Klein-Gordon eqn: why dismiss messages at phase velocity

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Discussion Overview

The discussion centers on the implications of superluminal phase velocities in the Klein-Gordon equation (KG eqn) and their relation to relativistic causality. Participants explore theoretical models, mathematical proofs, and conceptual clarifications regarding wave propagation and signal transmission in the context of the KG equation, including both massless and massive cases.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants express skepticism about the claim that superluminal phase velocities do not violate causality, suggesting that signals may be influenced by long waves traveling faster than light.
  • Others argue that wave fronts do not travel faster than light, citing a theorem that supports this notion, regardless of the mass-squared of the field.
  • A participant proposes a scenario involving a delta function and questions whether a spacelike separated observer could distinguish between different initial conditions, suggesting that the phase fronts may not cancel out as claimed.
  • Some participants mention the role of Green functions in demonstrating that solutions to the KG equation vanish outside the light cone, implying that superluminal signals do not carry information.
  • There is a discussion about the mathematical treatment of the KG equation, with references to causal Green functions and their properties, including the assertion that they can be constructed without introducing friction.
  • Concerns are raised regarding the applicability of quantum field theory concepts to classical systems, with calls for specific mathematical proofs to support claims about wave cancellation.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the implications of superluminal phase velocities and the validity of theorems related to wave propagation. While some assert that causality is preserved, others challenge this view, leading to an unresolved discussion on the topic.

Contextual Notes

Participants reference various mathematical frameworks and theorems, but there is no consensus on the interpretation or implications of these results. The discussion highlights the complexity of the KG equation and the nuances involved in its analysis.

  • #31
vanhees71 said:
Here Sommerfeld's argument cannot be applied ...
It would be nice if you could present details of an example were the Sommerfeld's argument can be applied.
 
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  • #32
Avodyne said:
I continue to claim that the inapplicability of Sommerfeld's argument is due to the form of the KG dispersion relation (with branch points on the real axis), and not due to the singular initial condition.
I'm not sure what exactly is the Sommerfeld's argument, but there is a very general proof in
A. Bers, R. Fox, C.G. Kuper and S.G. Lipson,
a chapter in the book C.G. Kuper and A. Peres (eds), "Relativity and Gravitation" (Gordon and Breach, 1971)
for which branch points on the real axis are not a problem. The contour of integration over frequency avoids all singularities on the real axis.
 
Last edited:
  • #33
Demystifier said:
@vanhees71, have you noticed that your proof in #28 is almost identical to my proof in #21?
Argh, I've not seen this posting. It's indeed identical. Sorry for that.
 

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