Troublesome infinities in physics

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SUMMARY

This discussion centers on the challenges physicists face when multiplying generalized functions, specifically the square Dirac delta function. Participants seek examples of both empirically consistent solutions and unresolved problems related to this issue. The conversation emphasizes the need for a solid understanding of the physics behind these equations, particularly in classical mechanics, field theory, and wave phenomena, rather than delving into quantum field theory (QFT) complexities.

PREREQUISITES
  • Understanding of generalized functions, particularly the Dirac delta functional.
  • Familiarity with classical mechanics concepts.
  • Basic knowledge of field theory principles.
  • Awareness of wave phenomena and their mathematical representations.
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  • Research the mathematical formulation of generalized functions in physics.
  • Study classical mechanics examples involving Dirac delta functions.
  • Explore field theory applications and their implications in physics.
  • Investigate unresolved problems in the multiplication of generalized functions.
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Physicists, students of physics, and researchers interested in the mathematical foundations of generalized functions and their applications in classical mechanics and field theory.

Akater
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Hello,

it is said that physicists sometimes need to multiply two generalized functions (square Dirac delta, for example) which makes little sense from the mathematical viewpoint. Could you please provide some examples where such an issue occurs?

Actually, I would be happy to see examples of two different kinds,

— those where (an empirical / magical) physically consistent solution is known;

— those which are still open problems.

I'd like to really understand the physics behind equations (their derivations, at least), but my physical background is not solid. So, please, if possible, don't stick to QFT problems. Classical mechanics / field theory / waves would be perfect. (Although, some example is definitely better than no example at all.)

________
Sorry for poor English.
 
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All such 'generalised' functions can be reformulated with mathematical rigour. The Dirac delta 'function', for instance, is really the Dirac delta functional.
 
Sure. I was talking about (supposed) situation when physicist need to multiply them and experience problems with the multiplication operation.
 

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