  We study the radio properties of 706 sub-millimeter galaxies (SMGs) selected
at 870$\mu$m with the Atacama Large Millimeter Array from the SCUBA-2 Cosmology
Legacy Survey map of the Ultra Deep Survey field. We detect 273 SMGs at
$>4\sigma$ in deep Karl G. Jansky Very Large Array 1.4 GHz observations, of
which a subset of 45 SMGs are additionally detected in 610 MHz Giant Metre-Wave
Radio Telescope imaging. We quantify the far-infrared/radio correlation through
parameter $q_\text{IR}$, defined as the logarithmic ratio of the far-infrared
and radio luminosity, and include the radio-undetected SMGs through a stacking
analysis. We determine a median $q_\text{IR} = 2.20\pm0.03$ for the full
sample, independent of redshift, which places these $z\sim2.5$ dusty
star-forming galaxies $0.44\pm0.04$ dex below the local correlation for both
normal star-forming galaxies and local ultra-luminous infrared galaxies
(ULIRGs). Both the lack of redshift-evolution and the offset from the local
correlation are likely the result of the different physical conditions in
high-redshift starburst galaxies, compared to local star-forming sources. We
explain the offset through a combination of strong magnetic fields
($B\gtrsim0.2$mG), high interstellar medium (ISM) densities and additional
radio emission generated by secondary cosmic rays. While local ULIRGs are
likely to have similar magnetic field strengths, we find that their
compactness, in combination with a higher ISM density compared to SMGs,
naturally explains why local and high-redshift dusty star-forming galaxies
follow a different far-infrared/radio correlation. Overall, our findings paint
SMGs as a homogeneous population of galaxies, as illustrated by their tight and
non-evolving far-infrared/radio correlation.
