Anomaly-free representations of the holonomy-flux algebra

We work on the uniqueness [1] of representations of the holonomy-flux algebra in loop quantum gravity. We argue that for analytic diffeomorphisms, the flux operators can be only constants as functions on the configuration space in representations with no anomaly, which are zero in the standard representation. In loop quantum gravity, the configuration variables are holonomies he[A] of a connection field and the momentum variables are surface integrals E(S, f) of a triad field. Quite interestingly, the Poisson brackets between the momentum variables do not vanish. The origin of this non-commutativity comes from the two-dimensional singular smearing of E(S, f) [3] and E(S, f) can be understood as some vector fields X(S, f) on the configuration space A. In the standard representation, every holonomy operator is multiplication and every flux operator is derivation on the Hilbert space L(Ā, μ) [4]. Representations of the holonomy-flux algebra were further investigated in [5]. It was motivated by the fact that the momentum variables E(S, f) are not constants on the configuration space A and proposed that E(S, f) can be functions F (S, f) on A π(E(S, f)) = X(S, f) + F (S, f) (1) where a map π is a representation of the holonomy-flux algebra. Later it was found that F (S, f) are real valued assuming that π is covariant with respect to the group of Among many nice introductions, [2] is good for understanding the uniqueness of the holonomyflux algebra. The Poisson brackets between two momentum functions are zero. the analytic diffeomorphisms [6]. Finally it was shown that F (S, f) vanish by proving that their norms which are obtained from a state via the GNS construction are zero requesting that they are invariant under the group of the semianalytic diffeomorphisms [1]. Whether this uniqueness holds for the analytic diffeomorphisms is an open question. In this paper, we argue that we only have freedom to take some constants besides zero as F (S, f) for the analytic diffeomorphisms provided that there is no anomaly in the representations of the holonomy-flux algebra. Considering the Poisson bracket between E(S1, f1) and E(S2, f2) is zero if S1 and S2 are disjoint, we request that π([E(S1, f1), E(S2, f2)]) = 0 for S1 ∩ S2 = φ. (2) In a GNS representation, it becomes X(S1, f1)(F (S2, f2))−X(S2, f2)(F (S1, f1)) = 0 for S1 ∩ S2 = φ. (3) From the definition of E(S, f), we request that F (S1 ∪ S2, f) = F (S1, f) + F (S2, f)− F (S1 ∩ S2, f). (4) We do not consider some difficulty with boundaries which is not essential in our purpose or we only consider surfaces which do not include their boundaries. Theorem 1. Suppose π is a representation of the holonomy-flux algebra and is covariant with respect to the group of the analytic diffeomorphisms. Suppose also, there is no anomaly in the representation. Then, all the L(Ā, μ) functions F (S, f) are constants on the configuration space A. To show this, we need two properties of Hilbert spaces [7]. Theorem 2. For any vector h and a given basis {ei} in a Hilbert space H, the expansion h = ∑ i aiei is unique. Theorem 3. For any vector h and a given basis {ei} in a Hilbert space H, < h, e > 6= 0 for at most a countable number of vectors e in {ei}. Theorem 2 is obvious and Theorem 3 comes from the condition that any vector has finite norm. 2 We are going to show that when we express F (S, f) in terms of the spin network basis [8][9], all the edges are trivial.