Constrained systems, generalized Hamilton-Jacobi actions, and quantization (original) (raw)
Related papers
Banach Center Publications, 2016
Inspired by problems arising in the geometrical treatment of Yang-Mills theories and Palatini's gravity, the covariant formulation of Hamiltonian dynamical systems as a Hamiltonian field theory of dimension 1 + 0 on a manifold with boundary is discussed. After a precise statement of Hamilton's variational principle in this context, the geometrical properties of the space of solutions of the Euler-Lagrange equations of the theory are analyzed. A sufficient condition is obtained that guarantees that the set of solutions of the Euler-Lagrange equations at the boundary of the manifold, fill a Lagrangian submanifold of the space of fields at the boundary. Finally a theory of constraints is introduced that mimics the constraints arising in Palatini's gravity.
Extended Hamiltonian Formalism of Field Theories: Variational Aspects and Other Topics
2012
We consider Hamiltonian systems in first-order multisymplectic field theories. In particular, we introduce Hamiltonian systems in the extended multimomentum bundle. The resulting extended Hamiltonian formalism is the generalization to field theories of the extended (symplectic) formalism for non-autonomous mechanical systems. In order to derive the corresponding field equations, a variational principle is stated for these extended Hamiltonian systems and, after studying the geometric properties of these systems, we establish the relation between this extended formalism and the standard one. + J J J J J J J J Ĵ ?
Semiclassical Quantization of Classical Field Theories
Mathematical Physics Studies, 2015
These lectures are an introduction to formal semiclassical quantization of classical field theory. First we develop the Hamiltonian formalism for classical field theories on space time with boundary. It does not have to be a cylinder as in the usual Hamiltonian framework. Then we outline formal semiclassical quantization in the finite dimensional case. Towards the end we give an example of such a quantization in the case of Abelian Chern-Simons theory.
The Hamiltonian analysis for the Euler and Second-Chern classes is performed. We show that, in spite of the fact that the Second-Chern and Euler invariants give rise to the same equations of motion, their corresponding symplectic structures on the phase space are different, therefore, one can expect different quantum formulations. In addition, the symmetries of actions written as a BF-like theory that lead to Yang-Mills equations of motion are studied. A close relationship with the results obtained in previous works for the Second-Chern and Euler classes is found.
Contact Symmetries in Non-Linear Mechanics: a preliminary step to (Non-Canonical) Quantization
arXiv: Mathematical Physics, 2014
In this paper we exploit the use of symmetries of a physical system so as to characterize algebraically the corresponding solution manifold by means of Noether invariants. This constitutes a necessary preliminary step towards the correct quantization in non-linear cases, where the success of Canonical Quantization is not guaranteed in general. To achieve this task "point symmetries" of the Lagrangian are generally not enough, and the notion of contact transformations is in order: the solution manifold can not be in general parametrized by means of Noether invariants associated with basic point symmetries. The use of the contact structure given by the Poincar\'e-Cartan form permits the definition of the symplectic form on the solution manifold, through some sort of Hamilton-Jacobi transformation. It also provides the required basic symmetries, realized as Hamiltonian vector fields associated with global functions on the solution manifold (thus constituting an inverse of...
Deformation Quantisation of Constrained Systems
We study the deformation quantisation (Moyal quantisation) of general constrained Hamiltonian systems. It is shown how second class constraints can be turned into first class quantum constraints. This is illustrated by the O(N) non-linear sigma\sigmasigma-model. Some new light is also shed on the Dirac bracket. Furthermore, it is shown how classical constraints not in involution with the classical Hamiltonian, can be turned into quantum constraints {\em in} involution with respect to the Hamiltonian. Conditions on the existence of anomalies are also derived, and it is shown how some kinds of anomalies can be removed. The equations defining the set of physical states are also given. It turns out that the deformation quantisation of pure Yang-Mills theory is straightforward whereas gravity is anomalous. A formal solution to the Yang-Mills quantum constraints is found. In the \small{ADM} formalism of gravity the anomaly is very complicated and the equations picking out physical states become i...
Converting classical theories to quantum theories by solutions of the Hamilton-Jacobi equation
Physical Review D, 2012
By employing special solutions of the Hamilton-Jacobi equation and tools from lattice theories, we suggest an approach to convert classical theories to quantum theories for mechanics and field theories. Some nontrivial results are obtained for a gauge field and a fermion field. For a topologically massive gauge theory, we can obtain a first order Lagrangian with mass term. For the fermion field, in order to make our approach feasible, we supplement the conventional Lagrangian with a surface term. This surface term can also produce the massive term for the fermion.
A pr 2 01 6 Structural aspects of Hamilton – Jacobi theory
2018
In our previous papers [11, 13] we showed that the Hamilton–Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how constants of the motion help to solve the Hamilton–Jacobi equation. Here we want to delve into this interpretation by considering the most general case: a dynamical system on a manifold that is described in terms of a family of dynamics (‘slicing vector fields’) on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamilton– Jacobi theory, by considering special cases like fibred manifolds and Hamiltonian dynamics, in the symplectic framework and the Poisson one. We also show how a set of functions on a tangent bundle can determine a second-order dynamics for which they are constants of the motion.
Hamiltonian Mechanics of Gauge Systems
2011
The principles of gauge symmetry and quantization are fundamental to modern understanding of the laws of electromagnetism, weak and strong subatomic forces, and the theory of general relativity. Ideal for graduate students and researchers in theoretical and mathematical physics, this unique book provides a systematic introduction to Hamiltonian mechanics of systems with gauge symmetry. The book reveals how gauge symmetry may lead to a non-trivial geometry of the physical phase space and studies its effect on quantum dynamics by path integral methods. It also covers aspects of Hamiltonian path integral formalism in detail, along with a number of related topics such as the theory of canonical transformations on phase space supermanifolds, non-commutativity of canonical quantization, and elimination of non-physical variables. The discussion is accompanied by numerous detailed examples of dynamical models with gauge symmetries, clearly illustrating the key concepts.