Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Block-diagonalization in second quantization

A. Tarantelli, L. S. Cederbaum

Published Apr 1, 1990• 3 citations
Publisher

Abstract

The unitary matrix that brings a Hermitian H into block-diagonal form can be uniquely determined under very simple and transparent conditions. Int his work the block-diagonalization problem is investigated in the framework of the second quantization formalism. Starting with an operator Ḩ which in any n-particle Fock space has a well-defined matrix representation an attemt was made to answer the question whether the transformation matrices T which can be separately given in the various n-particle spaces can be considered matrix representations of the same operator. T̂. Interestingly, the very important result was reached that the block-diagonalization operator T̂ exists and is unique. As a particular example, attention was concentrated on the case of an operator Ĥ given by a one-particle operator. In this case the block-diagonalization operator can be constructed and given in explicit from. This approach is applied to the theory of Green’s functions where the block-diagonalization of the Hamiltonian has interesting consequences that are illustrated in some details.

DOI
10.1063/1.528816

Related papers

Perturbative power series for block diagonalisation of Hermitian matrices

Ishan NH Mankodi, David Peter Divincenzo

Block diagonalisation of matrices by canonical transformation is important in various fields of physics. Such diagonalization is currently of interest in condensed matter physics, for modelling of gates in superconducting circuits and for studying isolated quantum many-body systems. While the block diagonalisation of a particular Hermitian matrix is not unique, it can be made unique with certain auxiliary conditions. It has been assumed in some recent literature that two of these conditions, ``least action" vs. block-off-diagonality of the generator, lead to identical transformations. We show that this is not the case, and that these two approaches diverge at third order in the small parameter. We derive the perturbative power series of the ``least action", exhibiting explicitly the loss of block-off-diagnoality.

arXiv:2408.14637•Aug 26, 2024•1 citation

Second Quantization and Representation Theory

Edward M. Corson

The formalism of the second quantization is developed entirely within the framework of the Dirac-Jordan representation theory. The equivalence of this formalism and the methods of ordinary coordinate-spin space is shown, and the equations of "connection" are fully developed. It is shown that the "creation" and "annihilation" operators, and indeed all operators which do not commute with the quantized Hamiltonian and hence whose representatives cannot be explicitly given in a configuration space of a fixed number of dimensions are easily treated in terms of the general representation theory and arise quite naturally. Both types of statistics are considered, and the Fock-Dirac density matrix and self-consistent field are treated as a simple illustration.

Physical Review•Nov 1, 1946•2 citations

Pseudounitary operators and pseudounitary quantum dynamics

Ali Mostafazadeh

We consider pseudounitary quantum systems and discuss various properties of pseudounitary operators. In particular we prove a characterization theorem for block-diagonalizable pseudounitary operators with finite-dimensional diagonal blocks. Furthermore, we show that every pseudounitary matrix is the exponential of i=−1 times a pseudo-Hermitian matrix, and determine the structure of the Lie groups consisting of pseudounitary matrices. In particular, we present a thorough treatment of 2×2 pseudounitary matrices and discuss an example of a quantum system with a 2×2 pseudounitary dynamical group. As other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics, demonstrate the coincidence of the symplectic group Sp(2n) with the real subgroup of a matrix group that is isomorphic to the pseudounitary group U(n,n), and elaborate on an approach to second quantization that makes use of the underlying pseudounitary dynamical groups.

arXiv:math-ph/0302050•Journal of Mathematical Physics•Feb 12, 2004•84 citations

Theory of transformation for the diagonalization of quadratic Hamiltonians

Ming-wen Xiao

A theory of transformation is presented for the diagonalization of a Hamiltonian that is quadratic in creation and annihilation operators or in coordinates and momenta. It is the systemization and theorization of Dirac and Bogoliubov-Valatin transformations, and thus provides us an operational procedure to answer, in a direct manner, the questions as to whether a quadratic Hamiltonian is diagonalizable, whether the diagonalization is unique, and how the transformation can be constructed if the diagonalization exists. The underlying idea is to consider the dynamic matrix. Each quadratic Hamiltonian has a dynamic matrix of its own. The eigenvalue problem of the dynamic matrix determines the diagonalizability of the quadratic Hamiltonian completely. In brief, the theory ascribes the diagonalization of a quadratic Hamiltonian to the eigenvalue problem of its dynamic matrix, which is familiar to all of us. That makes it much easy to use. Applications to various physical systems are discussed, with especial emphasis on the quantum fields, such as Klein-Gordon field, phonon field, etc..

arXiv:0908.0787•Aug 6, 2009•31 citations

Algebraic identities relating first- and second-quantized operators

W. E. Lawrence

Second-quantized operators are derived algebraically from first-quantized counterparts, thus providing an alternative to all existing methods, most of which rely upon demonstrating the equivalence of matrix elements of the two operators. This derivation establishes an operator identity between a second-quantized operator and a first-quantized operator whose domain of definition is extended explicitly from an N-particle Hilbert space to Fock space. The existence of such an identity eliminates the need to address the issue of the representations used for the two operators, which often obscures the underlying simple relationship between them.

American Journal of Physics•Feb 1, 2000•0 citations

Generalized Coherent States and Operator Descriptions in Quantum Mechanics

Narasimhaiengar Mukunda

"The possibility of describing noncommuting operators in quantum mechanics by classical type functions, and the associated expression of operator multiplication, is of considerable interest. The well known Wigner-Weyl-Moyal theory is an important example. Another is the diagonal representation of operators using standard coherent states. We develop general necessary and sufficient conditions for the existence of the diagonal representation in the context of a family of generalised coherent states associated with any unitary irreducible Lie group representation. Several examples illustrating these conditions, and interesting results in the Heisenberg-Weyl case, are presented".

Modern Physics Letters A•Nov 20, 2003•0 citations
The unitary matrix that brings a Hermitian H into block-diagonal form can be uniquely determined under very simple and transparent conditions. Int his work the block-diagonalization problem is investigated in the framework of the second quantization formalism. Starting with an operator Ḩ which in any n-particle Fock space has a well-defined matrix representation an attemt was made to answer the question whether the transformation matrices T which can be separately given in the various n-particle spaces can be considered matrix representations of the same operator. T̂. Interestingly, the very important result was reached that the block-diagonalization operator T̂ exists and is unique. As a particular example, attention was concentrated on the case of an operator Ĥ given by a one-particle operator. In this case the block-diagonalization operator can be constructed and given in explicit from. This approach is applied to the theory of Green’s functions where the block-diagonalization of the Hamiltonian has interesting consequences that are illustrated in some details.
Block diagonalisation of matrices by canonical transformation is important in various fields of physics. Such diagonalization is currently of interest in condensed matter physics, for modelling of gates in superconducting circuits and for studying isolated quantum many-body systems. While the block diagonalisation of a particular Hermitian matrix is not unique, it can be made unique with certain auxiliary conditions. It has been assumed in some recent literature that two of these conditions, ``least action" vs. block-off-diagonality of the generator, lead to identical transformations. We show that this is not the case, and that these two approaches diverge at third order in the small parameter. We derive the perturbative power series of the ``least action", exhibiting explicitly the loss of block-off-diagnoality.
The formalism of the second quantization is developed entirely within the framework of the Dirac-Jordan representation theory. The equivalence of this formalism and the methods of ordinary coordinate-spin space is shown, and the equations of "connection" are fully developed. It is shown that the "creation" and "annihilation" operators, and indeed all operators which do not commute with the quantized Hamiltonian and hence whose representatives cannot be explicitly given in a configuration space of a fixed number of dimensions are easily treated in terms of the general representation theory and arise quite naturally. Both types of statistics are considered, and the Fock-Dirac density matrix and self-consistent field are treated as a simple illustration.
We consider pseudounitary quantum systems and discuss various properties of pseudounitary operators. In particular we prove a characterization theorem for block-diagonalizable pseudounitary operators with finite-dimensional diagonal blocks. Furthermore, we show that every pseudounitary matrix is the exponential of i=−1 times a pseudo-Hermitian matrix, and determine the structure of the Lie groups consisting of pseudounitary matrices. In particular, we present a thorough treatment of 2×2 pseudounitary matrices and discuss an example of a quantum system with a 2×2 pseudounitary dynamical group. As other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics, demonstrate the coincidence of the symplectic group Sp(2n) with the real subgroup of a matrix group that is isomorphic to the pseudounitary group U(n,n), and elaborate on an approach to second quantization that makes use of the underlying pseudounitary dynamical groups.
A theory of transformation is presented for the diagonalization of a Hamiltonian that is quadratic in creation and annihilation operators or in coordinates and momenta. It is the systemization and theorization of Dirac and Bogoliubov-Valatin transformations, and thus provides us an operational procedure to answer, in a direct manner, the questions as to whether a quadratic Hamiltonian is diagonalizable, whether the diagonalization is unique, and how the transformation can be constructed if the diagonalization exists. The underlying idea is to consider the dynamic matrix. Each quadratic Hamiltonian has a dynamic matrix of its own. The eigenvalue problem of the dynamic matrix determines the diagonalizability of the quadratic Hamiltonian completely. In brief, the theory ascribes the diagonalization of a quadratic Hamiltonian to the eigenvalue problem of its dynamic matrix, which is familiar to all of us. That makes it much easy to use. Applications to various physical systems are discussed, with especial emphasis on the quantum fields, such as Klein-Gordon field, phonon field, etc..
Second-quantized operators are derived algebraically from first-quantized counterparts, thus providing an alternative to all existing methods, most of which rely upon demonstrating the equivalence of matrix elements of the two operators. This derivation establishes an operator identity between a second-quantized operator and a first-quantized operator whose domain of definition is extended explicitly from an N-particle Hilbert space to Fock space. The existence of such an identity eliminates the need to address the issue of the representations used for the two operators, which often obscures the underlying simple relationship between them.
"The possibility of describing noncommuting operators in quantum mechanics by classical type functions, and the associated expression of operator multiplication, is of considerable interest. The well known Wigner-Weyl-Moyal theory is an important example. Another is the diagonal representation of operators using standard coherent states. We develop general necessary and sufficient conditions for the existence of the diagonal representation in the context of a family of generalised coherent states associated with any unitary irreducible Lie group representation. Several examples illustrating these conditions, and interesting results in the Heisenberg-Weyl case, are presented".