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For the first time a rigorous quantum treatment of the Landau-Pomeranchuk-Migdal effect in QED and QCD is given. The rate of photon (gluon) radiation by an electron (quark) in medium is expressed through the Green's function of a two-dimensional Schrödinger equation with an imaginary potential. In QED this potential is proportional to the dipole cross section for scattering of e + e − pair off an atom, in QCD it is proportional to the cross section of interaction with color centre of the color singlet quark-antiquark-gluon system.

We review the properties of energetic parton propagation in hot or cold QCD matter, as obtained in recent works. Advances in understanding the energy loss -collisional and radiative -are summarized, with emphasis on the latter: it features very interesting properties which may help to detect the quark-gluon plasma produced in heavy ion collisions. We describe * preprint BI-TP 2000/08, LPT-Orsay 00-22 submitted to Annual Review of Nuclear and Particle Science two different theoretical approaches, which lead to the same radiated gluon energy spectrum.

The induced gluon radiation of a high energy quark in a finite-size QCD medium is studied. For a sufficiently energetic quark produced inside a medium we find the radiative energy loss ∆E q ∝ L 2 , where L is the distance passed by quark in the medium. It has a weak dependence on the initial quark energy E q . The L 2 dependence turns to L 1 as the quark energy decreases. Numerical calculations are performed for a cold nuclear matter and a hot quark-gluon plasma. For a quark incident on a nucleus we predict ∆E q ≈ 0.1E q (L/10 fm) β , with β close to unity.