# Is there a quatum of torsion in Cartan Einstein Gravity? What is its spin?

• MTd2
In summary, the Immirzi parameter, a constant used in loop quantization of gravity, appears in the equations of motion when fermions are present. The parameter determines the coupling constant of a four-fermion interaction and can lead to observable effects. The Nieh-Yan invariant and fermions in the Ashtekar-Barbero-Immirzi formalism can be described without affecting classical dynamics. Torsion fields in higher derivative quantum gravity possess states with different masses and spins. Quantum Regge Calculus of Einstein-Cartan theory provides a theoretical framework for studying quantum dynamics of the 4-simplices complex and fermions. The possibility of a theory for propagating torsion fields is discussed, but it is predicted
MTd2
Gold Member
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)

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.

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Naively I would say that in normal Einstein-Cartan theory you use the R(P)=0 constraint to interpret the gauge field of local Lorentz transformations as a spin connection and get rid of the local spacetime translations. This sets the torsion to zero via the vielbein postulate.

haushofer said:
Naively I would say that in normal Einstein-Cartan theory you use the R(P)=0 constraint to interpret the gauge field of local Lorentz transformations as a spin connection and get rid of the local spacetime translations. This sets the torsion to zero via the vielbein postulate.

Thanks for the very interesting articles. So far I understand correctly, the Einstein Cartan theory contains not only flat configurations. Or did I miss something important? The main question seems to be: do we have to introduce the torsion or not within the theory of gravity? Is it pertinent? does it bring something relavant? My conviction is yes.

I think the answer to this is the most obvious one. All your doing by introducing torsion is allowing the gravitational field to couple to spin. Therefore I'd expect that gravitons can then transfer spin via torsion at least at a perturbative level E<<Mpl. So the quanta is the spin-2 graviton. Of coarse when we have E ~ Mpl the notion of particles breaks down so we need some non-perturbative formulation.

Does anyone know how the spin-2 nature of gravity can be seen in spin foams or LQG? I guess this is related to the semi-classical limit?

Finbar said:
... All your doing by introducing torsion is allowing the gravitational field to couple to spin. Therefore I'd expect that gravitons can then transfer spin via torsion at least at a perturbative level E<<Mpl. So the quanta is the spin-2 graviton. Of coarse when we have E ~ Mpl the notion of particles breaks down so we need some non-perturbative formulation...

A naive question: "What would mean a quantized torsion? Something like Tabc = N. h (Planck) where T is the torsion and N = 1, 2...?"

afaik torsion is different from curvature as torsion is exactly zero in vacuum = outside matter distribution; the torsion constraint can be solved algebraically, therefore torsion becomes a function of matter variables which can be determined explicitly.

That means torsion does not propagate and therefore there are no torsion waves.

tom.stoer said:
afaik torsion is different from curvature as torsion is exactly zero in vacuum = outside matter distribution; the torsion constraint can be solved algebraically, therefore torsion becomes a function of matter variables which can be determined explicitly.

That means torsion does not propagate and therefore there are no torsion waves.

But how do you call the motion corresponding to a rotation of your arm around its own axix alternatively on the rigth and on the left? Isn't it a kind of trivial torsion wave? Perhaps is it just a question of language and semantic ...

Otherwise since the volumetric density of matter is nowhere exactly zero in the universe (= there is no perfect vacuum except in mathematics), following your way of thinking and calculating, there is nowhere no torsion.

Here's a link to an excellent review regarding Einstein-Cartan gravity

http://arxiv.org/abs/gr-qc/0606062
Einstein-Cartan Theory
Andrzej Trautman
(Submitted on 14 Jun 2006)
Abstract: The Einstein--Cartan Theory (ECT) of gravity is a modification of General Relativity Theory (GRT), allowing space-time to have torsion, in addition to curvature, and relating torsion to the density of intrinsic angular momentum. This modification was put forward in 1922 by Elie Cartan, before the discovery of spin. Cartan was influenced by the work of the Cosserat brothers (1909), who considered besides an (asymmetric) force stress tensor also a moments stress tensor in a suitably generalized continuous medium.

Blackforest said:
But how do you call the motion corresponding to a rotation of your arm around its own axix alternatively on the rigth and on the left? Isn't it a kind of trivial torsion wave? Perhaps is it just a question of language and semantic ... .
Consider a similar situation in classical electrodynamics. You can introduce charges and currents. You can ask for longitudinal photons - and will find that longitudinal photons are nothing else but functions of the charges and currents. So "yes", there are "longitudinal photons", but "no" they are not new physical degrees of frededom as they are expressed purely in terms of the matter fields.

In the same sense propagating torsion can always be expressed via propagating spin density of matter fields.

Look at Trautman's paper, eq. (25) and explanations below:

"Therefore, torsion vanishes in the absence of spin and then (23) is the classical Einstein field equation. In particular, there is no difference between the Einstein and Einstein–Cartan theories in empty space. Since practically all tests of relativistic gravity are based on consideration of Einstein’s equations in empty space, there is no difference, in this respect, between the Einstein and the Einstein–Cartan theories: the latter is as viable as the former."

The conclusion is interesting: we are not able to distinguish between Einstein and Einstein-Cartan gravity experimentally, as
a) in vacuum they are idetnical
b) inside matter the spin-density effects are too weak to be observed

For me Einstein-Cartan gravity seems to be the natural choice.

Interesting paper!

In principle one would have to investigate the whole theory space including curvature and torsion based on the 1st order connection formalism and try to construct a renormalization group approach in order to identify long-range forces, relevant and irrelevant operators and all that.

Are there some investigations in that directions as an extension of the asymptotic safety research program?

To uncover the high energy theory, sure.

But then that's kind of academic in nature and similar to asking where is the Landau pole of QED. In practise the theory is likely modified anyway.

The point is torsion seems to act like any other tensor field that you can imagine putting in the action, and unless you impose some sort of symmetry principle to constrain its couplings, there is nothing that prevents it from acquiring a large gauge invariant mass. Consequently it seems likely that in most theories you can simply integrate it out of the EFT, where observable consequences/differences end up being suppressed by powers of a heavy mass scale.

tom.stoer said:
Consider a similar situation in classical electrodynamics. You can introduce charges and currents. You can ask for longitudinal photons - and will find that longitudinal photons are nothing else but functions of the charges and currents. So "yes", there are "longitudinal photons", but "no" they are not new physical degrees of frededom as they are expressed purely in terms of the matter fields.

In the same sense propagating torsion can always be expressed via propagating spin density of matter fields.

Look at Trautman's paper, eq. (25) and explanations below:

"Therefore, torsion vanishes in the absence of spin and then (23) is the classical Einstein field equation. In particular, there is no difference between the Einstein and Einstein–Cartan theories in empty space. Since practically all tests of relativistic gravity are based on consideration of Einstein’s equations in empty space, there is no difference, in this respect, between the Einstein and the Einstein–Cartan theories: the latter is as viable as the former..."

Thanks so much for the explanations and for the links. Concerning the question of this thread, I am not (yet) able to answer. As a matter of fact the relativistic part of the Lorentz Einstein Law (LEL) stays unchanged if one adds an antisymmetric part to the connection (so long calculations take place on a commutative set K (e.g. real numbers). So it does not sound good for the torsion. But one could also interpret the LEL in saying that that relativistic part splits in a multitude of possible pairs ([F], acceleration) and then try to discover if the introduction of a antisymmetric part into the connection plays any role in these splits.

## 1. What is Cartan Einstein Gravity?

Cartan Einstein Gravity is a theory of gravity proposed by mathematician and physicist Elie Cartan and physicist Albert Einstein in the early 20th century. It combines the principles of general relativity and the mathematical framework of differential geometry to describe the curvature of space-time caused by the presence of matter and energy.

## 2. How does Cartan Einstein Gravity differ from other theories of gravity?

Cartan Einstein Gravity differs from other theories of gravity, such as Newton's theory of gravity and Einstein's general relativity, in its incorporation of torsion. Torsion is a measure of the twisting or rotation of space-time, which is not accounted for in other theories of gravity.

## 3. Is there a quantum of torsion in Cartan Einstein Gravity?

Yes, there is a quantum of torsion in Cartan Einstein Gravity. This means that torsion is not continuous, but rather exists in discrete units or packets. This is similar to the concept of quanta in quantum mechanics.

## 4. What is the role of spin in Cartan Einstein Gravity?

In Cartan Einstein Gravity, spin refers to the intrinsic angular momentum of particles. This is an important factor in the theory as it affects the curvature of space-time and thus the behavior of particles in the presence of gravity.

## 5. How does Cartan Einstein Gravity contribute to our understanding of the universe?

Cartan Einstein Gravity provides a more comprehensive and mathematically rigorous understanding of gravity and its effects on the universe. It has been used to explain phenomena such as the bending of light around massive objects and the expansion of the universe. It also plays a crucial role in the development of other theories, such as string theory and loop quantum gravity.

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