Lagrangian led to its replacement in a generating functional with some modified Lagrangian, depending on ghost fields and invariant under BRST transformations. These BRST transformations resulted from the replacement of parameter functions in gauge transformations with odd ghost fields, and their extension to action on these ghost fields. BRST theory was mainly developed in the framework of Hamiltonian formalism, but its Lagrangian variant also was under consideration. The main works in this direction were the articles of J.Gomis, J.Paris, S.Samuel in 1995 and G.Barnish, F.Brandt, M.Henneaux in 2000 in Physics Reports, as well as preceding works of these authors in the Communication in Mathematical Physics. These works, however, involved the so-called regularity condition which came from BRST theory of Hamiltonian systems with constraints, and which was not appropriate for Noether identities. The latter, in contrast to the algebraic constraint conditions, are the differential identities. Moreover, this BFST theory was developed for fields on .

I was interested in BRST theory, as a kind of prequantum field theory which is a necessary step in the procedure of BV-quantization of fields. Because a BRST operator is nilpotent, I spent the calculation of its relative and iterated cohomology on an arbitrary manifold X [95] in 2000. In 2005, I returned to BRST theory in connection with a consideration of a general type of gauge transformations, depending on the derivatives of fields of arbitrary order [118]. Later, when studying Noether identities, I gave up on the above-mentioned conditions of regularity and introduced a new cohomology condition. In 2008, after constructing a complete description of reducible degenerate Lagrangian systems, I began exploring their BRST extension. Such an extension was proved to be possible if the gauge operator continues to a nilpotent BRST operator acting on ghost fields. In this case, the above-mentioned cochain sequence becomes a complex, called the BRST complex, and an original Lagrangian admits the BRST extension, depending on original fields, antifields, indexing the zero and higher order Noether identities, and ghost fields, parameterizing  zero and higher order gauge symmetries [129]. The BRST extension of some basic field models was built.


G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical Field Theory  (WS, 2009)

Saturday, 5 May 2012

Classical mechanics and field theory admit comprehensive geometric formulation

Classical non-relativistic mechanics and classical field theory are adequately formulated in geometric terms of fibre bundles Y->X where dim X>1 in field theory and X=R in mechanics.

Configuration space of a classical non-relativistic mechanics is a fibre bundle Y->R over the time axis R. Its velocity space is a first order jet manifold JY of Y. Its phase space is the vertical tangent bundle VY of Y. Connections on a fibre bundle Y->R characterize non-relativistic reference frames.

Geometric formulation of classical field theory is based on a representation of classical fields by sections of fibre bundles Y->X where dim X>1. Their Lagrangians are densities on finite order jet manifolds of Y.  Connections on Y->X also are fields, e.g., gauge fields which are sections of the first order jet bundle JY->Y. In a very general setting, in order to include odd fields, e.g., fermions  and ghosts, field theory is formulated on a graded manifold whose body is a fibre bundle Y->X.

The formulation of relativistic mechanics generalizes that of non-relativistic mechanics. It is phrased in terms of one-dimensional submanifolds of a configuration space. In a case of two-dimensional submanifolds, we come to classical string theory.


G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical Field Theory (WS, 2009)
G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)
G.Sardanashvily, Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians, arXiv: 0908.1886
G.Sardanashvily, Advanced mechanics. Mathematical introduction arXiv: 0911.0411