A Note on Abelian Conversion of Constraints

We show that for a system containing a set of general second class constraints which are linear in the phase space variables, the Abelian conversion can be obtained in a closed form and that the first class constraints generate a generalized shift symmetry. We study in detail the example of a general first order Lagrangian and show how the shift symmetry noted in the context of BV quantization arises. (∗) Electronic mail: ift01001 @ ufrj (†) Permanent Adress: Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA 1 In an attempt to unify the method of quantization for systems containing both first class and second class constraints, Batalin, Fradkin and Tyutin ( BFT ) [1] have proposed a systematic method for converting all second class constraints in a theory to first class ones. The idea of BFT, in simple terms, is to introduce additional variables into the theory ( one for every second class constraint ) with a simple Poisson bracket structure and transform the second class constraints to a Taylor series in the new variables such that they become first class. The simplest case, which has been studied in detail [2] ( which also will be considered in this letter ) and where one assumes that the new constraints are strongly involutive ( not just first class ) is commonly referred as the Abelian conversion of the second class constraints. The Hamiltonian of the system is, similarly, expressed as a power series in the new variables and each term in the series is determined by requiring that the new Hamiltonian is in involution with the first class ( Abelian ) constraints. The original system is, of course, obtained if one chooses a gauge condition where all the new variables vanish. However, other gauge conditions may be more useful depending on the system under study [2]. The Abelian conversion is an iterative procedure and, in general, the new constraints and the Hamiltonian may not have a closed form. As a result, even though the first class constraints correspond to generators of symmetries [3] of the system, the question of symmetries cannot be studied in general. In this letter, therefore, we study a special class of second class constraints and the symmetries generated by their Abelian conversion. We show that for general second class constraints linear in the phase space variables, the Abelian conversion of the constraints and the Hamiltonian can be obtained in a closed form. We identify the general form of the operator which transforms any observable to its new form. The Abelian conversion, in this case, leads to a generalized shift symmetry in such systems and the first class constraints generate this symmetry. We show in detail, for a general first order Lagrangian, how the general results lead to the shift symmetry [4] which has come to play an important role in the BV quantization [5] of gauge theories and topological field theories. We would like to point out here that a shift in the original variables arising as a result of the Abelian conversion was already noted in the context of a first order Lagrangian in ref. [6] and that was the starting motivation for our detailed 2 examination of the question of symmetries. Let us consider a Hamiltonian system with phase space variables y, μ = 1, 2, ..., 2N and the canonical Hamiltonian Hc(y). We are considering here a system with a finite number of degrees of freedom for simplicity only and the discussion generalizes to continuum theories in a straight forward manner. Let us assume that the system has a set of second class constraints which are linear in the variables y and are denoted by χα(y) ≈ 0, α = 1, 2, ..., 2n, n ≤ N (1) (There may be other constraints first class and more complicated second class constraints in the system, but we will not be concerned with them. ) By assumption [3], therefore, {χα, χβ} = ∆αβ (2) defines a constant, antisymmetric and invertible matrix. We can, of course, write the Hamiltonian including the constraints as H = Hc(y) + λ χα , (3) where λ’s are the Lagrange multipliers which can be determined by requiring the constraints in eq. (1) to remain invariant under time evolution. In order to convert the constraints in eq. (1) to first class ( Abelian ) ones, we introduce [1] additional variables ψ, α = 1, 2, ..., 2n and assume that the Poisson bracket structure {ψ, ψ} = ω (4) defines a constant, antisymmetric and invertible matrix. Given this, we define new constraints χ̃α = χα +Xαβψ β (5) and require that 3 {χ̃α, χ̃β} = 0 (6) which leads to Xαβω Xδγ = −∆αδ , (7) . We note from eqs. (2), (4) and (6) that Xαβ can be chosen to be a constant, invertible matrix and the original constraints can be converted to Abelian ones. Note that when ψ ≈ 0, the constraints in eq. (5) reduce to the original ones in eq. (1). We can transform the Hamiltonian following the method of BFT so that it is in involution with χ̃α. However, let us note the following. Let B α =ωαβX {χγ , y } ỹ =ψB α (8) By definition, B α is a constant matrix of rank 2n and we note that H̃ = Hc(y − ỹ) (9) is in involution with χ̃α. In fact {χ̃α, H̃} ={χα, Hc(y − ỹ)}+ {Xαβψ , Hc(y − ỹ)} = ∂Hc(y − ỹ) ∂yμ {χα, y }+Xαβ ∂Hc(y − ỹ) ∂ỹμ {ψ, ỹ} = ∂Hc(y − ỹ) ∂yμ ({χα, y } −XαβB μ γ {ψ , ψ}) = ∂Hc(y − ỹ) ∂yμ ({χα, y } −Xαβω B γ ) =0 (10) It is needless to say that the Hamiltonian in eq. (9) coincides with the transformed Hamiltonian that will be obtained through the method of BFT. Let us note that the operator G = exp(−ỹ ∂ ∂yμ ) = exp(−ψB α ∂ ∂yμ ) (11)