Non-Standard Analysis  

In General > s.a. Hypernumbers; probability theory \ Continuum; Infinitesimal.
* History: Founded by A Robinson in the early 1960s; The theory has been made simpler by using internal set theory (E Nelson), but this gives only a partial approach.
* Idea: It takes over all results from standard analysis, but adds one more notion, the property of an object in a set of being standard or not; Infinitesimal and infinite quantities are treated like other numbers.
* Motivation: It simplifies many calculations, and gives a better understanding of the behavior of curves at non-differentiable points (like using a lens with infinite magnifying power), fractals, differential equations, ...
* Limitations: It can replace standard analysis to some extent, but not completely, since there is no unique non-standard enlargement of \(\mathbb R\).
> Online resources: see Wikipedia page.

And Physics > s.a. scalar.
* Idea: In most applications, only elementary facts and techniques of non-standard calculus seem to be necessary, and the advantages of a theory which includes infinitesimals rely more on the possibility of making new models than on the techniques used to prove results.
@ General references: Werner & Wolff PLA(95) [relationship classical-quantum mechanics]; Bagarello IJTP(99), IJTP(99) [variational principles in classical mechanics]; Ansoldi PhD(00)ht/04 [and strings]; Benci et al a0807 [elementary approach, and Fokker-Plank equation for brownian motion]; Fletcher et al RAE-a1703 [approaches].
@ Quantum physics: Gudder IJTP(94), FP(94) [quantum field theory and Fock space]; Almeida & Teixeira JMP(04) [space of pure states]; Raab JMP(04) [approach to quantm mechanics]; Bárcenas et al mp/06-wd [Casimir effect]; Fliess CRM(07)-a0704 [probabilities and fluctuations]; > s.a. path integrals.

Other References
@ Articles: Schmieden & Laugwitz MZ(58); Robinson PKNAW(61); Voros JMP(73); Machover BJPS(93).
@ Articles, I: Davis & Hersch SA(72)jun; Rech(83)oct; Diener Rech(88).
@ Books: Robinson 74; Hurd & Loeb 85; Benci & Di Nasso 15 [and counting infinite sets]; > s.a. functional analysis.
@ Internal set theory: Nelson BAMS(77); Robert 88.
@ Approaches: Cortizo fa/95, fa/95 ["virtual calculus"].
@ On the delta function: Laugwitz SBAW(59) [as regular function]; Ferreira Cortizo fa/95 [calculus].


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