595 Page 4 of 18 Eur. Phys. J. C (2019) 79 :595
action S() satisfies (4), which means that classical equa-
tions of motion, obtained by applying the variational princi-
ple to either S() or
˜
S(
˜
), are related by
δS[]
δ
a
=
b
∂
˜
b
∂
a
δ
˜
S
˜
δ
˜
b
. (5)
Provided that the Jacobian matrix
∂
˜
b
∂
a
is non-singular
when evaluated at each of the stationary solutions,
δS()
δ
a
= 0
implies
δ
˜
S(
˜
)
δ
˜
b
= 0 and stationary trajectories are in a one-
to-one correspondence. One can then say that two theories
where the variables are related to each other by (3) are classi-
cally equivalent if the dynamical shells corresponding to the
points
0
and
˜
0
, where the actions are stationary, are related
by
˜
0
=
˜
(
0
). Thus, any physical quantity will lead to the
same result in either frame when evaluated on-shell, as a
consequence of (4). Again, this just encloses the common
notion that one should be free to choose whatever variables
they prefer to perform a computation and, although particular
equations will be different, physical statements must remain
the same for any choice.
Although this statement is crystal clear in classical
mechanics, the situation is not so transparent in Quantum
Field Theory(QFT), where not only stationary trajectories
contribute to the dynamics of a given system. There, instead,
we are interested in objects formally obtained from a path
integral over all possible trajectories with the right boundary
conditions. In particular, we focus our interest on correlation
functions obtained from the Quantum Effective Action [Q],
which is defined in terms of the mean field Q by a Legendre
transform
[Q]=W [J ]−J · Q, (6)
where Q satisfies the effective equations of motion
∂
∂Q
a
=−J
a
, (7)
and W [J ] is given as a (Euclidean) path integral over the
field variables with a source J
a
Z[J ]=e
−W [J ]
=
[d]e
−S[]−J ·
. (8)
Here the dot product is assumed to represent sum over all
indices as well as integration over space-time coordinates
J · =
d
4
x J
a
a
. (9)
However,as noted by Vilkovisky in [1], the quantum effec-
tive action as defined here does not satisfy an analogous for-
mula to (5). In general
δ
[
Q
]
δQ
a
=
b
δ
˜
Q
b
δQ
a
δ
˜
˜
Q
δ
˜
Q
b
, (10)
and there is not a one-to-one correspondence of 1PI correla-
tion functions in different frames. Nevertheless, it was also
shown in [1] that the problematic pieces are proportional to
the equations of motion and they cancel on-shell, preserving
equivalence for those correlators that contribute to S-matrix
elements. The problem persists off-shell, and although there
is a way to covariantize [Q], arriving to what is known as
Unique Effective Action, it is not clear if this redefinition is
needed at all for most standard settings, since all dynamics
is presumably contained in the S-matrix.
4
One may summer-
ize the situation by noting that it is the non-covariance of
(9) which is responsible for (10) but that, since observables
are calculated for J = 0, this can only lead to disparities in
intermediate steps in the calculation of correlation functions
but not in observables (e.g. the S-matrix).
Even though things seem pretty clear from Vilkovisky’s
arguments, more concerns can be raised in the presence of
gravity as one of the dynamical fields in , even for on-shell
quantities. In particular, let us focus in the problem pointed
out in [2], where we consider non-linear redefinitions of the
fields. In those cases, the realization of gauge and global sym-
metries might differ in different frames. Therefore, it might
also happen that something which is an exact symmetry under
renormalization in one frame maps to an anomalous symme-
try in the other. Then, the anomaly to the current conservation
generates new S-matrix elements in one of the frames only,
through the expectation value
0|∇
μ
J
μ
|0=0| = 0, (11)
where |=∇
μ
J
μ
|0 is some state of the theory. This
amplitude, which it is not generated in the second frame,
where there is no anomaly, spoils the equivalence premise in
a strong way.
Although this is a quite general effect associated to field
redefinitions, let us here be explicit and show a realization
of this phenomenon by choosing a particular scale-invariant
scalar-tensor theory, where we couple the metric g
μν
to a
scalar field φ in the Jordan frame
S
J
[g
μν
,φ]=
d
4
x
|g|
−ξφ
2
R +
1
2
∂
μ
φ∂
μ
φ +
λ
4!
φ
4
,
(12)
where ξ and λ are dimensionless couplings. This action is
invariantunder diffeomorphismsas well as under global scale
transformations of the form
4
There is some discussion about the need of using the Unique Effective
Action in order to obtain gauge invariant beta functions for running
couplings [43,44].
123