Before turning to the formalism let us briefly comment on the status of the background field
configuration. First one should stress that the term “background field” in this context does not refer to a
solution of the classical field equations. It is an auxiliary device used to formulate covariance properties that
constrain the renormalization flow. The dependence on the background configuration is controlled by
splitting Ward identities that, roughly, ensure that nothing depends on the way how the integration variable
in the functional integral is decomposed into a reference configuration and a fluctuation field. Once a source
free condition is imposed the “background” gets related to the expectation value of the fluctuation field
through a consistency condition involving the full quantum effective action (see Equations (2.51)
and (B.53
) below).
When applied to quantum gravity, the background field formalism can be viewed as giving rise to a formulation with a state-dependent dynamically adjusted reference metric. The bare manifold is initially equipped with a reference geometry, but rather than being external it eventually gets related to the average of the quantum metric in a self-consistent manner. As an incomplete classical analogy one may take the variational principles for general relativity which in addition to the dynamical metric invoke a fiducial background metric (see e.g. [74] and references therein). The latter only provides the desired covariance properties and in the presence of generic boundary conditions serves to define conserved quantities relative to the background.
As a further clarification one should add that in the case of gauge theories the on-shell projections of the Green’s functions (e.g. S-matrix elements) computed from the background effective action are not assumed to be physical quantities from the outset. In particular in the case of gravity the precise role of the on-shell projections remains to be understood. The main reason for focusing on the background effective action here are its ‘benign’ properties under UV renormalization. The background covariance constrains the renormalization flow, both on the level of a Wilsonian action and on the the level of an effective average action.
The background field formalism comes in two main variants: one based on a linear background-fluctuation split and the other based on a geodesic background-fluctuation split. The latter is used to cope with field reparameterization symmetries. Both variants can be applied to non-gauge theories (regular Lagrangians) and gauge theories (singular Lagrangians). We briefly discuss these variants consecutively, later on the variants in Sections B.2.2 and B.2.3 will be used.
In this simple case the use of a background field formalism is ‘overkill’. However it is a convenient way to
introduce the relevant structures. The starting point is again the generating functional for the connected
Green’s functions, but with a modified coupling to the source . Specifically, one linearly decomposes the
quantum field, i.e. the integration variable
, according to
, where
is a
background configuration to be adjusted later and
is the fluctuation field, i.e. the new integration
variable. In the corresponding generating functional only
not the complete field
is coupled to the
source. For the sake of illustration we allow the action to depend explicitly on the background field
,
i.e.
. One introduces
The condition that the source in Equation (B.15
) vanishes is usually solved by formal power series
inversion and then gives
as the only solution,
. More generally the vanishing of
defines locally
as a function of
, say,
. For such configurations
becomes a
functional of a single field
We briefly mention two applications of the relations (B.65), both for actions
without explicit
background dependence. The relation (B.17
) is often used for constant background field configurations
.
It then implies that vertex functions with vanishing external momenta can be written as
derivatives of
vertex functions with a smaller number of legs
Good reasons to adopt such a split exist in theories with symmetries, which can be local gauge
symmetries, or field reparameterization symmetries, or both. In all situations the background field method
offers key advantages in that it can produce an effective action which is an invariant functional of its
argument. Via the above splitting principle this then greatly restricts the form of the (Wilsonian)
renormalized action. In a non-background field formalism these symmetries in contrast have to be imposed
by relating possibly noninvariant terms or pieces of the renormalized action via conventional Ward
identities, like Equation (A.9) in the case of field reparameterizations.
We first describe the background field technique for a non-gauge theory where reparameterization
invariance in field space is aimed at, and then for a diffeomorphism gauge theory (the case of Yang–Mills
theories runs completely parallel). Finally we mention the setting where both gauge and field
reparameterization invariance is aimed at. General references are [68, 49, 223
, 224, 129
, 178
].
Here reparameterization invariance in field space is aimed at; the original construction is due to
Honerkamp et al. [107, 109]. Since invariance under local field redefinitions is a hallmark of
physical quantities this field reparameterization invariant effective action is an object much
more intrinsic to the field theory under consideration. Morally speaking in this technique the
field reparameterization Ward identity (A.9) is built in, and does not have be imposed along
with the solution/definition of the functional integral. For the construction of the covariant
effective action the field configuration space is equipped with a metric and the associated metric
connection. The Lagrangian is assumed to be reparameterization in variant in the sense that
One now describes the configurations in terms of an arbitrary (off-shell) background configuration
and geodesic normal coordinates
, which are new dynamical fields. That is, a nonlinear
background-fluctuation split
is used. Here
is the function on
such that its
value
gives the endpoint of the (locally unique) geodesic in
connecting
to
and
having
as the tangent vector at
. The normal coordinate expansion of
is a power
series in
with coefficients built from the Christoffel symbols
of
evaluated at
. We shall
also need the inverse series
, defined by
. To quadratic order one has
For later reference let us also note that once in the source-free condition
is
imposed, with
as the only solution within formal power series inversion, one has
We now describe two types of Ward identities for these systems: diffeomorphism type Ward identities
and the nonlinear splitting Ward identities mentioned earlier. The former are really generalized Ward
identites relating different theories in the sense that a compensating change in the metric tensor is
needed. On the classical level an example is Equation (B.24
); only if
admits Killing vectors and one
takes for
one of the Killing vectors does Equation (B.24
) reduce to a conservation equation proper,
with
being the associated Noether current. In fact, Equation (B.24
) can be promoted to a
“Diffeomorphism Ward identity” in the quantum theory, at least perturbatively [201
, 162
] and
presumably also in a non-perturbative formulation. In perturbation theory the “equations of motion
operator” appearing on the right hand side of Equation (B.24
) is a finite operator, i.e. it is
the same when viewed as a functional of the bare fields and couplings, and when viewed as a
function of the renormalized fields and couplings. Once the second term on the left-hand-side has
been defined in terms of normal products, the diffeomorphism current
must be finite as
well.
For the background effective action a similar diffeomorphism Ward identity arises as follows.
Since a geodesic is a coordinate independent concept, transforming all of the ingredients in the definition of
by an infinitesimal diffeomorphism
,
, gives
This is different in the nonlinear splitting Ward identities, which control the dependence on the
background field configuration. If the metric in Equation (B.38) is kept fixed the splitting symmetry
,
becomes nonlinear (see Equation (B.27
)). The corresponding “nonlinear
splitting Ward identity” reads as follows:
Since we are interested here in diffeomorphism invariant theories we consider this type of gauge invariance.
The case of Yang–Mills fields is largely parallel (see [108, 1, 25
] for the latter). Let
be any
diffeomorphism invariant action of a Riemannian metric
. Infinitesimally the invariance
reads
, where
The symmetry variation can be decomposed in two different ways,
The background generating functional is formally defined by
The background effective action now is defined by
The extremizing source configurations It is convenient to regard as a functional of
and
instead of
and
. We thus set
Finally we have to switch off the sources. Since has ghost number zero
,
is always
a solution of Equation (B.49
). For the metric source
this is different. Within the realm of formal
power series inversions
is always a solution of
. Combined with the usual uniqueness
assumption it is the only solution, and the “self-consistent background determination” at which
the background field method aims at degenerates. Indeed, note that in
the
dependence drops out, as
is prescribed. The expectation value of the full quantum metric,
, say, just gives back the prescribed background
. This evidently
has a somewhat perturbative flavor, although no direct reference to perturbation theory is
made.
To go beyond that, we directly impose
as the source-free condition. It adjusts the background In view of Equation (B.53, B.54
) one would like to introduce a functional
of a single metric only,
whose one-point functions have stationary points. This can be done as follows. We define the final
“background effective action” [156] by
The background gauge invariant effective action in Section B.2.3 depend parametrically on the choice of the
background gauge condition . This dependence is a consequence of the field parameterization
dependence of the effective actions based on a linear fluctuation background split. The geometrical approach
of Vilkovisky and deWitt [68
, 223
] is designed to overcome this drawback and at least formally it produces
an off-shell effective action with all the desirable properties: It is gauge invariant with respect to the
background field, gauge invariant with respect to the mean of the fluctuation field, and independent of the
choice of the gauge fixing surface. In brief, the strategy is to project locally onto the gauge
invariant subspace and then apply the techniques of Section B.2.2. Since we shall not use this
formulation here it may suffice to refer to [129] for a brief survey and to [178, 68, 223] for detailed
expositions.
With these remarks we conclude our brief introduction to the background field formalism. In the context
of the asymptotic safety scenario the variant from Section B.2.2 has been used in [154, 155] (see Section 3)
and the variant from Section B.2.4 in [179, 133, 131] (see Section 4).
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