Table Of ContentA Conceptual History of Space and Symmetry
é
Pietro Giuseppe Fr
A Conceptual History
of Space and Symmetry
From Plato to the Superworld
123
Pietro Giuseppe Fré
University of Torino
Turin, Italy
ISBN978-3-319-98022-5 ISBN978-3-319-98023-2 (eBook)
https://doi.org/10.1007/978-3-319-98023-2
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This book is dedicated to my family, namely
to my beloved daughter Laura, to my darling
wife Olga, to my young son Vladimir and to
my former wife Tiziana, Laura’s mother, with
whom, notwithstanding our divorce, the ties
due to a common daughter and to more than
twenty five years spent together remain
strong. This work is also dedicated to my
dear Franca, to whom, since my mother’s
death, I am tied by a profound filial affection
and by the sharing of deep intellectual
interests. My feelings of love are hereby
transmitted to all of them, with all the hopes
and the worries of an old father who
stubbornly believes that culture and science
are just one thing and encode, for mankind,
the unique escape route from global
disasters.
Preface
Thisbookformsatwinpairwithanotherbook[90]bythesameauthor,whichisof
a different, thoroughly mathematical character, while the character of the present
volume is historically and philosophically oriented. These two pieces of work
constitute a twin pair since, notwithstanding their different profiles and contents,
they arise from the same vision and pursue complementary goals.
The vision, extensively discussed in the first chapter of the present book and
throughout several other chapters, consists of the following main conceptual
assessments:
1. Our current understanding of the Fundamental Laws of Nature is based on a
coherent,yetprovisional,setoffivemeta-theoreticalprinciples,listedbymeas
A)-E)anddubbedthecurrentepisteme.Thisepistemeisofgenuinegeometrical
nature and can be viewed as the current evolutionary state of Einstein’s ideas
concerning the geometrization of physics.
2. GeometryandSymmetryareinextricablyentangled,andtheircurrentconception
is the result of a long process of abstraction, traced back in the present work,
which was historically determined and makes sense only within the Analytic
System of Thought of Western Civilization, started by the ancient Greeks.
3. The evolution of Geometry and Symmetry Theory in the last 40 years has
been deeply and very much constructively influenced by Supersymmetry/
Supergravity and the allied constructions of Strings and Branes. My reader is
notsupposedtoknow whatsupersymmetric fieldtheoriesandsupergravity are,
since this book has the ambition to be written for a much more general public
than that formed by theoretical physicists specialized in the super-world. It
sufficestobealertedthat,since theseventiesofthetwentieth century,anentire
new theoretical world has opened up by the introduction of a new symmetry
principle,indeeddubbedsupersymmetry,thatcanbeenforcedonfieldtheories,
inparticularonEinsteinGravity,atthepriceofaddingtoeveryphysicalfielda
new copy of the same, named its superpartner with a spin shifted by 1. This
2
entrains that the superpartners of bosons are fermions and vice versa. In a
vii
viii Preface
nutshell, supersymmetry is the invariance of such new type of field theories
under the exchange of bosons with fermions.
4. FurtheradvancesinTheoreticalPhysicscannotbebasedsimplyontheGalilean
Method of Interrogatingfirst Nature and then formulating a testable theory that
explains the observed phenomena. As stated later on in the present work, one
ought to Interrogate also Human Thought, by this meaning frontier-line math-
ematics concerned with geometry and symmetry in order to find there the
threads of so far unobserved correspondences, reinterpretations and renewed
conceptions.
The complementary pursued goals are as follows:
(a) In the case of the present book
(cid:129) thehistoricalandconceptualanalysisoftheprocessmentionedinpoint(2).
of the above list which led to the current episteme.
(cid:129) thephilosophicalargumentation,onhistoricalbasis,oftheassessmentmade
in point (4). of the above list.
(b) In the case of book [90], the mathematical full-fledged illustration of the main
developments in geometry and symmetry theory that occurred under the fer-
tilizing influence of Supersymmetry/Supergravity and that would be incon-
ceivable without the latter.
Repeating in a slightly different from the arguments advocated particularly in
Chapter One of the present work, I think that what is currently practiced in the
whole world as Fundamental Physics or Mathematics is based on the Greek view
of the episteme and it is meaningful only inside the Analytic System of Thought
founded by the Ancient Greeks. To recuperate a full-conscience of this fact is
mandatory inorder tocontinue onthedifficult butexciting path we areconfronted
with.
The twin pair of which this book is a member is viewed by the author as his
limited, humble contribution to the promotion of a new season of more scholarly
teaching of physical-mathematics.
TO MY READER
Ideally, this book has been written for an audience wider than that composed by
mathematicians andtheoretical physicists.Myambitiousaimwasthat ofattracting
the attention of all finely educated persons with an interest in the history of Ideas,
mathematical conceptions being an integral part of the general episteme. Yet,
mathematicsisalsotechnicalanditisverydifficulttosingleoutthemainideasand
tell the story of their evolution without mentioning some fundamental definitions
and some formulae. Coping with this essential difficulty, well known to everyone
who tries to explain the theories of the world to wider audiences, I have tried my
best, where formulae and definitions were unavoidable, to convey the underlying
Preface ix
concepts by means of wordly explanations and with the help of numerous ad hoc
created images. I have tried to intertwine storytelling and philosophic argumenta-
tions with some essential technical material introduced according to the above-
mentioned method, hoping to keep the attention of all my dear readers alive
throughout the chapters and up to my epilogue that expresses the results of long
self-debating meditations.
Spes, ultima dea.
Turin, Italy Pietro Giuseppe Fré
June 2018
Acknowledgements
With great pleasure, I would like to thank my collaborators and colleagues Pietro
Antonio Grassi, and Mario Trigiante for the many suggestions and discussions we
hadduringthewritingofthepresentbook.Similarly,Iexpressmygratitudetothe
Editors of Springer-Verlag, in particular to Maria Bellantone, for their continuous
assistance, constructive criticism and suggestions.
Mythoughts,whilefinishingthewritingofthislongessay,thatmostlyoccurred
inMoscowinmylast2years(2016–2017)ofserviceasScientificCounsellorofthe
ItalianEmbassyinRussia,werefrequentlydirectedtomylateparents,whomImiss
verymuchandIwillneverforget.TothemIalsoexpressmygratitudeforallwhat
they taught me in their life, in particular to my father who, with his own example,
introduced me, since my childhood, to the great satisfaction and deep suffering of
writing books.
Furthermore, it is my pleasure to thank my very close friend and collaborator
Aleksander Sorin for his continuous encouragement and for many precious
consultations.
xi
Contents
1 The Episteme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Symmetry and Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Setting the Stage: Symmetry and Beauty . . . . . . . . . . . . . . . . . 9
2.2 Symmetry Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Geometry and Algebra up to the Birth of Group Theory. . . . . . 14
2.4 Galois and the Advent of Group Theory . . . . . . . . . . . . . . . . . 21
2.5 A New Conception of Symmetry. . . . . . . . . . . . . . . . . . . . . . . 23
2.5.1 Conceptual Analysis of Galois Results. . . . . . . . . . . . . 27
2.5.2 Symmetry After Galois. . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Symmetry, Geometry and Space . . . . . . . . . . . . . . . . . . . . . . . 29
3 How Group Theory Came into Being . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 The Essentials of Group Theory in Modern Parlance . . . . . . . . 33
3.1.1 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.2 Groups as Transformation Groups. . . . . . . . . . . . . . . . 35
3.1.3 Representations of a Group. . . . . . . . . . . . . . . . . . . . . 37
3.2 From Cayley and Sylvester’s Matrices to Vector Spaces
and Groups: A Long Gestation . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.1 Cayley and Sylvester: A Short Account
of Their Lives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.2 Matrices in the Middle 1850s . . . . . . . . . . . . . . . . . . . 42
3.2.3 Cayley and Sylvester: The Invariant Twins . . . . . . . . . 44
3.2.4 The Calculus of Operations: Cayley and George
Boole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.5 Grassmann, Peano and the Birth of Vector Spaces . . . . 51
3.3 Classification of Finite Groups. . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4 Group Representation and the Unhappy Life of the Man
of Two Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
xiii
