156
H. Wittig
One assigns a probability for the transition from
U μ (x)
i
to
U μ (x)
i+1
, which is
usually a function of the statistical weights of the two configurations, W i and W i+1 ,
respectively. For each individual configuration in the sequence one then evaluates
the observable, which yields the estimates i , i = 1, . . . , N cfg . The expectation
value is related to the mean value via
= lim
N cfg →∞
,
=
1
N cfg
N cfg
i=1
i .
(5.59)
In other words, in the limit of infinite statistics the mean value converges to
the ensemble average which is identical to the expectation value. An important
consequence of approximating the ensemble average by the sample average is a
non-zero value of the variance. Hence, in order to specify the results from a Monte
Carlo integration completely, one must also quote the statistical error which is given
by the square root of the variance.
In the standard algorithms that implement Markov processes (such as the
Metropolis algorithm [30]), the transition probabilities for going from one configuration to another are determined by comparing the statistical weights for local
variations in the field variables. This guarantees computational efficiency, since the
variation of individual link variables does not involve global information from the
entire lattice. In Eq. (5.55) the dynamical effects of the quark fields are incorporated
via the determinant of the lattice Dirac operator. The determinant, however, is a
non-local object, which is expensive to compute. When the first efforts were made
to compute observables in QCD in the 1980s, the available computer power did
not allow for the inclusion of the quark determinant. Instead, lattice physicists
resorted to what is known as the “quenched approximation”, which is based on the
assumption that the bulk of non-perturbative contributions is carried by the gauge
field, so that the determinant is set to a constant:
Quenched approximation:
det D lat = 1 ⇔ N f = 0.
(5.60)
The resulting gain in computer time amounts to several orders of magnitude. In the
quenched approximation the effects of virtual quark loops are entirely suppressed.
As a consequence, results for observables are afflicted with an unknown systematic
error. As we shall see later, there are several quantities (for instance, the masses
of the lightest hadrons) for which the quenching error amounts to just 10–15%.
Although this justifies the use of the quenched approximation to some extent, it is
clear that dynamical quark effects must be taken into account, in order to arrive at
reliable, non-perturbative predictions with a total accuracy at the percent level.
Modern algorithms for dynamical quarks, such as the Hybrid Monte Carlo algorithm [31], do not evaluate the quark determinant directly. Rather, one exploits the
property that the determinant can be rewritten as a functional integral over bosonic
fields, which is then evaluated stochastically. Thereby one avoids computing a
global object, but the computational effort involved in the stochastic estimation
H. Wittig
One assigns a probability for the transition from
U μ (x)
i
to
U μ (x)
i+1
, which is
usually a function of the statistical weights of the two configurations, W i and W i+1 ,
respectively. For each individual configuration in the sequence one then evaluates
the observable, which yields the estimates i , i = 1, . . . , N cfg . The expectation
value is related to the mean value via
= lim
N cfg →∞
,
=
1
N cfg
N cfg
i=1
i .
(5.59)
In other words, in the limit of infinite statistics the mean value converges to
the ensemble average which is identical to the expectation value. An important
consequence of approximating the ensemble average by the sample average is a
non-zero value of the variance. Hence, in order to specify the results from a Monte
Carlo integration completely, one must also quote the statistical error which is given
by the square root of the variance.
In the standard algorithms that implement Markov processes (such as the
Metropolis algorithm [30]), the transition probabilities for going from one configuration to another are determined by comparing the statistical weights for local
variations in the field variables. This guarantees computational efficiency, since the
variation of individual link variables does not involve global information from the
entire lattice. In Eq. (5.55) the dynamical effects of the quark fields are incorporated
via the determinant of the lattice Dirac operator. The determinant, however, is a
non-local object, which is expensive to compute. When the first efforts were made
to compute observables in QCD in the 1980s, the available computer power did
not allow for the inclusion of the quark determinant. Instead, lattice physicists
resorted to what is known as the “quenched approximation”, which is based on the
assumption that the bulk of non-perturbative contributions is carried by the gauge
field, so that the determinant is set to a constant:
Quenched approximation:
det D lat = 1 ⇔ N f = 0.
(5.60)
The resulting gain in computer time amounts to several orders of magnitude. In the
quenched approximation the effects of virtual quark loops are entirely suppressed.
As a consequence, results for observables are afflicted with an unknown systematic
error. As we shall see later, there are several quantities (for instance, the masses
of the lightest hadrons) for which the quenching error amounts to just 10–15%.
Although this justifies the use of the quenched approximation to some extent, it is
clear that dynamical quark effects must be taken into account, in order to arrive at
reliable, non-perturbative predictions with a total accuracy at the percent level.
Modern algorithms for dynamical quarks, such as the Hybrid Monte Carlo algorithm [31], do not evaluate the quark determinant directly. Rather, one exploits the
property that the determinant can be rewritten as a functional integral over bosonic
fields, which is then evaluated stochastically. Thereby one avoids computing a
global object, but the computational effort involved in the stochastic estimation