We measure the intrinsic relation between velocity dispersion ($sigma$) and luminosity ($L$) for massive, luminous red galaxies (LRGs) at redshift $z sim 0.55$. We achieve unprecedented precision by using a sample of 600,000 galaxies with spectra from the Baryon Oscillation Spectroscopic Survey (BOSS) of the third Sloan Digital Sky Survey (SDSS-III), covering a range of stellar masses $M_* gtrsim 10^{11} M_{odot}$. We deconvolve the effects of photometric errors, limited spectroscopic signal-to-noise ratio, and red--blue galaxy confusion using a novel hierarchical Bayesian formalism that is generally applicable to any combination of photometric and spectroscopic observables. For an L-$sigma$ relation of the form $L propto sigma^{beta}$, we find $beta = 7.8 pm 1.1$ for $sigma$ corrected to the effective radius, and a very small intrinsic scatter of $s = 0.047 pm 0.004$ in $log_{10} sigma$ at fixed $L$. No significant redshift evolution is found for these parameters. The evolution of the zero-point within the redshift range considered is consistent with the passive evolution of a galaxy population that formed at redshift $z=2-3$, assuming single stellar populations. An analysis of previously reported results seems to indicate that the passively-evolved high-mass L-$sigma$ relation at $zsim0.55$ is consistent with the one measured at $z=0.1$. Our results, in combination with those presented in Montero-Dorta et al. (2014), provide a detailed description of the high-mass end of the red sequence (RS) at $zsim0.55$. This characterization, in the light of previous literature, suggest that the high-mass RS distribution corresponds to the core elliptical population.
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