Mean-field theory based on the \mathfrak{Jacobi~hsp} := semi-direct sum \mathfrak{h}_N \rtimes \mathfrak{sp}(2N,$\mathbb{R}$)_$\mathbb{C}$ algebra of boson operators

In this paper, we give an expression for canonical transformation group with Grassmann variables, basing on the \mathfrak{Jacobi~hsp} !:= semi-direct sum \mathfrak{h}{N} \rtimes \mathfrak{sp}(2N,$\mathbb{R}$)$\mathbb{C}$ algebra of boson operators. We assume a mean-field Hamiltonian (MFH) linear in the \mathfrak{Jacobi} generators. We diagonalize the boson MFH. We show a new aspect of eigenvalues of the MFH. An excitation energy arisen from additional self-consistent field (SCF) parameters has never been seen in the traditional boson MFT. We derive this excitation energy. We extend the Killing potential in the \frac{Sp(2N,$\mathbb{R}$)$\mathbb{C}$}{U(N)} coset space to the one in the \frac{Sp(2N+2,$\mathbb{R}$)$\mathbb{C}$}{U(N+1)} coset space and make clear the geometrical structure of K"{a}hler manifold, a non-compact symmetric space \frac{Sp(2N+2,$\mathbb{R}$)$\mathbb{C}$}{U(N+1)}. The \mathfrak{Jacobi~hsp} transformation group is embedded into an Sp(2N+2,$\mathbb{R}$)$\mathbb{C}$ group and an \frac{Sp(2N+2,$\mathbb{R}$)$\mathbb{C}$}{U(N+1)} coset variable is introduced. Under such mathematical manipulations, extended bosonization of Sp(2N+2,$\mathbb{R}$)$\mathbb{C}$ Lie operators, vacuum function and differential forms for extended boson are presented by using integral representation of boson state on the \frac{Sp(2N+2,$\mathbb{R}$)_$\mathbb{C}$}{U(N+1)} coset variables.