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Spin Networks and Quantum Gravity CGPG Spin Networks and Quantum Gravity y Carlo Rovelli and Lee Smolin Department of Physics University of Pittsburgh Pittsburgh Pa y Center for Gravitational Physics and Geometry Department of Physics Pennsylvania State University University Park Pa and School of Natural Science Institute for Advanced Study Princeton NJ April Abstract We intro duce a new basis on the state space of nonp erturbative quantum gravity The states of this basis are linearly indep endent are well dened in b oth the lo op representation and the connection representation and are lab eled by a generalization of Penroses spin networks The new basis fully gr-qc/9505006 4 May 1995 reduces the spinor identities SU Mandelstam identities and simplies calculations in nonp erturbative quantum gravity In particular it allows a simple expression for the exact solutions of the Hamiltonian constraint WheelerDeWitt equation that have b een discovered in the lo op represen tation Since the states in this basis diagonalize op erators that represent the three geometry of space such as the area and volumes of arbitrary surfaces and regions these states provide a discrete picture of quantum geometry at the Planck scale y rovellivmscispittedu smolinphyspsuedu Intro duction The lo op representation is a formulation of quantum eld theory suit able when the degrees of freedom of the theory are given by a gauge eld or a connection This formulation has b een used in the context of contin uum and lattice gauge theory and it has found a particularly eective application in quantum gravity b ecause it allows a description of the dieomorphism invariant quantum states in terms of knot theory and at the same time b ecause it partially diagonalizes the quantum dynamics of the theory leading to the discovery of solutions of the dynamical constraints Recent results in quantum gravity based on the lo op representation include the construction of a nite physical Hamiltonian op erator for pure gravity and fermions the computation of the physical sp ectra of area and volume and the develop ement of a p erturbation scheme that may allow transition amplitudes to b e explicitely computed A mathematically rigorous formulation of quantum eld theories whose cong uration space is a space of connections inspired by the lo op representation has b een recently develop ed and the kinematics of the theory is now on a level of rigor comparable to that of constructive quantum eld theory This approach has also pro duced interesting mathematical spino s such as the construction of dieomorphism invariant generalised measures on spaces of connections and could b e relevant for a constructive eld theory approach to nonab elian YangMills theories Applications of the lo op representation however have b een burdened by complications arising from two technical nuisances The rst is given by the Mandelstam identities b ecause of which the lo op states are not indep endent and form an overcomplete basis The second is the presence of a certain sign n factor in the denition of the fundamental lo op op erators T for n This sign dep ends on the global connectivity of the lo ops on which the op erator acts and obstructs a simple lo cal graphical description of the op erators action In this work we describ e an elegant way to overcome b oth of these complications This comes from using a particular basis which we denote as spin network basis since it is related to the spin networks of Penrose The spin network basis has the following prop erties i It solves the Mandelstam identities n ii It allows a simple and entirely lo cal graphical calculus for the T op erators iii It diagonalises the area and volume op erators The spin network basis states b eing eigenstates of op erators that corresp ond to measurement of the physical geometry provide a physical picture of the three dimensional quantum geometry of space at the Planckscale level The main idea b ehind this construction long advo cated by R Loll is to identify a basis of indep endent lo op states in which the Mandelstam identities are completely reduced We achieve such a result by exploiting the fact that all irreducible representations of SU are built by symmetrized p owers of the fundamental representation We will show that in the lo op representation this translates into the fact that we can suitably antisym metrize all lo ops overlapping each other without lo osing generality More precisely the suitably antisymmetrized lo op states span but do not over span the kinematical state space of quantum gravity The indep endent basis states constructed in this way turn out to b e lab elled by Penroses spin networks and by a direct generalization of these A spin network is a graph whose links are colored by integers sat isfying simple relations at the intersections Roger Penrose intro duced spin networks in a context unrelated to the present one remarkably however his aim was to explore a quantum mechanical description of the geometry of space which is the same ambition that underlies the lo op representation construction The idea of using a spin network basis has app eared in other contexts in which holonomy of a connection plays a role including lattice gauge theory and top ological quantum eld theory The use of this basis in quantum gravity has b een suggested previously but its precise implementation had to await resolution of the sign diculties mentioned ab ove Here these diculties are solved by altering a sign in the relation b etween the graphical notation of a lo op and the corresp onding quantum state This mo died graphical notation for the lo op states allows us to reduce the lo op states to the indep endent ones by simply antisym metrizing overlapping lo ops The spinnetwork construction has already suggested several directions of investigation which are b eing pursued at the present time The fact that it diagonalizes the op erator that measures the volume of a spatial slice gives us a physical picture of a discrete quantum geometry and also makes the spin network basis useful for p erturbation expansions of the dynamics of general relativity as describ ed in It has also played a role in the mathematically rigorous investigations of refs Another intriguing suggestion is the p ossibility of considering q deformed spinnetworks on which we will comment in the conclusion The details of the application of the spin network basis to the diagonal ization of the volume and area op erators have b een describ ed in an earlier pap er The primary aim of this pap er is to give an intro duction to the spin network basis and to its use in nonp erturbative quantum gravity We emphasize the details of its construction at a level of detail and rigor that we hop e will b e useful for practical calculations in quantum gravity No claims are made of mathematical rigor for that we p oint the reader to the recent works by Baez and Thiemann where the spin network basis is put in a rigorous mathematical context Finally we note that in this pap er we work with SL C or SU spinors which are relevant for the application to quantum gravity but a spin network basis such as the one we describ e exists for all compact gauge groups This pap er is organized as follows In the next section we briey explain the two problems that motivate the use of the spin network basis This leads to section in which we provide the denition of spin network states in the lo op representation In section we describ e the spin network states as they app ear in the connection representation The pro of that the spin network states do form a basis of indep endent states may then b e given in section Following this in section we review the general structure of the transformation theory in the sense of Dirac b etween the lo op representa tion and the connection representation The use of the spin network basis considerably simplies the transformation theory as we show here Simi larly old results on the existence of solutions to the hamiltonian constraint and exact physical states of quantum gravity may b e expressed in a simpler way in terms of the spin network basis Its use makes it unnecessary to explicitly compute the extensions of characteristic states of nonintersecting 1 knots to intersecting lo ops as describ ed in Finally an imp ortant side result of the analysis ab ove is that it indicates how to mo dify the graphical calculus in lo op space in order to get rid of the annoying nonlo cality due to the dep endence on global ro oting The new notation that allows a fully lo cal calculus is dened in section The pap er closes with a brief summary of the results in section and with a short 1 These characteristic states were previously dened to b e equal to one on the knot class of a nonintersecting lo op zero on all other nonintersecting lo ops with an extension to the classes of intersecting lo ops dened by solving the Mandelstam identities Now they may b e succinctly describ ed as b eing equal to one on one element of the spin network basis and zero on all the others app endix in which we discuss the details of the construction of higher than trivalent vertices Denition of the problem The lo op representation is dened by the choice of a basis of bra states hj on the state space of the quantum eld theory These states are lab elled by lo ops By a lo op we mean here a set of a nite numb er of single lo ops 1 by a single lo op we mean a piecewise smo oth map from the circle S into the space manifold The lo op basis
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