Nonlinear regimes in spin dynamics of superfluid 3He

We study an algebra of Poisson brackets of the Hamiltonian system defined by the nonlinear Leggett equations of spin dynamics in the A- and the B-phases of superfluid ³He. For the A-phase the Poisson algebra results in a special case of the equations of motion of a rigid body in ideal fluid; for the B-phase, in the absence of magnetic field, it allows for a reduction to a smaller Poisson algebra that provides exact solutions for the Leggett equations.     

Algebraic model of a classical non-relativistic electron

An algebraic structure is constructed which serves as an algebraic analog of a phase space for a model of a non-relativistic classical electron. The structure consists of a type of Poisson bracket defined on the tensor product of a commutative algebra and a Grassmann algebra. The equivalent of Hamiltonian dynamics is defined and applied to specific models of an electron. A quantization procedure is introduced which leads to the usual quantum equivalents of the classical models.     

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.     

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.     

Vacuum structures in Hamiltonian light-front dynamics

Hamiltonian light-front dynamics of quantum fields may provide a useful approach to systematic nonperturbative approximations to quantum field theories. We investigate inequivalent Hilbert-space representations of the light-front field algebra in which the stability group of the light front is implemented by unitary transformations. The Hilbert space representation of states is generated by the operator algebra from the vacuum state. There is a large class of vacuum states besides the Fock vacuum which meet all the invariance requirements. The light-front Hamiltonian must annihilate the vacuum and have a positive spectrum. We exhibit relations of the Hamiltonian to the nontrivial vacuum structure.     

The Lie-Poisson structure of integrable classical non-linear sigma models

The canonical structure of classical non-linear sigma models on Riemannian symmetric spaces, which constitute the most general class of classical non-linear sigma models known to be integrable, is shown to be governed by a fundamental Poisson bracket relation that fits into ther-s-matrix formalism for non-ultralocal integrable models first discussed by Maillet. The matricesr ands are computed explicitly and, being field dependent, satisfy fundamental Poisson bracket relations of their own, which can be expressed in terms of a new numerical matrixc. It is proposed that all these Poisson brackets taken together are, representation conditions for a new kind of algebra which, for this class of models, replaces the classical Yang-Baxter algebra governing the canonical structure of ultralocal models. The Poisson brackets for the transition matrices are also computed, and the notorious regularization problem associated with the definition of the Poisson brackets for the monodromy matrices is discussed.Suported by the Deutsche Forschungsgemeinschaft, Contract No. Ro 864/1-1Supported by the Studienstiftung des Deutschen Volkes     

A generalized Hamiltonian formalism unifying classical and quantum mechanics

A Poisson bracket structure is defined on associative algebras which allows for a generalized Hamiltonian dynamics. Both classical and quantum mechanics are shown to be special cases of the general formalism.     

Fractional Fourier transform and geometric quantization

Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of the phase-space: no linear structure is necessary. It is shown that the “fractional Fourier transform” provides a simple example of this construction. As an application of this technique we show that for any linear Hamiltonian system, its quantum dynamics can be obtained exactly as the lift of the corresponding classical dynamics by means of the above transformation. Moreover, it can be deduced from the free quantum evolution. This way new, unknown symmetries of the Schrödinger equation can be constructed. It is also argued that the above construction defines in a natural way a connection in the bundle of quantum states, with the base space describing all their possible representations. The non-flatness of this connection would be responsible for the non-existence of a quantum representation of the complete algebra of classical observables.     

Oscillations of charged particles in an external magnetic field about steady motion

We develop a Hamiltonian formalism that can be used to study the particle dynamics near stable equilibria. The construction of an original canonical transformation allowed us to prove the conservation of the linear momentum P₃, which permitted the expansion of the Hamiltonian about a fixed point. The definition of the rotational variable h whose Poisson algebra properties played the essential role in the diagonalization of the quadratic Hamiltonian yielding two uncoupled oscillators with definite frequencies and amplitudes. It is through applying this variable near a fixed point that come to light Heisenberg's and Harmonic Oscillator equations of motion of the particles, leading thus the association of the fixed point trajectories with arbitrary trajectories in its immediate neighborhood. The present formalism succeeded to treat the problem of free-electron laser dynamics and may be applied to similar cases. Received 20 October 2001     

Matrix Algebras in Non-Hermitian Quantum Mechanics

In principle, non-Hermitian quantum equations of motion can be formulated using as a starting point either the Heisenberg's or the Schrödinger's picture of quantum dynamics. Here it is shown in both cases how to map the algebra of commutators, defining the time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The logic behind such a derivation is reversible, so that any Hermitian Hamiltonian can be used in the formulation of non-Hermitian dynamics through a suitable algebra of generalized (non-Hamiltonian) commutators.
These results provide a general structure (a template) for non-Hermitian equations of motion to be used in the computer simulation of open quantum systems dynamics.     

Consistent canonical quantization of general relativity in the space of vassiliev invariants

We present a quantization of the Hamiltonian and diffeomorphism constraint of canonical quantum gravity in the spin network representation. The novelty consists in considering a space of wave functions based on the Vassiliev invariants. The constraints are finite, well defined, and reproduce at the level of quantum commutators the Poisson algebra of constraints of the classical theory. A similar construction can be carried out in 2+1 dimensions leading to the correct quantum theory.     

Classical r-Matrices and Compatible Poisson Structures for Lax Equations on Poisson Algebras

Given a classical r-matrix on a Poisson algebra, we show how to construct a natural family of compatible Poisson structures for the Hamiltonian formulation of Lax equations. Examples for which our formalism applies include the Benny hierarchy, the dispersionless Toda lattice hierarchy, the dispersionless KP and modified KP hierarchies, the dispersionless Dym hierarchy, etc. Received: 10 February 1998 / Accepted: 9 December 1998     

Star exponentials of the elements of the inhomogeneous symplectic Lie algebra

We compute the star exponential of any element of the inhomogeneous symplectic Lie algebra on a 2l-dimensional phase space and show the existence of classical trajectories for a quantum system whose Hamiltonian belongs to this Lie algebra.     

Semiclassical Limits of Ore Extensions and a Poisson Generalized Weyl Algebra

We observe [Launois and Lecoutre, Trans. Am. Math. Soc. 368:755–785, 2016, Proposition 4.1] that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra A₁, considered as a Poisson version of the quantum generalized Weyl algebra, is constructed and its Poisson structures are studied. In particular, a necessary and sufficient condition is obtained, such that A₁ is Poisson simple and established that the Poisson endomorphisms of A₁ are Poisson analogues of the endomorphisms of the quantum generalized Weyl algebra.     

Classical and quantum q-deformed physical systems

On the basis of non-commutative q-calculus, we investigate a q-deformation of the classical Poisson bracket in order to formulate a generalized q-deformed dynamics in the classical regime. The obtained q-deformed Poisson bracket appears invariant under the action of the q-symplectic group of transformations. Within this framework we introduce the q-deformed Hamilton equations and we derive the evolution equation for some simple q-deformed mechanical systems governed by a scalar potential dependent only on the coordinate variable. It appears that the q-deformed Hamiltonian, which is the generator of the equation of motion, is generally not conserved in time but, in correspondence, a new constant of motion is generated. Finally, by following the standard canonical quantization rule, we compare the well-known q-deformed Heisenberg algebra with the algebra generated by the q-deformed Poisson bracket. PACS 02.45.Gh, 45.20.-d, 03.65.-w, 02.20.Uw     

Factorizations of one-dimensional classical systems

A class of one-dimensional classical systems is characterized from an algebraic point of view. The Hamiltonians of these systems are factorized in terms of two functions that together with the Hamiltonian itself close a Poisson algebra. These two functions lead directly to two time-dependent integrals of motion from which the phase motions are derived algebraically. The systems so obtained constitute the classical analogues of the well known factorizable one-dimensional quantum mechanical systems.     

Twisted Diff S 1-action on loop groups and representations of the virasoro algebra

A modified Hamiltonian action of Diff S ¹on the phase space LG ^C /G^C, where LG is a loop group, is defined by twisting the usual action by a left translation in LG. This twisted action is shown to be generated by a nonequivariant moment map, thereby defining a classical Poisson bracket realization of a central extension of the Lie algebra diff_C S ¹. The resulting formula expresses the Diff S ¹generators in terms of the left LG translation generators, giving a shifted modification of both the classical and quantum versions of the Sugawara formula.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the National Science Foundation.     

Dynamical interpretation for the quantum-measurement projection postulate

An apparatus model with discrete momentum space suitable for the exact solution of the problem is considered. The special Hamiltonian of its interaction with the object system under consideration is chosen. In this simple case it is easy to illustrate how difficulties in constructing the dynamical interpretation of selective collapse can be overcome without any limiting procedure. For this purpose one can apply either averaging with respect to a nonquantum parameter or reduce the algebra of joint-system operators (i.e., pass from an algebraA of operators to a subalgebraA ₀). The latter procedure implies averaging with respect to apparatus quantum variables not belonging toA ₀.On leave of absence from Physics Department, Moscow State University, 119899 Moscow, Russia.     

Hamiltonian Feynman-Kac and Feynman Formulae for Dynamics of Particles with Position-Dependent Mass

A Feynman formula is a representation of the semigroup, generated by an initial-boundary value problem for some evolutionary equation, by a limit of integrals over Cartesian powers of some space E, the integrands being some elementary functions. The multiple integrals in Feynman formulae approximate integrals with respect to some measures or pseudomeasures on sets of functions which take values in E and are defined on a real interval. Hence Feynman formulae can be used both to calculate explicitly solutions for such problems, to get some representations for these solutions by integrals over functions taking values in E (such representations are called Feynman-Kac formulae), to get approximations for transition probability of some diffusion processes and transition amplitudes for quantum dynamics and to get computer simulations for some stochastic and quantum dynamics. The Feynman formula is called a Hamiltonian Feynman formula if the space, Cartesian products of which are used, is the phase space of a classical Hamiltonian system; the corresponding Feynman-Kac formula is called a Hamiltonian Feynman-Kac formula. In the latter formula one integrates over functions taking values in the same phase space. In a similar way one can define Lagrangian Feynman formulae and Lagrangian Feynman-Kac formulae substituting the phase space by the configuration space.     

The Role of Time in Relational Quantum Theories

We propose a solution to the problem of time for systems with a single global Hamiltonian constraint. Our solution stems from the observation that, for these theories, conventional gauge theory methods fail to capture the full classical dynamics of the system and must therefore be deemed inappropriate. We propose a new strategy for consistently quantizing systems with a relational notion of time that does capture the full classical dynamics of the system and allows for evolution parametrized by an equitable internal clock. This proposal contains the minimal temporal structure necessary to retain the ordering of events required to describe classical evolution. In the context of shape dynamics (an equivalent formulation of general relativity that is locally scale invariant and free of the local problem of time) our proposal can be shown to constitute a natural methodology for describing dynamical evolution in quantum gravity and to lead to a quantum theory analogous to the Dirac quantization of unimodular gravity.