  The polynomials in the generators of a unitary representation of the
Poincar\'e group constitute an algebra which maps the dense subspace S of
smooth, rapidly decreasing wavefunctions to itself. This mathematical result is
highly welcome to physicists, who previously just assumed their algebraic
treatment of unbounded operators be justified. The smoothness, however, has the
side effect that a rough operator R, which does not map a dense subspace of S
to itself, has to be shown to allow for some other dense domain which is mapped
to itself both by R and all generators. Otherwise their algebraic product,
their concatenation, is not defined. Canonical quantization of the light cone
string postulates operators $-i X^1$ and $P^- = (P^0 - P^z)/2$ and as their
commutator the multiplicative operator R = P^1/(P^0 + P^z). This is not smooth
but rough on the negative z-axis of massless momentum. Using only the
commutation 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 $R$ is
inconsistent with a unitary representation of SO(D-1). This makes the algebraic
determination of the critical dimension, $D=26$, of the bosonic string
meaningless: if the massless states of the light cone string admit R then they
do not admit a unitary representation of the subgroup SO(D-1) of the Poincar\'e
group. With analogous arguments we show: Massless multiplets are inconsistent
with a translation group of the spatial momentum which is generated by a
self-adjoint spatial position operator $X$.
