Master Constraint Operator in Loop Quantum Gravity.pdf

a r X i v : g r - q c / 0 5 1 0 0 1 4 v 4 3 M a r 2 0 0 6 Master Constraint Operators in Loop Quantum Gravity Muxin Han 1,2? and Yongge Ma 1? 1. Department of Physics, Beijing Normal University, Beijing 100875, CHINA 2. Horace Hearne Jr. Institute for Theoretical Physics, Louisiana State University, Baton Rouge, LA 70803, USA February 5, 2008 Abstract We introduce a master constraint operator M? densely defined in the diffeomor- phism invariant Hilbert space in loop quantum gravity, which corresponds classi- cally to the master constraint in the programme. It is shown that M? is positive and symmetric, and hence has its Friedrichs self-adjoint extension. The same conclu- sion is tenable for an alternative master operator M? ′ , whose quadratic form coin- cides with the one proposed by Thiemann. So the master constraint programme for loop quantum gravity can be carried out in principle by employing either of the two operators. Keywords: loop quantum gravity, master constraint, quantum dynamics. PACS number(s): 04.60.Pp, 04.60.Ds 1 Introduction It is well known that the quantization programme of loop quantum gravity is based on the connection dynamics of general relativity [1][2][3]. The basic conjugate pairs in the phase space are su(2)-valued connections A i a and densitized triads P? a i on a 3-manifold Σ. In the case where Σ is a compact set without boundary, the Hamiltonian is a linear combination of constraints as follows: Htot = G(Λ) +V(~N) +H(N). (1) ? Email address: mhan1@lsu.edu ? Email address: mayg@bnu.edu.cn 1 As an infinite dimensional Poisson algebra, the constraints algebra is not a Lie algebra unfortunately, because the Poisson bracket between the two scalar (Hamiltonian) con- straints H(N) and H(M) has structure function depending on dynamical variables [1]. This character causes much trouble in solving the constraints quantum mechanically. On the other hand, the algebra generated by the Gaussian constraints G(Λ) forms not only a subalgebra but