Quantum Chromodynamics, B. Medeke, Univ. Wuppertal, Germany


Source:	    Bjoern Medeke, Department of Mathematics, Institute of Applied
Computer Science, University of Wuppertal, 42097 Wuppertal, Germany. Phone: +49
202 439-3776. Email: medeke at math.uni-wuppertal.de 
Discipline:	Physics
Accession (in Matrix Market collection):  December 2000

Background. Lattice gauge theory is a discretization of quantum chromodynamics
which is generally accepted to be the fundamental physical theory of strong
interactions among the quarks as constituents of matter. The most
time-consuming part of a numerical simulation in lattice gauge theory with
Wilson fermions on the lattice is the computation of quark propagators within a
chromodynamic background gauge field. These computations use up a major part of
the world's high performance computing power.

Quark propagators are obtained by solving the inhomogeneous lattice Dirac
equation Ax = b, where A = I - kD with 0 <= k < kc is a large but sparse
complex non-Hermitian matrix representing a periodic nearest-neighbour coupling
on a four-dimensional Euclidean space-time lattice.

From the physical theory it is clear that the matrix A should be positive real
(all eigenvalues lie in the right half plane) for 0 <= k < kc. Here, kc
represents a critical parameter which depends on the given matrix D. Denoting

\gamma_5 = \left( \begin{array}{cccc}
    0 & 0 & 1 & 0 \\
    0 & 0 & 0 & 1 \\
    1 & 0 & 0 & 0 \\
    0 & 1 & 0 & 0
 \end{array}\right)

 the Wilson fermion matrix A is Gamma-5 symmetric,

\Gamma_5 A = A^H \Gamma_5,\;\;\;\;\Gamma_5 = I \otimes ( \gamma_5 \otimes I_3 )

Due to the nearest neighbour coupling, the matrix A has 'property A'. This
means that with a red-black (or odd-even) ordering of the grid points the
matrix becomes

A = I - kD

with

D = \left( \begin{array}{cc}
    0 & D_{\rm oe} \\D_{\rm eo} & 0
 \end{array}\right)

Set of QCD Matrices. The QCD matrices provided in the set QCD consist of
realistic matrices D generated at different physical temperatures b.

matrix D    b	order	nonzeros    kc
conf5.0-00l4x4-1000.mtx	    5.0	3072	119808	0.20611
conf5.0-00l4x4-1400.mtx	    5.0	3072	119808	0.20328
conf5.0-00l4x4-1800.mtx	    5.0	3072	119808	0.20265
conf5.0-00l4x4-2200.mtx	    5.0	3072	119808	0.20235
conf5.0-00l4x4-2600.mtx	    5.0	3072	119808	0.21070
conf6.0-00l4x4-2000.mtx	    6.0	3072	119808	0.15968
conf6.0-00l4x4-3000.mtx	    6.0	3072	119808	0.16453
conf5.4-00l8x8-0500.mtx	    5.4	49152	1916928	0.17865
conf5.4-00l8x8-1000.mtx	    5.4	49152	1916928	0.17843
conf5.4-00l8x8-1500.mtx	    5.4	49152	1916928	0.17689
conf5.4-00l8x8-2000.mtx	    5.4	49152	1916928	0.17835
conf6.0-00l8x8-2000.mtx	    6.0	49152	1916928	0.15717
conf6.0-00l8x8-3000.mtx	    6.0	49152	1916928	0.15649
conf6.0-00l8x8-8000.mtx	    6.0	49152	1916928	0.15623

References. A survey of lattice gauge theory is given in

   1. M. Creutz: Quarks, Gluons, and Lattices,
     Cambridge University Press, (1986)

   2. I. Montvay and G. Munster: Quantum Fields on the Lattice,
     Cambridge University Press (1994) 

More background information can be found at various locations on the Web.
Search for "High Energy Physics" (HEP). A list of HEP Web sites is available at
CERN (European Laboratory for Particle Physics).

A PostScript version of this information is also available.