On the partner particles for black hole evaporation

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Due to the linearity of the field equations and the resulting bilinear structure of the Hamiltonian, quantum radiation effects such as black hole evaporation or particle creation in an expanding universe are typically described as (squeezing) processes where particles are created in pairs.
Here, we address the following question: given a mode (e.g., wave-packet) corresponding to a created particle (e.g., as part of Hawking radiation), what is its partner, i.e., the other particle of the pair?
After a general derivation of this partner mode, we will discuss some examples such as moving mirror radiation and speculate about possible implications for the black hole information puzzle.