arXiv subjects

Janusz Grabowski

Publications and source records attributed to Janusz Grabowski.

At least 91 records · Page 5Linked to original sources

arXiv2006

The Schroedinger operator in Newtonian space-time

The Schroedinger operator on the Newtonian space-time is defined in a way which is independent on the class of inertial observers. In this picture the Schroedinger operator acts not on functions on the space-time but on sections of certain one-dimensional complex vector bundle over space-time. This bundle, constructed from the data provided by all possible inertial observers, has no canonical trivialization, so these sections cannot be viewed as functions on the space-time. The presented framework is conceptually four-dimensional and does not involve any ad hoc or axiomatically introduced geometrical structures. It is based only on the traditional understanding of the Schroedinger operator in a given reference frame and it turns out to be strictly related to the frame-independent formulation of analytical Newtonian mechanics that makes a bridge between the classical and quantum theory.

math-ph↗

arXiv2006

Superposition rules, Lie theorem and partial differential equations

A rigorous geometric proof of the Lie's Theorem on nonlinear superposition rules for solutions of non-autonomous ordinary differential equations is given filling in all the gaps present in the existing literature. The proof is based on an alternative but equivalent definition of a superposition rule: it is considered as a foliation with some suitable properties. The problem of uniqueness of the superposition function is solved, the key point being the codimension of the foliation constructed from the given Lie algebra of vector fields. Finally, as a more convincing argument supporting the use of this alternative definition of superposition rule, it is shown that this definition allows an immediate generalization of Lie's Theorem for the case of systems of partial differential equations.

math-ph↗

arXiv2006

Quantum Bi-Hamiltonian Systems

We define quantum bi-Hamiltonian systems, by analogy with the classical case, as derivations in operator algebras which are inner derivations with respect to two compatible associative structures. We find such structures by means of the associative version of Nijenhuis tensors. Explicit examples, e.g. for the harmonic oscillator, are given.

math-ph↗

arXiv2006

Local Lie algebra determines base manifold

It is proven that a local Lie algebra in the sense of A. A. Kirillov determines the base manifold up to a diffeomorphism provided the anchor map is nowhere-vanishing. In particular, the Lie algebras of nowhere-vanishing Poisson or Jacobi brackets determine manifolds. This result has been proven for different types of differentiability: smooth, real-analytic, and holomorphic.

math.DG↗

arXiv2006

AV-differential geometry: Euler-Lagrange equations

A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize Lie algebras and Lie algebroids. This scheme covers and unifies various geometrical approaches to mechanics in the Lagrangian and Hamiltonian pictures, including time-dependent lagrangians and hamiltonians. In our approach, lagrangians and hamiltonians are, in general, sections of certain $\R$-principal bundles, and the solutions of analogs of Euler-Lagrange equations are curves in certain affine bundles. The correct geometrical and frame-independent description of Newtonian Mechanics is of this type.

math.DG↗

arXiv2006

Symmetries, group actions, and entanglement

We address several problems concerning the geometry of the space of Hermitian operators on a finite-dimensional Hilbert space, in particular the geometry of the space of density states and canonical group actions on it. For quantum composite systems we discuss and give examples of measures of entanglement.

math-ph↗

arXiv2006

Courant-Nijenhuis tensors and generalized geometries

Nijenhuis tensors $N$ on Courant algebroids compatible with the pairing are studied. This compatibility condition turns out to be of the form $N+N^*=aI$ for irreducible Courant algebroids, in particular for the extended tangent bundles $TM\oplus T^*M$. It is proved that compatible Nijenhuis tensors on irreducible Courant algebroids must satisfy quadratic relations $N^2-aN+bI=0$, so that the corresponding hierarchy is very poor. The particular case $N^2=-I$ is associated with Hitchin's generalized geometries and the cases $N^2=I$ and $N^2=0$ -- to other "generalized geometries". These concepts find a natural description in terms of supersymplectic Poisson brackets on graded supermanifolds.

math.DG↗

arXiv2005

Geometry of quantum systems: density states and entanglement

Various problems concerning the geometry of the space $u^*(\cH)$ of Hermitian operators on a Hilbert space $\cH$ are addressed. In particular, we study the canonical Poisson and Riemann-Jordan tensors and the corresponding foliations into Kähler submanifolds. It is also shown that the space $\cD(\cH)$ of density states on an $n$-dimensional Hilbert space $\cH$ is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space $\cD^k(\cH)$ of rank-$k$ states, $k=1,...,n$, is a smooth manifold of (real) dimension $2nk-k^2-1$ and this stratification is maximal in the sense that every smooth curve in $\cD(\cH)$, viewed as a subset of the dual $u^*(\cH)$ to the Lie algebra of the unitary group $U(\cH)$, at every point must be tangent to the strata $\cD^k(\cH)$ it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition $\cH=\cH^1\ot\cH^2$, an abstract criterion of entanglement is proved.

math-ph↗

arXiv2005

On quantum and classical Poisson algebras

Results on derivations and automorphisms of some quantum and classical Poisson algebras, as well as characterizations of manifolds by the Lie structure of such algebras, are revisited and extended. We prove in particular somehow unexpected fact that the algebras of linear differential operators acting on smooth sections of two real vector bundles of rank 1 are isomorphic as Lie algebras if and only if the base manifolds are diffeomorphic, independently whether the line bundles themselves are isomorphic or not.

math.DG↗

arXiv2005

Homology and modular classes of Lie algebroids

For a Lie algebroid, divergences chosen in a classical way lead to a uniquely defined homology theory. They define also, in a natural way, modular classes of certain Lie algebroid morphisms. This approach, applied for the anchor map, recovers the concept of modular class due to S. Evans, J.-H. Lu, and A. Weinstein.

math.DG↗

arXiv2005

Lie algebraic characterization of manifolds

Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlaying manifolds.

math.DG↗

arXiv2004

Isomorphisms of algebras of smooth functions revisited

It is proved that isomorphisms between algebras of smooth functions on Hausdorff smooth manifolds are implemented by diffeomorphisms. It is not required that manifolds are second countable nor paracompact. This solves a problem stated by A. Wienstein. Some related results are discussed as well.

math.DG↗

arXiv2004

AV-differential geometry: Poisson and Jacobi structures

Based on ideas of W. M. Tulczyjew, a geometric framework for a frame-independent formulation of different problems in analytical mechanics is developed. In this approach affine bundles replace vector bundles of the standard description and functions are replaced by sections of certain affine line bundles called AV-bundles. Categorial constructions for affine and special affine bundles as well as natural analogs of Lie algebroid structures on affine bundles (Lie affgebroids) are investigated. One discovers certain Lie algebroids and Lie affgebroids canonically associated with an AV-bundle which are closely related to affine analogs of Poisson and Jacobi structures. Homology and cohomology of the latter are canonically defined. The developed concepts are applied in solving some problems of frame-independent geometric description of mechanical systems.

math.DG↗

arXiv2004

Courant algebroid and Lie bialgebroid contractions

Contractions of Leibniz algebras and Courant algebroids by means of (1,1)-tensors are introduced and studied. An appropriate version of Nijenhuis tensors leads to natural deformations of Dirac structures and Lie bialgebroids. One recovers presymplectic-Nijenhuis structures, Poisson-Nijenhuis structures, and triangular Lie bialgebroids as particular examples.

math.DG↗

arXiv2003

Lie brackets on affine bundles

Natural analogs of Lie brackets on affine bundles are studied, based on natural examples from differential geometry and analytical mechanics. In particular, a close relation to Lie algebroids and, by a sort of duality, to affine analogs of Poisson structures is established as well as affine versions of the complete lift of a section and the Cartan exterior calculus.

math.DG↗

arXiv2003

Quasi-derivations and QD-algebroids

Axioms of Lie algebroid are discussed in order to review some known aspects for non-experts. In particular, it is shown that a Lie QD-algebroid (i.e. a Lie algebra bracket on the Functions(M)-module F of sections of a vector bundle E over a manifold M which satisfies [X,fY]=f[X,Y]+A(X,f)Y for all X,Y from F, all f from Functions(M), and for certain A(X,f) from Functions(M)) is a Lie algebroid if rank(E)>1, and is a local Lie algebra in the sense of Kirillov if E is a line bundle. Under a weak condition also the skew-symmetry of the bracket is relaxed.

math.DG↗

arXiv2002

The graded Jacobi algebras and (co)homology

Jacobi algebroids (i.e. `Jacobi versions' of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies varios concepts of graded Lie structures in geometry and physics. A method of describing such structures by classical Lie algebroids via certain gauging (in the spirit of E.Witten's gauging of exterior derivative) is developed. One constructs a corresponding Cartan differential calculus (graded commutative one) in a natural manner. This, in turn, gives canonical generating operators for triangular Jacobi algebroids. One gets, in particular, the Lichnerowicz-Jacobi homology operators associated with classical Jacobi structures. Courant-Jacobi brackets are obtained in a similar way and use to define an abstract notion of a Courant-Jacobi algebroid and Dirac-Jacobi structure. All this offers a new flavour in understanding the Batalin-Vilkovisky formalism.

math.DG↗

arXiv2002

Automorphisms of quantum and classical Poisson algebras

We prove Pursell-Shanks type results for the Lie algebra D(M) of all linear differential operators of a smooth manifold M, for its Lie subalgebra D^1(M) of all linear first-order differential operators of M, and for the Poisson algebra S(M)=Pol(T*M) of all polynomial functions on the cotangent bundle T*M, the symbols of the operators in D(M). Chiefly however we provide explicit formulas describing completely the automorphisms of the Lie algebras D^1(M), S(M), and D(M).

math.RA↗