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; Response Functions;
Special Functions.

**References** > s.a. computational
physics; programming languages [Mathematica].

@ __Texts, IIb__: Hassani 00;
Chow 00;
Seaborn 02;
Fischer-Cripps 05;
Blum & Lototsky 06;
Boas 06;
Kusse & Westwig 06;
Svozil a1203-ln;
Wong 13;
Herman 13;
Chu a1701-ln;
Altland & von Delft 19;
Svozil 20.

@ __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;
Fong et al 03 [esp. differential equations];
Vaughn 07;
Henner et al 09;
Stone & Goldbart 09;
Petrini et al 17,
18;
Cahill 19
[e1r CP(14)].

@ __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;
Shankar 95;
Cantrell 00;
McQuarrie 03;
Szekeres 04;
Riley et al 06;
Shima & Nakayama 10 [including wavelets];
Nair 11 [esp. differential and integral equations; Fourier and Laplace transforms];
Arfken et al 12;
Alastuey et al 16.

@ __Topology, differential geometry, group theory__:
Mukhi & Mukunda 10;
Marsh 18;
Starkovich 21.

@ __Other geometrical__:
Katanaev a1311
[geometrical methods, long monograph in Russian];
Aldrovandi & Pereira 16.

@ __Other theoretical physics__: Prakash 03;
Appel 07;
Prosperetti 11 [field theory];
Dutailly a1209 [long, comprehensive, detailed];
Lam 15 [quantum and statistical physics];
Niccolai a2105 [long notes].

@ __Handbooks, collections__: Fanchi 97 [refresher];
Françoise et al 06 [5-volume encyclopedia];
Anel & Catren ed-21 [new spaces].

@ __Problems__: Steeb 11;
Beloglazov et al a1110 [from the Students Training Contest Olympiad];
Cicogna 18 [ug].

@ __Special emphasis__: Barut ed-73;
Sneed 79 [logical structure];
Hassani 91, 13 [foundations];
Jackson 06 [quantum mechanics].

> __Online resources__: see
R Baretti's course site;
J Baez's news site; Encyclopedia
of Mathematical Physics articles;
Open Problems site;
Wikipedia page.

**Relationship between Mathematics and Physics**
> s.a. computability; 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 century; They separated in the XIX
century 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), but in the XX
century physics motivated many developments in mathematics, such as linear
operator theory on Hilbert spaces, rigged Hilbert spaces, distributions,
and unitary group representations.

* __Effectiveness of mathematics__:
It can be seen all the time in the way natural phenomena follow the laws of
physics, and it is highlighted by the predictions in theoretical physics of
unexpected phenomena that were subsequently verified experimentally –
Examples are antimatter, seen in the solutions of the Dirac equation; quarks,
predicted by Murray Gell-Mann; the expansion of the universe, black holes,
and the existence of gravitational waves, predicted by general relativity.

* __Effectiveness of physics__:
2020, Quantum entanglement can be used in vastly more complicated multiprover
interactive proofs than with classical provers.

* __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 mathematics__: Theory of brownian motion (analysis,
probability); Gauge theories and instantons (differential geometry).

* __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 \(\mathbb R^3\); Path
integrals.

**References**
> s.a. Geometric Algebra; history
of mathematics; physical theories [theory of resources].

@ __General references__:
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-fs;
Stewart 07;
Lax BAMS(08);
Aguirre et al ed-16.

@ __I__: Boudot Rech(89);
Oldershaw AJP(88)dec;
Irvine ed-89;
Lines 94.

@ __Physicists' point of view__: Dirac IJTP(82);
Oliver 94 [II];
Jackiw PT(96)feb;
Faddeev mp/00;
Witten BAMS(03);
Rohrlich GRG(11) [and logic];
Tegmark SA(14)jan;
Sakellariadou a1407-in [early-universe physics and quantum gravity];
Coley PS(17)-a1710 [open questions];
Aharonov et al a1902
[physical understanding and mathematical formalism];
Hartle a1909 [how nature is conformable to herself].

@ __And foundations__: Benioff JMP(70),
JMP(71) [mathematical logic];
Emch 84;
Nambiar m.GM/02;
Rios a1502-FQXi [quantum gravity as unified theory of mathematics];
Walker a1505-FQXi
[mathematics and the comprehensibility of the universe];
Aganagic BAMS-a1508-conf [and string theory];
Hartle a1612 [comprehensibility of the universe];
Svetlichny a1810 [canonical mathematical structures];
Willig Lima et al a1811 [articulating reality in theoretical physics];
> s.a. Comprehensibility; time [physics
based on intuitionist mathematics].

@ __Physics and geometry / topology__: Witten pr(86);
Atiyah in(88);
Lantsman mp/01;
Benioff SPIE(13)-a1306
[effects of local availability of mathematics on geometry and physics];
> s.a. geometry; topology and physics.

@ __Effectiveness of mathematics__: Wigner CPAM(60),
also here;
in Matthews 71, p122;
in Lightman 86, p132;
Rouet IJGMP(08);
Batterman BJPS(10)#1 [new approach to its applicability];
Plotnitsky FP(11);
Omnès FP(11) [Wigner revisited];
Harvey GRG(11)-a1212;
Livio SA(11)aug;
Benioff SPIE(12)-a1205 [local availability of mathematics];
Bueno & French BJPS(12);
Visser a1212-FQXi [alternatives to the real number system];
Das Gupta rg(14)-a1508;
Walker a1505-FQXi [and the structure of physical reality];
Smolin a1506-FQXi
[the reasonable effectiveness of mathematics in physics];
Mujumdar & Singh a1506-FQXi [and cognitive science];
Woit a1506-FQXi;
Yanofsky a1506-FQXi [from regularities, symmetries];
Foschini a1507-FQXi [no mysterious link];
Leifer a1508-FQXi;
Stoica a1512-FQXi;
Wise a1512-FQXi;
Sarma a1607 [and Leibniz's "universal calculus"];
McDonnell 17;
Visser a1703-FQXi [prosaic connection].

@ __Effectiveness of physics__:
news sn(20)feb [entanglement and proofs];
> s.a. computability.

@ __Related topics__: Ernest BJPS(90) [meaning of math expressions];
Liston PhSc(93)mar [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)dec [mathematical rigor];
Prykarpatsky et al a0902 ["quantum mathematics"];
Bailey et al CM(10)-a1005 [experimental mathematics results];
Miller pt(19)oct [the origin of equations];
Baez a2101 [This Week's Finds in Mathematical Physics (1-50)].

@ __Mathematical and physical cultures__:
Jaffe & Quinn BAMS(93)
+ responses BAMS(94);
Sinai BAMS(06).

@ __Cum granu salis__? Bartocci & Wesley 90;
Perchik mp/03.

"The book of nature is written in the language of mathematics" – Galileo (1564 – 1642).

"The Great Architect of the Universe now begins to appear as a pure mathematician,"
J Jeans (1877 – 1946) expressing his surprise that quantum theory works.

"Physics is mathematics not because we know a lot about the external world, but because
we know too little" – B Russell (1872 – 1970) as quoted by Sabato.

"The most incomprehensible thing about the universe is that it is comprehensible"
– A Einstein (1879 – 1955).

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