Relativistic harmonic oscillator revisited

The familiar Fock space commonly used to describe the relativistic harmonic oscillator, for example, as part of string theory, is insufficient to describe all the states of the relativistic oscillator. We find that there are three different vacua leading to three disconnected Fock sectors, all constructed with the same creation-annihilation operators. These have different spacetime geometric properties as well as different algebraic symmetry properties or different quantum numbers. Two of these Fock spaces include negative norm ghosts (as in string theory), while the third one is completely free of ghosts. We discuss a gauge symmetry in a worldline theory approach that supplies appropriate constraints to remove all the ghosts from all Fock sectors of the single oscillator. The resulting ghost-free quantum spectrum in $d+1$ dimensions is then classified in unitary representations of the Lorentz group $\mathrm{SO}(d,1)$. Moreover, all states of the single oscillator put together make up a single infinite dimensional unitary representation of a hidden global symmetry $\mathrm{SU}(d,1)$, whose Casimir eigenvalues are computed. Possible applications of these new results in string theory and other areas of physics and mathematics are briefly mentioned.

Figure 1

Parabolas in the spacelike region of $(x^0,x^1)$ at some fixed $x=\pm a$ and any $\theta$.

Figure 2

The dashed line is for $m=0$, and the solid line is for $m\ne 0$.