A Spin Network Primer Seth A

A spin network primer Seth A. Majora) Institut fu¨r Theoretische Physik, Der Universita¨t Wien, Boltzmanngasse 5, A-1090 Wien, Austria ͑Received 8 February 1999; accepted 15 April 1999͒ Spin networks, essentially labeled graphs, are ‘‘good quantum numbers’’ for the quantum theory of geometry. These structures encompass a diverse range of techniques which may be used in the quantum mechanics of ﬁnite dimensional systems, gauge theory, and knot theory. Though accessible to undergraduates, spin network techniques are buried in more complicated formulations. In this paper a diagrammatic method, simple but rich, is introduced through an association of 2ϫ2 matrices with diagrams. This spin network diagrammatic method offers new perspectives on the quantum mechanics of angular momentum, group theory, knot theory, and even quantum geometry. Examples in each of these areas are discussed. © 1999 American Association of Physics Teachers.

I. INTRODUCTION methods to model space–time. As the absolute space and time of Newton is a useful construct to apply in many every- Originally introduced as a quantum model of spatial day calculations, perhaps continuous space–time is simply geometry,1 spin networks have recently been shown to pro- useful as a calculational setting for a certain regime of phys- vide a basis for the states of quantum geometry—kinematic ics. 2 states in the Hamiltonian study of quantum gravity. At their Motivated by these difficulties, Penrose constructed a dis- roots, spin networks provide a description of the quantum crete model of space. The goal was to build a consistent mechanics of two-state systems. Even with this humble foun- model from which classical, continuum geometry emerged dation, spin networks form a remarkably diverse structure only in a limit. Together with John Moussouris, he was able which is useful in knot theory, the quantum mechanics of to show that spin networks could reproduce the familiar angular momentum, quantum geometry, and other areas. three-dimensional angles of space—a ‘‘theory of quantized Spin networks are intrinsically accessible to undergradu- directions.’’ 4 In this setting, spin networks were trivalent ates, but much of the material is buried in more complex graphs labeled by spins. For applications in quantum geom- formulations or lies in hard-to-find manuscripts. This article is intended to fill this gap. It presents an introduction to the etry it is better to work with spin networks with higher va- diagrammatic methods of spin networks, with an emphasis lence vertices. on applications in quantum mechanics. In so doing, it offers These suitably generalized spin networks have been undergraduates not only a fresh perspective on angular mo- shown to form the eigenspace of operators measuring geo- 5 mentum in quantum mechanics but also a link to leading metric quantities such as area and volume. These new spin edge research in the study of the Hamiltonian formulation of network techniques arose out of a powerful suite of methods quantum gravity. One quantum operator of geometry is pre- for background-independent quantization that has been de- sented in detail; this is the operator which measures the area veloped over the past few years. Spin networks are fantasti- of a surface. cally useful both as a basis for the states of quantum geom- The history of spin networks goes back to the early sev- etry and as a computational tool. Spin network techniques enties when Penrose first constructed networks as a funda- were used to compute the spectrum of area and volume mentally discrete model for three-dimensional space. Diffi- operators.6 Spin networks, first used as a combinatorial basis culties inherent in the continuum formulation of physics led for space–time, now find uses in quantum gravity, knot 3 Penrose to explore this possibility. These difficulties come theory, and group theory. from both quantum and gravitational theory as seen from This spin network primer begins by associating 2ϫ2 ma- three examples: First, while quantum physics is based on trices with diagrams. The first goal is to make the diagram- noncommuting quantities, coordinates of space are commut- matics ‘‘planar isotopic,’’ meaning the diagrams are invari- ing numbers, so it appears that our usual notion of space ant under smooth deformations of lines in the plane. It is conflicts with quantum mechanics. Second, on a more prag- analogous to the manipulations which one would expect for matic level, quantum calculations often yield divergent an- ordinary strings on a table. Once this is completed, the struc- swers which grow arbitrarily large as one calculates physical quantities on finer and smaller scales. A good bit of machin- ture is enriched in Sec. II C to allow combinations and inter- ery in quantum field theory is devoted to regulating and sections between lines. This yields a structure which includes renormalizing these divergent quantities. However, many of the rules of addition of angular momentum. It is further ex- these difficulties vanish if a smallest size or ‘‘cutoff’’ is in- plored in Sec. III with the diagrammatics of the usual angular troduced. A discrete structure, such as a lattice, provides momentum relations of quantum mechanics. ͑A reader more such a cutoff. Thus, were space–time built from a lattice or familiar with the angular momentum states of quantum me- network, then quantum field theory would be spared many of chanics may wish to go directly to this section to see how the problems of divergences. Third, there is a hint coming spin networks are employed in this setting.͒ In Sec. IV this from general relativity itself. Since regular initial data, say a connection to angular momentum is used to give a diagram- collapsing shell of matter, can evolve into a singularity, rela- matic version of the Wigner–Eckart theorem. The article fin- tivity demonstrates that the space–time metric is not always ishes with a discussion on the area operator of quantum grav- well-defined. This suggests that it is profitable to study other ity.

972 Am. J. Phys. 67 ͑11͒, November 1999 © 1999 American Association of Physics Teachers 972 II. A PLAY ON LINE ͑2͒ This section begins by building an association between the ϫ ͑ ␦B Kronecker delta functions, the 2 2 identity matrix or A), and a line. It is not hard to ensure that the lines behave like However, these ‘‘topological’’ difficulties are fixed by modi- elastic strings on a table. The association and this require- fying the definition of a bent line. One can add an i to the ment lead to a little bit of knot theory, to the full structure of antisymmetric tensors spin networks, and to a diagrammatic method for the quantum mechanics of angular momentum.

Since each of the two awkward features contains a pair of A. Line, bend, and loop ⑀’s, the i ﬁxes these sign problems. However, there is one more property to investigate. The Kronecker ␦B is the 2ϫ2 identity matrix in compo- ␦D␦C⑀ ϭϪ⑀ ͑ A On account of the relation A B˜ CD ˜ AB one has the nent notation. Thus, indices C and D are added to the diagram for clarity͒

10 ͑␦B͒ϭͩ ͪ A 01 —not what one would expect for strings. This final problem ␦0ϭ␦1ϭ ␦1ϭ␦0ϭ can be cured by associating a minus sign with each crossing. and 0 1 1 while 0 1 0. The indices A and B in this ⑀ expression may take one of two values, 0 or 1. The diagram- Thus, by associating an i with every and a sign with matics begins by associating the Kronecker ␦ with a line every crossing, the diagrams behave as continuously deformed lines in a plane. The more precise name of this concept is known as planar isotopy. Structures which can be moved about in this way are called topological. What this association of curves with ␦’s and ˜⑀’s accomplishes is that it allows one to perform algebraic calculations by moving lines The position of the indices on ␦ determines the location of in a plane. the labels on the ends of the line. Applying the definitions A number of properties follow from the above definitions. one has The value of a simple closed loop takes a negative value7

͑3͒

If a line is the identity then it is reasonable to associate a ⑀ ⑀ABϭϪ⑀ ⑀ABϭϪ curve with a matrix with two upper ͑or lower͒ indices. There since ˜ AB˜ AB 2; a closed line is a number. is some freedom in the choice of this object. As a promising This turns out to be a generic result in that a spin network ⑀ possibility, one can choose the antisymmetric matrix AB , which has no open lines is equivalent to a number. A surprisingly rich structure emerges when crossings are 01 considered. For instance the identity, often called the ͑⑀ ͒ϭ͑⑀AB͒ϭͩ ͪ ‘‘spinor identity,’’ links a pair of epsilons to products of AB Ϫ10 deltas, so that ⑀ ⑀BDϭ␦B␦DϪ␦D␦B AC A C A C .

Using the deﬁnitions of the ˜⑀ matrices one may show that, Similarly, diagrammatically, this becomes

͑4͒

As a bent line is a straight line ‘‘with one index lowered’’ ␦C⑀ ϭ⑀ Note that the sign changes, e.g., this choice fits well with the diagrammatics: A CB AB . After a bit of experimentation with these identifications, one discovers two awkward features. The diagrams do not match the expected moves of elastic strings in a plane. First, This diagrammatic relation of Eq. ͑4͒ is known as ‘‘skein ␦C⑀ ⑀DE␦Bϭ⑀ ⑀DBϭϪ␦B relations’’ or the ‘‘binor identity.’’ The utility of the relation since A CD E AD A , straightening a line yields a negative sign: becomes evident when one realizes that the equation may be applied anywhere within a larger diagram. One can also decorate the structure by ‘‘weighting’’ or ͑ ͒ 1 ‘‘tagging’’ edges.8 Instead of confining the diagrams to be simply a sum of products of ␦ ’s and ⑀’s, one can include other objects with a tag. For instance, one can associate a ⑀ ⑀ ⑀CDϭϪ⑀ ϫ ␺B Second, as a consequence of AD BC AB , tagged line with any 2 2 matrix such as A ,

973 Am. J. Phys., Vol. 67, No. 11, November 1999 Seth A. Major 973 These tags prove to be useful notation for angular momen- • Move III: One can perform planar deformations under tum operators and for the spin networks of quantum geom- ͑or over͒ a diagram etry. Objects with only one index can frequently be represented as Kronecker delta functions with only one index. For example,

With a finite sequence of these moves the projection of a knot may be transformed into the projection of any other The result of these associations is a topological structure knot which is topologically equivalent to the original. If one in which algebraic manipulations of ␦ ’s, ⑀’s, and other 2 knot may be expressed as another with a sequence of these ϫ2 matrices are encoded in manipulations of open or closed moves then the knots are called ‘‘isotopic.’’ Planar isotopy is lines. For instance, straightening a wiggle is the same as generated by all four moves with the significant caveat that simplifying a product of two ˜⑀’s to a single ␦. It also turns there are no crossings out that the algebra is ‘‘topological:’’ Any two equivalent algebraic expressions are represented by two diagrams which can be continuously transformed into each other. Making use only intersections of a result of Reidemeister and the identities above it takes a few lines of ␦⑀ algebra to show that the spin network diagrammatics is topologically invariant in a plane. Planar isotopy may be summarized as the manipulations one would expect for elastic, nonsticky strings on a table top—if they are infinitely thin. B. Reidemeister moves Move I on real strings introduces a twist in the string. This move is violated by any line which has some spatial extent in Remarkably, a knot9 in three-dimensional space can be the transverse direction, such as ribbons. Happily, there are continuously deformed into another knot, if and only if the diagrammatic spin networks for these ‘‘ribbons’’ as well.11 planar projection of the knots can be transformed into each other via a sequence of four moves called the ‘‘Reidemeister moves.’’ 10 Though the topic of this primer is mainly on C. Weaving and joining two-dimensional diagrams, the Reidemeister moves are ͑ ͒ given here in their full generality—as projections of knots in The skein relations of Eq. 4 show that given a pair of three-dimensional space. While in two dimensions one has lines, there is one linear relation among the three quantities: only an intersection,

when two lines cross, in three dimensions one has the ‘‘over and crossing,’’

So a set of graphs may satisfy many linear relations. It would and the ‘‘undercross,’’ be nice to select a basis which is independent of this identity. After some work, this may be accomplished by choosing the antisymmetric combinations of the lines—‘‘weaving with a as well as the intersection sign.’’ 12 The simplest example is for two lines,

͑5͒ There are four moves: • Move 0: In the plane of projection, one can make smooth deformations of the curve For more than two lines the idea is the same. One sums over permutations of the lines, adding a sign for each crossing. The general deﬁnition is

• Move I: As these moves are designed for one- ͑ ͒ dimensional objects, a curl may be undone, 6

in which a ␴ represents one permutation of the n lines and ͉␴͉ is the minimum number of crossings for this permutation. This move does not work on garden-variety string. The The boxed ␴ in the diagram represents the action of the ͑ ͒ ͑ string becomes twisted or untwisted . In fact, this is the permutation on the lines. It can be drawn by writing 12¯n, way yarn is made.͒ then permutation just above it, and connecting the same ele- •Move II: The overlaps of distinct curves are not knotted ments by lines.

974 Am. J. Phys., Vol. 67, No. 11, November 1999 Seth A. Major 974 In this deﬁnition, the label n superimposed on the edge With this vertex one can construct many more complex record the number of ‘‘strands’’ in the edge. Edge are usu- networks. After the loop, the next simplest closed graph has ally labeled this way, though I will leave simple 1-lines un- two vertices, labeled. Two other notations are used for this weaving with a sign The general evaluation, given in the appendix, of this diagram is signiﬁcantly more complicated. As an example I These antisymmetrizers have a couple of lovely proper- give the evaluation of ␪͑1,2,1͒ using Eq. ͑8͒, ties, retracing and projection: The antisymmetrizers are ‘‘irreducible,’’ or vanish when a pair of lines is retraced,

͑7͒ which follows from the antisymmetry. Using this and the binor identity of Eq. ͑4͒ one may show that the antisymmetrizers are ‘‘projectors’’ ͑the combination of two is equal to One can build ever more complicated networks. In fact, one one͒ can soon land a dizzying array of networks. I have collected a small zoo in the appendix with full definitions. Now all the elements are in place for the definition of spin networks. A spin network consist of a graph, with edges and vertices, and labels. The labels, associated edges, represent Making the simplest closed diagram out of these lines ⌬ the number of strands woven into edges. Any vertex with gives the loop value often denoted as n , more than three incident edges must also be labeled to specify a decomposition into trivalent vertices. The graphs of spin networks need not be confined to a plane. In a projection of a spin network embedded in space, the crossings The factor nϩ1 expresses the ‘‘multiplicity’’ of the number which appear in the projection may be shown as in the Re- of possible ‘‘A values’’ on an edge with n strands. Each line idemeister moves with over-crossing in the edge carries an index, which takes two possible values. To see this, note that for an edge with a strands the sum of the indices A,B,C,... is 0,1,2,...,a. So that the sum takes a and under-crossing ϩ1 possible values. One may show using the recursion re- ⌬ ͑ ͒ lations for n Ref. 13 that the loop value is equal to this multiplicity. As we will see in Sec. III, the number of possible combinations is the dimension of the representation. As an example of the loop value, the 2-loop has value 3. This is easily checked using the relations for the basic loop III. ANGULAR MOMENTUM REPRESENTATION value ͓Eq. ͑3͔͒ and the expansion of the 2-line using the skein relation As spin networks are woven from strands which take two values, it is well-suited to represent two-state systems. It is ͑8͒ perhaps not surprising that the diagrammatics of spin networks include the familiar ͉jm͘ representation of angular mo- Edges may be further joined into networks by making use mentum. The notations are related as of internal trivalent vertices,

͑Secretly, the ‘‘u’’ for ‘‘up’’ tells us that the index A only The dashed circle is a magniﬁcation of the dot in the diagram takes the value 1. Likewise ‘‘d’’ tells us the index is 0.͒ The on the left. Such dashed curves indicate spin network struc- inner product is given by linking upper and lower indices, for ture at a point. The ‘‘internal’’ labels i, j, k are positive instance, integers determined by the external labels a, b, c via ϭ͑ ϩ Ϫ ͒ ϭ͑ ϩ Ϫ ͒ ϭ͑ ϩ Ϫ ͒ 1 1 1 1 l i a c b /2, j b c a /2, k a b c /2. ͳ ͯ ʹ ϳ͉ϭ1. 2 2 2 2 As in quantum mechanics the external labels must satisfy the l triangle inequalities For higher representations,14 aϩbуc, bϩcуa, aϩcуb ͑9͒ and the sum aϩbϩc is an even integer. The necessity of these relations can be seen by drawing the strands through the vertex. in which

975 Am. J. Phys., Vol. 67, No. 11, November 1999 Seth A. Major 975 1 1/2 rϩs rϪs The total angular momentum z component is the sum of N ϭͩ ͪ , jϭ , mϭ . ͑10͒ individual measurements on each of the subsystems.18 In dia- rs r!s!͑rϩs͒! 2 2 ˆ grams, the action of the Jz operator becomes The parentheses in Eq. ͑9͒ around the indices indicate symmetrization, e.g., u(AdB)ϭuAdBϪϩuBdA. The normalization Nrs ensures that the states are orthonormal in the usual inner product. A useful representation of this state is in terms of the trivalent vertex. Using the notation for u and similarly for d I have

Angular momentum operators also take a diagrammatic 1 form. As all spin networks are built from spin- 2 states, it is 1 ϵប͉ ͘ worth exploring this territory first. Spin- 2 operators have a jm . representation in terms of the Pauli matrices The definition of the quantities r and s was used in the last Ϫ 01 0 i 10 line. ␴ ϭͩ ͪ , ␴ ϭͩ ͪ , ␴ ϭͩ ͪ , 1 10 2 i 0 3 0 Ϫ1 This same procedure works for the other angular momen- ˆ tum operators as well. The Jx operator is constructed from ␴ ˆ with the Pauli matrix 1 . When acting on one line the operator Jx matrix ‘‘flips the spin’’ and leaves a factor ប Sˆ ϭ ␴ i 2 i for iϭ1,2,3. One has The reader is encouraged to try the same procedure for Jˆ . ␴ y 3 1 1 1 1 1 ͯ ʹ ϭ ͯ ʹ , The raising and lowering operators are constructed with 2 2 2 2 2 2 these diagrams as in the usual algebra. For the raising opera- ˆ ϭ ˆ ϩ ˆ tor Jϩ J1 iJ2 one has which is expressed diagrammatically as

In a similar way one can compute Or, since Pauli matrices are traceless,

2 Jˆ ϪJˆ ϩ͉jm͘ϭប ͑rϩ1͒s͉jm͘,

from which one can compute the normalization of these op- 15 and using Eq. ͑8͒ one has erators: Taking the inner product with ͗jm͉ gives the usual normalization for the raising operator

Jˆ ϩ͉jm͘ϭបͱs͑rϩ1͉͒jm͘ϭបͱ͑ jϪm͒͑ jϩmϩ1͉͒jm͘. ͉ 1Ϫ 1 A similar relation holds for the states 2 2͘. The basic action of the spin operators can be described as a ‘‘hand’’ which acts on the state by ‘‘grasping’’ a line.16 The result, Note that since r and s are non-negative and no larger than Ϫ р р after using the diagrammatic algebra, is either a multiple of 2 j, the usual condition on m, j m j, is automatically ␴ satisﬁed. the same state, as for 3 , or a new state. If the operator acts on more than one line, a higher dimensional representation, Though a bit more involved, the same procedure goes then the total action is the sum of the graspings on each through for the Jˆ 2 operator. It is built from the sum of prod- edge.17 ˆ 2ϭ ˆ 2ϩ ˆ 2ϩ ˆ 2 ucts of operators J Jx Jy Jz . Acting once with the ap- ˆ ␴ The Jz operator can be constructed out of the 3 matrix. propriate Pauli operators, one ﬁnds

976 Am. J. Phys., Vol. 67, No. 11, November 1999 Seth A. Major 976 Acting once again, some happy cancellation occurs and the spective on the theorem. It can help those who feel lost in the result is mire of irreducible tensor operators, reduced matrix elements, and Clebsch–Gordon coefﬁcients. A general operator ប2 r2ϩs2 ˆ 2͉ ͘ϭ ͩ ϩ ϩ ϩ ͪ ͉ ͘ T j grasping a line in the j representation (2 j lines͒ to give J jm rs r s jm , 1 1 2 2 m a j2 representation is expressed as which equals the familiar j( jϩ1). Actually, there is a pretty identity which gives another route to this result. The Pauli matrices satisfy

͑ ͒ 11 Just from this diagram and the properties of the trivalent vertex, it is already clear that so the product is a 2-line. Similarly, the Jˆ 2 operator may be ͉ Ϫ ͉р р ϩ j1 j2 j j1 j2 . expressed as a 2-line. As will be shown in Sec. V this simplifies the above calculation considerably. Likewise it is also the case that ϭ ϩ m2 m1 m. IV. A BIT OF GROUP THEORY These results are the useful ‘‘selection rules’’ that are often given as a corollary to the Wigner–Eckart theorem. Notice As we have seen, spin networks, inspired by expressing that the operator expression is a diagram with the three legs simple ␦ and ⑀ matrices in terms of diagrams, are closely related to the familiar angular momentum representation of j, j1 , and j2 . This suggests that it might be possible to ex- press the operator as a multiple of the basic trivalent quantum mechanics. This section makes a brief excursion 19 into group theory to exhibit two results which take a clear vertex. Defining diagrammatic form, Schur’s lemma and the Wigner–Eckart theorem. Readers with experience with some group theory may have noticed that spin network edges are closely related to the irreducible representations of SU͑2͒. The key difference is that, on account of the sign conventions chosen in Sec. one can combine the two lower legs together with Eq. ͑21͒. II A, the usual symmetrization of representations is replaced Applying Schur’s lemma, one finds by the antisymmetrization of Eq. ͑6͒. In fact, each edge of the spin network is an irreducible representation. The tags on the edges can identify how these are generated—through the spatial dependence of a phase, for instance. Since this diagrammatic algebra is designed to handle the combinations of irreducible representations, all the familiar ͑12͒ results of representation theory have a diagrammatic form. where For instance, Schur’s lemma states that any matrix T which commutes with two ͑inequivalent͒ irreducible representa- Ј ϩ ϩ tions Dg and Dg of dimensions a 1 and b 1 is either zero or a multiple of the identity matrix 0 if aÞb This relation expresses the operator in terms of a multiple of TD ϭDЈT for all g෈G⇒Tϭͭ g g ␭ if aϭb. the trivalent vertex. It also gives a computable expression of the multiplicative factor. Comparing the first and last terms Diagrammatically, this is represented as with the usual form of the theorem,20 ͗ ͉ j ͉ ͘ϭ͗ ͉͉ j ͉͉ ͗͘ ͉ ͘ j2m2 Tm j1m1 j2 Tm j1 jmj1m1 j2m2 , one can immediately see that the reduced matrix element ͗ ͉͉ j ͉͉ ͘ ␻ ͑ ͒ j2 Tm j1 is the of Eq. 12 . In this manner, any invariant tensor may be represented as a labeled, trivalent graph. The constant of proportionality is given by ␭ which, being a closed diagram, is equivalent to a number. V. QUANTUM GEOMETRY: AREA OPERATOR The Wigner–Eckart theorem also takes a nice form in the In this final example of the spin network diagrammatic diagrammatic language, providing an intuitive and fresh per- algebra, the spectrum of the area operator of quantum gravity

977 Am. J. Phys., Vol. 67, No. 11, November 1999 Seth A. Major 977 is derived. Before beginning, I ought to remark that the hard work of defining what is meant by the quantum area operator is not done here. The presentation instead concentrates on the calculation of the spectrum. There are many approaches to constructing a quantum theory of gravity. The plethora of ideas arises in part from the lack of experimental guidance and in part from the com- Fig. 1. Two types of intersections of a spin network with a surface. ͑a͒ One pletely new setting of general relativity for the techniques of isolated edge e intersects the surface transversely. The normal nˆ is also quantization. One promising direction arises out of an effort shown. ͑b͒ One vertex of a spin network lies in the surface. All the nontan- to construct a background-independent theory which meets gent edges contribute to the area. Note that the network can be knotted. the requirements of quantum mechanics. This field may be called ‘‘loop quantum gravity’’ or ‘‘spin-net gravity.’’ 21 The key idea in this approach is to lay aside the perturbative methods usually employed and, instead, directly quantize the Hamiltonian theory. Recently this field has bloomed. There The ␶ is proportional to a Pauli matrix, ␶ϭ(i/2)␴. The ␬ is now a mathematically rigorous understanding of the kine- factor is a sign: It is positive when the orientations on the matics of the theory and a number of ͑in principle, testable͒ edge and surface are the same, negative when the edge is predictions of quantum geometry. One of the intriguing re- oriented oppositely from the surface, and vanishes when the sults of this study of quantum geometry is the discrete nature edge is tangent to the surface. The E operator acts like the of space. angular momentum operator Jˆ . Since the E operator van- In general relativity the degrees of freedom are encoded in ishes unless it grasps an edge, the operator only acts where the metric on space–time. However, it is quite useful to use the spin network intersects the surface. new variables to quantize the theory.22 Instead of a metric, in The square of the area operator is calculated first. Calling the canonical approach the variables are an ‘‘electric field,’’ which is the ‘‘square root’’ of the spatial metric, and a vector the square of the integrand of Eq. ͑13͒ Oˆ , the two-handed potential. The electric field E is not only vector but also operator at one intersection is takes 2ϫ2 matrix values in an ‘‘internal’’ space. This electric field is closely related to the coordinate transformation from curved to flat coordinates ͑a triad͒. The canonically Oˆ ͉s͘ϭϪ ͚ ␬ ␬ Jˆ Jˆ ͉s͘, ͑14͒ conjugate A, usually taken to be the configuration variable, is I J I• J eI ,eJ similar to the electric vector potential but is more appropri- ately called a ‘‘matrix potential’’ for A also is matrix valued. 1 It determines the effects of geometry on spin- 2 particles as ˆ they are moved through space.23 States of loop quantum where the sum is over edges eI at the intersection. Here, JI ˆ ϭ ˆ ϩ ˆ ϩ ˆ gravity are functions of the potential A. A convenient basis is denotes the vector operator J Jx Jy Jz acting on the edge ͉ ͘ ˆ ˆ 2 ␬ built from kets s labeled by spin networks s. In this appli- eI . This O is almost J , but for the sign factors I . The area cation of spin networks, they have special tags or weights on operator is the sum over contributions from all parts of the the edges of the graph. Every strand e of the gravitational spin network which thread through the surface. In terms of Oˆ spin network has the ‘‘phase’’ associated with it.24 An ori- over all intersections i, entation along every edge helps to determine these phases or weights. The states of quantum geometry are encoded in the knottedness and connectivity of the spin networks. G In classical gravity the area of a surface S is the integral ˆ ͉ ͘ϭ ˆ 1/2͉ ͘ AS s 3 ͚ Oi s , 4c i ϭ ͵ 2 ͱ AS d x g, S including the dimensional constants. in which g is the determinant of the metric on the surface.25 As a first step, one can calculate the action of the operator The calculation simplifies if the surface is specified by z Oˆ on an edge e labeled by n as depicted in Fig. 1͑a͒. In this ϭ ␬2 0 in an adapted coordinate system. Expressed in terms of case, the hands act on the same edge so the sign is 1, I E, the area of a surface S only depends on the z-vector ϭ1, and the angle operator squared becomes proportional to component26 Jˆ 2! In the calculation one may make use of the Pauli matrix identity of Eq. ͑11͒, ϭ ͵ 2 ͱ ͑ ͒ AS d x Ez Ez. 13 S •

The dot product is in the ‘‘internal’’ space. It is the same product between Pauli matrices as appears in Eq. ͑11͒. In the spin network basis, E is the momentum operator. As p →Ϫiប(d/dx) in quantum mechanics, the electric ﬁeld The edge is shown in the diagram so it is removed spin analogously becomes a ‘‘hand,’’ network s giving the state ͉(sϪe)͘. Now the diagram may be reduced using the recoupling identities. The bubble may be extracted with Eq. ͑18͒,

978 Am. J. Phys., Vol. 67, No. 11, November 1999 Seth A. Major 978 APPENDIX: LOOPS, THETAS, TETS, AND ALL THAT

This appendix contains the basic deﬁnitions and formulas of diagrammatic recoupling theory using the conventions of Kauffman and Lins29—a book written in the context of the more general Temperley–Lieb algebra. The function ␪(m,n,l) is given by

in which Eq. ͑17͒ was also used in the second line. Putting ͑16͒ this result into the area operator, one learns that the area coming from all the transverse edges is27 where aϭ(lϩmϪn)/2, bϭ(mϩnϪl)/2 and cϭ(nϩl Ϫm)/2. An evaluation which is useful in calculating the Gប n ͑n ϩ2͒ ˆ ͉ ͘ϭ ͱ i i ͉ ͘ϭ 2 ͱ ͑ ϩ ͉͒ ͘ spectrum of the area operator is ␪(n,n,2), for which aϭ1, AS s 3 ͚ s l P͚ ji ji 1 s . c i 4 i bϭnϪ1, and cϭ1, ͑15͒ ͑nϩ2͒!͑nϪ1͒! ប ␪͑ ͒ϭ͑Ϫ ͒͑nϩ1͒ The units , c, and G are collected into the Planck length n,n,2 1 2 ϭͱ ប 3ϳ Ϫ35 ͑2n!͒ l P G /c 10 m. The result is also reexpressed in terms of the more familiar angular momentum variables j ͑nϩ2͒͑nϩ1͒ ϭ ϭ͑Ϫ1͒͑nϩ1͒ . ͑17͒ n/2. 2n The full spectrum of the area operator is found by consid- ering all the intersections of the spin network with the sur- A ‘‘bubble’’ diagram is proportional to a single edge, face S, including vertices which lie on the surface as in Fig. ͑ ͒ 28 1 b . Summing over all contributions ͑18͒ l2 ˆ ͉ ͘ϭ P ͓ u͑ uϩ ͒ϩ d͑ dϩ ͒Ϫ t ͑ t ϩ ͔͒1/2͉ ͘ AS s ͚ 2 j v j v 1 2 j v j v 1 j v j v 1 s , 2 v The basic recoupling identity relates the different ways in which three angular momenta, say a, b, and c, can couple to u d ͑ ͒ in which j v( j v) is the total spin with a positive negative form a fourth one, d. The two possible recouplings are re- ␬ t sign and j v is the total spin of edges tangent to the surface lated by at the vertex v. This result is utterly remarkable in that the calculation predicts that space is discrete. Measurements of area can only take these quantized values. As is the case in many quantum systems there is a ‘‘jump’’ from the lowest possible ) 2 ͑19͒ nonzero value. This ‘‘area quanta’’ is ( /2)l p. In an analogous fashion, as for an electron in a hydrogen atom, surfaces where on the right-hand side is the 6 j symbol deﬁned below. make a quantum jump between states in the spectrum of the It is closely related to the Tet symbol. This is deﬁned by30 area operator.

VI. SUMMARY

This introduction to spin networks diagrammatics offers a view of the diversity of this structure. Touching on knot theory, group theory, and quantum gravity this review gives a glimpse of the applications. These techniques also offer a new perspective on familiar angular momentum representations of undergraduate quantum mechanics. As shown with the area operator in Sec. V, it is these same techniques which ͑20͒ are a focus of frontier research in the Hamiltonian quantization of the gravitational ﬁeld. in which ϭ 1͑ ϩ ϩ ͒ ϭ 1͑ ϩ ϩ ϩ ͒ a1 2 a d e , b1 2 b d e f , ϭ 1͑ ϩ ϩ ͒ ϭ 1͑ ϩ ϩ ϩ ͒ ACKNOWLEDGMENTS a2 2 b c e , b2 2 a c e f , ϭ 1͑ ϩ ϩ ͒ ϭ 1͑ ϩ ϩ ϩ ͒ It is a pleasure to thank Franz Hinterleitner and Johnathan a3 2 a b f , b3 2 a b c d , Thornburg for comments on a draft of the primer. I gratefully a ϭ 1͑cϩdϩ f ͒, mϭmax͕a ͖, Mϭmin͕b ͖. acknowledge support of the Austrian Science Foundation 4 2 i j ͑FWF͒ through a Lise Meitner Fellowship. The 6 j symbol is then deﬁned as

979 Am. J. Phys., Vol. 67, No. 11, November 1999 Seth A. Major 979 6 abi Roberto De Pietri and Carlo Rovelli, ‘‘Geometry eigenvalues and the sca- ͫ ͬ⌬ lar product from recoupling theory in loop quantum gravity,’’ Phys. Rev. Tet i abi cdj D 54 ͑4͒, 2664–2690 ͑1996͒. See also, Ref. 5. ͭ ͮ . 7 ª␪͑a,d,i͒␪͑b,c,i͒ This led Penrose to dub these ‘‘negative dimensional tensors.’’ In general cdj relativity, the dimension of a space is given by the trace of the metric, ␮␯ These satisfy a number of properties including the orthogo- g␮␯g , hence the name. 8 nal identity There is some redundancy in notation. Numerical ͑or more general͒ labels associated with edges are frequently called weights or labels. The term abl dai ‘‘tag’’ encompasses the meaning of these labels as well as operators on ͭ ͮͭ ͮ ϭ␦ j lines. ͚ i l cdj bcl 9A mathematical knot is a knotted, closed loop. One often encounters col- and the Biedenharn–Elliot or Pentagon identity lections of, possibly knotted, knots. These are called links. 10K. Reidemeister, Knotentheorie ͑Chelsea, New York, 1948͒, original dilabf afk printing ͑Springer, Berlin, 1932͒. See also Louis Kauffman, Knots and ͚ ͭ ͮͭ ͮͭ ͮ Physics, Series on Knots and Everything Vol. 1 ͑World Scientific, Sin- l emc eli ddl gapore, 1991͒,p.16. 11Louis H. Kauffman, in Ref. 10, pp. 125–130, 443–447. Also, Louis H. abk kbf Kauffman and So´stenes L. Lins, Temperley-Lieb Recoupling Theory and ϭͭ ͮͭ ͮ . cdi emc Invariants of 3-Manifolds, Annals of Mathematics Studies No. 134 ͑Princeton U.P., Princeton, 1994͒, pp. 1–100. Two lines may be joined via 12Note that, because of the additional sign associated with crossings, the ‘‘antisymmetrizer’’ symmetrizes the indices in the ␦⑀ world. 13 ⌬ ϭ ⌬ ϭϪ ⌬ ϭ Ϫ ⌬ Ϫ⌬ ͑ ͒ The loop value satisfies 0 1, 1 2, and nϩ2 ( 2) nϩ1 n . 21 14See Ref. 4, p. 17. 15Roberto De Pietri, ‘‘On the relation between the connection and the loop One also has occasion to use the coefficient of the ‘‘␭- representation of quantum gravity,’’ Class. Quantum Grav. 14, 53–70 move,’’ ͑1990͒. 16See Ref. 6, p. 2671. 17This may be shown by noticing that

͑ ͒ 22 as may be derived using Eqs. ͑4͒ and ͑7͒. 18 ˆ ϭប͚2 jϩ1 ␴ The operator is Jz iϭ1 1 ¯ ( 3/2)i ¯ 1 where the sum is over ͒ the possible positions of the Pauli matrix. a Electronic mail: [email protected] 19Since the Clebsch–Gordan symbols are complete, any map from a b to c 1Roger Penrose, ‘‘Angular momentum: An approach to combinatorial space–time,’’ in Quantum Theory and Beyond, edited by T. Bastin ͑Cam- must be a multiple of the Clebsch-Gordan or 3 j symbol. 20 ͑ bridge U.P., Cambridge, 1971͒; ‘‘Combinatorial quantum theory and See, for instance, Albert Messiah, Quantum Mechanics Wiley, New York, ͒ quantized directions,’’ in Advances in Twistor Theory, edited by L. P. 1966 , Vol. 2, p. 573 or H. F. Jones, Groups, Representations and Physics ͑ ͑Hilger, Bristol, 1990͒, p. 116. Hughston and R. J. Ward, Research Notes in Mathematics, Vol. 37 Pit- 21 man, San Francisco, 1979͒, pp. 301–307; and ‘‘Theory of Quantized Di- See Carlo Rovelli, ‘‘Loop Quantum Gravity,’’ in Living Reviews in Rela- rections’’ ͑unpublished notes͒. tivity, http://www.livingreviews.org/Articles/Volume1/1998-1rovelli for a 2John C. Baez, ‘‘Spin networks in gauge theory,’’ Adv. Math. 117, 253– recent review. 22 272 ͑1996͒, Online Preprint Archive: http://xxx.lanl.gov/abs/gr-qc/ Abhay Ashtekar, ‘‘New variables for classical and quantum gravity,’’ ͑ ͒ ͑ ͒ 9411007; ‘‘Spin networks in nonperturbative quantum gravity,’’ in The Phys. Rev. Lett. 57 18 , 2244–2247 1986 ; New Perspectives in Canoni- ͑ ͒ Interface of Knots and Physics, edited by Louis Kauffman ͑American cal Gravity Bibliopolis, Naples, 1988 ; Lectures on Non-perturbative Ca- Mathematical Society, Providence, RI, 1996͒, pp. 167–203, Online Pre- nonical Gravity, Advanced Series in Astrophysics and Cosmology Vol. 6 ͑ ͒ print Archive: http://xxx.lanl.gov/abs/gr-qc/9504036. World Scientific, Singapore, 1991 . 3 23For those readers familiar with general relativity the potential determines There are more philosophic motivations for this model as well. Mach 1 advocated an interdependence of phenomena: ‘‘The physical space I have the parallel transport of spin- 2 particles. 24 in mind ͑which already includes time͒ is therefore nothing but the depen- In more detail, every edge has a ‘‘holonomy,’’ or path ordered exponential ͑ P ͐ dence of the phenomena on one another. A completed physics that knew i.e., exp e dt e˙(t)•A(e(t))) associated with it.; See, for example, Ab- of this dependence would have no need of separate concepts of space and hay Ashtekar, ‘‘Quantum mechanics of Riemannian geometry,’’ http:// time because these would already have be encompassed’’ ͓Ernst Mach, vishnu.nirvana.phys.psu.edu/riem–qm/riem–qm.html. Fichtes Zeitschrift fu¨r Philosophie 49,227͑1866͒ cited by Lee Smolin in 25The flavor of such an additional dependence is already familiar in flat Conceptual Problems of Quantum Gravity, edited by A. Ashtekar and J. space integrals in spherical coordinates: Aϭ͐r2 sin(␪)d␾ d␪. Stachel ͑Birkhaüser, Boston, 1991͔͒—echoing Leibniz’s much earlier cri- 26See Ref. 5; Abhay Ashtekar and Jerzy Lewandowski, ‘‘Quantum Theory tique of Newton’s concept of absolute space and time. Penrose invokes of Geometry. I. Area operators,’’ Class. Quantum Grav. 14, A55–A81 such a Machian principle: A background space on which physical events ͑1997͒; S. Fittelli, L. Lehner, and C. Rovelli, ‘‘The complete spectrum of unfold should not play a role; only the relationships of objects to each the area from recoupling theory in loop quantum gravity,’’ ibid. 13,2921– other can have significance. 2932 ͑1996͒. 4John P. Moussouris, ‘‘Quantum models as spacetime based on recoupling 27See Ref. 5. theory,’’ Oxford Ph.D. dissertation, unpublished, 1983. 28See A. Ashtekar and J. Lewandowski in Ref. 26. 5Carlo Rovelli and Lee Smolin, ‘‘Discreteness of area and volume in quan- 29See L. Kauffman and S. Lins in Ref. 11. tum gravity,’’ Nucl. Phys. B 442, 593–622 ͑1995͒. 30See L. Kauffman and S. Lins in Ref. 11, p. 98.

980 Am. J. Phys., Vol. 67, No. 11, November 1999 Seth A. Major 980