Johannes Brunnemann and David Rideout 2008 Class. Quantum Grav. 25 065002 doi:10.1088/0264-9381/25/6/065002
Johannes Brunnemann1 and David Rideout2
Show affiliationsThe properties of the volume operator in loop quantum gravity, as constructed by Ashtekar and Lewandowski, are analyzed for the first time at generic vertices of valence greater than four. We find that the occurrence of a smallest non-zero eigenvalue is dependent upon the geometry of the underlying graph and is not a property of the volume operator itself. The present analysis benefits from the general simplified formula for matrix elements of the volume operator derived in Brunnemann and Thiemann (2006 Class. Quantum Grav. 23 1289), making it feasible to implement it on a computer as a matrix which is then diagonalized numerically. The resulting eigenvalues serve as a database to investigate the spectral properties of the volume operator. Analytical results on the spectrum at 4-valent vertices are included. This is a companion paper to Brunnemann and Rideout (2007 Properties of the volume operator in loop quantum gravity: I. Results Preprint 0706.0469), providing details of the analysis presented there.
Issue 6 (21 March 2008)
Received 24 July 2007, in final form 14 January 2008
Published 4 March 2008
Johannes Brunnemann and David Rideout 2008 Class. Quantum Grav. 25 065002
Jonathan Hackett and Simone Speziale 2007 Class. Quantum Grav. 24 1525
G Giampieri and A G Polnarev 1997 Class. Quantum Grav. 14 1521
Judah Levine 2008 Metrologia 45 S23
I V Mikityuk 2003 Russ. Math. Surv. 58 185
Kiwoon Choi et al JHEP11(2004)076
D Wirosoetisno and J Vanneste 2005 Nonlinearity 18 2657
Alex Bayliss et al 2004 Phys. Educ. 39 137
R Metzler et al 2001 J. Phys. A: Math. Gen. 34 317
Paul V Halstead and Gareth T James 1984 Phys. Educ. 19 275