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the star - product operation introduced by groenewold for phase - space functions @xcite permits formulation of quantum mechanics in phase space . it uses the weyl s association rule @xcite to establish one - to - one correspondence between phase - space functions and operators in the hilbert space . the wigner function @xcite appears as the weyl s symbol of the density matrix . the skew - symmetric part of the star - product , known as the moyal bracket @xcite , governs the evolution of symbols of heisenberg operators . refined formulation of the weyl s association rule is proposed by stratonovich @xcite . the weyl s association rule , star - product technique , star - functions , and some applications are reviewed in refs . @xcite . a one - parameter group of unitary transformations in the hilbert space @xmath0 with @xmath1 being hamiltonian , corresponds to a one - parameter group of canonical transformations in the classical theory @xcite , although canonical transformations provide a broader framework @xcite . weyl s symbols of time dependent heisenberg operators of canonical coordinates and momenta induce quantum phase flow . osborn and molzahn @xcite construct quantum hamilton s equations which determine quantum phase flow and analyze the semiclassical expansion for unconstrained quantum - mechanical systems . an earlier attempt to approach these problems is undertaken in ref . @xcite . the infinitesimal transformations induced by the evolution operator ( [ u7 ] ) in phase space coincide with the infinitesimal canonical transformations induced by the corresponding hamiltonian function @xcite . the quantum and classical finite transformations are , however , distinct in general , since the star- and dot - products as multiplication operations of group elements in quantum and classical theories do not coincide . the quantum phase flow curves are distinct from the classical phase - space trajectories . this fact is not well understood ( see e.g. refs . @xcite ) .    osborn and molzahn @xcite made important observation that quantum trajectories in unconstrained systems can be viewed as a `` basis '' to represent the evolution of quantum observables . such a property is usually assigned to characteristics appearing in a standard technique for solving first - order partial differential equations ( pde ) . the well known example is the classical liouville equation @xmath2 this equation is solved in terms of characteristic lines which are solutions of classical hamilton s equations @xmath3 with initial conditions @xmath4 . equations ( [ classham3 ] ) are characteristic equations . they represent a system of first - order ordinary differential equations ( ode ) for canonical variables . physical observables @xmath5 evolve according to @xmath6    it is remarkable that despite quantum liouville equation is an infinite - order pde its solutions are expressed in terms of solutions of the quantum hamilton s equations which are infinite - order pde also .    a technical advantage in using the method of characteristics in quantum mechanics stems from the fact that to any fixed order of the semiclassical expansion the quantum hamilton s equations can be viewed as a coupled system of first - order ode for quantum trajectories and generalized jacobi fields obeying certain initial conditions . the evolution can be considered , respectively , as going along a trajectory in an extended phase space endowed with auxiliary degrees of freedom ascribed to generalized jacobi fields . the evolution problem can be solved e.g. numerically applying efficient ode integrators . quantum characteristics can be useful , in particular , for solving numerically many - body potential scattering problems by semiclassical expansion of star - functions around their classical values with subsequent integration over the initial - state wigner function . among possible applications are transport models in quantum chemistry and heavy - ion collisions @xcite where particle trajectories remain striking but an intuitive feature . a covariant extensions of quantum molecular dynamics ( qmd ) transport models @xcite is based on the poincar invariant constrained hamiltonian dynamics @xcite . we show , in particular , that quantum trajectories exist and make physical sense in the constraint quantum systems also and play an important role similar to that in the quantum unconstrained systems . the paper is organized as follows : in sects . ii and iii , characteristics of unconstraint classical and quantum systems are discussed . sects . iv and v are devoted to properties of characteristics of constraint classical and quantum systems . quantum phase flows are analyzed using the star - product technique which we believe to be the most adequate tool for studying the subject . we give definitions and recall basic features of the method of characteristics in sect . ii .    in sect . iii , fundamental properties of quantum characteristics are derived . the weyl s association rule , the star - product technique , and the star - functions are reviewed based on the method proposed by stratonovich @xcite . we show , firstly , that quantum phase flow preserves the moyal bracket and does not preserve the poisson bracket in general . secondly , we show that the star - product is invariant with respect to transformations of the coordinate system , which preserve the moyal bracket . thirdly , non - local laws of composition for quantum trajectories and the energy conservation along quantum trajectories are found in sect . iii - d . applying the invariance of the star - product with respect to change of the coordinate system ( [ brinva ] ) and the energy conservation , we derive new equivalent representations of the quantum hamilton s equations eq.([qf2 ] ) - ( [ qf4 ] ) . in sect . iii - e , we derive using the star - product technique the semiclassical reduction of the quantum hamilton s equations to a system of first - order ode involving along with quantum trajectories their partial derivatives with respect to initial canonical variables . finally , we express the phase - space green function @xcite in terms of quantum characteristics and reformulate relation between quantum and classical time - dependent observables @xcite using the method of characteristics .    the possibility of finding quantum trajectories and generalized jacobi fields by solving a system of ode gives practical advantages because of the existence of efficient numerical ode integrators . it would be tempting to extend method of characteristics to constraint systems such as gauge theories , relativistic qmd transport models , etc . the skew - gradient projection method is found to be useful to formulate classical and quantum constraint dynamics @xcite . in sect . iv , we show that in classical constraint systems characteristic lines exist and the method of characteristics is efficient . the proof we provide does not presuppose that constraint equations can be solved . the phase flow is commutative with the phase flows generated by constraint functions . characteristic lines , if belong to the constraint submanifold at @xmath7 , belong to the constraint submanifold at @xmath8 also . v gives description of quantum characteristics in constraint systems . although the formalism is complete , we encounter unexpected difficulty to formulate simple geometric idea that quantum trajectory belongs to a constraint submanifold . using tools of the analytic geometry , any idea like that requires the use of composition of functions . in quantum mechanics , one has to use the star - composition . this calls for a modification of usual geometric relations `` belong '' , `` intersect '' , and others . in a specific quantum - mechanical sense , the hamiltonian and constraint functions can be said to remain constant along quantum trajectories , while in the usual geometric sense they obviously do nt . the problem of visualization of relations among quantum objects in phase space is discussed in sects . iii - d and v - b . conclusion summarizes results . the phase space of system with @xmath9 degrees of freedom is parameterized by @xmath10 canonical coordinates and momenta @xmath11 which satisfy the poisson bracket relations @xmath12 with @xmath13 where @xmath14 is the identity @xmath15 matrix . the phase space appears as the cotangent bundle @xmath16 of @xmath9-dimensional configuration space @xmath17 . the matrix @xmath14 imparts to @xmath18 a skew - symmetric bilinear form . the phase space acquires thereby structure of symplectic space .    in what follows , physical observables are time dependent , whereas density distributions remain constant . such a picture constitutes the classical analogue of the quantum - mechanical heisenberg picture .    in the classical unconstrained systems , phase flow : @xmath19 , is canonical and preserves the poisson bracket . the classical hamilton s equations ( [ classham3 ] ) are first - order ode . the energy is conserved along classical trajectories @xmath20 the classical hamilton s equations ( [ classham3 ] ) can be rewritten as first - order pde : @xmath21    the phase - space trajectories can be used to solve the liouville equation ( [ class ] ) which is the first - order pde . any observable @xmath5 is expressed in terms of @xmath22 , as indicated in eq.([charac ] ) . classical trajectories obey the dot - composition law : @xmath23 the stratonovich version of the weyl s quantization and dequantization @xcite is discussed in the next subsection and in more details in refs . @xcite . the phase - space variables @xmath24 correspond to operators @xmath25 acting in the hilbert space , which obey commutation rules @xmath26 = -i\hbar i^{kl}. \label{987897}\ ] ]    operators @xmath27 acting in the hilbert space admit multiplications by @xmath28-numbers and summations . the set of all operators constitutes a vector space . the basis of such a space can be labelled by @xmath24 . the weyl s basis looks like @xmath29 the association rule for a function @xmath30 and an operator @xmath27 has the form @xcite @xmath31 , \;\;\ ; \mathfrak{f } = \int \frac{d^{2n}\xi } { ( 2\pi \hbar ) ^{n}}f(\xi ) \mathfrak{b}(\xi ) . \label{inv}\ ] ] the value of @xmath32 can be treated as the @xmath33-coordinate of @xmath27 in the basis @xmath34 , while @xmath35 $ ] as the scalar product of @xmath34 and @xmath27 .    using eqs.([inv ] ) one gets an equivalent association rule @xmath36 the half - fourier transform , @xmath37 provides the inverse relation . the weyl - symmetrized functions of operators of canonical variables have representation @xcite @xmath38 where the subscripts indicate the order in which the operators act on the right .    given two functions @xmath39 $ ] and @xmath40 $ ] , one can construct a third function , @xmath41,\ ] ] called star - product . in terms of the poisson operator @xmath42 one has @xmath43 the star - product splits into symmetric and skew - symmetric parts , @xmath44 the skew - symmetric part is known under the name of moyal bracket . the wigner function is the weyl s symbol of the density matrix . in the heisenberg picture , the wigner function remains constant @xmath45 , whereas functions representing physical observables evolve with time in agreement with equation @xmath46 this equation is the weyl s transform of equation of motion for operators in the heisenberg picture . it is the infinite - order partial differential equation ( pde ) . the series expansions of @xmath47 over @xmath48 is given by @xmath49 where @xmath50 is the initial data function . using the @xmath51-adjoint notations of ref . @xcite , equation ( [ evol ] ) can be represented in the form @xmath52(\xi ) . \label{haki } \end{aligned}\ ] ] its formal solution , @xmath53 is equivalent to eq.([expa ] ) . if the target symbol @xmath54 is semiclassically admissible , the evolution operator has asymptotic expansion @xcite @xmath55    the power series expansion in @xmath56 is valid for semiclassically admissible symbols @xmath57 and @xmath58 . if , however , @xmath58 is a rapidly oscillating symbol , then ( [ haki3 ] ) fails and the solution of the evolution equation becomes of the wkb type whose exponential phase is a symplectic area ( see for details ref . @xcite ) . active transformations modify operators @xmath27 and commute with @xmath59 . passive transformations change the basis and keep operators fixed . these views are equivalent . we choose the former . consider transformations depicted by the diagram @xmath60 where @xmath61 is given by eq.([u7 ] ) . the operators of canonical variables are transformed as @xmath62 the coordinates @xmath63 of new operators @xmath64 in the old basis @xmath59 are given by @xmath65 . \label{uxit}\ ] ] since @xmath61 is the evolution operator , functions @xmath66 can be treated as the weyl s symbols of operators of canonical coordinates and momenta in the heisenberg picture . for @xmath7 , we have @xmath67    the set of operators of canonical variables is complete in the sense that any operator acting in the hilbert space can be represented as a function of operators @xmath68 . one can indicate it as follows : @xmath69 . the taylor expansion of @xmath70 permits the equivalent formulation of the weyl s association rule . transformations @xmath71 generate transformations of the associated phase - space functions : @xmath72 \nonumber \\ & = & \sum_{s=0}^{\infty } \frac{1}{s!}\frac{\partial ^{s}f(0)}{\partial \xi^{i_{1}} ... \partial \xi ^{i_{s } } } tr[\mathfrak{b}(\xi ) \mathfrak{u}^{+}\mathfrak{x}^{i_{1}} ... \mathfrak{x}^{i_{s}}\mathfrak{u } ]   \nonumber \\ & = & \sum_{s=0}^{\infty } \frac{1}{s!}\frac{\partial ^{s}f(0)}{\partial \xi ^{i_{1}} ... \partial \xi ^{i_{s}}}tr[\mathfrak{b}(\xi ) \acute{\mathfrak{x}}^{i_{1 } } ... \acute{\mathfrak{x}}^{i_{s } } ]   \nonumber \\ & = & \sum_{s=0}^{\infty } \frac{1}{s!}\frac{\partial ^{s}f(0)}{\partial \xi ^{i_{1}} ... \partial \xi ^{i_{s}}}u^{i_{1}}(\xi , \tau)\star ... \star u^{i_{s}}(\xi,\tau )   \nonumber \\ & = & \sum_{s=0}^{\infty } \frac{1}{s!}\frac{\partial ^{s}f(0)}{\partial \xi ^{i_{1}} ... \partial \xi ^{i_{s}}}u^{i_{1}}(\xi , \tau)\circ ... \circ u^{i_{s}}(\xi,\tau )   \nonumber \\ & \equiv & f(\star u(\xi,\tau ) ) . \label{tf}\end{aligned}\ ] ] last two lines define the star - composition . the star - function @xmath73 is a functional of @xmath74 . the @xmath75-product is not associative in general . however , the indices @xmath76 for @xmath77 are symmetrized , so the order in which the @xmath75-product is calculated is not important .    the antisymmetrized products @xmath78}$ ] of even number of operators of canonical variables are @xmath28-numbers as a consequence of the commutation relations . these products are left invariant by unitary transformations : @xmath79}\mathfrak{u}=\mathfrak{x}^{[i_{1}} ... \mathfrak{x}^{i_{2s}]}$ ] . in phase space , we get @xmath80}(\xi,\tau ) = \xi ^{[i_{1}}\star ... \star \xi ^{i_{2s } ] } $ ] and , in particular , @xmath81    phase - space transformations induced by @xmath61 preserve the moyal bracket and do not preserve the poisson bracket , so the evolution map @xmath82 , is not canonical . using eq.([teu ] ) , one can check e.g. that for @xmath83 where @xmath84 is the kronecker symbol functions @xmath85 do not satisfy the poisson bracket condition for canonicity to order @xmath86 .    for real functions @xmath66 satisfying eqs.([area ] ) one may associate hermitian operators @xmath64 which obey commutation rules for operators of canonical coordinates and momenta . as a result , functions @xmath66 appear in the coincidence with a unitary transformation relating @xmath68 and @xmath64 . the conservation of the moyal bracket for a one - parameter set of continuous phase - space transformations is the necessary and sufficient condition for unitary character of the associated continuous transformations in the hilbert space .      applying eq.([tf ] ) to product @xmath87 of two operators , we obtain function @xmath88 associated to operator @xmath89 and function @xmath90 associated to operator @xmath91 . these operators coincide , so do their symbols : @xmath92 the star - product is calculated with respect to @xmath93 and @xmath33 in the left- and right - hand sides , respectively . equation ( [ brinva ] ) is valid separately for symmetric and skew - symmetric parts of the star - product of the functions . the substantial content of eq.([brinva ] ) is that one can compute the star - product in the initial coordinate system and change variables @xmath94 , or equivalently , change variables @xmath94 and compute the star - product , provided eq.([area ] ) is fulfilled . the functions @xmath95 define quantum phase flow which represents quantum deformation of classical phase flow .    ) . the solid line stands for a quantum trajectory @xmath96 at @xmath97 . the dashed line is assigned to a trajectory @xmath98 which we would have at @xmath99 for the classical dot - composition law . the distance between the solid and dashed trajectories is of order of @xmath100 . , width=2 ]      in the usual geometric sense , quantum characteristics @xmath101 can not be considered as trajectories along which physical particles move . the reason lies , in particular , in the star - composition law @xmath102 which is distinct from @xmath103 , see fig . [ fig10 ] . in classical mechanics , the composition law has the form of eq.([compcl ] ) . the energy conservation in the course of quantum evolution implies @xmath104 where @xmath105 $ ] is hamiltonian function . @xmath106 is , however , not conserved along quantum trajectories in the usual geometric sense , so @xmath107 . in classical mechanics , the conservation law has the form ( [ clco ] ) . to express the idea that a point particle moves continuously along a phase - space trajectory , one has to use the star - composition ( [ comp ] ) . the dot - composition is not defined in quantum mechanics . similarly , @xmath108 does not make any quantum - mechanical sense . one has to work with @xmath109 . if so , the only way to express quantitatively the fact of the energy conservation along a phase - space trajectory is to use eq.([ec ] ) . the similar problem arises in constraint systems when we want to decide if quantum trajectories belong to a constraint submanifold . the analytic geometry provides tools to formulate relations among geometric objects . those relations which are expressed through composition of functions are modified . we discuss if possible to assign a geometric sense to formulas involving the star - composition in sect . v - b . quantum hamilton s equations can be obtained applying the weyl s transform to evolution equations for heisenberg operators of canonical coordinates and momenta @xmath110    to reach the step 2 , the energy conservation ( [ ec ] ) is used . going from ( [ qf2 ] ) to ( [ qf3 ] ) , the change of variables ( [ brinva ] ) is performed . to achieve ( [ qf4 ] ) , we exploit @xmath111 . the time derivative of @xmath112 can be computed classically using the poisson bracket . the substitution @xmath113 leads , however , to deformation of classical trajectories . equations ( [ qf3 ] ) and ( [ qf4 ] ) are the quantum analogues of eq.([classham3 ] ) , eq.([qf ] ) is the quantum analogue of eq.([classham2 ] ) , and eq.([qf2 ] ) is the quantum analogue of eq.([classham ] ) .    as distinct from the de broglie - bohm trajectories ( see e.g. @xcite ) , @xmath112 are not related to specific states in the hilbert space . the functional form of quantum hamilton s equations ( [ qf ] ) is left invariant by the change of variables @xmath114 provided the map @xmath115 : @xmath116 , preserves the moyal bracket . equations ( [ qf ] ) are not invariant under canonical transformations . consider e.g. canonical map : @xmath117 , with generating function @xmath118 such that @xmath119 and @xmath120 . one can compare @xmath121 and @xmath122 in the coordinate systems @xmath123 and @xmath124 . for functions @xmath125 and @xmath126 , one gets , respectively , @xmath127 and @xmath128 . the symmetric and skew - symmetric parts of the star - product are both not invariant under canonical transformations . coordinate systems in phase space if related by a canonical transformation provide non - equivalent quantum dynamics . this ambiguity is better known as the operator ordering problem . the quantum deformation of classical phase flow can be found by expanding @xmath129 the right - hand side of eqs.([qf ] ) @xmath130 is a function of @xmath131 ( i.e. functional of @xmath74 ) , so we have to expand @xmath132\ ] ] using e.g. the cluster - graph method @xcite . classical trajectories @xmath133 satisfy classical hamilton s equations @xmath134 and initial conditions @xmath135 . given @xmath133 , the lowest - order quantum correction @xmath136 can be found by solving first - order ordinary differential equations ( ode ) @xmath137 with initial conditions @xmath138 . the functions @xmath139 and @xmath140 entering eq.([qhe ] ) is a particular case of generalized jacobi fields @xmath141 given @xmath142 and @xmath143 for @xmath144 , the next corrections @xmath145 can be found from first - order ode involving generalized jacobi fields ( [ gj ] ) with @xmath144 . for a harmonic oscillator , @xmath146 for @xmath147 , in which case quantum phase flow is both canonical and unitary . the generalized jacobi fields ( [ gj ] ) satisfy ode also . the lowest order equations have the form : @xmath148    the first of these equations describes the evolution of small perturbations along the classical trajectories . being projected onto a submanifold of constant energy it becomes the jacobi - levi - civita equation @xcite . in stochastic systems , @xmath139 grow exponentially with time .    at any fixed level of accuracy of the semiclassical expansion , we have a coupled system of ode for @xmath149 and @xmath143 subjected to initial conditions @xmath150 where @xmath151 and @xmath151 or @xmath152 , respectively . the evolution problem can be solved e.g. numerically applying efficient ode integrators . a numerical computation of the semiclassical expansion of the quantum phase flow in the elastic scattering of atomic systems is performed in ref . @xcite . an alternative approach allowing to reduce the semiclassical quantum dynamics to a closed system of ode is proposed by bagrov with co - workers @xcite . the phase - space trajectories appearing in @xcite are connected to specific quantum states like in the de broglie - bohm theory . properties of quantum paths , localization of quantum systems , and a coherent - type representation of the quantum flow are discussed in ref . @xcite . the series expansions of @xmath112 and @xmath153 over @xmath48 are given by @xmath154    in general , quantum phase flow is distinct from classical phase flow . this feature holds in integrable systems also , as discussed in appendix a.    the lowest order operators @xmath155 entering eq.([haki3 ] ) can be found to be @xcite @xmath156 here , the derivatives of @xmath157 are calculated with respect to @xmath158 : @xmath159 the jacobi fields with the upper indices are defined by @xmath160    according to eq.([fina2 ] ) , time dependence of the zero order term @xmath161 is determined by time dependence of the classical phase - space trajectory and form of the function @xmath30 . the similar conclusion holds for @xmath162 : eq.([fina3 ] ) tells that time dependence enters through the classical trajectory , the first quantum correction to the classical trajectory , and the classical jacobi fields . the problem of convergence of a formal power series expansion is always a difficult subject . the convergence rate of the time series depends obviously on the system . generally , the series expansion in @xmath163 has a finite convergence radius . however , in the itegrable systems one has a truncated series expansion ( cf . eq.(iii.27 ) and eq.(a.3 ) ) .      using orthogonality condition @xmath164 = ( 2\pi \hbar)^{n}\delta^{2n}(\xi - \zeta)\ ] ] and eq.([tf ] ) , we express green function for the weyl s symbols @xcite in terms of the quantum characteristics : @xmath165 \nonumber \\ & = & ( 2\pi \hbar)^{n}\delta^{2n}(\star u(\xi,\tau ) - \zeta ) \nonumber \\ & = & ( 2\pi \hbar)^{n}\delta^{2n}(\xi - \star u(\zeta,-\tau ) ) . \label{gf}\end{aligned}\ ] ]    a compact operator relation between the classical and quantum time - dependent observables is established in ref . solutions of the quantum and classical liouville equations , @xmath5 and @xmath166 , with initial conditions @xmath167 are related through the product @xmath168 where @xmath169 is the classical green function @xmath170 in terms of the characteristics , we obtain @xmath171 it is assumed that classical and quantum hamiltonian functions coincide i.e. @xmath172 . given the green function is known , the quantum trajectories can be found from equation @xmath173    for @xmath174 where @xmath175 is an infinitesimal parameter , the associated transformations of canonical variables and phase - space functions are given by @xmath176 and @xmath177 . the transformations of canonical variables are canonical to order @xmath178 only . the infinitesimal transformations of symbols of operators are not canonical . any function @xmath179 can be used to generate classical phase flow or quantum phase flow , according as the dot - product or the star - product stands for multiplication operation in the set of phase - space functions . the analogue between unitary and canonical transformations is illustrated by dirac @xcite in terms of the generating function @xmath180 defined by @xmath181 . the evolution map @xmath182 , is canonical for @xmath183 and @xmath184 . the parallelism of the transformations is manifest , but trajectories are complex . the generating function defined by the phase of @xmath185 yields real trajectories . it is not clear , however , if time - dependent symbols of operators are entirely determined by such trajectories .    the weyl s symbols of operators of canonical variables @xmath186 are the genuine characteristics in the sense that they allow by equation @xmath187 the entire determination of the evolution of observables . the quantum dynamics is totally contained in @xmath186 , whereas the deformation of symbols of the operators calculated at @xmath188 has a kinematical meaning . we give first description of second - class constraints systems and of the skew - gradient projection formalism . the details are found elsewhere @xcite . second - class constraints @xmath189 with @xmath190 and @xmath191 have the poisson bracket relations which form a non - degenerate @xmath192 matrix @xmath193 if this would not be the case , it could mean that gauge degrees of freedom appear in the system . after imposing gauge - fixing conditions , we could arrive at the inequality ( [ nongen ] ) . alternatively , breaking the condition ( [ nongen ] ) could mean that constraint functions are dependent . after removing redundant constraints , we arrive at the inequality ( [ nongen ] ) . constraint functions are equivalent if they describe the same constraint submanifold . within this class one can make transformations without changing dynamics .    for arbitrary point @xmath33 of the constraint submanifold @xmath194 , there is a neighbourhood where one may find equivalent constraint functions in terms of which the poisson bracket relations look like @xmath195 where @xmath196 here , @xmath197 is the identity @xmath198 matrix , @xmath199 . the matrix @xmath200 is used to lift indices @xmath201 up . the basis ( [ sb ] ) always exists locally , i.e. , in a finite neighbourhood of any point of the constraint submanifold . this is on the line with the darboux s theorem ( see e.g. @xcite ) . all symplectic spaces are locally indistinguishable . the concept of the skew - gradient projection @xmath202 of canonical variables @xmath33 onto a constraint submanifold plays important role in the moyal quantization of constraint systems . geometrically , the skew - gradient projection acts along phase flows @xmath203 generated by constraint functions . these flows are commutative in virtue of eqs.([sb ] ) : using eqs.([sb ] ) and the jacobi identity , one gets @xmath204 for any function @xmath58 , so the intersection point with @xmath205 is unique .    to construct the skew - gradient projections , we start from equations @xmath206 which say that point @xmath207 is left invariant by phase flows generated by @xmath208 . using the symplectic basis ( [ sb ] ) for the constraints and expanding @xmath209 in the power series of @xmath210 , one gets @xmath211 similar projection can be made for function @xmath30 : @xmath212 one has @xmath213 the projected functions are in involution with the constraint functions : @xmath214 consequently , @xmath215 does not vary along @xmath216 , since @xmath217    the skew - gradient projection is depicted schematically in fig . [ fig1 ] .      in the classical second - class constraints systems , one has to start from constructing @xmath218 from @xmath219 . the evolution equation for phase - space functions can be converted then to the classical liouville equation : @xmath220    similarly , the canonical variables obey the classical hamilton s equations : @xmath221 with initial conditions @xmath222    equation @xmath223 tells that @xmath224 remain constant along @xmath225 : @xmath226    equations ( [ invo-2 ] ) show that trajectories do not leave level sets @xmath227 and therefore do not leave the constraint submanifold @xmath228    given @xmath218 is constructed , it becomes possible to extend standard theorems of the hamiltonian formalism to second - class constraints systems without modifications . the novel element is the interplay between the evolution and the skew - gradient projection . let the coordinate system @xmath229 is obtained from the coordinate system @xmath230 by the canonical transformation @xmath231 .     is commutative with classical projection @xmath232 onto constraint submanifold @xmath205 . , width=7 ]    eq.([fsg ] ) may be applied for @xmath225 . using eq.([invo-2 ] ) , we replace the arguments of the constraint functions to @xmath225 and replace everywhere @xmath225 with @xmath233 , as long as the poisson brackets are invariant and the constraint functions are scalars . we arrive at @xmath234 the first line is a consequence of eq.([fssf ] ) . the evolution is commutative with the skew - gradient projection . equation ( [ abba ] ) is illustrated on fig . [ fig4 ] . the liouville equation can be solved provided phase - space trajectories @xmath22 are known . in general , @xmath235 applying projection ( [ fsg ] ) , one gets @xmath236 the first line follows from eq.([fssf ] ) . equation ( [ 23456 ] ) shows how to use characteristics in order to solve evolution equations in the classical second - class constraint systems . the evolution depends on choice of the constraint functions up to a canonical transformation . suppose we found two sets of the constraint functions @xmath208 and @xmath237 describing the same constraint submanifold . each set can be transformed to the standard basis ( [ sb ] ) . such bases are related by canonical transformations , so one can find a canonical map : @xmath238 , such that @xmath239 . the inverse transform is @xmath240 . the skew - gradient projections @xmath232 and @xmath241 are related by : @xmath242 the skew - gradient projection depends on choice of the constraint functions up to a canonical transformation . the same is true for projected hamiltonian functions : @xmath243 where @xmath244 . two sets of the constraint functions @xmath208 and @xmath237 lead to the canonically equivalent hamiltonian phase flows . the constraint systems represent high interest since all fundamental interactions in the elementary particle physics are based on the principles of gauge invariance . gauge fixing turns gauge - invariant systems into constraint systems .    in the classical mechanics , the constraint systems can be treated as a limiting case @xmath245 of systems in a potential @xmath246 which rapidly increases when the coordinates @xmath247 go away from the constraint submanifold . in the limit of @xmath245 , @xmath248 if @xmath247 belongs to the constraint submanifold and @xmath249 when @xmath247 does not belong to the constraint submanifold . the classical systems obtained by imposing the constraints and by the limiting procedure have equivalent dynamic properties @xcite .    in the quantum mechanics , this is not the case . the limiting procedure applied to a particular system of ref . @xcite to model holonomic constraints , results to the quantum dynamics which depends on the way the constraint submanifold is embedded into the configuration space . from other hand , the quantization of constraint holonomic systems leads to the conclusion that the dynamics is determined by the induced metric tensor only @xcite . the limiting procedure and imposing the constraints are not equivalent schemes of the quantization . in what follows , we discuss the constraint dynamics as it appears in the gauge theories . the groenewold - moyal constraint dynamics has many features in common with the classical constraint dynamics . the projection formalism developed for constraint systems allows , from other hand , to treat unconstrained and constraint systems essentially on the same footing . we recall that classical hamiltonian function @xmath250 and constraint functions @xmath224 are distinct in general from their quantum analogues @xmath251 and @xmath252 . these dissimilarities are connected to ambiguities in quantization of classical systems . it is required only @xmath253 in what follows @xmath254    the quantum constraint functions @xmath252 satisfy @xmath255    the quantum - mechanical version of the skew - gradient projections is defined with the use of the moyal bracket @xmath256    the projected canonical variables have the form @xmath257 the quantum analogue of eq.([fsg ] ) is @xmath258 the function @xmath259 obeys equation @xmath260    the evolution equation which is the analogue of eq.([evol ] ) takes the form @xmath261 where @xmath262 is the hamiltonian function projected onto the constraint submanifold as prescribed by eq.([sgrad4 ] ) . any function projected quantum - mechanically onto the constraint submanifold can be represented in the form @xcite @xmath263 in the space of projected functions , the set of projected canonical variables @xmath264 is therefore complete . defined by eq.(4.21 ) . the submanifold @xmath265 does not coincide with the constraint submanifold @xmath266 . the variance is of order @xmath267 . the constraint submanifold @xmath205 can be parameterized by classical projection @xmath268 constructed with the use of the quantum constraint functions @xmath269 . , width=7 ]      the evolution equation in the quantum constraint systems has the form of eq.([pev2 ] ) which is essentially the same as in the quantum unconstrained systems . replacing @xmath106 by @xmath270 , one can work further with solutions @xmath271 of quantum hamilton s equations ( [ qf ] ) . it is not required for points @xmath272 to belong to the constraint submanifold , so phase - space trajectories @xmath271 occupy the whole phase space . the quantum phase flow preserves the constraint functions in the following sense : @xmath273    the alternative equation @xmath274 which would carry the conventional geometric meaning uses pre - conditionally the dot - composition law which is not allowed quantum - mechanically . it is obviously violated , so in the usual sense @xmath275 for @xmath8 even if @xmath276 ( see fig . [ fig12 ] ) .    any attempt to decide if @xmath277 involves the dot - composition e.g. @xmath278 statements involving the dot - composition are , however , forbidden . surprisingly , expressive means of the star - product formalism are not enough to formulate the simple geometric idea that a trajectory belongs to a submanifold . we wish to find statements admissible quantum - mechanically and from other hand which would support relations of belonging and intersection inherent for geometric objects . it is tempting to interpret eqs.([concon ] ) as the evidence that quantum trajectories @xmath101 do not leave , in a specific quantum - mechanical sense , level sets of constraint functions @xmath279 . such a statement has the invariant meaning with respect to unitary transformations : suppose the map @xmath280 : @xmath281 , corresponds to a unitary transformation in the hilbert space . the inverse unitary transformation generates the inverse map @xmath115 : @xmath282 , such that @xmath283 and , by virtue of eq.([brinva ] ) , @xmath284 . in the coordinate system @xmath285 , the constraint functions become @xmath286 equation ( [ brinva ] ) allows to change the variables @xmath287 in eq.([concon ] ) to give @xmath288 where @xmath289 represents the quantum phase flow in the coordinate system @xmath285 . equations ( [ concon ] ) and ( [ concon-2 ] ) are therefore equivalent . they show that `` do not leave '' represents a predicate invariant under unitary transformations . the non - local character of relations between the quantum phase flows is displayed in eq.([nonloc ] ) explicitly . one can conclude that quantum trajectories do not transform under unitary transformations as geometric objects .     and @xmath290 ( solid lines ) and quantum trajectories @xmath74 and @xmath291 ( dashed lines ) in unitary equivalent coordinate systems @xmath230 and @xmath285 , respectively . as shown , @xmath74 crosses @xmath205 twice , whereas its image @xmath291 crosses @xmath290 once . any counting of the intersections rests on an implicit use of the dot - composition , an operation which is forbidden quantum - mechanically . the property of the statements @xmath292 and @xmath293 be true or false depends on unitary transformations . from the viewpoints of eqs.([concon ] ) and ( [ concon-2 ] ) , @xmath74 and @xmath291 belong to the level sets of @xmath294 and @xmath295 , respectively . however , from condition @xmath296 it does not follow that @xmath297 and _ vice versa_. geometric relations among quantum objects , which use the dot - composition , do not have objective meaning . , width=2 ]    the coordinate transformation @xmath115 : @xmath282 does not superpose @xmath298 and @xmath299 assuming @xmath300 , we obtain @xmath301 and therefore @xmath302 in general . the constraint submanifold does not transform under unitary transformations as a geometric object also . we see that points of @xmath205 transform differently from @xmath205 . they are `` not attached to @xmath205 '' . in new coordinate system , @xmath205 represents a set of new points . to put it precisely , @xmath303    unitary transformations affect the visualization of trajectories and submanifolds . the relation `` do not leave '' supports , however , some features inherent to the usual geometric relations `` belong '' and `` intersect '' . one can show e.g. that if quantum trajectories do not leave the level sets of @xmath294 and each level set of @xmath294 is a subset of one of the level sets of @xmath304 then quantum trajectories do not leave the level sets of @xmath304 . one can not assign to quantum trajectories definite values of energy and constraint functions . in the coordinate system @xmath230 one has @xmath305 , whereas in the coordinate system @xmath285 one has @xmath306 where @xmath307 is defined by eq.([hprime ] ) . the constants @xmath308 and @xmath309 do not depend on time . however , @xmath310 in general even if trajectories are related by a unitary transformation . the same conclusion holds for constraint functions , as shown on fig . [ fig12 ] .    finally , the syntax of the star - product formalism is not rich enough to express the simple geometric idea that trajectory belongs to a submanifold . the star - product geometry admits the statement that quantum trajectories do not leave level sets of the constraint functions . the validity of this statement is not affected by unitary transformations and has the objective meaning . the quantum - mechanical relation `` do not leave '' is the remnant of usual relations of belonging and intersection inherent to geometric objects . it can not be completely visualized , however . the classical phase flow commutes with the classical skew - gradient projection , as discussed in sect . iv . we want to clarify if such a property holds for quantum systems .    given the quantum trajectories @xmath311 are constructed , the evolution of arbitrary function can be found with the help of eq.([tf ] ) and its projection can be computed using eq.([sgrad4 ] ) . the quantum projection applied to arbitrary function can not be expressed in terms of the same function of the projected arguments eq.([great ] ) , basically because the classical relation @xmath312 turns to the quantum inequality @xmath313 . in terms of a function @xmath314 defined for @xmath315 in eq.([great ] ) , the quantum analogue for eqs.([23456 ] ) reads @xmath316 the construction of @xmath314 from @xmath32 is a complicated task , so practical advantages of this equation are not seen immediately . equation ( [ 0000 ] ) accomplishes solution of the evolution problem for observable @xmath5 in terms of quantum characteristics . it remains to prove @xmath317    the first line is a consequence of the fact that the constraint functions @xmath294 are moyal commutative with the projected hamiltonian function @xmath318 . to arrive at the second line , it is sufficient to use eq.([concon ] ) to replace arguments of the constraint functions entering the skew - gradient projection .    the quantum phase flow commutes with the quantum projection , as illustrated on fig . [ fig6 ] . the composition law ( [ comp ] ) for quantum phase flow holds for the constraint systems . it holds for projected quantum trajectories also : @xmath319 as a consequence of eqs.([qabba ] ) .    . the phase - space trajectory @xmath320 does not belong to the submanifold @xmath265 except for @xmath7 , so the white planes on figs . [ fig5 ] and [ fig6 ] are distinct.,width=8 ] the method of characteristics for solving evolution equations in classical and quantum , unconstrained and constrained systems has been discussed . the analysis rests on the groenewold - moyal star - product technique . the classical method of characteristics applies to first - order pde and consists in finding characteristics which are solutions of first - order ode . for the classical liouville equation , the corresponding first - order ode are the hamilton s equations and the characteristics of interest are the classical phase - space trajectories . the quantum liouville equation is the infinite - order pde . nevertheless , it can be solved in terms of quantum characteristics which are solutions of the quantum hamilton s equations . these equations represent infinite - order pde also .    using the star - product formalism , we showed that to any fixed order in the planck s constant , quantum characteristics can be constructed by solving a closed system of ode for quantum trajectories and generalized jacobi fields . the quantum evolution becomes local in an extended phase space with new dimensions ascribed to generalized jacobi fields . this statement holds for constraint systems also . one - parameter continuous groups of unitary transformations in quantum theory represent the quantum deformation of one - parameter continuous groups of canonical transformations in classical theory . quantum phase flow , induced by the evolution in the hilbert space , does not satisfy the condition for canonicity and preserves the moyal bracket rather than the poisson bracket . the knowledge of quantum phase flow allows to reconstruct quantum dynamics . .solutions of evolution equations for functions ( second column ) and projected functions ( third column ) of classical systems ( first row ) and quantum systems ( second row ) in terms of characteristics . @xmath321 are solutions of classical hamilton s equations with hamiltonian function @xmath250 ( second column ) and projected hamiltonian function @xmath322 ( third column ) . @xmath323 are classical projections of @xmath321 . @xmath74 are solutions of quantum hamilton s equations with hamiltonian function @xmath251 ( second column ) and projected hamiltonian function @xmath262 ( third column ) . @xmath324 are quantum projections of @xmath74 . @xmath325 is defined in terms of @xmath326 by eq.([0000 ] ) . classical and quantum projections are defined by eqs.([fsg ] ) and ( [ sgrad4 ] ) , respectively . [ cols="<,^,^",options="header " , ]     the results reported in this work are valid for semiclassically admissible functions , i.e. for functions regular in @xmath56 at @xmath327 . physical observables are normally associated with classical devices and expressed as classical functions of classical variables . the quantum evolution turns , however , the set of classical functions into the set of semiclassically admissible functions . the use of the skew - gradient projection formalism allows to treat unconstrained and constraint systems essentially on the same footing . we showed that the skew - gradient projections of solutions of the quantum hamilton s equations onto the constraint submanifold comprise the complete information on quantum dynamics of constraint systems . the formalism we developed applies in particular to the dynamics of gauge - invariant systems which become second class upon gauge fixing . the quantum dynamics of charged particles in external gauge fields on flat and curved manifolds is discussed within the star - product formalism in a gauge - invariant manner in refs . @xcite . the evolution equations for semiclassically admissible functions admit solutions in terms of characteristics in all physical systems , as summarized in table [ lab3 ] . the analytic geometry uses the dot - product and rests on classical ideas how to arrange composition of functions . it is well known that all theorems of geometry can be reformulated using tools of the analytic geometry . given the dot - product is replaced with the star - product , we arrive at the star - product geometry with well defined coordinate systems , transformations of the coordinates and equations for functions of the coordinates . however , objects of the star - product geometry , defined algebraically , can hardly be visualized :    we found that quantum trajectories and constraint submanifolds do not transform as geometric objects . the statement `` quantum trajectory belongs to a constraint submanifold '' can be changed to the opposite by a unitary transformation . the star - composition law ( [ comp ] ) shows also that the quantum evolution can not be treated literally as moving along a quantum trajectory . we attempted to find statements whose validity can not be reverted by transformations of the coordinate system and which , from other hand , express relations similar to `` belong '' , `` intersect '' , etc . a weak but consistent geometric meaning can be attributed to the statement `` quantum trajectories do not leave level sets of constraint functions '' . the dot - product composition of linear functions coincides with the star - product composition of linear functions , so under linear transformations straight lines and hyperplanes turn to straight lines and hyperplanes . relations of the linear algebra , imbedded into the star - product geometry , preserve the consistent geometric meaning . finally , this work extended the method of characteristics to quantum unconstrained and constraint systems . from the point of view of applications , it is motivated by the fact of using classical phase - space trajectories in transport models and by the appearance of constraints in relativistic versions of qmd transport models . the method of quantum characteristics represents the promising tool for solving numerically many - body potential scattering problems . this work is supported by dfg grant no . 436 rus 113/721/0 - 2 , rfbr grant no . 06 - 02 - 04004 , and european graduiertenkolleg gr683 . a completely integrable classical system admits a canonical transformation which makes the hamiltonian function depending on half of the canonical variables only ( see e.g. @xcite ) . such variables if exist can be taken to be canonical momenta which usually referred to as actions . the canonically conjugate coordinates are referred to as angles .    in quantum mechanics , we search for a unitary transformation ( or a half - unitary transformation @xcite ) allowing to express the hamiltonian as an operator function of operators of canonical momenta . if such a transformation exists , the hamiltonian commutes with the operators of canonical momenta , so that the canonical momenta are integrals of motion , whereas the operators of canonical coordinates depend linearly on time . the quantum integrable systems admit an equivalent treatment in the framework of the groenewold - moyal dynamics @xcite . given the hamiltonian function @xmath179 is known , one has to search for a map @xmath280 : @xmath328 preserving the moyal bracket , for which the system admits a hamiltonian function @xmath329 depending on actions @xmath330 , i.e. , canonical momenta only . we restrict the discussion by unitary transformations ( [ unita ] ) , leaving aside more involved cases described in refs . @xcite . let @xmath66 and @xmath331 be solutions of eq.([qf ] ) with hamiltonian functions @xmath179 and @xmath307 , respectively . in the coordinate system @xmath285 , the series expansion ( [ teu ] ) is truncated at @xmath332 . the quantum hamilton s equations give a motion by inertia : @xmath333 the actions ( @xmath334 ) remain constant , whereas the angles ( @xmath335 ) evolve linearly with time . equations ( [ iner ] ) can be derived as the weyl s transform of the equations of motion for the heisenberg operators of the canonical coordinates and momenta obtained by a unitary transformation from the initial set of operators the canonical coordinates and momenta . the poisson bracket @xmath336 depends for any @xmath337 on the actions only , so one has @xmath338 the map @xmath339 : @xmath340 showing the evolution in the coordinate system @xmath285 is both canonical and unitary , as the left - 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the knowledge of quantum phase flow induced under the weyl s association rule by the evolution of heisenberg operators of canonical coordinates and momenta allows to find the evolution of symbols of generic heisenberg operators . 
 the quantum phase flow curves obey the quantum hamilton s equations and play the role of characteristics . at any fixed level of accuracy of semiclassical expansion 
 , quantum characteristics can be constructed by solving a coupled system of first - order ordinary differential equations for quantum trajectories and generalized jacobi fields . 
 classical and quantum constraint systems are discussed . 
 the phase - space analytic geometry based on the star - product operation can hardly be visualized . 
 the statement `` quantum trajectory belongs to a constraint submanifold '' can be changed e.g. to the opposite by a unitary transformation . some of relations among quantum objects in phase space are , however , left invariant by unitary transformations and support partly geometric relations of belonging and intersection . 
 quantum phase flow satisfies the star - composition law and preserves hamiltonian and constraint star - functions .