I am reading about the Velo-Zwanziger problem in which particles with spin higher than 1 propagates faster than light when the particles are coupled with an electromagnetic field.(adsbygoogle = window.adsbygoogle || []).push({});

In the original paper:G. Velo and D. Zwanziger, “Propagation And Quantization Of Rarita-Schwinger Waves In An External Electromagnetic Potential,” Phys. Rev. 186, 1337 (1969)the authors, to study the causal properties of the equation of motion for spin 3/2 particles coupled with electromagnetic field, use the "method of characteristic determinant" in which they replaces i∂_{μ}with n_{μ}, the normal to the characteristic hypersurfaces, in the highest-derivative terms of the equation of motion.

According to them the determinant ∆(n) of the resulting coefficient matrix determines the causal properties of the system: if the algebraic equation ∆(n) = 0 has real solutions for n_{0}for any ⃗n, the system is hyperbolic, with maximum wave speed n_{0}/|⃗n|. On the other hand, if there are time-like solutions n_{μ}for ∆(n) = 0, the system admits faster-than-light propagation.

My doubt about this is that I don't understand why can be faster-than-light propagation in the second case. If a partial differential equation is not hyperbolic does that meas that there can be faster-than-light propagation of waves?

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# Superluminal propagation of fields

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