Editing Bogoliubov transformation (section)

{{Short description|Mathematical operation in quantum optics, general relativity and other areas of physics}}

In [[theoretical physics]], the '''Bogoliubov transformation''', also known as the '''Bogoliubov–Valatin transformation''',<ref>{{Cite journal |last=Balian |first=R. |last2=Brezin |first2=E. |date=1969-11-01 |title=Nonunitary bogoliubov transformations and extension of Wick’s theorem |url=https://link.springer.com/article/10.1007/BF02710281 |journal=Il Nuovo Cimento B (1965-1970) |language=en |volume=64 |issue=1 |pages=37–55 |doi=10.1007/BF02710281 |issn=1826-9877|url-access=subscription }}</ref><ref>{{Cite book |last=Castro |first=Carlo Di |url=https://www.google.fr/books/edition/Statistical_Mechanics_and_Applications_i/2OWMCgAAQBAJ?hl=en&gbpv=0 |title=Statistical Mechanics and Applications in Condensed Matter |last2=Raimondi |first2=Roberto |date=2015-08-27 |publisher=Cambridge University Press |isbn=978-1-316-35198-7 |language=en}}</ref><ref>{{Cite book |last=Mattuck |first=Richard D. |url=https://www.google.fr/books/edition/A_Guide_to_Feynman_Diagrams_in_the_Many/1P_DAgAAQBAJ?hl=en&gbpv=1&dq=feynman+%22Valatin%22&pg=PA263&printsec=frontcover |title=A Guide to Feynman Diagrams in the Many-Body Problem: Second Edition |date=2012-08-21 |publisher=Courier Corporation |isbn=978-0-486-13164-1 |language=en}}</ref> was independently developed in 1958 by [[Nikolay Bogolyubov]] and [[John George Valatin]] for finding solutions of [[BCS theory]] in a homogeneous system.<ref>{{cite journal |last1=Valatin |first1=J. G. |title=Comments on the theory of superconductivity |journal=Il Nuovo Cimento |date=March 1958 |volume=7 |issue=6 |pages=843–857 |doi=10.1007/bf02745589|bibcode = 1958NCim....7..843V |s2cid=123486856 }}</ref><ref name=":0">{{cite journal |last1=Bogoljubov |first1=N. N. |title=On a new method in the theory of superconductivity |journal=Il Nuovo Cimento |date=March 1958 |volume=7 |issue=6 |pages=794–805 |doi=10.1007/bf02745585 |bibcode = 1958NCim....7..794B |s2cid=120718745 }}</ref> The Bogoliubov transformation is an [[isomorphism]] of either the [[canonical commutation relation algebra]] or [[canonical anticommutation relation algebra]]. This induces an autoequivalence on the respective representations. The Bogoliubov transformation is often used to diagonalize [[Hamiltonian (quantum mechanics)|Hamiltonian]]s, which yields the stationary solutions of the corresponding [[Schrödinger equation]]. The Bogoliubov transformation is also important for understanding the [[Unruh effect]], [[Hawking radiation]], Davies-Fulling radiation (moving mirror model), pairing effects in nuclear physics, and many other topics.

The Bogoliubov transformation is often used to diagonalize Hamiltonians, ''with'' a corresponding transformation of the state function. Operator eigenvalues calculated with the diagonalized Hamiltonian on the transformed state function thus are the same as before.