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.     

q-Deformation of algebraic structures of quantum and classical mechanics

There exists a coassociative and cocommutative coproduct in the linear space spanned by the two algebraic products of a classical Hamilton algebra (the algebraic structure underlying classical mechanics [1]). The transition from classical to quantum Hamilton algebra (the algebraic structure underlying quantum mechanics) is anħ-deformation which preserves not only the Lie property of the classical Hamilton algebra but also the coassociativity and cocommutativity of the above coproduct. By explicit construction we obtain the algebraic structures of theq-deformed Hamilton algebras which preserve the said properties of the coproduct. Some algorithms of these structures are obtained and their implications discussed. The problem of consistency of time evolution with theq-deformed kinematical structure is discussed. A characteristic distinction between the parametersħ andq is brought out to stress the fact thatq cannot be regarded as a fundamental constant.     

Star products and geometric algebra

The formalism of geometric algebra can be described as deformed super analysis. The deformation is done with a fermionic star product, that arises from deformation quantization of pseudoclassical mechanics. If one then extends the deformation to the bosonic coefficients of superanalysis one obtains quantum mechanics for systems with spin. This approach clarifies on the one hand the relation between Grassmann and Clifford structures in geometric algebra and on the other hand the relation between classical mechanics and quantum mechanics. Moreover it gives a formalism that allows to handle classical and quantum mechanics in a consistent manner.     

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.     

Twisting of Quantum Differentials and¶the Planck Scale Hopf Algebra

We show that the crossed modules and bicovariant differential calculi on two Hopf algebras related by a cocycle twist are in 1-1 correspondence. In particular, for quantum groups which are cocycle deformation-quantisations of classical groups the calculi are obtained as deformation-quantisations of the classical ones. As an application, we classify all bicovariant differential calculi on the Planck scale Hopf algebra . This is a quantum group which has an limit as the functions on a classical but non-Abelian group and a limit as flat space quantum mechanics. We further study the noncommutative differential geometry and Fourier theory for this Hopf algebra as a toy model for Planck scale physics. The Fourier theory implements a T-duality-like self-duality. The noncommutative geometry turns out to be singular when and is therefore not visible in flat space quantum mechanics alone. Received: 28 October 1998 / Accepted: 7 March 1999     

Orthomodular Lattices and a Quantum Algebra

We show that one can formulate an algebra with lattice ordering so as to contain one quantum and five classical operations as opposed to the standard formulation of the Hilbert space subspace algebra. The standard orthomodular lattice is embeddable into the algebra. To obtain this result we devised algorithms and computer programs for obtaining expressions of all quantum and classical operations within an orthomodular lattice in terms of each other, many of which are presented in the paper. For quantum disjunction and conjunction we prove their associativity in an orthomodular lattice for any triple in which one of the elements commutes with the other two and their distributivity for any triple in which a particular element commutes with the other two. We also prove that the distributivity of symmetric identity holds in Hilbert space, although whether or not it holds in all orthomodular lattices remains an open problem, as it does not fail in any of over 50 million Greechie diagrams we tested.     

Discrete Quantum Drinfeld–Sokolov Correspondence

We construct a discrete quantum version of the Drinfeld–Sokolov correspondence for the sine-Gordon system. The classical version of this correspondence is a birational Poisson morphism between the phase space of the discrete sine-Gordon system and a Poisson homogeneous space. Under this correspondence, the commuting higher mKdV vector fields correspond to the action of an Abelian Lie algebra. We quantize this picture (1) by quantizing this Poisson homogeneous space, together with the action of the Abelian Lie algebra, (2) by quantizing the sine-Gordon phase space, (3) by computing the quantum analogues of the integrals of motion generating the mKdV vector fields, and (4) by constructing an algebra morphism taking one commuting family of derivations to the other one. Received: 3 July 2001 / Accepted: 14 December 2001     

On the deformability of Heisenberg algebras

Based on the vanishing of the second Hochschild cohomology group of the Weyl algebra it is shown that differential algebras coming from quantum groups do not provide a non-trivial deformation of quantum mechanics. For the case of aq-oscillator there exists a deforming map to the classical algebra. It is shown that the differential calculus on quantum planes with involution, i.e., if one works in position-momentum realization, can be mapped on aq-difference calculus on a commutative real space. Although this calculus leads to an interesting discretization it is proved that it can be realized by generators of the undeformed algebra and does not possess a proper group of global transformations.     

On the Integrability of q-Oscillators Based on Invariants of Discrete Fourier Transforms

A discretization of the quantum mechanical phase space is presented in the context of q-noncommutative structures. We give two generalizations of the Heisenberg algebra in the arising lattice phase space. In contrast to ordinary quantum mechanics, there is, a priori, no systematic approach to an integrable oscillator Hamiltonian in lattice quantum mechanics. This is the central obstacle to deal with in this Letter. To do so, we show how in general the integrability of the harmonic oscillator is related to the Fourier transform between momentum and space variables. This will be done in both cases, the continuous and the discrete one. As an application, we finally obtain an integrable lattice Hamiltonian for the harmonic oscillator with generalized Hermite eigenfunctions.     

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.     

Convex Structures and Effect Algebras

Effect algebras have important applications inthe foundations of quantum mechanics and in fuzzyprobability theory. An effect algebra that possesses aconvex structure is called a convex effect algebra. Our main result shows that any convex effectalgebra admits a representation as a generating initialinterval of an ordered linear space. This result isanalogous to a classical representation theorem for convex structures due to M. H. Stone. We alsogive a relationship between a convex effect algebra anda statistical model called a convex effect-statespace.     

Geometric algebra and star products on the phase space

Superanalysis can be deformed with a fermionic star product into a Clifford calculus that is equivalent to geometric algebra. With this multivector formalism it is then possible to formulate Riemannian geometry and an inhomogeneous generalization of exterior calculus. Moreover, it is shown here how symplectic and Poisson geometry fit in this context. The application of this formalism together with the bosonic star product formalism of deformation quantization leads then on space and space-time to a natural appearance of spin structures and on phase space to BRST structures that were found in the path integral formulation of classical mechanics. Furthermore it will be shown that Poincaré and Lie-Poisson reduction can be formulated in this formalism.     

Nonstatistical quantum-classical correspondence in phase space

We study the properties of a causal quantum theory in phase space for which phase space classical mechanics is obtained as a limit. The causal quantum theory is obtained from a generalized coherent state representation. The behavior for the one particle case and the manyparticle case are illustrated for the harmonic oscillator. We also answer to the arguments against the possibility of constructing causal theories in phase space.     

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.     

Quantum mechanics,knot theory,and quantum doubles

In previous papers we have described quantum mechanics as a matrix symplectic geometry and showed the existence of a braiding and Hopf algebra structure behind our lattice quantum phase space. The first aim of this work is to give the defining commutation relations of the quantum Weyl-Schwinger-Heisenberg group associated with our ℜ-matrix solution. The second aim is to describe the knot formalism at work behind the matrix quantum mechanics. In this context, the quantum mechanics of a particle-antiparticle system (pˉp) moving in the quantum phase space is viewed as a quantum double.     

Variations on a theme of Schwinger and Weyl

The theme of doing quantum mechanics on all Abelian groups goes back to Schwinger and Weyl. This theme was studied earlier from the point of view of approximating quantum systems in infinite-dimensional spaces by those associated to finite Abelian groups. This Letter links this theme to deformation quantization, and explores the set of noncommutative associative algebra structures on the Schwartz-Weil algebra of any locally compact separable Abelian group. If the group is a vector space of even dimension over a non-Archimedean local fieldK, there exists a family of noncommutative (Moyal) structures parametrized by the local field and containing membersarbitrarily close to the classical one, although the classical algebra is rigid in the sense of deformation theory. The-products are defined by Fourier integral operators. The problem of constructing sucharithmetic Moyal structures on the algebra of Schwartz-Bruhat functions on manifolds that are locally likeK ²ⁿ is raised.In memory of Julian Schwinger     

Heisenberg algebra for noncommutative Landau problem

The Landau problem on non-commutative quantum mechanics is studied, where the Heisenberg algebra and the Landau energy levels as well as the non-commutative angular momentum are constructed in detail in non-commutative space and non-commutative phase space respectively.     

Dynamical symmetries in mechanics

The object of this review is to discuss methods that enable one to trace the origin of symmetries and conservation laws in mechanics to geometrical symmetries of space-time. Starting with the basic Newtonian assumptions on absolute space and time classical mechanics is developed in configuration space and phase space independently together with the related structures such as force-less mechanics. Heuristic considerations on geometric symmetries in configuration space reveal their intimate relation to conservation laws. Using the methods of differential geometry this relationship is put on a formal footing and symmetry groups of all spherically symmetric single term potentials are classified. The method of infinitesimal canonical transformations is presented as an alternative method of deducing dynamical symmetries of an arbitrary system in phase space. These methods also apply to non-relativistic quantum theory. Possible extension to special and general relatively is also discussed.