Bogoliubov Transformations  

In General > s.a. quantum field theory in curved spacetime.
* Idea: A transformation between one set of pure frequency modes (c.o.n.s. of solutions of a field equation, or Fock space structure for a quantum field theory) to another, in particular with respect to two timelike (Killing) vector fields in quantum field theory (in curved spacetime).
$ Def: If the set of modes {ui} is associated with operators ai and ai, and {vi} with bi and bi, then

ai = Σj (αji bj + βji* bj) ,   bj = Σi (αji* aiβji* ai) ,

where the coefficients are given by αij = (vi, uj), βij = –(vi, uj*), or

ui = Σj (αji* vjβji vj*) ,   vj = Σi (αji ui + βji ui*) .

* Properties of the coefficients: For a Bosonic field we get, from orthonormality of modes and preservation of commutation relations, respectively,

Σk (αik αjk* – βik βjk*) = δij ,   Σk (αik βjkβik αjk) = 0 .

* Fock spaces: The two Fock spaces are different if bij ≠ 0, and v-positive frequency modes contain u-negative frequency ones; e.g.,

\(\langle\) 0v | Nui | 0\(\rangle\) = ∑j |βji|2.

References
@ General: Bogoliubov JETP(58).
@ Bounds on coefficients: Visser PRA(99)qp [1D potential scattering]; Boonserm & Visser AP(08)-a0801; Boonserm PhD(09)-a0906.
@ In curved spacetime: Parker PR(69); Lapedes JMP(78); Ruijsenaars AP(78); Woodhouse PRS(81); in Birrell & Davies 82; Bombelli & Wyrozumski CQG(89).
@ Generalizations: Arraut & Segovia a1604 [q-deformed].
> Online resources: see Wikipedia page.


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