Chapter 6
Nonlocal Gravity
6.1 Motivations
As we have noted several times along this review, the main problem of various
gravity models is the development of a consistent quantum description. Indeed, the
Einstein gravity is non-renormalizable, and introduction of higher-derivative additive
terms implies in arising of ghosts. We have argued in the previous chapter that
the Horava–Lifshitz gravity seems to be a good solution since it is power-counting
renormalizable, and ghosts ate absent since the action involves only second time
derivatives. However, the HL gravity, first, is very complicated, second, breaks the
Lorentz symmetry strongly, third, displays a problem of extra degrees of freedom
whose solving, as we noted, requires special efforts. At the same time, the concept
of nonlocality developed originally within phenomenological context in order to
describe finite-size effects (see f.e. [107]), began to attract the interest. Besides of
this, the nonlocality enjoys also a stringy motivation since the factors like e
emerge
naturally within the string context [108]. The key idea of nonlocal field theories looks
like follows. Let us consider for example the free scalar field whose Lagrangian is
L =
1
2
φ f (//
2
)φ,
(6.1)
where f (z) is a some non-polynomial function (with is the characteristic nonlocality scale) which we choose to satisfy the following requirements.
First, at small arguments this function should behave as f (z) = a + z, in order
to provide the correct + m
2 IR asymptotic behavior. Second, this function must
decay rapidly at |z| → ∞ (in principle, we can consider only Euclidean space, so,
z is essentially positive), so that integrals like
∞
0 f (z)z
n dz are finite for any finite
non-negative n, to guarantee finiteness of the theory (in principle in some case this
requirement is weakened, if the theory is required to be not finite but only renormalizable). Third, the f (z) is required to be so-called entire function, i.e. it cannot be
presented in the form of a product of primitive multipliers like (z − a 1 )(z − a 2 ) . . .,
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
A. Petrov, Introduction to Modified Gravity, SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-52862-1_6
55
Nonlocal Gravity
6.1 Motivations
As we have noted several times along this review, the main problem of various
gravity models is the development of a consistent quantum description. Indeed, the
Einstein gravity is non-renormalizable, and introduction of higher-derivative additive
terms implies in arising of ghosts. We have argued in the previous chapter that
the Horava–Lifshitz gravity seems to be a good solution since it is power-counting
renormalizable, and ghosts ate absent since the action involves only second time
derivatives. However, the HL gravity, first, is very complicated, second, breaks the
Lorentz symmetry strongly, third, displays a problem of extra degrees of freedom
whose solving, as we noted, requires special efforts. At the same time, the concept
of nonlocality developed originally within phenomenological context in order to
describe finite-size effects (see f.e. [107]), began to attract the interest. Besides of
this, the nonlocality enjoys also a stringy motivation since the factors like e
emerge
naturally within the string context [108]. The key idea of nonlocal field theories looks
like follows. Let us consider for example the free scalar field whose Lagrangian is
L =
1
2
φ f (//
2
)φ,
(6.1)
where f (z) is a some non-polynomial function (with is the characteristic nonlocality scale) which we choose to satisfy the following requirements.
First, at small arguments this function should behave as f (z) = a + z, in order
to provide the correct + m
2 IR asymptotic behavior. Second, this function must
decay rapidly at |z| → ∞ (in principle, we can consider only Euclidean space, so,
z is essentially positive), so that integrals like
∞
0 f (z)z
n dz are finite for any finite
non-negative n, to guarantee finiteness of the theory (in principle in some case this
requirement is weakened, if the theory is required to be not finite but only renormalizable). Third, the f (z) is required to be so-called entire function, i.e. it cannot be
presented in the form of a product of primitive multipliers like (z − a 1 )(z − a 2 ) . . .,
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
A. Petrov, Introduction to Modified Gravity, SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-52862-1_6
55