Chapter 7
Summary
We discussed various modifications of gravity introduced within the framework of
the metric formalism. As we noted, in principle there are two fundamental problems
to be solved by desired modifications of gravity: first, explanation of the cosmic
acceleration, second, development of a theory consistent from the quantum viewpoint. Within the models we presented, different attempts to solve these problems
are taken. It turns out to be that the problem of cosmic acceleration is solved by
many extensions of gravity, and actually the main issue in this context consists in
finding the theory fitting better the observational results (for discussion of cosmological constraining of gravitational models, see f.e. [125] and many other papers).
At the same time, the problem of formulating a perturbatively consistent gravity
theory appears to be much more complicated. While the simplest way to construct
the renormalizable gravity model is based on introducing higher-derivative terms,
this manner suffers from the problem of arising ghost states. To solve the problem of
ghosts, one can follow two ways: either break Lorentz symmetry in a strong manner
introducing the HL gravity (effectively it means that we have a higher-derivative
regularization in a spatial sector only) paying a price of arising a very complicated
theory, and moreover, treating the Lorentz symmetry as an essentially low-energy
phenomenon, or introduce nonlocality which allows to achieve renormalizability or
even to rule out divergences, but, in this case, solving the problems of unitarity and
causality would require special efforts.
One more way to solve the problem of renormalizability of the gravity is based on
its supersymmetric extension. As it is well known, supersymmetric extension of any
field theory improves essentially its ultraviolet behavior since so-called “miraculous
cancellations” of UV divergences occur [126]. It is well known that the mechanism of
these cancellation is very simple—since fermionic contributions carry an extra minus
sign, under an appropriate relations between coupling constants occurring due to the
supersymmetry, some of fermionic divergent contributions cancel bosonic divergent
contributions (for example, while the φ
4 theory and Yukawa model display quadratic
divergences, the Wess-Zumino model involving these theories as ingredients displays
© 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_7
65
Summary
We discussed various modifications of gravity introduced within the framework of
the metric formalism. As we noted, in principle there are two fundamental problems
to be solved by desired modifications of gravity: first, explanation of the cosmic
acceleration, second, development of a theory consistent from the quantum viewpoint. Within the models we presented, different attempts to solve these problems
are taken. It turns out to be that the problem of cosmic acceleration is solved by
many extensions of gravity, and actually the main issue in this context consists in
finding the theory fitting better the observational results (for discussion of cosmological constraining of gravitational models, see f.e. [125] and many other papers).
At the same time, the problem of formulating a perturbatively consistent gravity
theory appears to be much more complicated. While the simplest way to construct
the renormalizable gravity model is based on introducing higher-derivative terms,
this manner suffers from the problem of arising ghost states. To solve the problem of
ghosts, one can follow two ways: either break Lorentz symmetry in a strong manner
introducing the HL gravity (effectively it means that we have a higher-derivative
regularization in a spatial sector only) paying a price of arising a very complicated
theory, and moreover, treating the Lorentz symmetry as an essentially low-energy
phenomenon, or introduce nonlocality which allows to achieve renormalizability or
even to rule out divergences, but, in this case, solving the problems of unitarity and
causality would require special efforts.
One more way to solve the problem of renormalizability of the gravity is based on
its supersymmetric extension. As it is well known, supersymmetric extension of any
field theory improves essentially its ultraviolet behavior since so-called “miraculous
cancellations” of UV divergences occur [126]. It is well known that the mechanism of
these cancellation is very simple—since fermionic contributions carry an extra minus
sign, under an appropriate relations between coupling constants occurring due to the
supersymmetry, some of fermionic divergent contributions cancel bosonic divergent
contributions (for example, while the φ
4 theory and Yukawa model display quadratic
divergences, the Wess-Zumino model involving these theories as ingredients displays
© 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_7
65