Symmetries of non-linear systems and non-canonical quantization

By Victor Aldaya (IAA, Granada)

We face a sound revision of the role of symmetries of a physical system so as to characterize the corresponding solution manifold by means of Noether invariants. This step is inescapable to achieve the correct quantization in non-linear cases, where the success of Canonical Quantization is not guaranteed. To this end, “point symmetries” of the Lagrangian are generally not enough, and the generalized use of contact symmetries will play a preponderant tool. They are defined in terms of the Poincaré-Cartan form, which permits us to find the symplectic structure on the Solution Manifold, through some sort of Hamilton-Jacobi transformation, as well as the required symmetries. They are realized as Hamiltonian vector fields, associated with functions on the solution manifold, lifted back to the Evolution Manifold through the inverse of this Hamilton-Jacobi mapping, that which constitutes an inverse of the Noether Theorem. In this framework, solutions and symmetries are somehow identified and this correspondence is also kept at a well-defined perturbative level. The next step in approaching the quantization consists in selecting the proper Poisson subalgebra to replace the standard Heisenberg-Weyl one, and proceeding to construct their unitary and irreducible representations according to an already well-established group-theoretical method. The paradigmatic example of Non-Linear Sigma Models will be considered in the context of Non-Abelian Stueckelberg approach to the Quantum Field Theory of Massive Gauge Theories.