Time Dependent Hartree–Bogoliubov Equation on the Coset Space  SO(2N + 2)/U(N + 1) and Quasi Anti-Commutation Relation Approximation

Journal of Mathematical Physics

2019

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

Section: Ghb Mean-field Hamiltonian and Its Diagonalizationmentioning

confidence: 99%

Mean-field theory based on the Jacobi hsp ≔ semidirect sum hN⋊sp(2N,R)C algebra of boson operators

Journal of Mathematical Physics

2019

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“…Successively using these identities, on the U(G)| 0 >, the operators c α and c † α are shown to satisfy exactly the anti-commutation relations of the fermion annihilation-creation operators [22]:…”

Section: Appendixmentioning

confidence: 99%

Extended supersymmetric σ-model based on the Lie algebra of the fermion operators

et al. 2008

Nuclear Physics B

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Extended supersymmetric σ-model is given, standing on the SO(2N+1) Lie algebra of fermion operators composed of annihilation-creation operators and pair operators. Canonical transformation, the extension of the SO(2N ) Bogoliubov transformation to the SO(2N + 1) group, is introduced. Embedding the SO(2N + 1) group into an SO(2N +2) group and using SO(2N+2) U (N+1) coset variables, we investigate a new aspect of the supersymmetric σ-model on the Kähler manifold of the symmetric space SO(2N+2) U (N+1) . We construct a Killing potential which is just the extension of the Killing potential in the SO(2N ) U (N ) coset space given by van Holten et al. to that in the SO(2N+2) U (N+1) coset space. To our great surprise, the Killing potential is equivalent with the generalized density matrix. Its diagonal-block matrix is related to a reduced scalar potential with a Fayet-Ilipoulos term. The reduced scalar potential is optimized in order to see the behaviour of the vacuum expectation value of the σ-model fields and a proper solution for one of the SO(2N + 1) group parameters is obtained. We give bosonization of the SO(2N+2) Lie operators, vacuum functions and differential forms for their bosons expressed in terms of the SO(2N +2) U (N +1) coset variables, a U (1) phase and the corresponding Kähler potential. * A preliminary version of this work has been presented by S. Nishiyama at the YITP workshop YITP-W-07-05 on String Theory and Quantum Field Theory held at Kinki University

“…S. N. has also proposed another type of the SO(2N+1) TDHB equation [15]. As the SO(2N+2) Lie OPs, operated onto functions on the SO(2N +2) U (N +1) coset manifold, are mapped into the regular representation consisting of those functions, we have reached an extended TDHBT (ETDHBT) on the coset space SO(2N +2) U (N +1) [16]. Embedding the SO(2N+1) group into an SO(2N+2) group and using the boson images of the fermion Lie OPs, we have obtained a new ETDHBT for fermionic (2N+2)-dimensional rotator.…”

Section: Introductionmentioning

confidence: 99%

“…Instead, we have tried to determine the parameters with the aid of the quasi anti-commutation relation approximation for the fermion OPs. A determination of the parameters is possible if we demand that expectation values of the anti-commutators by an SO(2N+1) HB WF satisfy the anti-commutation relations in the classical limit, i.e., the quasi anti-commutation relation approximation for the fermion OPs [16]. Under the approximation, the determination has been attempted but has unfortunately been executed incompletely [19].…”

Section: Introductionmentioning

confidence: 99%

Remarks on the mean-field theory based on the SO(2N + 1) Lie algebra of the fermion operators

Int. J. Geom. Methods Mod. Phys.

2019

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Toward a unified algebraic theory for mean-field Hamiltonian (MFH) describing pairedand unpaired-mode effects, in this paper, we propose a generalized HB (GHB) MFH in terms of the SO(2N+1) Lie algebra of fermion pair and creation-annihilation operators. We diagonalize the GHB-MFH and throughout the diagonalization of which, we can first obtain the unpaired mode amplitudes which are given by the SCF parameters appeared in the HBT together with the additional SCF parameter in the GHB-MFH and by the parameter specifying the property of the SO(2N +1) group. Consequently, it turns out that the magnitudes of these amplitudes are governed by such parameters. Thus, it becomes possible to make clear a new aspect of such the results. We construct the Killing potential in the coset space SO(2N ) U (N ) on the Kähler symmetric space which is equivalent with the generalized density matrix (GDM). We show another approach to the fermion MFH based on such a GDM. We derive an SO(2N+1) GHB MF operator and a modified HB eigenvalue equation. We discuss on the MF theory related to the algebraic MF theory based on the GDM and the coadjoint orbit leading to the non-degenerate symplectic form.