In this paper we achieve the quantization of a particle moving on the $SU(2)$ group manifold, that is, the three-dimensional sphere $S^{3}$, by using group-theoretical methods. For this purpose, a fundamental role is played by contact, non-point symmetries, i.e., symmetries that leave the Poincaré-Cartan form semi-invariant at the classical level, although not necessarily the Lagrangian. Special attention is paid to the role played by the basic quantum commutators, which depart from the canonical, Heisenberg-Weyl ones, as well as the relationship between the integration measure in the Hilbert space of the system and the non-trivial topology of the configuration space. Also, the quantization on momentum space is briefly outlined.
In this paper we outline the construction of semiclassical eigenfunctions of integrable models in terms of the semiclassical path integral for the Poisson sigma model with the target space being the phase space of the integrable system. The semiclassical path integral is defined as a formal power series with coefficients being Feynman diagrams. We also argue that in a similar way one can obtain irreducible semiclassical representations of Kontsevich's star product.
The classical quantization of a Lienard-type nonlinear oscillator is achieved by a quantization scheme (M.C. Nucci. Theor. Math. Phys., 168:997--1004, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger equation. This method straightforwardly yields the correct Schrödinger equation in the momentum space (V. Chithiika Ruby, M. Senthilvelan, and M. Lakshmanan. J. Phys. A: Math. Gen., 45:382002, 2012), and sheds light into the apparently remarkable connection with the linear harmonic oscillator.
We show that Novikov-Veselov hierarchy provides a complete family of commuting symmetries of two-dimensional $O(N)$ sigma model. In the first part of the paper we use these symmetries to prove that the Fermi spectral curve for the double-periodic sigma model is algebraic. Thus, our previous construction of the complexified harmonic maps in the case of irreducible Fermi curves is complete. In the second part of the paper we generalize our construction to the case of reducible Fermi curves and show that it gives the conformal harmonic maps to even-dimensional spheres. Remarkably, the solutions a…
The classical quantization of a family of a quadratic Liénard-type equation (Liénard II equation) is achieved by a quantization scheme (M.~C. Nucci. {\em Theor. Math. Phys.}, 168:994--1001, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger equation. This method straightforwardly yields the Schrödinger equation as given in (A.~Ghose~Choudhury and Partha Guha. {\em J. Phys. A: Math. Theor.}, 46:165202, 2013).
The classical quantization of the motion of a free particle and that of an harmonic oscillator on a double cone are achieved by a quantization scheme [M.C. Nucci, Theor. Math. Phys. 168 (2011) 994], that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schroedinger equation. The result is different from that given in [K. Kowalski, J.Rembielnski, Ann. Phys. 329 (2013) 146]. A comparison of the different outcomes is provided.
A general approach is presented for quantizing a metric nonlinear system on a manifold of constant curvature. It makes use of a curvature dependent procedure which relies on determining Noether symmetries from the metric. The curvature of the space functions as a constant parameter. For a specific metric which defines the manifold, Lie differentiation of the metric gives these symmetries. A metric is used such that the resulting Schrodinger equation can be solved in terms of hypergeometric functions. This permits the investigation of both the energy spectrum and wave functions exactly for this…
A procedure which obviates the constraint imposed by the conflict between consistent quantization and the invariance of the Hamiltonian description under nonlinear canonical transformation is proposed. This new quantization scheme preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger's equation. As an example, the quantization of the `goldfish' many-body problem extensively studied by Calogero et al. is presented.
We revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous, mathematical point of view. Our main contribution is to make precise the cohomological problem of eliminating potential anomalies that may arise when trying to preserve symmetries under quantization. The symmetries we consider are twofold: (i) diffeomorphism covariance for a general target manifold; (ii) a transitive group of isometries when the target manifold is a homogeneous space. We show that there are no anomalies in case (i) and that (ii) is also anomaly-free under additional…
We describe a covariant framework to construct a globalized version for the perturbative quantization of nonlinear split AKSZ Sigma Models on manifolds with and without boundary, and show that it captures the change of the quantum state as one changes the constant map around which one perturbs. This is done by using concepts of formal geometry. Moreover, we show that the globalized quantum state can be interpreted as a closed section with respect to an operator that squares to zero. This condition is a generalization of the modified Quantum Master Equation as in the BV-BFV formalism, which we …
Chern-Simons gauge theories in 3 dimensions and the Poisson Sigma Model (PSM) in 2 dimensions are examples of the same theory, if their field equations are interpreted as morphisms of Lie algebroids and their symmetries (on-shell) as homotopies of such morphisms. We point out that the (off-shell) gauge symmetries of the PSM in the literature are not globally well-defined for non-parallelizable Poisson manifolds and propose a covariant definition of them as left action of some finite-dimensional Lie algebroid. Our approach allows to avoid complications arising in the infinite dimensional super-…
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…
A phenomenon of classical quantization is discussed. This is revealed in the class of pseudoclassical gauge systems with nonlinear nilpotent constraints containing some free parameters. Variation of parameters does not change local (gauge) and discrete symmetries of the corresponding systems, but there are some special discrete values of them which give rise to the maximal global symmetries at the classical level. Exactly the same values of the parameters are separated at the quantum level, where, in particular, they are singled out by the requirement of conservation of the discrete symmetries…
We perform the momentum-space quantization of a spin-less particle moving on the $SU(2)$ group manifold, that is, the three-dimensional sphere $S^{3}$, by using a non-canonical method entirely based on symmetry grounds. To achieve this task, non-standard (contact) symmetries are required as already shown in a previous article where the configuration-space quantization was given. The Hilbert space in the momentum space representation turns out to be made of a subset of (oscillatory) solutions of the Helmholtz equation in four dimensions. The most relevant result is the fact that both the scalar…
We develop quantization techniques for describing the nonassociative geometry probed by closed strings in flat non-geometric R-flux backgrounds M. Starting from a suitable Courant sigma-model on an open membrane with target space M, regarded as a topological sector of closed string dynamics in R-space, we derive a twisted Poisson sigma-model on the boundary of the membrane whose target space is the cotangent bundle T^*M and whose quasi-Poisson structure coincides with those previously proposed. We argue that from the membrane perspective the path integral over multivalued closed string fields …
We extend to a non-Hermitian fermionic quantum field theory with PT symmetry our previous discussion of second quantization, discrete symmetry transformations, and inner products in a scalar field theory [arXiv:2006.06656]. For illustration, we consider a prototype model containing a single Dirac fermion with a parity-odd, anti-Hermitian mass term. In the phase of unbroken PT symmetry, this Dirac fermion model is equivalent to a Hermitian theory under a similarity transformation, with the non-Hermitian nature of the model residing only in the spinor structure, whereas the algebra of the creati…
We develop a new approach, based on quantization methods, to study higher symmetries of invariant differential operators. We focus here on conformally invariant powers of the Laplacian over a conformally flat manifold and recover results of Eastwood, Leistner, Gover and Šilhan. In particular, conformally equivariant quantization establishes a correspondence between the algebra of Hamiltonian symmetries of the null geodesic flow and the algebra of higher symmetries of the conformal Laplacian. Combined with a symplectic reduction, this leads to a quantization of the minimal nilpotent coadjoint o…
We construct a Chern-Simons type of theory using the $l_\infty$ algebra encoded by a Poisson structure on arbitrary Riemann surfaces with boundaries. A deformation quantization within the Batalin-Vilkovisky framework is performed by constructing propagators with Dirichlet boundary condition on Fulton-MacPherson compactified configuration space. Our results show that the BV quantization is independent of several gauge choices in propagators, which leads to global observables that are candidates for geometric invariants of Poisson structure and topological invariants for the worldsheet structure…
The Poisson--Weil sigma model, worked out by us recently, stems from gauging a Hamiltonian Lie group symmetry of the target space of the Poisson sigma model. Upon gauge fixing of the BV master action, it yields interesting topological field theories such as the 2--dimensional Donaldson-Witten topological gauge theory and the gauged A topological sigma model. In this paper, generalizing the above construction, we construct the Lie algebroid Poisson sigma model. This is yielded by gauging a Hamiltonian Lie groupoid symmetry of the Poisson sigma model target space. We use the BV quantization appr…
We present the effect of the quantum corrections on the Szekeres spacetime, a system important for the study of the inhomogeneities of the pre-inflationary era of the universe. The study is performed in the context of canonical quantisation in the presence of symmetries. We construct an effective classical Lagrangian and impose the quantum version of its classical integrals of motion on the wave function. The interpretational scheme of the quantum solution is that of Bohmian mechanics, in which one can avoid the unitarity problem of quantum cosmology. We discuss our results in this context.
We report briefly on an approach to quantum theory entirely based on symmetry grounds which improves Geometric Quantization in some respects and provides an alternative to the canonical framework. The present scheme, being typically non-perturbative, is primarily intended for non-linear systems, although needless to say that finding the basic symmetry associated with a given (quantum) physical problem is in general a difficult task, which many times nearly emulates the complexity of finding the actual (classical) solutions. Apart from some interesting examples related to the electromagnetic an…
This article studies large $N$ limits of a coupled system of $N$ interacting $Φ^4$ equations posed over $\mathbb{T}^{d}$ for $d=2$, known as the $O(N)$ linear sigma model. Uniform in $N$ bounds on the dynamics are established, allowing us to show convergence to a mean-field singular SPDE, also proved to be globally well-posed. Moreover, we show tightness of the invariant measures in the large $N$ limit. For large enough mass, they converge to the (massive) Gaussian free field, the unique invariant measure of the mean-field dynamics, at a rate of order $1/\sqrt{N}$ with respect to the Wasserste…
We show that SU(2)_L Yangian and q-deformed SU(2)_R symmetries are realized in a two-dimensional sigma model defined on a three-dimensional squashed sphere. These symmetries enable us to develop the two descriptions to describe its classical dynamics, 1) rational and 2) trigonometric descriptions. The former 1) is based on the SU(2)_L symmetry and the latter 2) comes from the broken SU(2)_R symmetry. Each of the Lax pairs constructed in both ways leads to the same equations of motion. The two descriptions are related one another through a non-local map.
We construct a two-dimensional topological sigma model whose target space is endowed with a Poisson algebra for differential forms. The model consists of an equal number of bosonic and fermionic fields of worldsheet form degrees zero and one. The action is built using exterior products and derivatives, without any reference to any worldsheet metric, and is of the covariant Hamiltonian form. The equations of motion define a universally Cartan integrable system. In addition to gauge symmetries, the model has one rigid nilpotent supersymmetry corresponding to the target space de Rham operator. Th…
We construct firstly the complete list of five quantum deformations of $D=4$ complex homogeneous orthogonal Lie algebra $\mathfrak{o}(4;\mathbb{C})\cong \mathfrak{o}(3;\mathbb{C})\oplus \mathfrak{o}(3;\mathbb{C})$, describing quantum rotational symmetry of four-dimensional complex space-time, in particular we provide the corresponding universal quantum $R$-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of $\mathfrak{o}(4;\mathbb{C})$: Euclidean $\mathfrak{o}(4)$, Lorentz $\mathfrak{o}(3,1)$, Kleinian $\ma…
The Painlevé equations can be written as Hamiltonian systems with affine Weyl group symmetries. A canonical quantization of the Painlevé equations preserving the affine Weyl group symmetries has been studied. While, the Painlevé equations are isomonodromic equations for certain second-order linear differential equations. In this paper, we introduce a canonical quantization of Lax equations for the Painlevé equations and construct symmetries of the quantum Lax equations. We also show that our quantum Lax equations are derived from Virasoro conformal field theory.
In this note we address the question whether one can recover from the vertex operator algebra associated with a four-dimensional N=2 superconformal field theory the deformation quantization of the Higgs branch of vacua that appears as a protected subsector in the three-dimensional circle-reduced theory. We answer this question positively if the UV R-symmetries do not mix with accidental (topological) symmetries along the renormalization group flow from the four-dimensional theory on a circle to the three-dimensional theory. If they do mix, we still find a deformation quantization but at differ…
A conditional symmetry is defined, in the phase-space of a quadratic in velocities constrained action, as a simultaneous conformal symmetry of the supermetric and the superpotential. It is proven that such a symmetry corresponds to a variational (Noether) symmetry.The use of these symmetries as quantum conditions on the wave-function entails a kind of selection rule. As an example, the minisuperspace model ensuing from a reduction of the Einstein - Hilbert action by considering static, spherically symmetric configurations and r as the independent dynamical variable, is canonically quantized. T…
Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces... In this work, we prove existence of an equivariant quantization for orbifolds. Our construction combines an appropriate desingularization of any Riemannian orbifold by a foliated smooth manifold, with the foliated equivariant quantization that we built in \cite{PoRaWo}. Further, we suggest definitions of the comm…
I construct lowest-energy representations of non-centrally extended algebras of Noether symmetries, including diffeomorphisms and reparametrizations of the observer's trajectory. This may be viewed as a new scheme for quantization. First classical physics is formulated as the cohomology of a certain Koszul-Tate (KT) complex, using not only fields and antifields but also their conjugate momenta. Then all fields are expanded in a Taylor series around the observer's present position, and terms of order higher than p are truncated. Finally, quantization is carried out by replacing Poisson brackets…
