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consider a massless dirac fermion interacting with a background gauge field . our present goal is to formulate a nonperturbatively regularized quantum field theory which at least satisfies the following physical constraints :    \(a ) in the classical limit , it reproduces the classical physics of the action , @xmath0 \psi(x ) $ ] . \(b ) for topologically trivial gauge backgrounds , and in the weak coupling limit , it agrees with the predictions of weak coupling perturbation theory of the action . \(c ) for topologically nontrivial gauge backgrounds , it possesses exact zero modes satisfying the atiyah - singer index theorem .    although wilson s idea @xcite of formulating gauge theories on the spacetime lattice is the most successful nonperturbative regularization for pure gauge fields , putting massless dirac fermions @xcite on the lattice has been a notorious problem for more than twenty years . the resolution of the lattice fermion problem first appeared in the context of the domain - wall fermion @xcite , and it motivated the overlap formalism @xcite which led to the construction of overlap - dirac operator @xcite in 1997 . we refer to ref . @xcite for a recent review of the domain - wall fermions , and to ref . @xcite for a recent review of the overlap . however , if we consider a weyl fermion interacting with a background gauge field , then a completely satisfactory nonperturbative regularization for chiral guage theories ( e.g. , the standard model ) has not yet been presented up to this moment .    in the following , we will concentrate our discussions on the general principles to construct chiral dirac fermion operators on a finite lattice , in particular , for vector - like gauge theories such as qcd . with the constraints imposed by the nielson - ninomiya no - go theorem@xcite , one can construct a gauge covariant dirac operator @xmath1 on a finite lattice such that :    ( i ) : :    @xmath1 breaks the chiral symmetry ( i.e. ,    @xmath2 ) at finite lattice spacing but    recovers the chiral symmetry in the continuum limit    @xmath3 . ( ii ) : :    @xmath1 is local . (    @xmath4 with    @xmath5 ; or @xmath6 for    @xmath7 , where @xmath8 is much less than    the size of the lattice . ) ( iii ) : :    in the free fermion limit , @xmath1 is free of species    doublings . ( the free fermion propagator @xmath9 has    only one simple pole at the origin @xmath10 in the    brillouin zone . ) ( iv ) : :    in the free fermion limit , @xmath1 has correct continuum    behavior . ( in the limit @xmath3 ,    @xmath11 around    @xmath10 . ) however , one can not push the property * ( i ) * any further , while maintaining properties * ( ii)-(iv)*. for example , if @xmath1 is chirally symmetric at finite lattice spacing , then it must violate at least one of the three properties * ( ii)-(iv)*. we note that these four properties * ( i)-(iv ) * form the necessary conditions to meet the requirements * ( a)-(c ) * , however , they are not sufficient to guarantee that * ( c ) * will be satisfied . an example satisfying * ( i)-(iv ) * is the standard wilson - dirac fermion operator@xcite [ eq : dw ] d_w = _ t_+ w where [ eq : tmu ] t_(x , y ) = [ u_(x ) _ x+,y - u_^(y ) _ x-,y ]  , [ eq : sigma ] _ _ ^ + _ _ ^ = 2 _  , [ eq : gamma ] _ & = & (    cc 0 & _ + _ ^ & 0    ) and @xmath12 is the wilson term [ eq : wilson ] w(x , y ) = _  . the color , flavor and spinor indices have been suppressed in ( [ eq : dw ] ) . the first term on the r.h.s . of ( [ eq : dw ] ) is the naive fermion operator which is chirally symmetric at any lattice spacing and satisfies properties * ( ii ) * and * ( iv ) * but violates * ( iii ) * since it has @xmath13 fermion doubled modes . the purpose of the wilson term @xmath12 is to give each doubled mode a mass of @xmath14 such that in the continuum limit ( @xmath3 ) , each doubled mode becomes infinitely heavy and decouples from the fermion propagator . however , the introduction of the wilson term has serious drawbacks . it causes @xmath15 artifacts and also leads to the notorious problems such as vector current renormalization , additive fermion mass renormalization , and mixings between operators in different chiral representations .    during the last two years , it has become clear that the proper way to break the chiral symmetry of @xmath1 at finite lattice spacing is to conform with the ginsparg - wilson relation @xcite [ eq : gwr ] d 5 + 5 d = 2 a d r _ 5 d where @xmath16 is a positive definite hermitian operator which is local in the position space and trivial in the dirac space . then the generalized chiral symmetry ( [ eq : gwr ] ) can ensure that the theory is free of above mentioned problems of the wilson - dirac operator @xcite .    the general solution to the ginsparg - wilson relation can be written as @xcite [ eq : gen_sol ] d = d_c ( + a r d_c ) ^-1 = ( + a d_c r ) ^-1 d_c where @xmath17 is any chirally symmetric ( @xmath18 ) dirac operator which must violate at least one of the three properties * ( ii)*-*(iv ) * above . now we must require @xmath17 to satisfy * ( iii ) * and * ( iv ) * , but violate * ( ii ) * ( i.e , @xmath17 is nonlocal ) , since ( [ eq : gen_sol ] ) can transform the nonlocal @xmath17 into a local @xmath1 on a finite lattice for @xmath19 with @xmath20 in the proper range @xcite , while the properties * ( iii ) * and * ( iv ) * are preserved . moreover , the zero modes and the index of @xmath17 are invariant under the transformation @xcite . that is , a zero mode of @xmath17 is also a zero mode of @xmath1 and vice versa , hence , [ eq : npm ] n_+ ( d_c ) = n_+ ( d ) , n_- ( d_c ) = n_- ( d ) , [ eq : index ] ( d_c ) = n_-(d_c ) - n_+(d_c ) = n_-(d ) - n_+(d ) = ( d )  . since the massless dirac fermion operator in continuum is antihermitan , we also require that @xmath17 is antihermitian ( @xmath21 ) even at finite lattice spacing . then the chiral symmetry of @xmath17 together with its antihermiticity implies that @xmath17 satisfies the @xmath22-hermiticity [ eq : hermit ] d_c^ = 5 d_c 5  . this implies that @xmath1 in the general solution ( [ eq : gen_sol ] ) also satisfies the @xmath23-hermiticity [ eq : hermit_d ] d^ = 5 d 5  . then the eigenvalues of @xmath1 are either real or come in complex conjugate pairs . furthermore , from ( [ eq : gen_sol ] ) , since @xmath16 is a positive definite hermitian operator , the lower bound of real eigenvalues of @xmath1 is zero , thus @xmath24 is real and nonnegative , and is amenable to hybrid monte carlo simulation for dynamical fermions with _ any _ number of fermion flavors .    in particular , for @xmath19 with @xmath25 , the analysis [ eqs . ( 24)-(41 ) ] in ref . @xcite goes through with trivial modifications . the main results are :    ( @xmath26 ) : :    the eigenvalues of @xmath1 fall on a circle with center at    @xmath27 , and radius @xmath27 , and have the    reflection symmetry with respect to the real axis . ( @xmath28 ) : :    the real eigenmodes ( if any ) at @xmath29 and    @xmath30 have definite chirality @xmath31 or    @xmath32 . ( @xmath33 ) : :    the chirality of any complex eigenmodes is zero . ( @xmath34 ) : :    total chirality of all eigenmodes must vanish . @xmath35    where @xmath36 denotes the number of zero modes of positive ( negative ) chirality , and @xmath37 the number of @xmath30 modes of positive ( negative ) chirality . from ( @xmath34 ) , we immediately see that _ any zero mode must be accompanied by a real @xmath30 mode with opposite chirality _ , and the index of @xmath1 is [ eq : index_d ] ( d ) n_- - n_+ = - ( n_- - n_+ )  . now the central problem is to construct the chirally symmetric @xmath17 which is nonlocal , and satisfies * ( iii ) * , * ( iv ) * , and ( [ eq : hermit ] ) . furthermore we also require that @xmath17 is topologically proper ( i.e. , satisfying the atiyah - singer index theorem ) for any prescribed smooth gauge background . these constitute the necessary requirements @xcite for @xmath17 to enter ( [ eq : gen_sol ] ) such that @xmath1 could provide a nonperturbative regularization for a massless dirac fermion interacting with a background gauge field . explicitly , these necessary requirements are :    ( a ) : :    @xmath17 is antihermitian ( hence    @xmath22-hermitian ) and it agrees with    @xmath38 in the    classical continuum limit . ( b ) : :    @xmath17 is free of species doubling . ( c ) : :    @xmath17 is nonlocal . ( d ) : :    @xmath17 is well defined in topologically trivial    background gauge field . ( e ) : :    @xmath17 has zero modes as well as simple poles in    topologically non - trivial background gauge fields ( each zero mode of    @xmath17 must be accompanied by a simple pole of    @xmath17 ) . furthermore , the zero modes of    @xmath17 satisfy the atiyah - singer index theorem for any    prescribed smooth gauge background . the general solution of @xmath17 satisfying these requirements had been investigated in ref . @xcite and was reported by the author at chiral99 . however , in general , given any lattice dirac operator @xmath1 , there exists a transformation @xmath39 for @xmath1 such that the transformed dirac operator @xmath40 $ ] is chirally symmetric @xcite . therefore , for the sake of completeness , the transformation @xmath39 is outlined in section 2 . then the construction in ref . @xcite is reviewed in section 3 . concluding remarks are briefly outlined in section 4 . given any lattice dirac operator @xmath1 , in general , there are many different ways to construct a chirally symmetric @xmath17 out of @xmath1 . for example , we can construct [ eq : ds ] d_s = ( d - _ 5 d _ 5 ) which is chirally symmetric ( @xmath41 ) . however , this transformation does not necessarily preserve the property * ( iii)*. for example , if we apply this transformation to the wilson - dirac operator ( [ eq : dw ] ) , we obtain @xmath42 , the naive fermion operator which suffers from the species doublings . although @xmath43 is free of species doublings in the continuum limit , the transformation ( [ eq : ds ] ) can not preserve this property since @xmath44 is local . therefore , we need a transformaton which preserves the properties * ( iii ) * , * ( iv ) * , ( [ eq : hermit_d ] ) and ( [ eq : npm ] ) , but exchanges the locality of @xmath1 for its chiral symmetry at finite lattice spacing . the transformation @xcite [ eq : tit ] ( r ) : d d= ( r)[d ] d ( + r d ) ^-1 = ( + d r ) ^-1 d which generalizes ( [ eq : gen_sol ] ) can serve our purposes . evidently , the set of these transformations , @xmath45 , form an abelian group with group parameter space @xmath46 @xcite .    for any @xmath1 satisfying @xmath22-hermiticity , ( [ eq : hermit_d ] ) , there exists a hermitian @xmath47 , [ eq : rc ] r_c = - ( d^-1 + _ 5 d^-1 _ 5 ) such that [ eq : d5d ] d_c = ( r_c ) [ d ] = 2 5 d ( 5 d - d 5 ) ^-1 d is chirally symmetric ( @xmath48 ) and antihermitian ( @xmath21 ) . it is evident that for any two lattice dirac operators @xmath49 and @xmath50 satisfying ( [ eq : hermit_d ] ) , their corresponding chiral limits obtained from ( [ eq : d5d ] ) , say , @xmath51 and @xmath52 , are in general different . however , they are related by the transformation [ eq : dct ] d_c^(1 ) = _ i t_i d_c^(2 ) t_i^ where each @xmath53 commutes with @xmath54 . in general , given any two @xmath51 and @xmath52 , it is a nontrivial task to obtain all @xmath53 in ( [ eq : dct ] ) .    for the wilson - dirac operator ( [ eq : dw ] ) , the transformation ( [ eq : d5d ] ) gives [ eq : dtwc ] d_c = _ t_- w  ( _ t_)^-1  w . it is antihermitian , nonlocal ( due to the second term ) , and satisfies * ( i ) , ( iii ) * and * ( iv)*. then we can substitute ( [ eq : dtwc ] ) into ( [ eq : gen_sol ] ) with @xmath19 to obtain a gw dirac operator [ eq : dw_gw ] d = where [ eq : c ] c = ( ^_t _ ) - w ( _ t_)^-1 w  .    on a finite lattice , the gw dirac operator ( [ eq : dw_gw ] ) can be constructed to be local but not highly peaked in the diagonal elements if the value of @xmath20 is within a proper range @xcite . in other words , if @xmath20 is too small , then @xmath1 must be nonlocal since @xmath1 is close to @xmath17 . on the other hand , if @xmath20 is too large , then @xmath1 becomes highly peaked in the diagonal elements . in these two extreme cases , @xmath1 does not respond properly to the background gauge field ( e.g. , the anomaly function of @xmath1 does not agree with the chern - pontryagin density of the prescribed gauge background ) . in the limit the number of sites in each dimension goes to infinity , the proper range of @xmath20 extends to @xmath55 .    the anomaly function of @xmath1 ( [ eq : dw_gw ] ) can be written [ eq : anxc ] = 2  ( x , x ) where @xmath56 denotes the trace over the color , flavor and spinor space . it is instructive to compare the anomaly function of the wilson - dirac operator @xmath43 ( [ eq : dw ] ) to that of the gw wilson - dirac operator @xmath1 ( [ eq : dw_gw ] ) , in a topologically trivial background gauge field , on a finite lattice , as shown in fig . 1 in ref . @xcite and fig . 1 in ref @xcite , respectively . the anomaly function of @xmath43 is very different from the chern - pontryagin density @xmath57 on a @xmath58 torus , while that of the gw dirac - wilson operator @xmath1 is in good agreement with @xmath57 at each site . this demonstrates that the transformation @xmath59 plays the important role in converting @xmath43 into a gw dirac operator @xmath60 $ ] which is free of @xmath15 lattice artifacts , thus @xmath1 can reproduce the correct anomaly function even on a _ finite _ lattice provided that the local fluctuations of the background gauge field are not too violent . however , a lattice dirac operator satisfying properties * ( i)-(iv ) * , ( [ eq : hermit_d ] ) and the gw relation ( [ eq : gwr ] ) does not guarantee that it has the correct anomaly function in topologically nontrivial gauge backgrounds . this can be seen as follows . it is well known that the sum of the axial anomaly over all sites of a finite lattice is equal to the index of the lattice dirac operator . therefore _ the necessary condition for a lattice dirac operator to have the correct anomaly function is that it possesses exact zero modes satisfying the atiyah - singer index theorem _ i.e. , @xmath61 . however , not every lattice dirac operator has exact zero modes in topologically nontrivial gauge backgrounds . for example , the wilson - dirac operator , @xmath43 in ( [ eq : dw ] ) , it does not have exact zero modes in topologically nontrivial sectors . then , according to ( [ eq : npm ] ) , @xmath60 $ ] also does not have any exact zero modes , even though @xmath1 satisfies the gw relation ( [ eq : gwr ] ) , and properties * ( i)-(iv ) * and ( [ eq : hermit_d ] ) . this implies that the anomaly function of @xmath43 or @xmath60 $ ] does not agree with the chern - pontryagin density for topologically nontrivial gauge fields on _ any _ finite lattices . consequently , _ the disagreement must persist in the continuum limit @xmath3_. this provides an example to illustrate that any lattice dirac operator @xmath1 must possess a nonperturbative attribute , the topological characteristics , @xmath62 $ ] @xcite . in general , @xmath62 $ ] is a rational number , a functional of @xmath1 and the gauge configuration . in a gauge background with nonzero topological charge @xmath63 , the index of a lattice dirac operator is [ eq : cd ] n_- - n_+ = c[d ]  q  . in the case of @xmath43 and @xmath60 $ ] ( [ eq : dw_gw ] ) , @xmath64 , thus the topological characteristics @xmath65 = c[d ] = 0 $ ] . for the vacuum sector with nonzero topological charge density ( i.e. , @xmath66 but @xmath67 ) , @xmath62 $ ] can not be defined by ( [ eq : cd ] ) , however , it can be defined unambiguously @xcite in the context of the anomaly function . we note in passing that for the overlap - dirac operator @xmath68 @xcite , the transformation ( [ eq : d5d ] ) gives @xmath69 , as expected . substituting @xmath17 into ( [ eq : gen_sol ] ) with @xmath19 , we obtain the generalized overlap - dirac operator [ eq : do_gw ] d = 2 ( + v ) [ - v + 2 r ( + v ) ] ^-1  . suppose that we do not have any lattice dirac operator to begin with . it is still possible for us to construct @xmath17 @xcite according to the necessary requirements * ( a)-(e ) * listed in section 1 . since @xmath17 is antihermitian , there exists one to one correspondence between @xmath17 and a unitary operator @xmath70 , d_c = ( + v ) ( - v ) ^-1 , v = ( d_c - ) ( d_c + ) ^-1 . [ eq : vdc ] where @xmath70 also satisfies the @xmath22-hermiticity , @xmath71 . then the unitary operator @xmath70 can be expressed in terms of a hermitian operator @xmath72 , [ eq : v5h ] v = _ 5 h = (    cc & 0 + 0 & -    ) (    cc h_1 & h_2 + h_2^ & h_3    ) = (    cc h_1 & h_2 + -h_2^ & -h_3    ) where @xmath73 and @xmath74 . using the unitarity condition @xmath75 , we have @xmath76 , [ eq : h^2 ] h^2 = (    cc h_1 ^ 2 + h_2 h_2^ & h_1 h_2 + h_2 h_3 + h_2^ h_1 + h_3 h_2^ & h_2^ h_2 + h_3 ^ 2    ) = (    cc & 0 + 0 &    ) then we obtain d_c [ eq : dc ] = where @xmath77 . the general solution to eq . ( [ eq : h^2 ] ) can be written as [ eq : h1 ] h_1 & = & + [ eq : h2 ] h_2 & = &  b  e^ i where @xmath78 is an arbitrary phase , and @xmath79 is any operator . in the following we will restrict @xmath80 to zero , and also pick the minus sign for @xmath81 . then the general solution for @xmath17 can be written in the following form [ eq : dl ] d_l & = & b^-1 + [ eq : dr ] d_r & = & ^-1 b due to the presence of the square root in ( [ eq : dl ] ) and ( [ eq : dr ] ) , @xmath17 is nonlocal for nontrivial @xmath82 , thus the requirement * ( b ) * is satisfied . in the following , we outline a construction of @xmath82 which satisfies constraints * ( a)-(d ) * but not * ( e)*.    in order to have @xmath17 satisfy the constraint * ( a ) * , we first try [ eq : b ] b = w^-1 _ _ t_where @xmath83 and @xmath84 are defined in ( [ eq : tmu ] ) and ( [ eq : sigma ] ) respectively , and @xmath12 is a non - singular hermitian operator which is trivial in the dirac space and goes to a constant in the classical continuum limit operator in this section is different from the wilson term in eq . ( [ eq : wilson ] ) . ] . ( [ eq : dl ] ) and ( [ eq : dr ] ) become [ eq : dlw ] d_l & = & ( t ) ^-1 + [ eq : drw ] d_r & = & - d_l^ where [ eq : sigma_t ] t & = & _ _ t _ + [ eq : t2 ] t^2 & = & - ( t ) ( ^ t ) however , @xmath83 suffers from species doublings . now our task is to construct a hermitian operator @xmath12 such that the doubled modes are completely decoupled from the fermion propagator @xmath85 , even at finite lattice spacing .    as discussed in ref . @xcite , we want to construct a hermitian @xmath12 such that in the free fermion limit , it satisfies the following condition , [ eq : wp ] w(p ) = \ {    ll > 0 & + < 0 & +    . a simplest solution to ( [ eq : wp ] ) is [ eq : wpc ] w(p ) = c - 2 _ ^2 ( p_a / 2 ) , c ( 0 , 2 ) note that the role of @xmath12 in the general solution of @xmath17 is quite different from the wilson term ( [ eq : wilson ] ) in the wilson - dirac operator ( [ eq : dw ] ) . in the general solution of @xmath17 , the chiral symmetry is always preserved , and _ the role of @xmath12 is to suppress the doubled modes completely at finite lattice spacing _ , while in the wilson - dirac operator , the wilson term breaks the chiral symmetry explicitly and gives a mass of order @xmath86 to the doubled modes such that they can be decoupled in the continuum limit ( @xmath3 ) . after the gauge links are restored , @xmath12 in the position space becomes [ eq : wxy ] w(x , y ) = c  _ x , y - _ this is one of the simplest solution of @xmath12 satisfying the requirement ( [ eq : wp ] ) in the free fermion limit . certainly , there exists other solutions to ( [ eq : wp ] ) .    substituting the @xmath17 [ eqs . ( [ eq : dlw])-([eq : drw ] ) with @xmath83 in ( [ eq : tmu ] ) , and @xmath12 in ( [ eq : wxy ] ) with @xmath87 , and the normalization constant @xmath88 ] into ( [ eq : gen_sol ] ) with @xmath19 , we obtain the gw dirac operator [ eq : twc ] d = where [ eq : bwt ] b & = & ( w - ) ^-1  ( t ) the anomaly function of @xmath1 is [ eq : anx ] = 2 r^2  ( x , x ) where @xmath56 stands for the trace over the color , flavor and spinor space .    since the @xmath17 of ( [ eq : twc ] ) in the free fermion limit is free of species doubling and has the correct continuum behavior , the perturbation calculation in ref . @xcite showed that @xmath89 has the correct chiral anomaly in topologically trivial gauge backgrounds . this has been verified explicitly on a finite lattice . an example is shown in fig . 2 of ref . @xcite . however , for topologically nontrivial sectors , the anomaly function of @xmath1 also depends on the topological characteristics , @xmath62 $ ] , which is a non - perturbative attribute of @xmath1 . since @xmath1 does not have any exact zero modes in nontrivial sectors , the anomaly function of @xmath1 does not agree with the chern - pontryagin density for nontrivial gauge backgrounds . now we have two examples of gw dirac operators , ( [ eq : dw_gw ] ) and ( [ eq : twc ] ) , which have the correct axial anomaly in the vacuum sector but not in the nontrivial sectors . in general , if @xmath17 is well defined in nontrivial sectors ( i.e. , not satisfying * ( e ) * ) , then @xmath90 = \mbox{index}[d ] = 0 $ ] @xcite ( i.e. , @xmath62 = c[d_c ] = 0 $ ] , @xmath1 and @xmath17 are _ topologically trivial _ ) . on the other hand , if @xmath17 has zeros and poles , then @xmath62 \ne 0 $ ] , however , it does not necessarily imply that @xmath62 = 1 $ ] . sometimes @xmath62 $ ] may even become a fraction ( see table 4 in ref . the important point is that in general we can not predict the value of @xmath62 $ ] in the topologically nontrivial sectors solely based on the properties of @xmath1 in the free fermion limit or in the vacuum sector . the necessary conditions for a lattice dirac operator @xmath1 to provide a nonperturbative regularization for massless fermion interacting with a background gauge field are * ( a)-(c ) * listed in section 1 . the conditions * ( a ) * and * ( b ) * can be satisfied if @xmath1 fulfils * ( i)-(iv ) * and ( [ eq : hermit_d ] ) , and the chiral symmetry is broken according to the ginsparg - wilson relation ( [ eq : gwr ] ) . since any gw dirac operator can be represented by the transformation ( [ eq : gen_sol ] ) [ preserving * ( iii ) , ( iv ) * , ( [ eq : hermit_d ] ) and ( [ eq : npm ] ) ] on a chirally symmetric @xmath17 , then the conditions for @xmath1 become the necessary requirements * ( a)-(e ) * for @xmath17 . it should be emphasized that the requirement * ( c ) * for @xmath1 ( or the requirement * ( e ) * for @xmath17 ) can not be expressed in terms of any conditions in the free fermion limit or in the vacuum sector . this naturally leads to the concept of topological characteristics @xcite associated with each lattice dirac operator @xmath1 . although we have no problems to construct @xmath1 to satisfy * ( i)-(iv ) * , ( [ eq : hermit_d ] ) and ( [ eq : gwr ] ) , it remains an interesting question whether one can construct a topologically proper @xmath1 on a four dimensional lattice without square root operations . if one has a lattice dirac operator @xmath1 satisfying * ( i)-(iv ) * , ( [ eq : hermit_d ] ) and ( c ) , then one can transform @xmath1 into a gw dirac operator through the transformation @xmath91 $ ] which not only preserves all essential physics of @xmath1 but also can restrict the quantum corrections to behave properly such that @xmath1 is free of @xmath15 lattice artifacts , additive mass renormalization , etc . recently it has been shown that the effective four - dimensional action ( @xmath92 ) of the domain wall fermion is local and satisfying the gw relation , for gauge fields with small field strength @xcite . however , it is possible to transform the domain - wall fermion operator into a local gw dirac operator even at finite @xmath93 @xcite , then the anomalous effects ( e.g. , the residual pion mass in domain - wall qcd ) due to chiral symmetry violations at finite @xmath93 could be suppressed even at moderate @xmath93 . further investigations are required before one can tell whether this scenario can be realized or not . i would like to thank all participants of chiral99 , for their inspiring talks , interesting questions , and enlightening discussions . this work was supported by the national science council , r.o.c . under the grant number nsc89 - 2112-m002 - 017 .
we discuss how to formulate dirac fermion operator on a finite lattice such that it can provide a nonperturbative regularization for massless fermion interacting with a background gauge field .    11.15.ha , 11.30.rd , 11.30.fs