Abstract:
The Einstein equations for spacetimes with two commuting spacelike Killing field symmetries are studied from a Hamiltonian point of view. The complexified Ashtekar canonical variables are used, and the symmetry reduction is performed directly in the Hamiltonian theory. The reduced system corresponds to the field equations of the SL(2,R) chiral model with additional constraints. On the classical phase space, a method of obtaining an infinite number of constants of the motion, or observables, is given. The procedure involves writing the Hamiltonian evolution equations as a single `zero curvature' equation, and then employing techniques used in the study of two dimensional integrable models. Two infinite sets of observables are obtained explicitly as functionals of the phase space variables. One set carries sl(2,R) Lie algebra indices and forms an infinite dimensional Poisson algebra, while the other is formed from traces of SL(2,R) holonomies that commute with one another. The restriction of the (complex) observables to the Euclidean and Lorentzian sectors is discussed. It is also shown that the sl(2,R) observables can be associated with a solution generating technique which is linked to that given by Geroch.

Abstract:
A compact solvmanifold of completely solvable type, i.e. a compact quotient of a completely solvable Lie group by a lattice, has a K\"ahler structure if and only if it is a complex torus. We show more in general that a compact solvmanifold $M$ of completely solvable type endowed with an invariant complex structure $J$ admits a symplectic form taming J if and only if $M$ is a complex torus. This result generalizes the one obtained in [7] for nilmanifolds.

Abstract:
In order to test the canonical quantization programme for general relativity we introduce a reduced model for a real sector of complexified Ashtekar gravity which captures important properties of the full theory. While it does not correspond to a subset of Einstein's gravity it has the advantage that the programme of canonical quantization can be carried out completely and explicitly, both, via the reduced phase space approach or along the lines of the algebraic quantization programme. This model stands in close correspondence to the frequently treated cylindrically symmetric waves. In contrast to other models that have been looked at up to now in terms of the new variables the reduced phase space is infinite dimensional while the scalar constraint is genuinely bilinear in the momenta. The infinite number of Dirac observables can be expressed in compact and explicit form in terms of the original phase space variables. They turn out, as expected, to be non-local and form naturally a set of countable cardinality.

Abstract:
By introducing the double spacetime manifold, the double gamma matrices and Dirac spinors, the action of the Dirac spinoral fields is doubled. Furthermore, the double coupling of the Dirac fields to the Ashtekar gravitational fields is studied.

Abstract:
We define the {\it rest-frame instant form} of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges. The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.

Abstract:
We use analytic (current) density-potential maps of time-dependent (current) density functional theory (TD(C)DFT) to inverse engineer analytically solvable time-dependent quantum problems. In this approach the driving potential (the control signal) and the corresponding solution of the Schr\"odinger equation are parametrized analytically in terms of the basic TD(C)DFT observables. We describe the general reconstruction strategy and illustrate it with a number of explicit examples. First we consider the real space one-particle dynamics driven by a time-dependent electromagnetic field and recover, from the general TDDFT reconstruction formulas, the known exact solution for a driven oscillator with a time-dependent frequency. Then we use analytic maps of the lattice TD(C)DFT to control quantum dynamics in a discrete space. As a first example we construct a time-dependent potential which generates prescribed dynamics on a tight-binding chain. Then our method is applied to the dynamics of spin-1/2 driven by a time dependent magnetic field. We design an analytic control pulse that transfers the system from the ground to excited state and vice versa. This pulse generates the spin flip thus operating as a quantum NOT gate.

Abstract:
Complexified spacetime algebra is defined as the geometric (Clifford) algebra of spacetime with complex coefficients, isomorphic $\mathcal{G}_{1,4}$. By resorting to matrix representation by means of Dirac-Pauli gamma matrices, the paper demonstrates isomorphism between subgroups of CSTA and SU(3). It is shown that the symmetry group of those subgroups is indeed $U(1) \otimes SU(2) \otimes SU(3)$ and that there are 4 distinct copies of this group within CSTA.

Abstract:
Dimensional reductions of various higher dimensional (super)gravity theories lead to effectively two-dimensional field theories described by gravity coupled G/H nonlinear sigma-models. We show that a new set of complexified variables can be introduced when G/H is a Hermitian symmetric space. This generalizes an earlier construction that grew out of the Ashtekar formulation of two Killing vector reduced pure 4d general relativity. Apart from giving some new insights into dimensional reductions of higher dimensional (super)gravity theories, these Ashtekar-type variables offer several technical advantages in the context of the exact quantization of these models. As an application, an infinite set of conserved charges is constructed. Our results might serve as a starting point for probing the quantum equivalence of the Ashtekar and the metric formalism within a non-trivial midi-superspace model of quantum gravity.

Abstract:
The presence of a constant background antisymmetric tensor for open strings or D-branes forces the space-time coordinates to be noncommutative. This effect is equivalent to replacing ordinary products in the effective theory by the deformed star product. An immediate consequence of this is that all fields get complexified. The only possible noncommutative Yang-Mills theory is the one with U(N) gauge symmetry. By applying this idea to gravity one discovers that the metric becomes complex. We show in this article that this procedure is completely consistent and one can obtain complexified gravity by gauging the symmetry $U(1,D-1)$ instead of the usual $SO(1,D-1)$. The final theory depends on a Hermitian tensor containing both the symmetric metric and antisymmetric tensor. In contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant. The results are then generalized to noncommutative spaces.