Geometry of Constrained Dynamical Systems

A lively, varied and topical presentation of this branch of theoretical physics.

Many of the fundamental theories of modern physics can be considered as descriptions of dynamical systems subjected to constraints. The study of these constrained dynamical systems, in particular the problems encountered in formulating them as quantum systems, has many profound links with geometry. These links were explored in the Symposium on Geometry and Gravity, held at the Newton Institute in 1994. This book arose from a conference held during that symposium, and is a collection of papers devoted to problems such as Chern–Simons theory, sigma-models, gauge invariance and loop quantization, general relativity and the notion of time and quantum gravity. They present a lively, varied and topical perspective on this important branch of theoretical physics, from some of the leading authorities in the subject.

Preface; The network; 1. The constraint algebra of higher dimensional Chern–Simons; 2. Non-relativistic Chern–Simons vortices from the constrained Hamiltonian formalism; 3. Classical solutions of gravitating Chern–Simons electrodynamics; 4. Exponentionally localised instantons in a hierachy of Higgs models; 5. Obstructions to gauging WZ terms: a symplectic curiosity; 6. Global aspects of symmetries in sigma models with torsion; 7. Canonical structure of the non-linear sigma model in a polynomial formulation; 8. A manifestly gauge-invariant approach to quantum theories of gauge fields; 9. On the Hamiltonian formulation of the higher dimensional Chern–Simons gravity; 10. An example of loop quantization; 11. Gauge fixing in constrained systems; 12. Light-cone formulation of gauge theories; 13. Hamiltonian constraints and Dirac observables; 14. Gauging kinematical and internal symmetry groups for extended systems; 15. On the harmonic interaction of three relativistic point particles; 16. Non-existence of static multi-black-hole solutions in 2+1 dimensions; 17. Spherically symmetric gravity and the notion of time in general relativity; 18. Canonical decomposition of Belinskii–Zakharov one-soliton; 19. Hamiltonian reduction and the R-matrix of the Calogero model; 20. Intrinsic approach to the Legendre transformation in super-mechanics; 21. Field-antifield description of the anomalous theories; 22.Transfer matrix quantization of general relativity, and the problem of time; 23. The W3-particle; 24. Pure geometric approach to singular Lagrangians with higher derivatives; 25. Dirac versus reduced phase space quantization; 26. Classical and quantum aspects of degenerate metric field; 27. BRST-antibracket cohomology in 2D conformal gravity; 28. Quantization of 2+1 gravity for g=1 and g=2; 29. Geometry and dynamics with time-dependent constraints; 30. Collective coordinates and BRST transformations, or Gauge theories without gauge fields; 31. Geometry of fermionic constraints in superstring theories; 32. BRST and new superstring states; 33. Generalized canonical quantization of gauge theories with polarized second-class constraints; 34. Radiation field on superspace.

Many of the fundamental theories of modern physics can be considered as descriptions of dynamical systems subjected to constraints. The collected papers deal with problems such as; Chern-Simons theory, sigma-models, gauge invariance, general relativity and the notion of time, and quantum gravity.