A Correction to the Immirizi Parameter of SU(2) Spin Networks

The elegant predictions of loop quantum gravity are obscured by the free Immirizi parameter. Dreyer (2003), considering the asymptotic quasinormal modes spectrum of a black hole, proposed that the Immirizi parameter may be fixed by letting the j=1 transitions of spin networks as the dominant processes contributing to the black hole area, as opposed to the expected j=1/2 transitions. This suggested that the gauge group of the theory might be SO(3) rather than SU(2). Corichi (2003), maintaining SU(2) as the underlying gauge group, and invoking the principle of local fermion-number conservation, reported the same value of the Immirizi parameter for j=1 processes as obtained by Dreyer. In this note, preserving the SU(2) structure of the theory, and considering j=1 transitions as the dominant processes, we point out that the value of the Immirizi parameter is in fact twice the value reported by these authors. We arrive at this result by assuming the asymptotic quasinormal modes themselves as dynamical systems obeying SU(2) symmetry.


Introduction
Loop quantum gravity (LQG) has produced results from first principle that geometry is discrete. LQG uses spin networks as a basis for its Hilbert space. Spin networks are graphs with edges that carry labels as j = 0, 1/2, 1, 3/2,…, that is, the representations of SU (2) group that serves as the gauge group of the theory. In LQG the area of a given region of space has a discrete spectrum in such a way that if a surface is punctured by an edge of the spin network carrying a label j, the surface acquires an area element [1,2,3]   Here 2 P l is the Planck area and  is the free undetermined Immirizi parameter [4] in the theory that remains obscured. The remarkable predictions of the theory are ambiguous up to this all-time present unfixed parameter. Nevertheless, indirect tools are used to fix the value of  .
In [5,6], it was shown that  can be fixed by the requirement that the quantum gravity results reproduce the Bakenstein-Hawking entropy [ where j min is the minimum (semi-integer) label for the representations of SU (2), responsible for the black hole entropy. Taking into consideration that statistically the most important contribution should come from j min =1/2,  is fixed as Dreyer [9], following a clue uncovered by Hod [10], fixed the value of  in an independent way by using a semi-classical argument based on the quasinormal mode (QNM) spectrum of a Schwarzschild black hole [11,12]. Dreyer's approach was based on Hod's conjecture that the real part of the highly damped (QNM) frequencies QNM  asymptotically tends to a fixed quantity This conjecture was proved analytically by Motl [13]. The argument used by Hod and also by Dreyer goes as follows: If we assume that the Bohr's correspondence principle is applicable to black holes, the radiation or absorption of such an asymptotic frequency of the quasinormal modes should be consistent with the variation in mass ∆M of the black hole, i.e. ln 3 8 Since the area A of the event horizon and the mass M of a Schwarzschild black hole are related by 2 16 , the variation in mass causes change ∆A in the quantized area of the event horizon. With the help of (5) and (6), equation (7) readily gives Dreyer considered that the most natural candidate for a transition of the quantum black hole, as described above, is the appearance or disappearance of a puncture with spin j min . The area of the black hole would then change by an amount given by equation (1), where j = j min , i.e.
Comparison of (8) and (9) In order to comply with the Bakenstein-Hawking entropy, Dreyer was forced to fix min 1 j  , and consequently, the Immirizi parameter as ln 3 22 At this point, in the absence of any other explanation why j min =1/2 does not comply with the correct entropy formula, Dreyer proposed that the true gauge group of the theory might be SO(3) rather than SU (2).
There have been various attempts to formulate a convincing explanation about why j=1 processes contribute dominantly to the black hole entropy. Corichi [14] has argued that one should maintain SU(2) as the gauge group of the theory if fermions are to be included in the theory. However, the j min =1/2 processes has to be highly suppressed. Corichi's argument goes as follows. Losing a j=1/2 representation would mean the edge becomes open in the bulk. In order to keep the local gauge invariance intact one has to attach a fermion to the open end; and this is not allowed if the fermion number is to be conserved locally. On the other hand, if an edge carries j = 1, one could attach a fermion-antifermion pair to the open end of the detached edge, preserving the local gauge invariance. Thus, even though punctures with j=1/2 edges are allowed kinemetically, the dominant contribution comes from processes for which the minimum allowed value of j is 1. Swain [15] proposed a generalized version of Pauli's exclusion principle applied to spin networks, which stated that "no more than two punctures of j=1/2, each with differing m values, may puncture a given surface". In this perspective, even though j=1/2 punctures are not forbidden, the dominant contribution appears to be coming from j=1 punctures. Astonishingly, exact agreement with correct entropy formula was achieved for j=1/2 as the dominant contributing processes by invoking supersymmetric extension of spin networks [16].
In the present note, working in the framework of SU(2) spin networks, we propose that the dynamics of an asymptotic QNM frequency be described by the Hamiltonian of a two-dimensional isotropic oscillator. It looks a natural choice because the 2D oscillator possesses the same group structure and symbols as those of the edges in loop quantum gravity. This picture clearly explains how the conversion of the quanta of geometry to matter quanta, and vice versa, at the horizon takes place in a consistent way. Further, as we will see, for SU(2) as the gauge invariance, and j=1 as the dominant processes, the value of the Immirizi parameter explicitly turns out to be twice the value reported by Dreyer and Corichi based on SO(3) and SU(2), respectively.
For relevance, we start in section 2 with a brief review of the Schwinger's scheme [17] of realizing a 2D isotropic oscillator as SU(2) system. In section 3 we work out the value of  for j=1 transitions in the SU(2) framework. The conclusions are presented in section 4.

2D Isotropic Oscillator and SU(2) Symmetry
In this section we briefly review Schwinger's method of establishing the relationship between a two-dimensional isotropic oscillator and the SU(2) symmetry. Consider a 2D isotropic oscillator described by the Hamiltonian (assuming 1  in this section)   where 0, 0 is the vacuum state such that 120 , 0 0, 0 0 aa  and n 1 , n 2 are nonnegative integers. The energy eigenvalue are     All the states , r n r  with r = 0,1,2,…n have the same energy, i.e. the energy eigenvalue E n is (n+1)-fold degenerate. Degeneracy in energy spectra indicate that there are symmetries associated with the system. In this case the symmetry in question is SU (2), that allows for transforming eigenstates with the same energy among themselves. It is worthwhile to work it out in some detail.
Let us construct new operators These are conserved quantities as one can readily verify ˆˆ, 0, The operators so defined in (17) The expressions (17) are known as Schwinger's bosonic realization of the usual angular momentum operators. One can interpret the action of the operators Ĵ  and Ĵ  in the following way. The motion of the system on a plane surface is composed of oscillations in two directions, say, x and y. The operator Ĵ  increases the oscillation amplitude in the x-direction and decreases the oscillation in the y-direction. It can continue doing so until the state of the particle consists only of oscillation in the x-direction. On the other hand, the operator Ĵ  just does the opposite; it squeezes the orbit of the particle towards oscillation in the y-direction.
The quadratic Casimir of the algebra  