56
6 Nonlocal Gravity
so, its propagator has no different poles (as we noted in the Chap. 2, namely presence
of such a set of poles implies in existence of ghost modes). The simplest example of
such a function is the exponential, f (z) = e
−z .
Another motivations for nonlocality are the loop quantum gravity dealing with
finite-size objects, and the noncommutativity, where the Moyal product is essentially
nonlocal by construction. At the same time, it is interesting to note that although the
so-called coherent states approach [109] has been motivated by quantum mechanics,
by its essence it represents itself as a natural manner to implement nonlocality, so
that all propagators carry the factor e
−θk
2 , with θ is the noncommutativity parameter.
Within the gravity context, use of the nonlocal methodology appears to be especially
promising since it is expected that the nonlocality, being implemented in a proper
manner, can allow to achieve renormalizability without paying the price of arising
the ghosts. The first step in this study has been done in the seminal paper [110].
6.2 Some Results in Non-gravitational Nonlocal Theories
Before embarking to studies of gravity, let us first discuss the most interesting results
in non-gravitational nonlocal theories, especially within the context of quantum corrections.
As we already noted, effectively the nonlocal methodology has been applied to
perturbative studies for the first time within the coherent states approach [109] which
includes Gaussian propagator guaranteeing convergence of quantum corrections.
Further, various other studies have been performed. An important role was played by
the paper [111] where the effective potential in a nonlocal theory has been calculated
for the first time. In that paper, the following theory has been introduced:
L = −
1
2
φ(exp(//
2
) + m
2
)φ − V (φ).
(6.2)
Here, is a characteristic nonlocality scale. For this theory, one can calculate the
one-loop effective potential given by the following integral:
V
(1)
=
1
2
d
4 k E
(2π) 4 ln
exp
−
k
2
E
2
k
2
E + m
2
+ V
.
(6.3)
It is clear that at k
2
2 , the theory is reduced to usual one. The exponential
factors guarantee finiteness. It is easy to see that there is no ghosts in the theory
since there is no different denominators + m
2
i in the propagator of the theory.
However, the integral (6.3) can be calculated only approximately for various limits,
and it is easy to see that it diverges as → ∞ (in [111], a some procedure to isolate
this divergence has been adopted). Further, this study has been generalized for the
superfield theories representing themselves as various nonlocal extensions of WessZumino model and super-QED, in [112]. It is clear that when, in these theories, one
6 Nonlocal Gravity
so, its propagator has no different poles (as we noted in the Chap. 2, namely presence
of such a set of poles implies in existence of ghost modes). The simplest example of
such a function is the exponential, f (z) = e
−z .
Another motivations for nonlocality are the loop quantum gravity dealing with
finite-size objects, and the noncommutativity, where the Moyal product is essentially
nonlocal by construction. At the same time, it is interesting to note that although the
so-called coherent states approach [109] has been motivated by quantum mechanics,
by its essence it represents itself as a natural manner to implement nonlocality, so
that all propagators carry the factor e
−θk
2 , with θ is the noncommutativity parameter.
Within the gravity context, use of the nonlocal methodology appears to be especially
promising since it is expected that the nonlocality, being implemented in a proper
manner, can allow to achieve renormalizability without paying the price of arising
the ghosts. The first step in this study has been done in the seminal paper [110].
6.2 Some Results in Non-gravitational Nonlocal Theories
Before embarking to studies of gravity, let us first discuss the most interesting results
in non-gravitational nonlocal theories, especially within the context of quantum corrections.
As we already noted, effectively the nonlocal methodology has been applied to
perturbative studies for the first time within the coherent states approach [109] which
includes Gaussian propagator guaranteeing convergence of quantum corrections.
Further, various other studies have been performed. An important role was played by
the paper [111] where the effective potential in a nonlocal theory has been calculated
for the first time. In that paper, the following theory has been introduced:
L = −
1
2
φ(exp(//
2
) + m
2
)φ − V (φ).
(6.2)
Here, is a characteristic nonlocality scale. For this theory, one can calculate the
one-loop effective potential given by the following integral:
V
(1)
=
1
2
d
4 k E
(2π) 4 ln
exp
−
k
2
E
2
k
2
E + m
2
+ V
.
(6.3)
It is clear that at k
2
2 , the theory is reduced to usual one. The exponential
factors guarantee finiteness. It is easy to see that there is no ghosts in the theory
since there is no different denominators + m
2
i in the propagator of the theory.
However, the integral (6.3) can be calculated only approximately for various limits,
and it is easy to see that it diverges as → ∞ (in [111], a some procedure to isolate
this divergence has been adopted). Further, this study has been generalized for the
superfield theories representing themselves as various nonlocal extensions of WessZumino model and super-QED, in [112]. It is clear that when, in these theories, one