Solution

Choose complete orthonormal sets of complex Klein-Gordon solutions

{u_{i}^{in}}

and

{u_{i}^{out}}

that have positive frequency with respect to the asymptotic timelike Killing fields in the remote past and future. Their normalization uses the conserved Klein-Gordon inner product:

(u_{i}, u_{j})_{K G} = δ_{ij}, (u_{i}^{*}, u_{j}^{*})_{K G} = - δ_{ij}, (u_{i}, u_{j}^{*})_{K G} = 0.

ϕ = \sum_{i} (a_{i}^{in} u_{i}^{in} + a_{i}^{in †} u_{i}^{in *}) = \sum_{i} (a_{i}^{out} u_{i}^{out} + a_{i}^{out †} u_{i}^{out *}) .

u_{i}^{out} = \sum_{j} (α_{ij} u_{j}^{in} + β_{ij} u_{j}^{in *}),

and hence

a_{i}^{out} = \sum_{j} (α_{ij}^{*} a_{j}^{in} - β_{ij}^{*} a_{j}^{in †}) .

α α^{†} - β β^{†} = I, α β^{T} = β α^{T} .

The past and future notions of positive frequency therefore define the generally different in-vacuum and out-vacuum. Nonzero

β

means that the in-vacuum is a squeezed state containing out-particles.

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