- #1

MTd2

Gold Member

- 2,028

- 25

So, is there a torsionon?

I tried to sort out something, but the only thing I could find was paranormal stuff, unfortunately. So, I at least I came up with this, although no mention to torsion as a particle:

http://arxiv.org/abs/gr-qc/0505081 (equation 4)

Alejandro Perez, Carlo Rovelli

(Submitted on 17 May 2005 (v1), last revised 19 Aug 2005 (this version, v2))

The Immirzi parameter is a constant appearing in the general relativity action used as a starting point for the loop quantization of gravity. The parameter is commonly believed not to show up in the equations of motion, because it appears in front of a term in the action that vanishes on shell. We show that in the presence of fermions, instead, the Immirzi term in the action does not vanish on shell, and the Immirzi parameter does appear in the equations of motion. It determines the coupling constant of a four-fermion interaction. Therefore the Immirzi parameter leads to effects that are observable in principle, even independently from nonperturbative quantum gravity.

http://arxiv.org/abs/gr-qc/0610026v3

Simone Mercuri

(Submitted on 6 Oct 2006 (v1), last revised 9 Mar 2007 (this version, v3))

In order to introduce an interaction between gravity and fermions in the Ashtekar-Barbero-Immirzi formalism without affecting classical dynamics a non-minimal term is necessary. The non-minimal term together with the Holst modification to the Hilbert-Palatini action reconstruct the Nieh-Yan invariant. As a consequence the Immirzi parameter, differently from the minimal coupling approach, does not affect the classical dynamics, which is described by the Einstein-Cartan action.

http://arxiv.org/abs/0710.2534v2

S.I. Kruglov

(Submitted on 12 Oct 2007 (v1), last revised 3 Jan 2008 (this version, v2))

We consider axial torsion fields which appear in higher derivative quantum gravity. It is shown, in general, that the torsion field possesses states with two spins, one and zero, with different masses. The first-order formulation of torsion fields is performed. Projection operators extracting pure spin and mass states are given. We obtain the Lagrangian in the framework of the first order formalism and energy-momentum tensor. The effective interaction of torsion fields with electromagnetic fields is discussed. The Hamiltonian form of the first order torsion field equation is given.

http://arxiv.org/abs/0912.2435

She-Sheng Xue

(Submitted on 12 Dec 2009 (v1), last revised 21 Feb 2010 (this version, v2))

This article presents detailed discussions and calculations of the recent letter "Quantum Regge Calculus of Einstein-Cartan theory" in Phys. Lett. B682 (2009) 300 [arXiv:0902.3407]. The Euclidean space-time is discretized by a 4-simplices complex. We adopt basic tetrad and spin-connection fields to describe the 4-simplices complex. Introducing diffeomorphism and local Lorentz invariant holonomy fields, we study a regularized Einstein-Cartan theory for the quantum dynamics of the 4-simplices complex and fermions. This regularized Einstein-Cartan action is shown to properly approaches to its continuum counterpart in the continuum limit. Based on the local Lorentz invariance, we derive the dynamical equations satisfied by invariant holonomy fields. In the mean-field approximation, we show the averaged size of 4-simplex, the element of the 4-simplices complex, has to be larger than the Planck length. This formulation provides a theoretical framework for analytical calculations and numerical simulations to study the quantum Einstein-Cartan theory.

http://arxiv.org/abs/0902.3407

She-Sheng Xue

(Submitted on 19 Feb 2009 (v1), last revised 12 Nov 2009 (this version, v4))

We study the Quantum Regge Calculus of Einstein-Cartan theory to describe quantum dynamics of Euclidean space-time discretized as a 4-simplices complex. Tetrad field e_\mu(x) and spin-connection field \omega_\mu(x) are assigned to each 1-simplex. Applying the torsion-free Cartan structure equation to each 2-simplex, we discuss parallel transports and construct a diffeomorphism and {\it local} gauge-invariant Einstein-Cartan action. Invariant holonomies of tetrad and spin-connection fields along large loops are also given. Quantization is defined by a bounded partition function with the measure of SO(4)-group valued \omega_\mu(x) fields and Dirac-matrix valued e_\mu(x) fields over 4-simplices complex.

http://arxiv.org/abs/hep-th/0103093v1

I.L. Shapiro

(Submitted on 13 Mar 2001)

We review many quantum aspects of torsion theory and discuss the possibility of the space-time torsion to exist and to be detected. The paper starts, in Chapter 2, with an introduction to the classical gravity with torsion, that includes also interaction of torsion with matter fields. In Chapter 3, the renormalization of quantum theory of matter fields and related topics, like renormalization group, effective potential and anomalies, are considered. Chapter 4 is devoted to the action of particles in a space-time with torsion, and to possible physical effects generated by the background torsion. In particular, we review the upper bounds for the background torsion which are known from the literature. In Chapter 5, the comprehensive study of the possibility of a theory for the propagating completely antisymmetric torsion field is presented. We show, that the propagating torsion may be consistent with the principles of quantum theory only in the case when the torsion mass is much greater than the mass of the heaviest fermion coupled to torsion. Then, universality of the fermion-torsion interaction implies that torsion itself has a huge mass, and can not be observed in realistic experiments. In Chapter 6, we briefly discuss the string-induced torsion and the possibility to induce torsion action and torsion itself through the quantum effects of matter fields.

http://arxiv.org/abs/0711.4209v1

Yakov Itin (Institute of Mathematics, Hebrew University of Jerusalem, and Jerusalem College of Technology

(Submitted on 27 Nov 2007)

The possible extensions of GR for description of fermions on a curved space, for supergravity and for loop quantum gravity require a richer set of 16 independent variables. These variables can be assembled in a coframe field, i.e., a local set of four linearly independent 1-forms. In the ordinary formulation, the coframe gravity does not have any connection to a specific geometry even being constructed from the geometrical meaningful objects. A geometrization of the coframe gravity is an aim of this paper.

We construct a complete class of the coframe connections which are linear in the first order derivatives of the coframe field on an $n$ dimensional manifolds with and without a metric. The subclasses of the torsion-free, metric-compatible and flat connections are derived. We also study the behavior of the geometrical structures under local transformations of the coframe. The remarkable fact is an existence of a subclass of connections which are invariant when the infinitesimal transformations satisfy the Maxwell-like system of equations. In the framework of the coframe geometry construction, we propose a geometrical action for the coframe gravity. It is similar to the Einstein-Hilbert action of GR, but the scalar curvature is constructed from the general coframe connection. We show that this geometric Lagrangian is equivalent to the coframe Lagrangian up to a total derivative term. Moreover there is a family of coframe connections which Lagrangian does not include the higher order terms at all. In this case, the equivalence is complete.

http://arxiv.org/abs/0711.1535v1

Friedrich W. Hehl, Yuri N. Obukhov

(Submitted on 9 Nov 2007)

We review the application of torsion in field theory. First we show how the notion of torsion emerges in differential geometry. In the context of a Cartan circuit, torsion is related to translations similar as curvature to rotations. Cartan's investigations started by analyzing Einsteins general relativity theory and by taking recourse to the theory of Cosserat continua. In these continua, the points of which carry independent translational and rotational degrees of freedom, there occur, besides ordinary (force) stresses, additionally spin moment stresses. In a 3-dimensional continuized crystal with dislocation lines, a linear connection can be introduced that takes the crystal lattice structure as a basis for parallelism. Such a continuum has similar properties as a Cosserat continuum, and the dislocation density is equal to the torsion of this connection. Subsequently, these ideas are applied to 4-dimensional spacetime. A translational gauge theory of gravity is displayed (in a Weitzenboeck or teleparallel spacetime) as well as the viable Einstein-Cartan theory (in a Riemann-Cartan spacetime). In both theories, the notion of torsion is contained in an essential way. Cartan's spiral staircase is described as a 3-dimensional Euclidean model for a space with torsion, and eventually some controversial points are discussed regarding the meaning of torsion.

I tried to sort out something, but the only thing I could find was paranormal stuff, unfortunately. So, I at least I came up with this, although no mention to torsion as a particle:

http://arxiv.org/abs/gr-qc/0505081 (equation 4)

**Physical effects of the Immirzi parameter**Alejandro Perez, Carlo Rovelli

(Submitted on 17 May 2005 (v1), last revised 19 Aug 2005 (this version, v2))

The Immirzi parameter is a constant appearing in the general relativity action used as a starting point for the loop quantization of gravity. The parameter is commonly believed not to show up in the equations of motion, because it appears in front of a term in the action that vanishes on shell. We show that in the presence of fermions, instead, the Immirzi term in the action does not vanish on shell, and the Immirzi parameter does appear in the equations of motion. It determines the coupling constant of a four-fermion interaction. Therefore the Immirzi parameter leads to effects that are observable in principle, even independently from nonperturbative quantum gravity.

http://arxiv.org/abs/gr-qc/0610026v3

**Nieh-Yan Invariant and Fermions in Ashtekar-Barbero-Immirzi Formalism**Simone Mercuri

(Submitted on 6 Oct 2006 (v1), last revised 9 Mar 2007 (this version, v3))

In order to introduce an interaction between gravity and fermions in the Ashtekar-Barbero-Immirzi formalism without affecting classical dynamics a non-minimal term is necessary. The non-minimal term together with the Holst modification to the Hilbert-Palatini action reconstruct the Nieh-Yan invariant. As a consequence the Immirzi parameter, differently from the minimal coupling approach, does not affect the classical dynamics, which is described by the Einstein-Cartan action.

http://arxiv.org/abs/0710.2534v2

**On Torsion Fields in Higher Derivative Quantum Gravity**S.I. Kruglov

(Submitted on 12 Oct 2007 (v1), last revised 3 Jan 2008 (this version, v2))

We consider axial torsion fields which appear in higher derivative quantum gravity. It is shown, in general, that the torsion field possesses states with two spins, one and zero, with different masses. The first-order formulation of torsion fields is performed. Projection operators extracting pure spin and mass states are given. We obtain the Lagrangian in the framework of the first order formalism and energy-momentum tensor. The effective interaction of torsion fields with electromagnetic fields is discussed. The Hamiltonian form of the first order torsion field equation is given.

http://arxiv.org/abs/0912.2435

**On Quantum Regge Calculus of Einstein-Cartan Theory**She-Sheng Xue

(Submitted on 12 Dec 2009 (v1), last revised 21 Feb 2010 (this version, v2))

This article presents detailed discussions and calculations of the recent letter "Quantum Regge Calculus of Einstein-Cartan theory" in Phys. Lett. B682 (2009) 300 [arXiv:0902.3407]. The Euclidean space-time is discretized by a 4-simplices complex. We adopt basic tetrad and spin-connection fields to describe the 4-simplices complex. Introducing diffeomorphism and local Lorentz invariant holonomy fields, we study a regularized Einstein-Cartan theory for the quantum dynamics of the 4-simplices complex and fermions. This regularized Einstein-Cartan action is shown to properly approaches to its continuum counterpart in the continuum limit. Based on the local Lorentz invariance, we derive the dynamical equations satisfied by invariant holonomy fields. In the mean-field approximation, we show the averaged size of 4-simplex, the element of the 4-simplices complex, has to be larger than the Planck length. This formulation provides a theoretical framework for analytical calculations and numerical simulations to study the quantum Einstein-Cartan theory.

http://arxiv.org/abs/0902.3407

**Quantum Regge Calculus of Einstein-Cartan theory**She-Sheng Xue

(Submitted on 19 Feb 2009 (v1), last revised 12 Nov 2009 (this version, v4))

We study the Quantum Regge Calculus of Einstein-Cartan theory to describe quantum dynamics of Euclidean space-time discretized as a 4-simplices complex. Tetrad field e_\mu(x) and spin-connection field \omega_\mu(x) are assigned to each 1-simplex. Applying the torsion-free Cartan structure equation to each 2-simplex, we discuss parallel transports and construct a diffeomorphism and {\it local} gauge-invariant Einstein-Cartan action. Invariant holonomies of tetrad and spin-connection fields along large loops are also given. Quantization is defined by a bounded partition function with the measure of SO(4)-group valued \omega_\mu(x) fields and Dirac-matrix valued e_\mu(x) fields over 4-simplices complex.

http://arxiv.org/abs/hep-th/0103093v1

**Physical Aspects of the Space-Time Torsion**I.L. Shapiro

(Submitted on 13 Mar 2001)

We review many quantum aspects of torsion theory and discuss the possibility of the space-time torsion to exist and to be detected. The paper starts, in Chapter 2, with an introduction to the classical gravity with torsion, that includes also interaction of torsion with matter fields. In Chapter 3, the renormalization of quantum theory of matter fields and related topics, like renormalization group, effective potential and anomalies, are considered. Chapter 4 is devoted to the action of particles in a space-time with torsion, and to possible physical effects generated by the background torsion. In particular, we review the upper bounds for the background torsion which are known from the literature. In Chapter 5, the comprehensive study of the possibility of a theory for the propagating completely antisymmetric torsion field is presented. We show, that the propagating torsion may be consistent with the principles of quantum theory only in the case when the torsion mass is much greater than the mass of the heaviest fermion coupled to torsion. Then, universality of the fermion-torsion interaction implies that torsion itself has a huge mass, and can not be observed in realistic experiments. In Chapter 6, we briefly discuss the string-induced torsion and the possibility to induce torsion action and torsion itself through the quantum effects of matter fields.

http://arxiv.org/abs/0711.4209v1

**Coframe geometry and gravity**Yakov Itin (Institute of Mathematics, Hebrew University of Jerusalem, and Jerusalem College of Technology

(Submitted on 27 Nov 2007)

The possible extensions of GR for description of fermions on a curved space, for supergravity and for loop quantum gravity require a richer set of 16 independent variables. These variables can be assembled in a coframe field, i.e., a local set of four linearly independent 1-forms. In the ordinary formulation, the coframe gravity does not have any connection to a specific geometry even being constructed from the geometrical meaningful objects. A geometrization of the coframe gravity is an aim of this paper.

We construct a complete class of the coframe connections which are linear in the first order derivatives of the coframe field on an $n$ dimensional manifolds with and without a metric. The subclasses of the torsion-free, metric-compatible and flat connections are derived. We also study the behavior of the geometrical structures under local transformations of the coframe. The remarkable fact is an existence of a subclass of connections which are invariant when the infinitesimal transformations satisfy the Maxwell-like system of equations. In the framework of the coframe geometry construction, we propose a geometrical action for the coframe gravity. It is similar to the Einstein-Hilbert action of GR, but the scalar curvature is constructed from the general coframe connection. We show that this geometric Lagrangian is equivalent to the coframe Lagrangian up to a total derivative term. Moreover there is a family of coframe connections which Lagrangian does not include the higher order terms at all. In this case, the equivalence is complete.

http://arxiv.org/abs/0711.1535v1

**Elie Cartan's torsion in geometry and in field theory, an essay**Friedrich W. Hehl, Yuri N. Obukhov

(Submitted on 9 Nov 2007)

We review the application of torsion in field theory. First we show how the notion of torsion emerges in differential geometry. In the context of a Cartan circuit, torsion is related to translations similar as curvature to rotations. Cartan's investigations started by analyzing Einsteins general relativity theory and by taking recourse to the theory of Cosserat continua. In these continua, the points of which carry independent translational and rotational degrees of freedom, there occur, besides ordinary (force) stresses, additionally spin moment stresses. In a 3-dimensional continuized crystal with dislocation lines, a linear connection can be introduced that takes the crystal lattice structure as a basis for parallelism. Such a continuum has similar properties as a Cosserat continuum, and the dislocation density is equal to the torsion of this connection. Subsequently, these ideas are applied to 4-dimensional spacetime. A translational gauge theory of gravity is displayed (in a Weitzenboeck or teleparallel spacetime) as well as the viable Einstein-Cartan theory (in a Riemann-Cartan spacetime). In both theories, the notion of torsion is contained in an essential way. Cartan's spiral staircase is described as a 3-dimensional Euclidean model for a space with torsion, and eventually some controversial points are discussed regarding the meaning of torsion.

Last edited: