In
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
, especially in
non-abelian gauge theories, global problems at
gauge fixing
In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct c ...
are often encountered. Gauge fixing means choosing a representative from each
gauge orbit, that is, choosing a
section of a fiber bundle. The space of representatives is a submanifold (of the bundle as a whole) and represents the gauge fixing condition. Ideally, every gauge orbit will intersect this submanifold once and only once. Unfortunately, this is often impossible globally for non-abelian gauge theories because of topological obstructions and the best that can be done is make this condition true locally. A gauge fixing submanifold may not intersect a gauge orbit at all or it may intersect it more than once. The difficulty arises because the gauge fixing condition is usually specified as a differential equation of some sort, e.g. that a divergence vanish (as in the Landau or
Lorenz gauge
In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ha ...
). The solutions to this equation may end up specifying multiple sections, or perhaps none at all. This is called a Gribov ambiguity (named after
Vladimir Gribov
Vladimir Naumovich Gribov (Russian Влади́мир Нау́мович Гри́бов; March 25, 1930, LeningradAugust 13, 1997, Budapest) was a prominent Russian theoretical physicist, who worked on high-energy physics, quantum field theory an ...
).
Gribov ambiguities lead to a
nonperturbative
In mathematics and physics, a non-perturbative function (mathematics), function or process is one that cannot be described by perturbation theory. An example is the function
: f(x) = e^,
which does not have a Taylor series at ''x'' = 0. Every c ...
failure of the
BRST symmetry, among other things.
A way to resolve the problem of Gribov ambiguity is to restrict the relevant functional integrals to a single Gribov region whose boundary is called a Gribov horizon.
Still one can show that this problem is not resolved even when reducing the region to the first Gribov region. The only region for which this ambiguity is resolved is the fundamental modular region (FMR).
Background
When doing computations in gauge theories, one usually needs to choose a gauge. Gauge degrees of freedom do not have any direct physical meaning, but they are an artifact of the mathematical description we use to handle the theory in question. In order to obtain physical results, these redundant degrees of freedom need to be discarded in a suitable way
In Abelian gauge theory (i.e. in
QED) it suffices to simply choose a gauge. A popular one is the Lorenz gauge
, which has the advantage of being
Lorentz invariant
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of v ...
. In non-Abelian gauge theories (such as
QCD
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a ty ...
) the situation is more complicated due to the more complex structure of the non-Abelian gauge group.
The Faddeev–Popov formalism, developed by
Ludvig Faddeev
Ludvig Dmitrievich Faddeev (also ''Ludwig Dmitriyevich''; russian: Лю́двиг Дми́триевич Фадде́ев; 23 March 1934 – 26 February 2017) was a Soviet and Russian mathematical physicist. He is known for the discovery of the ...
and
Victor Popov, provides a way to deal with the gauge choice in non-Abelian theories. This formalism introduces the Faddeev–Popov operator, which is essentially the
Jacobian determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...
of the transformation necessary to bring the gauge field into the desired gauge. In the so-called Landau gauge
[In quantum gauge theory, the term "Lorenz gauge" usually refers to more general gauges of the form , where the function is usually averaged over.] , this operator has the form
:
where
is the
covariant derivative in the adjoint representation. The determinant of this Faddeev–Popov operator is then introduced into the path integral using
ghost fields.
This formalism, however, assumes that the gauge choice (like
) is unique — i.e. for each physical configuration there exists exactly one
that corresponds to it and that obeys the gauge condition. In non-Abelian gauge theories of Yang–Mills type, this is not the case for a large class of gauges, though, as was first pointed out by Gribov.
Gribov's construction
Gribov considered the question of, given a certain physical configuration, how many different gauge copies of this configuration obey the Landau gauge condition
. No configurations without any representatives are known. It is perfectly possible, though, for there to be more than one.
Consider two gauge fields
and
, and assume they both obey the Landau gauge condition. If
is a gauge copy of
, we would have (assuming they are infinitesimally close to each other):
:
for some function
.
[The covariant derivative here contains the gauge field .] If both fields obey the Landau gauge condition, we must have that
:
and thus that the Faddeev–Popov operator has at least one zero mode. If the gauge field is infinitesimally small, this operator will not have zero modes. The set of gauge fields where the Faddeev–Popov operator has its first zero mode (when starting from the origin) is called the "Gribov horizon". The set of all gauge fields where the Faddeev–Popov operator has no zero modes (meaning this operator is positive definite) is called the "first Gribov region"
.
If gauge fields have gauge copies, these fields will be overcounted in the path integral. In order to counter that overcounting, Gribov argued we should limit the path integral to the first Gribov region. In order to do so, he considered the ghost propagator, which is the vacuum expectation value of the inverse of the Faddeev–Popov operator. If this operator is always positive definite, the ghost propagator cannot have poles — which is called the "no-pole condition". In usual perturbation theory (using the usual Faddeev–Popov formalism), the propagator does have a pole, which means we left the first Gribov region and overcounted some configurations.
Deriving a perturbative expression for the ghost propagator, Gribov finds that this no-pole condition leads to a condition of the form
:
with ''N'' the number of colors (which is 3 in QCD), ''g'' the gauge coupling strength, ''V'' the volume of space-time (which goes to infinity in most applications), and ''d'' the number of space-time dimensions (which is 4 in the real world). The functional