Revisiting EPRL: All Finite-Dimensional Solutions by Naimark’s Fundamental Theorem

Revisiting EPRL: All Finite-Dimensional Solutions by Naimark’s Fundamental Theorem In this paper, we research all possible finite-dimensional representations and corresponding values of the Barbero–Immirzi parameter contained in EPRL simplicity constraints by using Naimark’s fundamental theorem of the Lorentz group representation theory. It turns out that for each nonzero pure imaginary with rational modulus value of the Barbero–Immirzi parameter $$\gamma = i \frac{p}{q}, p, q \in Z, p, q \ne 0$$ γ = i p q , p , q ∈ Z , p , q ≠ 0 , there is a solution of the simplicity constraints, such that the corresponding Lorentz representation is finite-dimensional. The converse is also true—for each finite-dimensional Lorentz representation solution of the simplicity constraints $$(n, \rho )$$ ( n , ρ ) , the associated Barbero–Immirzi parameter is nonzero pure imaginary with rational modulus, $$\gamma = i \frac{p}{q}, p, q \in Z, p, q \ne 0$$ γ = i p q , p , q ∈ Z , p , q ≠ 0 . We solve the simplicity constraints with respect to the Barbero–Immirzi parameter and then use Naimark’s fundamental theorem of the Lorentz group representations to find all finite-dimensional representations contained in the solutions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales Henri Poincaré Springer Journals

Revisiting EPRL: All Finite-Dimensional Solutions by Naimark’s Fundamental Theorem

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Publisher
Springer International Publishing
Copyright
Copyright © 2017 by Springer International Publishing
Subject
Physics; Theoretical, Mathematical and Computational Physics; Dynamical Systems and Ergodic Theory; Quantum Physics; Mathematical Methods in Physics; Classical and Quantum Gravitation, Relativity Theory; Elementary Particles, Quantum Field Theory
ISSN
1424-0637
eISSN
1424-0661
D.O.I.
10.1007/s00023-017-0588-8
Publisher site
See Article on Publisher Site

Abstract

In this paper, we research all possible finite-dimensional representations and corresponding values of the Barbero–Immirzi parameter contained in EPRL simplicity constraints by using Naimark’s fundamental theorem of the Lorentz group representation theory. It turns out that for each nonzero pure imaginary with rational modulus value of the Barbero–Immirzi parameter $$\gamma = i \frac{p}{q}, p, q \in Z, p, q \ne 0$$ γ = i p q , p , q ∈ Z , p , q ≠ 0 , there is a solution of the simplicity constraints, such that the corresponding Lorentz representation is finite-dimensional. The converse is also true—for each finite-dimensional Lorentz representation solution of the simplicity constraints $$(n, \rho )$$ ( n , ρ ) , the associated Barbero–Immirzi parameter is nonzero pure imaginary with rational modulus, $$\gamma = i \frac{p}{q}, p, q \in Z, p, q \ne 0$$ γ = i p q , p , q ∈ Z , p , q ≠ 0 . We solve the simplicity constraints with respect to the Barbero–Immirzi parameter and then use Naimark’s fundamental theorem of the Lorentz group representations to find all finite-dimensional representations contained in the solutions.

Journal

Annales Henri PoincaréSpringer Journals

Published: May 17, 2017

References

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