# 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

, Volume 18 (9) – May 17, 2017
14 pages

/lp/springer_journal/revisiting-eprl-all-finite-dimensional-solutions-by-naimark-s-w8X2VUlIjp
Publisher
Springer Journals
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations