# Superluminal propagation of fields

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1. Feb 16, 2015

### Jesus

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.

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 nμ , 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 n0 for any ⃗n, the system is hyperbolic, with maximum wave speed n0/|⃗n|. On the other hand, if there are time-like solutions nμ 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?

2. Feb 16, 2015

### Staff: Mentor

Sort of. See the third paragraph at the top of this Wikipedia page:

http://en.wikipedia.org/wiki/Hyperbolic_partial_differential_equation

It contrasts hyperbolic PDEs with elliptic and parabolic PDEs; the first of these has a "limiting velocity" of propagation of disturbances, the others don't. But this "propagation of disturbances" may or may not correspond to "faster-than-light propagation of waves"; it depends on the specific PDE and what it's being used to describe.

I don't have access to the paper you refer to, so I can't say how the PDEs are classified there; but the general comment about PDEs in the Wikipedia page is valid.

3. Mar 20, 2015

### Jesus

ok, thank you for respond.