Categorified Symplectic Geometry and the Classical String

A Lie 2-algebra is a ‘categorified’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an ‘n-plectic manifold’: a finite-dimensional manifold equipped with a closed nondegenerate (n + 1)-form. Here we consider the case n = 2. For any 2-plectic manifold, we construct a Lie 2-algebra of observables. We then explain how this Lie 2-algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2-plectic structure for the string.     

Different Types of Conditional Expectation and the Lüders—von Neumann Quantum Measurement

In operator algebra theory, a conditional expectation is usually assumed to be a projection map onto a sub-algebra. In the paper, a further type of conditional expectation and an extension of the Lüders—von Neumann measurement to observables with continuous spectra are considered; both are defined for a single operator and become a projection map only if they exist for all operators. Criteria for the existence of the different types of conditional expectation and of the extension of the Lüders—von Neumann measurement are presented, and the question whether they coincide is studied. All this is done in the general framework of Jordan operator algebras. The examples considered include the type I and type II operator algebras, the standard Hilbert space model of quantum mechanics, and a no-go result concerning the conditional expectation of observables that satisfy the canonical commutator relation.     

Does accidental degeneracy imply a symmetry group?

The relation between the appearance of accidental degeneracy in the energy levels of a given Hamiltonian and its symmetry group is probed. This is done by analyzing the very simple problem of an oscillator to which a particular spin-orbit and centrifugal force are added. The operators that connect all the states of given energy as well as their corresponding observables in the classical limit are found. The Poisson bracket relations between these observables leads to a Lie algebra U(3) × SU(2), but it does not translate into a Lie algebra for the commutators of the corresponding operators, as some matrix elements of commutators, corresponding to Poisson brackets that are zero, do not vanish. Thus while accidental degeneracy in the quantum problem may lead to a larger group in the classical limit, it is not always given by the dimensions of the irreducible representations of this group.     

Deformation theory and quantization. I. Deformations of symplectic structures

We present a mathematical study of the differentiable deformations of the algebras associated with phase space. Deformations of the Lie algebra of C^∞ functions, defined by the Poisson bracket, generalize the well-known Moyal bracket. Deformations of the algebra of C^∞ functions, defined by ordinary multiplication, give rise to noncommutative, associative algebras, isomorphic to the operator algebras of quantum theory. In particular, we study deformations invariant under any Lie algebra of “distinguished observables”, thus generalizing the usual quantization scheme based on the Heisenberg algebra.     

Nonlinear integrable couplings of a nonlinear Schrdinger-modified Korteweg de Vries hierarchy with self-consistent sources

By means of the Lie algebra B 2,a new extended Lie algebra F is constructed.Based on the Lie algebras B 2 and F,the nonlinear Schro¨dinger-modified Korteweg de Vries(NLS-mKdV) hierarchy with self-consistent sources as well as its nonlinear integrable couplings are derived.With the help of the variational identity,their Hamiltonian structures are generated.     

Nonlinear integrable couplings of a nonlinear Schrödinger—modified Korteweg de Vries hierarchy with self-consistent sources

By means of the Lie algebra B₂, a new extended Lie algebra F is constructed. Based on the Lie algebras B₂ and F, the nonlinear Schrödinger-modified Korteweg de Vries (NLS-mKdV) hierarchy with self-consistent sources as well as its nonlinear integrable couplings are derived. With the help of the variational identity, their Hamiltonian structures are generated.     

Global Geometric Deformations of Current Algebras as Krichever-Novikov Type Algebras

We construct algebraic-geometric families of genus one (i.e. elliptic) current and affine Lie algebras of Krichever-Novikov type. These families deform the classical current, respectively affine Kac-Moody Lie algebras. The construction is induced by the geometric process of degenerating the elliptic curve to singular cubics. If the finite-dimensional Lie algebra defining the infinite dimensional current algebra is simple then, even if restricted to local families, the constructed families are non-equivalent to the trivial family. In particular, we show that the current algebra is geometrically not rigid, despite its formal rigidity. This shows that in the infinite dimensional Lie algebra case the relations between geometric deformations, formal deformations and Lie algebra two-cohomology are not that close as in the finite-dimensional case. The constructed families are e.g. of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory. The algebras are explicitly given by generators and structure equations and yield new examples of infinite dimensional algebras of current and affine Lie algebra type.     

Some non-abelian phase spaces in low dimensions

A non-abelian phase space, or a phase space of a Lie algebra, is a generalization of the usual (abelian) phase space of a vector space. It corresponds to a para-Kähler structure in geometry. Its structure can be interpreted in terms of left-symmetric algebras. In particular, a solution of an algebraic equation in a left-symmetric algebra which is an analogue of classical Yang–Baxter equation in a Lie algebra can induce a phase space. In this paper, we find that such phase spaces have a symplectically isomorphic property. We also give all such phase spaces in dimension 4 and some examples in dimension 6. These examples can be a guide for a further development.     

Statistical mechanics of quantum spin systems

The thermodynamic limit of a quantum spin system is considered. It is demonstrated that for a large class of interactions and a wide range of the thermodynamic parameters the equilibrium state of the system is describable by an extremalZ ^v -invariant state (a single phase state) over aC* algebra of local observables. It is further shown that the equilibrium state may be obtained as the solution of a variational problem involving the mean entropy. These results extend results previously obtained for classical spin systems byGallavotti, Miracle-Sole andRuelle.     

Selfconsistent approximations in Mori's theory

The constitutive quantities in Mori's theory, the residual forces, are expanded in terms of time-dependent correlation functions and products of operators at t = 0, where it is assumed that the time derivatives of the observables are given by products of them. As a first consequence the Heisenberg dynamics of the observables are obtained as an expansion of the same type. The dynamic equations for correlation functions result to be selfconsistent nonlinear equations of the type known from mode-mode coupling approximations. The approach yields a necessary condition for the validity of the presented equations. As a third consequence the static correlations can be calculated from fluctuation-dissipation theorems, if the observables obey a Lie algebra. For a simple spin model the convergence of the expansion is studied. As a further test, dynamic and static correlations are calculated for a Heisenberg ferromagnet at low temperatures, where the results are compared to those of a Holstein-Primakoff treatment.     

Dynamical algebras in classical mechanics

For a spectrum-generating algebra of classical observables, it is proven that the phase space dynamics simplifies to a Hamiltonian system on submanifolds of the algebra's dual. These submanifolds are coadjoint orbits if the algebra arises from a symplectic group action. If the Hamiltonian splits into the sum of a function of the algebra generators plus a commuting part, then the dynamics transfers to the dual space and an explicit formula is given for the flow vector field on the coadjoint orbits. A unique feature of the presentation is that all constructions are at the Lie algebra level.     

A Generalization of Entanglement to Convex Operational Theories: Entanglement Relative to a Subspace of Observables

We define what it means for a state in a convex cone of states on a space of observables to be generalized-entangled relative to a subspace of the observables, in a general ordered linear spaces framework for operational theories. This extends the notion of ordinary entanglement in quantum information theory to a much more general framework. Some important special cases are described, in which the distinguished observables are subspaces of the observables of a quantum system, leading to results like the identification of generalized unentangled states with Lie-group-theoretic coherent states when the special observables form an irreducibly represented Lie algebra. Some open problems, including that of generalizing the semigroup of local operations with classical communication to the convex cones case, are discussed. PACS: 03.65.Ud.     

The Operator Formulation of Classical Mechanics and Semiclassical Limit

The algebra of polynomials in operators that represent generalized coordinate and momentum and depend on the Planck constant is defined. The Planck constant is treated as the parameter taking values between zero and some nonvanishing h ₀. For the later of these two extreme values, introduced operator algebra becomes equivalent to the algebra of observables of quantum mechanical system defined in the standard manner by operators in the Hilbert space. For the vanishing Planck constant, the generalized algebra gives the operator formulation of classical mechanics since it is equivalent to the algebra of variables of classical mechanical system defined, as usually, by functions over the phase space. In this way, the semiclassical limit of kinematical part of quantum mechanics is established through the generalized operator framework.     

Wigner–Weyl–Moyal Formalism on Algebraic Structures

We first introduce theWigner–Weyl–Moyal formalism for a theorywhose phase space is an arbitrary Lie algebra. We alsogeneralize to quantum Lie algebras and to supersymmetrictheories. It turns out that the noncommutativity leads to a deformation ofthe classical phase space: instead of being a vectorspace, it becomes a manifold, the topology of which isgiven by the commutator relations. It is shown in fact that the classical phase space, for asemisimple Lie algebra, becomes a homogeneous symplecticmanifold. The symplectic product is also deformed. Wefinally make some comments on how to generalise to C*-algebras and other operator algebras,too.     

Graded Poisson Lie structures on classical complex Lie groups

The external algebra over holomorphic first order differential forms on a complex Lie groupG is endowed with the structure of a graded Poisson Lie algebra. This structure is introduced via graded bicovariant brackets that are shown to be in one to one correspondence withG-invariant tensors of special symmetry. Complete classification of graded Poisson Lie structures defined by homogeneous brackets is obtained for the case of classical complex Lie groups.     

Decomposition of symmetric tensor fields in the presence of a flat contact projective structure

Abstract

Let M be an odd-dimensional Euclidean space endowed with a contact 1-form α. We investigate the space of symmetric contravariant tensor fields over M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up of those vector fields that preserve the contact structure defined by a. If we consider symmetric tensor fields with coefficients in tensor densities (also called symbols), the vertical cotangent lift of the contact form a defines a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symbols. This generalized Hamiltonian operator on the space of symbols is invariant with respect to the action of the projective contact algebra sp(2n+2) the algebra of vector fields which preserve both the contact structure and the projective structure of the Euclidean space. These two operators lead to a decomposition of the space of symbols, except for some critical density weights, which generalizes a splitting proposed by V. Ovsienko in [18].     

Quantum Liouville Equation

The symmetrized product of quantum observables is defined. It is seen as consisting of ordinary multiplication followed by application of the superoperator that orders the operators of coordinate and momentum. This superoperator is defined in the way that allows obstruction free quantization of algebra of observables as well as introduction of operator version of the Poisson bracket. It is shown that this bracket has all properties of the Lie bracket and that it can substitute the commutator in the von Neumann equation leading to quantum Liouville equation.     

Galilean covariant Lie algebra of quantum mechanical observables

A Lie algebra unifying the noncanonical Lie algebra of quantum mechanical observables and the Lie algebra of the Galilei group is constructed. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Algebraical comparison of classical and quantum polynomial observables

The symplectic vector spaceE of theq andp's of classical mechanics allows a basis free definition of the Poisson bracket in the symmetric algebra overE. Thus the symmetric algebra overE becomes a Lie algebra, which can be compared with the quantum mechanical Weyl algebra with its commutator Lie structure. The universality of the Weyl algebra is used to study the well-known ‘classical’ Moyal realisation of the Weyl algebra in the symmetric algebra. Quantisations are defined as linear mappings of the underlying vector spaces of the two algebras. It is shown that the classical Lie algebra is −2 graded, whereas the quantum Lie algebra is not. This proves that they are not isomorphic, and hence there is no Dirac quantisation.