On the Canonical Treatment of Lagrangian Constraints

Exact procedures that follow Dirac's constraint quantization of gauge theories are usually technically involved and often difficult to implement in practice. We overview an "effective" scheme for obtaining the leading order semiclassical corrections to the dynamics of constrained quantum systems developed elsewhere. Motivated by the geometrical view of quantum mechanics, our method mimics the classical Dirac-Bergmann algorithm and avoids direct reference to a particular repre...

From the "vibrating string" and "Kepler's equation" theories to relativistic quantum fields, perturbation theory, (divergent) series resummations, KAM theory.

The Hamilton-Jacobi method of constrained systems is discussed. The equations of motion of a singular system with time dependent constraints are obtained as total differential equations in many variables. The integrability conditions for the relativistic particle in a plane wave lead us to obtain the canonical phase-space coordinates with out using any gauge fixing condition. As a result of the quantization, we obtain the Klein-Gordon theory for a particle in a plane wave. Th...

This is a collection of lectures given at the University of Heidelberg, especially but not exclusively for people who want to learn something about the canonical approach to quantum gravity, which is however not included in these lectures. They are about Dirac's general method to construct a quantum theory out of a classical theory, which has to be defined in terms of a Lagrangian. The classical Hamiltonian formalism is reviewed, with emphasis on the relation between constrai...

This work conducts a Hamilton-Jacobi analysis of classical dynamical systems with internal constraints. We examine four systems, all previously analyzed by David Brown: three with familiar components (point masses, springs, rods, ropes, and pulleys) and one chosen specifically for its detailed illustration of the Dirac-Bergmann algorithm's logical steps. Including this fourth system allows for a direct and insightful comparison with the Hamilton-Jacobi formalism, thereby deep...

In this note we present invariant formulation of the d'Alambert principle and classical time-dependent Lagrangian mechanics with holonomic constraints from the perspective of moving frames.

This is a substantially expanded version of a chapter-contribution to "The Springer Handbook of Spacetime", edited by Abhay Ashtekar and Vesselin Petkov, published by Springer Verlag in 2014. This contribution introduces the reader to the reformulation of Einstein's field equations of General Relativity as a constrained evolutionary system of Hamiltonian type and discusses some of its uses, together with some technical and conceptual aspects. Attempts were made to keep the pr...

These lecture notes were prepared as a basic introduction to the theory of constrained systems which is how the fundamental forces of nature appear in their Hamiltonian formulation. Only a working knowledge of Lagrangian and Hamiltonian formulation of mechanics is assumed. These notes are based on the set of eight lectures given at the {\em Refresher Course for College Teachers} held at IMSc during May-June, 2005. These are submitted to the arxiv for an easy access to a wider...

The systematization of the purely Lagrangean approach to constrained systems in the form of an algorithm involves the iterative construction of a generalized Hessian matrix W taking a rectangular form. This Hessian will exhibit as many left zero-modes as there are Lagrangean constraints in the theory. We apply this approach to a general Lagrangean in the first order formulation and show how the seemingly overdetermined set of equations is solved for the velocities by suitably...

We review the Dirac formalism for dealing with constraints in a canonical Hamiltonian formulation and discuss gauge freedom and display constraints for gauge theories in a general context. We introduce the Dirac bracket and show that it provides a consistent method to remove any gauge freedom present. We discuss stability in evolution of gauge theories and show that fixing all gauge freedom is sufficient to ensure well-posedness for a large class of gauge theories. Electrodyn...