DETLEV BUCHHOLZ, FABIO CIOLLI, GIUSEPPE RUZZI, and EZIO VASSELLI, in "The Universal C*-Algebra of the Electromagnetic Field", Lett Math Phys (2016) 106:269–285, https://arxiv.org/abs/1506.06603, construct a C*-Algebra of bounded operators effectively as \(\mathrm{e}^{\mathrm{i}\hat F_{\!f}}\), where \(f\) is a *real*-valued 2-form test function (the norm is \(\|\mathrm{e}^{\mathrm{i}\hat F_f}\| =\left\{\begin{array}{l}1,\|f\|=0\cr 0, \|f\|\not= 0 \end{array}\right.\)). That is, the vacuum sector is generated by complex superpositions of unitary coherent vector states. In quantum optics, however, the vacuum sector of the free EM field is constructed using both unitary and non-unitary operators, effectively generated by \(\mathrm{e}^{\mathrm{i}\hat F_{\!f}}\), where \(f\) is now a *complex*-valued 2-form test function.

So my question, "Is the space of coherent states generated by bounded operators empirically adequate?", asks whether the set of coherent states that are widely used in quantum optics is necessary for empirical adequacy or whether the C*-algebraic approach of using only bounded algebras is all that is needed. Is it just more convenient but unnecessary for physicists to use the larger algebra of unbounded operators, but more convenient for mathematicians to use only the algebra of bounded operators?