The Immirzi parameter and fermions with non-minimal coupling

Alexandrov, Sergei

doi:10.1088/0264-9381/25/14/145012

Cited by 57 publications

(60 citation statements)

References 18 publications

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“…Indeed the leading order in a in the expansion of h ABC vanishes, as expected from the curvature formula above (11). Second, we do want n a to be like a normal vector whose norm gives the area of the surface.…”

Section: B Non-abelian Normalsmentioning

confidence: 84%

“…The issue of the coarse-graining or of the continuum limit of the theory and of its dynamics are related, first because a consistent theory of quantum gravity should solve the problem of perturbative non-renormalizability and second because a good theory of quantum gravity should describe all scales * Electronic address: christoph.charles@ens-lyon.fr † Electronic address: etera.livine@ens-lyon.fr 1 This new set of variables corresponds to a canonical transformation at the classical level and depends on a new dimensionless parameter, the Immirzi parameter. Classically, this parameter does not play any role in the equation of motion, at least in the vacuum case [10,11]. 2 The torsion exists generally in the context of twisted geometries but does not reflect to a genuine torsion of the spin-connection.…”

Section: Introductionmentioning

confidence: 99%

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The closure constraint for the hyperbolic tetrahedron as a Bianchi identity

Charles

Livine

2017

Gen Relativ Gravit

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The closure constraint is a central piece of the mathematics of loop quantum gravity. It encodes the gauge invariance of the spin network states of quantum geometry and provides them with a geometrical interpretation: each decorated vertex of a spin network is dual to a quantized polyhedron in R 3 . For instance, a 4-valent vertex is interpreted as a tetrahedron determined by the four normal vectors of its faces. We develop a framework where the closure constraint is re-interpreted as a Bianchi identity, with the normals defined as holonomies around the polyhedron faces of a connection (constructed from the spinning geometry interpretation of twisted geometries). This allows us to define closure constraints for hyperbolic tetrahedra (living in the 3-hyperboloid of unit future-oriented spacelike vectors in R 3,1 ) in terms of normals living all in SU(2) or in SB(2, C). The latter fits perfectly with the classical phase space developed for q-deformed loop quantum gravity supposed to account for a non-vanishing cosmological constant Λ > 0. This is the first step towards interpreting q-deformed twisted geometries as actual discrete hyperbolic triangulations. IntroductionLoop quantum gravity (for monographs see [1][2][3]) is based on a reformulation of general relativity as an SU (2) gauge theory of the Ashtekar-Barbero connection [4][5][6][7] together with its canonically conjugated field, the densitized triad 1 . At the quantum level, the kinematics of the theory is well-understood: the space of quantum states of geometry are wave-functions of the Ashtekar-Barbero connection and admits a canonical basis labelled by spin networks [8]. Spin networks are (abstract) graphs (which might have knotting information in 3+1d) colored with spins on the edges (hence the name) and intertwiners, which are invariant representations of the gauge group, on the vertices. The spin network basis diagonalize the geometrical operators. The spins carried by the edges corresponding to quanta of surfaces [9]. The volume operator is a bit more involved but we still can find a basis of intertwiners that diagonalize the volume operator.The spin network states have a natural geometrical interpretation in the twisted geometry framework [12]. Each node of the graph is interpreted as a flat convex polyhedron. Each edge coming out of the vertex corresponds to a face of the polyhedron and the area of this face is given by the spin carried by the edge. The intertwiner should encode the remaining quantum degrees of freedom. Then, the polyhedra corresponding to the vertices are glued together on their faces. By construction, the area of two glued faces match. Still, the geometry is said to be twisted since the shapes of the faces do not have to match. In the continuum, this corresponds to a discontinuous metric [13]. There is a notable second geometrical parametrization that was proposed: spinning geometries [14]. The construction is roughly the same but the faces and edges of the polyhedra are not assumed to be (extrinsically) flat. As a consequence,...

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Section: B Non-abelian Normalsmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

The closure constraint for the hyperbolic tetrahedron as a Bianchi identity

Charles

Livine

2017

Gen Relativ Gravit

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“…In this article, we allow all possible real values of α as introduced in [23] (see also [53]). We emphasize that we have parity invariance for all real α, even though some torsion components such as those written below consist of contributions of different parity behavior.…”

Section: Discussionmentioning

confidence: 99%

Fermions in loop quantum cosmology and the role of parity

Bojowald

Das

2008

Class. Quantum Grav.

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Fermions play a special role in homogeneous models of quantum cosmology because the exclusion principle prevents them from forming sizable matter contributions. They can thus describe the matter ingredients only truly microscopically and it is not possible to avoid strong quantum regimes by positing a large matter content. Moreover, possible parity violating effects are important especially in loop quantum cosmology whose basic object is a difference equation for the wave function of the universe defined on a discrete space of triads. The two orientations of a triad are interchanged by a parity transformation, which leaves the difference equation invariant for ordinary matter. Here, we revisit and extend loop quantum cosmology by introducing fermions and the gravitational torsion they imply, which renders the parity issue non-trivial. A treatable locally rotationally symmetric Bianchi model is introduced which clearly shows the role of parity. General wave functions cannot be parity-even or odd, and parity violating effects in matter influence the microscopic big bang transition which replaces the classical singularity in loop quantum cosmology.

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“…Then it was noted that even in the classical theory of gravity, the new Holst term does influence field equations when fermions are minimally coupled [15]. This initiated discussion on the role played by the Immirzi parameter in classical gravity with fermions [16][17] [18][19] [20]. As originally observed in [15], the Immirzi parameter enters the coupling constant in front of the four-fermion interaction term of EC gravity.…”

Section: Introductionmentioning

confidence: 99%

“…It has been concluded that measuring the strength of this interaction can provide a tool to estimate the value of the Immirzi parameter independently from the quantum theory of gravity. Although the subsequent investigations [18] [20] showed that the Immirzi parameter can be "hidden" in the parameters of more general, non-minimal coupling procedures, it should be stressed that the minimal coupling scheme has been historically successful in constructing models, which could withstand the rigors of experimental testing whenever such tests were feasible. The experimental successes of the standard model of particle physics and general theory of relativity seem to support the minimal approach.…”

Section: Introductionmentioning

confidence: 99%

Nonuniqueness of gravity induced fermion interaction in the Einstein-Cartan theory

Kaźmierczak

2008

Phys. Rev. D

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The problem of nonuniqueness of minimal coupling procedure for Einstein-Cartan (EC) gravity with matter is investigated. It is shown that the predictions of the theory of gravity with fermionic matter can radically change if the freedom of the addition of a divergence to the flat space matter Lagrangean density is exploited. The well-known gravity-induced four-fermion interaction is shown to reveal unexpected features. The solution to the problem of nonuniqueness of minimal coupling of EC gravity is argued to be necessary in order for the theory to produce definite predictions. In particular, the EC theory with fermions is shown to be indistinguishable from usual General Relativity on the effective level, if the flat space fermionic Lagrangean is appropriately chosen. Hence, the solution to the problem of nonuniqueness of minimal coupling procedure is argued to be necessary if EC theory is to be experimentally verifiable. It could also enable experimental tests of theories based on EC, such as loop approach to quantisation of gravitational field. Some ideas of how the arbitrariness incorporated in EC theory could be restricted or even eliminated are presented.

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The Immirzi parameter and fermions with non-minimal coupling

Cited by 57 publications

References 18 publications

The closure constraint for the hyperbolic tetrahedron as a Bianchi identity

The closure constraint for the hyperbolic tetrahedron as a Bianchi identity

Fermions in loop quantum cosmology and the role of parity

Nonuniqueness of gravity induced fermion interaction in the Einstein-Cartan theory

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