Heisenberg versus the Covariant String

TL;DR

The paper demonstrates that a Poincaré multiplet of mass eigenstates cannot be represented within a space with a covariant position operator, challenging assumptions in relativistic quantum models like the covariant string.

Contribution

It provides a rigorous argument showing the incompatibility of covariant position operators with Poincaré multiplets, impacting the understanding of relativistic quantum theories.

Findings

Poincaré multiplets of definite mass cannot be subspaces with a covariant position operator.

The Stone-von Neumann theorem supports the non-existence of such subspaces.

The results exclude certain relativistic particle and string models from standard quantum frameworks.

Abstract

A Poincar\'e multiplet of mass eigenstates $\bigl(P^2 - m^2\bigr)\Psi = 0$ cannot be a subspace of a space with a $D$-vector position operator $X=(X_0,\dots X_{D-1})$: the Heisenberg algebra $[P^m, X_n] = i \delta^m{}_n$ implies by a simple argument that each Poincar\'e multiplet of definite mass vanishes. The same conclusion follows from the Stone-von Neumann theorem. In a quantum theory the constraint of an absolutely continuous spectrum to a lower dimensional submanifold yields zero even if Dirac's treatment of the corresponding classical constraint defines a symplectic submanifold with a consistent corresponding quantum model. Its Hilbert space is not a subspace of the unconstrained theory. Hence the operator relations of the unconstrained model need not carry over to the constrained model. Our argument excludes quantized worldline models of relativistic particles and the physical states of the covariant quantum string. We correct misconceptions about the generators of Lorentz transformations acting on particles.