Quantization of a Free Particle in Toric Geometry

Physics Letters B, 2005

Systems under holonomic constraints are classified within the generalized Hamiltonian framework as second-class constraints systems. We show that each system of point particles with holonomic constraints has a hidden gauge symmetry which allows to quantize it in the original phase space as a first-class constraints system. The proposed method is illustrated with quantization of a point particle moving on an n − 1-dimensional sphere S n−1 as well as its field theory analog the O(n) nonlinear sigma model.

Journal of Geometric Mechanics

Mechanical systems (i.e., one-dimensional field theories) with constraints are the focus of this paper. In the classical theory, systems with infinite-dimensional targets are considered as well (this then encompasses also higher-dimensional field theories in the hamiltonian formalism). The properties of the Hamilton–Jacobi (HJ) action are described in details and several examples are explicitly computed (including nonabelian Chern–Simons theory, where the HJ action turns out to be the gauged Wess–Zumino–Witten action). Perturbative quantization, limited in this note to finite-dimensional targets, is performed in the framework of the Batalin–Vilkovisky (BV) formalism in the bulk and of the Batalin–Fradkin–Vilkovisky (BFV) formalism at the endpoints. As a sanity check of the method, it is proved that the semiclassical contribution of the physical part of the evolution operator is still given by the HJ action. Several examples are computed explicitly. In particular, it is shown that th...

Czechoslovak Journal of Physics, 2000

We investigate the path-integral quantization of constrained systems with second-order Lagrangians using the Hamilton-Jacobi method. The path-integral quantization for two models is obtained using the canonical path-integral method.

International Journal of Modern Physics A, 1997

The quantization of isomonodromic deformation of a meromorphic connection on the torus is shown to lead directly to the Knizhnik–Zamolodchikov–Bernard equations in the same way as the problem on the sphere leads to the system of Knizhnik–Zamolodchikov equations. The Poisson bracket required for a Hamiltonian formulation of isomonodromic deformations is naturally induced by the Poisson structure of Chern–Simons theory in a holomorphic gauge fixing. This turns out to be the origin of the appearance of twisted quantities on the torus.

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 $\sigma$-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...

Journal of Mathematical Physics, 2017

It is natural to investigate if the quantization of an integrable or superintegrable classical Hamiltonian systems is still integrable or superintegrable. We study here this problem in the case of natural Hamiltonians with constants of motion quadratic in the momenta. The procedure of quantization here considered, transforms the Hamiltonian into the Laplace-Beltrami operator plus a scalar potential. In order to transform the constants of motion into symmetry operators of the quantum Hamiltonian, additional scalar potentials, known as quantum corrections, must be introduced, depending on the Riemannian structure of the manifold. We give here a complete geometric characterization of the quantum corrections necessary for the case considered. Stäckel systems are studied in particular details. Examples in conformally and non-conformally flat manifolds are given.

2006

We investigate the quantum mechanics of the doubled torus system, introduced by Hull [1] to describe T-folds in a more geometric way. Classically, this system consists of a world-sheet Lagrangian together with some constraints, which reduce the number of degrees of freedom to the correct physical number. We consider this system from the point of view of constrained Hamiltonian dynamics. In this case the constraints are second class, and we can quantize on the constrained surface using Dirac brackets. We perform the quantization for a simple T-fold background and compare to results for the conventional non-doubled torus system. Finally, we formulate a consistent supersymmetric version of the doubled torus system, including supersymmetric constraints.

Mathematische Annalen, 2015

Let P be a Delzant polytope. We show that the quantization of the corresponding toric manifold X P in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time t = √ −1s. We relate the quantization of X P in two different toric Kähler polarizations by taking the time-√ −1s Hamiltonian "flow" of strongly convex functions on the moment polytope P. By taking s to infinity, we obtain the quantization of X P in the (singular) real toric polarization. Recall that X P has an open dense subset which is biholomorphic to (C *) n. The quantization of X P in a toric Kähler polarization can also be described by applying the complexified Hamiltonian flow of the Abreu-Guillemin symplectic potential g, at time t = √ −1, to an appropriate finite-dimensional subspace of quantum states in the quantization of T * T n in the vertical polarization. By taking other imaginary times, t = k √ −1, k ∈ R, we describe toric Kähler metrics with cone singularities along the toric divisors in X P. For convex Hamiltonian functions and sufficiently negative imaginary part of the complex time, we obtain degenerate Kähler structures which are negative definite in some regions of X P. We show that the pointwise and L 2-norms of quantum states are asymptotically vanishing on negative-definite regions.

2003

Physical Review A, 1991

Many aspects of a model problem, the Lagrangian of which contains a term depending quadratically on the acceleration, are examined in the regime where the classical solution consists of two independent normal modes. It is shown that the techniques of conversion to a problem of Lagrange, generalized mechanics, and Dirac s method for constrained systems all yield the same canonical form for the Hamiltonian. It is also seen that the resultant canonical equations of motion are equivalent to the Euler-Lagrange equations. In canonical form, all of the standard results apply, quantization follows in the usual way, and the interpretation of the results is straightforward. It is also demonstrated that perturbative methods fail, both classically and quantum mechanically, indicating the need for the nonperturbative techniques applied herein. Finally, it is noted that this result may have fundamental implications for certain relativistic theories.