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 The introduction of the ket |Θ0i is just a mathematical trick so that we can analyze the action of exp[±iωˆ+t] in terms of an operator acting on some Hilbert-space. |Θ0i is not an element of a Hilbert space for quasi-particle modes. Rather, Θ0(x) is interpreted as a wavefunction in the Hilbert-space L2(R) of square-integrable functions so that we can use the usual operator algebra in the form hx|Oˆ(xˆ, pˆ)|Θ0i = Oˆ(x, −i¯h∂/∂x)hx|Θ0i.
 D.F. Walls and G.J. Milburn, Quantum Optics, SpringerVerlag, Berlin 1994.