Spin and torsion in the very early universe

In the very early universe with temperature T between 10²⁴ K and 10³² K the gravitational effect of torsion is dominant if particles with spin are sufficiently polarized. The source of the torsion is the spin density and the latter is usually described by a classical theory of Weyssenhoff and Raabe. In this article the spinning particles are described quantum mechanically, i.e. with a Dirac field and the spin density is defined as the source of the torsion. The macroscopic average of the spin density is obtained by the relativistic Wigner function formalism. The expression of the spin density, as derived in this article, is different from the classical one, except when both are zero.     

Spin and torsion in general relativity: I. Foundations

From physical arguments space-time is assumed to possess a connection \(\Gamma _{ij}^k = \left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\} + S_{ij}^{ k} - S_{j i}^{ k} + S_{ ij}^k = \left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\} - K_{ij}^{ k} \) . \(\left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\}\) is Christoffel's symbol built up from the metric g _ij and already appearing in General Relativity (GR). Cartan's torsion tensor \(S_{ij} ^k = \tfrac{1}{2}(\Gamma _{ij}^k - \Gamma _{ji}^k )\) and the contortion tensor K _ij ^k , in contrast to the theory presented here, both vanish identically in conventional GR. Using the connection introduced above in this series of articles, we will discuss the consequences for GR in the framework of a consistent formalism. There emerges a theory describing, in a unified way, gravitation and a very weak spin-spin contact interaction. In section 1 we start with the well-known dynamical definition of the energy-momentum tensor σ ^ij ～ δ?/δg _ij , where ? represents the Lagrangian density of matter (section1.1). In sections1.2,3 we will show that due to geometrical reasons, the connection assumed above leads to a dynamical definition of the spin-angular momentum tensor according to τ^k _ji ～ δ?/δK _ij ^k . In section1.4, by an ideal experiment, it will become clear that spin prohibits the introduction of an instantaneous rest system and thereby of a geodesic coordinate system. Among other things in section1.5 there are some remarks about the rôle torsion played in former physical theories. In section 2 we sketch the content of the theory. As in GR, the action function is the sum of the material and the field action function (sections2.1,2). The extension of GR consists in the introduction of torsion S _ij ^k as a new field. By variation of the action function with respect to metric and torsion we obtain the field equations in a general form (section2.3). They are also valid for matter described by spinors; in this case, however, one has to introduce tetrads as anholonomic coordinates and slightly to generalize the dynamical definition of energy-momentum (sections2.4,5).     

Spin and torsion in general relativity II: Geometry and field equations

From physical arguments space-time is assumed to possess a connection is Christoffel's symbol built up from the metric g _ij and already appearing in General Relativity (GR). Cartan's torsion tensor and the contortion tensor K _ij ^k , in contrast to the theory presented here, both vanish identically in conventional GR. Using the connection introduced above, in this series of articles we will discuss the consequences for GR in the framework of a consistent formalism. There emerges a theory describing in a unified way gravitation and a very weakspin-spin contact interaction. In Part I of this work† we discussed the foundations of the theory. In this Part II we present in section 3 the geometrical apparatus necessary for the formulation of the theory. In section 4 we take the curvature scalar (or rather its density) as Lagrangian density of the field. In this way we obtain in subsection 4.1 the field equations in their explicit form. In particular it turns out that torsion is essentially proportional to spin. We then derive the angular momentum and the energy-momentum theorems (subsections 4.2-4); the latter yields a force proportional to curvature, acting on any matter with spin. In subsection 4.5 we compare the theory so far developed with GR. Torsion leads to a universal spin-spin contact     

Spin forces in mesons made up of quarks of different mass

With the help of a variational method we apply the standard Coulomb + Linear potential to the analysis of mesons constructed from quarks of different mass. Fine and hyperfine splittings are discussed, with particular emphasis on their asymptotic behaviour. Striking differences in this behaviour are predicted to occur compared with the case of mesons built up from equal mass quarks, with the spin-orbit mixing force playing here a fundamental role. Such predictions may be tested within the bottom \((b\bar u)\) meson family. Our considerations are also extended to the strange and charmed mesons. Our model favours ak(O⁺) below theK ^* (1.430), and predicts a partial inversion in the ordering of theP-wave states of the charmed mesons \((c\bar u)\) .     

Spin q , Twistor and Spin c

Spin ^q structures induce (Spin ^q style) twistor spaces, which possess canonical Spin ^c structures. Such structures produce Dirac operators. Their indices for the even dimensional case, and the adiabatic limit of their reduced η-invariants for the odd dimensional case, are discussed. Received: 3 October 1995 / Accepted: 2 March 1997     

Representation of Spin Group Spin（p, q）

The representation η（P, q） of spin group Spin（p, q） in any dimensional space is given by induction, and the relation between two representations, which are obtained in two kinds of inductions from Spin（p, q） to Spin（p ＋ 1, q ＋ 1） are studied.     

Gravitation,torsion, and electromagnetism

A term bilinear in the derivative of the torsion is added to the Lagrangian of general relativity to produce torsion that propagates. Using standard variational techniques, field equations are derived with the torsion being interpreted as the electromagnetic potential and the antisymmetric part of the Ricci tensor as the electromagnetic field tensor. The equation of motion is derived from the field equations, and the results are compared to the Einstein-Maxwell formulation.     

Determinants,torsion, and strings

We apply the results of [BF1, BF2] on determinants of Dirac operators to String Theory. For the bosonic string we recover the “holomorphic factorization” of Belavin and Knizhik. Witten's global anomaly formula is used to give sufficient conditions for anomaly cancellation in the heterotic string (for arbitrary background spacetimes). To prove the latter result we develop certain torsion invariants related to characteristic classes of vector bundles and to index theory.     

Spin, gravity, and inertia

The gravitational effects in the relativistic quantum mechanics are investigated. The exact Foldy-Wouthuysen transformation is constructed for the Dirac particle coupled to the static spacetime metric. As a direct application, we analyze the nonrelativistic limit of the theory. The new term describing the specific spin (gravitational moment) interaction effect is recovered in the Hamiltonian. The comparison of the true gravitational coupling with the purely inertial case demonstrates that the spin relativistic effects do not violate the equivalence principle for the Dirac fermions.     

Gravitation,torsion, and string theory

The low energy effective Lagrangian of string theory presents us with not only gravity, but the dilaton and the antisymmetric field as well. It is shown that a Brans-Dicke generalization of a metric theory of gravity with torsion that is derived as the exterior derivative of a potential is equivalent to the low energy string theory Lagrangian. This gives all of the fields a physical interpretation in four dimensions and provides an indication that the dilaton representssmall corrections to general relativity at large distances.     

Planck's constant,torsion, and space-time defects

A possible way of building Planck's constant into the structure of space-time is considered. This is done by assuming that the torsional defect that intrinsic spin produces in the geometry is a multiple of the Planck length.     

Quantized space-time,torsion, and magnetic monopoles

Within the tetrad formalism we introduce quantized space-time in the curvilinear case by using general coordinate transformations with noncommuting terms. Fermion and boson fields are studied and the affine connection is also defined in this space. It is shown that space-time torsion and magnetic monopoles appear as consequences of the theory with quantized space-time at small distances. This method may open a new way of understanding topological structure of space-time.