Recently we have constructed a completely supersymmetric nonlinear action possessing the properties expected from multiple D0brane system. Its quantization should result in an interesting supersymmetric field theory in the superspace with additional matrix coordinates which can provide an important insights in the study of String Theory. As a first stage toward this aim in this paper we construct the Hamiltonian mechanics and perform covariant quantization of the simplest three dimensional counterpart of the ten dimensional multiple D0brane model. We obtain a supersymmetric system of equations in a superspacetime enlarged by bosonic and fermionic matrix coordinates which appears as a result of such quantization and discuss some of its properties.
['D-Branes', 'M-Theory', 'Superspaces', 'High Energy Physics - Theory', 'General Relativity and Quantum Cosmology', 'Mathematical Physics']
Towards field theory of multiple D0branes. Hamiltonian mechanics and quantization of simplest 3D prototype of multiple D0brane system
{'No Header': 'Towards field theory of multiple D0-branes. Hamiltonian mechanics and quantization of simplest 3D prototype of multiple D0-brane system  \nIgor Bandos †‡ and Unai D.M. Sarraga †  \n† Department of Physics and EHU Quantum Center, University of the Basque Country UPV/EHU,  \nP.O. Box 644, 48080 Bilbao, Spain  \n‡ IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009, Bilbao, Spain  \nAbstract: Recently we have constructed a completely supersymmetric nonlinear action possessing the properties expected from multiple D0-brane system. Its quantization should result in an interesting supersymmetric field theory in the (super)space with additional matrix coordinates which can provide an important insights in the study of String Theory. As a first stage toward this aim, in this paper we construct the Hamiltonian mechanics and perform covariant quantization of the simplest three dimensional counterpart of the ten dimensional multiple D0-brane model. We obtain a supersymmetric system of equations in a (super)spacetime enlarged by bosonic and fermionic matrix coordinates which appears as a result of such quantization and discuss some of its properties.', '1 Introduction': "The 'second (super)string revolution' of the middle 90th brought us understanding of an important role of supersymmetric extended objects, superp -branes, in String Theory (presently also known as M-theory). Here p is the number of spatial dimensions of the worldvolume of the brane which is a surface in spacetime (actually in superspace), so that p = 1 corresponds to string, p = 2 to membrane and p = 0 to particles. The bosonic degrees of freedom of simplest p -branes are described (in a suitably chosen coordinate system) by a D -p -1 coordinate functions which fix the position of the d = p +1 dimensional p -brane worldvolume in the D-dimensional spacetime. This is the case for ten dimensional (D = 10) fundamental string (F1-brane) [1, 2] and D = 11 supermembrane [3] (also known as M2-brane). However, there are branes which carry additional fields on their worldvolume.  \nIn particular, this is the case for Dirichlet p -branes or D p -branes which are the supersymmetric extended objects where the fundamental string can have its endpoints [4-6] which carry d = p + 1 dimensional gauge fields on their worldvolumes. Super-D p -branes can be described by supersymmetric solutions of 10D supergravity equations [7], by worldvolume actions [8-13] or by the superfield equations obtained in [14] (before the actions were known!) in the frame of superembedding approach [15-18].  \nIn type IIA superstring theory there exist supersymmetric D p -branes with p = 0 , 2 , 4 , 6 and 8, while the zoo of type IIB superstring theory is populated by super-D p -branes with p = 1 , 3 , 5 , 7 and 9. The D9-brane of type IIB theory is spacetime filling as far as superstring is consistent in spacetime of dimension D = 10, and this object was studies much earlier than its lower p cousins.  \nIts counterpart in the bosonic string theory, which is consistent in 26-dimensional spacetime, D = 26, is the D25-brane. A bosonic string ending on a D25-brane is actually an open string the endpoints of which are free; it was studied beginning the very first years of String Theory [19]. In fact, it was more interesting than closed string for considering string model as an effective description of QCD [20] with the aim to describe the confinement of quarks.  \nThe free endpoints of open strings can be coupled minimally to gauge fields. The consistency of such a coupling was studied by Fradkin and Tseytlin [21] who found that it requires the gauge field to obey equations which can be obtained from the Born-Infeld (BI) action [22]. In the modern language, this implies that the BI action is an effective action of spacetime filling D25-brane of the bosonic string theory. The spacetime filling D9-brane of type IIB superstring theory is described by a (nontrivial) supersymmetric extension of D = 10 BI action [10-12, 23].  \nThe discovery of lower p , not spacetime filling D p -branes was delayed till late 80th by believing that the open string with free endpoints and closed strings were the only possible boundary conditions which did not break the target space (i.e. D-dimensional spacetime) Lorentz invariance SO(1 , D -1). Indeed the string with endpoints located on d = p +1 dimensional planes in spacetime breaks SO(1 , D -1) down to its SO(1 , p ) ⊗ SO(D -p -1) subgroup. The breakthrough in [4-6] used the brilliant result of Scherk and Schwarz who discovered that closed string model actually describes a theory of gravity [24]. Since open string theory inevitably contains the closed string sector, it describes a theory of gravity too. In a theory of gravity one cannot imagine appearance of fixed planes so that the d = p +1 dimensional surfaces where the open string can have endpoints should be a dynamical objects and these objects were called Dirichlet p -branes in [5]. Then the breaking of the target space Lorentz symmetry SO(1 , D -1) down to SO(1 , p ) ⊗ SO(D -p -1) is a spontaneous breaking characteristic for any p -brane [25].  \nThe endpoints of an open string on the D p -brane can be coupled to d = p + 1 dimensional gauge field, and the consistency of such coupling was found to require the gauge field to obey the equations which follow from the Dirac-Born-Infeld (DBI) action [26]. The actions of super-D p -branes [8-12] are given by quite nontrivial supersymmetric generalization of DBI action which, in addition to a straightforward supersymmetric generalization of the DBI term, contains the so-called Wess-Zumino term describing the  \ncoupling of D p -brane to (a superspace generalization of) so-called Ramond-Ramond gauge fields of the supergravity multiplet. The characteristic property of these super-D p -brane actions is their invariance under local fermionic κ -symmetry. It is important because its existence guarantees that the ground state of the dynamical system preserves a half of target space supersymmetry 1 and thus is a stable 1/2 BPS state 2 [29] (see [30] for further development).  \nThe quest for the action of multiple D p -brane (mD p ) system, which should provide an effective description of the maximally supersymmetric system of N nearly coincident D p -branes and N 2 fundamental strings ending on these, was created by seminal paper [31] in which E. Witten argued that such a system exhibits the enhanced U( N ) gauge symmetry and, at very low energy, allows for the gauge fixed description by U( N ) Supersymmetric-Yang-Mills (SYM) action. Setting N = 1, we arrive at the Abelian SYM or super-Maxwell action with maximal supersymmetry which can be obtained as a weak field approximation from the gauge fixed version of the action for single super-D p -brane [8-12] which is given by the sum of Dirac-Born-Infeld (DBI) and Wess-Zumino (WZ) terms. Hence it was expected that the mD p action contains some non-Abelian generalization of the DBI term or Born-Infeld (BI) action in the case of spacetime filling D9-brane.  \nThis latter case deserves some comments. The set of worldvolume fields of single super-D9-brane consists of fermionic coordinate functions, a pair of 16-component fermionic spinors of the same chirality and an Abelian ten dimensional gauge field. Upon gauge fixing of κ -symmetry by reducing twice the number of fermionic fields, the worldvolume field content of D9-brane literally coincides with 10d SYM multiplet, the Wess-Zumino term vanishes and its action becomes a maximally supersymmetric extension of the BornInfeld functional with nonlinearly-realized supersymmetry 'a la Volkov-Akulov [11]. The multiple D9-brane action should contain a non-Abelian generalization of this latter and even in the purely bosonic case such a generalization requires some additional specification. In 1997 Tseytlin proposed to use the symmetric trace prescription for this purposes [32]. However, the supersymmetric generalization of Tseytlin's action, which could be considered as a complete action of multiple D9-brane action, is still unknown.  \nAlthough the problem of constructing supersymmetric actions for multiple D p -branes is still unsolved in its complete form, many interesting results were obtained during these years [32-62] (see [61] for a description of these results in some details). In particular, an action which possesses the properties expected from mD0 system in flat type IIA superspace was found in our [61].  \nMore specifically, the model in [61] is formulated in terms of bosonic and fermionic coordinate functions and the worldline fields of 1d SU( N ) SYM multiplet with N = 16 supersymmetry and it is invariant, besides the target space supersymmetry, under local worldline supersymmetry. This local worldline supersymmetry acts on the coordinate functions as κ -symmetry of single D0-brane and on the SYM fields as local version of the supersymmetry of SYM supermultiplet. Its existence guarantees that the ground state of the system is a supersymmetric 1/2 BPS state which can be identified with the BPS state of multiple D0-brane which can be described by supergravity solution. Upon gauge fixing of the worldline supersymmetry and reparametrization symmetry the field content reduces to that of 1d U( N ) SYM multiplet with N = 16 supersymmetry, or with the dimensional reduction of D = 10 SYM for U( N ) gauge group, as expected for the mD0 system. This allows us to conclude that in [61] we have found a good candidate for mD0-brane action.  \nA single D0-brane is just a massive superparticle in type IIA superspace [12] and its worldvolume is a one dimensional worldline. A one dimensional (1d) gauge field is known to be pure gauge, thus such is the D0-brane worldline gauge field. Furthermore its field strength cannot be defined so that in D0-brane  \ncase the DBI part of the action is reduced to just a standard superparticle kinetic term. Furthermore, the WZ term is constructed from fermionic coordinates of flat type IIA superspace and, hence, the D0-brane action is given by 10D generalization of the de Azc'arraga-Lukierski action for the D = 4 N = 2 massive superparticle [63, 64]. Thus, there is no reason to expect a kind of non-Abelian DBI-like contribution to the mD0 action, and our candidate mD0 action(s) in [61] do not contain such.  \nThe unexpected property of the action of [61] is the presence of an arbitrary positive definite function M ( H ) of an SU( N ) invariant function of matrix fields H in which one can recognize the Hamiltonian of SYM model 3 . Surprisingly, the action possesses local fermionic worldline supersymmetry, characteristic property expected from mD0 system, for arbitrary choice of M ( H ), so that we can speak about a family of candidate actions for the description of mD0. Of this family, the simplest representative with M ( H ) = m for constant m appearing also in the center of mass part of the action had been found before in [58]. However, the most promising candidate for the role of mD0 action has a particular nontrivial form of M ( H ) given in Eq. (2.44) below: as we showed in [62] it can be obtained by dimensional reduction from the 11D multiple M-wave (also called multiple M0 or mM0) action of [53].  \nThe quantization of mD0 system should result in an interesting field theory in a (super)spacetime extended by a number of bosonic and fermionic matrix coordinates which can bring us a new comprehension in String/M-theory. To argue in favour of such hope, let us recall the concept of p -brane democracy by Paul Townsend [65] which states that any of the String/M-theory p -branes can be considered as fundamental, thus taking on the role usually attributed to fundamental strings, while all other p ' -branes emerge as solitonic objects in such a theory of fundamental p -brane. From this point of view, the standard choice of string as the fundamental object is preferable just because of convenience to have equations which can be linearized in a suitable gauge and well defined perturbation theory. Furthermore, the interaction in this case is described by a topology of the worldsheet thus allowing one to use the free string action also in description of interacting strings. However the string field theory (see [66-70] and refs therein) is not easy at all, and is still under construction, at least in its complete form.  \nAmong other possibilities, the choice of D0-branes as fundamental objects was discussed as very intriguing opportunity, as it seemed to indicate a possibility to come back from extended objects to particles as basic building blocks for constructing a fundamental theory. However, to this end one should consider rather a theory of many D-particles interacting among themselves, which is to say the dynamical system which we call multiple D0-brane or mD0. This is what suggests us to expect that the field theory of mD0-brane system can give us new insights in String/M-theory.  \nThe study of mD0 system using the low energy effective action suggested by Witten, i.e. by using just the action of 10D U( N ) SYM dimensionally reduced to d = 1 [71], was one of the roots of the M(atrix)theory conjecture [72] which also used the 1d N = 16 U( N ) SYM action, presently also known as the action of BFSS matrix model 4 . The ground state wavefunction of the quantum mechanics obtained by quantization of BFSS action was studied in [73-79]; related studies can be found also in [80].  \nOur action [61, 62] can be considered as 10D Lorentz invariant and completely gauge invariant (supersymmetric and κ -symmetric) generalization of this BFSS action. Hence its quantization should provide us with the complete supersymmetric and manifestly Lorentz invariant mD0 field theory.  \nThe aim of this paper is to start the program of constructing the mD0-brane field theory. As this problem has happened to be quite complicated, we begin with quantization of a 3D counterpart of mD0 system (which we will roughly call 3D mD0) described by 3D version of 10D mD0 action found in our [60]. Moreover, we will restrict ourselves by the simplest case described by the action from [60] with constant M ( H ) = m (i.e. 3D counterpart of the action from [58]). This problem (which also appeared to be quite  \ncomplex) has provided us with a good toy model to approach 10D mD0 system: it has indicated where the problems can appear in 10D mD0 quantization and has suggested the method to resolve these.  \nFurthermore, the superfield theory obtained in this work may be interesting on its own as it provides a generalization (supersymmetric and 3D Lorentz covariant) of a simplest version of the problem of the quantization of Yang-Mills theory in Schrodinger representation (see recent [81] for the comprehensive review of D = 3 case and for references). To consider this just for the case of 1d dimensional reduction of SYM (of 3D N = 2 SYM in our case) can be relevant e.g. for the description of effective Hamiltonian of zero modes in this theory [82].  \nThe paper is organized according the Table of Contents. The brief description of the work can be found in the concluding Sec. 7.", '2.1 Physical and auxiliary fields of a 3D counterpart of mD 0 system': "The dynamical variables of 3D mD0 system consisting of N D0-branes and strings connecting the different D0's or ending on the same D0-brane, can be split into the center of mass variables and the matrix fields describing the relative motion of the mD0 constituents. This latter set contains the fields of D = 3 SU( N ) Yang-Mills model dimensionally reduced to d = 1 and thus includes traceless N × N bosonic matrix fields Z = || Z i j ( τ ) || and ¯ Z = || ¯ Z i j ( τ ) || related by Hermitian conjugation,  \nZ = ( ¯ Z ) † ⇔ Z i j = ( ¯ Z j i ) ∗ , tr Z = Z i i = 0 , i, j = 1 , ..., N , (2.1)  \ntwo Hermitian conjugate traceless N × N fermionic matrix fields Ψ = || Ψ i j ( τ ) || and ¯ Ψ = || ¯ Ψ i j ( τ ) ||  \nΨ = ¯ Ψ † ⇔ Ψ i j = ( ¯ Ψ j i ) ∗ , tr Ψ ≡ Ψ i i = 0 (2.2)  \nand anti-Hermitian 1-form  \nA = d τ A τ = d τ || A τ i j ( τ ) || = -A † ⇔ A τ i j = -( A τ j i ) ∗ . (2.3)  \nAll these fields depend on the proper time τ which parametrizes the center of mass worldline W 1 of our 3D mD0 system.  \nThe set of the center of mass variables is the same as one used for the description of single D0-brane in the spinor moving frame formulation. This contains real three-vector bosonic and two complex conjugate fermionic spinor coordinate functions  \nz M ( τ ) = ( x µ ( τ ) , θ α ( τ ) , ¯ θ α ( τ )) , ¯ θ α ( τ ) = ( θ α ( τ )) ∗ , µ = 0 , 1 , 2 , α = 1 , 2 , (2.4)  \nwhich define parametrically the worldline W 1 as a surface in flat D=3 N = 2 superspace Σ (3 | 4) with coordinates z M = ( x µ , θ α , ¯ θ α )  \nW 1 ⊂ Σ (3 | 4) : z M = z M ( τ ) , (2.5)  \nand the spinor moving frame variables which we will describe below.  \nWe have denoted the coordinate functions by the same symbol as coordinates believing that this cannot produce a confusion since only the coordinate functions will be used in the study of classical mechanics while only coordinates will be used then in the description of field theory obtained upon quantization.", "2.2 3D version of the de Azc'arraga-Lukierski action for single D0-brane": "When single D0-brane is considered, its action can be written in terms of the above described coordinate functions only. We present it here as a reference point which also allows us to fix our notation.  \nThe standard action for single D0-brane [12] is given by 10D counterpart of the D = 4 action by the de Azc'arraga and Lukierski [63, 64]. Here we will discuss its 3D counterpart given by  \nS A+L = -m ∫ d τ ( √ E a τ E b τ η ab + ˙ θ α ¯ θ α -θ α ˙ ¯ θ α ) (2.6)  \nwhere dot denotes derivative with respect to proper time τ and  \nE a τ = ˙ x a -i ˙ θγ a ¯ θ + iθγ a ˙ ¯ θ , (2.7)  \nis the (coefficient for d τ of the) pull-back of the Volkov-Akulov 1-form,  \nE a = d x a -i d θγ a ¯ θ + iθγ a d ¯ θ =: E a ( z ) , E a ( z ( τ )) = d τE a τ , a = 0 , 1 , 2 . (2.8)  \nIn it  \n˙ θγ a ¯ θ = ˙ θ α γ a αβ ¯ θ β , (2.9)  \nwith symmetric 2 × 2 matrices  \nγ a αβ = -iγ a α σ /epsilon1 σβ , ˜ γ aαβ = i/epsilon1 ασ γ a σ β (2.10)  \nconstructed from unit antisymmetric spin tensor  \n/epsilon1 αβ = iσ 2 = -/epsilon1 αβ = ( 0 1 -1 0 ) . (2.11)  \nand 3D Dirac matrices. These are imaginary in our mostly minus metric notation and obey  \nγ  \na  \nγ  \nb  \n=  \nη  \nab  \n+  \ni/epsilon1  \nabc  \nγ  \nc  \n,  \nη  \nab  \n= diag(+1  \n,  \n-  \n1  \n,  \n-  \n1)  \n.  \n(2.12)  \nIn all other cases the spinor indices are raised and lowered by the epsilon symbol (2.11), e.g.  \nθ α = /epsilon1 αβ θ β , θ α = /epsilon1 αβ θ β . (2.13)  \nThe characteristic property of the action (2.6) is its invariance under local fermionic κ -symmetry  \nδ  \nκ  \nx  \nµ  \n=  \niδ  \nκ  \nθγ  \na  \n¯  \nθ  \n-  \niθγ  \na  \nδ  \nκ  \n¯  \nθ ,  \nδ κ θ α = κ β ( /epsilon1 -i ˜ γ a E a τ / √ E c τ E τc ) βα , δ κ ¯ θ α = ¯ κ β ( /epsilon1 + i ˜ γ a E a τ / √ E c τ E τc ) βα . (2.14)  \nThis property is important because it implies that the ground state of the system preserves 1/2 of the spacetime supersymmetry [29] and hence is a BPS state, a 3D counterpart of the D0-brane BPS state of String Theory.  \nThus the moving frame and spinor moving frame variables are not obligatory to describe a single D0brane: in this case spinor moving frame formulation, which can be found in [60], is classically equivalent to the one provided by the above described de Azc'arraga-Lukierski action. However, the action for 3D mD0 system, which was found in [60], is known only in its form involving the spinor moving frame variables. We describe these in the next (sub)section.", '2.3 Spinor moving frame variables (Lorentz harmonics)': 'In [60] we have used the real spinor moving frame variables ( v 1 α , v 2 α ) constrained by v α 2 v 1 α = 1. 5 However, we later found it much more convenient to describe the spinor frame variables using complex spinors  \nw α = 1 √ 2 ( v 1 α -iv 2 α ) and its c.c. ¯ w α = 1 √ 2 ( v 1 α + iv 2 α ) (2.15)  \nwhich obey  \nso that  \nδ α β = i ¯ w α w β -iw α ¯ w β , /epsilon1 αβ = -i ¯ w α w β + iw α ¯ w β . (2.17)  \nTaking these constraints into account, we find that the derivatives of these complex spinors (providing a 3D version of the 4D Newman-Penrose diad [83]) are expressed by  \nD w α := d w α + iaw α = if ¯ w α , D¯ w α := d ¯ w α -ia ¯ w α = -i ¯ fw α , (2.18)  \nin terms of Cartan forms 6  \nf = w α d w α , ¯ f = ¯ w α d ¯ w α , a = w α d ¯ w α = ¯ w α d w α . (2.19)  \nThe first two of these provide a covariant basis of the space co-tangent to the coset SU(1 , 1) / U(1) = SL(2 , R ) / SO(2), while the third, a , transforms as a connection under U(1) symmetry of our construction. This allows us to introduce the U(1)-covariant derivatives D in (2.18). It is easy to check that these Cartan forms obey the following structure equations (Maurer-Cartan equations)  \nD f = d f +2 if ∧ a = 0 , D ¯ f = d ¯ f -2 i ¯ f ∧ a = 0 , d a = -if ∧ ¯ f . (2.20)  \nThe set of moving frame vectors (a 3D version of the complex 4D light-like Newman-Penrose tetrade [83]) is composed from the complex spinor w and its c.c. as  \nu (0) a = wγ a ¯ w , u a = wγ a w , ¯ u a = ¯ wγ a ¯ w . (2.21)  \nDue to the properties of D = 3 gamma matrices, these obey  \nu (0) a u a (0) = 1 , u (0) a u a = 0 , u (0) a ¯ u a = 0 , (2.22)  \nu a u a = 0 , ¯ u a ¯ u a = 0 , u a ¯ u a = -2 (2.23)  \nu (0) a γ a αβ = 2 w ( α ¯ w β ) , u a γ a αβ = 2 w α w β , ¯ u a γ a αβ = 2 ¯ w α ¯ w β . (2.24)  \nf = 1 2 ( f 11 -f 22 ) -if 12 , ¯ f = 1 2 ( f 11 -f 22 ) + if 12 , a = 1 2 ( f 11 + f 22 ) = 1 2 f qq .  \nand  \n¯ w α w α = i (2.16)  \nEqs. (2.22) and (2.23) imply that the real 3 × 3 matrix  \nu ( b ) a = ( u (0) a , 1 2 ( u a + ¯ u a ) , 1 2 i ( u a -¯ u a ) ) ∈ O(1 , 2) , (2.25)  \nbelongs to O(1 , 2) group. Furthermore, the way of constructing these vectors from bilinear of spinors implies that 7  \n/epsilon1 abc u (0) a u b ¯ u c = -2 i ⇔ u [ b ¯ u c ] = -i/epsilon1 abc u a (0) , (2.26)  \nwhich makes this matrix to be SO(1 , 2) valued,  \nu ( b ) a = ( u (0) a , 1 2 ( u a + ¯ u a ) , 1 2 i ( u a -¯ u a ) ) ∈ SO(1 , 2) , (2.27)  \nwhich is tantamount to saying that this set of vectors describes an oriented Lorentz frame.  \nThen (2.24), or equivalently (2.21), describes the relation of moving frame with spinor moving frame, i.e. to show as the former is constructed from the latter.  \nThe derivatives of the moving frame vectors are expressed through the Cartan forms by  \nd u (0) a = if ¯ u a -i ¯ fu a , D u a = d u a +2 iau a = 2 ifu (0) a , D¯ u a = d¯ u a -2 ia ¯ u a = -2 i ¯ fu (0) a , (2.28)  \nso that (2.19) are equivalent to  \nf = i 2 u a d u (0) a , ¯ f = -i 2 ¯ u a d u (0) a , a = -i 4 ¯ u a d u a = i 4 u a d¯ u a . (2.29)  \nWhen functions of the spinor frame variables f ( ¯ w,w ) are considered, due to the constraint (2.16), their differential (exterior derivative) can be decomposed in the basis of Cartan forms (2.19),  \nd f = d ¯ w α ∂f ∂ ¯ w +d w α ∂f ∂w = if ¯ D -i ¯ f D + ia D (0) , (2.30)  \nwhere at the second stage we have used Eqs. (2.18) and introduced the covariant derivatives  \nD = w α ∂ ∂ ¯ w α , ¯ D = ¯ w α ∂ ∂w α , D (0) = ¯ w α ∂ ∂ ¯ w α -w α ∂ ∂w α . (2.31)  \nThese have the characteristic property that their action on the l.h.s. of the constraint (2.16) vanish  \nD ( ¯ w α w α ) = 0 , ¯ D ( ¯ w α w α ) = 0 , D (0) ( ¯ w α w α ) = 0 (2.32)  \nand obey su (1 , 1) /similarequal su (2 , R ) algebra  \n[ D , ¯ D ] = D (0) , [ D (0) , D ] = -2 D , [ D , ¯ D ] = 2 ¯ D . (2.33)  \nUsing spinor moving frame and moving frame vectors we can split (the pull-back of) the supervielbein forms of D = 3 N = 2 superspace  \nE a = δ a µ d x µ -i d θγ a ¯ θ + iθγ a d ¯ θ , E α = d θ α , ¯ E α = d ¯ θ α (2.34)  \ninto the set of 3 bosonic and 4 fermionic Lorentz invariant one forms  \nE (0) = E a u (0) a , E = E a u a , ¯ E = E a ¯ u a , (2.35)  \nE w = d θ α w α , E ¯ w = d θ α ¯ w α , (2.36)  \n¯ E w = d ¯ θ α w α , ¯ E ¯ w = d ¯ θ α ¯ w α . (2.37)', '2.4 Lagrangian of the simplest 3D counterpart of the mD 0 system': 'The Lagrangian of the 3D counterpart of 10D mD0 action proposed in [61] contains two constants of dimension of mass, m and µ , as well as arbitrary positive definite function  \nM = M ( H ) (2.38)  \nof the following bosonic composite of the matrix fields  \nH = tr ( P ¯ P +[ Z , ¯ Z ] 2 -i 2 Z ΨΨ + i 2 ¯ Z ¯ Ψ ¯ Ψ ) . (2.39)  \nThis can be recognized as Hamiltonian of 3D SU( N ) SYM model dimensionally reduced to d = 1. Introducing for convenience also a pair of conjugate fermionic currents  \nν := tr( Ψ P + ¯ Ψ [ Z , ¯ Z ]) , ¯ ν := tr( ¯ Ψ ¯ P + Ψ [ Z , ¯ Z ]) , (2.40)  \nwhich can be recognized as the SYM supercurrents, we can write the Lagrangian from [61] as follows  \nL [ m,µ, M ( H )] := m E 0 τ + m ( ˙ θ α ¯ θ α -θ α ˙ ¯ θ α ) + 1 µ 6 tr ( ¯ P D τ Z + P D τ ¯ Z -i 8 D τ Ψ ¯ Ψ + i 8 Ψ D τ ¯ Ψ ) + + 1 µ 6 E 0 τ 2 M H + 1 µ 6 i √ M ˙ θ α w α ¯ ν + 1 µ 6 i √ M ˙ ¯ θ α ¯ w α ν + 1 µ 6 ˙ M M tr ( ¯ PZ + P ¯ Z ) . (2.41)  \nHere D τ are covariant derivatives which contain, besides the su ( N ) valued connection A = d τ A τ , also composite U(1) connection given by the Cartan form a = d τa τ defined in (2.19) (see also (2.29)),  \nD τ Z = ˙ Z +2 ia τ Z +[ A τ , Z ] , D τ Ψ = ˙ Ψ -ia τ Ψ +[ A τ , Ψ ] , D τ ¯ Z = ˙ ¯ Z -2 ia τ ¯ Z +[ A τ , ¯ Z ] , D τ ¯ Ψ = ˙ ¯ Ψ + ia τ ¯ Ψ +[ A τ , ¯ Ψ ] . (2.42)  \nTwo special members of this family of Lagrangians have this function (2.38) being constant  \nM = m = const , (2.43)  \nand of the following special form  \nM = m 2 + √ m 2 4 + H µ 6 . (2.44)  \nThe latter was obtained in [60] by dimensional reduction of the 4D counterpart of the multiple M0-brane (mM0) action [53] while the former is the simplest representative of the family which can be recognized as the 3D counterpart of the 10D action discussed previously in [58]. This simplest model with constant M = m and Lagrangian  \nL := L [ m,µ,m ] = m E 0 τ + m ( ˙ θ α ¯ θ α -θ α ˙ ¯ θ α ) + 1 µ 6 tr ( ¯ P D τ Z + P D τ ¯ Z -i 8 D τ Ψ ¯ Ψ + i 8 Ψ D τ ¯ Ψ ) + + 1 µ 6 E 0 τ 2 m H + 1 µ 6 i √ m ˙ θ α w α ¯ ν + 1 µ 6 i √ m ˙ ¯ θ α ¯ w α ν (2.45)  \nwill be the subject of this paper. We will elaborate it in Hamiltonian formalism and then quantize it, thus arriving at the equation of the simplest 3D counterpart of the mD0-brane field theory.', '3.1 Canonical momenta and Poisson brackets': 'Generically, the canonical momenta p A are defined as derivatives of the Lagrangian L ( q A , ˙ q A ) with respect to velocities ˙ q A = d d τ q A in direction of the coordinates q A ,  \np A = ∂L ∂ ˙ q A (3.1)  \nand are canonically conjugate to this coordinate. This statement implies the diagonal structure of the Poisson brackets algebra  \n[ p A , q B } PB = -( -) ε ( A ) ε ( B ) [ q B , p A } PB = -δ A B , [ q A , q B } PB = -( -) ε ( A ) ε ( B ) [ q B , q A } PB = 0 , [ p A , p B } PB = -( -) ε ( A ) ε ( B ) [ p A , p B } PB = 0 , (3.2)  \nwhere ε ( A ) is Grassmann (fermionic) parity of p A and q A which is equal to 0 for bosons and to 1 for fermions 8 .  \nThen the canonical Hamiltonian H 0 ( p A , q A ) is defined by Legendre transform of the Lagrangian 9  \nH 0 ( p A , q A ) = ˙ q A p A -L ( q A , ˙ q A ) (3.3)  \nand the evolution of the dynamical system is described by the Hamiltonian equations  \n˙ f ( q, p ) = [ f ( q, p ) , H 0 ] PB ⇔            ˙ q A = ∂H 0 ∂p A , ˙ p A = -∂H 0 ∂q A . (3.4)  \nIn our case we firstly denote the momenta conjugate to the bosonic and fermionic coordinate functions by  \np a = ∂L ∂ ˙ x a , ¯ Π α = ∂L ∂ ˙ θ α , Π α = ∂L ∂ ˙ ¯ θ α (3.5)  \nso that their nonvanishing Poisson brackets are  \n[ p a , x b ] PB = -[ x b , p a ] PB = -δ b a , { ¯ Π α , θ β } PB = { θ β , ¯ Π α } PB = -δ β α , { Π α , ¯ θ β } PB = { ¯ θ β , Π α } PB = -δ β α .  \n(3.6)', '3.2 Covariant momenta and Poisson/Dirac brackets in the spinor moving frame sector': 'Similarly, we can introduce the canonical momenta conjugate to spinor frame variables  \n¯ P α = ∂ L ∂ ˙ w α , P α = ∂ L ∂ ˙ ¯ w α (3.7)  \nwhich have nonvanishing Poisson brackets  \n[ ¯ P α , w β ] PB = -δ α β , [ P α , ¯ w β ] PB = -δ α β . (3.8)  \nIf working with these canonical momenta, we have to consider the condition (2.16) as a second class constraint (in terminology of Dirac [84], see below for more details and discussion), and find that this is conjugate to the constraint 10  \nw α ¯ P α + ¯ w α P α ≈ 0 , (3.9)  \nwhich will also appear in our model. Then we could resolve these by passing from Poisson brackets to corresponding Dirac brackets (see [85] and refs. therein for more discussion) after which both (2.16) and (3.9) can be considered as satisfied in the strong sense.  \nActually this stage can be overcame by introducing from the very beginning the so-called covariant momenta  \nd = w α P α , ¯ d = ¯ w α ¯ P α , d (0) = ¯ w α P α -w α ¯ P α , (3.10)  \nwhich have the following Poisson brackets with spinor frame variables  \n[  \n[ d , w α ] PB = 0 , [ d , ¯ w α ] PB = -w α , [ ¯ d , w α ] PB = -¯ w α , [ ¯ d , ¯ w α ] PB = 0 , d (0) , w α ] PB = w α , [ d (0) , ¯ w α ] PB = -¯ w α . (3.11)  \nIt is easy to check that these covariant momenta have vanishing Poisson brackets with the constraints (2.16). Hence, if using only the covariant momenta, we can consider the constraint (2.16) to be satisfied in the strong sense.  \nNotice that (3.11) imply the following brackets of the covariant momenta with moving frame vectors  \n[ d , u (0) a ] PB = -u a , [ d , u a ] PB = 0 , [ d , ¯ u a ] PB = -2 u (0) a , [ ¯ d , u (0) a ] PB = -¯ u a , [ ¯ d , u a ] PB = -2 u (0) a , [ ¯ d , ¯ u a ] PB = 0 , [ d (0) , u (0) a ] PB = 0 , [ d (0) , u a ] PB = 2 u a , [ d (0) , ¯ u a ] PB = -2¯ u a . (3.12)  \nOn the Poisson brackets, the covariant momenta represent the algebra so (1 , 2) /similarequal su (1 , 1)  \n[ d (0) , d ] PB = 2 d , [ d (0) , ¯ d ] PB = -2 ¯ d , [ d , ¯ d ] PB = d (0) .  \n(3.13)  \nTo find the expression for canonical Hamiltonian (3.3) in terms of Lagrangian and covariant momenta we start from the standard d τH 0 = d w α ¯ P α + d ¯ w α P α + ... -L , substitute there the expression for the derivatives of w α and ¯ w α in (2.18) valid when constraint (2.16) is satisfied, and use (3.10) arriving at  \nd τH 0 = ia d (0) + if ¯ d -i ¯ f d + ... -d τL ⇔ H 0 = ia τ d (0) + if τ ¯ d -i ¯ f τ d + ... -d τL . (3.14)  \nThis relation makes transparent that the covariant momenta can be identified with derivatives of the Lagrangian with respect to Cartan forms  \nd (0) = -i ∂L ∂a τ , d = i ∂L ∂ ¯ f τ , ¯ d = -i ∂L ∂f τ . (3.15)', '3.3 Simplifying Poisson/Dirac brackets in the sector of matrix fields': 'The canonical momentum of the bosonic matrix fields are  \n∂  \n∂  \n˙  \nZ  \nL  \ni  \nj  \n≈  \n1  \nµ  \n∂  \n∂  \n˙  \n¯  \nL  \nP  \nj  \ni  \n≈  \n∂ L ∂ ˙ ¯ Z i j ≈ 1 µ 6 P j i , ∂ L ∂ ˙ P j i ≈ 0 . (3.16)  \nThese relations have the form of resolved pairs of second class constraints allowing to exclude the variables P j i , ¯ P j i and their momenta from the consideration.  \nWe however prefer an equivalent but more convenient way consisting in removing the momenta conjugate to P j i and ¯ P j i and replacing the momenta conjugate to Z = Z j i and to ¯ Z = ¯ Z j i by ¯ P j i and P j i . Then the nonvanishing Poisson brackets for new phase space bosonic matrix variables (actually these are Dirac brackets) are 11  \n[ ¯ P j i , Z k l ] PB = -µ 6 ( δ l i δ j k -1 N δ j i δ l k ) , [ P j i , ¯ Z k l ] PB = -µ 6 ( δ l i δ j k -1 N δ j i δ l k ) . (3.17)  \nSimilar shortcut approach is usually applied when constructing the Hamiltonian mechanics of fermionic fields with canonical kinetic term in which case the conjugate of fermionic field is identified with its momentum. In our case the fermionic matrix fields have such a kinetic term.  \nThe canonical momentum for the fermionic matrix field Ψ = Ψ j i and its Hermitian conjugate ¯ Ψ = ¯ Ψ j i are given by  \nΠ ¯ Ψ = ∂ L ∂ ˙ ¯ Ψ ≈ -i 8 µ 6 Ψ , ¯ Π Ψ = ∂ L ∂ ˙ Ψ ≈ -i 8 µ 6 ¯ Ψ . (3.18)  \nThe canonical Poisson brackets with nonvanishing r.h.s. are  \n{ Π i j , ¯ Ψ k l } PB = { Ψ i j , ¯ Π k l } PB = -( δ l i δ j k -1 N δ j i δ l k ) (3.19)  \n6  \n¯  \nP  \nj  \ni  \n,  \n0  \n,  \nwhere for shortness we write Π i j := (Π ¯ Ψ ) i j and ¯ Π i j := ( ¯ Π ¯ Ψ ) i j . Using these we find that (3.18) are second class constraints. As they are not written in an explicitly resolved form, to treat them as equations which are satisfied in the strong sense, we have to replace Poisson brackets by Dirac brackets  \n[ ..., ... } D = [ ..., ... } PB -4 iµ 6 [ ..., (Π ¯ Ψ + i 8 µ 6 Ψ ) j i } PB [( ¯ Π Ψ + i 8 µ 6 ¯ Ψ ) i j , ... } PB --4 iµ 6 [ ..., ( ¯ Π Ψ i 8 µ 6 ¯ Ψ ) j i } PB [(Π ¯ Ψ + i 8 µ 6 Ψ ) i j , ... } PB . (3.20)  \nThen the complex fermionic matrix field and its Hermitean conjugate become also canonically conjugate with this Dirac brackets,  \n{ Ψ i j , ¯ Ψ k l } D = -4 iµ 6 ( δ l i δ j k -1 N δ j i δ l k ) , { Ψ i j , Ψ k l } D = 0 , { ¯ Ψ i j , ¯ Ψ k l } D = 0 . (3.21)  \nBelow, for simplicity, and also taking into account the custom to impose relations like (3.21) as canonical, we will refer to these Dirac brackets as Poisson brackets and write (3.21) as  \n{ Ψ i j , ¯ Ψ k l } PB = -4 iµ 6 ( δ l i δ j k -1 N δ j i δ l k ) , { Ψ i j , Ψ k l } PB = 0 , { ¯ Ψ i j , ¯ Ψ k l } PB = 0 . (3.22)  \nThis is also convenient because we have already used implicitly the passage to Dirac brackets in the spinor moving frame sector of our model.', '3.4 Primary constraints': 'With our Lagrangian (2.45), computing the canonical momenta (3.1) and covariant momenta (3.15) of the fields from center of mass sector and of the gauge field, we find the following set of primary constraints of the system  \nΦ a := p a -mu (0) a -2 µ 6 H m u (0) a ≈ 0 , (3.23)  \nd α := ¯ Π α + ip a ( γ a ¯ θ ) α -m ¯ θ α -i µ 6 √ m w α ¯ ν ≈ 0 , ¯ ν = tr( ¯ Ψ ¯ P + Ψ [ Z , ¯ Z ]) , (3.24)  \n¯ d α := Π α + ip a ( γ a θ ) α + mθ α -i µ 6 √ m ¯ w α ν ≈ 0 , ν = tr( Ψ P + ¯ Ψ [ Z , ¯ Z ]) , (3.25)  \nd ≈ 0 , (3.26)  \n¯  \nd  \n≈  \n0  \n,  \n(3.27)  \nU := d (0) -2 µ 6 B ≈ 0 , B := tr ( ¯ PZ -P ¯ Z + i 8 Ψ ¯ Ψ ) (3.28)  \nP A := ∂L ∂ ˙ A τ ≈ 0 . (3.29)  \nand  \nAlthough the latter relation is the only primary constraints imposed just on the matrix variables, the matrix fields also contribute to the constraints that appear from the definition of momenta of the center of mass variables through H of (2.39) in (3.23), through  \n¯ ν = tr( ¯ Ψ ¯ P + Ψ [ Z , ¯ Z ]) and ν = tr( Ψ P + ¯ Ψ [ Z , ¯ Z ]) (3.30)  \nin (3.24) and (3.25), as well as through  \nB := tr ( ¯ PZ -P ¯ Z + i 8 Ψ ¯ Ψ ) (3.31)  \nin (3.28).', '3.5 Canonical Hamiltonian and secondary constraints': 'Canonical Hamiltonian of our system is defined by the Legendre transform of the Lagrangian, which we write already in terms of SO(1 , 2) / SO(2) Cartan forms and covariant momenta (see (3.14)) as  \nd τH 0 = d x a p a +d θ α ¯ Π α +d ¯ θ α Π α + ia ˜ d (0) + if ¯ d -i ¯ f d +  \n+ 1 µ 6 tr(d Z ¯ P ) + 1 µ 6 tr(d ¯ ZP ) -i 8 µ 6 tr(d Ψ ¯ Ψ ) -i 8 µ 6 tr(d ¯ ΨΨ ) + tr(d AP A ) -L mD0 . (3.32)  \nFor our Lagrangian (2.45), after extracting the primary constraints, (3.32) reduces to  \nH 0 = E a τ Φ a + ˙ θ α d α + ˙ ¯ θ α ¯ d α + ia τ U + if τ ¯ ˜ d -i ¯ f τ ˜ d +tr(d AP A ) + tr( A τ G ) ≈ tr( A τ G ) , (3.33)  \nwhere G is N × N anti-Hermitian traceless matrix  \nG := 1 µ 6 ( [ ¯ Z , P ] + [ Z , ¯ P ] -i 4 { Ψ , ¯ Ψ } ) . (3.34)  \nThe last expression valid in the weak sense can be used to check whether the preservation of the primary constraints in the evolution, as defined by Hamiltonian equations (3.4), produce secondary constraints. In our case this happens: the requirement of preservation of the primary constraint (3.29), ˙ P A = [ P A , H 0 ] ≈ 0, leads to the secondary constraint  \nG = 1 µ 6 ( [ ¯ Z , P ] + [ Z , ¯ P ] -i 4 { Ψ , ¯ Ψ } ) ≈ 0 . (3.35)  \nThis reveals the role of the 1d gauge field A τ as a Lagrange multiplier for the constraint (3.35) which we will call the Gauss law keeping in mind its counterpart in higher dimensional supersymmetric gauge theories. Furthermore, as it can be shown that the constraint (3.29) is of the first class and generates a gauge symmetry consisting in an arbitrary shift of A τ , we will allow ourself to streamline the presentation by just skipping the conjugate variables A τ and P A from the consideration below 12 .  \nThen, the Gauss law (3.35) results in a vanishing of the canonical Hamiltonian in the weak sense  \nH 0 ≈ 0 . (3.36)', '3.6 First class constraints and gauge symmetries': "Eq. (3.36) implies that the total Hamiltonian, as introduced by Dirac in [84], for our system is equal to a linear combination of the constraints,  \nH = b a Φ a + κ α d α + ¯ κ α ¯ d α + ik ¯ d -i ¯ k d + ik (0) ( d (0) -2 µ 6 B ) +tr( YG ) . (3.37)  \nThe coefficients in this expression, the functions of proper time called Lagrange multipliers 13 , are restricted by the conditions of the preservation of all the constraints under the evolution [84],  \nd d τ (constraints) = [constraints , H ] PB ≈ 0 (3.38)  \n( cf. (3.4)) 14 . The solution of Eqs. (3.38) expresses the original Lagrange multipliers in terms of fewer fields serving as parameters for the so-called first class constraints, which generate gauge symmetries of the dynamical system under consideration on the Poisson brackets.  \nTo specify the system of equations, it is useful first to obtain the algebra on the constraints on the Poisson brackets. Let us begin from the Gauss law constraint (3.34) which is a(n anti-Hermitian traceless) matrix, G = ( G i j ), so that the brackets between different matrix elements of this matrix can be, and some are indeed, nonvanishing. Actually these provide a representation of su ( N ) algebra. To write it in a compact and comprehensive form, it is convenient to introduce 'reference' traceless matrices Y and Y ' and write the brackets of the traces of G with these matrices. In such a way we find the su ( N ) algebra in the form  \n[tr( YG ) , tr( Y ' G )] PB = tr ( [ Y , Y ' ] G ) . (3.39)  \nFurthermore, G has vanishing Poisson brackets with all other constraint, which just reflects their SU( N ) invariance (see Appendix A for technical details). Indeed G is thus the first class constraint which generates the SU( N ) gauge symmetry of the model.  \nThe algebra of the other constraints, given by Eqs. (3.23)-(3.29), is described by the following nonvanishing Poisson brackets (see Appendix A):  \n[ d , d (0) -2 µ 6 B ] PB = -2 d , [ d , Φ a ] PB = ( m + 2 µ 6 H m ) u a , (3.40)  \n[ d , ¯ d α ] PB = i µ 6 √ m w α ν , (3.41)  \n[ d , ¯ d ] PB = d (0) , (3.42)  \n[ ¯ d , d (0) -2 µ 6 B ] PB = 2 ¯ d , [ ¯ d , Φ a ] PB = ( m + 2 µ 6 H m ) ¯ u a , (3.43)  \n[ ¯ d , d α ] PB = i µ 6 √ m ¯ w α ¯ ν , (3.44)  \n[Φ a , d α ] PB = 2 i m √ m u (0) a w α tr( Ψ G ) , (3.45)  \n[Φ a , ¯ d α ] PB = -2 i m √ m u (0) a ¯ w α tr( ¯ Ψ G ) , (3.46)  \nand  \n{ d α , d β } PB = -8 i m w α w β tr( ¯ ZG ) , (3.47)  \n{ ¯ d α , ¯ d β } PB = 8 i m ¯ w α ¯ w β tr( ZG ) , (3.48)  \nd  \n{  \nα  \n,  \nd  \nβ  \n}  \nPB  \n=  \n-  \n2  \niP  \na  \nγ  \na  \nαβ  \n+2  \nm/epsilon1  \nαβ  \n+  \n4  \ni  \nµ  \n6  \nw  \nα  \n¯  \nw  \nβ  \nH  \nm  \n= -2 i Φ a γ a αβ -4 iw β ¯ w α ( m + 1 µ 6 H m ) . (3.49)  \nUsing this algebra, we find that the preservation of the constraints in the evolution, described by Eq. (3.38) with Hamiltonian (3.37), requires the Lagrange multipliers to obey a system of equations which is solved by  \nk = 0 , ¯ k = 0 , (3.50)  \nb a u a = 0 , b a ¯ u a = 0 ⇒ b a = bu (0) a , (3.51)  \n¯ κ β w β = 0 ⇒ ¯ κ β = ¯ κw β , (3.52)  \nβ  \nκ  \n¯  \nw  \nβ  \n= 0  \n⇒  \nβ  \nκ  \n=  \nκ  \n¯  \nw  \nβ  \n,  \n¯  \n(3.53)  \nso that the total Hamiltonian reads  \nH = b u a (0) Φ a + κ ¯ w α d α + ¯ κw α ¯ d α + ik (0) ( d (0) -2 µ 6 B ) +tr( YG ) . (3.54)  \nThe arbitrary Lagrange multipliers b , κ , ¯ κ , k (0) and Y j i obeying Y i i = 0 reflect the gauge symmetries of our model: reparametrization, local worldline supersymmetry ( κ -symmetry), U(1) and SU( N ) symmetries. In (3.54) they are multiplied by the constraints generating these symmetries,  \nΦ (0) := p a u a (0) -( m + 2 µ 6 H m ) ≈ 0 , (3.55)  \n¯ w α d α := ¯ w α ( ¯ Π α + iP a ( γ a ¯ θ ) α -m ¯ θ α ) + 1 µ 6 1 √ m ¯ ν ≈ 0 , (3.56)  \nw α ¯ d α := w α (Π α + iP a ( γ a θ ) α + mθ α ) -1 µ 6 1 √ m ν ≈ 0 , (3.57)  \nU (0) := d (0) -2 µ 6 B ≈ 0 , (3.58)  \nwith B defined in (3.31), and (3.35).", '3.7 Second class constraints': 'The remaining constraints are the second class ones. They can be collected in conjugate pairs  \nd ≈ 0 , ¯ d ≈ 0 , (3.59)  \nΦ := u a Φ a = u a p a ≈ 0 , ¯ Φ := ¯ u a Φ a = ¯ u a p a ≈ 0 , (3.60)  \nw α d α = w α ( ¯ Π α + ip a ( γ a ¯ θ ) α -m ¯ θ α ) ≈ 0 , ¯ w α ¯ d α = ¯ w α (Π α + ip a ( γ a θ ) α + mθ α ) ≈ 0 , (3.61)  \nas can be seen from their Poisson brackets, particularly  \n[ d , ¯ Φ] PB = -2Φ (0) -2 ( m + 2 µ 6 H m ) ≈ -2 ( m + 2 µ 6 H m ) , (3.62)  \n[ ¯ d , Φ] PB = -2Φ (0) -2 ( m + 2 µ 6 H m ) ≈ -2 ( m + 2 µ 6 H m ) , (3.63)  \n{ w α d α , ¯ w α ¯ d α } PB = -2 i Φ (0) -4 i ( m + 1 µ 6 H m ) ≈ -4 i ( m + 1 µ 6 H m ) . (3.64)', '3.8 The algebra of first and second class constraints': "Notice that there are other nonvanishing brackets of the second class constraints:  \n[ d , ¯ d ] PB = d (0) = ( d (0) -2 µ 6 B ) + 2 µ 6 B ≈ 2 µ 6 B , (3.65)  \n[ d , ¯ w α ¯ d α ] PB = -w α ¯ d α -1 µ 6 √ m ν ≈ -1 µ 6 √ m ν , (3.66)  \n[ ¯ d , w α d α ] PB = -¯ w α d α + 1 µ 6 √ m ¯ ν ≈ 1 µ 6 √ m ¯ ν . (3.67)  \nThe algebra of the I and II class constraints is resumed in the Table 1 where we used the notation  \nM ( H ) := m + 1 µ 6 H m . (3.68)  \nWe did not include in this Table 1 the first class constraints G i j (3.34), as these nonvanishing brackets with themselves, (3.39).  \n(This should not be confused with M ( H ) from [60] which was discussed in the first sections of this paper). The appearance of the positively definite additive contributions M ( H ) or M (2 H ) in the right hand side of the Poisson brackets clearly indicates the pairs of fermionic and bosonic second class constraints.  \nThe above algebra of constraints is complicated enough to make difficult to proceed with the BRST quantization along the scheme of 'abelization' of the second class constraints [98, 99]. The technical problem comes, in particular, from the non-canonical form of the bosonic second class constraints algebra in which, besides (3.62), (3.63) and (3.64), also the brackets (3.65), (3.66) and (3.67) do not vanish in the weak sense.  \n<!-- image -->  \nTable 1 . Algebra of the first and second class constraints on the Poisson brackets of the simplest 3D mD0 system; M ( H ) := m + 1 µ 6 H m is defined in (3.68).  \nFurthermore, when trying to apply the Gupta-Bleuler quantization scheme in its canonical forms, with this algebra of constraints we arrived (in supercoordinate representation) at a system of equations which has only trivial solution: we discuss this in the simplest case of vanishing all the matrix and fermionic fields in Appendix B. This is the only example known to the authors where the Gupta-Bleuler simplified method of quantization of dynamical systems fails.  \nHence the only option which remains, which is consistent and allows to overcome difficulties, is to use the Dirac brackets procedure in the bosonic sector of our dynamical system. Furthermore, this procedure can be significantly simplified, actually reduced to the explicit resolution of the bosonic second class constraints, after we pass to the so-called analytical basis in the center of mass sector of the configuration superspace of our system. We will make this change of coordinate basis in the next section.", '4.1 Constraints in the analytical basis': 'Let us begin by noticing that the center of mass sector of the configurational space of our dynamical system can be identified with the so-called Lorentz harmonic superspace, an extended enlarged superspace with coordinates 15  \nΣ (3+3 | 4) = { ( x a , θ α , ¯ θ α , w α , ¯ w α ) } =: { ( Z M , w α , ¯ w α ) } , ¯ w α w α = i . (4.1)  \nThe complete configuration superspace includes also 2( N 2 -1) bosonic and ( N 2 -1) fermionic matrix coordinates, and its basis can be chosen as  \nΣ (4+2 N 2 | 3+ N 2 ) = { ( x a , θ α , ¯ θ α , w α , ¯ w α ; Z , ¯ Z , Ψ ) } =: { ( Z M , w α , ¯ w α ; Z , ¯ Z , Ψ ) } . . (4.2)  \nThis implies that ¯ Ψ is considered to be the momentum conjugate to Ψ .  \nClearly, there is a possibility to choose another coordinate basis. In our case it is convenient to make a change of coordinates in the center of mass sector (4.1) of the whole configurational superspace choosing in it the analytical coordinate basis  \nΣ (3+3 | 4) = { (x (0) , x A , ¯ x A , θ w , θ ¯ w , ¯ θ w , ¯ θ ¯ w ; w α , ¯ w α ) } =: { (Z ( M ) An , w α , ¯ w α , Z , ¯ Z , Ψ , ¯ Ψ ) } (4.3)  \nwith  \nx (0) := x a u (0) a , (4.4)  \nx A := x -2 iθ w ¯ θ w , x := x a u a , (4.5)  \n¯ x A := ¯x + 2 iθ ¯ w ¯ θ ¯ w , ¯ x := x a ¯ u a , (4.6)  \nθ w := θ α w α , θ ¯ w := θ α ¯ w α , (4.7)  \n¯ θ w := ¯ θ α w α , ¯ θ ¯ w := ¯ θ α ¯ w α (4.8)  \n(see Appendix C, particularly Eqs. (C.5)-(C.7), cf. [86, 89, 90]).  \nIn this analytical basis our Lagrangian form, L = d τL with L from (2.45), reads  \nL = dx (0) ( m + 2 µ 6 H m ) -ifm [ ¯ x A ( 1 + 2 µ 6 H m 2 ) -4 i µ 6 H m 2 θ ¯ w ¯ θ ¯ w + i µ 6 1 m √ m θ ¯ w ¯ ν ] + + i ¯ fm [ x A ( 1 + 2 µ 6 H m 2 ) + 4 i µ 6 H m 2 θ w ¯ θ w + i µ 6 1 m √ m ¯ θ w ν ] --2 i ( d θ ¯ w -iaθ ¯ w ) ¯ θ w ( m + 1 µ 6 H m ) -2 i ( d ¯ θ w + ia ¯ θ w ) θ ¯ w ( m + 1 µ 6 H m ) --2 i (d θ w + iaθ w ) ( 1 µ 6 H m ¯ θ ¯ w -1 µ 6 1 2 √ m ¯ ν ) -2 i ( d ¯ θ ¯ w -ia ¯ θ ¯ w ) ( 1 µ 6 H m θ w -1 µ 6 1 2 √ m ν ) + + 1 µ 6 tr ( ¯ P D τ Z + P D τ ¯ Z -i 8 D τ Ψ ¯ Ψ + i 8 Ψ D τ ¯ Ψ ) . (4.9)  \nIt is convenient to introduce the notation  \n˜ H = 1 µ 6 H m , ˜ ν = 1 µ 6 1 2 √ m ν = ( ¯ ˜ ν ) ∗ (4.10)  \nand to write the above Lagrangian in a bit more compact form  \nL = dx (0) ( m +2 ˜ H ) -if [ ¯ x A ( m +2 ˜ H ) -4 i ˜ H θ ¯ w ¯ θ ¯ w +2 iθ ¯ w ¯ ˜ ν ] + + i ¯ f [ x A ( m +2 ˜ H ) + 4 i ˜ H θ w ¯ θ w +2 i ¯ θ w ˜ ν ] --2 i ( d θ ¯ w -iaθ ¯ w ) ¯ θ w ( m + ˜ H ) -2 i ( d ¯ θ w + ia ¯ θ w ) θ ¯ w ( m + ˜ H ) --2 i (d θ w + iaθ w ) ( ˜ H ¯ θ ¯ w -¯ ˜ ν ) -2 i ( d ¯ θ ¯ w -ia ¯ θ ¯ w ) ( ˜ H θ w -˜ ν ) + + 1 µ 6 tr ( ¯ P D τ Z + P D τ ¯ Z -i 8 D τ Ψ ¯ Ψ + i 8 Ψ D τ ¯ Ψ ) .  \nThen the canonical Hamiltonian is defined by  \nd τH 0 = dx (0) p (0) +dx A ¯ p +d¯x A p +d θ w Π θ w +d θ ¯ w Π θ ¯ w +d ¯ θ w ¯ Π ¯ θ w +d ¯ θ ¯ w ¯ Π ¯ θ ¯ w + ia ˜ d (0) + if ¯ ˜ d -i ¯ f ˜ d +  \n+ 1 µ 6 tr(d Z ¯ P ) + 1 µ 6 tr(d ¯ ZP ) -i 8 µ 6 tr(d Ψ ¯ Ψ ) -i 8 µ 6 tr(d ¯ ΨΨ ) + tr(d AP A ) -L mD0  \n.  \nIn (4.12) the momenta for the bosonic coordinate functions are related to their central basis counterparts by  \np (0) := u a (0) p a , p := -1 2 u a p a , ¯ p := -1 2 ¯ u a p a (4.13)  \nand have the nonvanishing Poisson brackets  \n[ p (0) , x (0) ] PB = -1 , [ p, ¯ x A ] PB = -1 , [¯ p, x A ] PB = -1 . (4.14)  \nThe momenta conjugate to the fermionic coordinate functions of the analytical basis obeying  \n{ Π θ w , θ w } PB = -1 , { Π θ ¯ w , θ ¯ w } PB = -1 , { ¯ Π ¯ θ w , ¯ θ w } PB = -1 , { ¯ Π ¯ θ ¯ w , ¯ θ ¯ w } PB = -1 (4.15)  \nare related to their central basis cousins by a bit more complicated relation  \nΠ θ w := -i ¯ w α Π α +2 i ¯ θ w ¯ p , Π θ ¯ w := iw α Π α -2 i ¯ θ ¯ w p , (4.16)  \n¯ Π ¯ θ w := -i ¯ w α ¯ Π α -2 iθ w ¯ p , ¯ Π ¯ θ ¯ w := iw α ¯ Π α +2 iθ ¯ w p .  \nThe covariant momenta of the analytical basis, distinguished by tilde from their central basis counterparts, obey  \n[ ˜ d (0) , Z ( M ) An ] PB = 0 , [ ˜ d , Z ( M ) An ] PB = 0 , [ ¯ ˜ d , Z ( M ) An ] PB = 0 , (4.17)  \nwhile  \n[ d (0) , Z M ] PB = 0 , [ d , Z M ] PB = 0 , [ ¯ d , Z M ] PB = 0 . (4.18)  \nEq. (4.12) also encodes the definitions of the canonical momenta as derivatives of the Lagrangian with respect to velocities, (3.1), as well as of the covariant momenta as derivatives of the Lagrangian with respect  \n(4.11)  \n(4.12)  \nto Cartan forms, (3.15). Calculating these with our Lagrangian (4.11) we find the primary constraints ( cf. (3.23)-(3.29))  \nΦ (0) := p (0) -( m +2 ˜ H ) ≈ 0 , (4.19)  \n-1 2 Φ := p ≈ 0 , (4.20)  \n-1 2 ¯ Φ := ¯ p ≈ 0 , (4.21)  \n˜ d +x A ( m +2 ˜ H ) + 4 i ˜ H θ w ¯ θ w +2 i ¯ θ w ˜ ν ≈ 0 , (4.22)  \n¯ ˜ d +¯x A ( m +2 ˜ H ) -4 i ˜ H θ ¯ w ¯ θ ¯ w +2 iθ ¯ w ¯ ˜ ν ≈ 0 , (4.23)  \n˜ U := ˜ d (0) -4 iθ ¯ w ¯ θ w ( m + ˜ H ) + 4 iθ w ¯ θ ¯ w ˜ H 2 iθ w ¯ ˜ ν +2 i ¯ θ ¯ w ˜ ν -2 ˜ B ≈ 0 , (4.24)  \nwhere ˜ B := 1 µ 6 B := 1 µ 6 tr ( ¯ PZ -P ¯ Z + i 8 Ψ ¯ Ψ ) , (4.25)  \nd w := Π θ w -2 i ( ¯ ˜ ν -¯ θ ¯ w ˜ H ) ≈ 0 , (4.26)  \nd ¯ w := Π θ ¯ w +2 i ( m + ˜ H ) ¯ θ w ≈ 0 , (4.27)  \n¯ d w := ¯ Π ¯ θ w +2 i ( m + ˜ H ) θ ¯ w ≈ 0 , (4.28)  \n¯ d ¯ w := ¯ Π ¯ θ ¯ w -2 i ( ˜ ν -θ w ˜ H ) ≈ 0 , (4.29)  \nand (3.29),  \nP A := ∂L ∂ ˙ A τ ≈ 0 . (4.30)  \nExactly as in the central basis calculations, the requirement of preservation of this latter leads to the secondary constraint (3.35),  \nG = 1 µ 6 ( [ ¯ Z , P ] + [ Z , ¯ P ] -i 4 { Ψ , ¯ Ψ } ) ≈ 0 (4.31)  \nand, as a result, the canonical Hamiltonian vanishes in the weak sense  \nH 0 ≈ 0 . (4.32)  \nThis means that the total Hamiltonian is given by linear combination of the constraints or, being more precise, of the first class constraints.  \nThe set of these first class constraints includes Eqs. (4.19), (4.26), (4.29),  \nΦ (0) ≈ 0 , d w ≈ 0 , ¯ d ¯ w ≈ 0 , (4.33)  \nthe following sum of (4.24) with the linear combination of the second class constraints (4.20), (4.21), (4.27) and (4.28),  \n˜ ˜ U (0) := ˜ U (0) -2x A ¯ p +2¯x A p -¯ θ w ¯ d w + θ ¯ w d ¯ w = = ˜ d (0) -2x A ¯ p +2¯x A p -¯ θ w ¯ Π ¯ θ w + θ ¯ w Π θ ¯ w +4 iθ w ¯ θ ¯ w ˜ H2 iθ w ¯ ˜ ν +2 i ¯ θ ¯ w ˜ ν -2 ˜ B ≈ 0 (4.34)  \nas well as (4.30) (just expressing the pure gauge nature of the 1d gauge field) and the Gauss constraint (4.31). The remaining constraints are of the second class.  \nBelow we will not need the complete knowledge of the algebra of the constraints on the Poisson brackets. Let us just notice that, instead of calculating it directly, we can use their relation with their central basis counterparts and knowledge of the algebra of these latter. In the case of first class constraint (3.55) and second class constraints (3.60), this relation is just a coincidence,  \nΦ (0) := p a u (0) -( m +2 ˜ H ) = p (0) -( m +2 ˜ H ) , (4.35)  \nΦ := Φ a u a = -2 p , (4.36)  \n¯ Φ := Φ a ¯ u a = -2¯ p . (4.37)  \nThe same is true for the Gauss constraints (4.31) and for (4.30) while for the remaining constraints the relations are nontrivial:  \nU (0) := d (0) -2 ˜ B = ˜ d (0) -2x A ¯ p +2¯x A p -θ w Π θ w -¯ θ w ¯ Π ¯ θ w + θ ¯ w Π θ ¯ w + ¯ θ ¯ w ¯ Π ¯ θ ¯ w -2 ˜ B = = ˜ ˜ U (0) -θ w d w + ¯ θ ¯ w ¯ d ¯ w , (4.38)  \n¯ w α d α := ¯ w α ( ¯ Π α + ip a ( γ a ¯ θ ) α -m ¯ θ α ) + 2 ¯ ˜ ν = id w -Φ (0) ¯ θ ¯ w , (4.39)  \nw α ¯ d α := w α (Π α + ip a ( γ a θ ) α + mθ α ) -2˜ ν = -i ¯ d ¯ w +Φ (0) θ w , (4.40)  \nw α d α := w α ( ¯ Π α + ip a ( γ a ¯ θ ) α -m ¯ θ α ) = -id ¯ w +4 p ¯ θ ¯ w +Φ (0) ¯ θ w , (4.41)  \n¯ w α ¯ d α := ¯ w α (Π α + ip a ( γ a θ ) α + mθ α ) = i ¯ d w -4 pθ w -Φ (0) θ ¯ w , (4.42)  \nd = ˜ d +2(x (0) + iθ ¯ w ¯ θ w + iθ w ¯ θ ¯ w ) p +(x A +2 iθ w ¯ θ w ) p (0) + ¯ θ w ¯ Π ¯ θ ¯ w + θ w Π θ ¯ w , (4.43)  \n¯ d = ¯ ˜ d +2(x (0) -iθ ¯ w ¯ θ w -iθ w ¯ θ ¯ w )¯ p +(¯x A -2 iθ ¯ w ¯ θ ¯ w ) p (0) + θ ¯ w Π θ w + ¯ θ ¯ w ¯ Π ¯ θ w . (4.44)  \nUsing these expressions, the algebra of the constraints in the analytical basis can be read off Table 1 and Eqs. (3.62)-(3.67). Below we will need a part of this algebra which is resumed (essentially) in Table 2 below.', '4.2 Resolving bosonic second class constraints': "Actually, as we have already stated, we do not need to know the complete algebra of the constraints in detail. Instead we can observe that the bosonic second class constraints in the analytical basis, Eqs. (4.21), (4.22) and (4.20), (4.23), are explicitly resolved (in terminology of Dirac [84]). This implies that in these pairs of constraints one element, (4.22) and (4.23), can be rewritten as expression for coordinate x A and ¯x A , respectively, in terms of other variables, while the other, conjugate element, (4.21) and (4.20), respectively, implies vanishing of the momenta conjugate to this dependent coordinate. In such cases, the procedure of changing the Poisson brackets by Dirac brackets can be replaced by just setting momenta to zero in the strong sense  \n¯ p = 0 , p = 0 , (4.45)  \nand substituting x A and ¯x A in all places when these appear by their expressions obtained by solving (4.22) and (4.23),  \nx A = -˜ d +4 i ˜ H θ w ¯ θ w +2 i ¯ θ w ˜ ν m +2 ˜ H , ¯ x A = -¯ ˜ d -4 i ˜ H θ ¯ w ¯ θ ¯ w +2 iθ ¯ w ¯ ˜ ν m +2 ˜ H . (4.46)  \nFortunately in our case x A and ¯x A do not contribute to other constraints so that the above prescription is tantamount to the reduction of configuration space by omitting the directions corresponding to these coordinates. The same effect can be reached by omitting (4.22) and (4.23) from the set of constraints and thus converting (4.21) and (4.20) into the first class constraints. Then x A and ¯x A can be gauged away using the symmetries generated by (4.21) and (4.20), thus arriving at the same result as described above.  \nAs far as the fermionic second class constraints are concerned, since the canonically conjugate constraints in this case are also complex conjugate, we can quantize these using the Gupta-Bleuler method. At the classical level this can be reflected by omitting one of the conjugate constraint, say d ¯ w , thus converting its conjugate, ¯ d w into an effectively first class constraint.  \n<!-- image -->  \nTable 2 . Closed algebra of the 'effective first class constraints' of the mD0 system. For compactness, we have omitted the lines and columns corresponding to the Gauss constraint (4.31) (which has only one nonvanishing element on their crossing) and to the first class constraint (4.30) (which are line and column of zeros), and have also renormalized the Gauss constraint and the fermionic matrix fields as follows: ˜ G := 1 m G , ˜ Ψ := 1 √ m Ψ , ¯ ˜ Ψ := 1 √ m ¯ Ψ .  \nThe essential part of the algebra of the effective first class constraints thus obtained is collected in Table 2, where we omitted the lines and columns with the constraints (4.30), which have vanishing brackets with all others, and with the Gauss constraint matrix (4.31) which has nonvanishing brackets only with  \nitself, (3.39). To make the r.h.s's of the brackets shorter, we use in Table 2 renormalized expressions for Gauss constraint and for the fermionic matrix fields:  \n˜ G := 1 m G ˜ Ψ := 1 √ m Ψ ¯ ˜ Ψ := 1 √ m ¯ Ψ . (4.47)  \nNotice that all the terms in the r.h.s. of the brackets resumed in Table 2 are proportional to the Gauss law constraints and hence vanish when this is taken into account.", '5 Quantization of multiple D 0 -brane system and its field theory equations': "Quantization implies the replacement of the phase space variables by operators the commutators and anticommutators of which can be determined through the Poisson brackets of their classical prototypes by the Dirac prescription  \n[ ..., ... } PB ↦→ 1 i [ ..., ... } . (5.1)  \nIn the supercoordinate representation the bosonic and fermionic configuration space variables maintain their (quasi-)classical description, i.e. they are numerical ( c -number) or Grassmann algebra ( a -number) valued, respectively, while the momenta are represented by differential operators such that the Dirac prescription (5.1) holds.  \nIn particular, in our case the bosonic and fermionic center of mass coordinate functions and their momenta are replaced by corresponding coordinates and differential operator as  \nˆ x (0) = x (0) , ˆ x A = x A , ˆ ¯ x A = ¯x A , ˆ p (0) = -i∂ x (0) , ˆ ¯ p A = -i∂ x A , ˆ p A = -i∂ ¯ x A , (5.2)  \nand  \nˆ θ w = θ w , ˆ θ ¯ w = θ ¯ w , ˆ ¯ θ w = ¯ θ w , ˆ ¯ θ ¯ w = ¯ θ ¯ w , ˆ Π θ w = -i∂ θ w , ˆ Π θ ¯ w = -i∂ θ ¯ w , ˆ ¯ Π ¯ θ w = -i∂ ¯ θ w , ˆ ¯ Π ¯ w ¯ θ = -i∂ ¯ θ ¯ w , (5.3)  \nwhere we have used the notation  \n∂ x (0) = ∂ ∂ x (0) , ∂ x A = ∂ ∂ x A , ∂ ¯ x A = ∂ ∂ ¯ x A , ∂ θ w = ∂ ∂θ w , ∂ θ ¯ w = ∂ ∂θ ¯ w , ∂ ¯ θ w = ∂ ∂ ¯ θ w , ∂ ¯ θ ¯ w = ∂ ∂ ¯ θ ¯ w . (5.4)  \nFor the complex spinors frame variables the covariant momenta are expressed in terms of covariant derivatives  \nD (0) = ¯ w α ∂ ∂ ¯ w α -w α ∂ ∂w α , D = w α ∂ ∂ ¯ w α , ¯ D = ¯ w α ∂ ∂w α , (5.5)  \nso that  \nˆ w α = w α , ˆ ¯ w α = ¯ w α , ˆ d (0) = -i D (0) , ˆ d = -i D , ˆ ¯ d = -i ¯ D . (5.6)  \nFor the bosonic matrix fields the quantization in coordinate representation is standard  \nˆ  \nZ  \ni  \nj  \n=  \nZ  \ni  \nj  \n,  \nˆ  \n¯  \nZ  \ni  \nj  \n=  \n¯  \nZ  \ni  \nj  \n,  \nˆ ¯ P i j = -iµ 6 ∂ ∂ Z j i , ˆ P i j = -iµ 6 ∂ ∂ ¯ Z j i , (5.7)  \nwhile for the fermionic matrix variables obeying  \n{ ˆ Ψ i j , ˆ ¯ Ψ k l } = 4 µ 6 ( δ l i δ j k -1 N δ j i δ l k ) , { ˆ Ψ j i , ˆ Ψ l k } = 0 , { ˆ ¯ Ψ j i , ˆ ¯ Ψ l k } = 0 (5.8)  \nwe should use the counterpart of the so-called holomorphic representation for Heisenberg algebra in which the annihilation operator is represented by derivative with respect to (classical counterpart of the) annihilation operator,  \nˆ Ψ j i = Ψ j i , ˆ ¯ Ψ j i = 4 µ 6 ∂ ∂ Ψ i j . (5.9)  \nThen the quantum counterpart of the SYM supercurrents and Hamiltonian are represented by the following differential operators  \nˆ ˜ ν = 1 µ 6 1 2 √ m ˆ ν = 1 2 √ m tr ( -i Ψ ∂ ∂ Z +4[ Z , ¯ Z ] ∂ ∂ Ψ ) , (5.10)  \nˆ ¯ ˜ ν = 1 µ 6 1 2 √ m ˆ ¯ ν = 1 2 √ m tr ( -4 iµ 6 ∂ ∂ Z ∂ ∂ Ψ + 1 µ 6 Ψ [ Z , ¯ Z ] ) , (5.11)  \nˆ ˜ H = 1 µ 6 ˆ H m = 1 m tr ( -µ 6 ∂ ∂ Z ∂ ∂ ¯ Z +8 iµ 6 ¯ Z ∂ ∂ Ψ ∂ ∂ Ψ + 1 µ 6 [ Z , ¯ Z ] 2 -i 2 1 µ 6 Z ΨΨ ) , (5.12)  \nand the quantum operator generating the U(1) transformations of matrix variables is  \nˆ ˜ B = 1 µ 6 ˆ B = i tr ( ¯ Z ∂ ∂ ¯ Z -Z ∂ ∂ Z + 1 2 Ψ ∂ ∂ Ψ ) . (5.13)  \nThe state vector is represented by a function dependent on the configuration space coordinates 16 ,  \nΞ = Ξ ( x (0) , ¯ w,w ; θ w , θ ¯ w , ¯ θ w , ¯ θ ¯ w ; Z , ¯ Z ; Ψ ) . (5.14)  \nThe physical states are represented by the state vectors obeying equations obtained as quantum version of the effective first class constraints which read  \n[ -i∂ x (0) -( m +2 ˆ ˜ H )] Ξ = 0 , (5.15)  \n[ ∂ θ w +2 ( ˆ ¯ ˜ ν -¯ θ ¯ w ˆ ˜ H )] Ξ = 0 , (5.17)  \ni ˆ ˜ ˜ U (0) Ξ = ( D (0) -¯ θ w ∂ ¯ θ w + θ ¯ w ∂ θ ¯ w -4 θ w ¯ θ ¯ w ˆ ˜ H +2 θ w ˆ ¯ ˜ ν -2 ¯ θ ¯ w ˆ ˜ ν -2 i ˆ ˜ B q ) Ξ = 0 , (5.16)  \n[ ∂ ¯ θ ¯ w +2 ( ˆ ˜ ν -θ w ˆ ˜ H )] Ξ = 0 , (5.18)  \n[ ∂ ¯ θ w -2 ( m + ˆ ˜ H ) θ ¯ w ] Ξ = 0 , (5.19)  \nwhere ˆ ˜ ν , ˆ ¯ ˜ ν and ˆ ˜ H are given in (5.10), (5.11) and (5.12), and the quantum SU( N ) Gauss law constraint  \ni ˆ G j i Ξ = ( ¯ Z k i ∂ ∂ ¯ Z k j -¯ Z j k ∂ ∂ ¯ Z i k + Z k i ∂ ∂ Z k j -Z j k ∂ ∂ Z i k + Ψ k i ∂ ∂ Ψ k j -Ψ j k ∂ ∂ Ψ i k ) Ξ = 0 , (5.20)  \nNotice that when calculating the quantum generator of SU( N ) transformations G in (5.20) we have to use the 'qp' ordering for the commutators and anticommutators of matrices,  \n[ ¯ Z , P ] i j ↦→ -iµ 6 ( ¯ Z i k ∂ ∂ ¯ Z j k -¯ Z k j ∂ ∂ ¯ Z i k ) , [ Z , ¯ P ] i j ↦→ -iµ 6 ( Z i k ∂ ∂ Z j k -Z k j ∂ ∂ Z k i ) , { Ψ , ¯ Ψ } i j ↦→ 4 µ 6 ( Ψ k i ∂ ∂ Ψ k j -Ψ j k ∂ ∂ Ψ i k ) , (5.21)  \nas otherwise the expression for G will not be traceless as it should be. Thus, Eqs. (5.15), (5.16), (5.17), (5.18), (5.19), (5.20) define the (super)field theory of simplest 3D counterpart of multiple D0-brane system in the superspace with coordinate presented in (5.14).  \nIt is convenient to allow the state vector to depend also on additional complex bosonic coordinate x A and its c.c. ¯x A ,  \nΞ = Ξ ( x (0) , x A , ¯ x A , ¯ w,w ; θ w , θ ¯ w , ¯ θ w , ¯ θ ¯ w ; Z , ¯ Z ; Ψ ) , (5.22)  \nbut impose on it the additional constraints removing this dependence:  \n∂ ∂ x A Ξ = 0 , ∂ ∂ ¯ x A Ξ = 0 . (5.23)  \nBeing equivalent to original, this way of describing state vector makes manifest that the state vector can be considered as dependent on the central basis coordinates,  \nΞ = ˜ Ξ ( x a ; θ α , ¯ θ α ; ¯ w,w ; Z , ¯ Z ; Ψ ) , (5.24)  \nand this is useful to understand better the obtained (super)field theory equations, and write this in more conventional form at least in some particular cases, like N = 1 case discussed in the next subsection. In particular, in this simple case the equations can be rewritten in terms of derivatives and covariant derivatives of the central basis,  \n∂ a = u (0) a ∂ x (0) + u a ∂ x A + ¯ u a ∂ ¯ x A , (5.25)  \nD α = ∂ α + i ( γ a ¯ θ ) α ∂ a = w α ( ∂ θ w + i ¯ θ ¯ w ∂ x (0) ) + ¯ w α ( ∂ θ ¯ w +4 i ¯ θ ¯ w ∂ ¯ x A + i ¯ θ w ∂ x (0) ) , (5.26)  \n¯ D α = ¯ ∂ α + i ( γ a θ ) α ∂ a = w α ( ¯ ∂ ¯ θ w +4 iθ w ∂ x A + iθ ¯ w ∂ x (0) ) + ¯ w α ( ¯ ∂ ¯ θ ¯ w + iθ w ∂ x (0) ) . (5.27)  \nand the spinor frame variables can be integrated out thus arriving to a more conventional description of field theory in terms of usual (central basis) spacetime coordinates.  \nBut before considering N = 1 case, let us notice the possibility, in general case of arbitrary N > 1, to change the variables in the relative motion sector by decomposing the matrix fields  \nˆ Z j i = √ 2 ˆ Z I T j Ii , ˆ ¯ Z j i = √ 2 ˆ ¯ Z I T j Ii , ˆ Ψ j i = √ 2 ˆ Ψ I T j Ii , ˆ ¯ P j i = √ 2 ˆ ¯ P I T j Ii , ˆ P j i = √ 2 ˆ P I T j Ii , ˆ ¯ Ψ j i = √ 2 ˆ ¯ Ψ I T j Ii , (5.28)  \non SU( N ) generators T j Ii which obey  \n[ T I , T J ] = if IJK T K , f IJK = f [ IJK ] , f IJK f I ' JK = Nδ II ' , I, J, K = 1 , . . . N 2 -1 , (5.29)  \ntr( T I T J ) = 1 2 δ IJ , T I T J = 1 2 ( 1 N δ IJ I +( d IJK + if IJK ) T K ) , d IJK = d ( IJK ) , (5.30)  \nT Ii j T Ik l = 1 2 ( δ i l δ k j -1 N δ i j δ k l ) ⇒ ( T I T I ) i j = C F δ i j , C F = N 2 -1 N (5.31)  \nand some other relations which can be found in e.g. [91]. In the simplest case of SU(2) ( N = 2),  \nSU(2) : T Ii j = 1 2 σ Ii j , f IJK = /epsilon1 IJK , d IJK = 0 , I, J, K = 1 , 2 , 3 . (5.32)  \nThe coefficients in (5.28) are chosen in such a way that the nontrivial commutation relations of the basic operators have (almost) the standard form  \n[ ˆ P I , ˆ ¯ Z J ] = -iµ 6 δ IJ , [ ˆ ¯ P I , ˆ Z J ] = -iµ 6 δ IJ , { ˆ Ψ I , ˆ ¯ Ψ J } = 4 µ 6 δ IJ , (5.33)  \nso that in the coordinate representation the above operators are realized by  \nˆ Z I = Z I , ˆ ¯ P I = -iµ 6 ∂ ∂Z I , ˆ ¯ Z I = ¯ Z I , ˆ P I = -iµ 6 ∂ ∂ ¯ Z I , (5.34)  \nˆ  \nΨ  \n= Ψ  \n,  \nΨ  \n= 4  \nµ  \n∂  \nΨ  \nI  \n,  \nwhere Z I = ( Z 1 , Z 2 , . . . , Z N 2 -1 ) are complex bosonic ( c -number) coordinates and Ψ I = (Ψ 1 , Ψ 2 , . . . , Ψ N 2 -1 ) are fermionic ( a -number) coordinates,  \nΨ I Ψ J = -Ψ J Ψ I . (5.35)  \nIn terms of these relative motion coordinates we have the following expressions for the quantum generator of SU( N ) transformations (5.20)  \nˆ G j i =: 2 ˆ G I T j Ii = 2 f IJK ( ¯ Z I ∂ ∂ ¯ Z J + Z I ∂ ∂Z J +Ψ I ∂ ∂ Ψ J ) T Ki j , (5.36)  \nfor the part of the U(1) generator acting on the matrix field (5.13)  \nˆ ˜ B = 1 µ 6 ˆ B = i ( ¯ Z I ∂ ∂ ¯ Z I -Z I ∂ ∂Z I + 1 2 Ψ I ∂ ∂ Ψ I ) , (5.37)  \nfor the SYM supercharges (5.10), (5.11)  \nˆ ˜ ν = 1 µ 6 1 2 √ m ˆ ν = -i 2 √ m ( Ψ I ∂ ∂ ¯ Z I -4 √ 2 f IJK Z I ¯ Z J ∂ ∂ Ψ K ) , (5.38)  \nI  \nI  \nˆ  \n¯  \nI  \n6  \n∂  \nˆ ¯ ˜ ν = 1 µ 6 1 2 √ m ˆ ¯ ν = -i 2 √ m ( 4 µ 6 ∂ ∂Z I ∂ ∂ Ψ I -√ 2 µ 6 f IJK Z I ¯ Z J Ψ K ) , (5.39)  \nand for the SYM Hamiltonian (5.12)  \nˆ ˜ H = 1 m ( -µ 6 ∂ ∂Z I ∂ ∂ ¯ Z I -4 √ 2 µ 6 ¯ Z I f IJK ∂ ∂ Ψ J ∂ ∂ Ψ K -2 µ 6 f IJM f KLM Z I ¯ Z J Z K ¯ Z L + √ 2 4 µ 6 f IJK Z I Ψ J Ψ K ) . (5.40)  \nThese expressions can be used in the equations (5.15)-(5.19) imposed on the state vector superfield ('wavefunction')  \nΞ = Ξ ( x (0) , x A , ¯ x A , ¯ w,w ; θ w , θ ¯ w , ¯ θ w , ¯ θ ¯ w ; Z I , ¯ Z I ; Ψ I ) , (5.41)  \nwhich obeys also (5.23). This is manifestly a superfield on superspace with 5 + N 2 bosonic and 3 + N 2 fermionic directions (9 bosonic and 7 fermionic directions in the simplest case N = 2).", '6.1 N = 1. Field theory of single D 0 -brane': "If we set the number N of D0-branes to unity, N = 1, the matrix fields disappear so that the state vector becomes superfield in a center of mass superspace, which is usual superspace enlarged by spinor frame variables (Lorentz harmonic superspace in terminology of [86])  \nΞ 0 := Ξ | N =1 = ˜ Ξ 0 ( x a ; θ α , ¯ θ α ; ¯ w,w ) = Ξ 0 ( x (0) , x A , ¯ x A , ¯ w,w ; θ w , θ ¯ w , ¯ θ w , ¯ θ ¯ w ) (6.1)  \nand the above system of equations for this vector reduces to  \n∂ θ w Ξ 0 = 0 , ∂ ¯ θ ¯ w Ξ 0 = 0 , (6.2)  \n[ ∂ ¯ θ w -2 mθ ¯ w ] Ξ 0 = 0 , (6.3)  \n∂ x A Ξ 0 = 0 , ∂ ¯ x A Ξ 0 = 0 , (6.4)  \n( -i∂ x (0) -m ) Ξ 0 = 0 , (6.5)  \n( D (0) -¯ θ w ∂ ¯ θ w + θ ¯ w ∂ θ ¯ w -q ) Ξ 0 = 0 . (6.6)  \nThis system of equations describes the free (super)field theory of single 3D counterpart of D0 brane, i.e. of a massive N = 2 superparticle in D = 3. In Appendix C we show how to obtain those straightforwardly by quantization of massive superparticle in its spinor moving frame formulation. While the quantization of such a simple system in the standard formulation described by the de Azc'arraga-Lukierski action (2.6) is much simpler, in our case to follow this more complicated quantization makes sense as it describes the N = 1 limit of our simplest counterpart of the multiple D0 system which is currently known only in its spinor moving frame formulation.  \nEqs. (6.4) and (6.2) result in independence of the state vector superfield Ξ on x A , ¯ x A ; θ w , ¯ θ ¯ w ,  \nΞ 0 = Ξ 0 ( x (0) , ¯ w,w ; θ ¯ w , ¯ θ w ) , (6.7)  \nEqs. (6.3) and (6.5) are easily solved by  \nΞ 0 = e im x (0) -2 mθ ¯ w ¯ θ w χ ( θ ¯ w , ¯ w α , w α ) ≡ e im x (0) (1 -2 mθ ¯ w ¯ θ w ) χ ( θ ¯ w , ¯ w α , w α ) (6.8)  \nand then (6.6) implies  \n( D (0) + θ ¯ w ∂ θ ¯ w -q ) χ ( θ ¯ w , ¯ w α , w α ) = 0 . (6.9)  \nDecomposing χ in the powers of (fermionic and hence nilpotent) θ ¯ w ,  \nχ ( θ ¯ w , ¯ w α , w α ) = φ ( ¯ w α , w α ) + iθ ¯ w ξ ( ¯ w α , w α ) (6.10)  \nwe find from (6.9) that the components of these superfields are bosonic and fermionic functions of complex normalized bosonic spinors (functions on SU(1 , 1) ≈ SL(2 , R )) with definite U(1) charges q and ( q -1),  \nφ = φ q ( ¯ w α , w α ) , ( D (0) -q ) φ q ( ¯ w α , w α ) = 0 . (6.11)  \nξ = ξ ( q -1) ( ¯ w α , w α ) , ( D (0) -( q -1) ) ξ ( q -1) ( ¯ w α , w α ) = 0 . (6.12)  \nThese definite U(1) charges make these fields to be functions on the coset SU(1 , 1) / U(1) 17 .  \nAs far as θ ¯ w carries charge one with respect to U(1) symmetry generated by the operator in (6.9), this latter equation implies that the superfield χ has definite charge q with respect to this U(1) symmetry,  \nχ = χ q ( θ ¯ w , ¯ w α , w α ) . (6.13)  \nEqs. (6.11) and (6.12) imply the following homogeneity properties of the bosonic and fermionic function under gauge transformations  \nφ q ( e -iϑ ¯ w α , e iϑ w α ) = e -iqϑ φ q ( ¯ w α , w α ) , ξ ( q -1) ( e -iϑ ¯ w α , e iϑ w α ) = e -i ( q -1) ϑ ξ ( q -1) ( ¯ w α , w α ) . (6.14)", '6.1.1 Spacetime interpretation I': 'To clarify the physical meaning of these functions, let us notice that, as a result of the constraints (2.16) and (4.19)-(4.21), which implies p a = mu (0) a and p a γ a αβ = mu (0) a γ a αβ = 2 m ¯ w ( α w β ) , we can write  \n¯ w α w β = 1 2 m p a γ a αβ + i 2 /epsilon1 αβ , p a p a = m 2 . (6.15)  \nAs a result, the dependence on the product ¯ w α w β can be identified as dependence on the on-shell momentum of the massive particle, and for integer q ≥ 1 we can write the following solution of Eqs. (6.11) and (6.12)  \nφ q ( ¯ w,w ) = ¯ w α 1 . . . ¯ w α q φ α 1 ...α q ( p ) | p 2 = m 2 , ξ q -1 ( ¯ w,w ) = ¯ w α 1 . . . ¯ w α q -1 ξ α 1 ...α q -1 ( p ) | p 2 = m 2 , q ≥ 1 . (6.16)  \nTo be in consonance with the generic spin-statistic theorem 18 , for even q , the field φ q ( ¯ w,w ) should be bosonic and ξ q -1 ( ¯ w,w ) should be fermionic while, for odd q , it should be the other way around. For q ≤ 0  \nthe similar solution is  \nφ q ( ¯ w,w ) = w α 1 . . . w α -q φ α 1 ...α -q ( p ) | p 2 = m 2 , ξ q -1 ( ¯ w,w ) = w α 1 . . . w α -q -1 ξ α 1 ...α -q -1 ( p ) | p 2 = m 2 , q ≥ 1 . (6.17)  \nFinally, for q = 1 we have  \nφ (1) ( ¯ w,w ) = ¯ w α φ α ( p ) | p 2 = m 2 , ξ (0) ( ¯ w,w ) = ξ ( p ) | p 2 = m 2 , q ≥ 1 . (6.18)  \nwith fermionic φ α ( p ) and bosonic ξ ( p ).  \nHowever, as it is not manifestly clear how/whether one can normalize the above analytic solutions, below we will describe the solution in terms of different type of functions.', '6.1.2 Spacetime interpretation II': "To pass to a more standard description in terms of central basis coordinates, let us first resume the above discussion by stating that the state vector superfield of single massive 3D N = 2 superparticle (3D counterpart of D0-brane) has the form  \nΞ 0 = Ξ q 0 = e im x (0) ( φ q ( ¯ w α , w α ) + iθ ¯ w ξ q -1 ( ¯ w α , w α ) -2 mθ ¯ w ¯ θ w φ q ( ¯ w α , w α ) ) . (6.19)  \nwith φ q ( ¯ w α , w α ) and ξ q -1 ( ¯ w α , w α ) obeying (6.11) and (6.12).  \nNow, using (4.4), (4.7), (4.8) (or (5.25)-(5.27)) and acting by covariant derivative in central basis on this superfield, we obtain  \n( ¯ D α + imθ α )Ξ q 0 = 0 , (6.20)  \nwhich can be obtained from a (generalized) dimensional reduction of the chirality constraint in D = 4 19 (see [60]), as well as  \n( ∂ a -imu (0) a )Ξ q 0 = 0 (6.21)  \nwhich implies the Klein-Gordon equation  \n( ∂ a ∂ a + m 2 )Ξ q 0 = 0 . (6.22)  \nLet us consider the leading component of the superfield (6.19), e im x (0) φ q ( ¯ w α , w α ). Roughly speaking (which is to say, following the line of (6.16)), for the case q = 0 this can be written in the form  \nΞ (0) 0 | θ =0 = e im x (0) φ (0) ( ¯ w α , w α ) = e ip a x a φ ( p ) | p 2 = m 2 , (6.23)  \nwhere we have used (4.19)-(4.21). In this form it becomes evident that our state vector depends on both momenta and coordinates. To have a usual state vector field ('wavefunction') in the coordinate representation, we should integrate over on-shell momentum. In the standard textbook notation this reads  \nφ ( x ) = ∫ d 2 /vector p (2 π ) 2 p 0 ( e ip a x a φ ( p ) + e -ip a x a φ ∗ ( p ) ) ∣ ∣ ∣ p 0 = √ /vector p 2 + m 2 . (6.24)  \n∣ Our twistor-like representation describes only one of two terms in the integrand of the above equation as the sign of p 0 is fixed by (3.55). This latter relation also allows to find the expression for integrating measure in terms of moving frame and spinor moving frame variables. Indeed, it implies (see (2.28))  \nand, after the use of (2.26),  \nHence, the natural correspondence is  \nd 2 /vector p ↔ ¯ f ∧ f (6.27)  \nand the standard coordinate representation of the state vector field with q = 0 is  \nφ ( x ) = ∫ ¯ f ∧ f e imx a u (0) a φ (0) ( ¯ w,w ) , q = 0 , (6.28)  \nwhere (let us recall)  \nu (0) a = ¯ wγ a w , f = w α d w α = ( ¯ f ) ∗ , ¯ w α w α = i . (6.29)  \nAll this line can be easily generalized for the case of nonvanishing q giving  \nand  \nφ α 1 ...α q ( x ) = ∫ ¯ f ∧ f w α 1 . . . w α q e imx a u (0) a φ q ( ¯ w,w ) , q > 0 , (6.30)  \nφ α 1 ...α -q ( x ) = ∫ ¯ f ∧ f ¯ w α 1 . . . ¯ w α -q e imx a u (0) a φ q ( ¯ w,w ) , q < 0 . (6.31)  \nNotice that generically, when we use equations involving φ q ( ¯ w,w ), we have to be careful about the class of functions we choose. In particular, we should not substitute (6.16)-(6.18) for φ q ( ¯ w,w ) in Eq. (6.31). We will not work out the precise mathematical formulation of the problem (see [86] and [100, 101] for a more rigorous treatment of D = 4 case) but rather state that Eqs. (6.30) and (6.31) describe the solutions of the wave equation assuming that φ q ( ¯ w,w ) are such that the integrals converge.  \nSimilarly, the standard superfield representation of the state vector can be obtained as  \nΞ α 1 ...α | q | ( x, θ, ¯ θ ) =        ∫ ¯ f ∧ f w α 1 . . . w α q e imx a u (0) a -2 mθ ¯ w ¯ θw χ q ( ¯ w,w,θ ¯ w ) , q ≥ 0 , ∫ ¯ f ∧ f ¯ w α 1 . . . ¯ w α -q e imx a u (0) a -2 mθ ¯ w ¯ θw χ q ( ¯ w,w,θ ¯ w ) , q < 0 , (6.32)  \nwhere θ ¯ w = θ α ¯ w α = ( ¯ θw ) ∗ , u (0) a = ¯ wγ a w . (6.33)  \nThese superfields obey Eqs. (6.20) and (6.22),  \n( ¯ D α + imθ α ) Ξ α 1 ...α | q | ( x, θ, ¯ θ ) = 0 , (6.34)  \n( ∂ a ∂ a + m 2 ) Ξ α 1 ...α | q | ( x, θ, ¯ θ ) = 0 . (6.35)  \nFurthermore, one can easily check that for q = 0 superfield (6.32) obeys the following Dirac equation  \n/negationslash  \n∂ αα 1 Ξ α 1 α 2 ...α | q | = -m q | q | Ξ α α 2 ...α | q | . (6.36)  \nd p a = imf ¯ u a -im ¯ fu a , (6.25)  \n1 2 /epsilon1 abc d p b ∧ d p c = -2 iu a (0) ¯ f ∧ f . (6.26)  \nThus the quantum state vector of D = 3 N = 2 superparticle can be identified with superfield in the standard D = 3 N = 2 superspace obeying Eqs. (6.34)-(6.36). This is equivalent to the description by a function of the coordinates of the analytical basis of Lorentz harmonic superspace (6.1) obeying (6.2)-(6.6).  \nUnfortunately, in the generic case of N > 1, the description of the first type is not available (or, probably better to say, not efficient). This is because the bosonic and fermionic matrix coordinates of the 3D mD0 configurational superspace are not inert under the U(1) symmetry. Thus we have to work with the description of the second type, in terms of superfield depending on the coordinates of the analytical basis of center of mass superspace (5.22) satisfying the set of equations (5.15)-(5.20).", '6.2 Dependence on Grassmann center of mass coordinates for generic case N > 1 .': 'The set of fermionic equations for the state vector (5.17)-(5.19) can be solved in the generic case of N > 1. To this end, let us first notice that the commutators and anticommutators of the SYM currents simplify essentially when applied to the state vector obeying Eq. (5.20),  \nˆ ˜ ν 2 Ξ = 0 , ˆ ¯ ˜ ν 2 Ξ = 0 , { ˆ ˜ ν, ˆ ¯ ˜ ν } Ξ = ˆ ˜ H Ξ , (6.37)  \n[ ˆ ˜ ν, ˆ ˜ H ] Ξ = 0 , [ ˆ ¯ ˜ ν, ˆ ˜ H ] Ξ = 0 , (6.38)  \nwhich, in particular, results in e -2 θ w ˆ ¯ ˜ ν ˆ ˜ ν = ˆ ˜ νe -2 θ w ˆ ¯ ˜ ν -2 θ w ˆ ˜ H .  \nThen it is easy to check that Eqs. (5.17)-(5.19) are solved (formally) by  \nΞ = e -2 θ w ˆ ¯ ˜ ν e -2 ¯ θ ¯ w ˆ ˜ ν e +2 θ w ¯ θ ¯ w ˆ ˜ H e -2 θ ¯ w ¯ θ w ( m + ˆ ˜ H ) ( B + θ ¯ w F ) = = ( 1 -2 θ w ˆ ¯ ˜ ν -2 ¯ θ ¯ w ˆ ˜ ν +2 θ w ¯ θ ¯ w ( ˆ ˜ H2 ˆ ¯ ˜ ν ˆ ˜ ν ) -2 θ ¯ w ¯ θ w ( m + ˆ ˜ H )+ +4 θ w θ ¯ w ¯ θ w ˆ ¯ ˜ ν ( m + ˆ ˜ H ) + 4 θ ¯ w ¯ θ w ¯ θ ¯ w ˆ ˜ ν ( m + ˆ ˜ H ) --4 θ w ¯ θ ¯ w θ ¯ w ¯ θ w ( ˆ ˜ 2 ˆ ¯ ˜ ν ˆ ˜ ν )( m + ˆ ˜ ) B + θ ¯ w F , (6.39)  \nHH ) ( )  \nB = B ( x (0) , ¯ w,w ; Z , ¯ Z ; Ψ ) , F = F ( x (0) , ¯ w,w ; Z , ¯ Z ; Ψ ) (6.40)  \nwhere  \nare functions (superfields) of opposite statistics (one bosonic another fermionic) dependent of spinor moving frame variables and of matrix coordinates of our enlarged superspace. They should obey  \nˆ G j i B = 0 , ˆ G j i F = 0 (6.41)  \nwith ˆ G j i from (5.20) (or, equivalently, (5.36)), which implies that they should be constructed from SU( N ) invariant combinations of the matrix fields, and  \n( D (0) -2 i ˆ ˜ B q ) B = 0 , ( D (0) -2 i ˆ ˜ B -( q -1) ) F = 0 , (6.42)  \nwhich fix the charge of the fields with respect to U(1) group acting on spinor frame variables and matrix fields,  \nB = B ( q ) ( x (0) , ¯ w,w ; Z , ¯ Z ; Ψ ) , F = F ( q -1) ( x (0) , ¯ w,w ; Z , ¯ Z ; Ψ ) . (6.43)  \nFurthermore, as a result of (5.15) these functions obey the Schrodinger equation with a shifted SYM Hamiltonian in which x (0) coordinate playing the role of time,  \n( i∂ x (0) + m +2 ˆ ˜ H ) B ( q ) = 0 , ( i∂ x (0) + m +2 ˆ ˜ H ) F ( q -1) = 0 . (6.44)  \nTo find (6.42) from (5.16) we have used the relations  \ni ˆ ˜ ˜ U (0) e -2 θ w ˆ ¯ ˜ ν = ( D (0) -¯ θ w ∂ ¯ θ w + θ ¯ w ∂ θ ¯ w -4 θ w ¯ θ ¯ w ˆ ˜ H +2 θ w ˆ ¯ ˜ ν -2 ¯ θ ¯ w ˆ ˜ ν -2 i ˆ ˜ B q ) e -2 θ w ˆ ¯ ˜ ν =  \n= e -2 θ w ˆ ¯ ˜ ν ( D (0) -¯ θ w ∂ ¯ θ w + θ ¯ w ∂ θ ¯ w +2 ¯ θ ¯ w ˆ ˜ ν -2 i ˆ ˜ B q ) , (6.45)  \n( D (0) -¯ θ w ∂ ¯ θ w + θ ¯ w ∂ θ ¯ w -2 ¯ θ ¯ w ˆ ˜ ν -2 i ˆ ˜ B q ) e -2 ¯ θ ¯ w ˆ ˜ ν = e -2 ¯ θ ¯ w ˆ ˜ ν ( D (0) -¯ θ w ∂ ¯ θ w + θ ¯ w ∂ θ ¯ w -2 i ˆ ˜ B q ) , (6.46)  \nwhich follow from the form of the commutators of the U(1) generators for matrix fields and SYM supercurrents  \n[ ˆ ˜ ν, B ] = i 2 µ 6 ˆ ˜ ν, [ ˆ ¯ ˜ ν, B ] = -i 2 µ 6 ˆ ¯ ˜ ν . (6.47)  \nOf course, we can easily write the formal solution of (6.44)  \nB ( q ) ( x (0) , ¯ w,w ; Z , ¯ Z ; Ψ ) = e i x (0) ( m +2 ˆ ˜ H ) ˇ B ( q ) ( ¯ w,w ; Z , ¯ Z ; Ψ ) ,  \nbut its usefulness is restricted: an explicit solution of the Schrodinger equation would carry much more information on the system.  \nF ( q -1) ( x (0) , ¯ w,w ; Z , ¯ Z ; Ψ ) = e i x (0) ( m +2 ˆ ˜ H ) ˇ F ( q -1) ( ¯ w,w ; Z , ¯ Z ; Ψ ) , (6.48)', '6.3 Solutions with BPS-type configurations of the SYM sector': "Let us consider the generic case N > 1 but impose the following set of conditions on the state vector superfield  \nˆ ˜ ν Ξ = 0 , (6.49)  \nˆ ¯ ˜ ν Ξ = 0 , (6.50)  \nˆ ˜ H Ξ = 0 , (6.51)  \nwhere ˆ ˜ ν , ˆ ¯ ˜ ν , ˆ ˜ H are given in (5.10), (5.11) and (5.12). Then Eqs. (5.15)-(5.19) reduce to (6.2)-(6.6) but imposed on the state vector superfield depending on matrix coordinates, (5.22).  \nFurthermore, in this case the center of mass part of the equations of motion is separated from the relative motion equations the set of which is given now by (6.49)-(6.51) and (5.20).  \nHence we can search for a solution in the form with factorized center of mass motion, i.e. in the form of sum of products  \nΞ q = ∑ q ' Ξ q -q ' 0 (x (0) , ¯ w,w ; θ ¯ w , ¯ θ w ) Ξ q ' ( Z I , ¯ Z I , Ψ I ) (6.52)  \nof functions of center of mass variables Ξ q -q ' 0 (x (0) , ¯ w,w ; θ ¯ w , ¯ θ w ) obeying (6.2), (6.3)-(6.5) and  \n( D (0) -¯ θ w ∂ ¯ θ w + θ ¯ w ∂ θ ¯ w -( q -q ' ) ) Ξ q -q ' 0 = 0 , (6.53)  \nand of functions of relative motion variables Ξ q ' ( Z I , ¯ Z I , Ψ I ) obeying (6.49)-(6.51), (5.20) and  \n( ˆ ˜ B i 2 q ' ) Ξ q ' ( Z I , ¯ Z I , Ψ I ) = 0 , (6.54)  \nwith operators defined in (5.36)-(5.39).  \nThe set of Eqs. (5.38)-(5.40) and (5.20) (with (5.36)) coincides with the equations for supersymmetric vacua of 1d reduction of 3D SYM model in the Schrodinger representation (see [81] for study of 3D YM in this representation). The related studies of the ground state of YM, SYM, and BFSS Matrix model were carried out in [75, 78, 79] and in refs. therein.  \nThe solutions of the equations for Ξ q -q ' 0 are the solution of the equations for single D0-brane and we have already discussed these in the previous (sub)section. Here we consider Eqs. (5.40), (5.38), (5.39) and (5.20) for the simplest case of N = 2.", '6.3.1 N = 2 case: field theory of simplest nontrivial 3D counterpart of mD 0 system': "Let us search for a solution of mD0 equations which obey the additional conditions (6.49) for the case of N = 2, i.e. for the system of two D0-branes and strings ending on the same and on different D0-branes.  \nIn this case the decomposition of Ξ q ' ( Z I , ¯ Z I , Ψ I ) superfield on the fermionic coordinates Ψ I = (Ψ 1 , Ψ 2 , Ψ 3 ) reads  \nΞ q ' ( Z I , ¯ Z I , Ψ I ) = f q ' ( Z, ¯ Z ) + Ψ I f q ' -1 I ( Z, ¯ Z ) + 1 2 Ψ I Ψ J f IJ q ' -2 ( Z, ¯ Z ) + Ψ ∧ 3 f q ' -3 ( Z, ¯ Z ) (6.55)  \nwhere  \nΨ ∧ 3 := 1 3! /epsilon1 IJK Ψ I Ψ J Ψ K = Ψ 1 Ψ 2 Ψ 3 (6.56)  \nis the basic monomial of maximal power constructed from three fermionic variables.  \nLet us substitute this decomposition in Eqs. (6.54), (6.51) , (6.49), (6.50) and (5.20) with (5.36).  \nFirst of all, Eq. (6.54) clearly implies that the coefficient functions in the decomposition (6.54) are the eigenfunctions of the operator  \n-2 i ˆ ˜ B 0 = ( 2 ¯ Z I ∂ ∂ ¯ Z I -2 Z I ∂ ∂Z I ) , (6.57)  \nwith eigenvalues (having the meaning of U(1) charges) written as superindices of these component fields.  \nNow, let us observe that the anticommutators of the operators in (6.49) and (6.50) are expressed in terms of the operators (6.51) and (5.20), so that it is sufficient to solve two former equations.  \nSubstituting (6.55) into Eq. (6.49) (and re-denoting q ' by q ) we find that this superfield equation implies the following set of four equations for the component functions  \nwhere (and below)  \n¯ ∂ I := ∂ ∂ ¯ Z I , ∂ I := ∂ ∂Z I . (6.62)  \nBy Poincar'e lemma, Eq. (6.61) is solved by  \nf ( q -2) IJ = ¯ ∂ [ I f J ] ( q ) , (6.63)  \nso that (6.59) acquires the form  \n¯ ∂ I f ( q ) = 4 √ 2 /epsilon1 JKL Z J ¯ Z K ¯ ∂ [ L f I ] ( q ) ⇔ ¯ ∂ I ( f ( q ) +2 √ 2 /epsilon1 JKL Z J ¯ Z K f L ( q ) ) = 2 √ 2 /epsilon1 JKL Z J ¯ Z K ¯ ∂ L f I ( q ) . (6.64)  \nSimilarly one can find that Eq. (6.50) for superfield (6.55) with (6.63) implies the following three equations (since the superfield equation (6.50) has the highest component identically equal to zero)  \nµ 6 ∂ I f ( q -1) I = 0 , (6.65)  \nµ 6 ∂ J ¯ ∂ [ I f ( q ) J ] = -1 2 √ 2 µ 6 /epsilon1 IJK Z J ¯ Z K f ( q ) , (6.66)  \nµ 6 ∂ I f ( q -3) = -1 √ 2 µ 6 Z [ I ¯ Z J ] f ( q -1) J . (6.67)  \nIn appendix D we discuss the Born-Oppenheimer-like method to search for asymptotic form of the solution of this system of equations in the line of [75] (see also [78]).  \n/epsilon1 IJK Z J ¯ Z K f ( q -1) I = 0 , (6.58)  \n¯ ∂ I f ( q ) = 4 √ 2 /epsilon1 JKL Z J ¯ Z K f ( q -2) LI , (6.59)  \n¯ ∂ [ I f J ] ( q -1) = 4 √ 2 Z [ I ¯ Z J ] f ( q -3) , (6.60)  \n¯ ∂ [ I f JK ] ( q -2) = 0 , (6.61)", '7 Conclusion and outlook': "In this paper we have initiated the program of construction of relativistic invariant (super)field theory of multiple-D0-brane (mD0) system by studying the quantization of its simplest 3D counterpart which we roughly call 3D mD0. This simplest system is described by the action which is the 3D counterpart of the 10D action constructed in [58] which was considered as a candidate action for mD0 since i) it is formulated in terms of variables which are expected to describe mD0 and ii) it possesses local fermionic κ -symmetry so that its ground state preserves a part (one half) of spacetime (superspace) supersymmetry.  \nActually, in [61] we found a family of the actions possessing these properties which includes an arbitrary positive definite function M ( H ) of the SYM Hamiltonian H constructed from the matrix fields of the model; the model of [58] corresponds to the choice M ( H ) = m with the constant m entering the center of mass part of the action. As we have shown in [62], the supersymmetric ground state configuration is the same for all the models with positive definite M ( H ), however the most promising candidate is given by M ( H ) of the form (2.44) since this can be obtained by dimensional reduction of 11D multiple M-wave (multiple M0 or mM0) model proposed in [53]. The 3D counterpart of the 10D model with arbitrary positive definite M ( H ) was constructed in [60] where it was shown that the model with M ( H ) of the form (2.44) can be obtained by dimensional reduction from the 4D counterpart of the 11D mM0 model.  \nAs we discussed in the Introduction, the quantization of mD0-brane system in a coordinate representation should result in equations of field theory of mD0 which might provide us with new insights in String/M-theory. However, as such a quantization appeared to be quite complicated, we decided to begin with a toy 3D model which has the properties characteristic of mD0 system, moreover we restricted ourselves by simplest of such models, with M ( H ) = m . As one can see from the main text, already the quantization of this toy model, which was the aim of this paper, was quite complex. It indicated some of the problems we will meet in quantization of 10D mD0 model(s), suggested the way to resolve these and allowed to develop and to check the methods for such resolution.  \nThe dynamical variables of our 3D mD0 system is split on the relative motion and center of mass sectors. The former is described by traceless N × N matrix fields of 3-dimensional N = 2 supersymmetric SU( N ) Yang-Mills model dimensionally reduced to d = 1: bosonic Z ( τ ) = ( ¯ Z ( τ )) † and fermionic Ψ ( τ ) = ( ¯ Ψ ( τ )) † , as well as auxiliary momenta P ( τ ) = ( ¯ P ( τ )) † and 1-form gauge field A = d τ A τ ( τ ). The center of mass sector contains bosonic and fermionic coordinate functions x µ ( τ ), θ α ( τ ), ¯ θ α ( τ ), describing the embedding of the center of mass of the system in N = 2 3D flat superspace, as well as spinor moving frame variables, bosonic spinors w α and ¯ w α constrained by ¯ w α w α = i and parametrizing SL(2 , R ) / U(1) coset. Although auxiliary, these bosonic spinor variables are essential to couple the center of mass sector to the relative motion one. In particular, the matrix fields of the latter are charged with respect to U(1) symmetry which is a defining symmetry of the spinor moving frame sector. As a result, their time derivative in the action are completed to the covariant time derivatives containing, besides the 1d SU( N ) gauge fields, also U(1) connection composed from spinor frame variables.  \nWe have constructed Hamiltonian mechanics of this model, one of the subtleties of which is the presence of manifestly constrained spinor frame variables which we treat using the method of Cartan forms and introducing the so-called covariant momenta conjugate to these Cartan forms. We discussed in detail the Hamiltonian approach to complex matrix fermionic fields Ψ and ¯ Ψ and the Dirac brackets which allows to treat ¯ Ψ as momentum for Ψ . The calculation of canonical momenta for center of mass variables (including covariant momenta for spinor frame variables) and for the SU( N ) gauge field has lead to identification of primary constraints. The SU( N ) gauge field does not carry physical degrees of freedom, but generate the non-Abelian counterpart of the Gauss law constraint which appears as the only secondary constraint in our system. The canonical Hamiltonian vanishes in the weak sense, i.e. when constraints are taken into account. This reflects the reparametrization invariance of our dynamical system.  \nWe have separated constraints into first class ones, generating gauge symmetries on Poisson brackets, and second class ones, the set of which can be split on canonically conjugate pairs. The Poisson brackets of these first and second class constraints represent a quite nontrivial algebra which makes the application of the procedure of the Abelization of the second class constraints quite difficult and thus hampered the way to Hamiltonian BRST quantization by Batalin-Fradkin-Fradkina method [99]. Since the Gupta-Bleuler method also failed in the sector of bosonic second class constraints (quite unexpectedly, at least for us, and even in N = 1 case corresponding to single D-particle; see Appendix B), the original Dirac brackets method seemed to be the only way to quantization of our simplest 3D mD0 system.  \nWe simplified the quantization by passing to the so-called analytical coordinate basis of the center of mass superspace, in which the bosonic second class constraints appeared to be resolved thus allowing us to remove 'unphysical' coordinates from the phase space without passing to the Dirac brackets. To deal with the fermionic second class constraints we applied the Gupta-Bleuler or, better to say, generalized conversion method. In such a way we arrive at a set of effective first class constraints which, upon quantization, can be imposed on the state vector.  \nWe used the (generalized) coordinate representation in which momenta are represented by differential operators over the configuration (super)space coordinates. Some subtleties appear in the spinor moving frame sector, where one should deal with covariant momenta and a kind of covariant derivatives, as well as in the case of two Hermitian conjugate fermionic matrix fields, where we have used the so-called holonomic representation in which ˆ ¯ Ψ operator is represented by derivative with respect to Ψ .  \nImposing the quantum constraints thus represented on the state vector Ξ we obtained equations of the field theory of simplest 3D counterpart of mD0 brane system. This field theory is defined on the superspace with additional matrix coordinates given by traceless N × N matrices Z = ( ¯ Z ) † and fermionic Ψ and ¯ Ψ . The superspace also contains the center of mass sector which can be identified with Lorentz harmonic D = 3 N = 2 superspace. That is parametrized by spinor moving frame variables and by the coordinate of usual D = 3 N = 2 superspace x a , θ α , ¯ θ α , but taken in the so-called analytical basis, e.g. x (0) = x a ¯ wγ a w , ¯ θ w = ¯ θ α w α , etc.  \nEven in the case of N = 1 corresponding to the simplest 3D counterpart of single D0-brane field theory, such a field theory looks unusual. We have discussed this case in detail and show how the equations for the state vector superfield looks like in the central basis of the Lorentz harmonic superspace, in which Ξ = Ξ( x a , θ, ¯ θ ; ¯ w,w ), as well as how one can pass to the standard 3d field theory equations in spacetime and to their solutions.  \nFor the generic N > 1 case we have presented the formal solution of the system of the equations specifying dependence of the state vector on fermionic coordinates of the center of mass sector. We have also discussed the solutions with BPS-like configuration of the relative motion sector in which this sector is decoupled from the center of mass one. Such type equations describe also the ground state of the BFSS matrix model [72] at finite N which was the subject of study in [73-79].  \nFor the simplest N = 2 case with SU(2) gauge invariance we have obtained the set of field theory equations which describes the ground state for the D = 3 N = 2 SU(2) SYM theory in Schrodinger-type representation 20 . We have also discussed, following the method of [75], the asymptotic form of the solution of these equations.  \nOne of the natural directions for further study is to search for the solution of our system of (super)field equations with coupled center of mass and relative motion sectors.  \nAnother interesting problem is the construction of Hamiltonian mechanics and quantization of the most general 3D counterpart of the mD0 system the action of which (2.41) includes an arbitrary positive definite function M ( H ) of the SYM Hamiltonian H . The presence of such a function makes the straightforward  \nelaboration of the Hamiltonian formalism quite technically involved. Probably a shortcut passes through first quantizing the 4D counterpart of multiple M-wave (mM0) system of [53], which was constructed in [60]. Indeed, as the dimensional reduction of this system to D = 3 was shown to produce generic 3D counterpart of the mD0 system described by the Lagrangian (2.41), it is natural to expect that the dimensional reduction of the field theory of this 4D mM0 would give the field theory for such a generic 3D mD0 model 21 .  \nOf course, all these studies, as well as the results of the present paper, will be preparatory steps toward the quantization of the 10D mD0 model proposed in [62]. Such a quantization will result in a field theory in 10D superspace extended by 9 bosonic and 16 fermionic matrix coordinates the study of which might give new insights in the structure of String/M-theory.", 'Acknowledgments': 'We are thankful to Dmitri Sorokin for useful comments and discussions, and to Padova Section of INFN for hospitality at Padova University where these discussion took place. IB thanks the Theoretical Physics Department of CERN for hospitality during his visit at the intermediate period of this work. The partial support by grant PID2021-125700NB-C21 funded by Spanish MCIN/ AEI and by EU ERDF, and by the Basque Government Grant IT-1628-22 is greatly acknowledged.', 'A Useful Poisson bracket relations': "Let us begin by presenting the Poisson brackets of the Gauss law constraint G (3.34) with the matrix variables. It is convenient to encode these in the brackets of the trace of the product of G with some 'reference' traceless matrix Y , i.e. of tr( YG ) = Y j i G i j . In such a way we arrive at simple and transparent expressions  \n[tr( YG ) , Z ] PB = [ Z , Y ] , [tr( YG ) , ¯ Z ] PB = [ ¯ Z , Y ] , (A.1)  \n[tr( YG ) , P ] PB = [ P , Y ] , [tr( YG ) , ¯ P ] PB = [ ¯ P , Y ] , (A.2)  \n[tr( YG ) , Ψ ] PB = [ Ψ , Y ] , [tr( YG ) , ¯ Ψ ] PB = [ ¯ Ψ , Y ] (A.3)  \nwhich clearly represent the infinitesimal SU( N ) transformations.  \nThese relations allow, in particular, to obtain the su ( N ) algebra (3.39) generated by the Gauss constraints  \n[tr( YG ) , tr( Y ' G )] PB = tr ( [ Y , Y ' ] G ) . (A.4)  \nIt is also easy to check that the Poisson brackets of the Gauss law with different currents that appear in our constraints,  \nν := tr( Ψ P + ¯ Ψ [ Z , ¯ Z ]) , ¯ ν := tr( ¯ Ψ ¯ P + Ψ [ Z , ¯ Z ]) , B := tr ( ¯ PZ -P ¯ Z + i 8 Ψ ¯ Ψ ) , (A.5)  \nH = tr ( P ¯ P +[ Z , ¯ Z ] 2 -i 2 Z ΨΨ + i 2 ¯ Z ¯ Ψ ¯ Ψ ) (A.6)  \nand  \nvanish,  \n[tr( YG ) , ν ] PB = 0 , [tr( YG ) , ¯ ν ] PB = 0 , [tr( YG ) , B ] PB = 0 , [tr( YG ) , H ] PB = 0 , (A.7)  \nwhich is just the reflection of SU( N ) invariance of these objects.  \nThe Poisson brackets of the fermionic and bosonic currents (A.5) among themselves and with the relative motion Hamiltonian (A.6) are  \n{  \nν, ν } PB = -8 i ( µ 6 ) 2 tr( ZG ) , { ¯ ν, ¯ ν } PB = 8 i ( µ 6 ) 2 tr( ¯ ZG ) , { ν, ¯ ν } PB = -4 iµ 6 H , [ ν, ] PB = 1 µ 6 ν, [¯ ν, ] PB = 1 µ 6 ¯ ν , (A.8)  \nand  \n[ ν, H ] PB = ( µ 6 ) 2 tr( ¯ Ψ G ) , [¯ ν, H ] PB = -( µ 6 ) 2 tr( Ψ G ) , [ B , H ] PB = 0 . (A.9)  \nIn terms of renormalized currents and matrix fields (4.10), (4.25) and (4.47) the above relations simplify to  \n{ ˜ ν, ˜ ν } PB = -2 i tr( Z ˜ G ) , { ¯ ˜ ν, ¯ ˜ ν } PB = 2 i tr( ¯ Z ˜ G ) , { ˜ ν, ¯ ˜ ν } PB = -i ˜ H , [˜ ν, ˜ B ] PB = 1 2 ˜ ν, [ ¯ ˜ ν, ˜ B ] PB = -1 2 ¯ ˜ ν , [ ˜ B , ˜ B ] PB ≡ 0 , [˜ ν, ˜ H ] PB = 1 2 tr( ˜ ¯ Ψ ˜ G ) , [ ¯ ˜ ν, ˜ H ] PB = -1 2 tr( ˜ Ψ ˜ G ) , [ ˜ B , ˜ H ] PB = 0 . (A.10)  \nB 2 B -2  \nThe above relations were obtained by using the brackets of currents with matrix fields which are  \n[ ν, Z ] PB = 0 , [ ν, P ] PB = 0 , { ν, Ψ } PB = -4 iµ 6 [ Z , ¯ Z ] , [ ν, ¯ Z ] PB = -µ 6 Ψ , [ ν, ¯ P ] PB = 0 , { ν, ¯ Ψ } PB = -4 iµ 6 P , [¯ ν, Z ] PB = -µ 6 ¯ Ψ , [¯ ν, P ] PB = 0 , { ¯ ν, Ψ } PB = -4 iµ 6 ¯ P , [¯ ν, ¯ Z ] PB = 0 , [¯ ν, ¯ P ] PB = 0 , { ¯ ν, ¯ Ψ } PB = -4 iµ 6 [ Z , ¯ Z ] , [ B , Z ] PB = -µ 6 Z , [ B , P ] PB = -µ 6 P , [ B , Ψ ] PB = 1 2 µ 6 Ψ , [ , ¯ Z ] PB = µ 6 ¯ Z , [ , ¯ P ] PB = µ 6 ¯ P , [ , ¯ Ψ ] PB = 1 µ 6 ¯ Ψ . (A.11)  \nB B B -2", 'B Problems with Gupta-Bleuler quantization scheme for massive particle in moving frame formulation': 'Let us consider the system of the first class constraints (3.55), (3.58) and second class constraints (3.59), (3.60) and set to zero all the fermionic and matrix fields. Then such a system of constraints describes, in the frame of Hamiltonian approach, just a massive bosonic particle in its spinor moving frame formulation.  \nIn this appendix we show that, surprisingly, the quantization of such a simple system by the GuptaBleuler method in its canonical form fails 22 .  \nImposing the quantum versions of the first class constraints (3.55), (3.58) and two of second class constraints (3.59), (3.60) which commute and are not related by Hermitian conjugation among themselves, and choosing coordinate representation for the state vector, Ξ 0 = Ξ 0 ( x a , ¯ w,w ), we can arrive at the following system of differential equations  \n( iu (0) a ∂ a + m )Ξ 0 = 0 , (B.1)  \n( D (0) -q ) Ξ 0 = 0 , (B.2)  \n¯ D Ξ 0 = 0 , (B.3)  \n¯ u a ∂ a Ξ 0 = 0 . (B.4)  \nThe x a -dependence of the state vector is fixed by Eqs. (B.1) and (B.4) to be  \nΞ 0 = e imx a u (0) a χ 0 ( x a ¯ u a , ¯ w α , w α ) . (B.5)  \nUsing this we find from (B.3) that χ 0 ( x a ¯ u a , ¯ w α , w α ) obeys  \n¯ D χ 0 ( x a ¯ u a , ¯ w α , w α ) := ¯ w α ∂ ∂w α χ 0 ( x a ¯ u a , ¯ w α , w α ) = -imx a ¯ u a χ 0 ( x a ¯ u a , ¯ w α , w α ) . (B.6)  \nThe second form of the l.h.s. of this equation makes manifest that its nontrivial solution, if existed, would be χ 0 = e -imx a u (0) a ˜ χ 0 ( x a ¯ u a , ¯ w α , w α ), but this form contradicts to the fact that in (B.5) χ 0 = χ 0 ( x a ¯ u a , ¯ w α , w α ) is independent on x a u (0) a .  \nThis example (actually the only presently known to the authors) indicates that the selfconsistency of the Gupta-Bleuler quantization method is not guaranteed and thus its should be used with precaution.  \nCuriously, if we modify the above procedure by imposing besides (B.1) and (B.2) the complex (Hermitian) conjugate but not canonically conjugate pair of second class constraints (3.59), which commute among themselves, we arrive at the system  \n( iu (0) a ∂ a + m )Ξ 0 = 0 , (B.7)  \n( D (0) -q ) Ξ 0 = 0 , (B.8)  \nu a ∂ a Ξ 0 = 0 , (B.9)  \n¯ u a ∂ a Ξ 0 = 0 (B.10)  \nwhich does have a nontrivial solution which is equivalent to the solution of the massive Klein-Gordon equation.  \nSuch a modification clearly does not feet in the standard Gupta-Bleuler prescription. However, as we show in the main text and also in Appendix C, its result is equivalent to the quantization with the method of Dirac brackets, which is further simplified and reduced to the explicit resolution of the constraints in the analytical basis. Such Dirac brackets/anaytical basis quantization gives a consistent field theory with nontrivial solution which, in the simplest case considered in this Appendix, coincides with the result of quantization in the standard formulation of the massive particle.', 'C Hamiltonian formalism of the 3D single D0-brane in the so-called analytical basis': "The Lagrangian of 3D counterpart of single D0-brane in spinor moving frame formulation reads ( cf. (2.6))  \nL D0 = d τL D0 = m E (0) + m (d θ α ¯ θ α -d ¯ θ α θ α ) (C.1)  \nwhere E (0) is defined in (2.35) with (2.8). The dynamical variables in this Lagrangian 1-form are coordinate functions which define parametrically the embedding of worldline in flat superspace  \nW 1 ⊂ Σ (3 | 4) : x a = x a ( τ ) , θ α = θ α ( τ ) , ¯ θ α = ¯ θ α ( τ ) (C.2)  \nand spinor frame variables (2.15), w α ( τ ) and its c.c. ¯ w α ( τ ) obeying (2.16).  \nIn terminology of [86], originating in the seminal papers on Harmonic superspaces [87-90], these latter variables can be called Lorentz harmonics thus reflecting the fact that they parametrize the double covering of 3D Lorentz group SL(2 , R ) and, if the requirement of the invariance under U(1) gauge transformations acting on w and ¯ w is imposed, as homogeneous coordinates of the coset SL(2 , R ) / U(1). Furthermore,  \nthe configuration space of our dynamical system, this is to say the target space of the spinor moving frame formulation of the 3D D0-brane, can be called, following the above terminology, Lorentz harmonic superspace , Σ (3+3 | 4) . This is the superspace with coordinates  \nΣ (3+3 | 4) = { ( x a , θ α , ¯ θ α , w α , ¯ w α ) } =: { ( Z M , w α , ¯ w α ) } , ¯ w α w α = i (C.3)  \nand in the spinor moving frame approach we are considering D0-brane as a particle moving in this extended and enlarged D = 3 superspace. Here 'extended' refers to N = 2 supersymmetry and 'enlarged' reflects the presence of additional bosonic spinor (spinor moving frame or Lorentz harmonic) coordinates.  \nThis perspective is convenient, in particular, because it suggests the possibility to choose an alternative coordinate basis of the Lorentz harmonic superspace, the analytical coordinate basis  \nΣ (3+3 | 4) = { (x (0) , x A , ¯ x A , θ w , θ ¯ w , ¯ θ w , ¯ θ ¯ w ; w α , ¯ w α ) } =: { (Z ( M ) An , w α , ¯ w α ) } , ¯ w α w α = i (C.4)  \ndefined by (see [86, 89, 90])  \nx (0) := x a u (0) a , x A := x -2 iθ w ¯ θ w = x a u a -2 iθ w ¯ θ w , ¯ x A := ¯x + 2 iθ ¯ w ¯ θ ¯ w = x a ¯ u a +2 iθ ¯ w ¯ θ ¯ w , (C.5)  \nθ w := θ α w α , θ ¯ w := θ α ¯ w α , (C.6)  \n¯ θ w := ¯ θ α w α , ¯ θ ¯ w := ¯ θ α ¯ w α . (C.7)  \nThe definition of x A = (¯x A ) ∗ is designed in such a way that in the analytical basis the Lagrangian form (C.1) simplifies to  \nL D0 = m dx (0) -imf ¯ x A + im ¯ f x A -4 maθ ¯ w ¯ θ w -2 im (d θ ¯ w ¯ θ w +d ¯ θ w θ ¯ w ) = (C.8)  \n= m dx (0) -2 im (d θ ¯ w -iaθ ¯ w ) ¯ θ w -2 im (d ¯ θ w + ia ¯ θ w ) θ ¯ w -imf ¯ x A + im ¯ f x A . (C.9)  \nIn the main text we will need also separate expressions for the kinetic and WZ terms of single 3D D0 action,  \nE 0 = dx (0) -i (d θ w + iaθ w ) ¯ θ ¯ w -i (d θ ¯ w -iaθ ¯ w ) ¯ θ w -i (d ¯ θ w + ia ¯ θ w ) θ ¯ w -i (d ¯ θ ¯ w -ia ¯ θ ¯ w ) θ w + if ¯ x -i ¯ f x , (C.10)  \nd θ α ¯ θ α -d ¯ θ α θ α = 2 f θ ¯ w ¯ θ ¯ w +2 ¯ f θ w ¯ θ w + i (d θ w + iaθ w ) ¯ θ ¯ w -i (d θ ¯ w -iaθ ¯ w ) ¯ θ w -  \n-i (d ¯ θ w + ia ¯ θ w ) θ ¯ w + i (d ¯ θ ¯ w -ia ¯ θ ¯ w ) θ w . (C.11)  \nThe canonical Hamiltonian H 0 in terms of the coordinates of the analytical basis is defined by  \nd τH 0 = dx (0) p (0) +dx A ¯ p +d¯x A p +d θ w Π θ w +d θ ¯ w Π θ ¯ w +d ¯ θ w ¯ Π ¯ θ w +d ¯ θ ¯ w ¯ Π ¯ θ ¯ w + ia ˜ d (0) + if ¯ ˜ d -i ¯ f ˜ d -L D0 (C.12)  \nwhere the momenta conjugate to the coordinate functions, i.e. having the nonvanishing Poisson brackets  \n[ p (0) , x (0) ] PB = -1 , [ p, ¯ x A ] PB = -1 , [¯ p, x A ] PB = -1 . (C.13)  \nare defined by (4.13),  \np (0) := u a (0) p a , p := -1 2 u a p a , ¯ p := -1 2 ¯ u a p a (C.14)  \nSimilarly, the momenta conjugate to the fermionic coordinate functions of the analytical basis, i.e. obeying  \n{ Π θ w , θ w } PB = -1 , { Π θ ¯ w , θ ¯ w } PB = -1 , { ¯ Π ¯ θ w , ¯ θ w } PB = -1 , { ¯ Π ¯ θ ¯ w , ¯ θ ¯ w } PB = -1 . (C.15)  \nare related to the ones of the central basis by (4.16)  \nΠ θ w := -i ¯ w α Π α +2 i ¯ θ w ¯ p , Π θ ¯ w := iw α Π α -2 i ¯ θ ¯ w p , ¯ Π ¯ θ w := -i ¯ w α ¯ Π α -2 iθ w ¯ p , ¯ Π ¯ θ ¯ w := iw α ¯ Π α +2 iθ ¯ w p . (C.16)  \nThe covariant momenta of the analytical basis are related to these of the central basis by  \nd := ˜ d +(x A +2 iθ w ¯ θ ¯ w ) p (0) +2(x (0) + iθ w ¯ θ ¯ w + iθ ¯ w ¯ θ w ) p + θ w Π θ ¯ w + ¯ θ w ¯ Π ¯ θ ¯ w ,  \n¯ d := ¯ ˜ d +(¯x A -2 iθ ¯ w ¯ θ ¯ w ) p (0) +2(x (0) -iθ w ¯ θ ¯ w -iθ ¯ w ¯ θ w )¯ p + θ ¯ w Π θ w + ¯ θ ¯ w ¯ Π ¯ θ w , (C.17)  \nd (0) := ˜ d (0) -2x A ¯ p +2¯x A p + θ ¯ w Π θ ¯ w + ¯ θ ¯ w ¯ Π ¯ θ ¯ w -θ w Π θ w -¯ θ w ¯ Π ¯ θ w .  \n[ ˜ d (0) , Z ( M ) An ] PB = 0 , [ ˜ d , Z ( M ) An ] PB = 0 , [ ¯ ˜ d , Z ( M ) An ] PB = 0 , (C.18)  \nThis implies that  \nwhile  \n[ d (0) , Z M ] PB = 0 , [ d , Z M ] PB = 0 , [ ¯ d , Z M ] PB = 0 . (C.19)  \nThe calculation of all the canonical and covariant momenta result in the (primary) constraints  \nΦ (0) := p (0) -m ≈ 0 , ¯ Φ := ¯ p ≈ 0 , Φ := p ≈ 0 , (C.20)  \n˜ d (0) -4 imθ ¯ w ¯ θ w ≈ 0 , ˜ d + m x A ≈ 0 , ¯ ˜ d + m ¯ x A ≈ 0 , (C.21)  \nd w := Π θ w ≈ 0 , d ¯ w := Π θ ¯ w +2 im ¯ θ w ≈ 0 , (C.22)  \n¯ d ¯ w := ¯ Π ¯ θ ¯ w ≈ 0 , ¯ d w := ¯ Π ¯ θ w +2 imθ ¯ w ≈ 0 . (C.23)  \nWe have written these in three columns in such a way that the lines correspond to different sectors of dynamical variables, from the second and third columns one can read the pairs of conjugate second class constraints, which in the case of bosonic constraints are explicitly solved, while the first column contains a prototypes of the first class constraints.  \nThe canonical Hamiltonian vanishes on the surface of primary constraints, this is to say  \nH 0 ≈ 0 . (C.24)  \nThe true first class constraints,  \nΦ (0) ≈ 0 , d w , ≈ 0 , ¯ d ¯ w ≈ 0 , and ˜ ˜ U (0) ≈ 0 , (C.25)  \nare given by the first equations in (C.20), (C.22), (C.23) and by the sum of the first equation in (C.21) with certain linear combination of the second class constraints,  \n˜ ˜ U (0) = ˜ d (0) -4 imθ ¯ w ¯ θ w -2x A ¯ p +2¯x A p -¯ θ w ¯ d w + θ ¯ w d ¯ w = = ˜ d (0) -2x A ¯ p +2¯x A p -¯ θ w ¯ Π ¯ θ w + θ ¯ w Π θ ¯ w ≈ 0 . (C.26)  \nAlso the set of second class constraints, to put in the canonical form their algebra, should be redefined a bit and written as  \n2nd class :  \n˜ ˜ d := ˜ d + m x A +2 iθ ¯ w ¯ θ w p , ¯ ˜ ˜ d := ¯ ˜ d + m ¯ x A -2 iθ ¯ w ¯ θ w ¯ p . (C.28)  \n{ ¯ Φ = ¯ p ≈ 0 ˜ ˜ d ≈ 0 ,    Φ = p ≈ 0 ¯ ˜ ˜ d ≈ 0 , { d ¯ w ≈ 0 ¯ d w ≈ 0 , (C.27)  \nwhere  \nThe algebra of the first class constraints (C.25) is Abelian (this is to say all the Poisson brackets of the first class constraints vanish). The only nonvanishing brackets of the first class constraints with the second class constraints appear when the first class constraint is given by the U(1) generator (C.26),  \n[ ˜ ˜ U (0) ,           ¯ p ˜ ˜ d p ¯ ˜ ˜ d d ¯ w ¯ d w           ] PB =           -2¯ p 2 ˜ ˜ d 2 p -2 ¯ ˜ ˜ d d ¯ w -¯ d w           . (C.29)  \nThe coefficients in the r.h.s. represent the charges of second class constraints with respect to U(1) gauge symmetry.  \nFinally, the nonvanishing brackets of the second class constraints are  \n[¯ p , ˜ ˜ d ] PB = m, [ p , ¯ ˜ ˜ d ] PB = m, { d ¯ w , ¯ d w } PB = 4 im , (C.30)  \nreflecting their second class nature, and weakly vanishing  \n[ ˜ ˜ d , ¯ ˜ ˜ d ] PB = ˜ ˜ U (0) +2x A ¯ p -2¯x A p + ¯ θ w ¯ d w -θ ¯ w d ¯ w ≈ 0 , (C.31)  \n[ ˜ ˜ d , d ¯ w ] PB = 2 i ¯ θ w p ≈ 0 , [ ˜ ˜ d , d ¯ w ] PB = -2 iθ ¯ w p ≈ 0 , (C.32)  \n[ ¯ ˜ ˜ d , d ¯ w ] PB = -2 i ¯ θ w ¯ p ≈ 0 , [ ¯ ˜ ˜ d , d ¯ w ] PB = 2 iθ ¯ w ¯ p ≈ 0 . (C.33)  \nResuming, the algebra of the constraints is represented in the following Table 3  \nNow the bosonic second class constraints in (C.27) are explicitly resolved with respect to x A , ¯x A and their conjugate momenta. Thus a consistent way is to use these to reduce the phase space of our dynamical  \nTable 3 . Closed algebra of the first and second class constraints of single D0-brane system.  \nsystem before quantization. An equivalent way is provided by (the classical counterpart of) the generalized/deformed Gupta-Bleuler procedure which implies that we impose on quantum system only half of the second class constraints p ≈ 0 and ¯ p ≈ 0. At the classical level this implies that we omit the conjugate constraints ¯ ˜ ˜ d ≈ 0 and ˜ ˜ d ≈ 0 from the consideration thus converting their conjugate p ≈ 0 and ¯ p ≈ 0 into the first class constraints 23 .  \nWe prefer this later treatment of the bosonic second class constraints because it will be in consonance with the (canonical) Gupta-Bleuler treatment of the fermionic second class constraints in (C.27). These are complex conjugates and, after quantization, we will impose on the state vector the quantum counterpart of only one of two fermionic second class constraints, ¯ d w ≈ 0. This is tantamount to omitting, at the classical level, the conjugate d ¯ w ≈ 0 constraints, which converts ¯ d w ≈ 0 into the first class constraints. The resulting algebra of the effective first class constraints appearing as a result of the implementation of the above classical counterpart of the generalized/deformed Gupta-Bleuler procedure is presented in Table 4.  \nTable 4 . Closed algebra of the effective first class constraints of single D0-brane system.  \nTo quantize our dynamical system in the supercoordinate representation we represent our effective first class constraints by differential operators,  \nˆ Φ (0) = -i∂ x (0) -m, (C.34)  \nˆ ¯ Φ = -i∂ x A , ˆ Φ = -i ¯ ∂ ¯ x A , (C.35)  \nˆ ˜ ˜ U (0) = -i ( D (0) +4 mθ ¯ w ¯ θ w -2x A ∂ x A +2¯x A ∂ ¯ x A -¯ θ w ¯ d w + θ ¯ w d ¯ w -q ) =  \n= -i ( D (0) -2x A ∂ x A +2¯x A ∂ ¯ x A -¯ θ w ∂ ¯ θ w + θ ¯ w ∂ θ ¯ w -q ) , (C.36)  \nˆ d w = -i∂ θ w , (C.37)  \nˆ ¯ d ¯ w = -i ¯ ∂ ¯ θ ¯ w , (C.38)  \nˆ ¯ d w = -i ( ¯ ∂ ¯ θ w -2 mθ ¯ w ) (C.39)  \nand impose them on the state vector. In (C.34)-(C.39) the derivatives and covariant derivatives are defined as follows  \n∂ x (0) = ∂ ∂ x (0) , ∂ x A = ∂ ∂ x A , ∂ ¯ x A = ∂ ∂ ¯ x A , (C.40)  \nD (0) = ¯ w α ∂ ∂ ¯ w α -w α ∂ ∂w α , D = w α ∂ ∂ ¯ w α , ¯ D = ¯ w α ∂ ∂w α . (C.41)  \nNotice the appearance of the ordering constant in the homogeneous differential operator (C.36).  \nTo understand the structure of the fermionic constraint (C.39) it is useful to keep in mind the relation of the derivatives and covariant derivatives in the central and analytical basis which can be obtained from the identity  \nd = d x a ∂ a +d θ α ∂ α +d ¯ θ α ¯ ∂ α + ia D (0) + if ¯ D -i ¯ f D = Π a ∂ a +d θ α D α +d ¯ θ α ¯ D α + ia D (0) + if ¯ D -i ¯ f D =(C.42)  \n= dx (0) ∂ x 0 +dx A ∂ x A +d¯x A ∂ ¯ x A +d θ w ∂ θ w +d θ ¯ w ∂ θ ¯ w +d ¯ θ w ∂ ¯ θ w +d ¯ θ ¯ w ∂ ¯ θ ¯ w + ia ˜ D (0) + if ¯ ˜ D -i ¯ f ˜ D . (C.43)  \nIt implies, in particular,  \n∂ a = u (0) a ∂ x (0) + u a ∂ x A + ¯ u a ∂ ¯ x A , (C.44)  \nD α = ∂ α + i ( γ a ¯ θ ) α = w α ( ∂ θ w + i ¯ θ ¯ w ∂ x (0) ) + ¯ w α ( ∂ θ ¯ w +4 i ¯ θ ¯ w ∂ ¯ x A + i ¯ θ w ∂ x (0) ) , (C.45)  \n¯ D α = ¯ ∂ α + i ( γ a θ ) α = w α ( ¯ ∂ ¯ θ w +4 iθ w ∂ x A + iθ ¯ w ∂ x (0) ) + ¯ w α ( ¯ ∂ ¯ θ ¯ w + iθ w ∂ x (0) ) . (C.46)  \nThe last equation can be used to write (C.39) in the form  \nˆ ¯ d w = -¯ w ¯ D+ imθ ¯ w -4 iθ w ˆ Φ -iθ ¯ w ˆ Φ (0) . (C.47)", 'D Born-Oppenheimer-like approach to asymptotic form of a solution for N = 2 3D mD 0 system': "Following [75] we can search for an asymptotic form of the solution of Eqs. (6.58)-(6.67) using the so-called tabular or endpoint coordinates (see [75] for references)  \nZ I = X I e iβ + | X | -1 / 2 z I , ¯ Z I = X I e -iβ + | X | -1 / 2 ¯ z I , (D.1)  \nin the neighborhood of the classical vacuum configuration Z I = X I e iβ , ¯ Z I = X I e -iβ , i.e. of the general solution of the conditions of vanishing of the YM potential which reduces to /epsilon1 IJK Z J ¯ Z K = 0.  \nIn (D.4) X I is real, X I = ( X I ) ∗ , | X | = √ X I X I , complex z I = (¯ z I ) ∗ are orthogonal to X I , and, besides, can be restricted by 2 conditions which we can choose to be /Rfractur e( z I e -iβ ) = 0,  \nX I z I = 0 = X I ¯ z I , z I e -iβ + ¯ z I e iβ = 0 . (D.2)  \nThe asymptotic regime corresponds to large value of | X | , formally  \nr = | X | = √ X I X I ↦→ ∞ , (D.3)  \nwith z I ¯ z I kept to be finite.  \nIn our case it is convenient to resolve (D.2) by z I = iy I e iβ in terms of real vector y I , so that ¯ z I = -iy I e -iβ and the above coordinate system is finally described by  \nZ I = ( rn I + ir -1 / 2 y I ) e iβ , ¯ Z I = ( rn I -ir -1 / 2 y I ) e -iβ , (D.4)  \nr := | X | = √ X I X I , n I = X I /r = X I / | X | ⇒ n I n I = 1 , (D.5)  \ny I = ( y I ) ∗ , n I y I = 0 . (D.6)  \nTo proceed, it is convenient to introduce also the complex null-vectors which complete the unit vector n I till complete SO(3) frame, i.e. which obey  \nn I U I = 0 , n I ¯ U I = 0 , U I U I = 0 , ¯ U I ¯ U I = 0 , U I ¯ U I = 1 . (D.7)  \nThen  \nδ IJ = n I n J + U I ¯ U J + ¯ U I U J , (D.8)  \nand the constraint (D.6) can be resolved by  \ny I = y U ¯ U I + y ¯ U U I , so that y I y I = | y U | 2 . (D.9)  \nThe orientation of our SO(3) frame is fixed by setting /epsilon1 IJK ¯ U I U J n K = i which implies  \n/epsilon1 IJK ¯ U J n K = -i ¯ U I , /epsilon1 IJK U J n K = iU I . (D.10)  \nUsing these relations, we find  \n/epsilon1 IJK Z J ¯ Z K = 2 √ r ( y U ¯ U I -y ¯ U U I ) = 2 i √ r/epsilon1 IJK y J n K . (D.11)  \nThe derivatives of the constrained vectors can be expressed in terms of three Cartan forms  \nΩ 1 = U I d n I , ¯ Ω 1 = ¯ U I d n I and Ω (0) 1 = ¯ U I d U I (D.12)  \nd n I = U I ¯ Ω 1 + ¯ U I Ω 1 , d U I = U I Ω (0) 1 -n I Ω 1 , d ¯ U I = -¯ U I Ω (0) 1 -n I ¯ Ω 1 . (D.13)  \nThese can be used to express the derivative of the y I coordinates as  \nd y I = (d y U -y U Ω (0) 1 ) ¯ U I +(d y ¯ U + y ¯ U Ω (0) 1 ) U I -n I ( y U ¯ Ω 1 + y ¯ U Ω 1 ) . (D.14)  \nNow, using (D.4), (D.9), (D.13), (D.14) we can calculate d Z I and d ¯ Z I and then decompose differential in the space of our bosonic coordinates written in two ways  \nd = d Z I ∂ I +d ¯ Z I ¯ ∂ I = d r∂ r +d β∂ ( β ) +d y I ∂ y I +d n I ∂ n I (D.15)  \non the basis of 6 independent 1-forms d r , d β , d y U , d y ¯ U , Ω 1 , ¯ Ω 1 24 . In such a way we find the following expression for the derivative with respect to the tabular coordinates  \n∂ r = n I ( e iβ ∂ I + e -iβ ¯ ∂ I ) -i 2 1 r 3 / 2 ( y U ¯ U I + y ¯ U U I ) ( e iβ ∂ I -e -iβ ¯ ∂ I ) , (D.16)  \n∂ ( β ) = irn I ( e iβ ∂ I -e -iβ ¯ ∂ I ) -1 r 1 / 2 ( y U ¯ U I + y ¯ U U I ) ( e iβ ∂ I + e -iβ ¯ ∂ I ) , (D.17)  \n¯ U I ( ∂ y I -i r 1 / 2 ( e iβ ∂ I -e -iβ ¯ ∂ I ) ) = 0 , (D.18)  \nU I ( ∂ y I -i r 1 / 2 ( e iβ ∂ I -e -iβ ¯ ∂ I ) ) = 0 , (D.19)  \n¯ U I ( ∂ n I -r ( e iβ ∂ I + e -iβ ¯ ∂ I ) ) -y ¯ U n I ( ∂ y I -i r 1 / 2 ( e iβ ∂ I -e -iβ ¯ ∂ I ) ) = 0 , (D.20)  \nU I ( ∂ n I -r ( e iβ ∂ I + e -iβ ¯ ∂ I ) ) -y U n I ( ∂ y I -i r 1 / 2 ( e iβ ∂ I -e -iβ ¯ ∂ I ) ) = 0 . (D.21)  \nThese relations are exact. But to solve them for ∂ I and ¯ ∂ I we need to use the conditions of the asymptotic regime r ↦→ ∞ . To this end let us first write Eqs. (D.16)-(D.21) in the form convenient for perturbative solution of the system for  \n∂ ± I = e iβ ∂ I ± e -iβ ¯ ∂ I . (D.22)  \nFirstly we find from (D.17), (D.18) and (D.19)  \nby  \nI  \nn I ∂ -I = -i r ∂ ( β ) -i r √ r ( y U ¯ U I + y ¯ U U I ) ∂ + I , (D.23)  \n¯ U I ∂ -I = -i √ r ¯ U I ∂ y I , U I ∂ -I = -i √ rU I ∂ y I , (D.24)  \nwhich, using (D.8), can be unified in one equation  \n∂ -I = -i √ r ( δ IJ -n I n J ) ∂ y J -i r n I ∂ ( β ) -i r √ r n I ( y U ¯ U J + y ¯ U U J ) ∂ + J . (D.25)  \nNow, using these equations, one can resolve formally Eqs. (D.16), (D.20) and (D.21) with respect to projections of ∂ + I :  \nn I ∂ + I = ∂ r + 1 2 r ( y U ¯ U J + y ¯ U U J ) ∂ y J , (D.26)  \n¯ U I ∂ + I = 1 r ¯ U I ∂ n I -1 r y ¯ U n I ∂ y I + 1 r 2 √ r y ¯ U ∂ ( β ) + 1 r 3 y ¯ U ( y U ¯ U J + y ¯ U U J ) ∂ + J , (D.27)  \nU I ∂ + I = 1 r U I ∂ n I -1 r y U n I ∂ y I ++ 1 r 2 √ r y U ∂ ( β ) + 1 r 3 y U ( y U ¯ U J + y ¯ U U J ) ∂ + J . (D.28)  \nNow notice that, according to (D.27) and (D.28), ( y U ¯ U J + y ¯ U U J ) ∂ + J ∝ 1 r and that the only place where ∂ r appears in the above equations (D.23)-(D.28) is the first term in the r.h.s. of (D.26). With this in mind and using (D.8) (several times) we arrive at the following approximate solution  \n∂ + I = 1 r [ n I ( r ∂ r + 1 2 y J ∂ y J ) +( δ IJ -n I n J ) ∂ n J + y I n J ∂ y J ] + O ( r -5 / 2 ) , (D.29)  \n∂ -I = -i √ r ( δ IJ -n I n J ) ∂ y J -i r n I ∂ ( β ) + O ( r -5 / 2 ) (D.30)  \nin which O ( r -5 / 2 ) terms do not involve ∂ r derivative.  \nNow from (D.22) we easily find  \n∂  \n=  \n1  \n2  \ne  \n-  \n[  \n-  \ni  \n√  \nr  \n(  \nδ  \n-  \nn  \nn  \n)  \n∂  \ny  \nJ  \n+  \n1  \nr  \n(  \nn  \n(  \nr ∂  \nr  \n-  \ni∂  \n(  \nβ  \n)  \n+  \n1  \n2  \ny  \n∂  \ny  \nJ  \n)  \n+  \n+( δ IJ -n I n J ) ∂ n J + y I n J ∂ y J ) ] + O ( r -5 / 2 ) , (D.31)  \n¯ ∂ I = 1 2 e iβ [ + i √ r ( δ IJ -n I n J ) ∂ y J + 1 r ( n I ( r ∂ r + i∂ ( β ) + 1 2 y J ∂ y J ) +  \n+( δ IJ -n I n J ) ∂ n J + y I n J ∂ y J ) ] + O ( r -5 / 2 ) . (D.32)  \nNotice that in the asymptotic region (D.3), r ↦→ ∞ , the leading order in the above decomposition of the derivatives is ∝ √ r ,  \niβ  \nIJ  \nI  \nJ  \nI  \nJ  \nso that  \nand  \n∂ I = -i 2 e -iβ √ r ( δ IJ -n I n J ) ∂ y J + O ( r -1 ) , (D.33)  \n¯ ∂ I = i 2 e iβ √ r ( δ IJ -n I n J ) ∂ y J + O ( r -1 ) , (D.34)  \nn I ∂ I = 0 + O ( r -1 ) , (D.35)  \nn I ¯ ∂ I = 0 + O ( r -1 ) . (D.36)  \nAccording to (D.11) the multiplier √ r is also present in all the nonvanishing r.h.s.'s of equations (6.58)-(6.63) and (6.65)-(6.67) which can be written in the form  \n2 i √ r/epsilon1 IJK y J n K f ( q -1) I = 0 ⇔ y U ¯ U I f ( q -1) I = y ¯ U U I f ( q -1) I , (D.37)  \n¯ ∂ I f ( q ) = 8 i √ 2 √ r/epsilon1 JKL y J n K ¯ ∂ [ L f I ] ( q ) , (D.38)  \n¯ ∂ [ I f J ] ( q -1) = 8 i √ 2 √ ry [ I n J ] f ( q -3) (D.39)  \n∂ I f ( q -1) I = 0 , (D.40)  \n∂ J ¯ ∂ [ I f ( q ) J ] = -i √ 2 µ 12 √ r /epsilon1 IJK y J n K f ( q ) , (D.41)  \n∂ I f ( q -3) = -i √ 2 µ 12 √ r y [ I n J ] f ( q -1) J . (D.42)  \nThe algebraic equation (D.37) can be easily solved by  \nf ( q -1) I = n I f ( q -1) ( n ) + y I f ( q -1) ( y ) . (D.43)  \nNow we can decompose our 'wavefunctions' in the inverse powers of r √ r and try to solve the equation order by order. To be consistent we set  \nf ( q... ) ... = r -k ( f ( q... ) ... [0] + 1 r √ r f ( q... ) ... [1] + 1 r 3 f ( q... ) ... [2] + ... ) , (D.44)  \nwith some k to be determined, for all but the leading components of the state vector superfield while for this we assume  \nf ( q ) = √ r f ( q ) [0] + 1 r f ( q ) [1] + 1 r 2 f ( q ) [2] + ... . (D.45)  \nHere we restrict our discussion by zero-th order in which we find (following the terms ∝ √ r ) the equations  \n( δ IJ -n I n J ) ∂ y J f ( q ) [0] = 8 i √ 2 /epsilon1 JKL y J n K ( δ [ L | P -n [ L n P ) ∂ y P f I ][0] ( q ) , (D.46)  \n( δ [ I | K -n [ I | n K ) ∂ y K f J ][0] ( q -1) = 16 √ 2 e -iβ y [ I n J ] f ( q -3) [0] , (D.47)  \n( δ IJ -n I n J ) ∂ y J f ( q -1) I [0] = 0 , (D.48)  \n( δ JK -n J n K ) ∂ y K ( δ [ I | L -n [ I | n L ) ∂ y L f ( q ) | J ][0] = -2 √ 2 i ( µ 6 ) 2 /epsilon1 IJK y J n K f ( q ) [0] , (D.49)  \n( δ IJ -n I n J ) ∂ y J f ( q -3) [0] = 2 √ 2 ( µ 6 ) 2 e iβ y [ I n J ] f ( q -1) J [0] . (D.50)  \nNotice that this system of equations contains derivatives with respect to variables y I only. With this in mind one easily finds that (D.46) implies  \n/epsilon1 JKL y J n K ( δ [ L | P -n [ L n P ) ∂ y P ( n I f I [0] ( q ) ) = 0 , (D.51)  \nwhich indicates that n I f I [0] ( q ) depends on y J only through its square | y | 2 = y J y J .  \nEq. (D.48) with leading order term of (D.43)  \nf ( q -1) I [0] = n I f ( q -1) ( n )[0] + y I f ( q -1) ( y )[0] (D.52)  \ndoes not impose any condition of f ( q -1) ( n )[0] but requires f ( q -1) ( y )[0] to obey ( y I ∂ y I +2) f ( q -1) ( y )[0] = 0. Nevertheless, as we will see in a moment, we have to choose the trivial solution of this equation.  \nIndeed, using (D.52) in (D.50), we find  \n( δ IJ -n I n J ) ∂ y J f ( q -3) [0] = √ 2 ( µ 6 ) 2 e iβ ( y I f ( q -1) ( n )[0] -n I ( y J y J ) f ( q -1) y [0] ) . (D.53)  \nAs both l.h.s. and the first term in the r.h.s. of this equation vanish when contracted with n I , the last term should vanish by itself,  \nf ( q -1) y [0] = 0 ⇒ f ( q -1) I [0] = n I f ( q -1) ( n )[0] . (D.54)  \nThen (D.53) reduces to  \n( δ IJ -n I n J ) ∂ y J f ( q -3) [0] = √ 2 ( µ 6 ) 2 e iβ y I f ( q -1) ( n )[0] . (D.55)  \nThis equation implies that both f ( q -3) [0] and f ( q -1) ( n )[0] depend on y I vector coordinate only through its length | y | = √ y I y I and that  \n∂ | y | 2 f ( q -3) [0] = 1 ( µ 6 ) 2 √ 2 e iβ f ( q -1) ( n )[0] . (D.56)  \nOn the other hand, with the above conclusion on dependence on y I vector only through its length | y | = √ y I y I , Eq. (D.39) reduces to ∂ | y | 2 f ( q -1) ( n )[0] = 8 √ 2 e -iβ f ( q -3) [0] . Using this and (D.56), we find a simple equation  \nwhich is solved by  \nf ( q -1) ( n )[0] = h ( q -1) ( r, β, /vectorn ) exp ( -2 √ 2 | y | 2 µ 6 ) (D.58)  \nwith some function h ( q -1) ( r, β, /vectorn ) depending on the remaining coordinates.  \nSimilarly one can study the case of other components of the state vector superfield, the statistics of which (bosonic if the superfield is bosonic) is opposite to the above discussed components (fermionic if the superfield is bosonic) which obey Eqs. (D.46) and (D.49). As the study of this case is a bit more involved then above, we will simplify it a bit by choosing a special frame in which n I = (0 , 0 , 1) = δ I 3 and hence y I = ( y 1 , y 2 , 0) = δ I r y r .  \nIn this SO(3) frame the solution of (D.51) can be written in the form f 3[0] ( q ) ( y J ) = f 3[0] ( q ) ( | y | 2 ) and the remaining components of (D.46) and (D.49) acquire the following simple form  \n∂ y r f ( q ) [0] = 8 i √ 2 /epsilon1 st y t ∂ y [ s f r ][0] ( q ) , (D.59)  \n∂ y s ∂ y [ r f ( q ) s ][0] = -i 2 √ 2 ( µ 6 ) 2 /epsilon1 rs y s f ( q ) [0] . (D.60)  \nFurthermore, as r, s, t = 1 , 2, ∂ y [ s f r ][0] ( q ) = /epsilon1 sr ˜ f ( q ) [0] (with ˜ f ( q ) [0] = 1 2 /epsilon1 sr ∂ y [ s f r ][0] ( q ) ) and the above equations further simplify to  \n∂ y r f ( q ) [0] = 8 i √ 2 y r ˜ f ( q ) [0] , ∂ y s ˜ f ( q ) [0] = -2 √ 2 i ( µ 6 ) 2 y s f ( q ) [0] . (D.61)  \nThen it is easy to conclude that both f ( q ) [0] and ˜ f ( q ) [0] depend on y I through its square | y | 2 = y I y I and obey the second order linear differential equation in | y | 2 , in particular  \n∂ | y | 2 ∂ | y | 2 f ( q ) [0] = 8 ( µ 6 ) 2 f ( q ) [0] (D.62)  \nwhich is solved by  \nf ( q ) [0] = h ( q ) [0] ( r, β, /vectorn ) exp ( -2 √ 2 µ 6 | y | 2 ) . (D.63)  \nThe sign of the expressions in the exponents of (D.58) and (D.63) is chosen to be such that the wavefunction is convergent in y I . We will not consider higher orders of the Born-Oppenheimer-like approximation, as this goes beyond the scope of this paper, but refer to [75] for such a study. Notice that there, besides the case of Matrix model obtained by dimensional reduction of D = 3 SYM model to d = 1 also the Matrix models proceeding from reduction of higher D SYM were studied and the exceptional properties of D = 10 case were noticed. This will appear in a setup similar to (6.49)-(6.51) from the field theory of our 10D mD0 model which we plan to quantize in the near future.  \n∂ ∂ | y | 2 ∂ ∂ | y | 2 f ( q -1) ( n )[0] = 8 ( µ 6 ) 2 f ( q -1) ( n )[0] (D.57)", 'References': '- [84] P.A.M. Dirac, Lectures on quantum mechanics , Academic Press, NY (1967).'}