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The polynomials in the generators of a unitary representation of thePoincar\'e group constitute an algebra which maps the dense subspace S ofsmooth, rapidly decreasing wavefunctions to itself. This mathematical result ishighly welcome to physicists, who previously just assumed their algebraictreatment of unbounded operators be justified. The smoothness, however, has theside effect that a rough operator R, which does not map a dense subspace of Sto itself, has to be shown to allow for some other dense domain which is mappedto itself both by R and all generators. Otherwise their algebraic product,their concatenation, is not defined. Canonical quantization of the light conestring postulates operators $-i X^1$ and $P^- = (P^0 - P^z)/2$ and as theircommutator the multiplicative operator R = P^1/(P^0 + P^z). This is not smoothbut rough on the negative z-axis of massless momentum. Using only thecommutation relations of P^m with the generators $-i M_iz$ of rotations in the$P^i$-$P^z$-plane we show that on massless states the operator $R$ isinconsistent with a unitary representation of SO(D-1). This makes the algebraicdetermination of the critical dimension, $D=26$, of the bosonic stringmeaningless: if the massless states of the light cone string admit R then theydo not admit a unitary representation of the subgroup SO(D-1) of the Poincar\'egroup. With analogous arguments we show: Massless multiplets are inconsistentwith a translation group of the spatial momentum which is generated by aself-adjoint spatial position operator $X$.

The Rough with the Smooth of the Light Cone String