Table Of ContentGraduate Texts in Mathematics
Jet Nestruev
Smooth
Manifolds and
Observables
Second Edition
Graduate Texts in Mathematics 220
Graduate Texts in Mathematics
Series Editors
Sheldon Axler
San Francisco State University, San Francisco, CA, USA
Kenneth Ribet
University of California, Berkeley, CA, USA
Advisory Board
Alejandro Adem, University of British Columbia
David Eisenbud, MSRI & University of California, Berkeley
Brian C. Hall, University of Notre Dame
Patricia Hersh, University of Oregon
J.F. Jardine, University of Western Ontario
Jeffrey C. Lagarias, University of Michigan
Eugenia Malinnikova, Stanford University
Ken Ono, University of Virginia
Jeremy Quastel, University of Toronto
Barry Simon, California Institute of Technology
Ravi Vakil, Stanford University
Steven H. Weintraub, Lehigh University
Melanie Matchett Wood, University of California, Berkeley
Graduate Texts in Mathematics bridge the gap between passive study and
creative understanding, offering graduate-level introductions to advanced topics in
mathematics. The volumes are carefully written as teaching aids and highlight
characteristic features of the theory. Although these books are frequently used as
textbooks in graduate courses, they are also suitable for individual study.
More information about this series at http://www.springer.com/series/136
Jet Nestruev
Smooth Manifolds
and Observables
Second Edition
123
Jet Nestruev
Lizzano inBelvedere, Italy
Moscow,Russia
ISSN 0072-5285 ISSN 2197-5612 (electronic)
Graduate Textsin Mathematics
ISBN978-3-030-45649-8 ISBN978-3-030-45650-4 (eBook)
https://doi.org/10.1007/978-3-030-45650-4
Mathematics Subject Classification: 58-01, 58Jxx, 55R05, 13Nxx, 08-01, 14F40, 58Axx, 17A70,
13B30,37J05
1stedition:©2003Springer-VerlagNewYork,Inc.
2ndedition:©SpringerNatureSwitzerlandAG2020
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Foreword
The author and his team undertook the revision of the book “Smooth
ManifoldsandObservables”,publishedin2002bySpringerVerlag,in2016at
the suggestion of the Editors of Springer. It immediately became clear that
the 2002 book should not only be revised, but must also be significantly
expanded, since the main underlying idea of the book—passing from the
differential geometry of smooth manifolds to the differential calculus in
commutativealgebras—hasbeensuccessfullydevelopedinthelast20yearsin
many new areas of mathematics and physics.
As a result, the 11 original chapters have been expanded to 21, the
number of pages has almost doubled (from 222 to 431), and the Preface has
become muchlonger. Nevertheless, themainprinciples ofthe firstedition, in
particular the “observability principle”, which determines the relationship of
mathematicstophysics,havebeenpreserved.Wehavetriedtomakethebook
as self-contained as possible, the only prerequisites being a general mathe-
maticalcultureontheleveloftwoyearsofgraduateschool.Inthisedition,we
havealsoincludedtheAfterword,aswellastheAppendixtothefirstedition,
written by A. M. Vinogradov, without any changes.
UnlikethefamousFrenchgeneral,JetNestruevisacivilian,andthenames
of the members of his team are not veiled in secrecy, as are those of the
Bourbaki team. For the first edition, they were: Alexander Astashov, Alexey
Bocharov, Sergei Duzhin, Alexandre Vinogradov, Mikhail Vinogradov, and
Alexey Sossinsky. For the present second edition, they are Alexander
Astashov, Alexandre Vinogradov, Mikhail Vinogradov, and Alexey
Sossinsky.ThefiguresforbotheditionswereexpertlypreparedbyAlexander
Astashov.
The author ackowledges the invaluable help of Joseph Krasil’shchik, who
read the entire first draft of this edition and noticed, besides misprints and
other minor snafus, numerous parts of the text that required essential chan-
ges; these changes have considerably improved the exposition. A. Sossinsky
acknowledges the partial support of a J. Simons Foundation grant.
Jet Nestruev
Lizzano in Belvedere, Italy and Moscow, Russia
December 2019
v
Alexandre Vinogradov
Александр Михайлович Виноградов
1938–2019
Alexandre Vinogradov, an extraordinary human being and outstanding
mathematician, deserves a more thorough and detailed life story than can fit
intothispage2,soherewewilllimitourselvestoaquotefromaletterfromhis
daughter Katya, who was at his bedside to the very end.
“AfewdaysagoFatherwasasleepinhisbedandIwassittingnexttohim.
Hewasnolongerabletoswallow,evenwater.Hesleptallthetime.Suddenly
he opened his eyes and tried to say something. Gently I said:
‘Daddy, do you want me to wet your lips?’
Unexpectedly he spoke, clearly and angrily:
‘Katya, damn, are you out of your mind? Lips, what lips? Don’t you
understand anything? I have finally figured out quantization! and here you
are wanting to wet my lips, stupid girl!’
Then I fully understood, once and for all, what my father is.”
2A necrologue with a short biography of A. M. Vinogradov and a sketch of his contribution to
mathematicsappearsinthesecondissuefor2020ofRussianMathematicalSurveys.
vii
Preface
Thelimitsof mylanguage
arethelimitsof myworld.
—L.Wittgenstein
Thisbookisaself-containedintroductiontosmoothmanifolds,fiberspaces,and
differential operators on them. It is not only accessible to graduate students
specializinginmathematicsandphysics,butalsointendedforreaderswhoare
alreadyfamiliarwiththesubject.Sincetherearemanyexcellenttextbooksin
manifold theory, the first question that should be answered is: Why another
bookonmanifolds?
The main reason is that the good old differential calculus is actually a
particularcaseofamuchmoregeneralconstruction,whichmaybedescribed
asthedifferentialcalculusovercommutativealgebras.Andthiscalculus,inits
entirety,isjusttheconsequenceofpropertiesofarithmeticaloperations.This
fact, remarkable in itself, has numerous applications, ranging from delicate
questions of algebraic geometry to the theory of elementary particles. Our
bookexplainsindetailwhythedifferentialcalculusonmanifoldsissimplyan
aspect of commutative algebra.
In the standard approach to smooth manifold theory, the subject is
developed along the following lines. First one defines the notion of smooth
manifold, say M. Then one defines the algebra F of smooth functions
M
on M, and so on. In this book this sequence is reversed: we begin with a
certain commutative R-algebra1 F, and then define the manifold M ¼MF
1Here and below R stands for the real number field. Nevertheless, and this is very important,
nothingpreventsusfromreplacingitbyanarbitraryfield(orevenaring)ifthisisappropriate
fortheproblemunderconsideration.
ix
x Preface
as the R-spectrum of this algebra. (Of course, in order that MF deserve the
title of a smooth manifold, the algebra F must satisfy certain conditions;
these conditions appear in Chapter 3, where the main definitions mentioned
here are presented in detail.)
This approach is by no means new: it is used, say, in algebraic geometry.
Oneofitsadvantagesisthatfromtheoutsetitisnotrelatedtothechoiceofa
specific coordinate system, so that (in contrast to the standard analytical
approach) there is no need to constantly check that various notions or
properties are independent of this choice. This explains the popularity of the
coordinate-free algebraic viewpoint among mathematicians attracted by
sophisticated algebra, but its level of abstraction discourages the more
pragmatically inclined applied mathematicians and physicists.
But what is really new in the book is the physical motivation of the alge-
braic approach. It is based, in particular, on the (physical) notion of observ-
able and the related observability principle, which asserts
what exists is only that which can be observed
As we all know, in developing scientific theories, homo sapiens gather the
necessarydatafromobservationandexperiment.Atthebeginning,thisdata
comes from our sensory organs, eventually replaced by accurate measuring
devices. An appropriate collection of these devices constitutes a physical
laboratory that studies some physical system. Let us formalize this notion,
havinginmindthatwearenotinterestedinhowthedevicesfunction,onlyin
what data from the physical system under study they register.
Measuring devices provide us with numerical data, i.e., their readings are
realnumbers.Theobtainednumbersarethenprocessed,mainlybyusingthe
fourarithmeticaloperations.NowiftwodevicesP andP givetworeadings,
1 2
instead of adding these numbers by hand, we can construct (or imagine) a
device, P þP that gives us the sum of the two readings. Since the choice
1 2
ofthescaleofthereadingsissubjective,forthesakeofobjectivity,insteadof
a measuring device P, we must use (virtual) devices ‚P, ‚2R. Further, in
order to fix the unit in our scales, we need a special device, I, whose only
possible reading is 1R. Indeed, as compared to P, the scale of the device
Pþ‚1R is shifted by ‚.
IfP ;...;P arerealmeasuringdevicesinthelaboratory,then,continuing
1 n
this process, we can associate to any polynomial pðx1;...;xnÞ the device
pðP1;...;PnÞ, whose reading is pða1;...;anÞ when a1;...;an are the readings
of the devices P1;...;Pn, respectively. Thus the operations of addition and
multiplication by real numbers defined above transform the set of all virtual
measuring devices into an algebra isomorphic to the polynomial algebra
R½x1;...;xn(cid:2).
But we can go even further and introduce virtual devices of the form
fðP1;...;PnÞ, where fðx1;...;xnÞ is a smooth (i.e., infinitely differentiable)
function, as it was done above for polynomial functions. Then the laboratory
Preface xi
consisting of all such virtual devices will be, from the mathematical point of
view, an algebra isomorphic to the algebra C1ðRnÞ of smooth functions in
Euclidean space Rn. The device I will be the unit of this algebra.
To summarize the above for future reference, let us agree that
the mathematical formalization of a physical laboratory
is an appropriate commutative algebra with unit overR.
In classical physics, we assess the state of a given physical system S only
from the readings of all the measuring devices. In that sense “state” and
“observation” are identical notions. Mathematically, an observation of the
stateofthesystemSmaybeunderstoodasamaph:A!RofthealgebraA
of all measuring devices, which to any device P assigns its reading hðPÞ in
that state. On the other hand, from the definition of arithmetical operations
on measuring devices, it obviously follows that h is a homomorphism of
unitary algebras, in particular, we have hðIÞ¼1R. It is important to under-
stand that not every such homomorphism is an observation of the system S.
To clarify this, let us consider the following construction.
Let the ideal I (cid:3)A be the intersection of the ideals KerðhÞ for all obser-
vations from S, and A def A=I . Then any such a homomorphism h may be
S S
represented in the form of the composition
A(cid:4)p!r A (cid:4)!~h R;
S
wherepristhenaturalprojectionand~hðamodI Þ¼hðaÞ.Conversely,under
S
certain restrictions on the algebra A (which we omit at this point), any
homomorphism A !R composed with the projection pr is an observation
S
ofthesystemS.ThusthefamilyofallstatesofthesystemScanbeidentified
with the set of unitary algebra homomorphisms from A to R. This set is
S
denoted by Spec A and is called the R-spectrum of the algebra A .
R S S
Summarizing the above, let us agree that
the algebraA isthemathematicalmodelofthephysical
S
system S, while Spec A is the set of its states
R S
The algebra A is sometimes called the algebra of observables, while its
S
elements are observable quantities or simply observables.
If we agree, as we did above, that A¼C1ðRnÞ, then Spec A=I, where
R
I (cid:3)C1ðRnÞistheidealnaturallyidentifiedwithsomeclosedsubsetofRn.On
the other hand, we know that the space of states of mechanical and other
physical systems with a finite number of degrees of freedom is a smooth
manifold.Fromthiswecanconcludethatthealgebraofobservablesofsucha
systemisnoneotherthanA ¼C1ðMÞ,thealgebraofsmoothfunctionsona
S
manifold M.
