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The Bekenstein Bound and Non-Perturbative Quantum Gravity

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The Bekenstein Bound and Non-Perturbative Quantum Gravity enstein b ound and View metadata, citation and similar papers at core.ac.uk The Bek brought to you by CORE provided by CERN Document Server non-p erturbative quantum gravity Kirill V.Krasnov Bogolyubov Institute for Theoretical Physics, Kiev 143, Ukraine (March 19, 1996) We study quantum gravitational degrees of freedom of an arbitrarily chosen spatial region in the framework of lo op quantum gravity . The main question concerned is if the dimension of the Hilb ert space of states corresp onding to the same b oundary area satis es the Bekenstein b ound. We show that the dimension of such space formed by all kinematical quantum states, i.e. with the Hamiltonian constraint b eing not imp osed, is in nite. We make an assumption ab out the Hamiltonian's action, which is equivalent to the holographic hyp othesis, an which lead us to physical states (solutions of the Hamiltonian constraint) sp eci ed by data on the b oundary. The physical states which corresp ond to the same b oundary area A form a nite dimensional Hilb ert space and we show that its dimension 2 , where lies in the interval 0:55 < < 1:01. We discuss in general the is equial to exp A=l p arising notion of statistical mechanical entropy.For our particular case, when the only parameter sp ecifying the system's macrostate is the b oundary area, this entropy is given by S (A)= A.We discuss the p ossible implications of the result obtained for the black hole entropy. Among the results of the lo op approach to non-p erturbative quantum gravity the are several which, if taken seriously, tell us that the picture of geometry on scales small comparatively with our usual scales (on Planckian scales) lo oks quite di erently from the habitual picture of Riemannian geometry, and that the usual classical picture arises only on a coarse-graining. The fundamental exitations of this quantum geometry are one-dimensional lo op exitations, and the whole quantum picture is of essentially discrete, combinatorial character [1]. One of the manifestations of this discretennes is that there exists a countable basis in the Hilb ert space of gravita- tional quantum states; this basis can b e realized [2] as the space of di eomorphism equivalence classes of emb eddings of the so called spin-networks. This means the striking di erence of the quantum geometry picture with its countable space of states from the classical Riemannian picture with the degrees of freedom of continuum. Moreover, not only the arising quantum geometry di ers profoundly from what the classical picture suggests, but it seems also to b e essentially di erent from what any usual p erturbative quantum eld theory might give. Namely, as it has b een argued by Smolin [3], there is a reason to exp ect that the numb er of quantum gravitational degrees of freedom p er a spatial region is, in fact, nite. If true, this fact would mean a fundamental di erence b etween non-p erturbative quantum theory of gravity and quantum eld theories in Minkowski space-time with their in nite numb er of degrees of freedom p er a spatial region. The reason whywe could exp ect that the dimension of the space of states of gravity p er an arbitrarily chosen spatial region is nite comes from the black hole physics. The Bekenstein b ound [4] asserts that the entropy of a system contained inside some region in space can not exceed the entropy of a black hole whose horizon surface just coincide with the b oundary of this region. Combining this with the Bekenstein-Hawking formula for the black hole entropy 1 S = A; (1) 4 where S is the entropy of a black hole and A is the area of its horizon surface (wework in Planckian units so that 1 the prop ortionality co ecient in (1) is just ), we obtain that the entropy of a system contained in a region of space 4 1 can not exceed A, where A is just the area of the b oundary under which our system resides 4 1 S A: (2) 4 1 This b ound immediately tells us that the dimension of the space of states H of our system do es not exceed exp A 4 1 A: (3) dim(H) exp 4 As discussed by Smolin in [3], it is essentially the second law of thermo dynamics which imp oses the Bekenstein b ound so this is a fundamental question for our quantum theory if it predicts that the b ound is satis ed. In this pap er we E-mail address: [email protected] c.org 1 investigate whether the answer is p ositive for the case of lo op quantum gravity. The other aim of this pap er is to try to reveal that connection b etween quantum gravity and thermo dynamics which is suggested by the Bekenstein argument. Accordingly, the pap er is divided in two parts. The rst section concerns the notion of geometrical entropy; we discuss how the macroscopic picture of continuous Riemannian geometry arises from an essentially discrete quantum picture and how the macroscopic description leads to the notion of entropy. In the second section, which contains the main result of this pap er, we compute the geometrical entropy of an arbitrarily chosen spatial region with a xed area of the b oundary and discuss the Bekenstein b ound. We conclude with the discussion of the results obtained. I. GEOMETRICAL ENTROPY: A DEFINITION Let us recall at the very b eginning how the standard notion of entropy in statistical mechanics arises. Tobe concrete, let us consider classical systems; the quantum case preserves the same main p oints. The concept of entropy in general arises with distinguishin g b etween macro- and micro-states of a system. Microstate describ es a system in its minute p ossible details; for a classical system microstate is represented by a p oint in its phase space. Macrostate, on the other hand, describ es a system only in its manifest macroscopic prop erties; this macroscopic description is usually far less precise. In particular, this means that there might b e quite a lot of microscopical states which are identical with regard to their macroscopic features, i.e. which corresp ond to the same macroscopic state. This gives rise to the notion of entropy.Entropy is a function which dep ends on macroscopic state of a system. It can b e thought of as a function which for each macroscopic state gives the (logarithm of the) numb er of di erent microscopic states corresp onding to this macrostate. More precisely, the space of states of a system (its phase space for a classical one) should b e divided into compartments, where all microstates b elonging to the same compartment are macroscopically 1 indistinguishable . Then the entropy of a macrostate is given by the logarithm of the volume of the compartment corresp onding to this macrostate. Several comments are in order: 1. If the space of states is discrete than instead of the logarithm of the volume of the compartmentwehaveto consider the logarithm of the numb er of di erent states corresp onding to the same macrostate. 2. The entropy discussed is the so-called informational entropy. The statistical mechanical entropyis X S= p log p ; (4) n n n where p is the probability to nd the system in the microstate n (the set of all p de nes a statistical state of n n the system) and the sum is taken over all microstates. The informational entropy coincides with the statistical mechanical one for such statistical states that all microstates b elonging to the same compartment of the phase space have an equial probabilityweight and the other microstates have the weight zero. 3. Quantum mechanical entropy, which is given by S = Tr^ log; ^ (5) with ^ b eing the density matrix op erator, takes in the basis of the eigenstates of ^ the form (4) and again coincide with the informational entropy for states when eigenvalues of ^ corresp onding to the same macrostate are equial. Let us now return to the lo op quantum gravity. As it is discussed in [1], kinematical quantum states of our gravity system are di eomorphism equivalence classes of con gurations of lo ops called spin-networks (this set forms a basis in the representation space) and their p ossible linear combinations. The basis spin-network states are eigenstates of op erators corresp onding to the area of di erent 2-surfaces in our space with a purely discreet sp ectra. It is imp ortant to understand how the continuous prop erties of the usual Riemannian geometry arise from such a discrete picture [5]. As wehave stressed at the very b eginning, the quantum gravity suggests that the usual picture of Riemannian geometry arises only as an approximation to the more fundamental quantum one. Therefore, wehave to b e able to 1 In fact, entropy is prop ortional to this numb er, but wework in the units in which this prop ortionality co ecientischosen to b e unity. 2 answer the twotyp es of questions: for a given quantum lo op state and a given precision wehave to tell which metrics it approximates with this precision (if any); conversely, for a given metric and a precision we should nd all quantum states approximating it with the precision chosen (again, if such quantum states exist). Let us assume for a moment that we can answer any such question and for any metric we know the set of quantum states approximating this metric.
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