Table Of ContentQUADERNDOE IG RUPP0I1 R ICERCA MATEMAPrICA
DEL CONSIGLNIAOZ IONADLEEL LREI CERCf-IE
E.T ONTI
ON �rHEF ORMAL STRUCTURE
OF PHYSICATLH EORIES
QUADERNO DEIG RUPPI0 1 RICERCAM ATEMA·rICA
DEL CONSIGLION AZIONALED ELLER ICERCI-IE
E.T ONTI
ON �rHEF ORMALS TRUC�rURE
OF PHYSICATLH EORIES
dijfusione
gratuita
Llvoreos egunietlol 'amdbeilGt rou ppNoa zionapleerl aF isiMcaat ematica
ISTITUTOD I MA TEMATICAD EL POLITECNICDOI MILANO
PIAZZAL EONARDO DA VINCI3 2
2013M3I LANO Oulia}
Tel2. 36166 3
PREFACE
"Thea im olM athematicaPhly .lc.
h noto nlyt ol acilitlaotret hep hyslcht
then umerical calculaotli coenr taicno n.t&nt.
or theI nte,ratloorn c ertain dUrereenqUuaalt i•o.n
It1 1b etide.i,t h abovea U, tor evealt oh im
theh iddenh aI' mo ny or thin,.
in makinh,i m .ee theImn a new way."
HenriP oincarf
The valueo fS cience.D,o ver.1 958
This report collectst he firstr esultso f a trial to
m�k� a syn thesis in physics from the point of view of formal
proprrteis . Undert he name "formal structure"w e lIean al«ebraic,
,
topolor,icnl,r. eometr1cnl, analytical and othersm athemAtical
str uctur�s.
Perhapst he most concrete resulti s the constructiono f
r! clr!r:!==ific::\tisocnh eme for the physical quantitiaensd the
f''lu·,lto n� of ('vpry physicnlt heory. A number of suchs chemes
nre quotp1 1n :8 . These schemesa re all written using a singl�
systf>1'I of phy:�,l ]� qunntlties. the ra tionalized M.K.Ss.yAst.e m.
Symb�ls nd� n�mps are those rec�mmended by the International
Union of Pure and Applied PhYSics( S.U.CoNmm.is sion).
Tr.isc ollectiono f schemes,denlingw ithm any physical
theol'iesm,� y fonn the core of an "Atlnso f the physicalt heo
rles" th�t under the nnme of rn.-'lny specialists of the various
field of phYSiCS,m ay be printed in Succp.ssivee ditions of in-
cre�Finrc ontent.
Detailedc ompariRons betweend ifferentp hYSical theor;��
have progressiveldyi splayed the existenceo f a common mathem�
ticnl stnl�ture,t hat in turn, come up from a common topolocicnl
I
"back-clotho"f ph]sicatlh eories.
Thiuc omparisowna s madea t firstu si�gc oncetps and
toolao f function spac.s,l ikeo peratorsan d bilinearf orms
(seer eCerence( 130S]uc)hp .o werfuld evices permitto go
overt he distinctionbse tween one or more Cunctionso f one or
morei ndependenvta ripbels. The usualan alogiesb etweend iCfe
r�ntt heoriesa re revealedb y the factt hatc orrespondinog
peratorsh avet he sames tructure.A typicala djointnesrse la
tionb etweenp airso f operatorasr isingi n every theory comes
intoa ppearencaes the main mathematicaplro �erty sharedb y
physicalt heories.In facti t is exhibitebdy linearan d n�n
lineart heoriesd,e alingw ithd iscreteo r continuum systems.
of classic,r elativistiacn d �uantum nature.
The existencoef such unityo f mathematicals tructure
(1291).
led us to constructa mathematicamlo del( seer eference
It is dictatedt rom the principloef economizingp ruofsi n the
singlep hysicalt heori�s,s huwingt he linksb etweenv arious
propertiecso mmonly usedi n physicalt heories.
As the comparison'wp8rso gressiveleyx tendedt o morea nd
moret heoriesw e hAd the pleasantf eelingt hats uchm athemaitcal
structureh as mored eepr oots.
The reasono r suchm :,th�maticsatlru cturec ome illtO8 .p
pearencew heni t was realizedt hati n every physical th�or.1
therea re physicalv ariabletsh ata re naturallays sociatetdo
simpleg eomerticalan d chronometricealle mentsl ikep oints,1 i-
I
nes, volumes, surfacest,i mei nst:mt and time intervals and
combniationo f them.
At this point a papero t P.J.B r�ninJ r. ofC eredu s the
key to translatet hisa ssociatiobne tweenp hYSicsa nd geometry
II
in mathematicallan guage: thisk ay is the a18��raict opolo�,
in partic ulart �ietao molog[ theory of cell-comlpexes.I ti s
shownt hat wit tlt he langut' .g.e of "chanis" and thato f "coboun
dary"o ne can describet he typicalo perationosf forming the
variousp hysicalv ariables.
The adjointnesbse tween opel' tthoartsw as previousyl
realizeda ppeareda s consequencoef thaed jointnesosf the
boundary and coboundar,y operatorsa, well known property of
homology of cell-compelxes.
The pointo f view undert akeni n this papeirs thato f
an experimentali�shti cht riest o infegre neral conclusinos
fl�m a body of experimentafla cts:t hese" facts"a re heret he
existingp hysicalt heori�sW.e have nott ried to invent "new
theories".O n U.ec ontraryw e h�ve ta.ken an inductivep oint
of view.
A word ab��t the styleo f the papre. It has heenw ritten
in such way to be understand�blbey the largesta udience:b y
experimp.ntaanld t heoretical physicists,by chemists, engin
eers,a nd also, by studentso f the last yearso f thesef aculties.
In order to makep ossiblet hisw e havee xplainefdr omt he start
the main algehraic and topologicalc onceptsg ivingan elementary
andc ompact presentatioo�f them and stressinRh ow theya rise
quiten aturallyi n th� descriptiono f physicals cienecs. These
mathematicatlo olsa re presentedf rom a physicispto into f view:
this impliest hats ometimet heym ay be lackineo f mathemntcial
rigour.
In dealingw ith this researcht he uathor was in an isolh
ted position:o ur time is eh?ra�teri7.ehdy over-specialitziaon
III
in all fieldo f sciencesT.h e authora sk for contactsc, ollabo
rationan d constructivec riticism.
The contento f thisr eport will be pubbilni sshoemde r e-
view.
Authora dressi s: Tonti E.,
Istitutod i Matematidceal Politecnico
PiazzaL eonardod a Vinci,3 2
201)) MILANO (Italy)
(phone2 ).66.16)
IV
CONTENTS
1. D'l'RODUCTIQI
2• .I. PRELIKllUllT CWSI1IC.lTIOB or PKrSIC.lLY lIIULE3
2.1 Conticuratioa &D4 .oure e �ri&b1e. )
2.2In d.pendentYar Iab1e. 5
2.)C oatI,uratloa-kind Y&riab1e. 6
2.4So urOe-kiD4 �iab1• • 6
2.,Pb .Do.. no1osloal .,uation. 1
2.6 Otherp h7aloa1 �1&b1e. 1
2.7 Oth.re quationa 8
).UA EBlU.IC TOPOLOCt'. CJ:U.DS .um COBOUNDART
).1 Intro4uotiOD 11
).: Ce11-oo.p1.x 12
).) �1 ce11-oo.ple% 14
).4 luaeratloont thec ell. 15
).5 Orientationo f theo ell. 18
).6C b.ain. 19
).7 Inoidenoe.. trioe. 2)
).8 Coboundar7 of a chain 26
).9 Poin0ar41 .... )0
.) .10P artioular chAln. )0
).11B o.oloao� chain. 31
).12C hain .paoe. 32
).1)C eneralizSetdo ke.T heore. 15
4.C lUIBS .um COBOUNDARIES IN PHYSICALTHEOR IES
4.1 lxa. p1es� f ohaine 19
4.2 lxa.p1e8 oto oboundarie. 41
4.) Balancee quationa 41
4.4 Cone.rv.atiloanvw 44
4.' Iquilibriua equatloc. «
4.6 Circuitall a". 45
4.1 Irrotatlonalit1 condition8 46
4.8 Th. gradient. 46
4.9 The coapatibiliootn1d itions 47
4.10Th e seneral. 01ution 41
4.11Ca uce in�iance 48
5. .I. CLASSlrIC.l.TISOIC HEME. DISCi£-rCASEE
5.1 Electrostatic. 49
5.2 Particl�em1 c.
53
5.) Classificat.icohne me
57
.,
6.
KtJLTIVECTCotl1 LCUWS
6.1 Introduction 61
6.2 Bl"otor 63
6.) Trlveotor 65
6.4 'J.'he exterioprr oduct 66
6.5 Soa1arp r04uct ot two p-Teotor. 67
6.6 Iuer product. 68
6.7 'J.'he • upp1e .etanr .r .ult l.. ..t oo r 70
6.8 Coaponento.t a .ultlTecto1r0 � 72
6.9 KultlTeotor1D. t10lt41e. ea.loaa1e paoe. 76
6.10B .a._1 ,..Teotor. 76
6.11Ix terlorpro4 uot 77
6.12S . ott '110. u1t lTe C tor. 78
6.13S l.p1aD.d co.poundb l"otor. 79
6.14Ba .e ,.."otor.
80
6.15Coa pooent. ot a .ultlTector 80
6.16'J.'h e t�.oriaaa1t ur. ot
the0 0.,.. 8at.o t a .ultlY8otor 81
6.17C oapoaenotlt thee xterioprr 04uct 82
6.18l>\a 1 .paoe 84
6.91 lNal 'ba.l. 85
6.20S calarpr oduot 86
6.21� er product 88
6.22S uppel.eLt&r.r. u1tl.. ..c tor. 91
6.23 cr. ..... ale-bra 93
24S calaru d inDer producte 95
&.
7.EX TEIlIOK PORMS .um EXTERIOR DIFElI'lEIl TI.U,s 1W PKTSIC!THiLOR liS
7.1 !b. rl.eat the. xteriotro r.. 97
7.2 Daa1 exteriotro rae 102
7.3 Couedoa. 104
.
7.4 Tbe rl• • ott hee %terlodri tterentla1 108
7.5 Bael0aD d .t.aad&rd c01UD1U 116
n
8• .1 Cu.sSlnC.l'tIOIf SCHEME, COIlTDWOUS SYSTEMS JJlD FIEI.JS 125
8.1 Bow to read the .ch.... 126
Partiol41Daea1 0. 129,r igid bo41 .otio1n) 5;a taticeo t
.tri� 1)6,� an41ac ot �ea .. 1)7,� �r&tiofto.t atringB,
ro4e 1)8; aNp fil.. 141,0 1.a.iNgr1a ntationa1t ie1d
142,OOD tiDUua atatio.1 4),p erfecftl uid IIOtion1 44.m e
ohanioa ot polar oODtinua1 45,l inear aoouatioa1,4 6.
flui41Daa1 o. 147, thermoetatic.1 .8,t hermal oonduction
lSI,i rr e.. raiblt.h .ra41Daoa1c. 152,. 1.ctroetat1i5c4.;
..... to.tatio1. 56,. 1.etr"'sn.ticf i.1d1 59.e 1.ctric
coDductlo1D6 0.' el.grapher'e.q uation1 61,n .twortkh eo
r7 162,ce o• •t rloalo ptic1.6 ).El .ia-Cordon .quatlon1 66;
Sckr8dJaser field1 70,P aull eq�tlon1 71.Pr oca tl.1d
172• • ca1ar&A d p• •u4 0.c.1a..r. OD ii.1d1.7 ).Di rac equa
tiOD1 74.t heq uantizatloprono ••• 115;r .1atiTi.tigrca&
ntatlou1 theor177 6,la t.ractlo1n7.9 ;. ca1ar'ft v. eq
uatl0D1 86,. ...t h. ...t loal.od.l 188. the cUlonical .qua
tion.1 90,t hepo t.ntial1.9 6,'ftZ' i.tioaa1 .tat• •ea t., 197.
aath. .. t1oal prop.rt•l• 202,e 1.�or.t..cdh• •••2 06.
9. CLIFf\lJlD J..LCBllRl .uD SPUOllS
9.1 Fro. Gra ••man n to Clifforadl gebra p. 215
9.2 !be ••t rio tanaor 216
9.) !tie Cllttorpdr oduot 22)
9.4 So• • proputi.. 221
9.5'C 11ttordn umb.r. ot p' 22&
9.5 Clifforadua ber. ot p2 226
9.1 CllttordD umber.o t EJ 228
9.8 B.o\a tioa.1m 1:.' 229
9.9 Clltford numbereo t epace-tl• • M4 2)1
9.10Id eal. 2)l
9.11ld ••po t.nte 2)5
9.12Xa trl% rapreaea.taidon of the Clifforadl gebra
2)1
9.11S plaor.
2.(2
9.14!b . 1JImag:lDalT un1 t 241
IEFERDCI:) 245
