Dirac Quantization of Parametrized Field Theory

Parametrized field theory (PFT) is free field theory on flat spacetime in a diffeomorphism invariant disguise. It describes field evolution on arbitrary foliations of the flat spacetime instead of only the usual flat ones, by treating the `embedding variables' which describe the foliation as dynamical variables to be varied in the action in addition to the scalar field. A formal Dirac quantization turns the constraints of PFT into functional Schrodinger equations which describe evolution of quantum states from an arbitrary Cauchy slice to an infinitesimally nearby one.This formal Schrodinger picture- based quantization is unitarily equivalent to the standard Heisenberg picture based Fock quantization of the free scalar field if scalar field evolution along arbitrary foliations is unitarily implemented on the Fock space. Torre and Varadarajan (TV) showed that for generic foliations emanating from a flat initial slice in spacetimes of dimension greater than 2, evolution is not unitarily implemented, thus implying an obstruction to Dirac quantization. We construct a Dirac quantization of PFT,unitarily equivalent to the standard Fock quantization, using techniques from Loop Quantum Gravity (LQG) which are powerful enough to super-cede the no- go implications of the TV results. The key features of our quantization include an LQG type representation for the embedding variables, embedding dependent Fock spaces for the scalar field, an anomaly free representation of (a generalization of) the finite transformations generated by the constraints and group averaging techniques. The difference between 2 and higher dimensions is that in the latter, only finite gauge transformations are defined in the quantum theory, not the infinitesimal ones.Comment: 33 page

A quantum logical and geometrical approach to the study of improper mixtures

We study improper mixtures from a quantum logical and geometrical point of view. Taking into account the fact that improper mixtures do not admit an ignorance interpretation and must be considered as states in their own right, we do not follow the standard approach which considers improper mixtures as measures over the algebra of projections. Instead of it, we use the convex set of states in order to construct a new lattice whose atoms are all physical states: pure states and improper mixtures. This is done in order to overcome one of the problems which appear in the standard quantum logical formalism, namely, that for a subsystem of a larger system in an entangled state, the conjunction of all actual properties of the subsystem does not yield its actual state. In fact, its state is an improper mixture and cannot be represented in the von Neumann lattice as a minimal property which determines all other properties as is the case for pure states or classical systems. The new lattice also contains all propositions of the von Neumann lattice. We argue that this extension expresses in an algebraic form the fact that -alike the classical case- quantum interactions produce non trivial correlations between the systems. Finally, we study the maps which can be defined between the extended lattice of a compound system and the lattices of its subsystems.Comment: submitted to the Journal of Mathematical Physic

Parametrizations of density matrices

This article gives a brief overview of some recent progress in the characterization and parametrization of density matrices of finite dimensional systems. We discuss in some detail the Bloch-vector and Jarlskog parametrizations and mention briefly the coset parametrization. As applications of the Bloch parametrization we discuss the trace invariants for the case of time dependent Hamiltonians and in some detail the dynamics of three-level systems. Furthermore, the Bloch vector of two-qubit systems as well as the use of the polarization operator basis is indicated. As the main application of the Jarlskog parametrization we construct density matrices for composite systems. In addition, some recent related articles are mentioned without further discussion.Comment: 31 pages. v2: 32 pages, Abstract and Introduction rewritten and Conclusion section added, references adde

Faster Algorithms for the Geometric Transportation Problem

Let R, B be a set of n points in R^d, for constant d, where the points of R have integer supplies, points of B have integer demands, and the sum of supply is equal to the sum of demand. Let d(.,.) be a suitable distance function such as the L_p distance. The transportation problem asks to find a map tau : R x B --> N such that sum_{b in B}tau(r,b) = supply(r), sum_{r in R}tau(r,b) = demand(b), and sum_{r in R, b in B} tau(r,b) d(r,b) is minimized. We present three new results for the transportation problem when d(.,.) is any L_p metric: * For any constant epsilon > 0, an O(n^{1+epsilon}) expected time randomized algorithm that returns a transportation map with expected cost O(log^2(1/epsilon)) times the optimal cost. * For any epsilon > 0, a (1+epsilon)-approximation in O(n^{3/2}epsilon^{-d}polylog(U)polylog(n)) time, where U is the maximum supply or demand of any point. * An exact strongly polynomial O(n^2 polylog n) time algorithm, for d = 2

Large quantum gravity effects: Unexpected limitations of the classical theory

3-dimensional gravity coupled to Maxwell (or Klein-Gordon) fields is exactly soluble under the assumption of axi-symmetry. The solution is used to probe several quantum gravity issues. In particular, it is shown that the quantum fluctuations in the geometry are large unless the number and frequency of photons satisfy the inequality $\N(\hbar G\omega)^2 << 1$. Thus, even when there is a single photon of Planckian frequency, the quantum uncertainties in the metric are significant. Results hold also for a sector of the 4-dimensional theory (consisting of Einstein Rosen gravitational waves).Comment: 8 pages, No figures, ReVTe

The parameterized complexity of some geometric problems in unbounded dimension

We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension $d$: i) Given $n$ points in \Rd, compute their minimum enclosing cylinder. ii) Given two $n$-point sets in \Rd, decide whether they can be separated by two hyperplanes. iii) Given a system of $n$ linear inequalities with $d$ variables, find a maximum-size feasible subsystem. We show that (the decision versions of) all these problems are W[1]-hard when parameterized by the dimension $d$. %and hence not solvable in ${O}(f(d)n^c)$ time, for any computable function $f$ and constant $c$ %(unless FPT=W[1]). Our reductions also give a $n^{\Omega(d)}$-time lower bound (under the Exponential Time Hypothesis)

The Spin-Statistics Theorem for Anyons and Plektons in d=2+1

We prove the spin-statistics theorem for massive particles obeying braid group statistics in three-dimensional Minkowski space. We start from first principles of local relativistic quantum theory. The only assumption is a gap in the mass spectrum of the corresponding charged sector, and a restriction on the degeneracy of the corresponding mass.Comment: 21 pages, 2 figures. Citation added; Minor modifications of Appendix

I/O-Efficient Algorithms for Contour Line Extraction and Planar Graph Blocking

For a polyhedral terrain C, the contour at z-coordinate h, denoted Ch, is defined to be the intersection of the plane z = h with C. In this paper, we study the contour-line extraction problem, where we want to preprocess C into a data structure so that given a query z-coordinate h, we can report Ch quickly. This is a central problem that arises in geographic information systems (GIS), where terrains are often stored as Triangular Irregular Networks (TINS). We present an I/O-optimal algorithm for this problem which stores a terrain C with N vertices using O(N/B) blocks, where B is the size of a disk block, so that for any query h, the contour ch can be computed using o(log, N + I&l/B) I/O operations, where l&l denotes the size of Ch. We also present en improved algorithm for a more general problem of blocking bounded-degree planar graphs such as TINS (i.e., storing them on disk so that any graph traversal algorithm can traverse the graph in an I/O-efficient manner), and apply it to two problms that arise in GIS