In computational physics, where the aim is to model physical
systems, not only a possible inaccuracy of the Gauss law has to
be conserved while solving the equations of motion approximately
but also the Gauss law itself has to be satisfied initially.
In order to achieve this goal the previous discussion about the
rate of change of the Gauss law is, however, of a great help:
we recognize that in order to get zero Noether charge on each
site only a construction like 
on the outgoing and like
on the incoming links sums up to zero.
Since by definition the color charge is alike sum with
and
respectivley, only an initialization of P being proportional
to V - minus the trace term - fulfills the zero charge initial condition.
This means that
is a proper initialization for zero color charge. Static color charges put on some lattice sites, on the other hand, can be simulated by adding (in three dimensions) one sixth of the color charge quaternion Q times the link group element U for outgoing links from the end of the actual link where we seek for the value of P and substracting UQ for links pointing to the beginning of the actual link
Here Q naturally has to be a traceless quaternion
- for being a color charge - otherwise it would spoil up the orthogonality between the canonical variables P and Q.
For second order schemes of Type III or Type IV, in particular for the
OET one needs to initialize the zeroth and first step values as well.
We use this fact for satisfying the unitarity and orthogonality
constraints locally at time 
.
The particular choice of 
with independent, random unit length
quaternions on each link simulates a hot gauge field system satisfying
locally.
Obtaining now the complement link variables V for this initial condition
all the canonical momenta 
and 
can be initialized in the
above described manner.
While these choices also fix the values of local Lagrange multipliers,
and 
, the 
parameters coping with the total
energy conservation have to be initialized independently.
A sensible choice is to take 
.
Using some more complicated update algorithms than the OET, which
still respect unitarity, orthogonality and Noether charge conservation,
the recursion of the 
parameters may be different.
Also the in continous time limit (
) marginal
stability of the constraint surfaces may deviate for one or the other
algorithm. The investigation of these properties, however,
becomes an increasingly
complex task so that only computational experiments can tell
us which is the best algorithm.
Generally - after a given physical
time t=Nh almost independently of the time step size h -
both the OET and the HIHEP algorithm proves to be unstable.
This instability occurs as an increasing deviation of the 
parameter, bringing the phase space point back onto the energy shell,
from the continous time limit value 1. Although in one step this
deviation is, of course, of order 
, it is monotonously
accumulating from the beginning of the calculation reaching sometimes
as much as 
and then oscillating around a level far from
the ideal case of 
.
While this undesired property is the most pronounced for a totally
unrestricted random choice of the initial link quaternions, at
initial states with lower potential energy (more aligned quaternions with
each other) the instability develops much slower and remains less
obvious. The reason of this behavior lies in the construction
principle of these algorithms: the kinetic energy is conserved
exactly once the unitarity and orthogonality conservation is
satisfied (see the discussion of the free motion on the unit circle).
Depending on the total energy of the lattice gauge field configuration under
investigation at a sufficiently small 
ratio, however, both
the OET and the HIHEP algorithm can be stabilized for a much longer
time. In order to find the best time and spatial resolution for a given
purpose numerical experimenting is necessary.
In conclusion, while these problems are somewhat discouraging and their solution requires some more computational research, the construction of Hamiltonian lattice gauge theory algorithms exactly satisfying the Gauss' law and fulfilling the necessary local unitarity constraints is a promising development in the investigation of the chaotic dynamics of nonabelian gauge fields[7] by means of computer simulations.
I thank Prof. János Polónyi for drawing my attention to the discrete time Noether theorem. The warm hospitality and the support of Physics Department of Bergen University and of Prof. László Csernai during my visit there is also gratefully acknowledged. This work was supported in part by the Collaboration Agreement between the Norvegian Research Council (NFR) and the Hungarian Academy of Science (MTA) (grant No. 422.92/001).