SU(2) Lattice Gauge Theory- Local Dynamics on Non-intersecting Electric flux Loops
SSU(2) Lattice Gauge Theory- Local Dynamics onNon-intersecting Electric flux Loops
Ramesh Anishetty ∗ and Indrakshi Raychowdhury † The Institute of Mathematical Sciences,CIT-Campus, Taramani, Chennai, IndiaAugust 28, 2018
Abstract
We use Schwinger Bosons as prepotentials for lattice gauge theory to define local linking oper-ators and calculate their action on linking states for 2 + 1 dimensional SU(2) lattice gauge theory.We develop a diagrammatic technique and associate a set of (lattice Feynman) rules to computethe entire loop dynamics diagrammatically. The physical loop space is shown to contain only non-intersecting loop configurations after solving the Mandelstam constraint. The smallest plaquetteloops are contained in the physical loop space and other configurations are generated by the actionof a set of fusion operators on this basic loop states enabling one to charaterize any arbitrary loopby the basic plaquette together with the fusion variables. Consequently, the full Kogut-SusskindHamiltonian and the dynamics of all possible non-intersecting physical loops are formulated interms of these fusion variables.
Lattice gauge theories, originally defined [1] within the Euclidean framework has found profoundapplicability for performing numerical computations using Monte Carlo simulation. The Hamiltonianapproach [2], although much less studied, has several important advantages over the Euclidean one.Both the Hamiltonian and path integral approach of lattice gauge theories are mostly studied in thestrong coupling limit, albeit the physical/continuum limit exists at weak coupling. Moreover, the mosteconomic and physical description of any gauge theory can only be in terms of gauge invariant degreesof freedom. Reformulation of gauge theories in terms of gauge invariant Wilson loops and stringscarrying fluxes is an old problem in physics [3, 4]. Formulation of gauge field theories on lattice [1]is indeed an important step towards the loop formulation as here one directly works with the linkvariables or holonomies (instead of the gauge field for continuum theories) which are gauge-covariant ∗ [email protected] † [email protected] a r X i v : . [ h e p - l a t ] N ov bjects and are the fundamental building blocks of gauge invariant Wilson loops. However, the gaugeinvariant wilson loops and strings form a over-complete basis for the physical Hilbert space of thetheory. Mandelstam constraints [5] indeed restricts the overcomplete Wilson loops to minimal loopswhich are also sufficiently complete to describe the physical Hilbert space. But that is not a trivial taskmostly because of the nonlocality of the Wilson loop states and their dynamics. This problem becomesmore and more tedious when one approaches the weak coupling limit of lattice gauge theory, whereall possible loops of arbitrary shapes and sizes start contributing. However, in the context of dualitytransformation [7], the electric flux loop and their dynamics has been shown to be manifestly local inthe continuum limit even for non-Abelian lattice gauge theories. Moreover, a recent development inthe formulation of Hamiltonian lattice gauge theory, namely the prepotential formulation [4, 8] hasshown a way to get rid of the problem of nonlocality and proliferation of loop states for any SU(N)gauge theory in arbitrary dimensions.The prepotential formulation is basically a reformulation of Hamiltonian lattice gauge theory interms of SU(N) Schwinger Bosons in which the loop operators and loop states are defined locally ateach site which cuts down the level of complications to a great extent. The Mandelstam constraintsare also local in this formulation which one can solve to find exact and local loop basis at each site.Thus this new local description of lattice gauge theory seems to provide the best framework for anypractical computation in the field of lattice gauge theory. Besides strong coupling calculations theweak coupling regime becomes much more amenable and easy to handle in terms of prepotentials.Using the Schwinger Boson representation of the gauge group at each lattice site, the originalKogut Susskind Hamiltonian [2] and its canonical conjugate variables are reconstructed. In terms ofSchwinger Bosons, the non-Abelian gauge group becomes ultra local at each site and the fluxes alongneighbouring sites flow following the new Abelian constraint, which is easy to handle. However, thefull Hamiltonian, even in terms of local gauge invariant operator is complicated enough while actingon an arbitrary loop state. In this work, exploiting the local description of loops in terms of SchwingerBosons, we calculate all possible action of local gauge invariant operators on any local gauge invariantstate of the theory with explicit realization for SU(2) lattice gauge theory defined on 2 + 1 dimensionallattice. Moreover, to realize the complicated actions and to perform computations (both analyticaland numerical) easily we develop a diagrammatic calculational technique. We describe the local gaugeinvariant state as well as the actions of the gauge invariant operators on those states by diagrams.Each diagram denotes the states together with a numerical coefficient, which can be read off fromit by a set ‘lattice Feynman rules’. We utilize this diagrammatic technique to compute the actionof full Kogut-Susskind Hamiltonian within loop states which is again expressed diagrammatically.Moreover, we improve the loop descriptions given in terms of local linking numbers in prepotentialformulation to a description in terms of fusion variables. The Abelian Gauss laws are solved by thesefusion variables by construction. The electric part of the Hamiltonian is simple in terms of the fusionvariables, which counts the units of flux flowing throughout the lattice and becomes dominant in thestrong coupling limit. The magnetic part of the Hamiltonian which is dominant in the weak couplingregime of the theory is quite complicated but have been written down entirely in terms of the shiftoperators corresponding to fusion variables. Both the diagrammatic representation as well as analyticexpression is given. 2he plan of the paper is as follows: we start with a brief review of the prepotential formulationand relate it to the Kogut-Susskind Hamiltonian formulation in section 2. In section 3, we discussall possible loop operators in prepotential formulation defined locally at each site, and calculate theiraction individually on any loop state characterized by prepotential linking numbers. In this section wedevelop the diagrammatic technique to handle loops. Next in section 4, we shift from linking numbersto fusion variables to characterize any arbitrary loop states within the theory. We also introducethe shift operators corresponding to fusion variables which are responsible for loop dynamics. Theassociated constraints on the states characterized by fusion quantum numbers are also discussed whichare there to define the loop states with only physical degrees of freedom. In section 5, we calculatethe action of the full Kogut-Susskind Hamiltonian in terms of diagrams as well as the fusion variables.In section 6 we briefly illustrate how to compute strong coupling perturbation expansion within ourformulation and compare our results for first few orders with available results. Finally we summarizeour results in section 6 and also discuss the future directions. The prepotential formulation of lattice gauge theory [8] provides us with a platform to work withgauge invariant operators and states defined locally at each site of the lattice. We briefly reviewthis particular formulation in this section for the sake of completeness. Note that, we keep ourselvesconfined to the gauge group SU(2) and 2+1 dimensional lattice in this work, although each of theseideas can be generalized to arbitrary gauge group and arbitrary dimensions as well.In Kogut-Susskind [2] formulation, the canonical conjugate variables in the theory are color electricfields E a L/R ( x, e i ) defined at each site x , for a = 1 , , L/R denotes that the left electric field islocated at the starting end of the link starting from x along e i and R denotes the electric field attachedat the ending point terminating at x + e i . The link operator U ( x, e i )’s are defined on a link originatingfrom site x along e i direction. The Hamiltonian of the theory is given by, H = g (cid:88) x (cid:88) a=1 E a ( x, e i ) E a ( x, e i ) − g (cid:88) plaquette T r (cid:16) U plaquette + U † plaquette (cid:17) (1)where, g is the coupling constant. In (1), U plaquette = U ( x, e ) U ( x + e , e ) U † ( x + e + e , e ) U † ( x + e , e ) is product over links around the smallest closed loop on a lattice, i.e a plaquette and a(= 1 , , T rU plaquette =
T rU † plaquette.The canonical conjugate variables, namely the color electric fields and the link operators satisfythe commutation relation:[ E a L ( x, e i ) , U αβ ( x, e i )] = − (cid:18) σ a U ( x, e i ) (cid:19) α β , (cid:2) E a R ( x + e i ) , U αβ ( x, e i ) (cid:3) = (cid:18) U ( x, e i ) σ a (cid:19) α β . (2)In (2), σ a are the Pauli matrices, satisfying: [ σ a , σ b ] = i(cid:15) abc σ c . The left and right electric fields are3enerators of the gauge transformation and hence follow SU(2) algebra:[ E a L ( x, e i ) , E b L ( x, e i )] = i(cid:15) abc E c L ( x, e i ) , (cid:2) E a R ( x, e i ) , E b R ( x, e i ) (cid:3) = i(cid:15) abc E c R ( x, e i ) , (3) (cid:2) E a L ( x, e i ) , E b R ( x, e i ) (cid:3) = 0 . Note that the left and right generators E a L ( x, e i ) and E a R ( x + e i , e i ) on the link ( x, e i ) are the paralleltransport of each other, i.e E R ( x + e i , e i ) = − U † ( x, e i ) E L ( x, e i ) U ( x, e i ), implying, (cid:88) a=1 E a ( x, e i ) E a ( x, e i ) ≡ (cid:88) a=1 E a L ( x, e i ) E a L ( x, e i ) = (cid:88) a=1 E a R ( x + e i , e i ) E a R ( x + e i , e i ) . (4)Hence the electric part of the Hamiltonian (1) contains either of the electric fields and we choose itto be the left electric field. Under gauge transformation, the left electric field and the link operatortransforms as: U ( x, e i ) → Λ( x ) U ( x, e i )Λ † ( x + e i ) ,E L ( x, e i ) → Λ( x, e i ) E L ( x, e i )Λ † ( x ) , E R ( x + e i , e i ) → Λ( x + e i ) E R ( x + e i , e i )Λ † ( x + e i ) . (5)Also note that, from (5), the SU (2) Gauss law constraint at every lattice site n is G ( n ) = d (cid:88) i =1 (cid:16) E a L ( x, e i ) + E a R ( x + e i , e i ) (cid:17) = 0 , ∀ x. (6)In the next subsection we briefly review how the SU(2) Hamiltonian lattice gauge theory is reformulatedin terms of prepotentials. Instead of associating electric fields and link operators to each link of the lattice as discussed before, letus associate a set of Harmonic oscillator doublets acting as Schwinger Bosons a α ( x, e i ; l ) and a † α ( x, e i ; l )with l = L, R, α = 1 ,
2. We call these oscillators as prepotentials since the electric field operators aswell as the link operators can be reconstructed solely in terms of these. Using the Schwinger BosonsFigure 1: Prepotentials on a linkconstruction of the angular momentum algebra (3), the left and the right electric fields on a link ( x, e i )can be written as:Left electric fields: E a L ( x, e i ) ≡ a † ( x, e i ; L ) σ a a ( x, e i ; L ) , (7)Right electric fields: E a R ( x + e i , e i ) ≡ a † ( x + e i , e i ; R ) σ a a ( x + e i , e i ; R ) . x, e i ) with the prepotential operators whenever we considerone single link at a time.Using (7), the electric field constraint (4) on any link becomes the following number operatorconstraints in terms of the prepotential operators:ˆ n ( L ) ≡ a † ( L ) · a ( L ) = ˆ n ( R ) ≡ a † ( R ) · a ( R ) ≡ ˆ n (8)In (8), ˆ n ≡ ˆ n ( x, e i ). Note that, this is indeed the most novel feature of prepotential formulation, wherethe non-Abelian fluxes can be absorbed locally at a site and the Abelian fluxes spread along the links.Both the gauge symmetries together lead to non-local (involving at least a plaquette) Wilson loopstates.In order to construct the Wilson loop states in terms of prepotentials, it is first necessary to constructlink operators on each link in terms of Schwinger Bosons. From SU (2) gauge transformations of thelink operator in (5) and SU (2) ⊗ U (1) gauge transformations properties of the Schwinger Bosons, wewrite the link operator of the form U αβ = 1 √ ˆ n + 1 (cid:16) ˜ a † α ( L ) a † β ( R ) + a α ( L ) ˜ a β ( R ) (cid:17) √ ˆ n + 1 , (9)The above link operators and electric field satisfies the same canonical commutation relations (3) and(2) together with the property U U † = U † U = 1 & Det U = 1 (10)The loop operators for a gauge theory are constructed by taking the trace of the path ordered productof link operators around any closed curve. Loop operators acting on strong coupling vacuum creates theloop states of the theory. The novel feature of the prepotential formulation is that, the loop operatorsaround any closed path, when re-expressed in terms of Schwinger Bosons turns out to be direct productof gauge invariant operators at each site. We call those local gauge invariant operators as the locallinking operators of the theory and linking states are created by the action of linking operators onstrong coupling vacuum. The linking variables together with the Abelian Gauss law constitutes theloop variables of the theory. In this section we explicitly illustrate all possible linking operators and linking states present at eachsite of a 2 dimensional spatial lattice. We also develop a diagrammatic prescription to illustrate thelinking operators and their actions on an arbitrary linking state, which turns out to be extremely usefulin the study of the Hamiltonian and its dynamics in later sections.We first concentrate at a particular site of a 2-dimensional spatial lattice, where, 4 links meet, eachlink carries its own link operator as given in (9). There exists four basic local gauge invariant operators5constructed by U αβ ( x, e i ) U βγ ( x + e i , e j ) at site ( x + e i )) which we list below:ˆ O i + j + ≡ a † β ( i ) 1 √ ˆ n i + 1 1 (cid:112) ˆ n j + 1 ˜ a † β ( j ) = 1 √ ˆ n i (cid:112) ˆ n j + 1 a † ( i ) · ˜ a † β ( j ) ≡ (cid:112) ˆ n i (ˆ n j + 1) k ij + (11)ˆ O i + j − ≡ a † β ( i ) 1 √ ˆ n i + 1 1 (cid:112) ˆ n j + 1 a β ( j ) = 1 √ ˆ n i a † ( i ) · a ( j ) 1 (cid:112) (ˆ n j + 2) ≡ √ ˆ n i κ ij (cid:112) (ˆ n j + 2) (12)ˆ O j + i − ≡ ˜ a β ( i ) 1 √ ˆ n i + 1 1 (cid:112) ˆ n j + 1 ˜ a † β ( j ) = 1 (cid:112) (ˆ n j + 1) a ( i ) · a † ( j ) 1 √ n i + 1 ≡ (cid:112) (ˆ n j + 1) κ ji √ n i + 1(13)ˆ O i − j − ≡ ˜ a β ( i ) 1 √ ˆ n i + 1 1 (cid:112) ˆ n j + 1 a β ( j ) = ˜ a ( i ) · a ( j ) 1 (cid:112) (ˆ n i + 1)(ˆ n j + 2) ≡ k ji − (cid:112) (ˆ n i + 1)(ˆ n j + 2) (14)where, the labels ( i/j ) associated with prepotential operators actually denote the prepotentials asso-ciated with the links along ( i/j ) directions at that site x . For d = 2, i, j can take values 1 , , ¯1 , ¯2 andeach direction contains a prepotential doublet a † ( i ) as shown in figure 2 . The maximally commutingFigure 2: A particular site on a 2 dimensional lattice and associated prepotentialsgauge invariant set of operators k ij + ’s are called linking operators and k ij − ’s are their conjugates. Thelinking states are constructed by the action of linking operators on strong coupling vacuum. Thus inthe prepotential formulation, defining the linking operators locally at each site enables us to define thelinking states also locally at each site as, | l ij (cid:105) = (cid:16) k ij + (cid:17) l ij l ij ! | (cid:105) . (15)In the prepotential approach, as defined in (15), the linking states are naturally characterized by thelinking quantum numbers l ij , which counts the flux along i − j direction. On a 2d lattice, four linksin direction i , (with i = 1 , , ¯1 , ¯2) meet at a site, each carrying its own prepotential a † ( i ). Note that k ji + = − k ij + by construction given in ((11)), makes the loop space in two spatial dimension, to becharacterized by six linking numbers l ij , for i < j with the convention that 1 < < ¯1 < ¯2. Thus themost general gauge invariant states at a particular site are characterized by the six liking quantumnumbers as follows: | l , l , l , l , l , l ¯1¯2 (cid:105) ≡ |{ l }(cid:105) = (cid:0) k (cid:1) l l ! (cid:0) k (cid:1) l l ! (cid:0) k (cid:1) l l ! (cid:0) k (cid:1) l l ! (cid:0) k (cid:1) l l ! (cid:0) k ¯1¯2+ (cid:1) l ¯1¯2 l ¯1¯2 ! | (cid:105) (16)From the definition of the state (16), one can relate the number of prepotential operators at each link6o the linking quantum numbers in the following way: n = l + l + l (17) n = l + l + l (18) n ¯1 = l ¯1¯2 + l + l (19) n ¯2 = l + l + l ¯1¯2 (20)These numbers are basically eigenvalues of the operators ˆ n i ≡ a † ( i ) · a ( i ). The linking quantum numbersare pictorially represented for a two dimensional lattice in figure 3.Figure 3: SU(2) fluxes: all possible linking at a site of a two dimensional latticeWe now illustrate the action of the linking operators defined in (11), (12) and (14) on the linkingstates defined in (16). We also prescribe a diagrammatic realization of these actions, which seems to bemuch more convenient than dealing with long mathematical expressions. The Mathematical expressioncan be read off from the diagrams by a set of rules given later in this section.The basic local gauge invariant operators arising at a particular site as given in (11-14) are ˆ O i + j + ,ˆ O i + j − and ˆ O i − j − . The first one acts trivially on the states (16), and increases the flux along i − j direction by one unit. With proper factors in the definition of the state in (16) as well as the linkingoperators in (11), the explicit action is obtained as:ˆ O i + j + |{ l }(cid:105) ≡ ( l ij + 1) (cid:112) ( n i + 1)( n j + 2) | l ij + 1 (cid:105) (21)Pictorially (21) is represented in figure 4. Note that, in figure 4, the left hand side contain solid dotsFigure 4: The left and right hand side of = denotes the respective sides of the equation (21) with allthe coefficients.on solid line. The solid dot denotes the operators acting on a state, more specifically dot on a solidline represents prepotential creation operator corresponding to that direction acts on a general state.The right hand side of the equation does not contain any dot and represents the state created. Anysolid linking line passing through i − j direction at a site, denotes that in the new state the flux along7hat particular direction has increased by one unit. Note that, in the pictures we are suppressing thesymbols for the state for brevity. The coefficients in (21) are all subsumed in figure 4. The algebratowards 21 is given in Appendix A.Next we consider the action of (12) on a general linking state. This action is a bit complicated asone need to use all the commutation relations between different k ij + , k ij − and κ ij to move the annihilationoperator towards the right. However, after the algebraic simplification (as shown in Appendix A) theaction of the operators defined in (12), on any arbitrary linking state is obtained as:ˆ O i + j − |{ l }(cid:105) ≡ (cid:112) ( n i + 1)( n j + 2) (cid:88) k (cid:54) = i,j ( − S ik ( l ik + 1) | l jk − , l ik + 1 (cid:105) (22)where, in any of the l ij ’s in the above equation (and also in any equation throughout the paper), theindices are by-default considered to be rearranged in such a way, that the first index is always less thanthe second one in accordance with the ordering convention 1 < < ¯1 < ¯2. The factor S ik is calculatedas, S ik = 1 if i > k & S ik = 0 if i < k. (23)We represent the action of the gauge invariant operator in (22) pictorially in figure 5. In the left handside of figure 5, the solid dot on solid line denotes prepotential creation operator along that directionand solid dot on dashed line denotes annihilation operator along that direction acts on the state. Inright hand side, the dashed line represents that the corresponding solid line in the state is removed ifit was already present in the state and it is zero if there were none already present. Note that as givenFigure 5: The left and right hand side of = denotes the respective sides of the equation (22) with allthe coefficients.in (22), each term comes with a particular coefficient which we absorb in the diagrams itself. This ispossible by providing with a set of rules (similar to the Feynman rules) for associating each diagramwith the coefficient. Having exhausted with all possible linking actions on a general linking states, wewill state all of the rules at the end of this section. These new lattice Feynman rules will enable us todo any loop computation diagrammatically.Let us next consider the remaining local gauge invariant operator ˆ O i − j − and its action on a generallinking state. This action is the most complicated one to calculate as both the annihilation operatorsare needed to move to right by using the commutation relations. A long calculation given in Appendix8 finally yields the following action,ˆ O i − j − |{ l }(cid:105) = 1 (cid:112) ( n i + 1)( n j + 2) (cid:34) ( n i + n j − l ij + 1) | l ij − (cid:105) + (cid:88) i (cid:48) ,j (cid:48) {(cid:54) = i,j } ( l i (cid:48) j (cid:48) + 1)( − S i (cid:48) j (cid:48) | l ii (cid:48) − , l jj (cid:48) − , l i (cid:48) j (cid:48) + 1 (cid:105) (cid:35) (24)To realize the action better, one can find its pictorial representation as in figure 6. The first term inthe right hand side of (24), is the simplest one and is given by the first diagram in the right handside of the figure 6. However, the terms within the summation in (24), gives rise to two terms for twodimensional spatial lattice as shown in figure 6.Figure 6: The left and right hand side of = denotes the respective sides of the equation (24) with allthe coefficients. Note that the usual vertex symbol denotes the unusual coefficient for the first term ofthe decomposition.The actions of local gauge invariant operators (constructed out of prepotential operators) on thelinking states characterized by linking quantum numbers in (16) are obtained in (21,22,24), and pic-torially represented in figures 4, 5, 6. Note that, the pictorial representation of the states contain theparticular coefficients appearing before the states in any of (21,22,24) along with the states producedcharacterized by the linking numbers. Hereby we prescribe a set of rules to read off the coefficient aswell as the state by just looking at a particular diagram! Hence a particular diagram would correspondto a state characterized by linking numbers with a coefficient sitting in front of it as shown in the tablein 7. Figure 7: The coefficients are explicitly given in the last column.Now, from the coefficients given above and the diagrams in 7, we can spell out the ‘lattice Feynman9ules’ as follows: • Any diagram with net flux increasing or decreasing along i − j direction (or increasing along i and decreasing along j directions together) contribute a factor of √ ( n i +1)( n j +2) , where n i , n j counts the flux of the state on which the loop operator has acted. • Each solid line crossing the site from direction i − j will contribute a factor of l ij + 1. • Each dotted line crossing the site from direction i − j , without having any overlap with any solidline on any of its arm, will contribute a factor of ( n i + n j − l ij + 1). • Each solid flux line along i − k direction with the link at k direction, having overlap with a dottedlink along k − j direction will contribute a factor of ( − S ik defined in (23). • Each solid flux line along i − j direction with the link at i direction, having overlap with a dottedlink along i − i (cid:48) direction and the link at j direction, having overlap with a dotted link along j − j (cid:48) direction will contribute a factor of ( − S ij defined in (23), where i (cid:48) < j (cid:48) .To make the above diagrammatic rules more clear, we tabulate all possible loop configurations that canoccur at each of the four vertices (namely a, b, c, d ) of a plaquette, by the action of local gauge invariantoperators at the same in the following table. Note that, these loop configurations are obtained in thedynamics of loops under the magnetic Hamiltonian, as discussed in detail in the next section.10ertex coefficient vertex coefficientd1: C + ¯2 + = l +1 √ ( n +1)( n ¯2 +2) a1: C + + = l +1 √ ( n +1)( n +2) d2: C − ¯2 − = ( n + n ¯2 − l +1) √ ( n +1)( n ¯2 +2) a2: C − − = ( n + n − l +1) √ ( n +1)( n +2) d3: (cid:0) C ¯2 + − (cid:1) ¯1 = − l ¯1¯2 +1 √ ( n ¯2 +1)( n +2) a3: (cid:0) C + − (cid:1) ¯1 = l +1 √ ( n +1)( n +2) d4: (cid:0) C ¯2 + − (cid:1) = − l +1 √ ( n ¯2 +1)( n +2) a4: (cid:0) C + − (cid:1) ¯2 = l +1 √ ( n +1)( n +2) d5: (cid:0) C + ¯2 − (cid:1) ¯1 = l +1 √ ( n +1)( n ¯2 +2) a5: (cid:0) C + − (cid:1) ¯1 = l +1 √ ( n +1)( n +2) d6: (cid:0) C + ¯2 − (cid:1) = l +1 √ ( n +1)( n ¯2 +2) a6: (cid:0) C + − (cid:1) ¯2 = l +1 √ ( n +1)( n +2) d7: C (1 − ) (¯2 − ) ¯1 = l +1 √ ( n +1)( n ¯2 +2) a7: C (1 − ) ¯2 (2 − ) ¯1 = − l ¯1¯2 +1 √ ( n +1)( n +2) d8: C (1 − ) ¯1 (¯2 − ) = − l +1 √ ( n +1)( n ¯2 +2) a8: C (1 − ) ¯1 (2 − ) ¯2 = l ¯1¯2 +1 √ ( n +1)( n +2) b1: C ¯1 + + = l ¯12 +1 √ ( n ¯1 +1)( n +2) c1: C ¯1 + ¯2 + = l ¯1¯2 +1 √ ( n ¯1 +1)( n ¯2 +2) b2: C ¯1 − − = ( n ¯1 + n − l ¯12 +1) √ ( n ¯1 +1)( n +2) c2: C ¯1 − ¯2 − = ( n ¯1 + n ¯2 − l ¯1¯2 +1) √ ( n ¯1 +1)( n ¯2 +2) b3: (cid:0) C + ¯1 − (cid:1) = − l +1 √ ( n +1)( n ¯1 +2) c3: (cid:0) C ¯2 + ¯1 − (cid:1) = − l +1 √ ( n ¯2 +1)( n ¯1 +2) b4: (cid:0) C + ¯1 − (cid:1) ¯2 = l +1 √ ( n +1)( n ¯1 +2) c4: (cid:0) C ¯2 + ¯1 − (cid:1) = − l +1 √ ( n ¯2 +1)( n ¯1 +2) b5: (cid:0) C ¯1 + − (cid:1) = − l +1 √ ( n ¯1 +1)( n +2) c5: (cid:0) C ¯1 + ¯2 − (cid:1) = − l +1 √ ( n ¯1 +1)( n ¯2 +2) b6: (cid:0) C ¯1 + − (cid:1) ¯2 = l ¯1¯2 +1 √ ( n ¯1 +1)( n +2) c6: (cid:0) C ¯1 + ¯2 − (cid:1) = − l +1 √ ( n ¯1 +1)( n ¯2 +2) b7: C (2 − ) (¯1 − ) ¯2 = l +1 √ ( n ¯1 +1)( n +2) c7: C (¯1 − ) (¯2 − ) = − l +1 √ ( n ¯1 +1)( n ¯2 +2) b8: C (2 − ) ¯2 (¯1 − ) = − l +1 √ ( n ¯1 +1)( n +2) c8: C (¯1 − ) (¯2 − ) = l +1 √ ( n ¯1 +1)( n ¯2 +2)
11t this point, we discuss the overcompleteness in the loop basis characterized by linking numbersin the next subsection.
We have already discussed that we can describe the local linking states on a 2d lattice, by a set of sixlinking numbers defined locally at each site. These set of linking variables form an over-complete basisof the theory as the physical degrees of freedom for SU(2) gauge theory on 2+1 dimensional lattice isonly 3 per lattice site. Hence, there must be three constraints at each lattice site among the linkingnumber variables, to obtain the exact physical degrees of freedom of the theory. Among these threeconstraints, two are the number operator constraint arising because of the fact that, E L = E R at eachsite (as given in (4)) and is realized in terms of prepotentials in (8). On two spatial dimensions thisconstraint ( U (1) constraint) reads as n ( x ) = n ¯1 ( x + e ) & n ( x ) = n ¯2 ( x + e ) (25)where, n , n , n ¯1 and n ¯2 are defined in (17,18,19,20), and e and e are unit vectors (in lattice units)along the two directions. In terms of linking numbers, the two number operator constraint reads as: l ( x ) + l ( x ) + l ( x ) = l ( x + e ) + l ( x + e ) + l ¯1¯2 ( x + e )& l ( x ) + l ( x ) + l ( x ) = l ( x + e ) + l ( x + e ) + l ¯1¯2 ( x + e ) (26)The other constraint in 2+1 dimension is the Mandelstam constraint which in Prepotential formulation,at a particular site of a 2-d lattice reads as the operator relation: k k = k k − k k ¯1¯2+ (27)Using the definitions (21) and (11), we can write (27) as: (cid:112) ( n + 1)( n ¯1 + 2) ˆ O + ¯1 + (cid:112) ( n + 1)( n ¯2 + 2) ˆ O + ¯2 + = (cid:112) ( n + 1)( n ¯2 + 2) ˆ O + ¯2 + (cid:112) ( n + 1)( n ¯1 + 2) ˆ O + ¯1 + − (cid:112) ( n + 1)( n + 2) ˆ O + + (cid:112) ( n ¯1 + 1)( n ¯2 + 2) ˆ O ¯1 + ¯2 + ⇒ ˆ O + ¯1 + ˆ O + ¯2 + = (cid:34) ˆ O + ¯2 + ˆ O + ¯1 + − (cid:115) ( n ¯1 + 1)( n + 2)( n + 1)( n ¯1 + 2) ˆ O + + ˆ O ¯1 + ¯2 + (cid:35) (28)The Mandelstam constraint is pictorially represented in figure 8, from which, we clearly find that, theFigure 8: Pictorial representation of Mandelstam constraint in terms of prepotentials as given in (27).linking states with two vertical and horizontal flux lines crossing each other a particular lattice siteare actually not independent states but are a combination of two different states where the flux lines12ouches each other at that site itself. Another useful way of solving the Mandelstam constraints is tonote that any local state generated by the combination k k can be replaced by the right hand sideof (27). That is in terms of linking numbers without any loss of generality, this amounts to choosing l and l linking numbers at any site, such that, l ( x ) · l ( x ) = 0 (29)Mandelstam constraint in terms of linking variables is given in (29). The U(1) constraint (26) and(29) define our physical space completely. More specifically, the Abelian U(1) constraint (26) impliesthat the physical states are closed electric flux loops while constraint (29) implies that these flux loopscannot intersect at any site while they can overlap over lines. Hence, our physical states are made ofnested electric flux loops which can overlap over portions but can never intersect. An example set ofphysically allowed loops are given in figure 9.Figure 9: Physical Loops: Nested and overlapped but non-intersecting onesFrom our construction of physical states of gauge theory, we have a norm on the states which isnot trivial and indeed our choice of basis states are not even orthogonal to each other. This norm isexplicitly spelled out in Appendix B. In this section we discuss enumeration of all physical loop states on the entire lattice. Naively thesenested loops can be of arbitrary size and shape, therefore their descriptions are non-local as well. Wewill show by defining the Fusion operators, the description does become local and complete. The keyidea follows from the fact that on a single plaquette, any arbitrary number of electric flux plaquetteloops are allowed in the physical space. Larger loops can be formed by suitable fusion of such basicplaquette loops, where the newly invented fusion operators play their roles.The simplest way of explaining this construction is by working with diagrammatic technique asgiven in figure 10. In each of these diagrams in figure 10, there is an explicit meaning in terms of thelinking operators and the corresponding linking states. To illustrate that clearly, let us understand thefollowing facts: 13igure 10: Variables defined locally at dual site (˜ x ), on a 2-dimensional lattice spanned by basis vectors e along + X axis, and e along + Y axis. • The basic plaquette, the first diagram (a) in figure 10, is the basic electric flux palquette loop andthis can be constructed by the action of four linking operators k ij + at the four vertices around theplaquette on the strong coupling vacuum | (cid:105) , which we denote as the creation operator Π + L (˜ x )acting on | (cid:105) and the inverse action, i.e annihilation of a plaquette loop by Π − L (˜ x ) . L (˜ x ) definesnumber of such plaquette loops at the dual sites ˜ x of the lattice. • Then we define the fusion operators Π ± N , Π ± N , Π ± D , Π ± D and the corresponding numbers N , N , D , D which construct larger loops by combining neighbouring smaller ones. These fusion variables canbe thought of as some operators which either merges two smaller loops to a bigger one, or an-nihilates the state if no such neighbouring loops are present. In explicit operator form, everysolid line in these fusion operators is the k ij + type of inking operator, while the dashed line is itspseudo-inverse in the sense that, when there is some nonzero flux (denoted by nonzero l ij ) orsolid line already present, the dashed line decreases that by one unit and if none were present, itannihilates. • To realize the action of the fusion operators in figure 10, let us consider the following examples: – If there exists a loop state with L (˜ x ) = 1 , L (˜ x + e ) = 1, then there can exist another loopstate with L (˜ x ) = 1 , L (˜ x + e ) = 1 , N (˜ x + e ) = 1, which is basically a rectangular loopwith horizontal length of two lattice units as shown in figure 11. Here, the second statecan be thought as created by the fusion operator Π + N (˜ x + e ) on the first state. However,applying fusion operator once again would annihilate the state implying no state to existwith L (˜ x ) = 1 , L (˜ x + e ) = 1 , N (˜ x + e ) = 2. – Similarly the fusion operator Π + N (˜ x − e ) combines vertical neighbouring plaquettes if theyare present. 14igure 11: Π + N (˜ x + e ) | L (˜ x ) = 1 , L (˜ x + e ) = 1 (cid:105) = | L (˜ x ) = 1 , L (˜ x + e ) = 1 , N (˜ x + e ) = 1 (cid:105) – The inverse action. i.e decoupling a bigger loop to two smaller loops with an overlap along avertical or horizontal link is performed by the fusion operators Π − N (˜ x + e ) and Π − N (˜ x − e )respectively. – The other two fusion operators Π ± D (˜ x − e − e ) combine the diagonal ones as shownin figure 12. Note that the individual Π ± D (˜ x − e − e ) operators contain intersectinghorizontal and vertical flux lines which is not a part of physical loop space. Hence theseparticular fusion operators should always come in a certain combination (like the Π + D Π − D )with other fusion variables such that, there exists no intersecting flux lines for the final loopstate produced.Figure 12: Π − D Π + D (˜ x − e + e ) | L (˜ x ) = 1 , L (˜ x − e + e ) = 1 (cid:105) = | L (˜ x ) = 1 , L (˜ x + e ) = 1 , D (˜ x − e =1 , D (˜ x − e = − (cid:105)• The quantum number L counts the flux around a plaquette, hence it is natural to assign thevariable L at the centre of each plaquette, i.e at each dual site by defining L (˜ x ), where, ˜ x = x + e + e . Similarly, as shown in figure 10, we can naturally assign the variable N to themidpoint of each for the vertical links, i.e N (˜ x − e ) and the variable N to the midpoint ofeach for the horizontal links, i.e N (˜ x − e ). The variables D are naturally assigned to eachoriginal lattice site, i.e D (˜ x − e − e ). This particular set of quantum numbers defined atand around a dual lattice site, is sufficient to characterize any loops in the theory, or in otherwords, each and every loops of the theory can be uniquely specified by specifying a set of fusionquantum numbers locally throughout the lattice. • We have already seen in the above example that, the basic loop variable L can take any positivevalue and is independent of others. However, the other variables can be both positive and negativebut are defined within a finite range. These new set of fusion variables are related to the linking15uantum numbers in the following way: l ( x ) = L (˜ x ) − N (˜ x − e − N (˜ x − e D (˜ x − e − e ≥ l ( x ) = N (˜ x − e N (˜ x − e − e ) − D (˜ x − e − e − D (˜ x − e − e ≥ l ( x ) = L (˜ x − e ) − N (˜ x − e − e ) − N (˜ x − e D (˜ x − e − e ≥ l ( x ) = L (˜ x − e ) − N (˜ x − e − N (˜ x − e − e D (˜ x − e − e ≥ l ( x ) = N (˜ x − e N (˜ x − e − e − D (˜ x − e − e − D (˜ x − e − e ≥ l ¯1¯2 ( x ) = L (˜ x − e − e ) − N (˜ x − e − e ) − N (˜ x − e − e D (˜ x − e − − e ≥ • For any arbitrary loop, the number of prepotentials on each link of the lattice are countedfollowing (17,18,19,20) as: n ( x ) = L (˜ x ) + L (˜ x − e ) − N (˜ x − e n ¯1 ( x + e ) (36) n ( x ) = L (˜ x ) + L (˜ x − e ) − N (˜ x − e n ¯2 ( x + e ) (37)Note that, the U(1) constraints are automatically satisfied in (36) and (37). • Note that, the description of local linking states in terms of five linking numbers provides acomplete description of loop states corresponding to only the physical degrees of freedom ofthe theory subject to the Mandelstam constrain together with the two U(1) constraint. Theequivalent description of loop states in terms of five fusion loop numbers are again complete.Here, the U(1) constraints are solved trivially by construction, hence after solving the Mandelstamconstraint one is left with four degrees of freedom implying that there exists another constraintin these variables which needs to be imposed to get the exact and complete loop basis. We willdiscuss that extra constraint later in this section.From these construction, we can label the loop states as | L, N , N , D , D (cid:105) , which are eigenstatesof the following operators with the corresponding eigenvalues:ˆ L (˜ x ) | L, N , N , D , D (cid:105) = L (˜ x ) | L, N , N , D , D (cid:105) ˆ N (˜ x − e | L, N , N , D , D (cid:105) = N (˜ x − e | L, N , N , D , D (cid:105) ˆ N (˜ x − e | L, N , N , D , D (cid:105) = N (˜ x − e | L, N , N , D , D (cid:105) ˆ D (˜ x − e − e | L, N , N , D , D (cid:105) = D (˜ x − e − e | L, N , N , D , D (cid:105) ˆ D (˜ x − e − e | L, N , N , D , D (cid:105) = D (˜ x − e − e | L, N , N , D , D (cid:105) (38)16nd the shift operators Π ± corresponding to each of the fusion variables are defined by,ˆ L (˜ x )Π ± L (˜ x ) | L, N , N , D , D (cid:105) = (cid:16) L (˜ x ) ± (cid:17) | L, N , N , D , D (cid:105) ˆ N (˜ x − e ± N (˜ x − e | L, N , N , D , D (cid:105) = (cid:16) N (˜ x − e ± (cid:17) | L, N , N , D , D (cid:105) ˆ N (˜ x − e ± N (˜ x − e | L, N , N , D , D (cid:105) = (cid:16) N (˜ x − e ± (cid:17) | L, N , N , D , D (cid:105) ˆ D (˜ x − e − e ± D (˜ x − e − e | L, N , N , D , D (cid:105) = (cid:16) D (˜ x − e − e ± (cid:17) | L, N , N , D , D (cid:105) ˆ D (˜ x − e − e ± D (˜ x − e − e | L, N , N , D , D (cid:105) = (cid:16) D (˜ x − e − e ± (cid:17) | L, N , N , D , D (cid:105) (39)It is evident from (36) and (37) that, the number operator constraints (8) present in prepotentialformulation are already solved by the fusion variables. However, the fusion variable are five in numberin contrast to only three physical degrees of freedom. This implies that there still exist two constraintsto be imposed on the Hilbert space of states characterized by fusion variables to obtain the physicalloop space. We will discuss about those constraints in the next section.The Mandelstam constraints are already solved when we consider our loop Hilbert space consistingof only non-intersecting loops by explicitly imposing: l ( x ) l ( x ) ≡ (cid:16) N (˜ x − e N (˜ x − e − e ) − D (˜ x − e − e − D (˜ x − e − e (cid:17)(cid:16) N (˜ x − e N (˜ x − e − e − D (˜ x − e − e − D (˜ x − e − e (cid:17) = 0 (40)As stated earlier, apart from the constraint (40), there still exists another constraint in the fusionquantum number characterization of loop state in order to obtain three physical degrees of freedom.This additional constraint, which we name “fusion constraint” and is given by:Π − D (˜ x − e − e − D (˜ x + e e − D (˜ x − e e − D (˜ x + e − e + N (˜ x − e + N (˜ x + e + N (˜ x − e + N (˜ x + e (cid:0) Π + L (˜ x ) (cid:1) = 1 (41)This fusion constraint is shown diagrammatically in figure 13. Note that, the fusion constraint and theFigure 13: Pictorial representation of the fusion constraint as given in (41).Mandelstam constraint discussed earlier are independent of each other and hence commutes amongstthemselves. 17n the next section we write the Hamiltonian in terms of the Fusion variables. The dynamics of loop states under Kogut-Susskind Hamiltonian given in (1), can be realized in termsof Fusion variables as well as the corresponding shift operators we have defined earlier. In this section,we consider the Hamiltonian operator and its action on loop states characterized by fusion variables.The Hamiltonian for lattice gauge theory given in (1) consists of two parts. The electric part of theHamiltonian which becomes dominant in the strong coupling limit of the theory, measures the fluxalong all the links of the lattice, whereas the magnetic part of the Hamiltonian, which is dominant inthe weak coupling limit of the theory, is responsible for the dynamics of the loop states.The electric part of the Hamiltonian counts the total SU(2) flux on all the links of the lattice, whichin terms of prepotential number operator is given by,ˆ H e = g (cid:88) links E links = g (cid:88) x (cid:20) n ( x )2 (cid:18) n ( x )2 + 1 (cid:19) + n ( x )2 (cid:18) n ( x )2 + 1 (cid:19)(cid:21) (42)where, n ( x ) and n ( x ) are the eigenvalues of the total number operator ˆ n counting the number ofprepotentials (left or right) on the links along 1 and 2 directions originating at the site x . In terms offusion operators, the total flux along the two links at each site are counted as given in (36) and (37).Using that, the electric part of the Hamiltonian is given by:ˆ H e = g (cid:88) ˜ x (cid:34) (cid:18) L (˜ x ) + L (˜ x − e ) − N (˜ x − e )2 (cid:19) (cid:18) L (˜ x ) + L (˜ x − e ) − N (˜ x − e )2 + 1 (cid:19) + (cid:18) L (˜ x ) + L (˜ x − e ) − N (˜ x − e )2 (cid:19) (cid:18) L (˜ x ) + L (˜ x − e ) − N (˜ x − e )2 + 1 (cid:19) (cid:35) (43)Now we concentrate on the magnetic part given by, H mag = 1 g (cid:16) Tr U plaquette + Tr U † plaquette (cid:17) (44)This is not as simple as the electric part even in terms of prepotentials or fusion variables. The magneticHamiltonian contains the gauge invariant loop operators. In previous sections we have already studiedthe actions of loop operators on loop states and have developed a diagrammatic technique to realizethese actions which we will utilize now to find the action of the magnetic part of the Hamiltonian onany arbitrary loop state. Note that, we will consider the loop Hilbert space to contain only those stateswhich solves the Mandelstam constraint, i.e satisfies (29).In terms of prepotentials each link operator breaks into 2 parts as given in (9). One of these twoparts contains only the creation operator and the other only annihilation, making U ≡ U + + U − .Hence, the prepotential formulation enables us to write down the gauge invariant plaquette operator,which is trace of the products of four link operators around a plaquette, as a sum of 2 = 16 operatorsas shown in figure 14. The constituent operators fall among different classes. We analyze each classseparately and calculate the dynamics of physical loop states in each case. Each of these plaquetteoperators are basically product of four different local gauge invariant operators at the four vertices.18igure 14: The Hamiltonian operator in terms of prepotential becomes a sum of sixteen operators asshown above diagrammatically. The solid line along a link denotes the presence of prepotential creationoperator on that link whereas, a dotted line denotes the annihilation operators on that. Clearly thewhole set is rotationally symmetric and hermitian. These set of operators again can be subdivided in sixclasses of operators as shown in (a), (b), (c), (d), (e) and (f) denoting by (a) ≡ H ++++ , (b) ≡ H +++ − ,(c) ≡ H ++ −− , (d) ≡ H + − + − , (e) ≡ H + −−− , (f) ≡ H −−−− .We have already studied these individual loop operators and have found their actions in (21,22,24).Now we exploit those calculations to compute the combinations of loop states produced by the actionof the Hamiltonian.Mandelstam constraint, (29) implies, that in the action of the loop operator O i − j − , as shownin figure 6, the last diagram of the right hand side would vanish. Hence within the loop space weconsider, we will have the reduced action for the loop operators. Let us now consider the actions ofeach plaquette operators individually:1. The operator in (a) of figure 14 is H ++++ . The local loop operators at each vertex are O i + j + which acts according figure 4, yielding only one loop state as shown in figure 15.2. The operator of type (b) are H +++ − , where at two adjacent vertices, the loop operators are O i + j + giving rise to only one state, and at the opposite two they are of the type O i + j − givingrise to 2 × × × × H ++++ Figure 16: Explicit action of type (b) or H +++ −
3. The (c) type operators, H ++ −− , where at one vertex the loop operator is O i + j + and at thediagonally opposite vertex it is O i − j − . The first one gives only one loop state whereas the secondone generates two following figure 6 (NOT 3 for loops which satisfy Mandelstam constraint).The other two vertices are of the type O i + j − giving rise to 2 × × × × H ++ −−
4. The action of operators of type (d), i.e H + − + − , are obtained by using figure 5 for the operators oftype O i + j − at all four vertices, yielding total of 2 = 16 terms for each of the two such operatorspresent in the class. The explicit states are given in figure 18.5. The operator of type (e) are H + −−− , where at two adjacent vertices, the loop operators are O i − j − giving rise to 2 states each following figure 6, and the opposite two they are of the type O i + j − again giving rise to 2 states each following figure 5. Hence each plaquette operators oftype (e) deforms the loop states on which it acts in 2 = 16 possible way as shown in figure 19.20igure 18: Explicit action of type (d) or H + − + − There are 4 such operators in type (e), which gives a total of 64 loop states.6. Finally, for type (f), i.e H −−−− at all the four vertices the loop operators are O i − j − giving riseto 2 states each following figure 6, yielding 2 = 16 loop states as shown in figure 20.The loop states produced by the action of the magnetic part of the Hamiltonian as discussed so far,can also be realized to be created by the actions of the shift operators Π ± corresponding to the fusionvariables as given in (39) together with a particular coefficient associated and fixed by each diagram.The action of the Hamiltonian on loop states has been described in figures 15, 16, 17, 18, 19, 20. Thesediagrams denotes that for each loop state created, the fusion new state can be realized by a new set offusion quantum numbers. Or in other words, the Hamiltonian can be represented by shift operators infusion variables together with the a certain coefficient which describes the new state created. Using thediagrammatic rules provided in figure 7 and the equations thereafter, we can calculate that coefficient.Now, from each diagram in figures 15, 16, 17, 18, 19, 20, one can read the constant coefficient in front ofit, and the change in fusion quantum numbers for each term. The action of the magnetic Hamiltonian21igure 19: Explicit action of type (e) or H + −−− on loop states | L, N , N , D , D (cid:105) is thus obtained as:Type (a): H = C + + a C + ¯1 + b C ¯1 + ¯2 + c C + ¯2 + d Π + L (˜ x ) (45)Type (b): H = C + + a C + ¯2 + d (cid:16) ( C ¯1 + ¯2 − c ) + ( C ¯1 + ¯2 − c ) Π + D (˜ x + e e (cid:17)(cid:16) ( C − ¯1 + b ) + ( C − ¯1 + b ) ¯2 Π + D (˜ x + e − e (cid:17) Π + N (˜ x + e + L (˜ x ) (46) H = C ¯1 + ¯2 + c C + ¯2 + d (cid:16) ( C − + a ) ¯2 + ( C − + a ) ¯1 Π + D (˜ x − e − e (cid:17)(cid:16) ( C + ¯1 − b ) ¯2 + ( C + ¯1 − b ) Π + D (˜ x + e − e (cid:17) Π + N (˜ x − e + L (˜ x ) (47) H = C + ¯1 + b C ¯1 + ¯2 + c (cid:16) ( C + − a ) ¯1 + ( C + − a ) ¯2 Π + D (˜ x − e − e (cid:17)(cid:16) ( C + ¯2 − d ) ¯1 + ( C + ¯2 − d ) Π + D (˜ x − e e (cid:17) Π + N (˜ x − e + L (˜ x ) (48) H = C + + a C + ¯1 + b (cid:16) ( C ¯1 − ¯2 + c ) + ( C ¯1 − ¯2 + c ) Π + D (˜ x + e e (cid:17)(cid:16) ( C − ¯2 + d ) + ( C − ¯2 + d ) ¯1 Π + D (˜ x − e e (cid:17) Π + N (˜ x + e + L (˜ x ) (49)22igure 20: Explicit action of type (f) or H −−−− Type (c): H = (cid:16) C + ¯2 + d Π − D (˜ x − e e (cid:17) (cid:16) ( C ¯1 + ¯2 − c ) + ( C ¯1 + ¯2 − c ) Π − D (˜ x + e e (cid:17)(cid:16) ( C − + a ) ¯2 + ( C − + a ) ¯1 Π − D (˜ x − e − e (cid:17) (50) (cid:16) C − ¯1 − b + C (2 − ) (¯1 − ) ¯2 b Π + D (˜ x + e − e − D (˜ x + e − e (cid:17) Π − N (˜ x − e − N (˜ x + e − L (˜ x ) H = (cid:16) C ¯1 + ¯2 + c Π − D (˜ x + e e (cid:17) (cid:16) ( C + ¯1 − b ) ¯2 + ( C + ¯1 − b ) Π − D (˜ x + e − e (cid:17)(cid:16) ( C + ¯2 − d ) ¯1 + ( C + ¯2 − d ) Π − D (˜ x − e e (cid:17) (51) (cid:16) C − − a + C (1 − ) ¯2 (2 − ) ¯1 a Π + D (˜ x − e − e − D (˜ x − e − e (cid:17) Π − N (˜ x + e − N (˜ x + e − L (˜ x ) H = (cid:16) C + ¯1 + b Π − D (˜ x + e − e (cid:17) (cid:16) ( C ¯1 + ¯2 − c ) + ( C ¯1 + ¯2 − c ) Π − D (˜ x + e e (cid:17)(cid:16) ( C − + a ) ¯2 + ( C − + a ) ¯1 Π − D (˜ x − e − e (cid:17) (52) (cid:16) C − ¯2 − d + C (1 − ) (¯2 − ) ¯1 d Π + D (˜ x − e e − D (˜ x − e e (cid:17) Π − N (˜ x + e − N (˜ x − e − L (˜ x ) H = (cid:16) C + + a Π − D (˜ x − e − e (cid:17) (cid:16) ( C + ¯2 − d ) ¯1 + ( C + ¯2 − d ) Π − D (˜ x − e e (cid:17)(cid:16) ( C + ¯1 − b ) ¯2 + ( C + ¯1 − b ) Π − D (˜ x + e − e (cid:17) (53) (cid:16) C ¯1 − ¯2 − c + C (¯1 − ) (¯2 − ) c Π + D (˜ x + e e − D (˜ x + e e (cid:17) Π − N (˜ x − e − N (˜ x − e − L (˜ x )23ype (d): H = (cid:16) ( C ¯1 + ¯2 − c ) + ( C ¯1 + ¯2 − c ) Π + D (˜ x + e e (cid:17)(cid:16) ( C − ¯1 + b ) + ( C − ¯1 + b ) ¯2 Π + D (˜ x + e − e (cid:17)(cid:16) ( C + − a ) ¯1 + ( C + − a ) ¯2 Π + D (˜ x − e − e (cid:17)(cid:16) ( C + ¯2 − d ) ¯1 + ( C + ¯2 − d ) Π + D (˜ x − e e (cid:17) Π + N (˜ x + e + N (˜ x − e + L (˜ x ) (54) H = (cid:16) ( C − + a ) ¯2 + ( C − + a ) ¯1 Π + D (˜ x − e − e (cid:17)(cid:16) ( C + ¯1 − b ) ¯2 + ( C + ¯1 − b ) Π + D (˜ x + e − e (cid:17)(cid:16) ( C ¯1 − ¯2 + c ) + ( C ¯1 − ¯2 + c ) Π + D (˜ x + e e (cid:17)(cid:16) ( C − ¯2 + d ) + ( C − ¯2 + d ) ¯1 Π + D (˜ x − e e (cid:17) Π + N (˜ x + e + N (˜ x − e + L (˜ x ) (55)24ype (e): H = (cid:16) ( C ¯1 + ¯2 − c ) + ( C ¯1 + ¯2 − c ) Π − D (˜ x + e e (cid:17)(cid:16) ( C + ¯1 − b ) ¯2 + ( C + ¯1 − b ) Π − D (˜ x + e − e (cid:17)(cid:16) C − − a + C (1 − ) ¯2 (2 − ) ¯1 a Π + D (˜ x − e − e − D (˜ x − e − e (cid:17) (56) (cid:16) C − ¯2 − d + C (1 − ) (¯2 − ) ¯1 d Π + D (˜ x − e e − D (˜ x − e e (cid:17) Π − N (˜ x + e − L (˜ x )(57) H = (cid:16) ( C − + a ) ¯2 + ( C − + a ) ¯1 Π − D (˜ x − e − e (cid:17)(cid:16) ( C + ¯1 − b ) ¯2 + ( C + ¯1 − b ) Π − D (˜ x + e − e (cid:17)(cid:16) C ¯1 − ¯2 − c + C (¯1 − ) (¯2 − ) c Π + D (˜ x + e e − D (˜ x + e e (cid:17)(cid:16) C − ¯2 − d + C (1 − ) (¯2 − ) ¯1 d Π + D (˜ x − e e − D (˜ x − e e (cid:17) Π − N (˜ x − e − L (˜ x )(58) H = (cid:16) ( C − + a ) ¯2 + ( C − + a ) ¯1 Π − D (˜ x − e − e (cid:17)(cid:16) ( C + ¯2 − d ) ¯1 + ( C + ¯2 − d ) Π − D (˜ x − e e (cid:17)(cid:16) C − ¯1 − b + C (2 − ) (¯1 − ) ¯2 b Π + D (˜ x + e − e − D (˜ x + e − e (cid:17)(cid:16) C ¯1 − ¯2 − c + C (¯1 − ) (¯2 − ) c Π + D (˜ x + e e − D (˜ x − e e (cid:17) Π − N (˜ x − e − L (˜ x )(59) H = (cid:16) ( C ¯1 + ¯2 − c ) + ( C ¯1 + ¯2 − c ) Π − D (˜ x + e e (cid:17)(cid:16) ( C + ¯2 − d ) ¯1 + ( C + ¯2 − d ) Π − D (˜ x − e e (cid:17)(cid:16) C − − a + C (1 − ) ¯2 (2 − ) ¯1 a Π + D (˜ x − e − e − D (˜ x − e − e (cid:17)(cid:16) C − ¯1 − b + C (2 − ) (¯1 − ) ¯2 b Π + D (˜ x + e − e − D (˜ x + e − e (cid:17) Π − N (˜ x + e − L (˜ x )(60)Type (f): H = (cid:16) C − − a + C (1 − ) ¯2 (2 − ) ¯1 a Π + D (˜ x − e − e − D (˜ x − e − e (cid:17)(cid:16) C − ¯1 − b + C (2 − ) (¯1 − ) ¯2 b Π + D (˜ x + e − e − D (˜ x + e − e (cid:17)(cid:16) C ¯1 − ¯2 − c + C (¯1 − ) (¯2 − ) c Π + D (˜ x + e e − D (˜ x + e e (cid:17)(cid:16) C − ¯2 − d + C (1 − ) (¯2 − ) ¯1 d Π + D (˜ x − e e − D (˜ x − e e (cid:17) Π − L (˜ x ) (61)In all the sixteen terms of the Hamiltonian, the coefficient C ’s with suffix a, b, c, d denotes them to bedefined at points (˜ x − e − e ) , (˜ x + e − e ) , (˜ x + e + e ) and (˜ x − e + e ) respectively.The matrix elements of this magnetic Hamiltonian within the loop states can be calculated followingappendix B. In Appendix B we compute the norm of loop states by noticing that this is itself productof four norms defined at the four corner sites of a plaquette. In appendix C, we briefly illustrate howthe strong coupling series in this new formalism, using the lattice Feynmann rules prescribed in thiswork matches exactly with the conventional approach [9]. Note that, our formulation is much moresimple as there is no need to deal with any complex 6j coeffiecient [7, 9] and is well suited for numericalcomputation. 25 Summary and Concusions
In this work, we have used the local loop description in prepotential formulation of lattice gaugetheory to construct all possible local gauge invariant operators or linking operators and found theirexplicit action on all possible local linking states defined locally at each lattice site. We develop a setof ‘lattice Feynman rules’ and hence a complete diagrammatic scheme to perform all computationsdiagrammatically bypassing long and tedious algebraic calculations.The linking number description of local gauge invariant operator and states is over-complete asthere exist the Mandelstam constraint. We have solved this constraint explicitly to find all the physicalloop configurations consisting of non-intersecting electric flux loops. The physical loop configurationscontain nested loops (all non-intersecting) which can overlap with neighbouring loops in one or moresegments as shown in figure 9. In order to characterize the physical loop Hilbert space we define abasic loop operator, i.e the smallest plaquette ones which solves the Mandelstam constraint and are apart of the physical loop configuration. We further show that, other configurations can be generatedfrom the basic plaquette loops by applying a set of fusion operators defined on the lattice locally. Infact arbitrary large loops can be generated by local action of these fusion operators. As a consequenceof this, the full lattice Hamiltonian is explicitly written in terms of the fusion operators. The completedynamics of arbitrary non-intersecting loops under this Hamiltonian is thus obtained.This diagrammatic tool to handle lattice gauge theories is extremely useful to proceed with latticecalculations analytically in both the strong and weak coupling limit of the theory. The works in thesedirections, specifically towards the analytic weak coupling expansion is in progress and will be reportedshortly. Moreover theses techniques can also find application in numerical simulation of Hamiltonianlattice gauge theories as one can enumerate the complete and physical loop configurations by justspecifying a set of integers locally throughout the lattice without any redundant degrees of freedomand their complete dynamics is already obtained in this work.The most novel feature of this approach is that all the steps computed in this work can be per-formed in any arbitrary dimension, more specifically for 3 + 1 dimension which is of physical interest.Addition of Fermions to the theory enlarges the physical configuration space with more local gaugeinvariant states or linking states but qualitatively the construction steps remain the same. This will beenumerated in a future publication. The recently developed tensor network approach to Hamiltonianlattice gauge theory [10, 11] should find this loop formulation most suitable to proceed with for nonAbelian gauge theories. This loop formulation and diagrammatic techniques should also be extremelyuseful towards the aim of the construction of quantum simulations [12] for lattice gauge theories.
Acknowledgement
The authors would like to thank Manu Mathur for many useful informal discussions at multiple stagesof this work. 26
Explicit action of loop operators on loop states
The basic local loop operators arising at a particular site are:ˆ O i + j + ≡ (cid:112) ( n i + 1)( n j + 2) k ij + (62)ˆ O i + j − ≡ (cid:112) ( n i + 1)( n j + 2) κ ij (63)ˆ O i − j − ≡ (cid:112) ( n i + 1)( n j + 2) k ij − (64)We now compute the action of these operators on a most general loop state locally characterized bylinking numbers as given in (16). Let us first consider the following action:ˆ O i + j + |{ l }(cid:105) ≡ (cid:112) ( n i + 1)( n j + 2) k ij + |{ l }(cid:105) = ( l ij + 1) (cid:112) ( n i + 1)( n j + 2) | l ij + 1 (cid:105) (65)where, | l ij + 1 (cid:105) denotes the state in (16) with the particular quantum number l ij increased by 1. Thisaction is simple and straightforward besides being applicable for any i, j . We represent the aboveaction pictorially in figure 4.Next we consider,ˆ O + − |{ l }(cid:105) ≡ (cid:112) ( n + 1)( n + 2) κ |{ l }(cid:105) = 1 (cid:112) ( n + 1)( n + 2) (cid:34) (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:104) κ , (cid:0) k (cid:1) l (cid:105) (cid:0) k (cid:1) l (cid:0) k ¯1¯2+ (cid:1) l ¯1¯2 l ! l ! l ! l ! l ! l ¯1¯2 ! | (cid:105) + (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:104) κ , (cid:0) k (cid:1) l (cid:105) (cid:0) k ¯1¯2+ (cid:1) l ¯1¯2 l ! l ! l ! l ! l ! l ¯1¯2 ! | (cid:105) (cid:35) (66)= 1 (cid:112) ( n + 1)( n + 2) (cid:34) ( l + 1) | l − , l + 1 (cid:105) + ( l + 1) | l − , l + 1 (cid:105) (cid:35) (67)In the above calculation we have used the relation: k − ( k + ) p | (cid:105) = [ k − , ( k + ) p ] | (cid:105) = (cid:34) [ k − , k + ]( k + ) p − + k + [ k − , k + ]( k + ) p − + ( k + ) [ k − , k + ]( k + ) p − + . . . + ( k + ) p − [ k − , k + ] (cid:35) | (cid:105) = (cid:34) (ˆ n a + ˆ n b + 2)( k + ) p − + (ˆ n a + ˆ n b + 2 − k + ) p − + (ˆ n a + ˆ n b + 2 − k + ) p − + . . . +(ˆ n a + ˆ n b + 2 − p − k + ) p − (cid:35) | (cid:105) = (cid:34) (ˆ n a + ˆ n b − p + 4) + (ˆ n a + ˆ n b − p + 6) + . . . + (ˆ n a + ˆ n b + 2) (cid:35) ( k + ) p − | (cid:105) = 12 p (2ˆ n a + 2ˆ n b + 6 − p )( k + ) p − | (cid:105) = p (ˆ n a + ˆ n b + 3 − p )( k + ) p − | (cid:105) ≡ p ( p + 1)( k + ) p − | (cid:105) (68)27n general these ˆ O i + j − operator acts in the following way:ˆ O i + j − |{ l }(cid:105) ≡ (cid:112) ( n i + 1)( n j + 2) κ ij |{ l }(cid:105) = 1 (cid:112) ( n i + 1)( n j + 2) (cid:88) k (cid:54) = i,j ( − S ik ( l ik + 1) | l jk − , l ik + 1 (cid:105) (69)where, in any l ij the indices are always ordered in a way such that the first index is always less thanthe first one, and S ik = 1 if i > k & S ik = 0 if i < k. We represent the above action pictorially in figure 5.The last but not the least complicated type of vertex operator is ˆ O i − j − which we calculate using(68). Let’s consider the action of the following operator on loop state:ˆ O − − |{ l }(cid:105) ≡ (cid:112) (ˆ n + 2)(ˆ n + 1) k − |{ l }(cid:105) = 1 (cid:112) (ˆ n + 2)(ˆ n + 1) (cid:34) { (cid:104) k − , (cid:0) k (cid:1) l (cid:105) + (cid:0) k (cid:1) l k − } (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k ¯1¯2+ (cid:1) l ¯1¯2 l ! l ! l ! l ! l ! l ¯1¯2 ! | (cid:105) (cid:35) = 1 (cid:112) (ˆ n + 2)(ˆ n + 1) (cid:34) l ( n + n − l + 1) (cid:0) k (cid:1) l − (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k ¯1¯2+ (cid:1) l ¯1¯2 l ! l ! l ! l ! l ! l ¯1¯2 ! | (cid:105) + (cid:0) k (cid:1) l (cid:16)(cid:104) k − , (cid:0) k (cid:1) l (cid:105) + (cid:0) k (cid:1) l k − (cid:17) (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k ¯1¯2+ (cid:1) l ¯1¯2 l ! l ! l ! l ! l ! l ¯1¯2 ! | (cid:105) (cid:35) = 1 (cid:112) (ˆ n + 2)(ˆ n + 1) (cid:34) ( n + n − l + 1) | l − (cid:105) + (cid:0) k (cid:1) l l (cid:0) k (cid:1) l − κ (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k ¯1¯2+ (cid:1) l ¯1¯2 l ! l ! l ! l ! l ! l ¯1¯2 ! | (cid:105) + (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:16)(cid:104) k − , (cid:0) k (cid:1) l (cid:105) + (cid:0) k (cid:1) l k − (cid:17) (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k ¯1¯2+ (cid:1) l ¯1¯2 l ! l ! l ! l ! l ! l ¯1¯2 ! | (cid:105) (cid:35) = 1 (cid:112) (ˆ n + 2)(ˆ n + 1) (cid:34) ( n + n − l + 1) | l − (cid:105) + (cid:0) k (cid:1) l l (cid:0) k (cid:1) l − (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:104) κ , (cid:0) k (cid:1) l (cid:105) (cid:0) k ¯1¯2+ (cid:1) l ¯1¯2 l ! l ! l ! l ! l ! l ¯1¯2 ! | (cid:105) + (cid:0) k (cid:1) l (cid:0) k (cid:1) l l (cid:0) k (cid:1) l − κ (cid:0) k (cid:1) l (cid:0) k (cid:1) l (cid:0) k ¯1¯2+ (cid:1) l ¯1¯2 l ! l ! l ! l ! l ! l ¯1¯2 ! | (cid:105) (cid:35) = 1 (cid:112) (ˆ n + 2)(ˆ n + 1) (cid:34) ( n + n − l + 1) | l − (cid:105) + ( l ¯1¯2 + 1)( − S ¯1¯2 | l − , l − , l ¯1¯2 + 1 (cid:105) +( l ¯1¯2 + 1)( − S ¯2¯1 | l − , l − , l ¯1¯2 + 1 (cid:105) (cid:35) (70)with, S ik = 1 if i > k & S ik = 0 if i < k. O i − j − operator, the action is:ˆ O i − j − |{ l }(cid:105) = 1 (cid:112) (ˆ n i + 2)(ˆ n j + 1) (cid:34) ( n i + n j − l ij + 1) | l ij − (cid:105) + (cid:88) ¯ i, ¯ j {(cid:54) = i,j } ( l ¯ i ¯ j + 1)( − S ¯ i ¯ j | l i ¯ i − , l j ¯ j − , l ¯ i ¯ j + 1 (cid:105) (cid:35) (71)We represent the above action pictorially in figure 6.29 Normalization of the Loop States
The linking states at a particular site of a two dimensional spatial lattice, are characterized by sixlinking numbers l , l , l , l , l , l ¯1¯2 . The SU(2) flux along each directions at a particular site arecounted as in (17,18,19,20). Moreover there exists the Mandelstam constraint given in (29), whichmust be solved in order to get independent loop states implying that at each site x , atleast either ofthe two quantum numbers l , l must be zero. Hence, after solving the Mandelstam constraint, onlyfive non-zero linking quantum number together with the two Abelian constraints are present at eachsite.Any linking state, characterized by five non-zero linking number is always orthogonal with respect tothe four number operators defined in (17)-(20) but there exists a fifth quantum number which makesthe orthogonality non-trivial as given below: (cid:104){ l (cid:48) ij }|{ l ij }(cid:105) = (cid:89) i =1 , , ¯1¯2 δ n (cid:48) i ,n i F ( { l (cid:48) ij } , { l ij } ) (72)Before determining the complicated function F ( { l (cid:48) ij } , { l ij } ), let us first realize the orthogonality oflinking states in terms of four quantum numbers. This can be realized trivially when one consider thelinking state which has only four non-zero linking number, such as for example with l = 0, besides l ( x ) l ( x ) = 0. The orthonormality of such states are obtained as: (cid:104) l (cid:48) = 0 | l = 0 (cid:105) = (cid:104) l (cid:48) , l (cid:48) , l (cid:48) ¯1¯2 , l (cid:48) , l (cid:48) | l , l , l ¯1¯2 , l , l (cid:105) = ( l + l + l ¯1¯2 + l + l + 1)! l ! ( l + l ¯1¯2 + l + l + 1)! δ l (cid:48) ,l × ( l + l ¯1¯2 + l + l + 1)! l ! ( l ¯1¯2 + l + l + 1)! δ l (cid:48) ,l × ( l ¯1¯2 + l + l + 1)! l ¯1¯2 ! ( l + l + 1)! δ l (cid:48) ¯1¯2 ,l ¯1¯2 × ( l + 1) ( l + 1) δ l (cid:48) ,l δ l (cid:48) ,l ≡ B p δ l (cid:48) ,l δ l (cid:48) ,l δ l (cid:48) ¯1¯2 ,l ¯1¯2 δ l (cid:48) ,l δ l (cid:48) ,l (73)(73) is obtained by extracting the k ij + operator from the bra state and acting that on the ket statefollowing (24) until it reaches l (cid:48) ij = 0 for all nonzero l ij , considering one by one.The next complicated orthogonality arises when either of the bra and ket state has 5 non-zerolinking numbers and the other one has only 4. For example, consider the following case: (cid:104) l (cid:48) = 0 |{ l ij }(cid:105) = 1 l (cid:104) l (cid:48) = 0 | k | l − (cid:105) = 1 l ( l − (cid:2) − ( l (cid:48) ¯1¯2 + 1) (cid:104) l (cid:48) = 0 , l (cid:48) − , l (cid:48) − , l (cid:48) ¯1¯2 + 1 | k | l − (cid:105) (cid:3) = A (cid:48) (1)1 (cid:104) l (cid:48) = 0 , l (cid:48) − , l (cid:48) − , l (cid:48) ¯1¯2 + 1 | k | l − (cid:105) = ......= A (cid:48) (1)1 A (cid:48) (2)1 . . . A (cid:48) ( l )1 (cid:104) l (cid:48) = 0 , l (cid:48) − l , l (cid:48) − l , l (cid:48) ¯1¯2 + l | l = 0 (cid:105) (74)30here, A (cid:48) ( i )1 = − l (cid:48) ¯1¯2 + il + i − . (75)(74) is also obtained by extracting the k operator from the ket state and acting that on the bra statefollowing (24) until it reaches l = 0. The orthogonality of the final state in (74) is already given in(73).Now moving further towards the most complicated and general situation where both the bra andket states has five non-zero linking numbers, the orthogonality of that state is again obtained in termsof the already calculated orthonormal states in (74) and (73). Let us consider the orthogonal linkingloop state defined at a site x , characterized by the set of 5 linking numbers as follows | l , l , l , l ¯1¯2 , ( l /l ) (cid:105) These loop states are trivially orthogonal with respect to n i ’s for i = 1 , , ¯1 , ¯2, but non-trivial orthonor-mality exists in terms of the linking quantum number. To calculate the orthogonality of loop states interms of the linking numbers , we take an iterative approach as discussed below: Let us consider thefollowing arbitrary overlap of the states (cid:104){ l (cid:48) ij }|{ l ij }(cid:105) = 1 l (cid:48) (cid:104) l (cid:48) − | k − |{ l ij }(cid:105) (76)Note that, in the right hand side of the above equation, in both the bra and ket states we havementioned the linking number, only which has been changed. We will maintain this approach in thelater part of this section as well by characterizing a newly produced state by the changed linkingnumbers only. Whenever, none of the linking numbers do change, we will characterize the state bythe whole set of linking numbers { l ij } . Now from the action given in (24) on the loop states whichsatisfies Mandelstam constraint (29), one obtain k − |{ l ij }(cid:105) = ( n + n − l + 1) | l − (cid:105) − ( l ¯1¯2+1 ) | l − , l − , l ¯1¯2 + 1 (cid:105) (77)with n = l + l + l and n = l + l + l . Note that, in the right hand side of the aboveequation we have suppressed the quantum numbers which remain unchanged. In this way, as donein (76), one can extract out a particular k − ij operator from the bra state or k + ij from the ket statestate and act that on the corresponding ket/bra state to increase or decrease the l ij quantum numbersby one unit until that particular l ij or l (cid:48) ij is exhausted. Or in other way, the iteration can stop at acertain value of l ij , (for example l and l as shown in the above example) which is being decreasedby one unit for each step of the iterations. Hence, clearly iteration will continue p times, where p = min ( l , l , l , l ¯1¯2 , l (cid:48) , l (cid:48) , l (cid:48) , l (cid:48) ¯1¯2 ). Continuing with the example discussed above in (76) and31onsidering the Mandelstam constraint at that particular site by putting l = 0, we finally get: (cid:104){ l (cid:48) ij }|{ l ij }(cid:105) = ( l + l + l + l + l + 1) l (cid:48) (cid:104) l (cid:48) − | l − (cid:105)− ( l ¯1¯2 + 1) l (cid:48) (cid:104) l (cid:48) − | l − , l − , l ¯1¯2 + 1 (cid:105)≡ A (1)0 (cid:104) l (cid:48) − | l − (cid:105) + A (2)1 (cid:104) l (cid:48) − | l − , l − , l ¯1¯2 + 1 (cid:105) (Repeating one more step of iteration for the two overlaps separately,)= A (1)0 (cid:34) A (2)0 (cid:104) l (cid:48) − | l − (cid:105) + A (2)1 (cid:104) l (cid:48) − | l − , l − , l − , l ¯1¯2 + 1 (cid:105) (cid:35) + A (1)1 (cid:34) A (2)0 (cid:104) l (cid:48) − | l − , l − , l − , l ¯1¯2 + 1 (cid:105) + A (2)1 (cid:104) l (cid:48) − | l − , l − , l − , l ¯1¯2 + 2 (cid:105) (cid:35) ≡ A (1)0 A (2)0 (cid:104) l (cid:48) − | l − (cid:105) + (cid:104) A (1)0 A (2)1 + A (1)1 A (2)0 (cid:105) (cid:104) l (cid:48) − | l − , l − , l − , l ¯1¯2 + 1 (cid:105) + A (0)1 A (1)1 (cid:104) l (cid:48) − | l − , l − , l − , l ¯1¯2 + 2 (cid:105) = ......(After p th iteration, for example if p = l (cid:48) ) ≡ p (cid:88) q =0 (cid:48) (cid:88) { s i } q (cid:16) A (1) s A (2) s . . . A ( p ) s p (cid:17) (cid:104) l (cid:48) = 0 | l − p + q, l − q, l − q, l ¯1¯2 + q (cid:105) (78)where, each s i can take values of either 1 or 0, and the (cid:80) (cid:48){ s i } q denotes that the sum is over allpermutations of the set { s i } q ≡ P , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) q times , , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) p − q times The coefficients A ( i ) s i ’s are given by, A (1)0 = ( l + l + l + l + l + 1) l (cid:48) , A (1)1 = − ( l ¯1¯2 + 1) l (cid:48) A (2)0 = ( l + l + l + l + l )( l (cid:48) − , A (2)1 = − ( l ¯1¯2 + 2)( l (cid:48) − A ( p )0 = ( l + l + l + l + l + 2 − p )( l (cid:48) − p + 1) , A ( p )1 = − ( l ¯1¯2 + p )( l (cid:48) − p + 1) (79)In this particular example, the iteration stops at p th level as the at the final step contain the overlap32iven below, (cid:104) l (cid:48) = 0 | l − p + q, l − q, l − q, l ¯1¯2 + q (cid:105) (80)Clearly, the ket state contain four nonzero l (cid:48) ij ’s whereas the bra state has five, the norm of which isgiven in (74) in terms of the norm given in (73). Using these, for our example case, after a few stepsof algebra we have, (cid:104) l (cid:48) = 0 | l − p + q, l − q, l − q, l ¯1¯2 + q (cid:105) = ˜ A (cid:48) (1) q ˜ A (cid:48) (2) q . . . ˜ A (cid:48) ( l − l (cid:48) ) q ˜ B ql δ l (cid:48) + l (cid:48) ,l + l δ l (cid:48) + l (cid:48) ,l + l δ l (cid:48) ¯1¯2 − l (cid:48) ,l − l δ l (cid:48) ,l δ l (cid:48) ,l (81)where , ˜ A (cid:48) iq = − l (cid:48) ¯1¯2 + il − l (cid:48) + q + i − , for i = 1 , , . . . , l − l (cid:48) + q (82)is obtained using (75) and˜ B ql = ( l + l + l ¯1¯2 + l + l + 1 − q )!( l − q )! ( l + l ¯1¯2 + l + l + 1)! × ( l + l ¯1¯2 + l + l + 1)!( l − q )! ( l ¯1¯2 + l + l + 1 + q )! × ( l ¯1¯2 + l + l + 1 + q )!( l ¯1¯2 + q )! ( l + l + 1)! × ( l + 1) ( l + 1) (83)is obtained using (73) for our case.Hence, the complete orthonormality relation of the states |{ l ij }(cid:105) as our example with p = l (cid:48) , canbe obtained combining the (78) and (81) as, p (cid:88) q =0 (cid:48) (cid:88) { s i } q (cid:16) A (1) s A (2) s . . . A ( p ) s p (cid:17) ( − l − p + q ( l (cid:48) ¯1¯2 + l − p + q )! l (cid:48) ¯1¯2 !( l − p + q )! ˜ B qp δ l (cid:48) + p,l + l δ l (cid:48) + p,l + l δ l (cid:48) ¯1¯2 − p,l ¯1¯2 − l δ l (cid:48) ,l δ l (cid:48) ,l (84)where, ˜ B qp are defined in (83).Moving away from this particular example, the most general case can have any of the l ij ’s asminimum and the same calculation will go through. The final expression of any arbitrary case (i.e forany arbitrary p) can be easily read off from the expression derived above just by replacing the role of l /l (cid:48) by the corresponding p . C Strong Coupling Perturbation Expansion
The unperturbed Hamiltonian in the limit g → H e given in(42). H e is solved exactly yielding the loop states as the strong coupling eigenstates with eigenvaluesmeasuring the total flux around the loop. The strong coupling vacuum satisgying H e | (cid:105) = 0 is thestate with no loop present and has unperturbed energy eigenvalue or the unperturbed vacuum energy E (0)0 = 0. We now calculate perturbative corrections to this vacuum energy for the first couple of ordersanalytically. Rayleigh-Schrdinger perturbation theory gives the corrections to the vacuum energy as: E = E (0)0 + 1 g E (1)0 + 1 g E (2)0 + 1 g E (3)0 + 1 g E (4)0 + . . . . . . (85)33he first order correction is given by (cid:104) | H I | (cid:105) = 0 for H I = H mag . Similarly all odd orders ofcorrections to vacuum energy do vanish implying the full correction to come only from even orders.The lowest order correction is of second order and is given by, E (2)0 = (cid:88) n (cid:54) =0 (cid:104) | H I | n (cid:105)(cid:104) n | H I | (cid:105)(cid:104) n | n (cid:105) ( E − E n ) = (cid:88) n (cid:54) =0 |(cid:104) n | H I | (cid:105)| (cid:104) n | n (cid:105) ( E − E n ) (86)where, H I ≡ H mag = 2Tr U plaquette for SU(2) case. In (86), | n (cid:105) is always the state created by a singleaction of Tr U plaquette on | (cid:105) , and it can only be a single plaquette state created by the first term H of the 16 terms figure 14 on vacuum. Obviously for a latice consisting of N number of plaquettes,there exists N such | n (cid:105) states which contributes to the perturbation expansion of vacuum energy.Note that, each of the loops contributing to the perturbation expansion which are eigenstates of theunperturbed Hamiltonian has its unperturbed energy given by, H el | n i (cid:105) = (cid:88) links E links | n i (cid:105) = (cid:88) links n (cid:16) n (cid:17) | n i (cid:105)∀ i (87)for a loop state with, n units of flux along a particular link. For example the single plaquette states | n (cid:105) will have E n = 4 × = 3. Hence, the second order correction is finally obtained as, E (2)0 = N |(cid:104) L (˜ x ) = 1 | U plaquette | (cid:105)| (cid:104) L (˜ x ) = 1 | L (˜ x ) = 1 (cid:105) × (cid:0) − × (cid:1) = N × − H mag as obtained in the earlier sectionsand the normalization of the state is obtained using Appendix B. This correction matches exactly tothe correction in [9] for this order. To confirm the viability of our formulation, we further proceed tocalculate the next order correction given by E (4)0 = (cid:88) { n i } (cid:54) =0 (cid:104) | H I | n (cid:105)(cid:104) n | H I | n (cid:105)(cid:104) n | H I | n (cid:105)(cid:104) n | H I | (cid:105)(cid:104) n | n (cid:105)(cid:104) n | n (cid:105)(cid:104) n | n (cid:105) ( E − E n ) ( E − E n ) ( E − E n ) − E (2)0 (cid:88) { n } (cid:54) =0 (cid:104) | H I | n (cid:105)(cid:104) n | H I | (cid:105)(cid:104) n | n (cid:105) ( E − E n ) (89)Note that, in the fourth order corrections | n (cid:105) as well as the | n (cid:105) are the single plaquette states, locatedanywhere on the lattice. E (4)0 involves another intermediate state | n (cid:105) which is a two plaquette state.Now there exists the following possibility for the two plaquette states:1. | n (cid:105) = H | n (cid:105) ≡ | L (˜ x ) = 1 , L (˜ x ) = 1 (cid:105) , i.e two decoupled plaquette loops located anywhere inthe lattice without any overlap or touch with the first plaquette. Clearly for each | n (cid:105) , there are N − | n (cid:105) with E n = 8 × (cid:0) + 1 (cid:1) = 6.2. The second plaquette can be created by the action of H but with complete overlap with thefirst one, i.e | n (cid:105) ≡ | L (˜ x ) = 2 (cid:105) . In this case, E n = 4 × (cid:0) + 1 (cid:1) = 8. The norm of such statecan be calculated from Appendix B.3. There exists four possibilities of the two plaquette state to be two separate plaquettes withoverlap along any of the link, i.e | n (cid:105) = H | n (cid:105) ≡ | L (˜ x ) = 1 , L (˜ x ± e ( ± e )) = 1 (cid:105) with E n = (cid:0) + 1 (cid:1) + 6 × (cid:0) + 1 (cid:1) = and respective norms. upto a factor of 2 , which is due to the mismatch of the Hamiltonian in (42) and that in [9].
34. The second plaquette can again be created by H in four other possible ways, where the twoplaquettes are touching each other at one of its four corners, i.e | n (cid:105) = H | n (cid:105) ≡ | L (˜ x ) =1 , L (˜ x ± e ± e ) = 1 (cid:105) . For those states E n = 6, but norm is different and can be calculatedeasily.5. By the action of type (b) terms in the Hamiltonian, the two plaquette state can be a loopcarrying unit flux with verical extension of two lattice units and horizontal extension of one,i.e | n (cid:105) = H / | n (cid:105) ≡ | L (˜ x ) = 1 , L (˜ x ± e ) = 1 , N (˜ x ± e ) = 1 (cid:105) . These two states are with E n = 6 × (cid:0) + 1 (cid:1) = and with certain norm.6. Similarly, by the action of type (b) terms in the Hamiltonian, the two plaquette state can be aloop carrying unit flux with verical extension of one lattice units and horizontal extension of two,i.e | n (cid:105) = H / | n (cid:105) ≡ | L (˜ x ) = 1 , L (˜ x ± e ) = 1 , N (˜ x ± e ) = 1 (cid:105) with E n = and norm to becaculated from Appendix B.Explicit calculation incorporating all the coefficients given in table 1 for the Hamiltonian actions andthe norm of each state calculated using the appendix we finally obtain, E (4)0 = N × × ≡ N × × with [9]. In the same waythe strong coupling perturbation correction to any loop state can be performed within this scheme andnote that this scheme is independent of any cluster size or lattice size.Besides making strong coupling perturbation expansion viable upto any arbitrary order our for-mulation is also suitable to approach towards weak coupling limit. It seems that the fusion variablesbecome extremely important to work with in this regime. The work in this direction is in progress andwill be reported shortly. References [1] K. Wilson, “Confinement of quarks,” Phys. Rev.
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