A classical treatment of the radiation of super-phase velocity of scalar and vector fields is presented in this paper with a hope that it could be extended to the case of multiple meson production in a nucleus by nucleons of a very high energy.
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Yang, C. (1992). Radiation of Fast Charged Particles in Media. Chinese Journal of Physics, 30(3), 321-338. https://www.airitilibrary.com/Article/Detail?DocID=05779073-199206-201304150004-201304150004-321-338
Tsui, Y. K. (1992). Quantum phase operators [master's thesis, The University of Hong Kong]. Airiti Library. https://www.airitilibrary.com/Article/Detail?DocID=U0029-1812201200005195