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This appears fine "mathematically". But what of the interpretation. Does there exist quantities like s^+s and what do the mean? There exist eigenoperators \begin{equation}  s^+s|n>=\sqrt{n+\frac{1}{2}}|n> \\  r^+r|n>=\sqrt{n}|n> \\  ss^+|n>=\sqrt{n+1}|n> \\  rr^+|n>=\sqrt{n+\frac{1}{2}}|n>  \end{equation}  Which means a double application of r^+r is equal to the number operator N=a^+a. N=a^+a, a double application of ss^+ is equivalent to the operator $aa^$ and it implies that $s^+s \equiv rr^+$.  But, from the old definitions \begin{equation} N=a^+a \\  N=r^+s^+sr=r^+rr^+r