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 wrt two timelike (Killing) vector fields in quantum field theory (in curved spacetime).
$ Def: If the set of modes {u_i} is associated with operators a_i and a_i, and {v_i} with b_i and b_i, then

a_i = _j (_ji b_j + _ji* b_j) ,   b_j = _i (_ji* a_i – _ji* a_i) ,

where the coefficients are given by _ij = (v_i, u_j), _ij = –(v_i, u_j*), or

u_i = _j (_ji* v_j – _ji v_j*) ,   v_j = _i (_ji u_i + _ji u_i*) .

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

_k (_{ik jk}* – _{ik jk}*) = _ij ,   _k (_{ik jk} – _{ik jk}) = 0 .

* Fock spaces: The two Fock spaces are different if b_ij 0, and v-positive frequency modes contain u-negative frequency ones; e.g.,  0_v | N_ui | 0_v  = _j |_ji|².

References
@ General: Bogoliubov JETP(58); Boonserm & Visser a0801 [bounds on coefficients].
@ In curved spacetime: Parker PR(69); Lapedes JMP(78); Ruijsenaars AP(78); Woodhouse PRS(81); in Birrell & Davies 82; Bombelli & Wyrozumski CQG(89).

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