Eigenvalues of the volume operator in loop quantum gravity.pdf

a r X i v : g r - q c / 0 5 0 9 0 4 9 v 2 1 3 J a n 2 0 0 6 Eigenvalues of the volume operator in loop quantum gravity Krzysztof A. Meissner1 Institute of Theoretical Physics, Warsaw University Hoz?a 69, 00-681 Warsaw, Poland and CERN, Theoretical Division Abstract We present a simple method to calculate certain sums of the eigen- values of the volume operator in loop quantum gravity. We derive the asymptotic distribution of the eigenvalues in the classical limit of very large spins which turns out to be of a very simple form. The results can be useful for example in the statistical approach to quantum gravity. 1 Introduction Volume operators were introduced by Rovelli and Smolin [1] and by Ashtekar and Lewandowski [2] (the latter will be used in the current paper). The volume operator is much less understood than the area operator, one of the reasons being that it is not diagonal in any reasonable basis and there are ambiguities in defining its action on a given graph (see [3] and references therein; clarification of the issue is given in [4]). The matrix elements of the volume operator were obtained in [5], [6] (in the context of cosmology [7]); however, the formulae are very complicated and it is extremely difficult to get analytic expressions for the eigenvalues of the volume operator, except for very small values of the momenta (for some bounds on the eigenvalues see [8]). Therefore it seems worthwhile to try to find some alternative way of describing the eigenvalues. In this paper we show that relatively simple formulae can be obtained for the sums of squares (or higher even powers) of the eigenvalues, with the most important “quantum numbers” in the vertex 1E-mail address: Krzysztof.Meissner@fuw.edu.pl 1 fixed: j, j1, j2, j3, where j is the total spin and j1, j2, j3 are spins of the incoming legs. We also calculate the sums of higher powers (4, 6 and 8) of the eigenvalues with fixed j1, j2, j3. At the end of the paper we derive (in the form of an integra