Reducible second-class constraints of order L: An irreducible approach

An irreducible canonical approach to second-class constraints reducible of an arbitrary order is given. This method generalizes our previous results from [Europhys. Lett. 50 (2000) 169, J. Phys. A: Math. Theor. 40 (2007) 14537] for first- and respectively second-order reducible second-class constraints. The general procedure is illustrated on Abelian gauge-fixed p-forms.


Introduction
The canonical approach to systems with reducible second-class constraints is quite intricate, demanding a modification of the usual rules as the matrix of the Poisson brackets among the constraint functions is not invertible. Thus, it is necessary to isolate a set of independent constraints and then construct the Dirac bracket [1,2] with respect to this set. The split of constraints may however lead to the loss of important symmetries, so it should be avoided. As shown in [3,4,5,6,7,8], it is however possible to construct the Dirac bracket in terms of a noninvertible matrix without separating the independent constraint functions. A third possibility is to substitute (by an appropriate extension of the phase-space) the reducible second-class constraints with some irreducible ones, defined in the extended phase-space, and further work with the Dirac bracket based on the irreducible constraints. This idea, suggested in [9] mainly in the context of 2-and 3-form gauge fields, has been developed in a general manner only for first-and respectively second-order reducible second-class constraints [10,11]. Other interesting contributions to reducible second-class constrained systems (including the split involution formalism) can be found in [12,13,14,15,16]. The idea of extending the phase-space is not new. It has been used previously, for instance in the context of the conversion approach exposed in [17], where some supplementary variables are added in order to convert a set of irreducible second-class constraints into a first-class one.
In this paper we give an irreducible approach to third-order reducible second-class constraints and then generalize the results to an arbitrary order of reducibility, L. Our strategy includes three main steps. First, we express the Dirac bracket for the reducible system in terms of an invertible matrix. Second, we construct an intermediate reducible second-class system (of the same reducibility order like the original one) on a larger phase-space and establish the (weak) equality between the original Dirac bracket and that corresponding to the intermediate theory. Third, we prove that there exists an irreducible second-class constraint set equivalent to the intermediate one, such that the corresponding Dirac brackets coincide (weakly). These three steps enforce the fact that the fundamental Dirac brackets derived within the irreducible and original reducible settings coincide (weakly). The equality between the fundamental Dirac brackets associated with the original phasespace variables in the reducible and respectively irreducible formulations has major implications on the relationship between the reducible and irreducible systems: i) the two systems exhibit the same number of physical degrees of freedom, which is precisely the rank of the induced symplectic form (since the Dirac bracket restricted to the constraint surface is determined by the inverse of the induced symplectic form, see Theorem 2.5 from [7]); ii) the physical content of the two theories is the same from the perspective of quantization as they display the same fundamental observables; iii) the original, reducible system can be equivalently replaced with the irreducible one. It is important to remark that the irreducible approach is useful mainly in field theory because it does not spoil the important symmetries of the original system, such as the spacetime locality of second-class field theories.
The present paper is organized into six sections. In Section 2 we briefly review the procedure for second-class constraints that are reducible of order one and respectively two. Sections 3 and 4 define the 'hard core' of the paper. We initially approach second-class constraints reducible of order three in Section 3 by implementing the three main steps mentioned above, and then generalize these results to an arbitrary order of reducibility in Section 4. In Section 5 we exemplify in detail the general procedure from Section 4 on gauge-fixed Abelian p-form gauge fields. Section 6 ends the paper with the main conclusions.
2 First-and second-order reducible secondclass constraints: a brief review

Dirac bracket for first-and second-order reducible second-class constraints
We start with a system locally described by N canonical pairs z a = (q i , p i ) and subject to the constraints For simplicity, we take all the phase-space variables to be bosonic. However, our analysis can be extended to fermionic degrees of freedom modulo including some appropriate phase factors. We choose the scenario of systems with a finite number of degrees of freedom only for notational simplicity, but our approach is equally valid for field theories. In addition, we presume that the functions χ α 0 are not all independent, but there exist some nonvanishing functions Z α 0 α 1 such that Moreover, we assume that the functions Z α 0 α 1 are all independent and (2) are the only reducibility relations with respect to the constraints (1). These constraints are purely second class if any maximal, independent set of M 0 − M 1 constraint functions χ A (A = 1, M 0 − M 1 ) among the χ α 0 is such that the matrix C is invertible. Here and in the following the symbol [, ] denotes the Poisson bracket. In terms of independent constraints, the Dirac bracket takes the form where M (1)AB C (1) In the previous relations we introduced an extra index, (1), having the role to emphasize that the Dirac bracket given in (4) is based on a first-order reducible second-class constraint set. We can rewrite the Dirac bracket expressed by (4) without finding a definite subset of independent second-class constraints as follows. We start with the matrix which clearly is not invertible because Ifā α 1 α 0 is a solution to the equation then we can introduce a matrix [6] of elements M (1)α 0 β 0 through the relation with defines the same Dirac bracket like (4) on the surface (1). We remark that there exist some ambiguities in defining the matrix of elements M (1)α 0 β 0 since if we make the transformation with q α 1 β 1 some completely antisymmetric functions, then equation (8) is still satisfied. Relations (6) and (8) show that which ensures the fact that the rank of the matrix of elements is equal to the number of independent second-class constraints in the presence of the first-order reducibility. Let us extend the previous construction to the case of second-order reducible second-class constraints. This means that not all of the first-order reducibility functions Z α 0 α 1 are independent. Beside the first-order reducibility relations (2), there appear also the second-order reducibility relations We will assume that the reducibility stops at order two, so all the functions Z α 1 α 2 are by hypothesis taken to be independent. It is understood that the functions Z α 1 α 2 define a complete set of reducibility functions for Z α 0 α 1 . In this situation, the number of independent second-class constraints is equal to M 0 − M 1 + M 2 . As a consequence, we can work with a Dirac bracket of the type (4), but in terms of It is obvious that the matrix of elements C satisfies the relations so its rank is equal to M 0 − M 1 + M 2 . LetĀ α 2 α 1 be a solution of the equation We define an antisymmetric matrix, of elementsω α 1 β 1 , through the relation Taking (17) into account, it results thatω α 1 β 1 contains some ambiguities, namely it is defined up to the transformation with q α 2 β 2 some arbitrary, antisymmetric functions. On the other hand, simple computation shows that the matrix of elements D α 1 γ 1 satisfies the proper-tiesĀ Based on the latter formula from (20), we infer an alternative expression for for some functionsĀ α 1 α 0 . From the former relation in (21) and (22) we deduce that where At this stage, we can rewrite the Dirac bracket given in (13) without separating a specific subset of independent constraints. In view of this, we introduce an antisymmetric matrix, of elements M (2)α 0 β 0 , through the relation such that formula defines the same Dirac bracket like (13) on the surface (1). It is simple to see that M (2)α 0 β 0 also contains some ambiguities, being defined up to the transformation withq α 1 β 1 some antisymmetric, but otherwise arbitrary functions. Relations (12) and (23) ensure that so the rank of the matrix of elements M (2)α 0 β 0 C β 0 γ 0 is equal to the number of independent second-class constraints also in the presence of the second-order reducibility.
Direct manipulations emphasize that the Dirac bracket in each case, (9) and (26) respectively, satisfies the relations (where the index (1) corresponds to (9) and the index (2) to (26) respectively), so the property [χ α 0 , G] (1,2) * = 0 (for any G) indeed holds on the surface of first-or second-order reducible second-class constraints respectively. In the meanwhile, each of the Dirac brackets (9) or (26) satisfies the Jacobi identity, but only in the weak sense.

Irreducible analysis of first-and second-order reducible second-class constraints
As it has been shown in [10], first-order reducible second-class constraints can be approached in an irreducible manner. To this end, one starts from the solution to equation (7)ā where a γ 1 α 0 are some functions chosen such that andD β 1 γ 1 stands for the inverse of Z α 0 α 1 a γ 1 α 0 . In order to develop an irreducible approach, it is necessary to enlarge the original phase-space with some new variables, (Y α 1 ) α 1 =1,M 1 , endowed with the Poisson brackets where Γ α 1 β 1 are the elements of an invertible, antisymmetric matrix that may depend on the newly added variables. Consequently, one constructs the constraintsχ which are second-class and, essentially, irreducible. Following the line exposed in [10] it can be shown that the Dirac bracket associated with the irreducible constraints (33) takes the form and it is (weakly) equal to the original Dirac bracket (9) In (34) the quantities µ (1)α 0 β 0 are the elements of an invertible, antisymmetric matrix, expressed by with Γ β 1 γ 1 the inverse of Γ α 1 β 1 . Formula (35) is essential in our context because it proves that one can indeed approach first-order reducible secondclass constraints in an irreducible fashion.
In the case of second-order reducible second-class constraints, one constructs the irreducible constraints where withÊ ρ 1 α 1 the elements of an invertible matrix [11]. Following the line exposed in [11] it can be shown that the Dirac bracket associated with the irreducible constraints (37) takes the form where andê α 1 σ 1 are the elements of the inverse of the matrix with the elementsÊ γ 1 α 1 . In (39) the quantities denoted by A τ 2 λ 1 are some functions chosen such that andD β 2 τ 2 stand for the elements of the inverse of the matrix with the elements Moreover, according to the general proof from [11], one has which shows that second-order reducible second-class constraints can also be approached in an irreducible fashion.
3 Third-order reducible second-class constraints 3.1 Reducible approach 3.1.1 Dirac bracket for third-order reducible second-class constraints In this section we will consider third-order reducible second-class constraints. This means that, beside the first-order reducibility relations (2), the following relations also hold They are known as the reducibility relations of order two and three, respectively. In addition, all the third-order reducibility functions Z α 2 α 3 are assumed to be independent. Under these circumstances, the number of independent second-class constraint functions is equal to M ≡ M 0 − M 1 + M 2 − M 3 . As a consequence, we can work again with a Dirac bracket of the type (4), but written in terms of M independent functions χ A , i.e.
where C It is clear that the matrix of elements C also satisfies the relations and, actually, its rank is equal to M.
LetĀ α 3 α 2 be a solution to Then, we can introduce an antisymmetric matrix, of elementsω β 2 γ 2 , defined through the relationω If we take into account equation (50), then it can be checked thatω β 2 γ 2 are defined up to the transformation whereq β 3 γ 3 are some arbitrary, antisymmetric functions. On the other hand, simple computation shows that the matrix of elements D γ 2 α 2 satisfies the relations Based on the latter formula from (53), we find that D γ 2 α 2 can alternatively be expressed as for some functionsĀ γ 2 α 1 . We notice that the above mentioned functions are defined up to the transformations with µ γ 2 α 0 some arbitrary functions. Using now the former relation from (54) and (55), we deduce that where Relations (57) and (58) ensure that D γ 1 α 1 is a 'projection' (idempotent) in the weak sense With D γ 1 α 1 of the form (58) at hand, from (44) it follows that Formula (57) emphasizes an alternative expression for D γ 1 for some functionsĀ γ 1 α 0 . Accordingly, from (60) and (61) we find that where Just like before, from relations (62) and (63) we obtain that D γ 0 α 0 is also a 'projection' in the weak sense At this stage, we can rewrite the Dirac bracket expressed by (46) in terms of all the second-class constraint functions. In view of this, we add an antisymmetric matrix, of elements M (3)α 0 β 0 , through the relation such that the formula defines the same Dirac bracket like (46) on the surface (1). It is simple to see that the elements M (3)α 0 β 0 are defined up to the transformation with p α 1 β 1 some arbitrary, antisymmetric functions. We notice that relations (44), (45), and (62) ensure that and hence the rank of the matrix of elements C α 0 β 0 M (3)β 0 γ 0 is equal to the number of independent second-class constraints in the case of the reducibility of order three. Meanwhile, we have that so [χ α 0 , G] (3) * = 0, for any G, on the surface of third-order reducible secondclass constraints.

Expressing the Dirac bracket in terms of an invertible matrix
Initially, we will establish some useful properties of the functionsĀ α 1 α 0 ,Ā α 2 α 1 , andĀ α 3 α 2 . We introduce (55) in the former relation from (53) and infer which implies the existence of some smooth functions M γ 3 α 0 such that On the other hand, the functionsĀ α 2 α 1 contain the ambiguities given in (56), which can be speculated via choosing µ α 2 α 0 = −M γ 3 α 0 Z α 2 γ 3 such that these functions satisfy the conditionsĀ Using definition (51) and relations (58) and (72), we obtain By inserting now (74) in (61), we deduce the relation which enables us, by means of equations (58) and (63), to establish the formulas Before expressing the Dirac bracket in terms of an invertible matrix, let us analyze equations (49) and (50). The solution to (49) may be set under the formĀ where A β 3 α 2 are some functions taken such that the matrix of elements The notationsD α 3 β 3 stand for the elements of the inverse of D γ 3 while (78) and the former relation from (53) lead to Inserting D β 2 α 2 given by (81) in (73), we deducē Employing now the latter relation from (53), we get that the solution to equation (50) reads asω withω γ 2 δ 2 the elements of an antisymmetric matrix. Multiplying (51) with A γ 3 γ 2 and taking into account (82), we infer the equation 1 Strictly speaking, the solution to equation (49) has the general formĀ α3 and v λ2α3 are some arbitrary functions. If we make the redefinitions u α3 α1 =û λ3 α1D α3 λ3 and v λ2α3 =v λ2λ3Dα3 λ3 , withû λ3 α1 andv λ2λ3 some arbitrary functions, then we can bringĀ α3 α2 to the formĀ α3 λ3 . On the other hand, the quantities A λ3 α2 taken such that the rank of (79) is maximum are defined up to the transformation A λ3 , with τ λ3 α1 and λ λ2λ3 also arbitrary. Thus, we can absorb the quantity Z α1 α2û λ3 α1 +ω α2λ2v λ2λ3 fromĀ α3 α2 through a redefinition of A λ3 α2 and finally obtain solution (78). whose solution isω Since the matrix of elementsω β 2 γ 2 is defined up to transformation (52), we are free to make the choiceq β 3 γ 3 ≈ −Q β 3 λ 3D γ 3 λ 3 , which brings the solution to equation (85) at the formω which further impliesω withω ρ 2 σ 2 the elements of an antisymmetric matrix. Under these conditions, the next theorem can be proved to hold.
(b) Simple computation outputs which further implyω such that (b) is also proved. Letω α 1 β 1 = −ω β 1 α 1 be a solution to the equation Then, one can introduce an antisymmetric matrix, of elementsω β 1 γ 1 , through the relationω Due to (99), we conclude that the elementsω β 1 γ 1 are defined up to the transformationω withq β 2 γ 2 some antisymmetric, but otherwise arbitrary functions. Recalling relation (57), we obtain that the solution to (99) can be expressed as withω γ 1 δ 1 the elements of an antisymmetric matrix. Acting withĀ γ 2 γ 1 on (100) and taking into account the result given by (74), we infer the equation whose solution reads asω Due to the fact that the matrix of elementsω β 1 γ 1 is defined up to transformation (101), we are free to make the choiceq β 2 γ 2 ≈ −Q β 2 γ 2 , which brings equation (103) at the formω such that its solution can be taken as 2 withω λ 1 ρ 1 the elements of an antisymmetric matrix. Except from being antisymmetric, the matrices of elementsω γ 1 δ 1 and respectivelyω λ 1 ρ 1 are arbitrary at this stage. The next theorem shows that they are in fact related.
(b) By straightforward computation, it results which further yieldsω and proves (b).
With these elements at hand, the next theorem is shown to hold.
Proof. First, we observe that D α 0 γ 0 given in (63) satisfies the relations Multiplying (65) byĀ γ 1 γ 0 and using (76), we obtain the equation which then leads to for some functions f β 1 γ 1 . Acting with D τ 0 β 0 on (129) and employing (62), we find the relation with the help of which (via formula (76)) we can write for some λ β 0 τ 0 . Acting now with D τ 0 β 0 on (65) and taking into account (131), On the other hand, relation (127) implies such that, on behalf of (132) and (133), we have − C Comparing (134) with (65) and using the fact that the functions M (3)α 0 β 0 are defined up to transformation (67), we infer the relation which substituted in (131) provides the equation Using one more time the fact that the elements M (3)α 0 β 0 are defined up to (67), from (136) we get where µ (3)λ 0 σ 0 is an antisymmetric matrix. Due to formula (76) and relation (137) we can write Inserting the former relation from (59) in (137), we deduce which further yields for an antisymmetric matrix, of elements ν λ 1 σ 1 . Now, we show that the matrix of elements µ (3)λ 0 σ 0 can be taken to be invertible. If we take ν λ 1 σ 1 under the form ν λ 1 σ 1 =ω λ 1 σ 1 , whereω λ 1 σ 1 are precisely the elements of the invertible matrix given in (111), then we find directly Next, we show that the matrix of elements whereω ρ 1 τ 1 determines the invertible matrix given in (110) Taking into account the results of Theorem 2 (see (107)) and (60)), we arrive at the relationĀ which substituted into (143) leads us to the formula proving that the matrix of elements µ (3)λ 0 σ 0 given by (141) where ω α 1 β 1 and ω α 3 β 3 are the elements of some antisymmetric, invertible matrices, and consider a system subject to the reducible second-class constraints In what follows we will call the system subject to constraints (147) "intermediate system". The Dirac bracket on the phase-space locally described by (z a , y α 1 , y α 3 ) constructed with respect to the above second-class constraints reads as where the Poisson brackets from the right-hand side of (148) contain derivatives with respect to all the variables z a , y α 1 , and y α 3 . The notations ω α 1 β 1 and ω α 3 β 3 denote the elements of the inverses of the matrices of elements ω α 1 β 1 and ω α 3 β 3 respectively. The most general form of a function defined on the phase-space of coordinates (z a , y α 1 , y α 3 ) is given by where y A = (y α 1 , y α 3 ), F 0 (z a ) = F (z a , 0), and By inserting (149) in (148) we obtain where the previous weak equality holds on the surface (147). Moreover, equations (1) and (147) describe the same surface, but embedded in two phase-spaces of different dimensions. In other words, equations (1) and (147) represent equivalent descriptions of one and the same constraint surface. For this reason, we will maintain the symbol of weak equality with respect to both descriptions 5 . Substituting (149) in (148) and taking into account (150), we infer We recall that the Dirac bracket [F, G] (3) * contains only derivatives with respect to the variables z a .

Irreducible system
Letê α 2 σ 2 be the elements of an invertible matrix, taken such that with where σ α 1 λ 1 and σ α 2 β 2 determine some invertible matrices. From (152) it is easy to see that withÊ α 2 σ 2 the elements of the inverse of the matrix of elementsê α 2 σ 2 . Substituting (152) in (75) and taking into account the invertibility of the matrix of elementsê α 2 σ 2 , we obtainĀ Next, we add an invertible matrix, whose elements will be denoted byÊ γ 1 α 1 , through the relationsω and define the functions Then, it is clear thatω withê α 1 σ 1 the elements of the inverse ofÊ γ 1 α 1 , while (157) produces In this context the next theorem is shown to hold.

Theorem 4
The elementsê α 1 σ 1 andÊ τ 1 β 1 can be taken such that Proof. We takeÊ α 1 β 1 andê α 1 β 1 such that the following relations are satisfied: where the matrix of elements σ α 0 β 0 is taken to be invertible and σ β 1 α 1 are the elements of the inverse of the matrix of elements σ α 1 λ 1 . By 'solving' (153) and (161) with respect to the reducibility functions of order one and two where σ α 0 β 0 and σ λ 2 τ 2 are the elements of the inverses of the matrices of elements σ α 0 β 0 and σ α 2 β 2 respectively, we can write From (164) and taking into account (159) and (162), we deduce the relation Inserting now (155) in (165), we arrive at Based on the results expressed by (155) and (166), we are able now to prove the validity of (160). If we make the notation then it is easy to see thatD α 1 β 1 is a 'projection' On the other hand, with the help of relations (153) and (161), we deduce that A α 1 α 0 A α 2 α 1 ≈ 0, which further implies and hence we findĀ Applying Z α 0 α 1 on (167) and relying on (166), we get Multiplying (170) with Z α 0 ρ 1 and (171) withĀ α 1 α 0 , we are led tô The general solution to equations (172) is of the form for an arbitrary matrix of elements M λ 2 τ 2 . Direct computation yieldŝ Comparing (174) with (168) and employing (173), we obtain that the elements M λ 2 τ 2 are subject to the equations It is easy to see that equations (175) possess two types of solutions, namely and If we employ solution (176) 6 , from (173) we infer such that (160) is valid. This proves the theorem.
Replacing (156) and (158) in (107) and recalling (160), it is easy to obtain the relation On the other hand, formulae (156)-(158) imply that µ (3)λ 0 σ 0 and µ σ 0 ρ 0 given by (141) and (142) respectively can be expressed as At this point, we construct the constraints Under these considerations, we are able to prove the following key theorem.
(ii) With the help of formulae (182) and (183), we find the expressions of the Poisson brackets among the functionsχ ∆ as: where µ α 0 β 0 reads as in (181). Then, the matrix of their Poisson brackets, of elements C ∆∆ ′ , takes the concrete form where ∆ = (α 0 , α 2 ) indexes the line, ∆ ′ = (β 0 , β 2 ) the column, and φ In order to prove the invertibility of the matrix (192), we will give its inverse. Direct computation shows that the matrix with µ (3)β 0 ρ 0 given by (181) and ψ (3)β 2 ρ 2 of the form satisfies the relations so it is indeed the inverse of (194). This proves (ii).
(iii) Since matrix (192) is invertible, it follows that it possesses no nontrivial null vectors and hence the functionsχ ∆ are independent, which is equivalent to the fact that the constraint set given by (182) and (183) is irreducible. This proves (iii).
Taking into account the result (194), the Dirac bracket built with respect to the irreducible second-class constraint set (182) and (183) takes the concrete form Theorem 6 The Dirac bracket with respect to the irreducible second-class constraints (198) coincides with that of the intermediate system Proof. In order to prove this theorem, we start from the right-hand side of (198) and show that it is (weakly) equal with the right-hand side of (148).
Substituting the previous results in (198), we arrive precisely at (199), which proves the theorem.

Basic result for L = 3
Combining (151) and (199), we are led to the result The last formula proves that we can indeed approach third-order reducible second-class constraints in an irreducible fashion.
4 Generalization to an arbitrary reducibility order L

Reducible approach
In the sequel we generalize the previous results to the case of a system of second-class constraints, reducible of an arbitrary order L with α k = 1, M k for each k = 1, L. In addition, the reducibility functions of maximum order (L), Z α L−1 α L , are assumed to be all independent. Consequently, the number of independent second-class constraints is equal to where C The matrix of the Poisson brackets among the constraint functions is not invertible due to the relations but its rank is equal to M. Just like in the case of order three of reducibility, we introduce some functions Ā α k α k−1 k=1,L , subject to the relations such that defines the same Dirac bracket like (206) on the surface (1). Similar to the case of third-order reducible second-class constraints, the Dirac bracket for L-order reducible constraints can be expressed in terms of a noninvertible matrix.
The relationship between the invertible matrix µ (L) and the matrix M (L) is given by a relation similar to that from the third-order reducible case where ω α i β j are the elements of an antisymmetric, invertible matrix, and consider the system subject to the reducible second-class constraints The system constrained to satisfy (214) will be called , constructed with respect to the above second-class constraints, reads as where the Poisson brackets from the right-hand side of (215) contain derivatives with respect to all the variables z a and y α 2k+1 k=0,[ L−1 2 ] and ω α 2k+1 β 2k+1 denote the elements of the inverse of the matrix of elements ω α 2k+1 β 2k+1 . In this case the most general form of a function defined on the phase-space locally parameterized by z a , y α 2k+1 k=0, is given by , F 0 (z a ) = F 0 (z a , 0), and If we introduce (216) in (215), then we obtain where the previous weak equality takes place on the surface defined by (214). Moreover, equations (1) and (214) describe the same surface, but embedded in phase-spaces of different dimensions, such that (1) and (214) are equivalent descriptions of one and the same constraint surface. This is why we will maintain the same sign of weak equality related to both descriptions 8 . Replacing (216) in (215) and making use of (217), we infer the result We recall the fact that the Dirac bracket [F, G] (L) * contains only derivatives with respect to the original phase-space variables z a .

Irreducible system
In order to construct the irreducible system in the general case, we act in a manner similar to that exposed in subsection 3.2.2 and start by adding the constraints: 8 It is understood that for the functions defined on the phase-space locally parameterized by the variables z a we use (1) and for those defined on the larger phase-space, of coordinates , we employ representation (214).
These constraints are defined on the larger phase-space, locally parameterized by z a , y α 2k+1 k=0, . The functions A α 2k+1 α 2k appearing in the above are defined by the relations: The elementsê α 2k+1 β 2k+1 determine an invertible matrix andD α L β L are the elements of the inverse of the matrix of elements In the following we show that (219) and (220) (or (221)-(223)) display all the desired properties: equivalence with the intermediate system (214), second-class behaviour, irreducibility and, most important, the fact that associated Dirac bracket (weakly) coincides with the original one, corresponding to the second-order reducible second-class constraints. The proof of all these properties is contained within the next two theorems.
(ii) Now, we employ formulae (219) and (220) (or (221)-(223)) and find the concrete form of the Poisson brackets among the constraint functionsχ ∆ as: with k = 1, L 2 − 1 in (241) and k = 1, L 2 in (242). Accordingly, the matrix of elements given in (228) reads as where The last block on the main diagonal of (244) is of the type (245), with k = L 2 for L odd or respectively of the form for L even. The invertibility of C ∆∆ ′ is obtained by constructing its inverse, which can be checked to have the expression The last block on the main diagonal of (247) is given by (248), with k = L 2 for L odd or respectively for L even. Indeed, simple computation yields such that (244) is indeed invertible and its inverse is expressed by (247). This proves (ii).
(iii) As (244) is invertible, it follows that it displays no null vectors and hence the functionsχ ∆ are all independent or, in other words, the constraint set (219) and (220) (or (221)-(223)) is irreducible. This proves (iii).
Taking into account the result given by (247), it follows that the Dirac bracket built with respect to the irreducible second-class constraints (219) and (220) (or (221)-(223)) takes the particular form Theorem 9 The Dirac bracket with respect to the irreducible second-class constraints (252) coincides with that of the intermediate system Proof. We start from the right-hand side of (252) and show that it is (weakly) equal to the right-hand side of (215). By direct computation, we obtain that: with k = 0, L 2 − 1. Also direct computation provides: with k = 0, L 2 − 2. Further computation finally gives: Inserting the last formulae in (252) we arrive at (253), which proves the theorem.

Main result
Based on (218) and (253), we are led to the relation which expresses the fact that second-class constraints reducible of an arbitrary order L can be systematically approached in an irreducible manner. This is the key result of the present paper.

Geometrical interpretation of the irreducible approach
Let us denote by P the original phase-space and by P ′ the phase-space of the intermediate system, and hence also of the irreducible theory. Both are symplectic manifolds endowed with symplectic two-forms whose coefficients are in each case the elements of the inverse of the matrix having as elements the fundamental Poisson brackets. We denote by Σ and respectively Σ ′ the second-class constraint surface for the original system and respectively for the intermediate theory. By Theorem 8 it follows that the second-class constraint surface of the irreducible system, given by equations (219) and (220) (or (221)-(223)), is nothing but an equivalent representation of Σ ′ . Let j and respectively j ′ be the injective immersions of Σ in P and respectively of Σ ′ in P ′ . The second-class property of Σ and respectively of Σ ′ is equivalent to the fact that the induced symplectic two-forms j * ω and respectively j ′ * ω ′ are non-degenerate [7], which is the same with [18] It is easy to argue now the preservation of the original number of physical degrees of freedom with respect to the intermediate and irreducible systems. The dimensions of the original and respectively of the intermediate or irreducible phase-space are valued as while the dimensions of the corresponding submanifolds Σ and respectively Σ ′ are equal by construction Because the induced symplectic two-forms j * ω and respectively j ′ * ω ′ are non-degenerate, from (263) we deduce that and therefore all the three systems, original, intermediate, and irreducible, possess the same number of physical degrees of freedom, N, defined as half of the rank of the induced two-forms. Moreover, the induced symplectic two-forms j * ω and j ′ * ω ′ can be brought to exactly the same form in some conveniently chosen charts. For instance, if we (locally) parameterize the submanifolds Σ and Σ ′ (having the same dimension 2N) by the coordinates (ξ α ) α=1,2N , then the local expressions of the immersions j and respectively j ′ read as and respectively z a = z a (ξ) , a = 1, 2N, Obviously, related to the local expressions of (265) and (266) we have that One of the main benefits enabled by our irreducible construction is the computation in a standard manner of the coefficients of the induced symplectic two-form (267) as the elements of the inverse of the matrix having as elements the fundamental Dirac brackets (see Theorem 2.5 from [7]). By 'standard' we mean without need to take any specific parametrization of the second-class constraint surface and, implicitly, to perform any separation into dependent and independent constraint functions.
We have seen that the matrices D γ k α k (with k > 0) are some intermediate steps required by the irreducible procedure, which serve to the construction of the projection D γ 0 α 0 , which projects the system of local generators of the space T Σ ⊥ into a local basis of the same space.

Example
Let us exemplify the general theory on gauge-fixed Abelian p-form gauge fields. Abelian p-forms are described by the Lagrangian action where the field strength of A µ 1 ...µp is defined in the standard manner by . Furthermore, we take the spacetime dimension D to satisfy D ≥ p + 1, since otherwise the number of physical degrees of freedom would be strictly negative. Everywhere in the sequel the notation [µ . . . ν] signifies antisymmetry with respect to all the indices between brackets without normalization factors (i.e., the independent terms appear only once and are not multiplied by overall numerical factors). We will briefly expose the canonical analysis of Abelian p-forms. For more details, see [19] and [20]. From the definitions of canonical momenta 9 on the one hand one obtains the primary constraints and, on the other hand, one expresses the time derivatives of A i 1 ···ip aṡ The canonical Hamiltonian in defined in the standard manner for constrained systems [7] and reduces to where we made the notation x = (x 0 , x). Dirac's algorithm (the consistency conditions for the primary constraints (271)) provides the secondary constraints H, χ which show that (273) is also a first-class Hamiltonian for Abelian p-form gauge fields. The primary first-class constraints are irreducible, while the secondary first-class ones are off-shell reducible (meaning that the null eigenvector equations for the constraint functions and for all the higher-order reducibility functions hold strongly, everywhere on the phase-space, and not only on the first-class surface) of order (p − 1). The associated reducibility functions are given below. It is known that the first-class constraints produce some local transformations of the canonical variables, which do not affect the physical state of the system. They are called Hamiltonian gauge transformations. Although only the primary first-class constraints can be shown to generate gauge transformations, we accept Dirac's conjecture, according to which all first-class constraint generate Hamiltonian gauge transformations. The dynamics of first-class systems is thus not fixed in the sense that for some fixed initial set of canonical variables, the solution to the Hamiltonian equations of motion in the presence of first-class constraints is not unique. In other words, a given physical state of a first-class system is expressed by more than one set of canonical variables (any two such sets are related by a Hamiltonian gauge transformation). In practice, it is useful to eliminate this ambiguity and restore a one-to-one correspondence between physical states and values of the independent canonical variables. This is realized via the so called 'gauge-fixing procedure' by means of imposing further restrictions on the canonical variables, known as 'canonical gauge conditions'. These must be 'good' canonical gauge conditions in the sense of [7], subsection 1.4.1. It is easy to see that a set of good canonical gauge conditions with respect to the first-class constraints (271) and (274) reads as The overall constraint set formed with the first-class constraints (271)  Due to the fact that the second-class constraints (271) and (277) are independent, we will eliminate them from the theory by means of the Dirac bracket built with respect to themselves and will treat along the irreducible approach exposed in the main body of this paper only the reducible secondclass constraints (274) and (278) It is useful to organize these second-class constraints in a column vector Constraints (279) are (p − 1)-order reducible, with the reducibility functions of the form which clearly exhibits the invertibility of µ (p−1)α 0 β 0 . By computing the fundamental Dirac brackets with the help of (211) (with µ (p−1)α 0 β 0 given by (289)), we reobtain precisely (286) and (287). In order to construct the irreducible system of second-class constraints that is equivalent to the original one (like in subsection 4.2.2), we need to enlarge the phase-space by the independent variables y α 2k+1 k=0,[ with the Poisson brackets ω α 2k+1 β 2k+1 = 0 − strategy includes three main steps. First, we express the Dirac bracket for the reducible system in terms of an invertible matrix. Second, we establish the equality between this Dirac bracket and that corresponding to the intermediate theory, based on the constraints (214). Third, we prove that there exists an irreducible second-class constraint set equivalent with (214) such that the corresponding Dirac brackets coincide. These three steps enforce the fact that the fundamental Dirac brackets with respect to the original variables derived within the irreducible and original reducible settings coincide. Moreover, the newly added variables do not affect the Dirac bracket, so the canonical approach to the initial reducible system can be developed in terms of the Dirac bracket corresponding to the irreducible theory. The general procedure was exemplified on Abelian gauge-fixed p-form gauge fields. It is important to mention that our procedure does not spoil other important symmetries of the original system, such as spacetime locality of second-class field theories.