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Hamiltonian Gauge Gravity Surveyor (HiGGS). Tools for Hamiltonian constraint, canonical and Dirac-Bergmann analysis of gravity theories with spacetime curvature and torsion.

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Hamiltonian Gauge Gravity Surveyor (HiGGS)

Version 1.2.3

  • Feature: abstract indices representing gauge-fixed theory given cleaner formatting with Gothic script, extended indices a1, b1, c1 etc. denoted by primes.
  • Feature: provide more intuitive formatting of Poisson brackets in terms of smearing functions, as seen in output of PoissonBracket and ViewTheory.
  • Patch: fix shadowing error messages during the Needs or Get package call preamble.
  • Patch: fix Issue #1, loading of binaries on Windows.
  • Patch: fix errors produced by PoissonBracket with the option "Surficial"->True. This error follows from sign errors in line two of Equation (E3) in 2205.13534, and generates nonphysical surface terms. My particular thanks to Manuel Hohmann for identifying this.

License

Copyright © 2022 Will E. V. Barker

HiGGS is distributed as free software under the GNU General Public License (GPL).

HiGGS is provided without warranty, or the implied warranty of merchantibility or fitness for a particular purpose.

Users of HiGGS, including authors of derivative works as defined by the GPL, are kindly requested to cite the 2022 HiGGS papers (2206.00658 and 2205.13534) in their resulting publications.

These conditions apply to all software in this repository, including "HPC-EEG" visualisation tools.

About

HiGGS is an (unofficial) part of the xAct bundle. It provides tools for the Hamiltonian constraint analysis (canoncical analysis or Dirac-Bergmann algorithm) of gravity with spacetime curvature and torsion. HiGGS can be used on a desktop PC, but it is parallelised for theory surveys on clusters and supercomputers.

Installation

Requirements

HiGGS has been tested in the following environment(s):

  • Linux x86 (64-bit), specifically Manjaro, Arch, CentOS, Scientific Linux and Ubuntu
  • Windows 10, as of HiGGS v 1.2.3
  • Mathematica v 11.3.0.0
  • xAct v 1.2.0

Install

  1. Make sure you have installed xAct.
  2. Download HiGGS:
    git clone https://github.com/wevbarker/HiGGS
    cd HiGGS
  3. Place the ./xAct/HiGGS directory relative to your xAct install. A global install might have ended up at:
    /usr/share/Mathematica/Applications/xAct

Quickstart

The package loads just like any other part of xAct, just open a fresh notebook and run:

Needs["xAct`HiGGS`"];

This loads the package (i.e. the names of the functions provided), along with its dependencies in the xAct bundle. However it does not load the physics. To construct the HiGGS environment, one must run:

BuildHiGGS[];

The build process may take about a minute or so. When it has concluded, you should be able to proceed to science. For example, try evaluating the Poisson bracket between the spin-parity 2+ irreducible component of the foliation-projected momentum of the translational gauge field, and the 1- irrep of the foliation-projected torsion tensor, without first defining a constraint shell for a particular theory, type:

PoissonBracket[PiPB2p[-a, -b], TP1m[-c], "ToShell" -> False];

If you want to try something more ambitious, build the constraint structure for Einstein-Cartan theory:

DefTheory[{Alp1 == 0, Alp2 == 0, Alp3 == 0, Alp4 == 0, Alp5 == 0, 
   Alp6 == 0, Bet1 == 0, Bet2 == 0, Bet3 == 0, cAlp1 == 0, cAlp2 == 0,
    cAlp3 == 0, cAlp4 == 0, cAlp5 == 0, cAlp6 == 0, cBet1 == 0, 
   cBet2 == 0, cBet3 == 0}, "Export" -> "EinsteinCartan"];

That output is less easy to show.

Installation test

More general examples can be found in the notebook ./xAct/HiGGS/Documentation/Examples/tutor.nb. This notebook also acts as an install test.

  1. Move the test files into your working directory, e.g. for a global install:
    cp /usr/share/Mathematica/Applications/xAct/HiGGS/Documentation/Examples/tutor.nb ./
    cp -r /usr/share/Mathematica/Applications/xAct/HiGGS/Documentation/Examples/svy ./
    mkdir ./fig
  2. Open Mathematica and run ./tutor.nb in a notebook front end. Make sure you run all the initialisation cells, from the beginning, to the end.

What's in the box?

The HiGGS package has the following structure:

xAct
└── HiGGS
    ├── bin
    │   └── build
    │       ├── CanonicalPhiToggle.mx
    │       ├── CDPiPToCDPiPO3.mx
    │       ├── ChiPerpToggle.mx
    │       ├── ChiSingToggle.mx
    │       ├── CompleteO3ProjectionsToggle.mx
    │       ├── GeneralComplementsToggle.mx
    │       ├── NesterFormIfConstraints.mx
    │       ├── NonCanonicalPhiToggle.mx
    │       ├── O13ProjectionsToggle.mx
    │       ├── ProjectionNormalisationsToggle.mx
    │       └── VelocityToggle.mx
    ├── COPYING
    ├── Documentation
    │   ├── Examples
    │   │   ├── appcg.job.m
    │   │   ├── appcg.job.nb
    │   │   ├── appcg.job.sh
    │   │   ├── appcg.plt.py
    │   │   ├── appcg.plt.sh
    │   │   ├── peta4.job.m
    │   │   ├── peta4.job.nb
    │   │   ├── peta4.job.sh
    │   │   ├── peta4.job.slm
    │   │   ├── peta4.plt.py
    │   │   ├── peta4.plt.sh
    │   │   ├── peta4.rdm.png
    │   │   ├── peta4.svy.m
    │   │   ├── peta4.svy.nb
    │   │   ├── svy
    │   │   │   ├── EinsteinCartan.thr.mx
    │   │   │   ├── EinsteinCartan_vel.thr.mx
    │   │   │   ├── simple_spin_1p.thr.mx
    │   │   │   └── simple_spin_1p_vel.thr.mx
    │   │   ├── tutor.nb
    │   │   └── tutor.png
    │   ├── HiGGS.pdf
    │   └── HiGGS_sources.pdf
    ├── HiGGS.m
    ├── HiGGS.nb
    ├── HiGGS_smearing_functions_global.m
    ├── HiGGS_smearing_functions.m
    ├── HiGGS_SO3.m
    ├── HiGGS_SO3.nb
    ├── HiGGS_sources.m
    ├── HiGGS_sources.nb
    ├── HiGGS_variations.m
    └── Kernel
        └── init.wl

The license is in COPYING.

The file init.wl is called when the package is invoked, and points to HiGGS.m, a small Wolfram language file and main package file sourced by the notebook HiGGS.nb.

When the HiGGS environment is actually built, HiGGS.m is actually running HiGGS_sources.m - the larger "physics package" sourced by HiGGS_source.nb.

During the course of the build, the binaries ./xAct/HiGGS/bin/build/*.mx are incorporated; these contain some heavy expressions.

The sub-package HiGGS_variations.m incorporates elements of Cyril Pitrou's code from this repository.

The files HiGGS.pdf and HiGGS_sources.pdf are carbon copies of the source notebooks.

The notebook tutor.nb contains some more basic examples, and it relies on the *.thr.mx files in the svy directory.

The Wolfram Language files which refer to smearing functions are patches in version 1.2.2.

What are peta4 and appcg?

The files ./xAct/HiGGS/Documentation/Examples/peta4.* and ./xAct/HiGGS/Documentation/Examples/appcg.* refer to the jobs which implement the HiGGS Commissioning Survey and various unit tests. HiGGS does not need these files to function. The names refer to two computing services:

  1. Peta-4 is a supercomputer, the CPU component of the heterogeneous CSD3 facility belonging to the University of Cambridge.
  2. appcg is a small, private compute server belonging to the Cavendish Laboratory Astrophysics Group.

These sources are included to give inspiration to users who which to perform HPC surveys, though the user's architecture may well differ.

Contribute

Please do! I'm always responsive to emails (about science), so be sure to reach out at wb263@cam.ac.uk. I will also do my best to get your code working if you are just trying to use HiGGS.

Acknowledgements

This work was performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (www.csd3.cam.ac.uk), provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP/T022159/1), and DiRAC funding from the Science and Technology Facilities Council (www.dirac.ac.uk).

I am grateful for the kind hospitality of Leiden University and the Lorentz Institute, and am supported by Girton College, Cambridge.

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Hamiltonian Gauge Gravity Surveyor (HiGGS). Tools for Hamiltonian constraint, canonical and Dirac-Bergmann analysis of gravity theories with spacetime curvature and torsion.

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