Constraint structure and Hamiltonian treatment of Nappi-Witten model
aa r X i v : . [ h e p - t h ] O c t Constraint structure and Hamiltonian treatment of Nappi-Witten model
M. Dehghani A. Shirzad Department of Physics, Isfahan University of TechnologyP.O.Box 84156-83111, Isfahan, IRAN,School of Physics, Institute for Research in Fundamental Sciences (IPM)P.O.Box 19395-5531, Tehran, IRAN.
Abstract
We investigate the Hamiltonian analysis of Nappi-Witten model (WZW actionbased on non semi simple gauge group) and find a time dependent non-commutativityby canonical quantization. Our procedure is based on constraint analysis of the modelin two parts. A first class analysis is used for gauge fixing the original model followingby a second class analysis in which the boundary condition are treated as Diracconstraints. We find the reduced phase space by imposing our second class constraintson the variables in an extended Fourier space.
Keywords :Noncommutativity, constraint analysis
Treating boundary conditions as Dirac constrains has been considered in the recent yearsby so many authors [1, 2, 3, 4]. This approach has been used first in studying the Polyakovstring coupled to a B-field. The common feature of all works is non commutativity of thecoordinate fields on the boundaries which may lie on some brains, as first predicted by[5]. However, there are different approaches in defining the constraints and investigatingtheir consistency in time. We have reviewed the whole subject in our previous work [6]and showed if we impose the set of constraints on the Fourier expansions of the fields, theredundant modes will be omitted in a natural way.For simple physical models obeying linear equations of motion, the ordinary Fourierexpansion gives appropriate coordinates to reach the reduced phase space. In other words,the infinite set of second class constraints emerging as the result of boundary conditions,forces us to omit a number of Fourier modes. However, ordinary Fourier transformationis not essential for quantization; it is just one tool that acts well for most physical modelsat hand. In the general case one should search for ”appropriate coordinates”, in whichimposing the set of second class constraints is equivalent to omitting some canonical pairsfrom the theory.In this paper we study the constraint structure of the Nappi-Witten model in the Hamil-tonian formalism. This model acquires complicated boundary conditions so that the or-dinary Fourier expansion seems inadequate to impose the whole set of constraints which [email protected] [email protected] merge from the boundary conditions. Nevertheless, the Nappi-Witten model, on its owngrands, is an attractive one since it describes a non semi simple gauge group as well as givingtime dependent non commutativity in some gauges [7]. Our next interest is to emphasizethat solving the equations of motion is not necessarily needed for quantizing a theory; theonly necessity is finding the dynamics of the constraints and construct their algebra withthe Hamiltonian such that they remain consistent with time on the constraint surface.We give a precise Hamiltonian treatment of the model in which the constraint structureis followed step by step from the initial action to the final reduced phase space. In section 2we introduce the model and find primary and secondary constraints of the system. Section3 is devoted to fixing the gauge by introducing appropriate gauge fixing conditions. Insection 4 we follow our strategy of treating the boundary conditions as primary Diracconstraints and follow their consistencies. The boundary conditions which come from theoriginal action, in fact, make the system more complicated. So, it is not possible to writedown the solutions in a closed form similar to a simple Fourier expansion (see reference[8]). We try to find a basis which is appropriate for imposing the infinite set of constraintsin section 5. In section 6 we will give our concluding remarks and will compare our resultswith parallel approaches. The Nappi-Witten model describes a 4-component bosonic string X a = ( a , a , u, v ) livingin the background metric G ab ( X ) and coupled to a B -field. The action is given as: S = Z d σ (cid:20) √− gg ij G ab ∂ i X a ∂ j X b + B ab ǫ ij ∂ i X a ∂ j X b (cid:21) , (1)where G ( X ) = a
00 1 − a a − a b
10 0 1 0 , B ( X ) = u − u . (2)The special form of G ( X ) and B ( X ) are chosen so that the gauge group of the model isnon semi-simple [8]. The metric field can be written in terms of the following variables: N = g √− g , N = − g g , N = √− g = √ ( g ) − g g . (3)In terms of the variables X a and N α the action becomes: S = Z d σ (cid:20) N G ab ( X )( ˙ X a ˙ X b − N ˙ X a X ′ b + ( N − N ) X ′ a X ′ b ) + 2 B ab ˙ X a X ′ b (cid:21) , (4)where dot and prime means temporal and spatial derivatives, respectively. The canonicalmomenta π α and p a conjugate to N α and X a are: π α = 0 α = 1 , , p i = N (2 ˙ a i + ˙ uǫ ij a j ) − N N (2 a ′ i + u ′ ǫ ij a j ) + 2 uǫ ij a ′ j p u = N (2 b ˙ u + 2 ˙ v + ǫ ij ˙ a i a j ) − N N (2 bu ′ + 2 v ′ + ǫ ij a ′ i a j ) p v = uN − N N u ′ . (5)2he Canonical Hamiltonian reads: H = Z d σ N G ab ( F a F b − ( N − N ) X ′ a X ′ b ) , (6)where F a = ˙ X a = N ( G − ) ab ( p b − B bc X ′ c ) + N B ab X ′ b (7)In terms of component fields a i , u and v we have H = Z d σ ( N Ψ + N Ψ ) (8)where Ψ = p i + ǫ ij p v a i p j + p u p v − bp v + a i p v + u ′ ǫ ij a ′ i a j + ǫ ij ua ′ i p j + up v a ′ i a i + (1 + u ) a ′ i + bu ′ + 2 u ′ v ′ Ψ = a ′ i p i + u ′ p u + v ′ p v , (9)As can be seen from Eqs. (5) the momenta π α are primary constraints. The dynamics ofthe system is achieved by the total Hamiltonian: H T = H + Z dσλ α π α ( σ, τ ) , (10)in which λ α are Lagrange multipliers. As usual we should impose the consistency conditionson the constraints so that they remain valid during the time. For this reason we demand˙ π α ≈
0, where ≈ means weak equality i.e. equality on the constraint surface. Using Eqs.(10) and (6) we have: ˙ π = { π , H T } = − Ψ ˙ π = { π , H T } = − Ψ ˙ π = { π , H T } = 0 , (11)Therefore, the consistency of three primary constraints π α gives two second level constraintsΨ and Ψ . In this way we have so far two levels of constraints as π π π Ψ Ψ . (12)In order to investigate the consistency of second level constraints, we need to calculate thePoisson brackets of Ψ ( σ, τ ) and Ψ ( σ, τ ) at different points. Direct calculation, using thefundamental Poisson brackets among the four conjugate pairs ( u, p u ), ( v, p v ) and ( a i , p i )gives: { Ψ ( σ, τ ) , Ψ ( σ ′ , τ ) } = (Ψ ( σ, τ ) ∂ σ − Ψ ( σ ′ , τ ) ∂ σ ′ ) δ ( σ − σ ′ ) { Ψ ( σ, τ ) , Ψ ( σ ′ , τ ) } = Ψ ( σ, τ ) ∂ σ δ ( σ − σ ′ ) { Ψ ( σ, τ ) , Ψ ( σ ′ , τ ) } = (Ψ ( σ, τ ) ∂ σ − Ψ ( σ ′ , τ ) ∂ σ ′ ) δ ( σ − σ ′ ) , (13)where δ ′ ( σ − σ ′ ) ≡ ∂∂σ δ ( σ − σ ′ ). It should be noted that each of the above Poisson bracketsleads to a set of terms at different points σ and σ ′ multiplied by ∂∂σ δ ( σ − σ ′ ) or ∂∂σ ′ δ ( σ − σ ′ )which equals to − ∂∂σ δ ( σ − σ ′ ). However, since these terms have only non vanishing valuewhen σ ′ approaches to σ , one can consider all of them at the same point. Then they addup to give the above results. The algebra (13) shows that Ψ ( σ, τ ) and Ψ ( σ, τ ) are firstclass constraints. Moreover, from (8) we see that: { Ψ , H } = N ′ Ψ + N ′ Ψ + N Ψ ′ ≈ { Ψ , H } = N ′ Ψ + N ′ Ψ + N Ψ + N Ψ ′ ≈ ( σ, τ ) and Ψ ( σ, τ ) does not give any new constraint,and we are left with the five first class constraints given in (12).In this way we have derived three constraint chains (cid:18) π Ψ (cid:19) , (cid:18) π Ψ (cid:19) and (cid:0) π (cid:1) inthe terminology of reference [9]. In fact, the chain relation { π α , H } = Ψ α holds for all ofthe chains. However the first two chains are correlated, since the Poisson bracket of thelast element of each chain with the Hamiltonian contains the other constraint. This meansthat it is not possible to construct closed algebra within each chain. The last chain containsjust one element and is not correlated to other chains, since it commutes with all of themas well as with Hamiltonian.As in ordinary Polyakov string one can show that π generates the Weyl symmetry ofthe model which affects only the components of the world-sheet metric. In terms of thevariables N α we have N → N + ǫ under Weyl transformation. On the other hand the con-straint chains (cid:18) π Ψ (cid:19) , (cid:18) π Ψ (cid:19) can be shown that generate the effect of reparametrizationinvariance on the metric variables N and N as well as the variables X a . We began the theory with 14 field variables in the phase space, i.e. X a , N α and theircorresponding momentum fields p a and π α . Then we derived 5 first class constraints givenin (12). As is well known from Dirac theory the first class constraints are generators ofgauge transformations [10]. One needs to consider additional conditions to fix the gauges.These ”gauge fixing conditions” are functions of phase space variables which should vanishto fix the gauges. The gauge fixing conditions should fulfill two conditions. First, theyshould constitute a system of second class constraints when added to the original firstclass constraints of the system. This condition is necessary to fix the values of variableswhich vary under the action of gauge generators [12]. Second, they should have a closedalgebra under the consistency conditions, i.e. under the successive Poisson brackets withthe Hamiltonian.For a ”complete gauge fixing” the number of independent gauge fixing conditions shouldbe equal to the number of first class constraints [13]. In this way, we should suggest 5 gaugefixing conditions to fix the gauges generated by the constraints given in (12), and reacha ”reduced phase space” of 4 field variables. Since the momenta π α are generators oftransformations in N α , we fix the corresponding gauge by choosing the values of N α as N ≈ , N ≈ N ≈
1. These values are chosen such that g ij = η ij . In this way wehave so far introduced three gauge fixing conditionsΩ ≡ N − , Ω ≡ N , Ω ≡ N − . (15)It can easily seen that the system of 6 constraints π α and Ω α are second class. The consis-tency of Ω α ’s by the use of total Hamiltonian (10) determines the lagrange multipliers λ α to be zero and does not give any new constraint. This makes us sure that the two criterionsof a good gauge mentioned above are satisfied. In fact, by the above gauge fixing threedegrees of freedom N α are removed completely from the theory. This gauge has fixed the4eyl symmetry as well as the effect of the reparametrization on the metric variables N and N . On the other hand, we are still left with the remaining gauges generated by Ψ and Ψ which generate the effect of reparametrization on the variables X a . In fact, since we havefixed the gauge from the middle of the constraint chains, the gauge is fixed partially in thelanguage of reference [13]. In partial gauge fixing the Lagrange multipliers are determinedwhile the variations generated by some of the gauge generators are not fixed.To fix the effect of the parametrization of the world-sheet on X a ’s, as in so manymodels in string theory we need to determine some definite combinations of fields as thetime variable in target space. Taking a look on the form of the constraints Ψ and Ψ in (9) shows that the choice u = µτ is more economical in the sense that simplifies theconstraints better. Here µ is a parameter with dimension of (length) − . We recall that allof the dynamical variables in the action are dimensionless. Hence, we consider the gaugefixing condition Ω = u − µτ. (16)To fulfill the second criterion of a good gauge we choose the last gauge fixing condition asΩ ≡ ˙Ω = { Ω , H T } + ∂ Ω ∂τ (17) ≈ p v − µ This new constraint should also be valid during the time. Since˙Ω = 2 µ ( − N N + N ′ ) ≈ , (18)the chosen gauges are consistent and make a closed algebra with the Hamiltonian. It is alsoclear that Ω and Ω make a second class system with Ψ and Ψ . Imposing strongly theconstraints (16) and (17) on the system, simplifies the constraints Ψ and Ψ asΨ → ¯Ψ = p i + ǫ ij µa i p j + ǫ ij µτ a ′ i p j + (1 + µ τ ) a ′ i + µp u − bµ + µ a i + µ τ a i a ′ i , Ψ → ¯Ψ = a ′ i p i + 2 µv ′ , (19)This shows that p u and v can be derived on the constraint surface, i.e. from identities¯Ψ = 0 and ¯Ψ = 0, in terms of the physical variables a i and p i . In this way the reducedphase space is just the four dimensional space of ( a i , p i ) whose original Poisson bracketsserve as the Dirac brackets in the remaining physical space. The terms µp u and µ b in theexpressions of ¯Ψ have nothing to do with the dynamics of ( a i , p i ) and can be dropped. Theparameter b has in fact no important role in the theory and only shifts the spectrum of theenergy with a constant value.As in reference [8] we consider the dimensionless quantity µl as a small parameter whichshould be considered only in the first order. Therefore, in all of the foregoing calculationswe keep only linear terms with respect to µ , assuming that l is finite. Therefore, theHamiltonian (8) in the reduced phase space can be written in terms of the Hamiltoniandensity: H GF = 14 p i + 12 ǫ ij µa i p j + ǫ ij µτ a ′ i p j + a ′ i . (20)Since B ( X ) in (2) is linear with respect to u one may think of µ as the order of magnitudeof the B -field. This assumption is equivalent to considering the effect of the B -field onlyup to the first order. 5 Boundary conditions as constraints
From now on we forget about the original theory and suppose we are given a theory withtwo degrees of freedom a i and the corresponding momenta p i whose dynamics is given by thefinal Hamiltonian (20). We make a change of variables from ( a i , p i ) to ( A i = ǫ ij a j , P i = p i ).Then the the fundamental Poisson brackets which is the same as the final Dirac bracket ofthe original theory read { A i ( σ, τ ) , P j ( σ ′ , τ ) } = ǫ ij δ ( σ − σ ′ ) , { A i ( σ, τ ) , A j ( σ ′ , τ ) } = { P i ( σ, τ ) , P j ( σ ′ , τ ) } = 0 (21)The Hamiltonian equation of motion for the remaining fields, can be written as˙ A i = ǫ ij ( P j − µτ A ′ j − µA j )˙ P i = − ǫ ij ( µP j − µτ P ′ j + 2 A ′′ j ) (22)The only things that should be brought from the original theory are the boundaryconditions. Using the original action (4) the boundary condition after gauge fixing emergein terms of phase space variables as:Φ (1) i = µτ P i − A ′ i = 0 at σ = 0 , l (23)We have shown in the appendix that the boundary condition (23) can also be derived fromthe parallel approach as the equations of motion of the end points in the discretized version.As mentioned in the introduction we do not want to find the general solution of thedynamical equations of motion. On the other hand, we are interested to follow the dynamicsof the boundary conditions which means investigating the consistency of primary constraintsΦ (1) i ( σ ) | σ =0 and Φ (1) i ( σ ) | σ = l . Using the gauge fixed Hamiltonian of the previous section (20)the total Hamiltonian at this stage is H T = Z l dσ [ 14 P i P i − µA i P i − µτ A ′ i P i + A ′ i A ′ i ] + Λ i Φ (1) i ( σ ) | σ =0 + Λ i Φ (1) i ( σ ) | σ = l . (24)The consistency of primary constraints for instance at σ = 0 gives0 = (cid:2) µP i − ǫ ij P ′ j + µǫ ij A ′ j (cid:3) σ =0 + Λ j n Φ (1) i | σ =0 , Φ (1) j | σ =0 o (25)Similar equations should be written at the end-point σ = l . As discussed in details in [14]the first term in the LHS of Eq. (25) has not the same order as the coefficient of Λ i (and Λ i )in the second term when regularizing the Dirac delta function. Therefore this condition canbe fulfilled identically only if Λ i , as well as the first term vanish simultaneously. In this waywe have used the consistency conditions of the constraints for simultaneously determiningthe undetermined Lagrange multiplier and finding the next level of constraints as Φ (2) i (0)and Φ (2) i ( l ) where Φ (2) i ( σ ) = P i − ǫ ij P ′ j + µǫ ij A ′ j . (26)Then we should consider the consistency of second level constraints by using the Hamilto-nian H = Z l dσ [ 14 P i P i − µA i P i − µτ A ′ i P i + A ′ i A ′ i ] (27)6hich is the same as the total Hamiltonian (24) after imposing Λ i , = 0. This gives thethird level of constraints. Subsequent levels of constraints can be derived in the same way.Using the relations: { A ( n ) i , H } = ǫ ij ( P ( n ) j − µA ( n ) j − µτ A ( n +1) j ) + O ( µ ) { P ( n ) i , H } = − ǫ ij ( µP ( n ) j − µτ P ( n +1) j + 2 A ( n +2) j ) + O ( µ ) , (28)where A ( n ) i = ∂ nσ A i and P ( n ) i = ∂ nσ P i one can inductively show that the full set of constraintsare Φ ( N ) i (0) ≈ ( N ) i ( l ) ≈ (2 n +1) i = − nµP (2 n − i + µτ P (2 n ) i − nµǫ ij A (2 n ) j − A (2 n +1) i + O ( µ ) , Φ (2 n +2) i = ( n + 1) µP (2 n ) i − ǫ ij P (2 n +1) j + (2 n + 1) µǫ ij A (2 n +1) j + O ( µ ) n = 0 , , , · · · (29)For practical calculations we write the constraints as ordinary functions in the bulk of thestring and then integrate them with the use of δ ( σ ) and δ ( σ − l ) respectively.Now we want to investigate whether the constraints are first or second class. For thisreason one should calculate the Poisson brackets of the constraints. Since the constraintscontain different orders of derivatives of A i ( σ, τ ) and P i ( σ, τ ), the Poisson brackets C k,k ′ ij ≡{ Φ ki , Φ k ′ j } contain derivatives of orders k + k ′ , k + k ′ −
1, etc, of the Dirac delta function,which are highly divergent and independent of each other. One way of treating the matrixof Poisson brackets is regularizing the delta functions as gaussian functions of width ε andlet ε → C m +1 , n +1 ij = − µǫ ij √ π ε − m + n +1) ( ε ( m + n ) H m +2 n (0) − τ H m +2 n +1 (0)) + O ( µ ) C m +2 , n +1 ij = − √ π ε − m + n +1) − ( nµεǫ ij H m +2 n +1 (0) + δ ij H m +2 n +2 (0)) + O ( µ ) ,C m +2 , n +2 ij = µǫ ij √ π ε − m + n +1) − H m +2 n +2 (0) + O ( µ ) (30)where H n ( x ) are Hermite polynomials. Similar expressions should be considered with H n (1)at the end-point σ = l . The non vanishing elements on each row are located such that novanishing linear combination of rows may be found. This means that the infinite dimen-sional matrix C k,k ′ ij is not singular and can in principle be inverted. Therefore, all of theconstraints are second class. However, it is not practically possible to find the inverse of C k,k ′ ij . The problem is how we can find the Dirac brackets of the fields which need to have C − . As stated before, we seek for appropriate coordinates in which imposing the constraints(29) lead to omitting a set of canonical pairs. Here we have a difficult problem in which theordinary Fourier expansion does not do this job. However, in the limit µ → A i and P i that include at most linear corrections with respectto the parameter µ and go to the ordinary Fourier transformation in the limit µ →
0. Since µτ and µσ are the only dimensionless quantities that can be used for this correction, whatcan we do is correcting the Fourier coefficients by correction terms linear in τ or σ . The7inear term in τ , however, is not needed at this stage, since it can be considered as part ofthe solution of the equations of motion. Adding all these points up together we suggest thefollowing extended Fourier transformations for the fields A i ( σ, τ ) = 1 √ π Z ∞−∞ dk [( A i ( k, τ ) + µσα i ( k, τ )) cos kσ + ( B i ( k, τ ) + µσβ i ( k, τ )) sin kσ ] , (31) P i ( σ, τ ) = − ǫ ij √ π Z ∞−∞ dk [( C j ( k, τ ) + µσγ j ( k, τ )) cos kσ + ( D j ( k, τ ) + µσδ j ( k, τ )) sin kσ ] . (32)In ordinary Fourier expansions the coefficients A i ( k, τ ), B i ( k, τ ), C i ( k, τ ) and D i ( k, τ ) con-tain the same amount of data as the original fields A i ( σ, τ ) and P i ( σ, τ ). Comparing theexpansions (31) and (32) with ordinary Fourier expansions shows that we have used a du-plicated basis including sin’s, cos’s, σ times sin’s and σ times cos’s for expanding our fields.This basis is complete but its elements are not independent. Mathematically it is allowedto use a basis which is ”larger than necessary”. However, the essential point is that oneshould assume appropriate Poisson brackets among the extended Fourier modes such thatthe desired fundamental Poisson brackets (21) remain valid. In other words, we shouldtune their brackets in such a way that our physical phase space variables, which are half ofthe extended phase space variables, do obey the right Poisson brackets. Direct calculationshows that the following Poisson brackets lead to the standard Poisson algebra (21) for thephysical fields, { A i ( k, τ ) , C j ( k ′ , τ ) } = { B i ( k, τ ) , D j ( k ′ , τ ) } = δ ij δ ( k − k ′ ) , { α i ( k, τ ) , D j ( k ′ , τ ) } = { γ i ( k, τ ) , B j ( k ′ , τ ) } = δ ij ∂ k ′ δ ( k − k ′ ) . (33)All other Poisson brackets are assumed to vanish. Specially the modes β i and δ i havevanishing Poisson brackets with all other variables in the extended Fourier space and sodecouple from the theory. This means that we can put them away and write down theexpansions only with linear terms in the cosine modes. We will see on the other handthat omitting the modes β i and δ i does not disturb our analysis of imposing the boundaryconditions. We have, up to this point, 6 sets of real variables in the extended Fourier spacewhich depend on real, continues and positive variable k .Now we want to impose the full set of constraints (29) on the fields. Using the expansions(31) and (32) the constraints at the end-point σ = 0 lead to R ∞−∞ dkk n h µτ ǫ ij C j + 2 nǫ ij A j + (4 n + 2) α i + 2 k ˜ B i i + O ( µ ) = 0 R ∞−∞ dkk n − h ( n + 1) ǫ ij C j + (2 n + 1) γ i + k ˜ D i i + O ( µ ) = 0 (34)where B i = µ ˜ B i and D i = µ ˜ D i . Since these conditions should be satisfied for arbitraryvalues of n we have µτ ǫ ij C j + 2 nǫ ij A j + (4 n + 2) α i + 2 k ˜ B i = 0 , ( n + 1) ǫ ij C j + (2 n + 1) µγ i + k ˜ D i = 0 . (35)The difficulty arises here since the integer n , which shows the level of constraints, hasappeared in the form of relations among the Fourier modes. This means that it is notpossible to satisfy the constraints of all levels just by considering simple linear relations8mong the Fourier modes of a given k as can be done in ordinary Dirichlet, Neumann, oreven mixed boundary conditions [6]. In fact, this phenomenon is the reason which makesthe ordinary Fourier expansion inadequate for realizing the constraints. However, we havethe opportunity of existence of extra variables in the extended phase space, which providesus additional tools for satisfying the constraints. In this way we are allowed to assume thatthe coefficients of n besides the terms independent of n in (35) vanish. This gives α i = − ǫ ij A j + O ( µ ) ˜ B i = k ǫ ij ( A j − τ C j ) + O ( µ ) γ i = − ǫ ij C j + O ( µ ) ˜ D i = − k ǫ ij C j + O ( µ ) (36)Hence the main fields A i ( σ, τ ) and P i ( σ, τ ) can be written in terms of two remaining setsof Fourier modes A i ( k, τ ) and C i ( k, τ ) as A i ( σ, τ ) = 1 √ π Z ∞−∞ dk (cid:20) ( δ ij − µσǫ ij ) A j cos kσ + µ k ǫ ij ( A j − τ C j ) sin kσ (cid:21) , (37) P i ( σ, τ ) = − √ π Z ∞−∞ dk (cid:20) ( ǫ ij + 12 µσδ ij ) C j cos kσ + µ k C i sin kσ (cid:21) . (38)As expected, the zeroth order (with respect to µ ) of the Eqs. (37) and (38) is the expansionof a simple bosonic string with Neumann boundary condition at the end point σ = 0. Thelinear term with respect to σ in cosine modes as well as the sin term itself are appeared asthe first order corrections.Next we should impose the constraints (29) at the end-point σ = l on the fields derivedrecently in Eqs. (37) and (38). Hence we find R ∞−∞ dkk n − ( − n [ nµǫ ij C j + 2 k ( A i − µσA j )] sin( kl ) + O ( µ ) = 0 , R ∞−∞ dkk n +1 ( − n [( δ ij − µσǫ ij ) C j − (2 n + 1) µǫ ij A j ] sin( kl ) + O ( µ ) = 0 . (39)The above constraints are satisfied identically for kl = mπ . However, for k = mπl there isno way for satisfying the constraints for arbitrary n except assuming that A i ( k, τ ) = C i ( k, τ ) = 0 for k = mπl (40)This leads to descritizing the Fourier modes.Before writing the final form of the fields in terms of the set of enumerable Fouriermodes, care is needed to write the zero modes. The contributions due to cosine modescome out automatically by letting k = 0. However, contributions to zero modes originatingfrom sine terms should be derived by taking the following limits:lim k → ˜ B i sin kσ = 12 σǫ ij ( A j (0 , τ ) − τ C j (0 , τ )) , lim k → ˜ D i sin kσ = − σǫ ij C j (0 , τ ) , (41)which follow from Eqs.(36). Adding these two contributions the zero mode part of the fieldsare so far as follows A i ( σ, τ ) = A i ( τ ) − µστ ǫ ij C j ( τ ) P i ( σ, τ ) = − ( ǫ ij + µσδ ij ) C j ( τ ) (42)At this point we want to notice the reader to a global symmetry of the gauged fixedLagrangian. If we turn off the B-field we would have an ordinary bosonic string in which9nly the derivatives of the A-fields are present in the Lagrangian. This allows one to shiftthe fields by a constant amount without any change in the Lagrangian. When the B-fieldis on, Eq. (20) shows that the A-field itself is present in the gauged fixed Hamiltonian.However, the relevant term, i.e. the second term in Eq. (20), is proportional to µ . Thisshows that the theory is symmetric, up to second order terms with respect to µ , under thefollowing transformation A i ( σ, τ ) → A i ( σ, τ ) + µf ( τ ) (43)where f ( τ ) is an arbitrary function of time. This symmetry leads to an ambiguity in thezero mode of the A-field. Hence we should correct the first row of Eq. (42) in the mostgeneral case as follows A i ( σ, τ ) = A i ( τ ) − µστ ǫ ij C j ( τ ) + µl [( a ij A j ( τ ) + b ij C j ( τ )] (44)Note that µl is the only relevant dimensionless quantity which is first order in µ . Theunknown coefficients a ij and b ij should be determined upon suitable assumptions aboutthe algebra of the fields. The best assumption seems to be keeping the standard algebra(21) in the bulk of the string and letting all changes in the algebra of the fields lay on theboundaries. If we make this choice the final form of the physical fields in terms of the setof discrete Fourier modes A mi ( τ ) ≡ A i ( mπl , τ ) and C mi ( τ ) ≡ C i ( mπl , τ ) are as follows A i ( σ, τ ) = √ l (cid:20) A i ( τ ) − µτ ( σ − l ) ǫ ij C j ( τ ) − µlǫ ij A j ( τ ) (cid:21) + q l P ∞ m =1 (cid:20) ( A mi ( τ ) − µσǫ ij A mj ( τ )) cos mπσl + µl mπ ǫ ij ( A mj ( τ ) − τ C mj ( τ )) sin mπσl (cid:21) (45) P i ( σ, τ ) = − √ l (cid:20) ǫ ij C j ( τ ) + µσC i ( τ ) (cid:21) − q l P ∞ m =1 (cid:20) ( ǫ ij C mj ( τ ) + µσC mi ( τ )) cos mπσl + µl nπ C mi ( τ ) sin mπσl (cid:21) (46)The normalization factor √ π is replaced by q l for oscillatory modes and √ l for zeromode upon going from the continues parameter k to the discrete parameter m . With thisnormalization the brackets of the discrete modes should also be given in terms of Kroneckerdelta as { A mi , C m ′ j } = δ ij δ mm ′ , (47) { A mi , A m ′ j } = { C mi , C m ′ j } = 0 . (48)In fact, the remaining canonical pairs A mi and C mi as a small part of the original phasespace are natural coordinates of the reduced phase space. On the other hand, a great partof the initial phase space variables are omitted due to the constraints.Remember that if one is able to omit the redundant variables due to all kinds of con-straints and write down the relevant fields in terms of final canonical coordinates of thereduced phase space, then there is no need to find the Dirac brackets. In other words, wepay the expense of using the Dirac brackets whenever it is not possible to find a canonical Since another length scale, i.e. µ − , exists in the model, one may suppose that the normalizationfactors should differ from the ordinary Fourier series. However, it can be shown that such corrections onlychanges the observables by amounts of O ( µ ) which is not important A i ( σ, τ ) and P i ( σ, τ ) if we calculate their brackets by using the brackets (47)and (48).Eq. (46) shows that the momentum-fields P i ( σ, τ ) just include the variables C mi andhave vanishing brackets: { P i ( σ, τ ) , P j ( σ ′ , τ ) } = 0 . (49)Straightforward calculations gives the brackets of coordinate and momentum fields as { A i ( σ, τ ) , P j ( σ ′ , τ ) } = ǫ ij δ N ( σ, σ ′ ) , (50)where δ N ( σ, σ ′ ) ≡ δ ( σ − σ ′ ) + δ ( σ + σ ′ ) . Since both σ and σ ′ lie in the interval [0 , l ] their sum never vanishes. So the seconddelta function does not have any role and Eq. (50) reduces to the usual form of Eq.(21). However, since in the expansion of A -fields both variables A mi and C mj are present,the interesting phenomenon appears in the bracket of coordinate fields at different points.Direct calculation gives { A i ( σ, τ ) , A j ( σ ′ , τ ) } = 12 µτ ǫ ij σ + σ ′ l − π ∞ X n =1 n sin nπl ( σ + σ ′ ) ! . (51)This result is similar to what derived in [6] for a string coupled to constant backgroundB-field. The right hand side of Eq. (51) vanishes in the bulk of the string, i.e. when σ or σ ′ does not lie on the end points. It gives (-2) when σ = σ ′ = 0 and (+2) when σ = σ ′ = l . However, as the B-field itself, the amount of non commutativity grows linearlywith time. Our result here defers from reference [7] with a term proportional to µτ whichis the same on both boundaries as well as in the bulk of the string. If, however, we add aterm − µτ ǫ ij C j ( τ ) to the zero mode part of the field A i ( σ, τ ) in Eq. (45), our result willcoincide with reference [7]. This correction is allowed according to the global symmetry ofEq. (43). This means that we have forgiven our previous assumption that the componentsof the A-field commute in the bulk of the string. With this assumption the resulted bracketscan be summarized as follows { A i ( σ, τ ) , P j ( σ ′ , τ ) } = ǫ ij δσ, σ ′ ) , { P i ( σ, τ ) , P j ( σ ′ , τ ) } = 0 { A i ( σ, τ ) , A j ( σ ′ , τ ) } = µτ ǫ ij l σ = 0 , l or σ ′ = 0 , lµτ ǫ ij (1 + τ l ) σ = σ ′ = 0 µτ ǫ ij ( − τ l ) σ = σ ′ = l (52)This shows that the fundamental characters of the A -fields and P -fields as coordinate andmomentum fields are remained almost as before and the time dependent B-field leads to atime dependent non commutativity in the coordinate fields all over the string. In this paper we gave a complete Hamiltonian treatment of the Nappi-Witten model (WZWmodel based on non semi simple gauge group) as an interesting and non trivial system in11hich complicated boundary conditions make the physical subset of variables far fromreaching. The initial dynamical variables in this model are 4 components of a bosonicstring, X a = ( a , a , u, v ), and the components of world-sheet metric. We used appropriatevariables to find 3 primary and 2 secondary first class constraints. It can be shown thatthese constraints are generators of reparametrizations as well as Weyl transformations.Then we fixed the gauge such that the world-sheet metric is flat and u = µτ where thesmall parameter µ is proportional to the strength of the B-field. In this way the componentsof the world-sheet metric and the variables u and v disappeared as the result of constraintsand gauge fixing conditions. Hence, we derived a smaller theory with two coordinate fields a and a and their corresponding momentum fields.The most important part of the problem seems to be the boundary conditions whichshould be brought from the original theory. Considering the boundary condition as Diracconstraints and following their consistency, we found two infinite chains of second classconstraints at the end-points which restricted the space of physical variables to a muchsmaller set. Due to complicated form of the boundary conditions, it is not an easy taskto impose them on the space of the physical variables. In fact, with an ordinary Fourierexpansion the constraints do not lead simply to omitting some Fourier modes as in Dirichletor Neumann boundary conditions.To overcome this difficulty we extended the phase space to a larger one which is given byan extended Fourier expansion in which the Fourier modes are replaced by linear functionsof the variables. In this basis the infinite set of constraints can be imposed more easilyby using the arbitrariness due to extra variables. This results to disappearing of so manycanonical pairs among the used extended Fourier basis and finally a set of discrete modesremain which act as the canonical coordinates of the reduced phase space. Then all physicalobjects including the original coordinate and momentum fields can be expanded in termsof these modes.Using these expansions we found that the commutation relations of the coordinate andmomentum fields are almost as usual, except that the coordinate fields do not commuteat the boundaries, with an amount proportional to time and/or B-field but with oppositesigns at two boundaries. We showed that it is allowed to insert a term which gives noncommutativity proportional to τ throughout the string. This correction may make ourresults consistent with those of reference [7] in which the authors have given iterativesolutions for the equations of motion.We think that our method here has two main advantages in two different areas. First,we do not solve the equation of motion. Therefore, in our final result the time dependenceof remaining modes are not specified. However, this time dependence is not essential forquantization of the model. If needed, one can use the Hamiltonian written in terms of thefinal modes and then derive their time dependence. In fact, our main objective is that forquantizing a theory, i.e. investigating the algebraic structure of the observables, it is notneeded to follow the full dynamics of the system; it is just enough to study the dynamics ofconstraints. As a matter of fact, for simple models it may seem more simple and economicto solve the equations of motion and then quantize the theory, since this procedure containsthe dynamics of the constraints within itself. But this may not be the case for a complicatedmodel such as the model considered in this paper.The next advantage is in the context of constraint systems. As we see in the literature[1, 14] the main difficulty in considering the infinite set of constraints due to boundaryconditions is deriving the Dirac brackets. In this paper, as in our previous work [6] we12howed that if one is able to find a set of canonical variables describing the reduced phasespace, then there is naturally no need to calculate the Dirac brackets. In fact, this was themain brilliant idea of Dirac [11], who gave his famous formula of Dirac brackets in such away that it is equivalent to calculating the Poisson brackets only in the space of canonicalvariables describing the reduced phase space. References [1] C.S Chu, P.M Ho, Nucl. Phys. B 550 (1999) 151.[2] F. Ardalan, H. Arfaei, M.M. Sheikh-Jabbari, Nucl. Phys. B 576 (2000) 578.[3] T. Lee, Phys. Rev. D 62 (2000) 024022.[4] R. Banerjee, B. Chakraborty, K. Kumar, Nucl. Phys. B 668 (2003) 179.[5] N. Seiberg, E. Witten, JHEP 09 (1999) 032.[6] M. Dehghani and A. Shirzad, Eur. Phys. J. C48 (2006) 315.[7] L. Dolan and C. R. Nappi, Phys. Lett. B 551 (2003) 369.[8] C.R. Nappi , E. Witten, Phys. Rev. Lett. 71 (1993) 3751.[9] F. Loran and A. Shirzad, Int. J. Mod. Phys. A 17 (2002) 625.[10] M. Henneaux, C. Teitelboim, J. Zanelli, Nucl. Phys. B 332 (1990) 169.[11] P.A.M. Dirac,
Lecture Notes on Quantum Mechanics , Yeshiva University New York,1964. Also see P.A.M. Dirac, Proc. Roy. Soc. London. ser. A,246