International Journal of Geometric Methods in Modern Physics

We highlight aspects of equivariant bundles with invariant connections over the quantum projective line and mention their use for equivariant reduction. We also study holomorphic bundles whose sections lead to a quantum homogeneous coordinate ring.

This paper is a terse highlight of the remarkable similarities and differences among two matrix ODEs — one solvable and one integrable — and the many-body problems related to them.

We shall investigate the consequences of non-trivial Weyl geometries on conservation laws of a fluid. In particular we shall obtain a set of properties which allow to obtain in this generalized setting the standard relation between conservation of the energy-momentum tensor and number of particles.

We use the conditions of linearizability by point and by Hopf–Cole transformations, introduced recently by the authors, to classify all real multilinear equations on a square lattice with at most quadratic nonlinearity. We find that, up to a linear transformation of the dependent variable and an exchange of the independent variables, only two equations in this class are linearizable by point transformations and none by Hopf–Cole transformations.

A classical problem in axiomatic quantum mechanics is deducing a Hilbert space realization for a quantum logic that admits a vector space coordinatization of the Piron–McLaren type. Our aim is to show how a theorem of M. Solér [Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra23 (1995) 219–243.] can be used to get a (partial) solution of this problem. We first derive a generalization of the Wigner theorem on symmetry transformations that holds already in the Piron–McLaren frame. Then we investigate which conditions on the quantum logic allow the use of Solér's theorem in order to obtain a Hilbert space solution for the coordinatization problem.

The quantum description of non-linear systems finds a deep obstruction in the Canonical Quantization framework and Non-Linear Sigma Models constitute the best representatives. In this paper, we face the quantization of such systems on the grounds of a Group Approach to Quantization, and extend the algorithm to the specific case of massive Non-Abelian gauge theories. The basic geometric structures behind are the so-called "jet-gauge groups".

This work presents a newly renovated approach to the analysis of second-order Riccati equations from the point of view of the theory of Lie systems. We show that these equations can be mapped into Lie systems through certain Legendre transforms. This result allows us to construct new superposition rules for studying second-order Riccati equations and to reduce their integration to solving (first-order) Riccati equations.

I review my proposal about how to extend the geometric Hamilton–Jacobi theory to higher derivative field theories on fiber bundles.

Using a non-canonical braiding over the three-dimensional left covariant calculus we present a family of Hodge operators on SU_q(2) and on its homogeneous quantum space .

The study of isolated system is of little interest to physicists since it is not applicable to practical situations. Analytical mechanics should be brought closer to control theory. Behavior of a system coupled to external controlling devices should be examined. There is a class of controlling devices representable by external forces. We proposed a framework for describing the dynamics of mechanical devices subject to control by external forces. The original version of the theory was presented in G. Marmo, W. Tulczyjew and P. Urbański, Dynamics of autonomous systems with external forces, Acta Phys. Polon. B33 (2002) 1181–1240. The geometric constructions used in original paper are clarified. The Legendre–Fenchel transformation is described and the concept of external force is introduced. Liouville structures are defined.

The recently proposed quantum Arnold transformation is revisited and extended and its relation with other methods is emphasized. Possible applications are also outlined.

Resorting to the Lagrange–Souriau 2-form formalism, a wide class of systems are derived in non-commuting and/or non-canonical variables, nor the Darboux theorem can be helpful, because of the gauge character of all phase-space variables. As a paradigmatic example, the motion of a charged particle in a magnetic monopole field in the presence of a momentum space monopole is considered.

Recent results in Local Regge Calculus are confronted with Spin-Foam Formalism. Introducing Barrett–Crane Quantization in Local Regge Calculus makes it possible to associate a unique Spin j_h with an hinge h, fulfilling one of the requirements of Spin-Foam definition. It is shown that inter-twiner terms of Spin-Foam can follow from the closure constraint in Local Regge Calculus.

In this paper we discuss some constructive procedures which can be used in characterization of irreducible quantum operations and quantum channels. Our method makes use of an explicit form of a fixed operation in its Kraus representation. A generalization of the Shemesh criterion is discussed.

Based on a geometric picture, the example of free particle motion for both classical and quantum domains is considered in the tomographic probability representation. Wave functions and density operators as well as optical and symplectic tomograms are obtained as solutions of kinetic classical and quantum equations for the state tomograms. The difference of tomograms of free particle for classical and quantum states is discussed.

We argue that Snyder's algebra has two distinct representations and they yield a lattice description of space. We also argue that particle dynamics can be defined on such a lattice in a manner which is consistent with Poincaré invariance.

We analyze the role of boundaries in the infrared behavior of quantum field theories. By means of a novel method we calculate the vacuum energy for a massless scalar field confined between two homogeneous parallel plates with the most general type of boundary properties. This allows the discrimination between boundary conditions which generate attractive or repulsive Casimir forces between the plates. In the interface between both regimes we find a very interesting family of boundary conditions which do not induce any type of Casimir force. We analyze the effect of the renormalization group flow on these boundary conditions. Even if the Casimirless conformally invariant conditions are physically unstable under renormalization group flow they emerge as a new set of conformally invariant boundary conditions which are anomaly free.

We discuss the similarities and differences between the formalism of Hamiltonian Classical Mechanics and of Quantum Mechanics and exemplify the differences through an analysis of tracks in a cloud chamber.

We analyze two criteria of (non)Markovianity: one based on the mathematical concept of divisibility of the dynamical map and the other one based on distinguishability of quantum states. We illustrate these concepts from the geometric perspective using simple examples of qubit dynamics.

We explore the conceptual usefulness of Riemannian geometric tools induced by the statistical concept of distinguishability in quantifying the effect of a depolarizing channel on quantum states. Specifically, we compare the geometries of the interior of undeformed and deformed Bloch spheres related to density operators on a two-dimensional Hilbert space. We show that randomization emerges geometrically through a smaller infinitesimal quantum line element on the deformed Bloch sphere while the uniform contraction manifests itself via a deformed set of geodesics where the spacial components of the deformed four-Bloch vector are simply the contracted versions of the undeformed Bloch vector components.

Correlations between states in a general quantum ensemble can be encoded in a Gram matrix that depends on a prescribed joint purification. This allows to establish an upper bound for the Holevo quantity χ. Optimizing over joint purifications is a hard problem. For small ensembles the connection with matrices of fidelities is investigated. General bounds are obtained for a linearized version of χ.

Bipartite entanglement between two parties of a composite quantum system can be quantified in terms of the purity of one party and there always exists a pure state of the total system that maximizes it (and minimizes purity). When many different bipartitions are considered, the requirement that purity be minimal for all bipartitions gives rise to the phenomenon of entanglement frustration. This feature, observed in quantum systems with both discrete and continuous variables, can be studied by means of a suitable cost function whose minimizers are the maximally multipartite-entangled states (MMES). In this paper we extend the analysis of multipartite entanglement frustration of Gaussian states in multimode bosonic systems. We derive bounds on the frustration, under the constraint of finite mean energy, in the low- and high-energy limits.

We introduce on physical grounds a new measure of multipartite entanglement for pure states. The function we define is discriminant and monotone under LOCC; moreover, it can be expressed in terms of observables of the system.

The quantum Zeno effect is described in geometric terms. The quantum Zeno time (inverse standard deviation of the Hamiltonian) and the generator of the quantum Zeno dynamics are both given a geometric interpretation.

We will present a short review of some work we have done in the last ten years with Giuseppe Marmo, on the attempt to formulate some interesting physical problems — such as the Quantum Inverse Problem, Alternative Structures and Berry Phase — in a geometrical setting.

Geometry and Physics developed independently, until the past twentieth century, where physicists realized geometry is rather flexible and can adapt itself to the needs and characteristics of modern physics. Besides the use of Riemannian manifolds to describe General Relativity, classical mechanics encounters symplectic geometry, not to speak of the bundle connection ingredient of modern gauge theories; even Quantum Mechanics, after the initial Hilbert space period, is seeking nowadays to adapt itself better to a geometrical interpretation, by imperatives of the path integral description and also to incorporate more clearly the symplectic aspects of its classical antecedent.

There are many examples of quantum anomalies of continuous and discrete classical symmetries. Examples come from chiral anomalies in the Standard Model and gravitational anomalies in string theories. They occur when classical symmetries do not preserve the domains of quantum operators like the Hamiltonian. Here we show by a simple example, a particle on a circle, that anomalous symmetries can often be implemented at the expense of working with mixed states.

The reduction of a quantum system ("folding" a quantum system) is described as the reduction of its Lie–Jordan Banach algebra of observables with respect to Lie–Jordan Banach subalgebras and Jordan ideals. The space of states of the reduced Lie–Jordan Banach algebra is described in terms of unreduced states ("unfolding" states) as well as the GNS construction. A few examples are discussed.