My Library: Time-Dependent Mechanics

Classical and quantum non-autonomous mechanics with respect to different reference frames is formulated in terms of fibre bundles over the time axis R.

The file Library2.pdf (7 Mb) contains the attached PDF files of my main works in classical and quantum time-dependent (non-autonomous) mechanics.

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Geometric and Algebraic Topological Methods in Quantum Mechanics (World Scientific, Singapore, 2005)

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

G.Sardanashvily, Hamilton time-dependent mechanics, J. Math. Phys. 39 (1998) 2714-2729

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Non-holonomic constraints in time-dependent mechanics, J. Math. Phys. 40 (1999) 1376-1390

L.Mangiarotti and G.Sardanashvily, On the geodesic form of second order dynamic equations, J. Math. Phys. 41 (2000) 835-844

L.Mangiarotti and G.Sardanashvily, Constraints in Hamiltonian time-dependent mechanics,

J. Math. Phys. 41 (2000) 2858-2876

G.Sardanashvily, Classical and quantum mechanics with time-dependent parameters, J. Math. Phys. 41 (2000) 5245-5255

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Covariant geometric quantization of nonrelativistic time-dependent mechanics, J. Math. Phys. 43 (2002) 56-68

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Geometric quantization of mechanical systems with time-dependent parameters, J. Math. Phys. 43 (2002) 2882-2894

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Geometric quantization of completely integrable Hamiltonian systems in action-angle coordinates, Phys. Lett. A 301 (2002) 53-57

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Action-angle coordinates for time-dependent completely integrable Hamiltonian systems, J. Phys. A 35 (2002) L439-L445

E.Fiorani, G.Giachetta and G.Sardanashvily, Geometric quantization of time-dependent completely integrable Hamiltonian systems, J. Math. Phys. 43 (2002) 5013-5025

E.Fiorani, G.Giachetta and G.Sardanashvily, The Liouville -- Arnold -- Nekhoroshev theorem for non-compact invariant manifolds, J. Phys. A 36 (2003) L101-L107

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Jacobi fields of completely integrable systems, Phys. Lett. A 309 (2003) 382-386

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Bi-Hamiltonian partially integrable systems, J. Math. Phys. 44 (2003) 1984-1987

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Nonadiabatic holonomy operators in classical and quantum completely integrable systems, J. Math. Phys. 45 (2004) 76-86

E.Fiorani and G.Sardanashvily, Noncommutative integrability on noncompact invariant manifolds, J. Phys. A 39 (2006) 14035-14042

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Quantization of noncommutative completely integrable systems, Phys. Lett. A 362 (2007) 138-142

E.Fiorani and G.Sardanashvily, Global action-angle coordinates for completely integrable systems with noncompact invariant submanifolds, J. Math. Phys. 48 (2007) 032901

L.Mangiarotti and G.Sardanashvily, Quantum mechanics with respect to different reference frames, J. Math. Phys. 48 (2007) 082104

G.Sardanashvily, Superintegrable Hamiltonian systems with noncompact invariant submanifolds. Kepler system, Int. J. Geom. Methods Mod. Phys. 6 (2009) 1391-1420

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

Review on our book "Advanced Classical Field Theory" in Mathematical Reviews

MR2527556 (2010h:70028)
Giachetta, Giovanni; Mangiarotti, Luigi; Sardanashvily, Gennadi
Advanced classical field theory. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009. x+382 pp. ISBN: 978-981-283-895-7; 981-283-895-3

Unlike quantum field theory, classical field theory is a theory that can be dealt with in a purely mathematical way. This book aims at providing a complete mathematical basis of classical Lagrangian field theory and its BRST extension as a preliminary step towards quantum field theory. Lagrangian field theory is thereby treated in a very general framework relying, among other things, on a geometric approach to the theory of nonlinear differential operators.
  

The first chapter is devoted to an overview of the basic facts about the geometry of fibre bundles, the Frölicher-Nijenhuis calculus of vector-valued forms, jet manifolds (of finite and infinite order), connections on fibre bundles and differential operators. Chapter 2 deals with Lagrangian field theory on fibre bundles, with a discussion of the variational bicomplex and of Lagrangian symmetries and gauge symmetries. Special attention is thereby paid to the case of first-order Lagrangian field theories. Chapter 3 passes to the Lagrangian theory of even and odd fields by means of the Grassmann-graded variational bicomplex. First, an introduction to Grassmann-graded algebraic and differential calculus and to the geometry of graded manifolds is given.
  

Chapter 4 deals with Lagrangian BRST theory. Degenerate Lagrangians are characterised by a family of nontrivial Noether identities. These form a hierarchy which, under certain conditions, can be described by the exact Koszul-Tate complex. By means of a formulation of the inverse second Noether theorem in homology terms, this complex is associated to a cochain sequence of ghosts with an ascent operator, called a gauge operator. The components of this operator represent nontrivial gauge and higher-stage gauge symmetries. Whereas the gauge operator itself in general is not nilpotent, in some cases it may admit a nilpotent extension, which is called the BRST operator and which turns the cochain sequence of ghosts into the BRST complex.
  

Gauge theory on principal bundles is the topic of Chapter 5 with, among others, a study of Yang-Mills gauge theory and supergauge theory, and a discussion of matter fields and Higgs fields. Chapters 6, 7 and 8 are devoted to a complete treatment of gravitation theory on natural bundles, the theory of spinor fields (Dirac spinor and universal spinor structure) and topological field theories (Chern-Simons topological field theory), respectively. Finally, in Chapter 9, some aspects of covariant Hamiltonian field theory are described.
  

To make the exposition as self-contained as possible, the book ends with ten appendices devoted to several mathematical topics, such as differential operators on modules, homology and cohomology theory, sheaf cohomology, local-ringed spaces, leafwise and fibrewise cohomology. In addition, almost every chapter ends with an appendix in which some specific mathematical concept, relevant to the chapter under consideration, is further elucidated.
  

In conclusion, this is a very interesting book which contains a wealth of information regarding the mathematics underlying classical field theories. It is primarily oriented towards a mathematical audience: although the treatment is fairly self-contained, the reader is nevertheless supposed to have a solid background in differential geometry. In the beginning one gets a bit overwhelmed by the rapid succession of definitions, properties and notational conventions, but the effort of struggling through it is definitely rewarding.

Reference:

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

My Library: Advanced Classical Field Theory

Fibre bundles and jet manifolds provide the adequate mathematical formulation of classical field theory and its prequantum BRST extension.

The file Library1.pdf (12 Mb) contains the attached PDF files of my main works on geometric formulation of classical field theory

Contents

G.Giachetta, L.Mangiarotti and G.Sardanashvily, New Lagrangian and Hamiltonian Methods in Field Theory (World Scientific, Singapore, 1997)

L.Mangiarotti and G.Sardanashvily, Connections in Classical and Quantum Field Theory (World Scientific, Singapore, 2000)

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Advanced Classical Field Theory (World Scientific, Singapore, 2009)

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Covariant Hamilton equations for field theory, J. Phys. A 32 (1999) 6629-6642

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Iterated BRST cohomology, Lett. Math. Phys. 53 (2000) 143-156

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Cohomology of the infinite-order jet space and the inverse problem, J. Math. Phys. 42 (2001) 4272-4282

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Lagrangian supersymmetries depending on derivatives. Global analysis and cohomology, Commun. Math. Phys. 259 (2005) 103-128

D.Bashkirov, G.Giachetta, L.Mangiarotti and G.Sardanashvily, The antifield Koszul-Tate complex of reducible Noether identities, J. Math. Phys. 46 (2005) 103513

G.Sardanashvily, Graded infinite order jet manifolds, Int. J. Geom. Methods Mod. Phys. 4 (2007) 1335-1362

D.Bashkirov, G.Giachetta, L.Mangiarotti and G.Sardanashvily, The KT-BRST complex of a degenerate Lagrangian system, Lett. Math. Phys. 83 (2008) 237-252

G.Sardanashvily, Classical field theory. Advanced mathematical formulation, Int. J. Geom. Methods Mod. Phys. 5 (2008) 1163-1189

G.Giachetta, L.Mangiarotti, G.Sardanashvily, On the notion of gauge symmetries of generic Lagrangian field theory, J. Math. Phys. 50 (2009) 012903

G.Sardanashvily, Gauge conservation laws in a general setting: Superpotential, Int. J. Geom. Methods Mod. Phys. 6 (2009) 1047-1056

My Library: Gauge Gravitation Theory

Gravitation theory is formulated as gauge theory on natural bundles where a gravitational field is a Higgs field responsible for spontaneous breaking of space-time symmetries.

The file Library0.pdf (14 Mb) contains the attached PDF files of my main works on gauge gravitation theory.

Contents

G.Sardanashvily, Gravity as a Goldstone field in the Lorentz gauge theory, Phys. Lett. A 75 (1980) 257-258

D.Ivanenko and G.Sardanashvily, The gauge treatment of gravity, Physics Reports 94 (1983) 1-45

G.Sardanashvily, The gauge model of the fifth force, Acta Phys. Polon. B 21 (1990) 583-587

G.Sardanashvily and O. Zakharov, Gauge Gravitation Theory (World Scientific, Singapore, 1992)

G.Sardanashvily, Stress-energy-momentum conservation law in gauge gravitation theory, Class. Quant. Grav. 14 (1997) 1371-1386

G.Sardanashvily, Covariant spin structure, J. Math. Phys. 39 (1998) 4874-4890

G.Sardanashvily, Classical gauge theory of gravity, Theor. Math. Phys. 132 (2002) 1163-1171

G.Sardanashvily, Gauge gravitation theory from the geometric viewpoint, Int. J. Geom. Methods Mod. Phys. 3 (2006) N1, v-xx

G.Sardanashvily, Supermetrics on supermanifolds, Int. J. Geom. Methods Mod. Phys. 5 (2008) 271-286

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Advanced Classical Field Theory (World Scientific, Singapore, 2009)

G.Sardanashvily, Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys. 8 (2011) 1869-1895