Quantum Theory: A Mathematical Approach

460 Pages·2015·6.619 MB·English

by  Peter Bongaarts (auth.)

Peter Bongaarts
Quantum
Theory
A Mathematical Approach
Quantum Theory
Peter Bongaarts
Quantum Theory
A Mathematical Approach
123
Peter Bongaarts
InstituutLorentz
Universityof Leiden
Leiden
The Netherlands
ISBN 978-3-319-09560-8 ISBN 978-3-319-09561-5 (eBook)
DOI 10.1007/978-3-319-09561-5
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La filosofia è scritta in questo grandissimo libro che continuamente ci sta aperto
innanzi a gli occhi (io dico l’universo), ma non si può intendere se prima non
s’imparaaintenderlalingua,econoscericaratteri,ne’qualiscritto.Eglièscritto
in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure
geometriche, senza i quali mezi è impossibile a intenderne umanamente parola;
senza questi è un aggirarsi vanamente per un oscuro laberinto.
(Philosophy is written in this vast book, which continuously lies open before our
eyes (I mean the universe). But it cannot be understood unless you have first
learned to understand the language and recognise the characters in which it is
written. It is written in the language of mathematics, and the characters are
triangles, circles, and other geometrical figures. Without such means, it is
impossible for us humans to understand a word of it, and to be without them is to
wander around in vain through a dark labyrinth.)
Galileo Galilei: Il Saggiatore (The Assayer), 1623
Translation by George MacDonald Ross
http://www.philosophy.leeds.ac.uk/GMR/hmp/texts/modern/galileo/assayer.html
Voor Piek
To the memory of John Cox, my teacher
Three mathematicians and one mathematical physicist who inspired the author of this
book
Irving Segal (1918–1998)
John von Neumann (1903–1957) Source1964SummerSchool,
Cargese, France
Alain Connes b.1947
Huzihiro Araki b.1932
Source Workshop “Seminal Interaction Source Workshop “Seminal
between Mathematics and Physics.
InteractionbetweenMathematics
Rome 2010” and Physics. Rome 2010”
Note The photographs of Irving Segal, Huzihiro Araki and Alain Connes were
taken by the author. Professors Araki and Connes gave permission to use their
pictures forthisbook.TheauthorwasunabletotracethedescendantsofProfessor
Segal who died in 1998.
Preface
Historical Remarks: Mathematics and Physics
Physics,asweknowit,beganinthesixteenthandseventeenthcentury,wheninthe
studyofnaturalphenomenaempiricalobservationandmathematicalmodelingwere
for the first time systematically and successfully combined. This is exemplified in
thepersonofGalilei,andevenmoresoinNewton,wholaidthefoundationsofour
pictureofthephysicalworldandwhowasequallygreatasamathematicianandas
an empirical observer and investigator.
For a long time mathematics and physics formed in an obvious way a single
integrated subject. Think of Archimedes, Newton, Lagrange, Gauss, more recently
Riemann, Cartan, Poincaré, Hilbert, von Neumann, Weyl, Birkhoff and many
others.Lorentz,thegreatDutchphysicist,wasofferedachairinmathematicsatthe
university of Utrecht, simultaneously with a chair in physics at a second Dutch
university.Hechosethelatterandbecamein1878ProfessorofTheoreticalPhysics
at Leiden university, the first in this subject in Europe.
All this isa thing ofthe past. Fromthe 1950s onward physicsandmathematics
partedcompany,orrathermathematicsunderwentadrasticchangeinthewayitwas
formulated, largely due to the Bourbaki movement. It became more abstract, more
“formal”.Inthisrespect,itshouldbenotedthatnoneofthegreatBourbakimathe-
maticians,Weil,Dieudonné,Grothendieck,forexample,hadtheslightestinterestin
physics. Another cause was the growing specialization in all of science and, more
recently,theincreasingpublicationpressureleadingtomuchnarrowlyfocusedshort-
termresearch.Thelanguageofmathematicsisnowverydifferentfromthatofphysics.
Inthegapbetweenphysicsandmathematics,mathematicalphysicsasadistinct
discipline came into being. Journals were established: Journal of Mathematical
Physics (1960), Communications in Mathematical Physics (1965), Letters in
Mathematical Physics (1975). Separate conferences were held. The International
AssociationofMathematicalPhysicswasfoundedin1976.Allthiswasandstillis
very welcome, not because it might lead to a new specialization, which would not
ix
x Preface
be a good thing, but as bridges between mathematics and physics. This book is a
modest attempt to contribute to this.
Notwithstandingthepresentdistancebetweenthefields,theinteractionbetween
mathematics and physics remains of great importance. Modern theoretical physics
cannot exist without advanced mathematics. On the other hand, many ideas in
mathematics still have their origin in physics—often in a heuristic form.
Theimportanceoftheconnectionbetweenmathematicsandphysicsisnolonger
reflected in the curriculum of most universities—certainly not in the Dutch uni-
versities. Physics students have to learn a great deal of standard mathematics in
their first and second year, mainly calculus and linear algebra, but they are not
exposed to the more advanced parts of modern mathematics; its abstract language
remainsstrangeinspirittothem,eventhoughtheysometimespickupandlearnto
use methods based on it. Mathematics, on the other hand, is taught as a self-
containedsubject,whichcanbestudiedforitsownsake,withoutanyreferenceto,
or knowledge of physics. Words like “classical mechanics” or “quantum mechan-
ics”havenomeaningformostpresent-daymathematicsgraduates.Moreover,after
afewyearsoftraininginrigorouslyformulatedmathematicstheywillfindtheloose
language of standard physics textbooks very hard to understand.
Comparable Books
Therehasalwaysbeenastrangeasymmetry.Numerousbookshavebeenwrittenin
the past, explaining to physicists advanced topics from mathematics, such as
functional analysis, differential geometry, Lie groups and Lie algebras, and alge-
braic topology. There has been much less in the other direction; books on physics
written for mathematicians are relatively rare, even though one would think that
there is a need for such books.
However,inthelastdecadeanumberofbooksofthissorthaveappeared.There
is one book that may not have been written with this intent, but it nevertheless
clearly stands out in this field. It is Roger Penrose’s The Road to Reality [1], an
amazing book which gives in almost 1,100 pages a wonderful overview of the
physical world seen through mathematical eyes. The books that I am about to
mention, as well as my own book, can in a certain sense all be seen as providing
additional material or more detailed or slightly alternative versions of subjects or
aspects discussed in the book of Penrose.
Books that have recently appeared:
– L.A. Takhtajan (2008)
Quantum Mechanics for Mathematicians [2].
– L.D. Faddeev, O.A. Yakubovskii (2009)
Lectures on Quantum Mechanics for Mathematics Students [3].
– K. Hannabuss (1997)
An Introduction to Quantum Theory [4].
Preface xi
– F. Strocchi (2005)
An Introduction to the Mathematical Structure of Quantum Mechanics [5].
– G. Teschl (2009)
Mathematical Methods in Quantum Mechanics. With Applications to Schrö-
dinger Operators [6].
– Jonathan Dimock (2011)
Quantum Mechanics and Quantum Field Theory. A Mathematical Primer [7].
– Brian C. Hall (2013)
Quantum Theory for Mathematicians [8].
Allthese bookshave some overlap with each other and also with mybook. Let
me give here the principal points in which my book is different from the books in
this list.
(cid:129) Oneofthemainideasputforwardinmybookisthat,byusingforthedescription
ofaphysicalsystemthealgebraofitsobservablesasbasicnotion,onecangivea
uniform formulation of both classical and quantum physics. The algebra of
classical observables is commutative, that of quantum observables noncommu-
tative. The states of a system are in both cases the positive normalized linear
functionalsonthealgebraofobservables.Thisideaisalsoabasisfor[3,5].Book
[3]expressesitinanelegantbutverygeneralmanner;itis,however,ashortbook
with not much explicit mathematical detail. Book [5] has more such details, but
the mathematical framework used for this basic idea is much too narrow. My
discussion in Chap. 12 has most of the mathematical details, as far as they are
known in the literature at the present time. I, moreover, apply this algebraic
frameworkin Chap. 13 toconnectingthe variousapproaches toquantization.
(cid:129) Most modern applications of quantum theory depend on statistical quantum
physics.In[4,7]thereareshortandexcellentsectionsonthis,butthereisnothing
intheotherbooks.MybookgivesinChap.11anextensivediscussionofquantum
statistical physics,after areviewof classical statistical physicsin Chap. 10.
(cid:129) Book [2] describes the mathematics of quantum theory. A superb book,
advanced, too difficult even for the average graduate student in mathematics. It
explainsthemathematics,butnotthewayitisusedinquantumtheory.Book[6],
another excellent book, gives an account of the mathematics of the Schrödinger
equation, an important special topic in quantum theory. My book gives, some-
what different from [2, 6], quantum theory as seen through mathematical eyes,
but still as a complete physical theory.
(cid:129) “Quantization”,isthenonuniqueprocedurewhich,startingfromagivenclassical
theory, constructs a corresponding quantum theory. Historically, there is, for
example,Born-JordanquantizationandWeylquantization,inprincipledifferent,
butforwhichatpresentnoexperimentalsituationsexistthatdistinguishbetween
the two. Nevertheless, the theoretical problem of the nonuniqueness of quanti-
zation remains.
Aspecificquantumtheorymaybeseenasadeformationofaclassicaltheory,
withPlanck’s(cid:1)hasdeformationparameter.Notethat(cid:1)hisaconstantofnature,but