Chryssomalis Chryssomalakos

GENERALIZED QUANTUM RELATIVISTIC KINEMATICS: A STABILITY POINT OF VIEW

We apply Lie algebra deformation theory to the problem of identifying the stable form of the quantum relativistic kinematical algebra. As a warm up, given Galileos conception of spacetime as input, some modest computer code we wrote zeroes in on the Poincare-plus-Heisenberg algebra in about a minute. Further ahead, along the same path, lies a three-dimensional deformation space, with an instability double cone through its origin. We give physical as well as geometrical arguments supporting our view that moment, rather than position operators, should enter as generators in the Lie algebra. With this identification, the deformation parameters give rise to invariant length and mass scales. Moreover, standard quantum relativistic kinematics of massive, spinless particles corresponds to non-commuting moment operators, a purely quantum effect that bears no relation to spacetime non-commutativity, in sharp contrast to earlier interpretations.

Hamiltonians for curves

We examine the equilibrium conditions of a curve in space when a local energy penalty is associated with its extrinsic geometrical state characterized by its curvature and torsion. To do this we tailor the theory of deformations to the Frenet–Serret frame of the curve. The Euler–Lagrange equations describing equilibrium are obtained; Noether’s theorem is exploited to identify the constants of integration of these equations as the Casimirs of the Euclidean group in three dimensions. While this system appears not to be integrable in general, it is in various limits of interest. Let the energy density be given as some function of the curvature and torsion, f( κ, τ ) .I ff is a linear function of either of its arguments but otherwise arbitrary, we claim that the first integral associated with rotational invariance permits the torsion τ to be expressed as the solution of an algebraic equatio ni n terms ofthe bending curvature, κ. The first integral associated with translational invariance can then be cast as a quadrature for κ or for τ .

Area-constrained planar elastica.

We determine the equilibria of a rigid loop in the plane, subject to the constraints of fixed length and fixed enclosed area. Rigidity is characterized by an energy functional quadratic in the curvature of the loop. We find that the area constraint gives rise to equilibria with remarkable geometrical properties; not only can the Euler-Lagrange equation be integrated to provide a quadrature for the curvature but, in addition, the embedding itself can be expressed as a local function of the curvature. The configuration space is shown to be essentially one dimensional, with surprisingly rich structure. Distinct branches of integer-indexed equilibria exhibit self-intersections and bifurcations-a gallery of plots is provided to highlight these findings. Perturbations connecting equilibria are shown to satisfy a first-order ODE which is readily solved. We also obtain analytical expressions for the energy as a function of the area in some limiting regimes.

On the Geometrical Character of Gravitation

The issue of whether some manifestations of gravitation in the quantum domain, are indicative or not of a non-geometrical aspect in gravitation is discussed. We focus on gedanken experiments, involving generalizations of the flavor-oscillation clocks of Ahluwalia and Burgard, and provide a critical analysis of previous interpretations. A detailed quantum mechanical description of the inner workings of these clocks reveals that there are, at this time, no indications of any departure of this nature from the geometrical character of gravitation.

Classical and quantum sl(1‖2) superalgebras, Casimir operators and quantum chain Hamiltonians

We examine in this paper the two parameter deformed superalgebra Uqs(sl(1‖2)) and use the results in the construction of quantum chain Hamiltonians. This study is done both in the framework of the Serre presentation and in the R‐matrix scheme of Faddeev, Reshetikhin, and Takhtajan (FRT). We show that there exists an infinite number of Casimir operators, indexed by integers p≥2 in the undeformed case and by p∈Z in the deformed case, which obey quadratic relations. The construction of the dual superalgebra of functions of SLqs(1‖2) is also given and higher tensor product representations are discussed. Finally, we construct quantum chain Hamiltonians based on the Casimir operators. In the deformed case we find two Hamiltonians which describe deformed t−J models.

Spin-1/2 particle on a cylinder with radial magnetic field

We study the motion of a quantum charged particle, constrained on the surface of a cylinder, in the presence of a radial magnetic field. When the spin of the particle is neglected, the system essentially reduces to an infinite family of simple harmonic oscillators, equally spaced along the axis of the cylinder. Interestingly enough, it can be used as a quantum Fourier transformer, with convenient visual output. When the spin-1/2 of the particle is taken into account, a non-conventional perturbative analysis results in a recursive closed form for the corrections to the energy and the wavefunction, for all eigenstates, to all orders in the magnetic moment of the particle. A simple two-state system is also presented, the time evolution of which involves an approximate precession of the spin perpendicularly to the magnetic field. A number of plots highlight the findings while several three-dimensional animations have been made available on the web.

What Basis for Genetic Dynamics

We present a covariant form for genetic dynamics and show how different formulations are simply related by linear coordinate transformations. In particular, in the context of the simple genetic algorithm, we show how the Vose model, in either the string or Walsh bases, is related to recent coarse-grained formulations that are naturally interpreted in terms of the Building Block basis (BBB). We also show that the latter is dual to the Taylor basis. The tensor product structure of the dynamical equations is analyzed, permitting the factorization of the N-bit operators in 1-bit factors.

STABILITY OF LIE SUPERALGEBRAS AND BRANES

The algebra of the generators of translations in superspace is unstable, in the sense that infinitesimal perturbations of its structure constants lead to non-isomorphic algebras. We show how superspace extensions remedy this situation (after arguing that remedy is indeed needed) and review the benefits reaped in the description of branes of all kinds in the presence of the extra dimensions.The algebra of the generators of translations in superspace is unstable, in the sense that infinitesimal perturbations of its structure constants lead to non-isomorphic algebras. We show how superspace extensions remedy this situation (after arguing that remedy is indeed needed) and review the benefits reaped in the description of branes of all kinds in the presence of the extra dimensions.

OPERATIONAL GEOMETRY ON DE SITTER SPACETIME

Traditional geometry employs idealized concepts like that of a point or a curve, the operational definition of which relies on the availability of classical point particles as probes. Real, physical objects are quantum in nature though, leading us to consider the implications of using realistic probes in defining an effective spacetime geometry. As an example, we consider de Sitter spacetime and employ the centroid of various composite probes to obtain its effective sectional curvature, which is found to depend on the probes internal energy, spatial extension, and spin. Possible refinements of our approach are pointed out and remarks are made on the relevance of our results to the quest for a quantum theory of gravity.

Center of mass in special and general relativity and its role in an effective description of spacetime

In this contribution, we suggest the approach that geometric concepts ought to be defined in terms of physical operations involving quantum matter. In this way it is expected that some (presumably nocive) idealizations lying deep within the roots of the notion of spacetime might be excluded. In particular, we consider that spacetime can be probed only with physical (and therefore extended) particles, which can be effectively described by coordinates that fail to commute by a term proportional to the spin of the particles.