Bosonization in the path integral formulation

We establish the direct d = 2 on-shell bosonization psi(L)(x(+)) = e(i xi(x+)) and psi(+)(R)(x(-)) and e(i xi(x-)) in path integral formulation by deriving the off-shell relations psi(L)(x)psi(+)(R)(x) = exp[i xi(x)] and psi(R)(x)psi(+)(L)(x) = exp[i xi(x)]. Similarly, the on-shell bosonization of the bosonic commuting spinor, phi(L)(x(+)) = ie-i xi(x(+))partial derivative(+) e(-ix(x+)), phi(+)(R)(x(-)) = e(-i xi(x-)-ix(x-)) and phi(R)(x(-)) = ie(i xi(x-))partial derivative(-) e(-ix(x-)), phi(+)(L)(x(+)) = e(-i xi(x+)+ix(x+),) is established in path integral formulation by deriving the off-shell relations phi(L)(x)phi(+)(R)(x) = ie-i xi(x)partial derivative(+)e(-ix(x)) and phi(R)(x)phi(+)(L)(x) = ie(i)xi(x)partial derivative(+)e(-ix(x))