Bogoliubov transformation
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| Bogoliubov transformation | |
|---|---|
| Name | Bogoliubov transformation |
| Field | Theoretical physics |
| Introduced | 1947 |
| Introduced by | Nikolay Bogoliubov |
| Related | Bardeen–Cooper–Schrieffer theory, Hartree–Fock, Quasiparticles |
Bogoliubov transformation The Bogoliubov transformation is a linear canonical change of basis for creation and annihilation operators introduced to diagonalize quadratic Hamiltonians in quantum field theory and many-body physics. It provides a framework connecting ground states, quasiparticle excitations, and symmetry-breaking phenomena across models studied by Nikolay Bogoliubov, Lev Landau, John Bardeen, Leon Cooper, and Robert Schrieffer. The transformation underlies methods used in treatments by Richard Feynman, Julian Schwinger, Pieter van Nieuwenhuizen, and Freeman Dyson.
Definition and motivation
The transformation was proposed by Nikolay Bogoliubov to address interacting systems that can be approximated by quadratic forms, motivated by problems studied in the context of Superfluidity, Fermi liquid theory, Bose–Einstein condensation, and the emergent quasiparticles of Bardeen–Cooper–Schrieffer theory. It serves to diagonalize Hamiltonians originally examined in works by Lev Landau on collective excitations, by Lev Pitaevskii on condensates, and by Lev Gor'kov on superconductivity, thereby clarifying the relation between bare particles treated in perturbative expansions by Sin-Itiro Tomonaga and dressed quasiparticles appearing in the analyses of Enrico Fermi and Paul Dirac.
Mathematical formulation
In algebraic form the transformation mixes operators in a manner akin to canonical linear transformations used by Paul Dirac in quantum mechanics and by Andrey Kolmogorov in linear algebraic methods, preserving commutation relations that echo symplectic structures encountered in work by Élie Cartan and Hermann Weyl. For fermionic systems the matrices implement a unitary rotation related to representations studied by Eugene Wigner and Hermann Weyl, while for bosonic systems the maps correspond to Bogoliubov transformations respecting positivity conditions treated in studies by John von Neumann and Israel Gelfand. The formalism is closely connected to techniques advanced by Julian Schwinger in operator algebra and by Arthur Wightman in axiomatic field theory.
Applications in quantum many-body physics
Bogoliubov transformations are central in analyses conducted by Lev Landau and Ilya Lifshitz on superfluids, by John Bardeen and Leon Cooper on superconductors, and by Carl Anderson on localization phenomena. They appear in treatments by Philip Anderson of broken symmetry, by Giovanni Jona-Lasinio in modeling spontaneous symmetry breaking, and by Yoichiro Nambu in the context of mass generation. The approach is used in large-N methods favored by Kenneth Wilson in renormalization group studies, in random-matrix contexts pursued by Freeman Dyson, and in cold-atom experiments interpreted using theory from Wolfgang Ketterle and Eric Cornell. Implementations rely on algebraic structures similar to those in scattering theory developed by Lev Landau and Evgeny Lifshitz and on mean-field approximations used by John Slater and Douglas Hartree.
Bogoliubov–de Gennes equations and superconductivity
The Bogoliubov–de Gennes formalism, extended by Pierre-Gilles de Gennes to spatially inhomogeneous systems, casts the mean-field superconducting problem into coupled equations whose diagonalization uses Bogoliubov transformations as in the original proposals by Nikolay Bogoliubov. This machinery is applied in analyses by Brian Josephson of tunneling phenomena, in microscopic derivations by Lev Gor'kov, and in topological superconductivity work by Alexei Kitaev, Andrey Kitaev-related research, and investigations by Roman Lutchyn and Jay Sau. The equations are employed in studies of vortex states promoted by Abrikosov and in proximity effects explored experimentally by Pieter de Gennes and Clifford Beenakker.
Canonical transformations and symplectic structure
Viewed as canonical transformations, Bogoliubov maps preserve commutation or anticommutation brackets analogous to symplectic maps studied in classical mechanics by Henri Poincaré and in geometric quantization by Jean-Marie Souriau. The mathematical properties connect to symplectic groups analyzed by Élie Cartan and to representation theory developed by Harish-Chandra and Hermann Weyl. Applications in quantum information draw on structure analyzed by Asher Peres and John Preskill, and rigorous operator-algebraic treatments follow lines traced by Israel Gelfand and Masamichi Takesaki.
Examples and solvable models
Canonical examples include the quadratic Hamiltonian solved in the original work by Nikolay Bogoliubov for weakly interacting Bose gases, models of superconductivity developed by John Bardeen, Leon Cooper, and Robert Schrieffer, and the exactly solvable Kitaev chain advanced by Alexei Kitaev. Other solvable instances appear in spin-wave theory formulated by Felix Bloch and Tsuneyoshi Kambe, in the Dicke model analyzed by Robert Dicke, and in the Luttinger model studied by J. M. Luttinger and JR Mattis. Exactly diagonalizable quadratic forms also play roles in studies by Bethe on integrable systems, in Sachdev–Ye–Kitaev explorations by Subir Sachdev and Alexei Kitaev, and in quantum optics treatments by Roy Glauber.