Photons and nonstandard analysis

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Discussion Overview

The discussion explores the application of nonstandard analysis (NSA) to the behavior of photons, particularly considering the implications of treating the mass of a photon as infinitesimal. Participants inquire about existing references and the potential benefits of such an approach within the context of physics.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether there have been attempts to apply NSA to photons, specifically regarding the concept of infinitesimal mass.
  • Another participant asserts that there have been no such applications and provides a reference to a Wikipedia article on NSA.
  • A participant suggests that Brownian motion might be the closest application of NSA to physics, although they note that NSA was not mentioned in their mathematical exposure to Brownian motion.
  • Concerns are raised about the utility of considering an infinitesimal photon mass, with one participant proposing that regularization of infrared divergences might be a potential benefit, but questioning whether standard cutoff approaches are preferable due to gauge invariance.
  • Another participant argues that conventional methods are generally simpler than NSA, although they acknowledge that NSA might provide a more intuitive understanding.
  • A later reply mentions the utility of NSA in handling the Dirac Delta Function, contrasting it with distribution theory.
  • One participant asks whether there is any merit to using NSA in calculations and if it simplifies any processes.

Areas of Agreement / Disagreement

Participants express differing views on the applicability and utility of nonstandard analysis in physics, particularly regarding photons. There is no consensus on whether NSA offers significant advantages over conventional methods.

Contextual Notes

Limitations in the discussion include the lack of specific references to applications of NSA in photon behavior and the unresolved nature of the benefits of NSA compared to traditional approaches.

phoenixthoth
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Hi PF contributors,

Have there been attempts to apply nonstandard analysis to the behavior of photons? For instance, considering the mass of a photon to be infinitesimal.

I'm guessing the answer is yes so if you could provide references, I would be much obliged.

phoenixthoth
 
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Thanks for that. It looks like Brownian motion is the closest thing to an application of NSA to physics.
 
phoenixthoth said:
For instance, considering the mass of a photon to be infinitesimal.
What is gained by considering the infinitesimal photon mass? Perhaps regularization of IR divergences? For that purpose, I guess the standard cutoff approach is much better, because it does not violate the gauge invariance.

Anyway, the cutoff is supposed to be "infinitely big", so it can be viewed as 1/infinitesimal, so it could also be treated by non-standard analysis.
 
phoenixthoth said:
Thanks for that. It looks like Brownian motion is the closest thing to an application of NSA to physics.
In all my exposure to Brownian motion (mostly mathematical), NSA was never mentioned.
 
mathman said:
In all my exposure to Brownian motion (mostly mathematical), NSA was never mentioned.
Anything that can be done by NSA, can also be done by more conventional limits. Moreover, in the technical sense the conventional methods are usually simpler. A possible advantage of NSA is a more intuitive picture, but intuition is a matter of practice.
 
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Is there actually any merit to doing things with NSA? Does it make any calculations simpler?
 

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