Group-theoretic Approach for Symbolic Tensor Manipulation: II. Dummy Indices

Computational Group Theory is applied to indexed objects (tensors, spinors, and so on) with dummy indices. There are two groups to consider: one describes the intrinsic symmetries of the object and the other describes the interchange of names of dummy indices. The problem of finding canonical forms for indexed objects with dummy indices reduces to finding double coset canonical representatives. Well known computational group algorithms are applied to index manipulation, which allow to address the simplification of expressions with hundreds of indices going further to what is needed in practical applications.Comment: 14 pages, 1 figure, LaTe

Hiding canonicalisation in tensor computer algebra

Simplification of expressions in computer algebra systems often involves a step known as "canonicalisation", which reduces equivalent expressions to the same form. However, such forms may not be natural from the perspective of a pen-and-paper computation, or may be unwieldy, or both. This is, for example, the case for expressions involving tensor multi-term symmetries. We propose an alternative strategy to handle such tensor expressions, which hides canonical forms from the user entirely, and present an implementation of this idea in the Cadabra computer algebra system

Core foundations, algorithms, and language design for symbolic computation in physics

This thesis presents three contributions to the field of symbolic computation, followed by their application to symbolic physics computations. The first contribution is to interfacing systems. The Notation package, which is developed in this thesis, allows the entry and the creation of advanced notations in the Mathematica symbolic computation system. In particular, a complete and functioning notation for both Dirac's BraKet notation as well as a full tensorial notation, are given herein. The second part of the thesis introduces a prototype based rule inheritance language paradigm that is applicable to certain advanced pattern matching rewrite rule language models. In particular, an implementation is presented for Mathematica. After detailing this language extension, it is adopted throughout the rest of the thesis. Finally, the third major contribution is a highly efficient algorithm to canonicalize tensorial expressions. By an innovative technique this algorithm avoids the dummy index relabeling problem. Further algorithmic optimizations are then presented. The complete algorithm handles linear symmetries such as the Bianchi identities. It also fully accommodates partial derivatives as well as mixed index classes. These advances in language and notations are extensively demonstrated on problems in quantum mechanics, angular momentum, general relativity, and quasi-spin. It is shown that the developments in this thesis lead to an extremely flexible, extensible, and powerful working environment for the expression and ensuing calculation of symbolic physics computations

Lecture Notes of Tensor Network Contractions

Tensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum information science, statistical physics, and so on. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarse-graining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multi-linear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches.Comment: 134 pages, 68 figures. In this version, the manuscript has been changed into the format of book; new sections about tensor network and quantum circuits have been adde

Computational General Relativity in the Wolfram Language using Gravitas I: Symbolic and Analytic Computation

We introduce a new, open-source computational general relativity framework for the Wolfram Language called Gravitas, which boasts a number of novel and distinctive features as compared to the many pre-existing computational and numerical relativity frameworks currently available within the open-source community. These include, but are not limited to: seamless integration of its powerful symbolic and numerical subsystems, and, by extension, seamless transition between analytic/continuous representations and numerical/discrete representations of arbitrary spacetime geometries; highly modular, general and extensible representations of spacetime geometries, spacetime topologies, gauge conditions, coordinate systems, matter fields, evolution equations and initial data; ability to set up and run complex numerical relativity simulations, and to perform 2D and 3D visualizations, symbolic computations and numerical analysis (including the extraction of gravitational wave signals) on the resulting data, all from within a single notebook environment; and a totally-unstructured adaptive refinement scheme based on hypergraph rewriting, allowing for exceedingly efficient discretization and numerical evolution of Cauchy initial data for a wide range of challenging computational problems involving strong relativistic field dynamics. In this first in a series of two articles covering the framework, we focus on the design and capabilities of Gravitas's symbolic subsystem, including its general and flexible handling of arbitrary geometries parametrized by arbitrary curvilinear coordinate systems (along with an in-built library of standard metrics and coordinate conditions), as well as its various high-level tensor calculus and differential geometry features. We proceed to show how this subsystem can be used to solve the Einstein field equations both analytically and numerically.Comment: 86 pages, 74 figure

Routines and Applications of Symbolic Algebra Software

Computing has become an essential resource in modern research and has found application across a wide range of scientific disciplines. Developments in symbolic algebra tools have been particularly valuable in physics where calculations in fields such as general relativity, quantum field theory and physics beyond the standard model are becoming increasing complex and unpractical to work with by hand. The computer algebra system Cadabra is a tensor-first approach to symbolic algebra based on the programming language Python which has been used extensively in research in these fields while also having a shallow learning curve making it an excellent way to introduce students to methods in computer algebra. The work in this thesis has been concentrated on developing Cadabra, which has involved looking at two different elements which make up a computer algebra program. Firstly, the implementation of algebraic routines is discussed. This has primarily been focused on the introduction of an algorithm for detecting the equivalence of tensorial expressions related by index permutation symmetries. The method employed differs considerably from traditional canonicalisation routines which are commonly used for this purpose by using Young projection operators to make such symmetries manifest. The other element of writing a computer algebra program which is covered is the infrastruc- ture and environment. The importance of this aspect of software design is often overlooked by funding committees and academic software users resulting in an anti-pattern of code not being shared and contributed to in the way in which research itself is published and promulgated. The focus in this area has been on implementing a packaging system for Cadabra which allows the writing of generic libraries which can be shared by the community, and interfacing with other scientific computing packages to increase the capabilities of Cadabra

Probes of strong-field gravity

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 221-234).In this thesis, I investigate several ways to probe gravity in the strong-field regime. These investigations focus on observables from the gravitational dynamics, i.e. when time derivatives are large: thus I focus on sources of gravitational waves. Extreme mass-ratio inspirals (EMRIs) can be very sensitive probes of strong-field physics. Predicting observables from EMRIs must be done numerically, so accurate numerical methods are required to ensure that any comparison with measurement is not spoiled by numerical artefacts. The first investigation of this thesis is a spectral (in the angular sector), pseudospectral (in the radial sector) time-domain PDE solver for perturbations of a Kerr black hole (i.e. solving the Teukolsky equation). The method exhibits good convergence and prompts much future investigation. A second approach to probing strong gravity is to consider theories which are general relativity (GR) with a few small corrections and investigate the effect of these corrections on observables. Since gravitational waves are the prime observable and they control the long-term evolution of dynamical systems, I investigate their properties in almost-GR theories. The second investigation of this thesis is a study of the propagation and energy content of gravitational waves in these theories. I find that in a large class of theories, approaching the asymptotically at part of spacetime, gravitational waves propagate in the same fashion as in GR and have the same effective stress-energy tensor as in GR. Next, I study the strong-field correction to the structure of a Schwarzschild black hole in a class of theories. Finally, with these ingredients, I investigate the leading corrections to the dynamics and observables of a comparable mass-ratio inspiral using post-Newtonian techniques. The main result is the appearance of dipolar scalar radiation in this class of theories. The dipolar radiation has a frequency dependence which does not arise in GR and is a distinct signature of corrections. Such signatures should be testable using gravitational wave detection and pulsar timing.by Leo Chaim Stein.Ph.D

Applications of Symbolic Computation to the Calculus of Moving Surfaces

In the physical world, objects change shape over time. A soap bubble blowing in the wind changes shape and density as it floats through the air. Red blood cells change shape to carry oxygen through our veins. Modeling these problems requires deforming manifolds. The Calculus of Moving Surfaces (CMS) is an analytical framework for studying deforming manifolds. The CMS is an extension of tensor calculus. Both approach problems from a geometric perspective, without reference to specific coordinate systems. To evaluate a specific realization of a problem, a coordinate system is chosen and a CMS expression is converted to a series of n-dimensional array calculations using standard calculus. This generality has many costs. The length of expressions grows quickly, in many cases exponentially. Although it is applicable to a wide range of problems, calculations quickly become intractable. The expressions generated are not only long and difficult to work with, evaluating them on a specific coordinate system introduces an entirely different set of challenges. We present the first compute algebra system designed specifically for the CMS. Our system, the Symbolic Computation of Moving Surfaces (SCMS) supports the derivation of CMS expressions and the evaluation of expressions on specific coordinate systems. Although large expressions are inherent in the framework, computer automation allows for the application of the CMS to significantly larger problems then can be done by hand and allows the CMS to be applied in an error free way to non-trivial problems. We have developed two libraries making up the SCMS. The first is a term rewrite system, CMSTRS, developed in Java. This library automates the analytic framework of the CMS. Expressions are kept at a high level, retaining the generality of the CMS. The second, CMSTensor, is for evaluation on specific coordinate systems. It is implemented using the Maple computer algebra system. It leverages the power of this computer algebra system to evaluate CMS expressions as a combination of n-dimensional array manipulations and standard calculus operations. We have applied our system to a non-trivial boundary variation problem: the symbolic series expansion of the Laplace Eigenvalues on the N-sided regular polygon under Dirichlet boundary conditions. This series is computed up to N^(-6), two orders higher then previous results. Our calculations confirm previous hand calculations and extend the series beyond what was previously known.Ph.D., Computer Science -- Drexel University, 201