Spectral Theory and Quantum Mechanics: Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation

962 Pages·2018·9.49 MB·English

by  Valter Moretti

UNITEXT 110
Valter Moretti
Spectral Theory
and Quantum
Mechanics
Mathematical Foundations
of Quantum Theories,
Symmetries and Introduction
to the Algebraic Formulation
Second Edition
UNITEXT - La Matematica per il 3+2
Volume 110
Editor-in-chief
A. Quarteroni
Series editors
L. Ambrosio
P. Biscari
C. Ciliberto
C. De Lellis
M. Ledoux
V. Panaretos
W.J. Runggaldier
More information about this series at http://www.springer.com/series/5418
Valter Moretti
Spectral Theory
and Quantum Mechanics
Mathematical Foundations of Quantum
Theories, Symmetries and Introduction
to the Algebraic Formulation
Second Edition
123
ValterMoretti
Department ofMathematics
University of Trento
Povo, Trento
Italy
Translated by: Simon G. Chiossi, Departamento de Matemática Aplicada (GMA-IME),
Universidade FederalFluminense
ISSN 2038-5714 ISSN 2532-3318 (electronic)
UNITEXT- La Matematica peril3+2
ISSN 2038-5722 ISSN 2038-5757 (electronic)
ISBN978-3-319-70705-1 ISBN978-3-319-70706-8 (eBook)
https://doi.org/10.1007/978-3-319-70706-8
LibraryofCongressControlNumber:2017958726
TranslatedandextendedversionoftheoriginalItalianedition:V.Moretti,TeoriaSpettraleeMeccanica
Quantistica,©Springer-VerlagItalia2010
1stedition:©Springer-VerlagItalia2013
2ndedition:©SpringerInternationalPublishingAG2017
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To Bianca
Preface to the Second Edition
In this second English edition (third, if one includes the first Italian one), a large
number of typos and errors of various kinds have been amended.
I have added more than 100 pages of fresh material, both mathematical and
physical,inparticularregardingthenotionofsuperselectionrules—addressedfrom
severaldifferentangles—themachineryofvonNeumannalgebrasandtheabstract
algebraic formulation. I have considerably expanded the lattice approach to
QuantumMechanicsinChap.7,whichnowcontainsprecisestatementsleadingup
to Solèr’s theorem on the characterization of quantum lattices, as well as gener-
alised versions of Gleason’s theorem. As a matter offact, Chap. 7 and the related
Chap.11havebeencompletelyreorganised.Ihaveincorporatedavarietyofresults
on the theory of von Neumann algebras and a broader discussion on the mathe-
matical formulation of superselection rules, also in relation to the von Neumann
algebraofobservables.Thecorrespondingpreparatorymaterialhasbeenfittedinto
Chap.3.Chapter 12hasbeendevelopedfurther, inorder toincludetechnical facts
concerning groups of quantum symmetries and their strongly continuous unitary
representations.IhaveexaminedindetailtherelationshipbetweenNelsondomains
and Gårding domains. Each chapter has been enriched by many new exercises,
remarks, examples and references. I would like once again to thank my colleague
Simon Chiossi for revising and improving my writing.
For having pointed out typos and other errors and for useful discussions, I am
grateful to Gabriele Anzellotti, Alejandro Ascárate, Nicolò Cangiotti, Simon G.
Chiossi, Claudio Dappiaggi, Nicolò Drago, Alan Garbarz, Riccardo Ghiloni, Igor
Khavkine, Bruno Hideki F. Kimura, Sonia Mazzucchi, Simone Murro, Giuseppe
Nardelli, Marco Oppio, Alessandro Perotti and Nicola Pinamonti.
Povo, Trento, Italy Valter Moretti
September 2017
vii
Preface to the First Edition
Imusthavebeen8or9whenmyfather,amanoflettersbutwell-readineverydiscipline
andwithacuriousmind,toldmethisstory:“AgreatscientistnamedAlbertEinstein
discoveredthatanyobjectwithamasscan'ttravelfasterthanthespeedoflight”.Tomy
bewildermentIreplied,boldly:“Thiscan'tbetrue,ifIrunalmostatthatspeedandthen
acceleratealittle,surelyIwillrunfasterthanlight,right?”Myfatherwasadamant:“No,
it'simpossibletodowhatyousay,it'saknownphysicsfact”.AfterawhileIadded:“That
bloke,Einstein,must'vecheckedthisthingmanytimes…howdoyousay,hedidmany
experiments?”TheanswerIgotwasutterlyunexpected:“NotevenoneIbelieve.Heused
maths!”
Whatdidnumbersandgeometricalfigureshavetodowiththeexistenceofanupperlimitto
speed?Howcouldonestandbysuchanapparentlynonsensicalstatementastheexistence
of amaximum speed,althoughcertainly true (Itrustedmy father),just based onmaths?
Howcouldmathematicshavesuchbigacontrolontherealworld?And Physics?Whaton
earthwasit,andwhatdidithavetodowithmaths?Thiswasoneofthemostbeguilingand
irresistiblethingsIhadeverheardtillthatmoment…Ihadtofindoutmoreaboutit.
ThisisanextendedandenhancedversionofanexistingtextbookwritteninItalian
(and published by Springer-Verlag). That edition and this one are based on a
common part that originated, in preliminary form, when I was a Physics under-
graduate at the University of Genova. The third-year compulsory lecture course
called Theoretical Physics was the second exam that had us pupils seriously
climbingthewalls(thefirstbeingthefamousPhysicsII,coveringthermodynamics
and classical electrodynamics).
QuantumMechanics,taughtinInstitutions,elicitedanovelandinvolvedwayof
thinking,atruechallengeforcravingstudents:formonthswehesitantlyfalteredon
a hazy and uncertain terrain, not understanding what was really key among the
notions we were trying—struggling, I should say—to learn, together with a com-
pletelynewformalism:linearoperatorsonHilbertspaces.Atthattime,actually,we
did not realise we were using this mathematical theory, and for many mates of
mine, the matter would have been, rightly perhaps, completely futile; Dirac's bra
vectorswerewhattheywere,andthat’sit!Theywerecertainlynotelementsinthe
topological dual of the Hilbert space. The notions of Hilbert space and dual
topologicalspacehadnorightofabodeinthemathematicaltoolboxofthemajority
ix
x PrefacetotheFirstEdition
of my fellows, even if they would soon come back in through the back door, with
the course Mathematical Methods of Physics taught by Prof. G. Cassinelli.
Mathematics, and the mathematical formalisation of physics, had always been my
flagshiptoovercomethedifficultiesthatstudyingphysicspresentedmewith,tothe
point that eventually (after a Ph.D. in Theoretical Physics) I officially became a
mathematician. Armed with a maths’ background—learnt in an extracurricular
courseofstudythatIcultivatedovertheyears,inparalleltoacademicphysics—and
eagertobroadenmyknowledge,ItriedtoformaliseeverynotionImetinthatnew
andrivetinglecturecourse.Atthesametime,Iwascarryingalongasimilarproject
forthemathematicalformalisationofGeneralRelativity,unawarethattheworkput
into Quantum Mechanics would have been incommensurably bigger.
Theformulationofthespectraltheoremasitisdiscussed inx8,9isthesameI
learnt when taking the Theoretical Physics exam, which for this reason was a
dialogueofthedeaf.LatermyinterestturnedtoQuantumFieldTheory,asubjectI
still work on today, though in the slightly more general framework of QFT in
curved spacetime. Notwithstanding, my fascination with the elementary formula-
tionofQuantumMechanicsneverfadedovertheyears,andtimeandagainchunks
were added to the opus I begun writing as a student.
Teaching this material to master’s and doctoral students in mathematics and
physics, thereby inflicting on them the result of my efforts to simplify the matter,
hasprovedtobecrucialforimprovingthetext.ItforcedmetotypesetinLaTeXthe
pile of loose notes and correct several sections, incorporating many people’s
remarks.
Concerning this, I would like to thank my colleagues, the friends from the
newsgroupsit.scienza.fisica,it.scienza.matematicaandfree.it.scienza.fisica,andthe
many students—some of which are now fellows of mine—who contributed to
improve the preparatory material of the treatise, whether directly or not, in the
courseoftime:S.Albeverio,G.Anzellotti,P.Armani,G.Bramanti,S.Bonaccorsi,
A. Cassa, B. Cocciaro, G. Collini, M. Dalla Brida, S. Doplicher, L. Di Persio,
E. Fabri, C. Fontanari, A. Franceschetti, R. Ghiloni, A. Giacomini, V. Marini,
S. Mazzucchi, E. Pagani, E. Pelizzari, G. Tessaro, M. Toller, L. Tubaro,
D. Pastorello, A. Pugliese, F. Serra Cassano, G. Ziglio and S. Zerbini. I am
indebted, for various reasons also unrelated to the book, to my late colleague
Alberto Tognoli. My greatest appreciation goes to R. Aramini, D. Cadamuro and
C. Dappiaggi, who read various versions of the manuscript and pointed out a
number of mistakes.
I am grateful to my friends and collaborators R. Brunetti, C. Dappiaggi and N.
Pinamontiforlastingtechnicaldiscussions,forsuggestionsonmanytopicscovered
in the book and for pointing out primary references.
At last, I would like to thank E. Gregorio for the invaluable and on-the-spot
technical help with the LaTeX package.
In the transition from the original Italian to the expanded English version, a
massive number of (uncountably many!) typos and errors of various kinds have
been corrected. I owe to E. Annigoni, M. Caffini, G. Collini, R. Ghiloni,
A.Iacopetti,M.OppioandD.Pastorellointhisrespect.Freshmaterial wasadded,
PrefacetotheFirstEdition xi
both mathematical and physical, including a chapter, at the end, on the so-called
algebraic formulation.
In particular, Chap. 4 contains the proof of Mercer’s theorem for positive
Hilbert–Schmidt operators. The analysis of the first two axioms of Quantum
Mechanics in Chap. 7 has been deepened and now comprises the algebraic char-
acterisationofquantumstatesintermsofpositivefunctionalswithunitnormonthe
C(cid:2)-algebra of compact operators. General properties of C(cid:2)-algebras and (cid:2)-morph-
isms are introduced in Chap. 8. As a consequence, the statements of the spectral
theoremandseveralresultsonfunctionalcalculusunderwentaminorbutnecessary
reshapinginChaps.8and9.IincorporatedinChap.10(Chap.9inthefirstedition)
abriefdiscussiononabstractdifferentialequationsinHilbertspaces.Animportant
example concerning Bargmann’s theorem was added in Chap. 12 (formerly
Chap.11).Inthesamechapter,afterintroducingtheHaarmeasure,thePeter–Weyl
theoremonunitaryrepresentationsofcompactgroupsisstatedandpartiallyproved.
This is then applied to the theory of the angular momentum. I also thoroughly
examined the superselection rule for the angular momentum. The discussion on
POVMsinChap.13(exChap.12)isenrichedwithfurthermaterial,andIincludeda
primer on the fundamental ideas of non-relativistic scattering theory. Bell’s
inequalities (Wigner’s version) are given considerably more space. At the end
of the first chapter, basic point-set topology is recalled together with abstract
measure theory. The overall effort has been to create a text as self-contained as
possible. I am aware that the material presented has clear limitations and gaps.
Ironically—my own research activity is devoted to relativistic theories—the entire
treatise unfolds at a non-relativistic level, and the quantum approach to Poincaré’s
symmetry is left behind.
I thank my colleagues F. Serra Cassano, R. Ghiloni, G. Greco, S. Mazzucchi,
A. Perotti and L. Vanzo for useful technical conversations on this second version.
For the same reason, and also for translating this elaborate opus into English,
I would like to thank my colleague S. G. Chiossi.
Trento, Italy Valter Moretti
October 2012