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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