Heisenberg Algebra and String Theory

TL;DR

This paper explores the incompatibility of the Heisenberg algebra with string theory constraints, showing that certain spacetime operators cannot coexist with physical string states, questioning the Lorentz invariance and the critical dimension of the bosonic string.

Contribution

It demonstrates fundamental limitations in combining position operators with string physical states, challenging the algebraic derivation of the bosonic string's critical dimension.

Findings

Position operators cannot be defined with physical string states.

The algebraic approach to critical dimension is invalidated.

Light cone string lacks Lorentz invariance due to algebraic constraints.

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.