where c {\displaystyle ~c~} and s {\displaystyle ~s~} are c-numbers and b ^ † {\displaystyle ~{\hat {b}}^{\dagger }~} is a creation operator characterizing the amplifier. Without loss of generality, it may be assumed that c {\displaystyle ~c~} and s {\displaystyle ~s~} are real. The commutator of the field operators is invariant under unitary transformation U ^ {\displaystyle ~{\hat {U}}~} :