Abelian Subset of Second Class Constraints

We show that after mapping each element of a set of second class constraints to the surface of the other ones, half of them form a subset of abelian first class constraints. The explicit form of the map is obtained considering the most general Poisson structure. We also introduce a proper redefinition of second class constraints that makes their algebra symplectic.


introduction
When Dirac introduced constrained systems [1], he classified constraints as first class and second class. First class constraints have been interesting since they turned out to be generators of gauge transformation. These constraints introduced a new class of symmetries, which for example, lead to Ward identities in the context of renormalization [2]. The main requirement in quantization of first class constraints is the covariance of observables under gauge transformations. In Dirac quantization, this requirement is satisfied by considering physical states as null eigen states of the generator of gauge transformation. The same idea is followed in BRST where a nilpotent BRST-charge generates BRST-transformation [3].
There are two major difficulties in both Dirac quantization and BRST. In general, first class constraints satisfy a closed algebra in which the structure coefficients are some functions of phase space coordinates. Consistency of these methods of quantization depends on the possibility of a definite operator ordering; the structure coefficients should stand on the left side of first class constraints. Another problem is obtaining the explicit form of the generator of gauge transformation or BRST-charge. Both difficulties can be overcome by making the first class constraints abelian [4].
Second class constraints were thought to be redundant degrees of freedom that one should get rid of them before quantization, for example by using Dirac bracket instead of Poisson bracket. But second class constraints are more important. For example, in reference [5], the gauge theory of second class systems is discussed. Or in closed string theory, it is claimed that boundary conditions lead to a set of second class constraints which give rise to non-commutativity of space-time [6]. On the other hand, covariant quantization, in general, is not consistent with classification of constraints as first and second class [7]. Consequently, we need a general method of quantization which treats both classes on the same footing.
1. One possibility is to convert second class constraints to first class [8]. Given a constraint system possessing second class constraints, in principle, one can consider an extended phase space and redefine second class constraints and the Hamiltonian to find an equivalent first class system. There are two difficulties in doing so. Firstly, it is not so easy to find out such redefinitions in general cases. Secondly, assuming that the conversion is done, one may still encounter the above mentioned difficulties in quantization of first class constraints. As is well known, all these problems can be remedied most easily provided one makes the algebra of second class constraints symplectic.
2. Another possibility is to consider half of second class constraints as first class constraints and the remaining ones as gauge fixing conditions [2]. This method, for example, is used to study gauge invariance in the Proca model [9].
In this paper, we prove that after mapping each element of a set of second class constraints to the surface of the other ones, half of them form an abelian subset. In addition we present a general method for redefining second class constraints to make their algebra symplectic.
Although this method may not preserve covariance but it is still interesting since it works globally, and for a general Poisson structure. Therefore it provides a simple conversion of second class constraints to first class ones.
In reference [4] it is shown that first class constraints become abelian when they are mapped to the surface of each other. Thus, it seems that in this way, one can obtain abelian subset of a given set of constraints in the most simple way.
The organization of paper is as follows. In section 2, we introduce necessary definitions and lemmas. The method is introduced in section 3. We conclude our results in section 4.

Definitions and Lemmas
In this section we provide some general tools necessary for arguments of the next section.
Consider a phase space defined by a set of coordinates z µ satisfying the Poisson algebra, in which J µν (z) is a full rank anti-symmetric tensor, e.g. the symplectic two form: Assume a pair of conjugate functions φ(z) and ω(z) in F , satisfying the relation, where F stands for the set of real analytic functions of the phase space coordinates. In fact for a given φ ∈ F , using the Cauchy-Kowalevski theorem [10], one can show that there exist at least one function ω ∈ F that satisfies the relation, Corresponding to each ξ ∈ F an operatorξ : F → F can be defined as follows, It is easy to verify that,ξ Considering the operators (φ,ω) where {φ, ω} = 1, from Eq. (3) one can show that these operators satisfy the following relations: These properties can be easily verified. For example, for an arbitrary function ξ ∈ F , we have, where in the second equality we have used the Jaccobi identity. Considering the operatorŝ one can use Eqs. (7,8) to show that [φ,P φ ] = [ω,P ω ] = 0, and consequently, Lemma 1. The operatorsP φ andP ω satisfy the following properties: Proof. We proof the first equality. The second equality can be proved in the same way. Using Eq. (9), one can show that [ω, φ n ] = −nφ n−1 . Thus, Lemma 2. Given conjugate functions ω, φ ∈ F , the operatorP φ is the projection map to the subspace of the phase space defined by φ = 0.
The above results become practically interesting if conjugate to a given φ ∈ F , one can obtain explicitly a function ω that satisfies Eq. (3). This can be easily done if there exist a function H ∈ F , such thatφH = 0 butφ M +1 H = 0, for an integer M ≥ 1. Since, in principle, ω exists and satisfies Eq. (3), using the coordinate transformation (16), H can be written as a polynomial in ω, Comparing Eq. (20) with Eq. (21), one can verify ω as the coefficient ofφ M H inφ M −1 H.
As an example suppose, A simple calculation shows that M = 2 and Comparing Eq. (23) with Eq.(24) one reads ω = e −x p x . This method can be used to obtain gauge fixing conditions conjugate to first class constraints [4].
Corollary 3. For arbitrary functions ξ and χ in F , Corollary 3 can be proved using the Jaccobi identity to show that the Poisson brackets of the LHS of Eqs. (25,26) with ω is vanishing.
Proof. Since there exist a function ω conjugate to φ, one can write ψ as a polynomial in φ (similar to Eq.(19)), Thus if ψ = 0 then a i 's should vanish and ψ = ψ| φ .
Proof. It is sufficient to prove that [(φ n 1ω n 1 ), (φ m 2ω m 2 )] = 0. Using the Jaccobi identity and lemma 4, one can show that, From lemma 5 and Eq.(28) one verifies that This can be proved using lemma 3 and corollary 2.

Proof.
Writing ω as a polynomial in ψ, one verifies that, Thus, where in the third equality we have used Eq.(25).
Using Eq.(26), the proof is similar to the proof of lemma 7.
In this section we show that the subspace M of the phase space, defined by a set of irreducible second class constraints, which satisfy the relation, can be equivalently determined by a set of constraintsφ i ,ω i , i = 1, · · · , k, satisfying the symplectic algebra, For this reason, we consider the following lemmas.
Consider the constraints φ 1 and φ k+1 and the definition, where ω 1 ∈ F is some function conjugate to φ 1 .
When we have found second class constraints satisfying the symplectic algebra, we can convert them to first class constraints by extending the phase space to include new coordinates η i 's and π i 's, where and redefine constraints as follows: It can be easily verified that the constraints Φ a , a = 1, · · · , 2k are abelian, Another interesting result is that, the operatorP defined by the relation, is the projection map to the constraint surface M and the projected coordinates z µ p ≡ P z µ = z µ | M , are the coordinates of the constrained surface M. In addition, from corollary 2 it is clear that, where { , } DB stands for Dirac bracket respective to the constraintsφ's.
Assume one maps each constraintφ i to the surface of its conjugateω i , i.e.
As an example see reference [9], where gauge invariance in the Proca model is studied considering the abelian subset of second class constraints.

Conclusion
The main purpose of this article is to show that there exist an abelian subset of second class constraints that can be obtained by mapping each constraint to the surface of other constraints. In addition we introduced a method that can be practically used to transform a given set of second class constraints to an equivalent set satisfying the symplectic algebra.
In this way, second class constraints can be simply converted to abelian first class ones.
In reference [4], it is proved that first class constraints become abelian when they are mapped to the surface of each other. Therefore one can conclude that, using the same technique, the abelian subset of a given set of constraints, can be found independent of the details of their algebra.
Assuming the most general Poisson structure, we have found the projector operators that map functions of phase space to the constraint surface. It is shown that the Poisson brackets of these mapped functions with other functions are equivalent to the corresponding Dirac brackets.