Quantization of Dieomorphism-Invariant Theories with Fermions John C. Baez Department of Mathematics, University of California Riverside, California 9

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1 Quantization of Diomorphism-Invariant Thoris with Frmions John C. Baz Dpartmnt of Mathmatics, Univrsity of California ivrsid, California 92521, USA Kirill V. Krasnov Cntr for Gravitational Physics and Gomtry, Pnnsylvania Stat Univrsity Univrsity Park, PA 16802, USA and Bogolyubov Institut for Thortical Physics, Kiv 143, Ukrain mail: March 16, 1997 Abstract W xtnd idas dvlopd for th loop rprsntation of quantum gravity to diomorphism-invariant gaug thoris coupld to frmions. Lt P! b a principal G-bundl ovr spac and lt F b a vctor bundl associatd to P whos br is a sum of continuous unitary irrducibl rprsntations of th compact connctd gaug group G, ach rprsntation apparing togthr with its dual. W considr thoris whos classical conguration spac is A F, whr A is th spac of connctions on P and F is th spac of sctions of F, rgardd as a collction of Grassmannvalud frmionic lds. W construct th `quantum conguration spac A F as a compltion of A F. Using this w construct a Hilbrt spac L 2 (A F) for th quantum thory on which all automorphisms of P act as unitary oprators, and dtrmin an xplicit `spin ntwork basis' of th subspac L 2 ((A F)G) consisting of gaug-invariant stats. W rprsnt obsrvabls constructd from holonomis of th connction along paths togthr with frmionic lds and thir conjugat momnta as oprators on L 2 ((A F)G). W also construct a Hilbrt spac H di of diomorphism-invariant stats using th group avraging procdur of Ashtkar, Lwandowski, Marolf, Mour~ao and Thimann. 1

2 1 Introduction In this papr w addrss th problm of th quantization of a wid class of diomorphism-invariant thoris. W considr thoris whos dgrs of frdom ar dscribd in th Hamiltonian framwork by a connction on som principal G-bundl on spac togthr with a st of frmionic lds on spac taking valus in rprsntations of th compact connctd gaug group G. Thoris of this class can b thought of as giving a diomorphism-invariant dscription of frmionic mattr intracting by mans of gaug lds. On of th main motivations for considring such thoris is that gnral rlativity can b rformulatd as a diomorphism-invariant thory having as canonically conjugat variabls a connction on spac and a dnsitizd triad ld [1]. This mans that th gravitational forc coupld to Yang-Mills gaug lds and frmionic mattr ts into our framwork. Our approach builds upon xisting work on th loop rprsntation of quantum gravity and othr diomorphism-invariant thoris of connctions [2]. Th loop quantization of thoris of connctions is basd on an assumption that Wilson loop functionals of connction bcom wll-dnd quantum oprators. Similarly, our approach is basd on an assumption that crtain frmionic analogs of Wilson obsrvabls bcom wll-dnd oprators. An important fatur of our schm is that w do not introduc closd Wilson loops as fundamntal obsrvabls, as typical in th loop rprsntation of a pur gaug thory. Instad, w us opn Wilson paths with frmions or thir conjugat momnta at th ndpoints, which w call `frmionic path obsrvabls'. Closd loops aris as scondary quantitis. Th basic stratgy of th papr can b summarizd as follows. W introduc a st of path obsrvabls on th classical phas spac of our thory that forms a closd algbra with rspct to Poisson brackts. Th main ida thn is to construct a `kinmatical' Hilbrt spac for th thory and rprsnt our algbra by oprators on this Hilbrt spac. This is not th spac of physical stats, bcaus th constraints gaug-invarianc, spatial diomorphism-invarianc, and th Hamiltonian constraint hav not yt bn implmntd. Th quantization procdur w adopt satiss som natural rquirmnts; for xampl, gaug-invariant classical quantitis bcom oprators that prsrv th gauginvariant subspac of th kinmatical Hilbrt spac, `ral' quantitis bcom slf-adjoint oprators, and so on. It turns out, howvr, that rprsntation of our algbra is, in gnral, rducibl vn on th subspac of gaug-invariant stats. An important issu in quantizing diomorphism-invariant gaug thoris coupld to frmions is that of rspcting th classical rality conditions on th quantum lvl. In ld thoris on Minkowski spactim th oprator y of Hrmitian conjugation of frmionic lds snds conguration frmionic lds into thir canonically conjugat momnta. Howvr, in diomorphism-invariant thoris th y opration is mor complicatd. Considr for xampl a thory in 2

3 which th only gaug ld is th gravitational ld, dscribd by th connction ld A a canonically conjugat to th dnsitizd triad ld [1]. Th conguration and momntum frmionic lds (that w shall dnot hr by and ~ rspctivly) hav dirnt dnsity wights, and th action of y oprator involvs th squar root of th dtrminant of th mtric: ~ y?i: (1) Hr is th squar root of th dtrminant of th mtric, dnd in trms of th dnsitizd triad ld. In th quantum thory, on wants to impos this `rality condition' in an appropriat form. Howvr, th following dicultis may arris. In gnral rlativity th triad ld is a dynamical variabl. In th quantum thory, thrfor, th quantity constructd from th triad ld bcoms an oprator. On might rquir that th adjoint of th oprator ^~ corrsponding to th frmionic momntum b givn by: (^~) y?i^ ^: Lt us considr th commutator d n x ^~(x)f(x); d n x Tr( ^Aa (x)g a (x)) of oprators ^~ and ^Aa smard with tst functions. This commutator should vanish, sinc th corrsponding Poisson brackt is zro in th classical thory. Howvr, if on assums that th connction A a is ral, and applis th y opration to this commutator, on obtains d n x(?i ^)^(x); d n x Tr( ^Aa (x)g a (x)) which appars to b nonzro, sinc th oprator ^ dos not commut with ^A a. This inconsistncy, howvr, may not aris if on dos not assums th connction to b ral. In fact, it is known [3] that in th cas whn frmionic lds ar coupld to th slf-dual gravity th connction must b complx, and its ral and imaginary parts dpnd on th frmionic lds. Unfortunatly, th functional analysis rquird to quantiz thoris of complx connctions is not fully dvlopd. Th stratgy w adopt in this papr for daling with th issu of rality conditions is as follows. W assum that th thoris of intrst can b cast into a form in which th connction bcoms ral by an appropriat canonical transformation, as it happns, for instanc, in th cas of pur gravity [4]. On can thn impos th rality conditions on th quantum lvl by rquiring that classical ral quantitis ar rprsntd by slf-adjoint oprators at th quantum lvl. On has a satisfactory quantum thory if a sucint numbr of physically intrsting ral obsrvabls ar rprsntd by slf-adjoint oprators. 3

4 Th quantization schm adoptd in this papr rspcts th classical rality conditions in th sns that a crtain st of ral obsrvabls is rprsntd by slf-adjoint oprators. Not, howvr, that in this papr only a rathr small st of ral obsrvabls is tratd. For instanc, th quantization of th Hamiltonian constraint, which is itslf a ral functional on th phas spac of th thory (in its `ral' formulation usd in [5]), is not considrd in this papr. Not that our way of daling with th rality conditions (1) dos not allow us to prov that th innr product w construct is slctd by ths rality conditions. Indd, w trat only a rathr small st of ral obsrvabls, whil to prov that th innr product is slctd by th classical rality conditions on has to show that a sucintly larg st of ral obsrvabls bcom slf-adjoint oprators. Th organization of this papr is as follows. In Sc. 2 w start by rviwing th main rsults of th loop rprsntation approach to quantizing diomorphism-invariant thoris of connctions. In Sc. 3 w spcify th class of thoris which ar of intrst for us hr and giv som xampls of thoris blonging to this class. In Sc. 4 w dscrib th path obsrvabls which ar th basic building blocks of our quantum thory. In Sc. 5 w dscrib th classical conguration spac and a crtain compltion of it which w call th `quantum conguration spac'. W thn construct th kinmatical Hilbrt spac as a spac of functions on th quantum conguration spac. In Sc. 6 w construct th Hilbrt spac of gaug-invariant stats, and dscrib a basis of this spac givn by `opn spin ntworks' with nds lablld by frmionic lds. In Sc. 7 w dn oprators rprsnting th path obsrvabls. Th issu of diomorphism-invariant stats is discussd in Sc. 8. W conclud with a discussion of th rsults obtaind, focusing on th issu of suprslction sctors. Dspit its grat importanc, w do not discuss dynamics in this papr, as th corrct tratmnt of th Hamiltonian constraint is still a mattr of controvrsy in th loop rprsntation of pur gravity, dspit rcnt progrss on this front [5]. Aftr this work was compltd, paprs by Thimann [6] appard in which both kinmatical and dynamical aspcts of diomorphism-invariant thoris with frmions ar tratd. Thimann proposs a dirnt stratgy to dal with th frmionic rality conditions (1) on th quantum lvl. His work also contains a proposal for th quantum Hamiltonian constraint for thoris with frmions. 2 Quantization of diomorphism-invariant thoris of connctions This sction rviws th main stps of quantization of diomorphism-invariant thoris of connctions. S rfrncs [2, 7, 8] for mor dtails. In th Hamiltonian formalism, th kinmatical phas spac of such a thory consists of pairs (A; ~ E) satisfying suitabl rgularity conditions. Hr A is a 4

5 connction on a principal G-bundl P ovr a manifold which rprsnts `spac', and th ld E ~ is th canonically conjugat momntum (th tild mans that it is a dnsitizd vctor ld). For tchnical rasons w assum that is ralanalytic and G is a compact connctd Li group. Actually th assumption that is analytic is unncssary [9]; much of what w do in this papr gnralizs to th smooth cas, but analyticity maks things a bit simplr. Th classical conguration spac of our thory is th th spac of smooth connctions on P, which w dnot as A. In th cas of canonical quantization of a thory with a nit numbr of dgrs of frdom on normally rprsnts quantum stats by squar-intgrabl functions on th classical conguration spac. Th Hilbrt spac of stats is thn constructd as th spac of such functions with th innr product bing dnd by th usual intgral ovr th conguration spac, h j i d, whr d is som masur on th conguration spac. Howvr, in th cas of thoris with an innit numbr of dgrs of frdom, which is of intrst for us hr, thr oftn dos not xist an appropriat masur on th classical conguration spac. For xampl, thr is no `Lbsgu masur' on an innit-dimnsional vctor spac. Howvr, on can oftn complt th classical conguration spac in som topology and construct a suitabl masur on th rsulting `quantum conguration spac'. In ld thoris, this largr spac usually includs distributional lds in addition to smooth lds. Also, it is common in ld thoris for th classical conguration spac to hav zro masur with rspct to th masur on th quantum conguration spac. Th quantum conguration spac dpnds on a choic of a functions on th classical conguration spac, which play th rol of distinguishd obsrvabls. In our cas w work with (smooth) `cylindr functions' on A, that is, functions dpndin smoothly on th holonomy of th connction A along nitly many analytic paths. In othr words, is a cylindr function if it is of th form (A) (P xp 1 A; : : : ; P xp n A) for som analytic paths 1 ; : : : ; n in and som smooth function on G n. Th most wll-known functions of this form ar th Wilson loops, introducd as obsrvabls for quantum gravity by ovlli and Smolin [10] in thir original papr on th loop rprsntation. Unlik th Wilson loops, th abov cylindr functions ar not ncssarily gaug-invariant. This allows thm to srv as a complt st of functions on A. On can complt th algbra of cylindr functions with rspct to th sup norm to obtain a commutativ C*-algbra Fun(A) for which th -opration is just pointwis complx conjugation. Th Glfand-Naimark thorm thn tlls us that this C*-algbra is isomorphic to th algbra of all continuous functions on its spctrum. W tak th spctrum of Fun(A) to b th quantum conguration spac of our thory. W dnot this spac by A, bcaus it contains th classical conguration spac A as a dns subst. 5

6 Elmnts of th quantum conguration spac ar calld `gnralizd connctions', and can b dscribd as follows. First, dn a `transportr' from th point p to th point q to b a map from P p to P q that commuts with th right action of G on th bundl P. If w trivializ th bundl ovr p and q, w can think of such a transportr simply as an lmnt of G. A `gnralizd connction' A is a map assigning to ach orintd analytic path in a paralll transportr A : P p! P q, whr p is th initial point of th path and q is th nal point. W rquir that A satisfy crtain obvious consistncy conditions: A should assign th sam transportr to two paths that dir only by an orintation-prsrving rparamtrization, it should assign to th invrs of any path th invrs transportr, and it should assign to th composit of two paths th composit transportr. An ordinary smooth connction A givs a gnralizd connction whr th paralll transportr A along any path is simply th holonomy of A along this path, so A A. Also, any cylindr function xtnds to a continuous function on A by stting (A) (A 1 ; : : : ; A n ) for any gnralizd connction A. This lts us think of cylindr functions as functions on th quantum conguration spac. Thr is a natural masur 0 on A, which coms from Haar masur on th group G. On can show that any rasonabl masur on A is dtrmind by th valus of th intgrals of all cylindr functions. Thus, to dn th masur 0 w spcify th valus of th intgrals d 0 for all cylindr functions givn as abov. Such an intgral is dnd as th intgral ovr th corrsponding copis of th group G: d 0 (g 1 ; : : : ; g n ) dg 1 dg n whr dg dnots th normalizd Haar masur on G. Th masur 0 has thr important proprtis. First, it is gaug-invariant. Mor prcisly, not that any gaug transformation g acts on A 2 A to giv a gnralizd connction A 0 with A 0 g(q)?1 A g(p), whr p; q ar th nal and th initial point of th path, rspctivly. This givs an action of th group G of gaug transformations of P on th spac A, and this action prsrvs th masur 0. Scond, 0 is diomorphism-invariant. Mor prcisly, 0 is invariant undr all automorphisms of th bundl P, not ncssarily acting as th idntity on th bas spac. Third, 0 is strictly positiv, maning that d 0 > 0 for any nonngativ intgrabl function on A xcpt th function 0. W dn th kinmatical Hilbrt spac L 2 (A) to b th spac of functions on A that ar squar-intgrabl with rspct to th masur d~ 0. This spac is not th physical stat spac, sinc it contains stats that ar not invariant undr th `gaug' symmtris of our thory, that is, gaug transformations and diomorphisms of spactim. On way to try to nd physical stats is to 6

7 look for solutions of quantum constraints in L 2 (A). In gnral, th solutions may not liv in L 2 (A), but in som compltion throf, but for th Gauss law constraint this problm dos not occur: thr is a larg subspac of L 2 (A), th `gaug-invariant Hilbrt spac', consisting of gaug-invariant squar-intgrabl functions on A. Altrnativly, sinc th masur 0 is gaug-invariant, it givs ris to a wll-dnd masur on th spac AG of gnralizd connctions modulo gaug transformations. Thn w may quivalntly dn th gauginvariant Hilbrt spac to b th spac L 2 (AG) of squar-intgrabl functions on AG. W can construct an xplicit basis of L 2 (AG) using `spin ntworks' [8, 11]. A spin ntwork is a tripl (; ; ) consisting of: (1) a graph analytically mbddd in, (2) a lablling of ach dg of with an irrducibl rprsntation of G, (3) a lablling of ach vrtx v of with an intrtwining oprator v. Hr th dgs of th graph ar assumd to b unparamtrizd but orintd, and v is an intrwining oprator from th tnsor product of th rprsntations corrsponding to th incoming dgs at th vrtx v to th tnsor product of th rprsntations labling th outgoing dgs. (For mor dtails s rfrncs [7, 8]. Hr w allow graphs with isolatd vrtics, i.., vrtics that ar not vrtics of any dg. This is unimportant now but will b important whn w com to frmions.) Without loss of gnrality, spin ntworks? ar always assumd to satisfy a fourth non-dgnracy condition: (4) All rprsntations ar nontrivial and is a `minimal' graph, in th sns that it connot b obtaind from anothr graph 0 by subdividing dgs of 0. Th `spin ntwork stat'? is a gaug-invariant cylindr function on A constructd from th spin ntwork? as follows: O O?(A) (A ) whr `' stands for contracting, at ach vrtx v of, th uppr indics of th matrics corrsponding to th incoming dgs, th lowr indics of th matrics assignd to th outgoing dgs, and th corrsponding indics of th intrtwining oprator v. Bing gaug invariant, such stats li in L 2 (AG). W obtain an orthonormal basis of L 2 (AG) if w us spin ntwork stats corrsponding to all possibl choics of and and a choic of an orthonormal basis of intrtwining oprators v for ach vrtx v and ach choic of rprsntations labling incidnt dgs. Th stats in L 2 (AG) ar still not invariant undr th action of diomorphisms of. Thr is a natural unitary rprsntation of Di() on L 2 (AG). For instanc, th action of th oprator U() corrsponding to a diomorphism v v ; 7

8 of on any spin ntwork stat is givn by U() (;;) (;;): Hr is th imag of th graph undr th diomorphism, and ; ar th corrsponding rprsntations and intrtwining oprators associatd with th nw graph. To nd diomorphism-invariant stats on has to impos th quantum diffomorphism constraint. Thr ar vry fw diomorphism-invariant stats in L 2 (AG), so hr w hav to follow th stratgy of looking for solutions in som largr spac. On oftn nds to do this whn solving quantizd constraint quations, and th largr spac is usually chosn to b a spac of linar functionals on som dns subspac of th initial Hilbrt spac. That is, w should choos som dns subspac C of th gaug-invariant Hilbrt spac L 2 (AG), quip it with som topology in which it is complt, and look for solutions in th topological dual C. Not that C L 2 (AG) C. Sinc thr is a simpl gomtrical action of diomorphisms on cylindr functions, it is natural to choos th subspac of gaug-invariant cylindr functions as C. This has a natural topology in which it is complt, th inductiv limit topology. To solv th diomorphism constraint, w thn sk diomorphism-invariant vctors in C. W obtain ths by avraging gaug-invariant cylindr functions ovr th action of th diomorphism group, following th procdur of Ashtkar, Lwandowski, Marolf, Mour~ao and Thimann [2]. It is asist to avrag a spcial sort of gaug-invariant cylindr function, namly a spin ntwork stat?, whr? (; ; ) is som spin ntwork. For tchnical rasons w assum is `typ I', i.., that for vry dg of thr is an analytic function on that vanishs only on th maximal analytic curv xtnding that dg. Lt S() b any st of diomorphisms of with th following proprty: for any graph 0 which quals for som 2 Di(), thr is a uniqu diomorphism 2 S() with 0. Also lt GS() b th group of `graph symmtris' of, that is, th group Iso()TA(), whr Iso() is th group of diomorphisms mapping to itslf, and TA() is th subgroup xing ach dg of. W may dn an lmnt? 2 C by:?() X X 12S() [ 2]2GS() h ( 1 2)? j i; whr 2 C, and whr w choos on rprsntativ 2 for ach quivalnc class [ 2 ] 2 GS(). It is asy to chck that? is diomorphism-invariant. Mor gnrally, suppos th cylindr function P is a (possibly innit) linar combination of such spin ntwork stats? a??. Thn w dn 2 C by: () X? a??(): 8

9 It is asy to s that th sum convrgs and dns a diomorphism-invariant lmnt 2 C. W may dn th innr product of diomorphism-invariant vctors of this form by h j i (): Complting this spac of such vctors in this innr product w thus obtain a Hilbrt spac, th `diomorphism-invariant Hilbrt spac' H di. Th diomorphism-invariant spin ntwork stats? form an orthogonal (but not orthonormal) basis of H di as w lt? rang ovr a st of spin ntworks containing on from ach diomorphism quivalnc class (but rstricting ourslvs to thos whos graph is typ I). For any spin ntwork?, th diomorphisminvariant stat? taks a zro valu on spin ntworks that do not blong to th sam quivalnc class, and a nonzro valu on spin ntworks from th sam quivalnc class. In short, th main stps of th quantization procdur includ: (i) construction of th kinmatical Hilbrt spac of stats L 2 (A), which rquirs th spcication of a quantum conguration spac togthr with an appropriat masur on this spac; (ii) construction of th gaug-invariant Hilbrt spac L 2 (AG) (and, convnintly, an xplicit basis for this spac), and (iii) construction of th diomorphism-invariant Hilbrt spac H di. In what follows w gnraliz all ths stps to th cas of thoris involving frmions. Finally, thr is th problm of imposing th Hamiltonian constraint and th problm of nding solutions to this constraint. In this papr w do not ntr into this all-important problm. In th nxt sction w dscrib th class of thoris which ar considrd in this papr and giv svral xampls of physically intrsting thoris blonging to this class. Thir quantization will b discussd in th following sctions. 3 Th class of thoris In this papr w considr a spcial class of thoris with frmionic dgrs of frdom. Each thory blonging to this class is spcid, rst, by: (i) a principal G-bundl P!, whr th gaug group G is a compact connctd Li group and is a ral-analytic manifold. W also nd to spcify (ii) a nit list I of irrducibl continuous unitary rprsntations of G. Each rprsntation in I corrsponds to an lmntary frmion apparing in th thory. Using any rprsntation 2 I w may associat to P a vctor bundl P G ovr. It will b convnint to lump ths togthr by forming a singl vctor bundl F P G?M sctions of which simultanously dscrib all th frmionic lds in th thory. 2I ; 9

10 From this data w can construct th classical conguration spac and phas spac of th thory. W hav alrady dscribd th classical conguration spac for th gaug lds; this is th spac A of smooth connctions on P. Th corrsponding classical phas spac is th spac of pairs (A; E) ~ whr A is a smooth connction on P and E is a smooth AdP -valud vctor ld of dnsity wight 1. W dnot this classical phas spac as T A, sinc w can think of ~E as a cotangnt vctor to A satisfying crtain smoothnss conditions. Both A and T A bcom innit-dimnsional smooth manifolds in a natural way. Similarly, th classical conguration spac for th frmionic lds is th spac F of smooth sctions of F, and th classical phas spac for th frmionic lds is spac of pairs consisting of a smooth sction of F togthr with a dnsitizd smooth sction of F with dnsity wight 1. W dnot this classical phas spac as T F. Both F and T F bcom innit-dimnsional topological vctor spacs in a natural way. Th classical conguration spac for th whol thory is thus th product A F, whil th classical phas spac is T A T F. In down-to-arth trms, a point in th classical phas spac is simply a list of pairs (A; E); ~ (; ~); : : : ; (; ~!). Hr A is th connction on P and E ~ is its canonically conjugat momntum. Similarly, th lds ; : : : ; ar frmionic conguration lds corrsponding to th rprsntations in I, whil th lds ~; : : : ; ~! ar thir canonically conjugat momnta. In what follows w shall trat th frmionic lds as Grassmann-valud, in ordr to guarant that th Pauli principl holds for our frmions. This amounts to trating F as a suprmanifold with all odd coordinats [12]. Thus w dn th algbra C 1 (F) of `smooth functions' on F to b th xtrior algbra F gnratd by th continuous linar functionals on F. Not that ths ar not functions on F in th standard sns, but only in th sns of suprmanifold thory. Similarly, w dn th algbra C 1 (T F) of `smooth functions' on th frmionic classical phas spac T F to b th xtrior algbra gnratd by th continuous linar functions on T F. W also dn `smooth functions' on th conguration spac and phas spac of th whol thory as follows: C 1 (A F) C 1 (A) C 1 (F); C 1 (T A T F) C 1 (T A) C 1 (T F): Th lattr algbra is th algbra of classical obsrvabls of th thory. Thoris may hav th sam phas spac and dir only in th form of th Hamiltonian, so a thory is nally dtrmind by: (iii) th Hamiltonian. Sinc w shall not actually trat dynamics in this papr, w shall not b vry prcis about th class of allowd Hamiltonians, but w ar intrstd in thos for which th action can b writtn as follows: S dt d 3 x? Tr ~ E a L t A a + ~L t + ~!L t + + N ~ H +N a ~ Ha +Tr N ~ G: (2) 10

11 Hr L t stands for th tim drivativ and th quantitis N; N a ; and N ar Lagrang multiplirs. Th quantitis ~ H; ~ Ha ; and ~ G ar functionals on th phas spac corrsponding to th Hamiltonian, diomorphism and Gauss law constraints. Hr tilds ovr th momntum lds kp track of dnsity wights: a singl tild ovr a symbol stands for a dnsity of wight 1, a doubl tild ovr a symbol stands for a dnsity of wight 2, and a singl tild undr a symbol dnots a dnsity of wight?1. Onc th canonical variabls ar chosn (or in othr words, onc th phas spac is chosn), a thory is dtrmind solly by th Hamiltonian. Th othr two constraint functionals ar dtrmind by th rquirmnt that thy gnrat diomorphisms and gaug transformations on th phas spac of th thory. Lt us giv svral xampls. An xampl of such a thory is Husain-Kuchar modl with frmions [13]. In its simplst vrsion it dscribs a masslss SU(2) frmionic ld intracting with th gravitational ld, with th 4-mtric bing dgnrat in a vry spcial way [14]. In th canonical formulation th Lagrangian of this modl is of th form (2), without th Hamiltonian constraint trm. Thus, th only constraints in th thory ar Gauss and diomorphism constraints. All lds in th modl ar ral, so thr ar no rality conditions to dal with. Thus th quantization schm dvlopd in th prsnt papr givs a complt quantization of this modl. Anothr xampl was considrd in [15]. It dscribs a masslss frmionic ld intracting with a gravitational ld in four-dimnsional (imannian) spactim, with th gravitational ld bing dscribd using a chiral spin connction. Th gaug group is SU(2), and th gravitational dgs of frdom ar dscribd by th SU(2) connction A a and its canonically conjugat momntum ld ~ Ea. Th only frmionic ld of th thory,, taks valus in th spin-12 rprsntation. Th Hamiltonian constraint of th thory consists of two parts: ~H 1 2 Tr( ~ E a ~ E b F ab ) + ~ E a B A D a B ~ A ; whr th rst part is th Hamiltonian of th gaug ld and th scond part is rsponsibl for th dynamics of th frmionic ld. Hr F ab is th curvatur of th connction ld A a, and D a + A a is th covariant drivativ oprator. This thory is th simplst thory of mattr coupld to Einstin gravity. On mor xampl is th thory dscribing massiv frmions intracting both gravitationally and lctromagntically, which was considrd in [16]. Th gaug group hr is SU(2) U(1). Th gravitational and lctromagntic dgrs of frdom ar dscribd togthr by a SU(2) U(1) connction ld A a and canonically conjugat momntum ld Ea ~. Th frmionic dgrs of frdom 11

12 ar dscribd by two Grassmann-valud lds ; taking valus in th rprsntations of spin 12 and charg 1. Th Hamiltonian of th thory consists of th following thr parts. Th part gnrating th dynamics of th gravitational ld is ~H grav 1 2 abc Tr( ~ E a ~ E b ~ B c ); whr th magntic ld ~ Ba is th dual of th curvatur F ab abc ~ Bc, abc bing th totally antisymmtric tnsor of wight?1. Th part gnrating th dynamics of th lctromagntic ld is ~H m 1 h 32?2 ab cdf Tr( E ~ a E ~ c ) Tr( E ~ b E ~ d ) Tr( B ~ ) Tr( B ~ f )? Tr( E ~ b E ~ d ) Tr( E ~ ) Tr( E ~ f )? Tr( ~ E b ) Tr( ~ E d ) Tr( ~ B ) Tr( ~ B f ) whr is th squar root of th dtrminant of th mtric, as dnd using th ~ E ld. Finally, th part of th Hamiltonian rsponsibl for th frmionic lds is ~H frm ~ E a B A? 1 2 Tr( ~ E a ) B A whr m is th frmion mass.? Da B ~ A + D a A ~! B + im ~ A ~! A? () 2 A A In all th thoris abov particls appar in th thory with thir antiparticls. Mathmatically this is manifstd by th fact that ach irrducibl rprsntation that appars in th thory appars with its dual. Howvr, this coms about in vry dirnt ways for th two dirnt thoris abov. Th SU(2) thory in xampls a) and b) contains only on frmionic ld, but this ld transforms undr a rprsntation that is isomorphic to its dual. It dscribs a particl that is its own antiparticl. In xampl c) w hav two dirnt frmionic lds transforming undr dual rprsntations of th group SU(2) U(1). This thory dscribs a particl that is distinct from its antiparticl. On can s that rprsntations must appar with thir duals in ordr to construct a gaug-invariant mass trm in th Hamiltonian, or any othr gauginvariant xprssion bilinar in th conguration frmionic lds. It is snsibl thrfor to limit our attntion to thoris for which ach irrducibl rprsntation of G appars in th list I togthr with its dual. Actually, to stramlin th xposition, w shall considr only th cas whr particls ar distinct from thir antiparticls. That is, w assum: (iia) th list I of rprsntations of G is of th form ( 1 ; 1 ; : : : ; n; n). It is not dicult to xtnd our analysis to th cas of particls that ar thir own antiparticls, and w mntion a fw of th changs that nd to b mad as w com to thm. i 12

13 Having dscribd th class of thoris which ar th subjct of this papr w ar now rady to procd with th quantization program. For this w nd to construct frmionic path obsrvabls, which ar th basic building blocks of th quantum thory. 4 Frmionic path obsrvabls W now introduc `path obsrvabls', which play in our thoris a rol similar to that of th standard Wilson loop obsrvabls in thoris of connctions. W start with by dscribing som path obsrvabls that ar functions on th classical conguration spac AF. Ths obsrvabls ar built from th holonomy of th connction along a path in spac togthr with frmion lds at th ndpoints of this path. Not rst that givn two frmionic lds ; transforming undr dual rprsntations and of th gaug group, on can construct a gaug-invariant quantity from ths lds using th G-invariant bilinar pairing btwn and. This quantity dpnds on th valus p ; p of ths lds at a point p 2 and is givn simply by ( p ; p ). Mor gnrally, givn points p and p 0 and a path from p to p 0, on can paralll transport p to p 0 along th path using th connction A and thn pair it with p 0, obtaining th gaug-invariant quantity ( p jj p 0) ((P xp A) p ; p 0) which w call a `congurational path obsrvabl'. It is also hlpful to hav a graphical notation for a congurational path obsrvabl, in which w rprsnt th orintd path by a lin with an arrow on it, and draw th frmionic lds and as dots at th ndpoints: ( p jj p 0) W may think of th dot lablld by as a particl and th th dot lablld by as its antiparticl. Congurational path obsrvabls can b introducd for any rprsntation in th list I togthr with its dual, or in othr words, for any particl in th th thory togthr with its antiparticl. Not that th congurational path obsrvabls ar vn sinc thy involv a product of two frmion lds. As a rsult thy commut. W may also think of thm as functions on th classical phas spac, but sinc thy involv no momnta thy all Poisson-commut. Lt us now introduc th basic path obsrvabls involving frmionic momntum lds and dscrib th Poisson algbra ths obsrvabls gnrat. call that th momntum ld ~ canonically conjugat to transforms undr th dual rprsntation. This mans that on can apply th abov construction to th ; ~ lds. Namly, givn two points p; p 0 and a path conncting (3) 13

14 thm, on can construct th quantity ( p jj~ p 0) ([P xp A] p ; ~ p 0); which is gaug invariant and dpnds only on th valus of lds ; ~ at th ndpoints p; p 0 of th path and on th holonomy of th connction ld along. Th quantity (jj~) is almost th on which w nd. Howvr, th Poisson brackt of this quantity with a congurational path obsrvabl (jj) is a distribution. Following [16], lt us introduc som `avragd' momntum obsrvabls in such a way that th Poisson brackt of a congurational path obsrvabl with a momntum obsrvabl is again a congurational path obsrvabl. In othr words, lt us introduc momntum obsrvabls in such a way that th rsulting Poisson algbra contains no distributions. To do this w choos an arbitrary rul which spcis a path pp 0 from p to p 0 givn points p; p 0 in som rgion, an opn st whos closur is compact. Having this rul at our disposal, w can intgrat (j pp 0j~) ovr as a function of p 0. W obtain th quantity d n p 0 ( p j pp 0j~ p 0) (4) Hr in our graphical notation w us a dot as bfor to stand for th frmionic conguration ld, but us a littl circl to rprsnt th avragd frmionic momntum ld. Th rsult is a gaug-invariant quantity, which dpnds on valus of th momntum ld ~ and th connction ld A ovr a rgion of th spatial manifold, on th valu of at th point p, and on a rul (p; p 0 ) 7! pp 0. Similarly, on can introduc a quantity d n p 0 (~! p 0j p0 pj p ) dpnding on th momntum ld ~! canonically conjugat to (or, physically, th momntum ld of th antiparticl). On can also construct quantitis involving two momntum lds d n p 0 d n p 0 (~! p j pp 0j~ p 0); Again w rprsnt avragd momntum lds by littl circls. Not that wr w to considr a thory in which coincidd with its antiparticl, thr would b only on path obsrvabl linar in th momntum ld, instad of two dirnt quantitis (4) and (5). Som of th Poisson brackts blow would b dirnt in this cas, but th ncssary modications ar not dicult. (5) (6) 14

15 Th quantitis w hav dscribd so far ar not invariant undr th Hrmitian conjugation opration `y'. This opration is dnd whn th gaug lds of our thory includ gravity, and th crucial proprty it satiss is givn by quation (1). Lt us now introduc a st of `ral' obsrvabls, by which w man functions on th classical phas spac that ar prsrvd by this y opration. Sinc Hrmitian conjugation snds frmionic momntum lds into conguration lds, ral obsrvabls should involv both lds. Lt us introduc th following obsrvabls which ar linar in th frmionic momnta: d n p ( p ; ~ p ); d n p (~! p ; p ); whr (; ) is th bilinar pairing btwn th dual rprsntations and. As w shall s, whn quantizd ths obsrvabls play th rol of numbr oprators masuring th numbr of particls of th appropriat sort in th rgion. Th graphical rprsntation that w us for ths obsrvabls is as shown abov. Th quantitis w hav introducd form a st of functions on th phas spac of th thory. Lt us now dscrib th Poisson algbra which ths quantitis gnrat (w also giv a graphical dscription). First, th Poisson brackts of th obsrvabls linar in th momntum (4-5) with th congurational path obsrvabls (3) ar again congurational path obsrvabls. Th following idntitis can b vrid [16]: ( ( d n p 0 ( p j pp 0j~ p 0) ; ( q jfj q 0) ; f d n p 0 (~! p 0j p0 pj p ) ; ( q jfj q 0) ; f ) ) ( p j pq fj q 0); f ( q jf q0 pj p ): f (7) (8) (9) (10) Hr w us `' to dnot composition of paths. Also, it is assumd in (9) that q 2. If th point q lis outsid of th rgion th Poisson brackt is zro. Similarly, in (10) w assum q 0 2 ; othrwis th Poisson brackt is zro. Not that if th rsult is not zro, in both cass it is givn by a congurational path obsrvabl with th path bing th composit of th paths ; f compatibl with thir orintation. 15

16 Th Poisson brackt of a momntum obsrvabl (4) with anothr obsrvabl of this sort givs ( d n p 0 ( p j pp 0j~ p 0) ; d n q 0 ( q jf qq 0j~ q 0) 0 0 d n q 0 ( p j pp0 f p0q0j~ q0) + d n p 0 ( q jf qq 0 q0 p 0j~ p0): (11) ; Similarly, for (5) w gt ( f ) d n p 0 (~! p 0j p0 pj p ) ; ; f d n p 0 (~! p 0j p0 p f pq j q ) + f ) + 0 d n q 0 (~! q 0jf q0 qj q ) f f 0 d n q 0 (~! q 0jf q0 q qp j p ): (12) + f whr w assum that th point whr th conguration ld in ach of th obsrvabls is valuatd lis insid th rgion ovr which th momntum ld in th othr is smard; othrwis on or both trms of th Poisson brackt vanishs. Th Poisson brackt of (6) and a conguration obsrvabl also consists of two trms: ( ; f d n p 0 d n p 0 (~! p j pp 0j~ p 0) ; ( q jfj q 0) d n p(~! p j pq fj q 0) + ) f d n p 0 ( q jf q0 p 0j~ p0): (13) 0 + f whr again w assum q 2 0 and q 0 2. For th Poisson brackts btwn (6) and obsrvabls linar in momntum w hav ( d n p 0 d n p 0 (~! p j pp 0j~ p 0) ; ; d n p f d n q 0 ( q jf qq 0j~ q 0) 00 d n q 0 (~! p j pq f qq 0j~ q 0); (14) ) f

17 ( d n p 0 d n p 0 (~! p j pp 0j~ p 0) ; ; 00 d n q 0 f 00 d n q 0 (~! q 0jf q0 qj q ) 0 d n p 0 (~! q 0jf q0 q qp 0j~ p 0): (15) ) f whr in (14) w assum q 0 2 0, and in (15) w assum q 2 ; othrwis th brackts vanish. On can also writ down Poisson brackts btwn (3-6) and th ral quantitis (7-8). For xampl, th Poisson brackts btwn congurational path obsrvabls and ths ral quantitis ar givn by ( ( d n p( p ; ~ p ) ; ( q jj q 0) ; d n p(~! p ; p ) ; ( q jj q 0) ; ) ( q jj q 0); ( q jj q 0): Hr in (16) w assum q is containd in, and in (17) w assum q 0 is containd in ; othrwis th Poisson brackts vanish. A nic fatur of th graphical notation is that it givs a simpl mnmonic for th Poisson brackt formulas (9-17) abov. Not that in vry cas, th trms apparing in th Poisson brackt of two of our obsrvabls corrspond to th ways of attaching thm togthr by lling littl mpty circls (i.. momntum lds) of on obsrvabl with dots (i.. congurational lds) of th othr, in a mannr compatibl with th orintations of th paths. (In thoris for which particls coincidd with thir antiparticls, w would not includ orintation arrows on th paths in our graphical notation, and not rquir compatibility of orintations whn attaching paths togthr. This would giv prcisly th xtra trms in th Poisson brackt rlations that actually appar in such thoris.) W s that th obsrvabls w hav introducd ar closd with rspct to Poisson brackts: a Poisson brackt of any two our obsrvabls is a linar combination of ths obsrvabls. Not, howvr, that th path obsrvabls involving frmionic momnta dpnd on a rul assigning to any two points p; p 0 2 a path pp 0. Th ruls that appar on th right hand sid of (9)- (15) ar of th form pq f qq 0, i.., thy ar formd by composition from th ruls w startd with, so if w x a rul w do not obtain a Poisson algbra. Instad, w work with th `big' algbra gnratd by all obsrvabls of th form ) (16) (17) 17

18 (3-8), allowing all possibl choics of a rul pp 0 and all possibl rgions (opn sts whos closur is compact). W dnot this algbra as B. W may summariz by saying that w hav introducd a Poisson algbra B of smooth functions on th classical phas spac T A T F. This algbra is not closd undr th Hrmitian conjugation oprator y. Howvr, it has a distinguishd subalgbra of obsrvabls prsrvd by th y opration, namly th ral-linar combinations of products of th obsrvabls (7-8). Our aim now is to quantiz th obsrvabls (3-8), nding oprators on som Hilbrt spac whos commutators mimic th classical Poisson brackt rlations. W procd in svral stps. W rst construct a kinmatical Hilbrt spac. Thn w dscrib a subspac of gaug invariant stats in this spac, which can b thought of as a Hilbrt spac of solution of Gauss law constraint, and rprsnt th obsrvabls in B as oprators on this spac. 5 Quantum conguration spac and kinmatical Hilbrt spac call that for diomorphism-invariant thoris of connctions, th quantum conguration spac was most cintly obtaind by complting th algbra of cylindr functions to obtain a commutativ C*-algbra, and taking th spctrum of this C*-algbra. W would lik to follow a similar stratgy for our thoris with frmions. Howvr, in our cas th Glfand-Naimark thorm is not applicabl bcaus of th prsnc of Grassmann-valud lds. Anothr stratgy is ndd. call that in th cas of thoris of connctions, w could also dscrib th quantum conguration spac as a crtain compltion of th classical conguration spac. W can implmnt this ida in our cas and construct th frmionic quantum conguration spac F as a compltion of th classical conguration spac F. Namly, w dn F to b th spac of all (not ncssarily smooth) sctions of th vctor bundl F ovr. Th spac F is dns in F in th topology of pointwis convrgnc. Altrnativly, on can dscrib F as th projctiv limit of conguration spacs of crtain frmionic systms with nitly many dgrs of frdom. Namly, givn any nit subst of points V w may considr th conguration spac of a systm of frmions living at th points of V ; this is th product F V Y p2v F p of copis of brs of F, on for ach point in V. Givn nit substs V V 0 of points in thr is a natural projction from F 0 to F V V, and th projctiv limit of all ths spacs F V is F. W also wish to hav a kinmatical Hilbrt spac for th frmion lds. Although w shall dnot this Hilbrt spac by L 2 (F), it is not a spac of 18

19 squar-intgrabl functions on F in th standard sns. W us th notation L 2 (F) just to mphasiz th similarity in notations btwn th frmionic and th gaug lds dgrs of frdom. Th spac L 2 (F) can b constructd in a varity of quivalnt ways. Prhaps th asist is to us th fact that th brs F p ar nit-dimnsional Hilbrt spacs L and lt L 2 (F) b th frmionic Fock spac ovr th Hilbrt spac dirct sum p2 F p. This is th natural Hilbrt spac compltion of th xtrior algbra M ( F p ): p2 Altrnativly, w can think of L 2 (F) as th innit `groundd' tnsor product O p2 F p of th frmionic Fock spacs F p, on for ach point p in spac. (For an introduction to th mathmatics of frmionic Fock spacs and innit groundd tnsor products, s [17].) In th cas of connction lds w can dn th kinmatical Hilbrt spac as a compltion of an algbra of cylindr functions. An analogous fact holds for frmion lds if w dn a `cylindr function' on F to b a smooth function on F dpnding only on th valu of th frmion lds at nitly many points of. Th spac of cylindr functions is thus? M p2 F p F ; th xtrior algbra ovr th algbraic dirct sum L p2 F p. Sinc ach F p is a Hilbrt spac, this spac is naturally isomorphic to? M p2 F p ; and by th abov rmarks w s that it can b compltd to obtain L 2 (F). Finally, w may also dscrib L 2 (F) in th following physically appaling way. For ach nit st V of points of w may start with th thory of frmions living at ths points. Th conguration spac for this thory is th nitdimnsional spac F V dscribd abov. Quantizing this systm w obtain th frmionic Fock spac ovr F V, which is just th xtrior algbra F V quippd with its standard innr product. Givn nit substs V V 0 of points in thr is a natural projction from F V 0 to F V, and th projctiv limit of ths Hilbrt spacs is th kinmatical Hilbrt spac L 2 (F). Putting togthr th quantum conguration spacs for th gaug ld and frmionic dgrs of frdom, w obtain th quantum conguration spac A F 19

20 for th whol thory. Similarly, th kinmatical Hilbrt spac for th full thory is L 2 (A) L 2 (F), which w dnot by L 2 (A F). Not that cylindr functions on A ar dns in L 2 (A), and cylindr functions on F ar dns in L 2 (F). Thus if w dn th algbra of `cylindr functions' on AF to b th tnsor product of th algbras of cylindr functions on A and cylindr functions on F, it follows immdiatly that cylindr functions on A F ar dns in L 2 (A F). In simpl trms, a cylindr function on A F is a wavfunction dpnding on th connction only via its holonomis along nitly many paths in and on th frmionic lds only via thir valus at nitly many points of. As w shall s in th nxt sction, cylindr functions may b thought of as stats of gaug thoris coupld to frmions living on graphs in with nitly many dgs and vrtics. Cylindr functions play an important rol in th quantum thory, for it is natural to dn quantum oprators rst on th dns subspac of cylindr functions, and thn to xtnd thm to th ntir Hilbrt spac L 2 (A F). Th Hilbrt spac L 2 (A F) srvs only as an auxiliary Hilbrt spac of th thory, sinc its stats ar typically not invariant undr gaug transformations, diomorphisms of spac, or tim volution. To nd physical stats w must impos constraints corrsponding to invarianc undr ths symmtris. In th nxt sction w dscrib a basis of solutions to th Gauss law constrant, that is, gaug-invariant lmnts of L 2 (A F). 6 Gaug-invariant stats Thr is a natural unitary rprsntation of th group G of smooth gaug transformations on th Hilbrt spac of quantum stats L 2 (A F). In this sction w dscrib th subspac of L 2 (A F) consisting of that ar invariant undr ths gaug transformations. W dnot this spac by L 2 ((A F)G). W also dscrib an orthonormal basis for L 2 ((A F)G) that is a gnralization of th spin ntwork basis of L 2 (AG). Givn any graph analytically mbddd in, lt E b its st of dgs and lt V b its st of vrtics. Any dg 2 E bgins at som vrtx s() calld its `sourc' and nds at som vrtx t() calld its `targt'. W dn th spac of connctions on to b A Y 2E A whr A is th spac of transportrs from s() to t(). If w trivializ P at all th vrtics of w can think of A as a product of copis of th gaug group G, on for ach dg. Similarly, w dn th classical conguration spac for frmion lds on to b Y F F p : p2v 20

21 W dn L 2 (A ) using normalizd Haar masur on G, dn L 2 (F ) to b th frmionic Fock spac ovr F, and st L 2 (A F ) L 2 (A ) L 2 (F ): W may think of this as a subspac of L 2 (A F). For any graph thr is a unitary rprsntation of G on L 2 (A F ), and w dnot th subspac consisting of gaug-invariant lmnts by L 2 ((A F )G). As in th cas of pur gaug lds [7, 8], th union of ths spacs as rangs ovr all graphs is dns in L 2 ((A F)G). Thus to dscrib L 2 ((A F)G) it sucs to dscrib ths spac for arbitrary graphs. Trivializing P at all th vrtics of w hav L 2 (A F ) O 2E L 2 (G) O v2v F v : (18) Lt us dcompos th Grassmann algbra F v at ach L vrtx as an orthogonal dirct sum of irrducibl rprsntations of G: F v 2S, whr w hav dnotd by S th list of irrducibl rprsntations apparing in F v. Similarly, th Ptr-Wyl thorm says that L 2 (G) L 2p(G), whr th st p(g) contains on irrducibl continuous unitary rprsntation of G from ach quivalnc class, so that (18) implis L 2 (A F ) O 2E? M 2p(G) O v2v?m 2S : (19) Th right hand sid of (19) can b rwrittn as follows: M O?? O O L 2 (A F ) 2p(G); v2s v2v t()v s()v v (20) Th sum hr runs ovr all lablings of dgs by irrducibl rprsntations of th group G, and by all lablings of vrtics by irrducibl rprsntations from th list S. It follows that M O O L 2 ((A F )G) v (21) 2p(G); v2s v Inv?? O t()v s()v whr Inv?N dnots th G-invariant N subspac of th givn rprsntation. Not that Inv t()v s()v v has a natural innr product, and is iso- N N morphic to th spac of intrtwining oprators from t()v to ( s()v ) v. W dnot this spac of intrtwining oprators by? O? O Hom ; v : t()v s()v 21

22 As in th cas of pur gaug lds, th abov discussion also givs an orthonormal basis of spin ntwork stats for L 2 ((AF)G). Unlik spin ntworks for thoris dscribing only gaug lds, th spin ntworks for thoris with frmions can contain dgs with `opn nds'. Such opn nds rprsnt frmionic dgrs of frdom. W dscrib th spin ntwork basis for L 2 ((A F)G) in th following thorm summarizing th rsults of this sction: Thorm 1 Th spac L 2 ((A F)G) has an orthonormal basis of `frmionic spin ntwork stats', ach such stat bing spcid by a choic of: 1. a graph analytically mbddd in, 2. a lablling of ach dg of by an irrducibl rprsntation 2 p(g), 3. a lablling of ach vrtx v of by an irrducibl rprsntation v 2 S, 4. a lablling of ach vrtx v of by an intrtwining oprator v 2 Hom? O t()v ;? O s()v v chosn from an orthonormal basis of such intrtwining oprators. W call th data? (; ; ) a `frmionic spin ntwork', and dnot th frmionic spin ntwork stat corrsponding to? as abov by?. It is intrsting to not th similarity btwn ths frmionic spin ntworks and th xtndd spin ntworks introducd by Ashtkar and Lwandowski [18] as a basis of L 2 (A). Not, howvr, that unlik xtndd spin ntworks of thoris of connctions, frmionic spin ntworks dscrib frmionic dgrs of frdom and ar gaug invariant. 7 Quantization of frmionic path obsrvabls W hav now constructd th kinmatical Hilbrt spac L 2 (A F) as a compltion of th spac of cylindr functions on A F, and found a spin ntwork basis for th gaug-invariant subspac L 2 ((A F)G). Our construction so far has nothing dirctly to do with th frmionic path obsrvabls introducd in Sc. 4. Thus, w hav to mak sur that thr is a way to quantiz ths obsrvabls and obtain oprators on L 2 ((A F)G). In fact, w shall obtain oprators on L 2 (A F) that commut with gaug transformations, and thus map L 2 ((A F)G) to itslf. Thr is an obvious way to promot th congurational path obsrvabls into oprators. W rprsnt th quantitis (jj) by multiplication oprators (j^j) on th spac of cylindr functions on A F as follows: (j^j) (jj) (22) 22

23 whr (jj) on th right hand sid is considrd as a cylindr function on AF. Not that if is a cylindrical function so is (jj). On has, thrfor, a wlldnd action of (j^j) on th spac of cylindr functions. Th obsrvabls (4-8) involving frmionic momnta can also b rst promotd to oprators on th spac of cylindr functions. Th natural way to do this is to rplac th frmionic momntum ld ~ in such obsrvabls by th corrsponding Brzin drivativ. Brzin drivativs [19] ar usually usd in th contxt of ld thoris in Minkowski spactim. Howvr, w can also mak sns of this notion in th contxt of diomorphism-invariant thoris. W introduc Brzin drivativs as oprators on th spac of cylindr functions as follows. call that th algbra of cylindr functions on F is th xtrior algbra? M p2 F p : For any point p choos an orthonormal basis (p) of th subspac of F p corrsponding to th rprsntation 2 I undr which th ld transforms. Not that (p) is a cylindr function on F, and th algbra of cylindr functions on F is gnratd by functions of this form. Thus, formally spaking, w may dn th Brzin drivativ oprator to b th suprdrivation (i.., gradd drivation) of th algbra of cylindr functions such that (p) (p 0 ) ~ n (p; p 0 ); (23) whr is th Kronkr dlta. Howvr, th right hand sid is distributional, so strictly spaking w should smar this quation with a `tst function'. In othr words, for any sction f of P G, w dn d n p f (p) (p) to b th uniqu suprdrivation on th algbra of cylindr functions such that d n p f (p) (p) (p 0 ) f (p 0 ): To dscrib th action of this oprator on an arbitrary cylindr function, lt us introduc th intrior product oprator i[] (p), namly th adjoint of xtrior multiplication by (p). W thn hav d n p f (p) (p) X p2 f (p) i[] (p) ; whr th right hand sid, for any cylindr function, contains only a nit numbr of non-zro trms. 23

24 Using this dnition of Brzin drivativ on can procd with th quantization of obsrvabls (4-6). Howvr, thr is an oprator ordring ambiguity that nds to b rsolvd in this procdur. Sinc th obsrvabls (4-6) involv both frmion lds and thir canonically conjugat momnta, which do not anticommut, w must spcify which acts rst. Not that on of our classical quantitis, namly dn p( p ; ~ p ), has a natural intrprtation as th numbr of particls of typ within th rgion. To obtain this numbr oprator whn w quantiz, w nd to choos th oprator ordring in which th Brzin drivativ corrsponding to th momntum ld ~ p acts bfor th multiplication oprator corrsponding to th conguration ld p. Lt us thrfor dmand that drivativs act bfor multiplication oprators for all our obsrvabls, which amounts to a `normal ordring' prscription. Following this choic of oprator ordring w can now quantiz all our frmionic obsrvabls, obtaining oprators on th spac of cylindr functions. Mor prcisly, w dn th quantum analogs of th obsrvabls (4-6) as follows: X d n p 0 ( p j^ pp 0j~ p 0) (p) (P A) i[] (p 0 ) (24) p 0 2 d n p 0 (~! p 0j^ p0 pj p ) X X p2 d n p X p 0 2 (p) (P 0 d n p 0 (~! p j^ pp 0j~ p 0) p i[] (p) (P d n p ( p ; ~ p ) X p2 d n p (~! p ; p ) X p2 A) i[] (p 0 ) (25) A) i[] (p 0 ) (26) (p) i[] (p) (27) (p) i[] (p) (28) Th oprators (22-28) map cylindr functions to cylindr functions, and xtnd uniquly to boundd linar oprators on L 2 (AF). Sinc thy commut with gaug transformations thy may also b thought of as oprators on L 2 ((A F)G). Finit linar combinations of frmionic spin ntwork stats ar dns in L 2 ((A F)G), so th actions of ths oprators on L 2 ((A F)G) ar dtrmind by thir actions on frmionic spin ntwork stats. On such stats th action is simply dscribd in our graphical notation, which is vry nic for xplicit computations. Lt us giv svral simpl xampls; gnralization to arbitrary spin ntwork stats is straightforward. W considr th action of our oprators on a frmionic spin ntwork stat containing a singl path, that is, a cylindr function of th form (jj). First, th frmionic path obsrvabls 24