Extended Theories of Gravity

Over recent decades, the role of torsion in gravity has been extensively investigated along the main direction of bringing gravity closer to its gauge formulation and incorporating spin in a geometric description. Here we review various torsional constructions, from teleparallel, to Einstein-Cartan, and metric-affine gauge theories, resulting in extending torsional gravity in the paradigm of f (T) gravity, where f (T) is an arbitrary function of the torsion scalar. Based on this theory, we further review the corresponding cosmological and astrophysical applications. In particular, we study cosmological solutions arising from f (T) gravity, both at the background and perturbation levels, in different eras along the cosmic expansion. The f (T) gravity construction can provide a theoretical interpretation of the late-time universe acceleration, alternative to a cosmological constant, and it can easily accommodate with the regular thermal expanding history including the radiation and cold dark matter dominated phases. Furthermore, if one traces back to very early times, for a certain class of f (T) models, a sufficiently long period of inflation can be achieved and hence can be investigated by cosmic microwave background observations-or, alternatively, the Big Bang singularity can be avoided at even earlier moments due to the appearance of non-singular bounces. Various observational constraints, especially the bounds coming from the large-scale structure data in the case of f (T) cosmology, as well as the behavior of gravitational waves, are described in detail. Moreover, the spherically symmetric and black hole solutions of the theory are reviewed. Additionally, we discuss various extensions of the f (T) paradigm. Finally, we consider the relation with other modified gravitational theories, such as those based on curvature, like f (R) gravity, trying to illuminate the subject of which formulation, or combination of formulations, might be more suitable for quantization ventures and cosmological applications.

f(T) teleparallel gravity and cosmology

Over the past decades, the role of torsion in gravity has been extensively investigated along the main direction of bringing gravity closer to its gauge formulation and incorporating spin in a geometric description. Here we review various torsional constructions, from teleparallel, to Einstein-Cartan, and metric-affine gauge theories, resulting in extending torsional gravity in the paradigm of f(T) gravity, where f(T) is an arbitrary function of the torsion scalar. Based on this theory, we further review the corresponding cosmological and astrophysical applications. In particular, we study cosmological solutions arising from f(T) gravity, both at the background and perturbation levels, in different eras along the cosmic expansion. The f(T) gravity construction can provide a theoretical interpretation of the late-time universe acceleration, and it can easily accommodate with the regular thermal expanding history including the radiation and cold dark matter dominated phases. Furthermore, if one traces back to very early times, a sufficiently long period of inflation can be achieved and hence can be investigated by cosmic microwave background observations, or alternatively, the Big Bang singularity can be avoided due to the appearance of non-singular bounces. Various observational constraints, especially the bounds coming from the large-scale structure data in the case of f(T) cosmology, as well as the behavior of gravitational waves, are described in detail. Moreover, the spherically symmetric and black hole solutions of the theory are reviewed. Additionally, we discuss various extensions of the f(T) paradigm. Finally, we consider the relation with other modified gravitational theories, such as those based on curvature, like f(R) gravity, trying to enlighten the subject of which formulation might be more suitable for quantization ventures and cosmological applications.

There is presented a particularly simple transformation of the Schwarzschild metric into new coordinates, whereby the "spherical singularity" is removed and the maximal singularity-free extension is clearly exhibited.

Maximal Extension of Schwarzschild Metric

Even and odd coherent states and excitations of a singular oscillator

Use of polar coordinates is examined in performing summation over all Feynman histories. Several relationships for the Lagrangian path integral and the Hamiltonian path integral are derived in the central‐force problem. Applications are made for a harmonic oscillator, a charged particle in a uniform magnetic field, a particle in an inverse‐square potential, and a rigid rotator. Transformations from Cartesian to polar coordinates in path integrals are rather different from those in ordinary calculus and this complicates evaluation of path integrals in polars. However, it is observed that for systems of central symmetry use of polars is often advantageous over Cartesians.

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Summation over Feynman Histories in Polar Coordinates

The Lagrangian path integral for the hydrogen atom is calculated exactly by rescaling paths and performing the Kustaanheimo-Stiefel transformation in each short time integral.

Exact-path-integral treatment of the hydrogen atom

An exact path-integral treatment of the s states for the Hulth\'en potential is presented. A procedure of the nontrivial change of variable accompanied by the local time rescaling is given in detail. The dimensional extension trick is applied to convert the radial path integral into a path integral in the SU(2) manifold. The exact energy spectrum and the normalized s-state eigenfunctions are obtained from the poles of the Green function and their residues, respectively. The Yukawa and the Coulomb limits are also discussed.

Path-integral treatment of the Hulthén potential

Alternative exact-path-integral treatment of the hydrogen atom

Path integrals with a periodic constraint ∫θ ds =Θ+2πn (n=integer) are studied. In particular, the path integral for a string entangled around a singular point in two dimensions is evaluated in polar coordinates. Applications are made for the entangled polymers with and without interactions, the Aharonov–Bohm effect, and the angular momentum projection of a spinning top.

Akira Inomata

Papers

Summation over Feynman Histories in Polar Coordinates

Exact-path-integral treatment of the hydrogen atom

Path-integral treatment of the Hulthén potential

Alternative exact-path-integral treatment of the hydrogen atom

Path integrals with a periodic constraint: Entangled strings