Quantum Mechanics Bootstrap and the BFSS Conjecture

This repository contains the code for the implementation of my Master's Thesis supervised by Prof. Dr. Vasilis Niarchos.

Bellow I outline which files are needed to replicate the results I got for the bootstraps I showcased in the text.

Project Structure

Zero Temperature

These are the files one would need to recreate the results I present in the thesis for the zero temperature cases.

zero-temperature_harmonic_oscillator.nb

Mathematica implementation of the harmonic oscillator bootstrap at T=0.
Constructs and simplifies the Hankel moment matrix, solves the SDP, and exports results to JSON.

plotting_harmonic_oscillator.ipynb

Reads exported JSON data and generates plots.

zero_temperature_anharmonic_oscillator.py

Python implementation of the T=0 anharmonic oscillator bootstrap using sympy, cvxpy, and
SDPA-Multiprecision through the Python wrapper sdpa-multiprecision. Note that to use cvxpy with SDPA-Multiprecision one would have to build the sdpa-multiprecision from source and link against the SDPA-Multiprecision libraries. The specifics are outlined here. Solves the SDP and exports results.

A sister file, zero-temperature_harmonic_oscillator.nb, contains the same implementation in a Mathematica notebook.

plotting_anharmonic_oscillator.ipynb

Loads output data and produces plots.

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Thermal Bootstrap

These are the files needed to replicate my results for the thermal bootstrap.

matrix_maker.nb

Mathematica code generating operator bases, Schwinger–Dyson relations, moment matrices, and Gauss–Radau quadrature data.
Exports all matrices and parameters to JSON for the thermal SDP. Works with any 1 dimensional hamiltonian modulo constants which would have to be taken care of by hand. The following files depend on the user having exported the appropriate files using this notebook.

thermal_bootstrap_header_file.py

Core Python module that:

• Converts exported matrices into CVXPY expressions
• Constructs and solves the thermal SDP
• Performs temperature scans
• Exports results to HDF5
  The solver used has to be hardcoded in this header file. Includes numerical diagonalization for anharmonic thermal energy comparison.

thermal_harmonic_bootstrap.py

Runs the thermal SDP for the harmonic oscillator (upper and lower bounds). I used the CLARABEL solver for the harmonic oscillator which comes out of the box with cvxpy. I didn't tweak any parameters.

thermal_anharmonic_bootstrap.py

Runs the thermal SDP for the quartic oscillator (upper and lower bounds). I used the 'SDPA-Multiprecision` solver through the wrapper as outlined above. I didn't tweak any parameters.

plotting_harmonic.py

Plots harmonic thermal bootstrap results from HDF5 output.

plotting_anharmonic.py

Plots quartic thermal bootstrap results and compares with numerical thermal energy.

Extra Files

deprecated_python_class_psd_hermitian

Contains psdhermitian.py which is a headerfile I was working on that I didn't end up using and is not fully finished. I made it for treating Hermitian SDPs symbolically in code and have the ability to directly export them in a format that the SDPA-GMP solver can read from the terminal. I include it because someone might find it useful. Instrumental to understanding how to interface with the solver directly is the [manual for the SDPA solvers]{https://usermanual.wiki/Document/sdpa711manual20080618.114778884.pdf}.

wolfram_output/python_output

This folders include the json files from the matrix_maker.nb that contain the exported matrices and the HDF5 files that are outputed from the corresponding python bootstrap file. This, last mentioned files are the ones I used to make the plots.

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If you stumbled upon this repository and need help using this code feel free to contact me either through GitHub or at pripkand@protonmail.com