Abstract: Noether theorems in a general setting

This is an Abstract of my invited lecture: Noether theorems in a general setting. Reducible graded Lagrangians, in the Conference: Geometry of Jets and Fields (10-16 May 2015, Bedlewo, Poland).

Noether theorems are formulated in a general case of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. A problem is that any Euler-Lagrange operator satisfies Noether identities, which therefore must be separated into the trivial and non-trivial ones. These Noether identities can obey first-stage Noether identities, which in turn are subject to the second-stage ones, and so on. Thus, there is a hierarchy of non-trivial Noether and higher-stage Noether identities. This hierarchy is described in homology terms. If a certain homology regularity conditions holds, one can associate to a reducible degenerate Lagrangian the exact Koszul-Tate chain complex possessing the boundary operator whose nilpotentness is equivalent to all complete non-trivial Noether and higher-stage Noether identities. Since this complex is necessarily Grassmann-graded, Lagrangian theory on graded bundles is considered from the beginning, and is formulated in terms of the Grassmann-graded variational bicomplex. Its cohomology defines a first variational formula whose straightforward corollary is the first Noether theorem. Second Noether theorems associate to the above mentioned  Koszul-Tate complex a certain cochain sequence whose ascent operator consists of the gauge and higher-order gauge symmetries of a Lagrangian system. If gauge symmetries are algebraically closed, this ascent operator is prolonged to the ilpotent BRST operator which brings the gauge cochain sequence into a BRST complex, and thus provides a BRST extension of an original Lagrangian system. [G.Sardanashvily,  arXiv: 1411.2910]

Archive

Foundations of Modern Physics 8: Relativistic mechanics

Non-relativistic mechanics (FMP-7) as like as classical field theory (FMP-3) is adequately formulated in the terms of fiber bundles Q->R over the time axis R and jet manifolds of their sections.

If a configuration space Q of a mechanical system has no preferable fibration Q->R, we obtain a general formulation of relativistic mechanics, including Special Relativity on the Minkowski space Q=R^4. This fomulation involves a more sophisticated technique of jets of one-dimensional submanifolds. In the framework of this formalism, submanifolds of a manifold Q are identified if they are tangent to each other at points of Q with some order. Jets of sections are particular jets of submanifolds when Q->R is a fiber bundle and these submanifolds are its sections. In contrast with jets of sections, jets of submanifolds in relativistic mechanics admit arbitrary transformations of time t’= t(q) including the Lorentz transformations, but not only t’=t+const. in the non-relativistic case.

Note that jets of two-dimensional submanifolds provide a formulation of classical string theory.

A velocity space of relativistic mechanics is the first-order jet manifold J^1Q of one-dimensional submanifolds of the configuration space Q. The jet bundle J^1Q → Q is projective, and one can think of its fibers as being spaces of the three velocities of a relativistic system. The four velocities of a relativistic system are represented by elements of the tangent bundle TQ of a configuration space Q. Lagrangian formalism of relativistic mechanics on the jet bundle J^1Q → Q is developed.

References:

G. Giachetta, L. Mangiarotti, G. Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010)

G. Sardanashvily, Relativistic mechanics in a general setting, Int. J. Geom. Methods Mod. Phys. 7 (2010) 1307-1319

WikipediA: Relativistic system (mathematics)

Archive