Θ-Term AND COSMOLOGICAL CONSTANT FROM CJD ACTION

Abstract

In the gravity without metric formalism of Capovilla, Jacobson and Dell, the topological θ-term appears through a canonical transformation.The origin of this canonical transformation is probed here. It is shown here that when θ-term appears cosmological λ-term also appears simultaneously.PACS numbers : 04.20.Cv, 04.20.Fy Key words : Canonical transformation, Torsion, θ-term, Cosmological constant Einstein originally formulated the theory of gravitation as a set of differential equations obeyed by the metric tensor of space-time. Ashtekar 1,2 has rewritten Einstein's theory, in its hamiltonian formulation as a set of differential equations obeyed by an SO(3) connection and its canonically conjugate momentum. A Lagrangian formulation, in which the variables are the space-time tetrads and the self-dual spin connection, was given by Samuel 3 , Jacobson and Smolin 4 .If tetrad is invertible, one may eliminate the spin connection to obtain the conventional Hilbert action, plus a (complex) surface term, so that the Lagrangian formulation is given entirely in terms of the tetrad. However, it is clearly natural to enquire whether one can give a Lagrangian formulation of Ashtekar's theory in which the metric or tetrad, has been completely eliminated in favour of the connection. This has been done by Capovilla, Jacobson and Dell 5,6 .Bengtsson and Peldan 7 have shown that if one performs a particular canonical transformation involving Ashtekar variables and corresponding SO(3) gauge fields, the expression for the Hamiltonian constraint changes when other constraints remain unaffected. This corresponds precisely to the addition of a "CP-violating" θ-term to the CJD Lagrangian. Mullick and Bandyopadhyay 8 have shown that this "CPviolating" θ-term is responsible for non-zero torsion. This θ-term effectively corresponds to the chiral anomaly when a fermion chiral current interacts with a gauge field. In a recent paper 9 , it has been shown that chiral anomaly gives rise to the mass of a fermion which implies that for a massive fermion the divergence of the axial vector current (which is associated with torsion via θ-term) is non-vanishing.

Gravitational Constant and Torsion

“…So this approach is very much akin to our present approach if we consider that torsion is connected to internal space which makes the spacetime noncommutative such that the gravitational constant fixes the scale of noncommutativity. This supports the findings of some other works, when the gauge group is SL(2, C) which is the covering group of SO (3,1), where torsion has been shown to be linked with CP-violation 27 and fermion mass 28 .…”

Torsion, Dirac Field, Dark Matter and Dark Radiation

“…In this context we may mention a recent work [30] where, in the gravity without metric formalism of gravity, a particular canonical transformation of the field variables causes the simultaneous appearance of the CP-violating θ-term and the cosmological term. Here we also like to consider the finding of some other works, where the gauge group is SL(2, C) , where torsion has been shown to be linked with CP-violation [41] and fermion mass [42].…”

“…In some earlier works [29,30,15] φ has been treated as an inverse length constant. In a recent paper [16] φ has been associated, either in laboratory scale or in cosmic sale, with a local energy scale.…”

de Sitter group and Einstein-Hilbert Lagrangian

“…It is important to note that, in our present formalism, the only assumption is that the torsion is represented by a null axial vector and the corresponding Lagrangian is a particular In a recent paper [45] it has been shown that, in the gravity without metric formalism of gravity, when one performs a particular canonical transformation of the field variables, CP-violating θ-term appears in the Lagrangian together with the cosmological term. This supports the finding of this note when we consider that the torsion, being an axial vector, has a certain role to play in CP-violation.…”