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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send feedback and suggestions to bombelli at olemiss.edu – modified
2 mar 2017