Characteristic States of the Electromagnetic Radiation Field

It has been argued that the positive-frequency part of the quantized electromagnetic field is the "observable" that one would most naturally associate with field measurements using quantum photodetectors. However, since it is possible in principle to make field measurements via the process of stimulated emission, the question of the possible solutions of the characteristic-value equation for the creation operator $a†$ is examined. Various proofs are given to demonstrate that the characteristic kets of $a†$ are not physically admissible states of the radiation field. The possible existence of other useful basis states besides $|n〉$, $|\alpha 〉$, and states generated from these by unitary transformations is then considered. It is shown that when certain restrictions are placed on the correspondence between Hermitian combinations of the arbitrary non-normal operators $b$ and $b†$ and the harmonic-oscillator variables $x$, $p$, and $H$, then the only possible basis states are the coherent states $|\alpha 〉$ and the number states $|n〉$. A $\lambda$-dependent variation on the photon annihilation operator $a$ is also considered. Its characteristic states for $-1\mathrm{}\le \lambda \le 1$ are derived, and shown to form a complete set.