Conjecture on real-space creation and annihilation operators

Let $a^{(+)}(k)$ denote real-space creation or annihilation variables, and let $a^{(+)}(k,\beta)$ and $\tilde{a}^{(+)}(k,\beta)$ denote temperature-dependent primordial variables. Real-space operator conjecture. The real-space creation and annihilation operators can be written in the proposed temperature-dependent form; the following displayed relation is given in the surrounding text:

$$a(k)=\cosh c_k\,a(k,\beta)+\sinh c_k\,\tilde{a}^+(k,\beta).$$

Here $a(k)$, $a(k,\beta)$, $\tilde{a}(k,\beta)$ and their adjoints are assumed to satisfy the same bosonic canonical commutation relations, including $[a(k),a^+(l)]=\delta(k-l)$; $a(k,\beta)$ annihilates a mode above the Fermi surface, while $\tilde{a}^+(k,\beta)$ creates a hole of energy-momentum $(-\omega_k,-k)$. The claim is an approximate physical picture of collective excitations and holes in a Dirac sea, motivated by the KMS condition and modular theory. It is not proved in the paper.