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. Infinitesimal; scalar field theory.

* __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 18 [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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