11 Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism where π$_{λ}$ and π$_{θ}$ are the canonical momenta conjugated to λ and θ , respectively. The corresponding Dirac ma- trix is singular. However, there are nonvanishing Poisson brackets among some constraints, indicating that there are both second-class and first-class constraints. This problem is solved separating the second-class constraints from the first-class ones through constraint analysis. The set of first-class constraints is where ε is an infinitesimal parameter. It is easy to ver- ify that the Hamiltonian (4.35) is invariant under these transformations because σ$_{a}$ are eigenvectors of the phase space metric ( M$_{ij}$ ) with null eigenvalues. In this section we reproduce the results originally obtained in [36] from an alternative point of view. while the set of second-class constraints is It has been shown over the last decade that anoma- lous gauge theories in two dimensions can be consistently and unitarily quantized for both Abelian [7, 37, 38] and non-Abelian [39, 40] cases. In this scenario, the two di- mensional model that has been extensively studied is the CSM. We start with the following Lagrangian density of the bosonized CSM with a > 1, Since the second-class constraints are assumed to be equal to zero in a strong way, and using the Maskawa- Nakajima theorem [35], the Dirac brackets are con- structed as Hence, the gauge invariant Hamiltonian is rewritten as Here, F$_{µν}$ = ∂$_{µ}$A$_{ν}$ − ∂$_{ν}$A$_{µ}$ , g$_{µν}$ = diag(+1 , − 1) and ϵ 01 = − ϵ 10 = ϵ$_{10}$ = 1. The symplectic method will be used here to quantize the original second-class model. Then, the Dirac brackets and the respective reduced Hamiltonian will be determined as well. In order to implement the symplectic method, the original Lagrangian written in second order in velocity, given in (5.1), is reduced into its first-order as follows, where the phase space metric M$_{ij}$ , given by where the zeroth-iterative symplectic potential U (0) is which is a singular matrix. The set of first-class con- straints becomes Note that the constraint χ$_{2}$ , originally a second-class con- straint, becomes the generator of gauge symmetries, sat- isfying the first-class property where dot and prime represent temporal and spatial derivatives, respectively. The zeroth-iterative symplectic variables are ξ (0) α = ( φ, π$_{φ}$, A$_{0}$, A$_{1}$, π$_{1}$ ) with the following one-form canonical momenta A$_{α}$ , Due to this result, the infinitesimal gauge transforma- tions are computed as A (0) φ = π$_{φ}$, A (0) A$_{1}$ = π$_{1}$, (5.4) A (0) π$_{φ}$ = A (0) A$_{0}$ = A (0) π$_{1}$ = 0 . { χ$_{2}$, ˜ H } = 0 . (4.38) χ$_{1}$ = π$_{λ}$, χ$_{2}$ = − 1 2 ( σ 2 a − 1) . (4.37) L (0) = π$_{φ}$ ˙ φ + π$_{1}$ ˙ A$_{1}$ − U $^{(0)}$, (5.2) M$_{ij}$ = δ$_{ij}$ − σ$_{i}$σ$_{j}$ σ 2 k , (4.36) ˜ H = 1 2 π 2 a + 1 2 σ ′ 2 a − 1 2 ( σ$_{a}$π $^{a}$) 2 σ$_{a}$σ a + λ 2 ( σ 2 a − 1) = 1 2 π$_{i}$M$_{ij}$π$_{j}$ + 1 2 σ ′ 2 a + λ 2 ( σ 2 a − 1) , (4.35) { σ$_{i}$ ( x ) , σ$_{j}$ ( y ) } ∗ = 0 , { σ$_{i}$ ( x ) , π$_{j}$ ( y ) } ∗ = δ$_{ij}$ δ ( x − y ) , (4.34) { π$_{i}$ ( x ) , π$_{j}$ ( y ) } ∗ = 0 . L = − 1 4 F$_{µν}$ F µν + 1 2 ∂$_{µ}$φ ∂ $^{µ}$φ + q ( g µν − ϵ $^{µν}$) ∂$_{µ}$φ A$_{ν}$ + 1 2 q $^{2}$a A$_{µ}$ A µ . (5.1) ς$_{1}$ = π$_{θ}$, ς$_{2}$ = σ$_{a}$π$_{a}$ − σ 2 $_{a}$θ. (4.33) χ$_{1}$ = π$_{λ}$, χ$_{2}$ = − 1 2 ( σ 2 a − 1) + π$_{θ}$, (4.32) δσ$_{a}$ = ε { σ$_{a}$, χ$_{2}$ } = 0 , δπ$_{a}$ = ε { π$_{a}$, χ$_{2}$ } = εσ$_{a}$, (4.39) δλ = 0 . V. THE GAUGE INVARIANT BOSONIZED CHIRAL SCHWINGER MODEL