Abstract
Starting from the coadjoint Poincaré algebra we construct a point particle relativistic model with an interpretation in terms of extra-dimensional variables. The starting coadjoint Poincaré algebra is able to induce a mechanism of dimensional reduction between the usual coordinates of the Minkowski space and the extra-dimensional variables which turn out to form an antisymmetric tensor under the Lorentz group. Analysing the dynamics of this model, we find that, in a particular limit, it is possible to integrate out the extra variables and determine their effect on the dynamics of the material point in the usual space time. The model describes a particle in D dimensions subject to a harmonic motion when one of the parameters of the model is negative. The result can be interpreted as a modification to the flat Minkowski metric with non trivial Riemann, Ricci tensors and scalar curvature.
Highlights
The signature of the Minkowski space time metric is mostly plus, the total space time is a space with more that one time
We will integrate out the variables ξμν at the first non vanishing order in 1/R. obtaining an effective action which, at the lowest order describes a free particle in D dimensions, whereas at the order the equations of motion describe a particle in a quadratic potential, that could be interpreted as a correction to the Minkowski flat metric
In this paper we have considered a dynamical model based on a non-linear representation of the coadjoint Poincare group, which has a natural interpretation in terms of extra dimensions
Summary
The coadjoint Poincare algebra is an extension of the Poincare algebra with a vector and an antisymmetric rank-two tensor, Zμ and Zμν respectively, satisfying the commutation relations [7]. [Mμν , Zρ] = i(ημρZν − ηνρZμ), [Mμν , Zρσ] = i(ημρZνσ + ηνσZμρ − ημσZνρ − ηνρZμσ). [Zμν , Pρ] = i(ημρZν − ηνρZμ), [Zμ, Pν] = 0, [Zμ, Zν] = 0, [Zμν, Zρ] = 0, μ, ν = 0, 1 · · · D − 1. Let us consider the quotient space of the group generated by this algebra with respect to the Lorentz group. A local parameterisation of the coset is given by g. It is easy to verify that the MC forms are invariant under the following transformations: Pμ : Zμν : Zμ : δxμ = aμ δξμν = μν , δημ = μ
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