N=1 Supersymmetric Quantum Mechanics: SU(2) gauge theory.
Here I make available mathematica code and simple Fortran programs.
The most important program is Generate.ma, which is used for making the
gauge invariant spherical harmonics for (N=1 supersymmetric) SU(2) gauge
theory. The other programs deal with the radial sectors and with finding
the lowest eigenvalues and other properties of the Hamiltonian.
The non-supersymmetric (F=0) case reproduces old results by M. Lüscher
and G. Münster (Nucl. Phys. B232 (1984) 445), and those I
obtained in collaboration with
Jeff Koller. See also another paper
with him, which forms essential background for this study. The very first
time I discussed in writing (with little success) the problem of the
adiabatic approximation for determining the Witten Index was in
September 1991. Writing a recent
review on QCD in a finite volume,
prompted me to come back to this problem. The background material for the
supersymmetric analysis can be found in the paper
"The Witten index beyond the adiabatic
approximation" [hep-th/0112072], which is dedicated to the memory of
Here is the Mathematica code for generating the basis and matrix elements.
I did not use Mathematica Notebook features, so that the available
code should run on any platform. It has been tested with Mathematica
versions 2.0, 3.0 and 4.0. Use each mathematica program in a separate
session, such that global definitions do not interfere!
The following deals with diagonalization using Mathematica (slow).
- Generate.ma, creates spherical
harmonics and makes tables of reduced matrix elements (Htab0r, HHtab0r, Htab2r,
and HHtab2r) in a format suitable for input in Fortran programs.
- Besj.ma, computes radial matrix
elements and makes tables (Rtab2 and RRtab2) suitable for input in Fortran
- Bjtab.ma, makes table (Bjtab) of
spherical Bessel functions in a format suitable for input in Fortran programs
and used to reconstruct the wave function.
- roots.out, roots of the
derivative of the spherical Bessel functions needed in Besj.ma, to make
radial matrix elements.
J. Wosiek has implemented symbolically the Hamiltonian for supersymmetric
quantum mechanics, including SU(2) gauge theory, using the full Fock
space defined in terms of harmonic oscillators. See
as well as more recent work with M. Campostrini, in particular
hep-th/0407021 (with a link
to supplementary material).
For F=0, our basis' agree (take w=1) and a direct comparison (for J=0) gives
perfect agreement. Small changes in H0t.ma were made to facilitate
this comparison for both parities (run EB[1,Ncut,Em] for the first Em
eigenvalues). The following Mathematica program implements the harmonic
basis for F=2 and J=0, also showing perfect agreement. I modified diag.f
slightly to deal with this case as well.
- H0t.ma, builds the full matrix
for the F=0 (bosonic) case, allows for diagonalization with mathematica
and building the wave function (slow but manageable). Also makes a table
for the full matrix (Htab0) in a format suitable for input in Fortran
- H2t.ma, builds the full matrix
for the F=2 case, allows for diagonalization and Temple's inequality with
mathematica (not advised, except for testing). Also makes tables for the
full matrix (Htab2 and HHtab2) in a format suitable for input in Fortran
(also not advised, except for testing).
Here is the Fortran program, which uses the
IMSL libraries to
find eigenvalues (I used an old Fortran77 version).
- H2h.ma, builds the full matrix
for the F=2 case with harmonic oscillator basis, allows for diagonalization
and makes a table (Htab2h) for the full matrix in a format similar to Htab0
and suitable for input in Fortran.
- diag.f, used for diagonalizing
H(r), computing Veff, diagonalizing H and reconstructing the groundstate
wave function. It allows for automatic pruning. Read the extensive comments.
See also here.
Page initially created on 10 December 2001. Any changes in the programs can
be detected from their version date listed in the beginning.
Disclaimer: Users themselves are responsible for the results obtained
with these programs.
This is free code. Of course it has been tested
extensively, but I can give no guarantees.
Pierre van Baal, last updated
14 September 2004.