Heisenberg Algebra and String Theory

Abstract

If the algebra of the Poincar\'e generators is enlarged by the spacetime position operator $X=(X_0,\dots, X_{D-1})$ then the spectra of the momentum $P$ and the mass $P^2$ are unbounded and continuous. In particular, the constraint $(P^2 - m^2)\Psi_{\text{phys}}=0$ of the covariant string has no solution in the space which admits $X$: All physical states vanish, $\Psi_{\text{phys}}=0$. Vice versa, a space spanned by mass eigenstates does not admit the position operator $X$ in $D$ dimensions. A massless particle does not allow a spatial position operator $\vec X$. The domain of Heisenberg pairs $X^i$ and $P^j$, $i,j\in \{1,\dots D-2\}$, $D > 2$, which commute with $P^+=(P^0 + P_z)/\sqrt{2}$, $[P^+,X^i] = 0$, does not allow for a space with massless or tachyonic states, which is mapped to itself by rotations, leave alone Lorentz transformations. This is true in all dimensions and makes the algebraic calculation of the critical dimension, $D=26$, of the bosonic string meaningless: the light cone string is not Lorentz invariant.