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In this paper we intend to present another quite simple and the number of microscopic quantum states con- but elegant scheme to fix the Immirzi parameter by con- tributes the black hole an entropy with sidering the supersymmetric extension of loop quantum gravity, but more importantly, we find in this case Jmin is S = N ln(4Jmin + 1). (9) forced to take the minimum value of the representation of gauge group Osp(1 2) which is perfectly consistent with where we have used the fact that given a black hole with the most probable| value as we expect based on the sta- fixed horizon, the most probable distribution of spins as- tistical consideration. sociated to the punctures is the configuration in which all the punctures are labelled by the minimum value of spin, denoted as Jmin. Combining Eq.(8) and Eq.(9), we II. AREA SPECTRUM IN N = 1 find the black hole entropy is proportional to the area of SUPERGRAVITY AND BLACK HOLE ENTROPY the event horizon, i.e.

ln(4Jmin + 1) N = 1 supergravity in Ashtekar-Sen variables was orig- S = A. (10) 2 1 inally given by Jacobson in [10] and its canonical quan- 8πγl˜ p Jmin(Jmin + 2 ) tization has been extensively studied in[11, 12]. In this q section we only recall the crucial results that we need Furthermore, comparing this with Bekenstein-Hawking here; for more details on this subject please see[11, 12]. formula (2) yields the following relation between the Im- Similar to the case in quantum general relativity, a gen- mirzi parameter and the minimum value of spin, eralized notion of supersymmetric spin networks can be ln(4J + 1) 1 introduced to construct the Hilbert space of canonical min = . (11) supergravity, while a key different ingredient from the or- 1 4 8πγ˜ Jmin(Jmin + 2 ) dinary one is that the edges in spin networks now are la- q belled by the representations of the supergroup Osp(1 2) | Next we follow the strategy advocated in [6] to fix the rather than SU(2). Such a replacement gives rise to a Immirzi parameter by considering the asymptotic quasi- different expression for the area spectrum at quantum normal modes of black holes, but present a more reason- mechanical level. More explicitly, if a surface is inter- able value of Jmin. sected by an edge of the network carrying the label J, then its area is given by [12] III. FIXING THE IMMIRZI PARAMETER BY 1 A(J)=8πγl˜ 2 J(J + ), (6) QUASINORMAL MODE SPECTRUM pr 2 where J still takes half-integer but labels Osp(1 2) rep- It is argued in [5, 6] that according to Bohr’s corre- resentation andγ ˜ denotes the Immirzi parameter| in su- spondence principle, the increase of the black hole mass pergravity. If we simply set γ =γ ˜ and compare it with should be ascribed to a quanta with the energy ¯hωQNM the area spectrum (1) in ordinary loop quantum grav- absorbed by the black hole, i.e. ity, we find that the discrepancy is so tiny that it can be ignored for large J, while for small values of J these ∆M =¯hωQNM . (12) two sorts of spectra in principle should be distinguishable and perhaps yield quite different physics. We will soon On the other hand, since see it does so for the statistical entropy of black holes as A = 16πM 2, (13) discussed below. The derivation of statistical entropy of black holes in such an increase in the mass of the black hole induces a loop quantum gravity has been thoroughly studied and corresponding increase in the area of the event horizon we refer to [4, 13] for details. Here we simply argue 2 that the strategy can be directly extended to consider the ∆A = 32πM∆M = 4 ln 3lp. (14) entropy of black holes (at least for Schwarzschild black holes) in the context of supergravity. The only thing that At quantum mechanical level, this kind of increase in should be stressed is that the dimension of the Hilbert the area of the horizon corresponds to the appearance of space associated to a puncture with spin J on the event another edge labelled with Jmin in the supersymmetric horizon now is [12] spin network, thus

D(J)=4J +1. (7) ∆A = A(Jmin). (15)

Therefore, the area of its event horizon with N punctures Plugging it into Eq.(14) we have may be obtained as ln 3 2 1 γ˜ = . (16) A = N8πγl˜ J (J + ), (8) 1 pr min min 2π Jmin(Jmin + 2 ) 2 q 3

Therefore, Eq.(16) together with Eq.(11) uniquely fixes previous intuition is not quite correct. As a matter of the value of the Immirzi parameter to be fact, two Immirzi parameters are not equal but ln 3 γ˜ = . (17) γ˜ =2γ. (19) √2π At the same time it also determines Therefore, the area spectrum in canonical supergravity can be rewritten as 1 Jmin = . (18) 2 sg 2 2 A (J)=8πγlp 2J(2J + 1) 8πγlp Jeff (Jeff + 1). ≡ q This is a remarkable result because it is consistent with p (20) our argument based on statistical principle, stating that Now it’s evident that the area spectrum is the same ex- the most probable distribution should be the set of punc- pression as the one in quantum general relativity except tures on the horizon uniformly labelled by the fundamen- that Jeff here can only take integer! That is to say, tal representation of supergroup Osp(1 2) which takes all the area eigenvalues with half-integers in loop quan- the minimum positive value of J. Here| we do not need tum gravity now are forbidden by the supersymmetry in any other extra assumption or condition. It only seems N = 1 supergravity. implying that the supersymmetry might be a favorable Corichi and Swain have suggested to suppress the con- nature of spacetime. tribution from J = 1/2 edges in loop quantum gravity by considering the coupling of fermions or Pauli exclu- sion principle respectively. From this point of view, we IV. DISCUSSION may also argue that we have presented a very elegant scheme to do so, but the reason is simply due to the We conclude this paper by pointing out a very inter- supersymmetry. Interestingly enough, all these sugges- esting relation between two area spectra respectively ob- tions share one common point, namely the involvement tained in quantum general relativity and the simplest of fermions. As a result, it’s quite desirable to investigate supergravity, namely Eq.(1) and Eq.(6). Note in quan- their relations and show more insight into this subject. tum gravity, we do not have any reason to argue that So far, we have only focused on Schwarzschild-like those two Immirzi parameters should be related in any black holes. We expect the similar consideration will be manner. Therefore these two spectra in principle have applicable to more general black holes. At the same time, nothing to do with each other. One naive but natural investigating the supersymmetric extension of loop quan- choice is just setting two Immirzi parameters equal as we tum gravity with N > 1 is also under progress. assumed in section two such that at each level labelled by the same J the spectrum has only a small shift after the supersymmetry is concerned. In particular, such a shift is vanishing as J approaches to infinity. However, this Acknowledgement picture has greatly changed when the semi-classical limit of these two theories is concerned. The asymptotic quasi- It’s our pleasure to thank Olaf Dreyer, Yongge Ma and normal modes of black holes do provide us a strategy to Lee Smolin for correspondence and helpful discussions. fix the Immirzi parameter so that these two sorts of spec- Y. Ling is partly supported by NSFC (No.10205002) and tra are comparable. Quite strikingly, here we found our K.C.Wong Education Foundation, Hong Kong.

[1] L. Smolin, arXiv:hep-th/0303185; T. Thiemann, [9] J. Swain, gr-qc/0305073. arXiv:gr-qc/0110034; A. Ashtekar, arXiv:gr-qc/0112038. [10] T. Jacobson, Class. Quant. Grav. 5, 923 (1988). [2] C. Rovelli and L. Smolin, Nucl. Phys. B 442, 593 (1995) [11] R. Gambini, O. Obregon and J. Pullin, Nucl. Phys. [Erratum-ibid. B 456, 753 (1995)] [arXiv:gr-qc/9411005]. B 460, 615 (1996) [arXiv:hep-th/9508036]; H. Ku- [3] G. Immirzi, Nucl. Phys. Proc. Suppl. 57, 65 (1997) nitomo and T. Sano, Int. J. Mod. Phys. D 1, 559 [arXiv:gr-qc/9701052]. (1993) [Prog. Theor. Phys. Suppl. 114, 31 (1993)]; [4] K. Krasnov, Phys. Rev. D. 55,3505(1997); C. Rovelli, T. Sano and J. Shiraishi, Nucl. Phys. B 410, 423 (1993) Phys. Rev. Lett. 77, 3288 (1996) [arXiv:gr-qc/9603063]; [arXiv:hep-th/9211104]; T. Sano, arXiv:hep-th/9211103; A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. K. Ezawa, Prog. Theor. Phys. 95, 863 (1996) Rev. Lett. 80, 904 (1998) [arXiv:gr-qc/9710007]. [arXiv:hep-th/9511047]. [5] S. Hod, Phys. Rev. Lett. 81, 4293(1998). [12] Y. Ling and L. Smolin, Phys. Rev. D 61, 044008(2000). [6] O. Dreyer, Phys. Rev. Lett. 90, 081301(2003). [13] L. Smolin, J. Math. Phys. 36, 6417 (1995) [7] L. Motl, Adv. Theor. Math. Phys. 6, 1135 (2003). [arXiv:gr-qc/9505028]. [8] A. Corichi, Phys. Rev. D 67, 087502(2003).