Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism 19 ˜ H = 1 2 ( π i $_{a}$) 2 − A a $_{0}$Ω$_{a}$ + 1 2 m $^{2}$A a i A i a + 1 2 m $^{2}$A a $_{0}$A 0 a + 1 4 F a $_{kj}$F a kj − gC b $_{fa}$∂ x i ( A a $_{0}$( x ) π i $_{b}$( x )) θ $^{f}$( x ) − m $^{2}$( ∂ x i A i $_{f}$) θ $^{f}$( x ) − 1 2 ∫ y ∂ y i ( F a $_{kj}$( x ) ∂F kj a ( x ) ∂A f i ( y ) ) θ $^{f}$( y ) + 1 2 m $^{2}$( ∂ $^{i}$θ$^{f}$( x )) 2 + 1 2 θ$_{f}$ ( w ) ∫ w ∂ w i ∫ y [ ( ∂ y l A il $_{fb}$) θ $^{b}$( y ) ] − 1 2 ∫ θ $^{g}$( z ) ∫ z ∂ z n ∫ θ$_{f}$ ( x ) ∫ w ∂ w k ∫ y ∂ y l ∂ A kl fb ∂A n $_{g}$( z ) θ $^{b}$( y ) − 1 2 ∫ θ $^{h}$( v ) ∫ v ∂ v i ∫ θ$_{g}$ ( z ) ∫ z ∂ z n ∫ θ$_{f}$ ( x ) ∫ w ∂ w k ∫ y ∂ y l ∂ $^{2}$A kl fb ∂A i $_{h}$( v ) ∂A n $_{g}$( z ) θ $^{b}$( y ) . (6.28) C = ( 0 − m $^{2}$δ cd m $^{2}$δ dc B cd ) δ $^{(3)}$( ⃗ x − ⃗ y ) , (6.33) B cd = 2 gC cd b χ b $_{2}$( x ) − 2 gm $^{2}$C cd b A b $_{0}$( x ) . (6.34) Ω a 1 = ∂ $^{i}$πa i + π a $_{θ}$, χ a 1 = π a $_{0}$. (6.29) C ( − 1) = 1 m 2 ( B cd m 2 δ cd − δ dc 0 ) δ $^{(3)}$( ⃗ x − ⃗ y ) . (6.35) χ a 2 = Ω a − gC ba f π i $_{b}$∂$_{i}$ θ $^{f}$. (6.30) H = ˜ H + λ 1 $_{a}$Ω a 1 + ζ 1 $_{a}$χ a 1 + ζ 2 $_{a}$χ a $_{2}$, (6.31) { A a $_{i}$( x ) , A b $_{j}$( y ) } ∗ = 0 , { A a $_{i}$( x ) , π b $_{j}$( y ) } ∗ = δ $^{ab}$δ $^{(3)}$( ⃗ x − ⃗ y ) , { A a $_{i}$( x ) , A b $_{0}$( y ) } ∗ = − 1 m $_{2}$∂ x i δ $^{(3)}$( ⃗ x − ⃗ y ) + g m $_{2}$C ab f A f i ( x ) δ $^{(3)}$( ⃗ x − ⃗ y ) , { π a i ( x ) , A b $_{0}$( y ) } ∗ = − 1 m $_{2}$gC ab e π e i δ $^{(3)}$( ⃗ x − ⃗ y ) , { π a i ( x ) , π b $_{j}$( y ) } ∗ = 0 , { A a $_{0}$( x ) , A b $_{0}$( y ) } ∗ = − g m $_{2}$C ab e A e $_{0}$( x ) δ $^{(3)}$( ⃗ x − ⃗ y ) , { A a $_{i}$( x ) , θ $^{b}$( y ) } ∗ = 0 , (6.36) { π a i ( x ) , θ $^{b}$( y ) } ∗ = 0 , { A a $_{0}$( x ) , θ $^{b}$( y ) } ∗ = 0 , { A a $_{i}$( x ) , π b $_{θ}$( y ) } ∗ = 0 , { π a i ( x ) , π b $_{θ}$( y ) } ∗ = 0 , { A a $_{0}$( x ) , π b $_{θ}$( y ) } ∗ = g m $_{2}$C ab e ∂ x i π e $_{(}$x ) δ $^{(3)}$( ⃗ x − ⃗ y ) , { θ $^{a}$( x ) , π b $_{θ}$( y ) } ∗ = δ $^{ab}$δ $^{(3)}$( ⃗ x − ⃗ y ) , { θ $^{a}$( x ) , θ $^{b}$( y ) } ∗ = 0 , { π a $_{θ}$( x ) , π b $_{θ}$( y ) } ∗ = 0 . { Ω a $_{1}$( x ) , Ω b $_{1}$( y ) } = 0 , { Ω a $_{1}$( x ) , χ b $_{1}$( y ) } = 0 , { Ω a $_{1}$( x ) , χ b $_{2}$( y ) } = 0 , (6.32) { χ a $_{1}$( x ) , χ b $_{2}$( y ) } = − m $^{2}$δ $^{ab}$δ $^{(3)}$( ⃗ x − ⃗ y ) , { χ a $_{2}$( x ) , χ b $_{2}$( y ) } = 2 gC ab d χ d $_{2}$( x ) δ $^{(3)}$( ⃗ x − ⃗ y ) − 2 gm $^{2}$C ab d A d $_{0}$( x ) δ $^{(3)}$( ⃗ x − ⃗ y ) , Finally, the infinitesimal gauge transformations are ob- tained, namely, no more constraints arise. Notice that some brack- ets above are null, indicating that there are both first and second-class constraints. Indeed, the first-class con- straint is Ω a 1 and the second-class are χ a 1 and χ a $_{2}$. In agreement with Dirac’s procedure, the second-class con- straints can be taken equal to zero in a strong way, this allows us to compute the primary Dirac brackets. Due to the Maskawa-Nakajima theorem [35], the primary Dirac brackets among the phase space fields are canonical. To demonstrate this, the brackets are computed explicitly. The Dirac matrix is where λ 1 $_{a}$, ζ 1 a and ζ 2 a are Lagrange multipliers. Since the Poisson brackets among those constraints are Due to this, the total Hamiltonian is In accordance with the Dirac process, the Dirac brackets among the phase space fields are obtained as The inverse of the Dirac matrix is with For the first set of constraints, the time stability con- dition is satisfied ( ˙ Ω a 1 = 0). For the second one, the following secondary constraints are required, At this stage, we would like to disclose the hidden sym- metry present inside the model using the Dirac point of view. To this end, we start with the set of primary con- straints, This completes our proposal.