Four Lectures on Poincaré Gauge Field Theory

Abstract

The Poincaré (inhomogeneous Lorentz) group underlies special relativity. In these lectures a consistent formalism is developed allowing an appropriate gauging of the Poincaré group. The physical laws are formulated in terms of points, orthonormal tetrad frames , and components of the matter fields with respect to these frames. The laws are postulated to be gauge invariant under local Poincaré transformations. This implies the existence of 4 translational gauge potentials e α ("gravitons") and 6 Lorentz gauge potentials Γ αβ ("rotons") and the coupling of the momentum current and the spin current of matter to these potentials, respectively. In this way one is led to a Riemann-Cartan spacetime carrying torsion and curvature, richer in structure than the spacetime of general relativity. The Riemann-Cartan spacetime is controlled by the two general gauge field equations (3.44) and (3.45), in which material momentum and spin act as sources. The general framework of the theory is summarized in a table in Section 3.6. -Options for picking a gauge field lagrangian are discussed (teleparallelism, ECSK). We propose a lagrangian quadratic in torsion and curvature governing the propagation of gravitons and rotons. A suppression of the rotons leads back to general relativity.Author's note. This work was originally published as a chapter in the book that is now long out of print. § The purpose of this arXiv version is to make these lectures more accessible to the current generation of students and researchers. I am extremely grateful to my colleague and friend Milutin Blagojević (Belgrade) for arranging the republication of my Erice lectures. Moreover, I'd like to thank his secretary Vanja Mihajlović for putting the text most carefully into latex. For a more modern look at that subject, see M.B. & F.W.H. (eds.), Gauge Theories of Gravitation, Imperial College Press, London (2013).

Andrei Borisovich Shidlovskii (on his eightieth birthday)

“…Generalizing this ansatz to v = 0, the simplest choice f c = 0 with an arbitrary function v(φ, Y ) (deformed rigid supersymmetry, DRS) yields 20) and thus for P IJ P ab = v ab , (5.21) Of course, in order to describe flat space-time, corresponding to the Poisson tensor of rigid supersymmetry, one has to set v(φ, Y ) = 0. Then the singularity at Y = 0 in the extended Poisson tensor disappears.…”

Teleparallel Poincaré cosmology and ΛCDM model

“…Moreover, since a proper tetrad field used to construct a teleparallel structure cannot be uniquely specified by a given metric, the lack of local Lorentz symmetry will cause difficulties in formulating a satisfactory cosmological model to account for the late time accelerated expansion [22]. In light of the fact that the Weitzenboöck connection is simply a special choice of the affine structure to make the curvature tensor identically null, to avoid those defects of the f (T ) gravity while retaining the notion of absolute parallelism, we thus resort to the more general gravitational theory: the Poincaré gauge theory (PGT) of gravity [23][24][25].…”

Heat Invariant E 2 for Nonminimal Operator on Manifolds with Torsion

“…The torsion arises naturally in the different (based on Poincaré and affine groups) gauge theories of gravity developed in the recent years (see Refs. [19,20]). Moreover, all kinds of modern superstring theories [21] (for the recent review see e. g. [22]), which allow to deduce the properties of space-time, also predict, along with the metric, the existence of torsion.…”