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 {u_i} is associated with operators a_i and a_i^†, and {v_i} with b_i and b_i^†, then

a_i = Σ_j (α_ji b_j + β_ji* b_j^†) ,   b_j = Σ_i (α_ji* a_i − β_ji* a_i^†) ,

where the coefficients are given by α_ij = (v_i, u_j), β_ij = −(v_i, u_j*), or

u_i = Σ_j (α_ji* v_j − β_ji v_j*) ,   v_j = Σ_i (α_ji u_i + β_ji u_i*) .

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

Σ_k (α_ik α_jk* − β_ik β_jk*) = δ_ij ,   Σ_k (α_ik β_jk − β_ik α_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.,

\(\langle\) 0_v | N_ui | 0_v \(\rangle\) = ∑_j |β_ji|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 PLA(18)-a1604 [q-deformed].
> Online resources: see Wikipedia page.

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