The BFT Method With Chain Structure

We have constructed a modified BFT method that preserves the chain structure of constraints. This method has two advantages: first, it leads to less number of primary constraints such that the remaining constraints emerge automatically; and second, it gives less number of independent gauge parameters. We have applied the method for bosonized chiral Schwiger model. We have constructed a gauge invariant embedded Lagrangian for this model.

chain structure, as fully discussed in [7], is that it is possible to derive the constraints as commuting distinct chains such that within each chain the following iterative relation where Φ a 0 stand for primary constraints. The advantages of this method will be discussed afterward.
Consider a second class constraint system described by the Hamiltonian H 0 and a set of second class constraints Φ α ; α = 1, ...N satisfying the algebra where ∆ is an antisymmetric and invertible matrix. For simplicity and without loss of generality we suppose that the second class constraints Φ α are elements of one chain.
The results can be extended to multi-chain system by adding the chain index (like the superscript a in Eq. (1)) to the constraints Φ α . For converting this second class system into a gauge system, one can enlarge the phase space by introducing auxiliary variables η where we assume their algebra to be We demand that the embedded constraints τ α (q, p, η) and HamiltonianH(q, p, η) in the extended phase space satisfy the following algebra This gives an abelian first class chain such that its terminating element commute with the Hamiltonian. We call this system a semi-strongly involutive system; compared with strongly involutive one in which the constraints commute with the Hamiltonian.
As discussed in [2,3,4], considering the power series in which τ (0) α = Φ α and H (0) = H c (q, p), one can show that these may be solutions to Eqs. (4) and (5) if in which and χ αβ (q, p) should satisfy the following equation For α = N, due to Eq. (6), the last terms in Eqs. (14-16) are absent. The above results for semi-strongly involutive system is different from our previous results for strongly involutive one [2,3,4] in terms τ in Eqs. (14-16) respectively. We remind that [6], the aim is to choose a suitable solution for χ αβ in Eq. (17) such that the series for τ α andH truncates after a few steps. For example when ∆ is the symplectic matrix J, the choice ω = −J and χ = J solves(17); similarly when ∆ is a constant (antisymmetric) matrix, the choice ω = −∆ and χ = 1 is appropriate.
Now let see what is the advantage of the chain structure in our modified BFT method.
We emphasize on two points: 1) Suppose we are given a singular Lagrangian L which leads to one primary constraint It should be added that sometimes it is possible to reconstruct a singular Lagrangian from a given canonical Hamiltonian and primary constraints, even though it is not guarantied generally. However, the less the number of primary constraints, the more the chance to find the original Lagrangian which gives the desired primary constraints and Hamiltonian. In the following we will give an example to show this point. We think that our modified BFT method improves the chance of finding a corresponding Lagrangian yielding the embedded primary constraints.
To be more precise, when the Hamiltonian is quadratic and the primary constraints 2)When a chain structure exists the number of independent gauge parameters is much less than when we lack it. By "gauge parameters" we mean arbitrary functions of time that appear in the solutions of equations of motion. To be more precise, in the traditional BFT method where we have ultimately the strongly involutive constraints τ α and Hamil-tonianH, it is clear that the following function acts as the generating function of gauge transformation G = ε α (t)τ α (q, p, η).
As is apparent, here the number of gauge parameters ε α (t) is equal to the total number of constraints.
However, in the presence of the chain structure it can be shown that the number of gauge parameters is just equal to the number of distinct chains [8]. There exist two equivalent methods to construct gauge generating function [9,10]. In the special case, where all the constraints are abelian and the terminating elements of each chain commute strongly with Hamiltonian, the gauge generating function can be written as where ζ i (t) are infinitesimal arbitrary functions of time, m is the total number of first class chains and N i is the length of the i-th chain.
It should be remembered that the generating function (18) gives the symmetries of the extended action, while G in (19) gives the symmetries of the total action. The former formalism (i.e. the extended Hamiltonian), however, may be shown to give the correct results for physical (non-gauged) observables.
Now it is instructive to apply the general idea discussed above to a specific model.
The bosonized chiral Schwinger model in (1+1) dimensions with regularization parameter a = 1 is described by the Lagrangian density [11,12]: in which φ is a scalar and A µ is a vector field. There is one second class chain including four second class constraints as follows: where π, π 0 and E are momenta conjugate to φ, A 0 , and A 1 respectively. The canonical Hamiltonian density corresponding to Eq. (20) is Eqs. (21) represent a second class constrained system with the algebra where Let define four auxiliary fields η α (x) with the algebra given by ω = −∆. As discussed above, the choice χ = 1 satisfy Eq. (17). Then following the instructions given in Eqs. (9) and (10), the new set of first class constraint are found to be From Eqs. (14-16) the embedded Hamiltonian which preserves the chain structure (except for the last element) isH where Here it can be checked that the constraints in Eqs. (25) satisfy the chain structure relation (1) where τ 1 is the primary constraint. The chiral Schwinger model has also been discussed in [6] where the set of first class constraints are the same as (25) The infinitesimal gauge variations of the original and auxiliary fields generated by G are as follows δφ = ζ −ζ δA 0 =˙ζ It can be directly checked that the total action is invariant under these variations.
Fortunately all the terms in Hamiltonian (26) are quadratic. This make it easy to reconstruct the following Lagrangian L = L N + η(φ ′′ + π 1 + 2A 1 + 2E) + (η 2 − η ′2 −ηη ′ +ηξ + 2ηξ + The first term is the original Lagrangian (20), the second and third terms are due to H (1) and H (2) in Eqs. (27) and (28) respectively and the last term (−ηȦ 0 ) is the crucial term which converts the primary constraint π 0 into π 0 + η of the embedded system. One can directly check that beginning with the Lagrangian (32) the first class system given by HamiltonianH in Eq. (26) and τ 1 to τ 4 in Eq. (25) would be obtained.