existe uma fronteira entre a física e a matemática? · “matemática = linguagem da natureza.”...

Existe uma Fronteira entre aFísica e a Matemática?

Prof. Dr. Jayme Vaz Jr.

Departamento de Matemática AplicadaIMECC

Jayme Vaz Jr. (DMA-IMECC) Existe fronteira entre física e matemática? 11-04-2017 1 / 35

“Matemática = linguagem da natureza.”

Pythagoras, as everyone knows, said that “all things are numbers."This statement, interpreted in a modern way, is logical nonsense,but what he meant was not exactly nonsense. He discovered the

importance of numbers in music and the connection which heestablished between music and arithmetic survives in the

mathematical terms “harmonic mean"and “harmonic progression."He thought of numbers as shapes, as they appear on dice

or playing cards. We still speak of squares or cubes of numbers,which are terms that we owe to him.

Bertrand RussellA History of Western Philosophy


“Matemática = linguagem da natureza."

La filosofia è scritta in questo grandissimo libro, che continuamente cista aperto innanzi agli occhi (io dico l’Universo), ma non si può

intendere, se prima non il sapere a intender la lingua, e conoscer icaratteri ne quali è scritto. Egli è scritto in lingua matematica, e i

caratteri son triangoli, cerchi ed altre figure geometriche, senza i qualimezzi è impossibile interderne umanamente parola; senza questi è un

aggirarsi vanamente per un oscuro labirinto.

Galileu GalileiIl saggiatore


filosofiafisicomatemática


A Revolução Científica – Séculos XVI e XVII

Copérnico (1473-1543)Kepler (1571-1630)Galileu (1564-1642)Descartes (1596-1650)Fermat (1601-1665)Pascal (1623-1662)Newton (1642-1727)Leibniz (1646-1716)


Newton – Philosophiæ Naturalis Principia Mathematica (1687)

As três leis do movimento

Lei da atração universal

“Fluxões e fluentes"

“Derivadas e funções"

df (x)dx


Leibniz – Nova Methodus pro Maximis et Minimis (1684)

Teoria das mônodas

“Substâncias simples”

Sistema filosófico

Infinitesimais

dx


Revolução

Francesa

Início da

Revolução

Industrial

1700 1800

O ILUMINISMO

Newton

Leibniz

1643-1727

1646-1716

1667-1748 Johann Bernoulli

1717-1783 d’Alembert

Lagrange

1749-1827 Laplace

1752-1833 Legendre

1768-1830 Fourier

1777-1855 Gauss

1707-1783 Euler

1736-1813


Euler (1707-1783)

Matemático, físico,astrônomo, lógico eengenheiro (wikipedia)

Cálculo infinitesimal, cálculode variações, teoria degrafos, teoria dos números,topologia, etc.

Mecânica, mecânica dosfluidos, óptica, astronomia,teoria musical, etc.


1700 1800

1707 1720 1727 1738 1741 1766 1783

5 % 10 %

19 % 14 %

18 %

34 %

http://eulerarchive.maa.org


A Revolução Industrial – ∼ 1760 ∼ 1830

Máquinas a vapor = impulsona R.I.

James Watt (1736 - 1819)

1777: Watt “consertou” amáquina a vapor deNewcomen

Carnot (1796-1832):Réflexions sur la puissancedu feu et sur les machinespropres à développer cettepuissance (1824)


Fourier (1768-1830): Théorie analytique de la chaleur (1822)

Continuidade + lei de Fourier= Equação de difusão docalor

∂u∂t

= κ∂2u∂x2

Séries de FourierTransformadas de Fourier


filosofia–física–matemática


LegendreLaplaceLagrangeFourierCauchy

GaussRiemannGrassmannMöbiusWeierstrass

HamiltonCayleyBooleMaxwellStokes


Einstein independently discovered Riemann’s original program, to givea purely geometric explanation to the concept of “force". ... To Riemann,the bending and warping of space causes the appearance of a force.Thus forces do not really exist; what is actually happening is that spaceitself is being bent out of shape. The problem with Riemann’s appro-ach... was that he had no idea specifically how gravity or electricity andmagnetism caused the warping of space. ...Here Einstein succeededwhere Riemann failed.

Michio KakuHyperspace: A Scientific Odyssey

Through Parallel Universes,Time Warps, and the 10th Dimension


“O último da espécie": Poincaré (1854-1912)


Matemático, físico e filósofo da ciênciaEquações diferenciais, topologia, topologia algébrica, geometriaalgébrica, geometria hiperbólica, teoria das funções analíticascom várias variáveis complexas, etcMecânica celestre, mecânica dos fluidos, óptica, eletricidade,telegrafia, capilaridade, elasticidade, termodinâmica, mecânicaquântica, teoria da relatividade e cosmologia (wikipedia)“A Ciência e a Hipótese", “Ensaios Fundamentais", “O Valor daCiência"


filosofia física matemática


matemática puraX

matemática aplicada

física teóricaX

física aplicada


No discovery of mine has made, or is likely to make,directly or indirectly, for good or ill,

the least difference to the amenity of the world.

G. H. HardyA Mathematician’s Apology


New J. Phys. 17 (2015) 013036 doi:10.1088/1367-2630/17/1/013036

PAPER

Are physicists afraid of mathematics?

JonathanEKollmer1,2, Thorsten Pöschel1,2,3 and JasonACGallas1,2,3,4

1 Institute forMultiscale Simulations, Friedrich-Alexander-Universität, Erlangen, Germany2 Max-Planck Institute for the Physics of Complex Systems, Dresden, Germany3 Departamento de Física, Universidade Federal da Paraíba, João Pessoa, Brazil4 Instituto deAltos Estudos da Paraíba, Rua InfanteDomHenrique 100-1801, 58039-150 João Pessoa, Brazil

E-mail: [email protected]

Keywords: social physics, citation strategies, science communication

AbstractA recent study claimed that heavy use of equations impedes communication among biologists, asmeasured by the ability to attract citations frompeers. It was suggested that to increase the probabilityof being cited one should reduce the density of equations in papers, that equations should bemoved toappendices, and thatmath training among biologists should be improved.Here, we report a detailedstudy of the citation habits among physicists, a community that has traditionally strong training anddependence onmathematical formulations. Is it possible to correlate statistical citation patterns andfear ofmathematics in a community whosework strongly depends on equations? By performing asystematic analysis of the citation counts of papers published in one of the leading journals in physicscovering all its disciplines, wefind striking similarities with distribution of citations recorded in biolo-gical sciences. However, based on the standard deviations in citation data of both communities, biolo-gists and physicists, we argue that trends in statistical indicators are not reliable to unambiguouslyblamemathematics for the existence or lack of citations.We digress briefly about other statisticaltrends that apparently would also enhance citation success.

1. Introduction

In a recent work, Fawcett andHigginson reported an interesting statistical analysis of the citation habitsprevalent among the community of biologists working in the fields of ecology and evolution [1]. Their goal wastomeasure the communication efficiency among these researchers. In their report, communication efficiencywastaken as tantamount to accruing paper citations, so that their taskwas then reduced to quantifying the ability of apaper to attract a large number of citations, regardless of the possible content of the papers, the perceivedauthority of the authors, or any other relevant aspect of the paper. To quantify citations, the authors consideredtrends in statistical indicators of citation patterns as observed on a sample containing 649 papers published in1998 in the top three journals specialized in the aforementioned fields.

Communication efficiencywas assessed by studying citation habits of twonon-overlapping groups of papersreferred to as theoretical and non-theoretical papers. Fawcett andHigginson considered theoretical papers to bethose containing certain variations of thewordmodel in their title or abstract, while equationswere defined asmathematical expressions on lines set apart from the text, with two ormore such equations written on the sameline considered as separate.Mathematical papers are presumed to generate testable predictions andinterpretations of observationsmade by empirical studies, i.e. by non-theoretical studies. The rationale for thismutually exclusive divisionwould be that sincemost research in biology is empirical, despite theinterdependence between empirical andmathematical papers,many empirical studies build largely on otherempirical studies with little reference to relevant theory. Such dichotomic tendencywas taken as suggestive of afailure of communication that ultimately ended up hindering scientific progress.

Based on their statistical analysis, Fawcett andHigginson argued that heavy use ofmathematical equationsimpedes communication among biologists. To counter this undesired effect, the authors offered threerecommendations intended to restore effective communication (i.e. citations) among theoretical and non-

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New J. Phys. 18 (2016) 118003 doi:10.1088/1367-2630/18/11/118003

COMMENT

Comment on ‘Are physicists afraid of mathematics?’

AndrewDHigginson1 andTimWFawcettCentre for Research in Animal Behaviour, College of Life and Environmental Sciences, University of Exeter, Exeter EX4 4QG,UK1 Author towhomany correspondence should be addressed.

E-mail: [email protected] [email protected]

AbstractIn 2012, we showed that the citation count for articles in ecology and evolutionary biology declineswith increasing density of equations. Kollmer et al (2015New J. Phys. 17 013036) claim this effect is anartefact of themanner inwhichwe plotted the data. They also present citation data from PhysicalReview Letters and argue, based on graphs, that citation counts are unrelated to equation density. Herewe show that both claims aremisguided.We identified the effects in biology not by visualmeans, butusing themost appropriate statistical analysis. Since Kollmer et al did not carry out any statisticalanalysis, they cannot draw reliable inferences about the citation patterns in physics.We show thatwhen statistically analysed their data actually do provide evidence that in physics, as in biology, citationcounts are lower for articles with a high density of equations. This indicates that a negative relationshipbetween equation density and citationsmay extend across the breadth of the sciences, even those inwhich researchers arewell accustomed tomathematical descriptions of natural phenomena.Werestate our assessment that this is a genuine problem and discuss what we think should be doneabout it.

1.Mathematics plays a vital role in the sciences

Mathematical theory is an indispensable part of scientific research, capturing the essence of fundamentalphysical, chemical and biological processes with greater clarity, precision, rigor, and brevity than verbalarguments can achieve. In a range of disciplines, efficient dialogue between theoretical developments andempirical testing is critical to driving science forwards [1–8]. Reports of a barrier to communication betweentheoretical and empirical research [9–11] should therefore arouse concern.

In a recent study [12], we showed that the citation counts of articles in ecology and evolutionary biology arenegatively associatedwith the density ofmathematical equations presented in themain text. In a paperpublished in theNew Journal of Physics, Kollmer et al [13]—hereafter KPG—attacked our interpretation of thecitation patterns in ecology and evolutionary biology, criticized ourmethodology and argued that there is noevidence for a negative impact of high equation density on citation counts. Responding to our call for similaranalyses in otherfields [14], KPG also investigated the relationship between equation density and citation countin a leadingmultidisciplinary physics journal,Physical Review Letters. As they did for the biology data, theyargued that there was no evidence for a relationship. Herewe show that the conclusions drawn byKPG areincorrect.

2.Our original analysis is objective and valid

KPGpresent our data in alternative graphical formats (theirfigure 1) and suggest that the effect we found issimply an artefact of howwe binned the data. This is incorrect: our conclusions were based on a formal statisticalanalysis that did not involve any binning of data, but instead treated equation density as a continuous covariate.This statistical analysis—a generalized linearmodel (GLM)with a negative binomial error function—isappropriate for count data that are extremely over-dispersed (in this case, themost cited papers aremuchmore

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J. DIFFERENTIAL GEOMETRY17 (1982) 661-692

SUPERSYMMETRY AND MORSE THEORY

EDWARD WITTEN

Abstract

It is shown that the Morse inequalities can be obtained by consideration of a certain supersymmet-ric quantum mechanics Hamiltonian. Some of the implications of modern ideas in mathematics forsupersymmetric theories are discussed.

1. Introduction

Supersymmetry is a relatively recent development in theoretical physicswhich has attracted considerable interest and has been actively developed inseveral different directions [17], [18].

A number of concepts in modern mathematics have significant applicationsto supersymmetric quantum field theory [22]. Conversely, as we will see in thispaper, supersymmetry has some interesting applications in mathematics. Thepurpose of this paper is to describe some of those applications and to make thenotions of "supersymmetric quantum mechanics" and "supersymmetric quan-tum field theory" accessible to a mathematical audience.

The mathematical applications in §§2 and 3 will be self-contained. However,it may be useful to first make a few remarks about some of the relevant aspectsof supersymmetry.

In any quantum field theory, the Hubert space % = %+ Θ3C~, where %+

and %~ are the spaces of "bosonic" and "fermionic" states respectively. Asupersymmetry theory is by definition a theory in which there are (Hermitian)symmetry operators Qi9 i = 1, ,N, which map %+ into %~ and vice-versa.

Let us define the operator (-1)F which distinguishes %+ from %~ (andcounts the number of fermions modulo two). Thus we define (-l)Fψ = ψ forψ E %+ , and (-l)Fχ = -χ for χ G %~. The first basic condition which mustbe satisfied by the supersymmetry operators Qt is that they each anticommutewith(-l)F:

(i) (- i) F β, + β,(- i) F = o.

Received March 26, 1982. Supported in part by the National Science Foundation under Grant

No. PHY80-19754. The author would like to thank R. Bott, W. Browder, S. Graffi, J. Milnor, and

D. Sullivan for discussions.


Edward Witten (1951)

Teoria MTeoria topológica quântica decamposGeometria não comutativaTeoria de gaugesupersimétricaGravitação quânticaPrimeiro e único físico aganhar Medalha Fields


Física = inspiração da matemática.

Although he is definitely a physicist (as his list of publications clearlyshows) his command of mathematics is rivaled by few mathematicians,and his ability to interpret physical ideas in mathematical form is quiteunique. Time and again he has surprised the mathematical communityby a brilliant application of physical insight leading to new and deepmathematical theorems.

Michael AtiyahOn the Work of Edward Witten

Proceedings of the ICM, Kyoto, 1990


Física = inspiração da matemática.

From this very brief summary of Witten’s achievements it should beclear that he has made a profound impact on contemporary mathema-tics. In his hands physics is once again providing a rich source of inspi-ration and insight in mathematics. Of course physical insight does notalways lead to immediately rigorous mathematical proofs but it frequen-tly leads one in the right direction, and technically correct proofs canthen hopefully be found. This is the case with Witten’s work. So far hisinsight has never let him down and rigorous proofs, of the standard wemathematicians rightly expect, have always been forthcoming. There istherefore no doubt that contributions to mathematics of this order arefully worthy of a Fields Medal.

Michael AtiyahOn the Work of Edward Witten

Proceedings of the ICM, Kyoto, 1990


JOURNAL OF MATHEMATICAL PHYSICS 56, 112101 (2015)

Quantum mechanical derivation of the Wallis formula for πTamar Friedmann1,a) and C. R. Hagen2,b)1Department of Mathematics and Department of Physics and Astronomy,University of Rochester, Rochester, New York 14627, USA2Department of Physics and Astronomy, University of Rochester,Rochester, New York 14627, USA

(Received 13 August 2015; accepted 14 August 2015; published online 10 November 2015)

A famous pre-Newtonian formula for π is obtained directly from the variationalapproach to the spectrum of the hydrogen atom in spaces of arbitrarydimensions greater than one, including the physical three dimensions.

C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4930800]

The formula for π as the infinite product

π

2=

2 · 21 · 3

4 · 43 · 5

6 · 65 · 7 · · · (1)

was derived by Wallis in 16551 (see also Ref. 2) by a method of successive interpolations. Whileseveral mathematical proofs of this formula have been put forth in the past (many just in the lastdecade) using probability,3 combinatorics and probability,4 geometric means,5 trigonometry,6,7 andtrigonometric integrals,8 there has not been in the literature a derivation of Eq. (1) that originates inphysics, specifically in quantum mechanics.

It is the purpose of this paper to show that this formula can in fact be derived from a variationalcomputation of the spectrum of the hydrogen atom. The existence of such a derivation indicatesthat there are striking connections between well-established physics and pure mathematics9 that areremarkably beautiful yet still to be discovered.

The Schrödinger equation for the hydrogen atom is given by

Hψ =(− ~

2

2m∇2 − e2

r

)ψ = Eψ,

with the corresponding radial equation obtained by separation of variables being

H(r)R(r) =− ~

2

2m

(d2

dr2 +2r

ddr− ℓ(ℓ + 1)

r2

)− e2

r

R(r) = ER(r).

Using the trial wave function

ψαℓm = rℓe−αr2Ymℓ (θ,φ), (2)

where α > 0 is a real parameter and the Ymℓ (θ,φ) are the usual spherical harmonics, the expectation

value of the Hamiltonian is found to be given by

⟨H⟩αℓ ≡ ⟨ψαℓm|H(r)|ψαℓm⟩⟨ψαℓm |ψαℓm⟩

=~2

2m

(ℓ +

32

)2α − e2 Γ(ℓ + 1)

Γ(ℓ + 32 )√

2α.

As a consequence of the Ymℓ (θ,φ) in Eq. (2), the function ψαℓm is orthogonal to any energy and

angular momentum eigenstate that has a value for angular momentum different from ℓ. Therefore

a)Email address: [email protected])Email address: [email protected]

0022-2488/2015/56(11)/112101/3/$30.00 56, 112101-1 © 2015 AIP Publishing LLC

π

2=

2 · 21 · 3

4 · 43 · 5

6 · 65 · 7

. . .


Hamiltonian for the Zeros of the Riemann Zeta Function

Carl M. Bender,1 Dorje C. Brody,2,3 and Markus P. Müller4,51Department of Physics, Washington University, St. Louis, Missouri 63130, USA

2Department of Mathematics, Brunel University London, Uxbridge UB8 3PH, United Kingdom3Department of Optical Physics and Modern Natural Science, St. Petersburg National Research University

of Information Technologies, Mechanics and Optics, St. Petersburg 197101, Russia4Departments of Applied Mathematics and Philosophy, University of Western Ontario, Middlesex College,

London, Ontario N6A 5B7, Canada5The Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

(Received 23 September 2016; revised manuscript received 17 February 2017; published 30 March 2017)

A Hamiltonian operator H is constructed with the property that if the eigenfunctions obey a suitableboundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zetafunction. The classical limit of H is 2xp, which is consistent with the Berry-Keating conjecture. While His not Hermitian in the conventional sense, iH isPT symmetric with a broken PT symmetry, thus allowingfor the possibility that all eigenvalues of H are real. A heuristic analysis is presented for the construction ofthe metric operator to define an inner-product space, on which the Hamiltonian is Hermitian. If the analysispresented here can be made rigorous to show that H is manifestly self-adjoint, then this implies that theRiemann hypothesis holds true.

DOI: 10.1103/PhysRevLett.118.130201

The Riemann zeta function ζðzÞ is conventionallyrepresented as the sum or the integral

ζðzÞ ¼X∞

k¼1

1

kz¼ 1

ΓðzÞZ

∞

0

dttz−1

et − 1:

(The integral reduces to the sum if the denominator of theintegrand is expanded in a geometric series.) Both repre-sentations converge and define ζðzÞ as an analytic functionwhen ReðzÞ > 1. These representations diverge whenz ¼ 1 because the zeta function has a simple pole atz ¼ 1. Substituting z ¼ −2n (n ¼ 1; 2; 3;…) in the reflec-tion formula

ζðzÞ ¼ 2zπz−1 sinðπz=2ÞΓð1 − zÞζð1 − zÞ

shows that the zeta function vanishes when z is a negative-even integer. These zeros of ζðzÞ are called the trivial zeros.The Riemann hypothesis [1] states that the nontrivial

zeros of ζðzÞ lie on the line ReðzÞ ¼ 12. This hypothesis has

attracted much attention for over a century because there isa deep connection with number theory and other branchesof mathematics. However, the hypothesis has not beenproved or disproved. Any advance in understanding the

zeta function would be of great interest in mathematicalscience, whether or not one succeeds in finally proving orfalsifying the hypothesis.In this Letter, we examine the Riemann hypothesis by

constructing and studying an operator H that plays the roleof a Hamiltonian. The conjectured property of H is that itseigenvalues are exactly the imaginary parts of the nontrivialzeros of the zeta function. The idea that the imaginary partsof the zeros of ζðzÞ might correspond to the eigenvalues ofa Hermitian, self-adjoint operator (assuming the validity ofthe Riemann hypothesis) is known as the Hilbert-Pólyaconjecture. Research into this connection has intensifiedfollowing the observation that the spacings of the zerosof the zeta function on the line ReðzÞ ¼ 1

2and the spacings

of the eigenvalues of a Gaussian unitary ensemble ofHermitian random matrices have the same distribution[2–4]. Berry and Keating conjectured that the classicalcounterpart of such a Hamiltonian would have the formH ¼ xp [5,6]. However, a Hamiltonian possessing thisproperty has hitherto not been found (see [7] for a detailedaccount of the Berry-Keating program and its extensions).We propose and consider the Hamiltonian

H ¼ 11 − e−ip

ðx pþp xÞð1 − e−ipÞ: ð1Þ

Our main findings are as follows. (i) The non-HermitianHamiltonian H in (1) formally satisfies the conditions ofthe Hilbert-Pólya conjecture. That is, if the eigenfunctionsof H are required to satisfy the boundary conditionψnð0Þ ¼ 0 for all n, then the eigenvalues fEng have the

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.

PRL 118, 130201 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

31 MARCH 2017

0031-9007=17=118(13)=130201(5) 130201-1 Published by the American Physical Society


Physics Reports 315 (1999) 95}105

Is theoretical physics the same thing as mathematics?q

George Chapline

Lawrence Livermore National Laboratory, Livermore, CA 94551, USA

Abstract

The growing realization that the fundamental mathematical structure underlying superstring models isclosely related to Langlands' program for the uni"cation of mathematics suggests that the relationshipbetween theoretical physics and mathematics is more intimate than previously thought. We show thatquantum mechanics can be interpreted as a canonical method for solving pattern recognition problems,which suggests that mathematics is really just a re#ection of the fundamental laws of physics. ( 1999Elsevier Science B.V. All rights reserved.

PACS: 03.65.!w

1. Introduction

One of the perennial mysteries of theoretical physics is why the laws of physics should so oftenhave an elegant mathematical formulation } a circumstance often referred to as the `unreasonablee!ectiveness of mathematicsa [5]. In fact, natural phenomena very often exhibit regularities thatfrom the mathematical point of view seem to involve especially unique and beautiful mathematicalstructures. A good example of this is provided by the apparently strong likelihood [1] that theparity violation observed in the weak interactions has its origins in the naturally chiral natureof a supersymmetric E

8gauge theory in 10 dimensions. Indeed the implied involvement in

elementary particle physics of the exceptional Lie group E8

provides a connection betweenfundamental physics and various remarkable mathematical structures including octonians, self-dual lattices, perfect error correcting codes, Kummer surfaces, and non-standard Euclidean

qBased on a talk given at the Dick Slansky Memorial Symposium, May 21, 1998.E-mail address: [email protected] (G. Chapline)

0370-1573/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved.PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 5 - 0


Mathematics is a part of physics.Physics is an experimental science,

a part of natural science.Mathematics is the part of physics

where experiments are cheap.

V. I. ArnoldParis, 7-3-1997


existe uma fronteira entre a física e a matemática? · “matemática = linguagem da natureza.”...

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