Thermal excitation spectrum from entanglement in an expanding quantum string

A surprising result in $e^+ e^-$ collisions is that the particle spectra from the string formed between the expanding quark-antiquark pair have thermal properties even though scatterings appear not to be frequent enough to explain this. We address this problem by considering the finite observable interval of a relativistic quantum string in terms of its reduced density operator by tracing over the complement region. We show how quantum entanglement in the presence of a horizon in spacetime for the causal transfer of information leads locally to a reduced mixed-state density operator. For very early proper time $\tau$, we show that the entanglement entropy becomes extensive and scales with the rapidity. At these early times, the reduced density operator is of thermal form, with an entanglement temperature $T_\tau=\hbar/(2\pi k_B \tau)$, even in the absence of any scatterings.