Mathematical Physics  

In General > s.a. field theory; gauge theory.
* Goal: Use a well-defined framework, a precise language, and the standards of rigor of modern mathematics to
(1) Determine consequences of physical theories, by proving theorems about physically motivated concepts;
(2) Derive properties and numbers of physical interest for model systems through calculation.
> Areas: see differential equations, integral equations; operator theory; group theory; differential geometry; set theory; and, more recently, topology and algebraic topology (including homotopy and homology).
> Specific topics: see Banach Space; chern-simons theory; distributions; hilbert space; Special Functions.

References > s.a. computational physics; programming languages [with Mathematica].
@ Texts, IIb: Hassani 00; Chow 00; Seaborn 02; Fischer-Cripps 05; Boas 06; Kusse & Westwig 06.
@ Texts, III: Whittaker & Watson 27; Courant & Hilbert 53-62; Morse & Feschbach 53; Von Neumann 55; Reed & Simon 72–79; Thirring 78; Richtmyer 78-81; Wilf 78; Arfken 85; Geroch 85; Vaughn 07.
@ Texts: Mathews & Walker 65; Dennery & Krzywicki 67; Butkov 68; Byron & Fuller 69; Cushing 75; Wyld 76; Bender & Orszag 78; Harper & Weaire 85; Bamberg & Sternberg 88; Carroll 88; Dettman 88; Battaglia & George 90; Chattopadhyay 90; Wong 91; Shankar 95; Cantrell 00; Arfken & Weber 01; Riley et al 02; McQuarrie 03; Szekeres 04.
@ For theoretical physics: Lam 03; Prakash 03; Appel 07.
@ Handbooks: Fanchi 97 [refresher]; Françoise et al 06 [encyclopedia].
@ Special emphasis: Barut ed-73; Sneed 79 [logical structure]; Hassani 91, 98 [foundations]; Steeb 03 [problems].

Relationship between Mathematics and Physics > s.a. Models; physics [laws and ultimate theories]; proofs.
* Idea: It is a fact that we need math to investigate physical laws and to express them, since the laws are only understandable mathematically; In this view, one can see mathematics as a language invented by humans, or mathematicians, to model nature; But this is a partial view, the relationship is deeper and many believe that mathematics is nature.
* History: There was a close symbiosis through all of the XVIII cy; They separated in the XIX cy because of developments in pure math (number theory ...; Gauss ...) and new math-independent physics (Faraday ... J W von Goethe – see Ritter, Oersted – with his romantic natural philosophy).
* Quote: C N Yang: "What surprised me is not that gauge field is the connection of fiber bundles, but more so that mathematicians can create it without touching the world of physics. I was shocked and puzzled, because you mathematicians can create these ideas from nothing"; Reply by S S Chern: "No, no, these ideas are not just imagination, they are natural and real".]
* Areas developed together: Calculus, differential equations, variational theory, differential geometry.
* Areas of physics that have contributed to math: Theory of brownian motion (analysis, probability); Gauge theories and instantons (dg).
* Differences: Essentially, mathematics has no external constraints on what is interesting or relevant; Theoretical physics uses units/dimensions, and its results are subject to verification by experiment.
* Conjectures/rigor: Examples are Kepler's conjecture on close packing of spheres in R3; Path integrals.
@ References: Stewart 07; Lax BAMS(08).

Quotations and Opinions
* Galileo: "The book of nature is written in the language of mathematics".
* J Jeans: "God is a mathematician".
* B Russell: Physics is math not because we know a lot about the external world, but because we know too little (quoted by Sabato).

References > s.a. Geometric Algebra; history of mathematics.
@ General: Poincaré BAMS(06), reprinted BAMS(00) [status]; Birkhoff BAMS(27), reprinted BAMS(00); Dirac PRSE(39); Manin 81; Gross PNAS(88); Glimm et al ed-90; Squires PW(90)aug; Chapline PRP(99); Morrison 00; Colyvan 01; Benioff FP(02)qp, FP(05)qp/04-in.
@ I: Boudot Rech(89); Oldershaw AJP(88); Irvine ed-89; Lines 94.
@ Physicists' point of view: Dirac IJTP(82); Oliver 94 [II]; Jackiw PT(96)feb; Faddeev mp/00-in; Witten BAMS(03).
@ And mathematical logic: Benioff JMP(70), JMP(71).
@ And foundations of physics: Emch 84; Nambiar m.GM/02.
@ Physics and geometry / topology: Witten pr(86); Atiyah in(88); Lantsman mp/01; > s.a. geometry, topology and physics.
@ Effectiveness of mathematics: Wigner CPAM(60); in Matthews 71, p122; in Lightman 86, p132.
@ Related topics: Ernest BJPS(90) [meaning of math expressions]; Liston PhSc(93) [reliability]; Davey BJPS(03) [on mathematical rigor]; Bueno SHPMP(05) [and Dirac's delta function]; Anderson & Joshi phy/06 [example of SU(2)]; Gelfert PhSc(05) [mathematical rigor].
@ Mathematical and physical cultures: Jaffe & Quinn BAMS(93) + responses BAMS(94); Sinai BAMS(06).
@ Cum granu salis? Bartocci & Wesley 90; Perchik mp/03.

"The most incomprehensible thing about the universe is that it is comprehensible" – A Einstein

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