Physical Interpretation of Oscillatory Terms in the Non-Relativistic Spectrum

Characterize the physical meaning of the oscillatory terms in the non-relativistic spectrum I(ω) that regularize the infrared divergence in particle number, determining whether they correspond to soft particle production without net energy or to compensating positive and negative energy contributions.

Background

In the non-relativistic regime, the spectrum comprises a one-dimensional Planck term (which alone yields an infrared-divergent particle count) plus oscillatory terms that regularize the divergence. The total radiated energy is insensitive to the oscillations, but the particle number depends crucially on them.

The authors note two physical possibilities: the oscillations might produce soft particles that carry negligible energy, or their contributions could include positive and negative energy parts that cancel. Since particle numbers do not reveal negative energy flux in prior work, the precise physical interpretation of these oscillations remains unsettled.