Dynamics of a rigid test body in curved space-time

A covariant canonical formulation of the motion of a rigid test body in a curved space-time is obtained from a suitable Cartan form on the tangent bundleT of the bundle of Lorentz frames over the space-time manifoldV. The form (essentially equivalent to a Lagrangean) is chosen in close analogy to the corresponding 1-form in the classical Newtonian model of a rigid body and is very simple in terms of the natural geometrical structure of . The presymplectic manifold (T,d) then serves as evolution manifold of the system. One obtain the equations of motion and also a uniquely defined Poisson bracket on the set of observables defined as real valued functions on the manifold of motions. The rigid body interacts with the space-time curvature only via its spin in the same way as a spinning particle. Quadrupole and higher multiple interactions with the space-time curvature are not considered in this model.Supported in part by the National Research Council of Canada.     

Transquantum Dynamics

Segal proposed transquantum commutation relations with two transquantum constants , besides Planck's quantum constant and with a variable i. The Heisenberg quantum algebra is a contraction—in a more general sense than that of Inönü and Wigner—of the Segal transquantum algebra. The usual constant i arises as a vacuum order-parameter in the quantum limit ,0. One physical consequence is a discrete spectrum for canonical variables and space-time coordinates. Another is an interconversion of time and energy accompanying space-time meltdown (disorder), with a fundamental conversion factor of some kilograms of energy per second.     

The Phase of a Quantum Mechanical Particle in Curved Spacetime

We investigate the quantum mechanical wave equations for free particles of spin 0, 1/2, 1 in the background of an arbitrary static gravitational field in order to explicitly determine if the phase of the wavefunction is S/ = p dx /, as is often quoted in the literature. We work in isotropic coordinates where the wave equations have a simple manageable form and do not make a weak gravitational field approximation. We interpret these wave equations in terms of a quantum mechanical particle moving in medium with a spatially varying effective index of refraction. Due to the first order spatial derivative structure of the Dirac equation in curved spacetime, only the spin 1/2 particle has exactly the quantum mechanical phase as indicated above. The second order spatial derivative structure of the spin 0 and spin 1 wave equations yield the above phase only to lowest order in . We develop a WKB approximation for the solution of the spin 0 and spin 1 wave equations and explore amplitude and phase corrections beyond the lowest order in . For the spin 1/2 particle we calculate the phase appropriate for neutrino flavor oscillations.     

Bohm's quantum potentials and quantum gravity

A generally covariant theory, written in the spirit of Bohm's theory of quantum potentials, which applies to spinless, non interacting, gravitating systems, is formulated. In this theory the quantum state is coupled to the metric tensor g, and the effect of the quantum potential is absorbed in the geometry. At the same time, satisfies a covariant wave equation with respect to the very same g. This provides sufficient constraints to derive 11 coupled equations in the 11 unknowns: and the components of the metric tensor g_µv. The states of stable localized particles are identified, and vacuum-state solutions for both the Euclidean and the Lorentzian case are explicitly presented.     

Mean field theory for Coulomb systems

We study a classical charge symmetric system with an external charge distributionq in three dimensions in the limit that the plasma parameter zero. We prove that ifq is scaled appropriately then the correlation functions converge pointwise to those of an ideal gas in the external mean field(x) where is given by-+ 2z sinh() =q This is the mean field equation of Debye and Hückel. The proof uses the sine-Gordon transformation, the Mayer expansion, and a correlation inequality.Work partially supported by NSF Grant MCS 82-02115.     

Finitary, Causal, and Quantal Vacuum Einstein Gravity

We continue recent work (Mallios and Raptis, International Journal of Theoretical Physics 40, 1885, 2001; in press) and formulate the gravitational vacuum Einstein equations over a locally finite space-time by using the basic axiomatics, techniques, ideas, and working philosophy of Abstract Differential Geometry. The main kinematical structure involved, originally introduced and explored in (Mallios and Raptis, International Journal of Theoretical Physics 40, 1885, 2001), is a curved principal finitary space-time sheaf of incidence algebras, which have been interpreted as quantum causal sets, together with a nontrivial locally finite spin-Loretzian connection on it which lays the structural foundation for the formulation of a covariant dynamics of quantum causality in terms of sheaf morphisms. Our scheme is innately algebraic and it supports a categorical version of the principle of general covariance that is manifestly independent of a background -smooth space-time manifold M. Thus, we entertain the possibility of developing a fully covariant path integral-type of quantum dynamical scenario for these connections that avoids ab initio various problems that such a dynamics encounters in other current quantization schemes for gravity—either canonical (Hamiltonian) or covariant (Lagrangian)—involving an external, base differential space-time manifold, namely, the choice of a diffeomorphism-invariant measure on the moduli space of gauge-equivalent (self-dual) gravitational spin-Lorentzian connections and the (Hilbert space) inner product that could in principle be constructed relative to that measure in the quantum theory—the so-called inner product problem, as well as the problem of time that also involves the Diff(M) structure group of the classical -smooth space-time continuum of general relativity. Hence, by using the inherently algebraico—sheaf—theoretic and calculus-free ideas of Abstract Differential Geometry, we are able to draw preliminary, albeit suggestive, connections between certain nonperturbative (canonical or covariant) approaches to quantum general relativity (e.g., Ashtekar's new variables and the loop formalism that has been developed along with them) and Sorkin et al.'s causal set program. As it were, we noncommutatively algebraize, differential geometrize and, as a result, dynamically vary causal sets. At the end, we anticipate various consequences that such a scenario for a locally finite, causal and quantal vacuum Einstein gravity might have for the obstinate (from the viewpoint of the smooth continuum) problem of -smooth space-time singularities.     

Mesoscopic spin-flip transport through a quantum dot system responded by ac magnetic fields

We investigate mesoscopic spin transport through a quantum dot (QD) responded by a rotating and an oscillating magnetic fields. The rotating magnetic field rotates with the angular frequency 0 around the z-axis with the tilt angle , while the time-oscillating magnetic field is located in the z-axis with the angular frequency . The spin flip is caused by the rotating magnetic field, and it is the major source of spin current. The Zeeman effect is contributed by the two field components, and it is important as the magnetic fields are strong. The oscillating magnetic field takes significant role due to the spin-photon pumping effect, and the spin current can be generated by it even as 00 for the tilt angle 0. The peak and valley structure appears with respect to the frequency of oscillating field. The generation of spin current is companying with charge current. Spin current displays quite different appearance between the cases in the absence of source-drain bias (eV=0) and in the presence of source-drain bias (eV0). The symmetric spin current disappears to form asymmetric spin current with a negative valley and a positive plateau. The charge current is mainly determined by the source-drain bias, photon absorption, and spin-flip effect. This system can be employed as an ac charge-spin current generator, or ac charge-spin field effect transistor.     

Buoyancy forces in magnetic fluids

The ponderomotive force on a macroscopic body in a magnetic fluid is calculated by a hydrodynamic approach. The resulting equations are generally valid, neither small susceptibilities nor stationarity are assumed. The simple and widely-used formulaV(M-M ^bg )H is recoverd in linear order of ; magnetostrictive effects are shown to contribute in the order ³. The expressions derived here are definite and unambiguous, they do not depend on whether one starts from a theory in terms ofH, or in terms ofB: the correct evaluation of the contribution dV[-p] resolves the apparent contradiction between the force density expressions ₀ MH, orMB.     

The singular holonomy group

The fibre of the extension of the frame-bundle of a space-time over ab-boundary pointp is a homogeneous space /G _p . It is shown thatG _p can be found by a construction like that for a holonomy group, and that it contains a subgroup determined by the Riemann tensor. Near a curvature singularity one would expectG _p =      

Noncommutative Ricci Curvature and Dirac Operator on C q [SL2] at Roots of Unity

We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group C _q [ SL₂], using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for q an odd rth root of unity that its eigenvalues are given by q-integers [m] _q for m=0,1...,r–1 offset by the constant background curvature. We fully solve the Dirac equation for r=3.     

The eightfold way to color geometrodynamics

Color models of strong interactions are generalized to aGL(8,) ^f GL(8,) ^c gauge theory incorporating space-time curvature and Cartan's torsion. Following Salam, the dynamics is determined by an Einstein-Dirac-type Lagrangian. The resulting field equations are anonlinear (due to the torsion) Heisenberg-Pauli-Weyl equation for the fundamental spinor fields and a generalized Einstein equation for the background metric of hadronic dimensions. According to this model baryonic quarks are confined ingeon (black soliton)-type objects by the tensor gluons ofstrong gravity. This approach also leads to a black soliton mass formula which is in qualitative agreement with part of the baryon spectrum. Hadronic mesons are interpreted as gluon strings trapped in a multiconnected space-time. Interrelations of color geometrodynamics with other bag models are pointed out. Finally, the conceptual origin of this space-time foundation of quark confinement is presented.     

Correction to Einstein's perihelion precession formula from a traceless,anisotropic vacuum energy

Quantum field theory in curved space-time implies that the strong equivalence principle is violated outside a spherically symmetric, static star. Here we assume that quantum gravity effects restore the strong equivalence principle. Together with the assumption that the effective vacuum polarization energy-momentum tensor is traceless, this leads to a specific algebraic form of the energy-momentum tensor for which an exact solution of Einstein's field equations is found. The solution gives the post-Newtonian parameters=1 and=1+3, where is a dimensionless constant which determines the energy density of the anisotropic vacuum. The vacuum energy changes the perihelion precession by a factor of 1-.     

Quantum nonlocality as an axiom

In the conventional approach to quantum mechanics, indeterminism is an axiom and nonlocality is a theorem. We consider inverting the logical order, making nonlocality an axiom and indeterminism a theorem. Nonlocal superquantum correlations, preserving relativistic causality, can violate the CHSH inequality more strongly than any quantum correlations.     

New super-selection sectors (“soliton-states”) in two dimensional Bose quantum field models

A rigorous construction of new super-selection selectors — so-called soliton-sectors — for the quantum sine-Gordon equation and the (·)²-quantum field models with explicitly broken isospin symmetry in two space-time dimensions is presented. These sectors are eigenspaces of the chargeQdx(grad )(x) with non-zero eigenvalue. The scattering theory for quantum solitons is briefly discussed and shown to have consequences for the physics in the vacuum sector. A general theory is developed which explains why soliton-sectors may exist for theories in two but not in four space-time dimensions except possibly for non-abelian Yang-Mills theories.Supported in part by the National Science Foundation under Grant NSF-GP-39048 and by the ETH, Zürich.     

General relativity is a gauge type theory

It is shown that the Einstein-Maxwell theory of interacting electromagnetism and gravitation, can be derived from a first-order Lagrangian, depending on the electromagnetic field and on the curvature of a symmetric affine connection on the space-time M. The variation is taken with respect to the electromagnetic potential (a connection on a U(1) principal fiber bundle on M) and the gravitational potential (a connection on the GL(4, R) principal fiber bundle of frames on M). The metric tensor g does not appear in the Lagrangian, but it arises as a momentum canonically conjugated to . The Lagrangians of this type are calculated also for the Proca field, for a charged scalar field interacting with electromagnetism and gravitation, and for a few other interesting physical theories.     

A variational principle giving gravitational “superpotentials,” the affine connection,Riemann tensor,and Einstein field equations

A first-order Lagrangian is given, from which follow the definitions of the fully covariant form of the Riemann tensorR _k in terms of the affine connection and metric; the definition of the affine connection in terms of the metric; the Einstein field equations; and the definition of a set of gravitational superpotentials closely connected with the Komar conservation laws [7]. Substitution of the definition of the affine connection into this Lagrangian results in a second-order Lagrangian, from which follow the definition of the fully covariant Riemann tensor in terms of the metric, the Einstein equations, and the definition of the gravitational superpotentials.Dedicated to Achille Papapetrou on the occasion of his retirement.     

A cosmology without big bang

In contrast to standard ECSK theory with the Brans-Dicke scalar field () nonminimally coupled to the curvature scalar, an additional new pseudo scalar term ⁿ E R (contraction between Levi-Civita pseudo tensor and curvature tensor) has been included in the Lagrangian. The new term is non-zero due to the non-symmetric nature of the connection and vanishes identically in the general theory of relativity. We show that there exists a nonsingular cosmological solution for a spatially flat (k=0) Robertson-Walker line element in the radiation era; therefore our model has no big bang.     

Stiff magnetic field lines. I. A geometrical foundation

It is shown that the foliation of a space-time manifold of codimension 2 provides a basis for the study of the deformation of magnetic field lines. It is found that the fluid flow vector and the curvature vector of a nongeodesic stiff magnetic field line are always orthogonal. Further, it is shown that the metric tensor of the 2-space orthogonal to the Maxwellian string is Lie-transported along the magnetic field lines when the magnetic field lines are stiff. If there exists a spacelike Killing vector field parallel to the magnetic field, then the magnetic field lines must be stiff.     

Quasi-regular singularities based on null planes

We examine quasi-regular singularities that take the form of invariant two-dimensional null planes in Minkowski space-time, thus extending earlier studies of conical singularities based on timelike and spacelike planes. The result is described in terms of a deficit parameter. We also examine the form of the Riemann curvature tensor at the singularity.