Perfect discretization of path integrals

In order to obtain a well-defined path integral one often employs discretizations. In the case of General Relativity these generically break diffeomorphism symmetry, which has severe consequences since these symmetries determine the dynamics of the corresponding system. In this article we consider the path integral of reparametrization invariant systems as a toy example and present an improvement procedure for the discretized propagator. Fixed points and convergence of the procedure are discussed. Furthermore we show that a reparametrization invariant path integral implies discretization independence and acts as a projector onto physical states.Comment: 4 pages, 1 figure, based on a talk given at Loops '11, Madrid, to appear in Journal of Physics: Conference Series (JPCS

Perfect discretization of path integrals

In order to obtain a well-defined path integral one often employs discretizations. In the case of General Relativity these generically break diffeomorphism symmetry, which has severe consequences since these symmetries determine the dynamics of the corresponding system. In this article we consider the path integral of reparametrization invariant systems as a toy example and present an improvement procedure for the discretized propagator. Fixed points and convergence of the procedure are discussed. Furthermore we show that a reparametrization invariant path integral implies discretization independence and acts as a projector onto physical states.Comment: 4 pages, 1 figure, based on a talk given at Loops '11, Madrid, to appear in Journal of Physics: Conference Series (JPCS

Perfect discretization of reparametrization invariant path integrals

To obtain a well defined path integral one often employs discretizations. In the case of gravity and reparametrization invariant systems, the latter of which we consider here as a toy example, discretizations generically break diffeomorphism and reparametrization symmetry, respectively. This has severe implications, as these symmetries determine the dynamics of the corresponding system. Indeed we will show that a discretized path integral with reparametrization invariance is necessarily also discretization independent and therefore uniquely determined by the corresponding continuum quantum mechanical propagator. We use this insight to develop an iterative method for constructing such a discretized path integral, akin to a Wilsonian RG flow. This allows us to address the problem of discretization ambiguities and of an anomaly--free path integral measure for such systems. The latter is needed to obtain a path integral, that can act as a projector onto the physical states, satisfying the quantum constraints. We will comment on implications for discrete quantum gravity models, such as spin foams

Perfect discretization of path integrals

In order to obtain a well-defined path integral one often employs discretizations. In the case of General Relativity these generically break diffeomorphism symmetry, which has severe consequences since these symmetries determine the dynamics of the corresponding system. In this article we consider the path integral of reparametrization invariant systems as a toy example and present an improvement procedure for the discretized propagator. Fixed points and convergence of the procedure are discussed. Furthermore we show that a reparametrization invariant path integral implies discretization independence and acts as a projector onto physical states.Comment: 4 pages, 1 figure, based on a talk given at Loops '11, Madrid, to appear in Journal of Physics: Conference Series (JPCS

A Discrete and Bounded Envy-free Cake Cutting Protocol for Four Agents

We consider the well-studied cake cutting problem in which the goal is to identify a fair allocation based on a minimal number of queries from the agents. The problem has attracted considerable attention within various branches of computer science, mathematics, and economics. Although, the elegant Selfridge-Conway envy-free protocol for three agents has been known since 1960, it has been a major open problem for the last fifty years to obtain a bounded envy-free protocol for more than three agents. We propose a discrete and bounded envy-free protocol for four agents

Knaster's problem for (Z2)k(Z_2)^k-symmetric subsets of the sphere S2k−1S^{2^k-1}

We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in $S^{2^k-1}$, calculating the Euler class obstruction. Among the consequences are: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbb R^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$ by $(Z_2)^{k+1}$-symmetric convex fans

From the discrete to the continuous - towards a cylindrically consistent dynamics

Discrete models usually represent approximations to continuum physics. Cylindrical consistency provides a framework in which discretizations mirror exactly the continuum limit. Being a standard tool for the kinematics of loop quantum gravity we propose a coarse graining procedure that aims at constructing a cylindrically consistent dynamics in the form of transition amplitudes and Hamilton's principal functions. The coarse graining procedure, which is motivated by tensor network renormalization methods, provides a systematic approximation scheme towards this end. A crucial role in this coarse graining scheme is played by embedding maps that allow the interpretation of discrete boundary data as continuum configurations. These embedding maps should be selected according to the dynamics of the system, as a choice of embedding maps will determine a truncation of the renormalization flow.Comment: 22 page

The Role of Pressure in Inverse Design for Assembly

Isotropic pairwise interactions that promote the self assembly of complex particle morphologies have been discovered by inverse design strategies derived from the molecular coarse-graining literature. While such approaches provide an avenue to reproduce structural correlations, thermodynamic quantities such as the pressure have typically not been considered in self-assembly applications. In this work, we demonstrate that relative entropy optimization can be used to discover potentials that self-assemble into targeted cluster morphologies with a prescribed pressure when the iterative simulations are performed in the isothermal-isobaric ensemble. By tuning the pressure in the optimization, we generate a family of simple pair potentials that all self-assemble the same structure. Selecting an appropriate simulation ensemble to control the thermodynamic properties of interest is a general design strategy that could also be used to discover interaction potentials that self-assemble structures having, for example, a specified chemical potential.Comment: 29 pages, 8 figure

GA4GH Phenopackets: A Practical Introduction.

The Global Alliance for Genomics and Health (GA4GH) is developing a suite of coordinated standards for genomics for healthcare. The Phenopacket is a new GA4GH standard for sharing disease and phenotype information that characterizes an individual person, linking that individual to detailed phenotypic descriptions, genetic information, diagnoses, and treatments. A detailed example is presented that illustrates how to use the schema to represent the clinical course of a patient with retinoblastoma, including demographic information, the clinical diagnosis, phenotypic features and clinical measurements, an examination of the extirpated tumor, therapies, and the results of genomic analysis. The Phenopacket Schema, together with other GA4GH data and technical standards, will enable data exchange and provide a foundation for the computational analysis of disease and phenotype information to improve our ability to diagnose and conduct research on all types of disorders, including cancer and rare diseases

GA4GH Phenopackets: A Practical Introduction

The Global Alliance for Genomics and Health (GA4GH) is developing a suite of coordinated standards for genomics for healthcare. The Phenopacket is a new GA4GH standard for sharing disease and phenotype information that characterizes an individual person, linking that individual to detailed phenotypic descriptions, genetic information, diagnoses, and treatments. A detailed example is presented that illustrates how to use the schema to represent the clinical course of a patient with retinoblastoma, including demographic information, the clinical diagnosis, phenotypic features and clinical measurements, an examination of the extirpated tumor, therapies, and the results of genomic analysis. The Phenopacket Schema, together with other GA4GH data and technical standards, will enable data exchange and provide a foundation for the computational analysis of disease and phenotype information to improve our ability to diagnose and conduct research on all types of disorders, including cancer and rare diseases