Editing Bogoliubov transformation (section)

== Fermionic mode ==

For the [[Commutator|anticommutation]] relations
:<math>\left\{ \hat{a}, \hat{a}\right\} = 0, \left\{ \hat{a}, \hat{a}^\dagger \right\} = 1,</math>
the Bogoliubov transformation is constrained by <math>uv=0, |u|^2+|v|^2=1</math>. Therefore, the only non-trivial possibility is <math>u=0, |v|=1,</math> corresponding to particle–antiparticle interchange (or particle–hole interchange in many-body systems) with the possible inclusion of a phase shift. Thus, for a single particle, the transformation can only be implemented (1) for a [[Dirac fermion]], where particle and antiparticle are distinct (as opposed to a [[Majorana fermion]] or [[Chirality_(physics)|chiral fermion]]), or (2) for multi-fermionic systems, in which there is more than one type of fermion.

===Applications===
The most prominent application is again by Nikolai Bogoliubov himself, this time for the [[BCS theory]] of [[superconductivity]].<ref name="Kittel" /><ref name="NMTS1">{{cite journal |last1=Bogoliubov |first1=N. N. |date=1 Jan 1958 |title=A new method in the theory of superconductivity. I |journal=Soviet Physics (U.S.S.R.) JETP |volume=7 |issue=1 |pages=41–46}}</ref><ref name="NMTS3">{{cite journal |last1=Bogoliubov |first1=N. N. |title=A new method in the theory of superconductivity III |journal=Soviet Physics (U.S.S.R.) JETP |date=July 1958 |volume=34 |issue=7 |pages=51–55 |url=http://www.jetp.ac.ru/files/Bogolubov_007_01_0051.pdf |access-date=2017-04-27 |archive-date=2020-07-27 |archive-url=https://web.archive.org/web/20200727153421/http://jetp.ac.ru/files/Bogolubov_007_01_0051.pdf |url-status=dead }}</ref><ref name="BTS">{{cite journal |last1=Bogolyubov |first1=N. N. |last2=Tolmachev |first2=V. V. |last3=Shirkov |first3=D. V. |title=A new method in the theory of superconductivity |journal=Fortschritte der Physik |date=November 1958 |volume=6 |issue=11–12 |pages=605–682 |doi=10.1002/prop.19580061102|bibcode = 1958ForPh...6..605B }}</ref> The point where the necessity to perform a Bogoliubov transform becomes obvious is that in mean-field approximation the Hamiltonian of the system can be written in both cases as a sum of bilinear terms in the original creation and destruction operators, involving finite <math>\langle a_i^+a_j^+\rangle</math> terms, i.e. one must go beyond the usual [[Hartree–Fock method]]. In particular, in the mean-field [[Bogoliubov–de Gennes Hamiltonian]] formalism with a superconducting pairing term such as <math>\Delta a_i^+a_j^+ + \text{h.c.}</math>, the Bogoliubov transformed operators <math>b, b^\dagger</math> annihilate and create quasiparticles (each with well-defined energy, momentum and spin but in a quantum superposition of electron and hole state), and have coefficients <math>u</math> and <math>v</math> given by eigenvectors of the Bogoliubov–de Gennes matrix. Also in [[nuclear physics]], this method is applicable, since it may describe the "pairing energy" of nucleons in a heavy element.<ref>{{cite journal |last1=Strutinsky |first1=V. M. |title=Shell effects in nuclear masses and deformation energies |journal=Nuclear Physics A |date=April 1967 |volume=95 |issue=2 |pages=420–442 |doi=10.1016/0375-9474(67)90510-6 |bibcode = 1967NuPhA..95..420S }}</ref>