Quantum: The Quantum Theory of Particles, Fields and Cosmology

606 Pages·1998·16.003 MB·English

by  Professor Dr. Edgard Elbaz (auth.)

Texts and Monographs in Physics
Series Editors: R. Balian W. Beiglbock H. Grosse E. H. Lieb
N. Reshetikhin H. Spohn W Thirring
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Texts and Monographs in Physics
Series Editors: R. Balian W Beiglbock H. Grosse E. H. Lieb
N. Reshetikhin H. Spohn W Thirring
From Microphysics to Macrophysics Supersymmetric Methods in Quantum
I + II Methods and Applications of and Statistical Physics By G. Junker
Statistical Physics By R. Balian
CP Violation Without Strangeness
Variational Methods in Mathematical Electric Dipole Moments of Particles,
Physics A Unified Approach Atoms, and Molecules
By P. Blanchard and E. Briining By I. B. Khriplovich and S. K. Lamoreaux
Quantum Mechanics: Inverse SchrOdinger Scattering
Foundations and Applications in Three Dimensions
3rd enlarged edition By A. Bohm By R. G. Newton
The Early Universe Scattering Theory of Waves
Facts and Fiction 3rd corrected and Particles 2nd edition
and enlarged edition By G. Borner By R. G. Newton
Operator Algebras and Quantum Quantum Entropy and Its Use
Statistical Mechanics I + II 2nd edition By M. Ohya and D. Petz
By O. Bratteli and D. W. Robinson
Generalized Coherent States
Geometry of the Standard Model and Their Applications
of Elementary Particles By A. Perelomov
By A. Derdzinski
Essential Relativity Special, General,
Scattering Theory of Classical and Cosmological Revised 2nd edition
and Quantum N-Particle Systems By W. Rindler
By J. Derezinski and C. Gerard
Path Integral Approach
Effective Lagrangians to Quantum Physics An Introduction
for the Standard Model 2nd printing By G. Roepstorff
By A. Dobado, A. Gomez-Nicola,
Finite Quantum Electrodynamics
A. L. Maroto and J. R. Pelaez
The Causal Approach 2nd edition
Quantum By G. Scharf
The Quantum Theory of Particles, Fields,
From Electrostatics to Optics
and Cosmology By E. Elbaz
A Concise Electrodynamics Course
Quantum Relativity By G. Scharf
A Synthesis of the Ideas of Einstein
The Mechanics and Thermodynamics
and Heisenberg
of Continuous Media By M. Silhavy
By D. R. Finkelstein
Large Scale Dynamics of Interacting
Quantum Mechanics I + II
Particles By H. Spohn
By A. Galindo and P. Pascual
The Theory of Quark and Gluon
The Elements of Mechanics
Interactions 2nd completely revised
By G. Gallavotti
and enlarged edition By F. J. Yndurain
Local Quantum Physics
Relativistic Quantum Mechanics and
Fields, Particles, Algebras
Introduction to Field Theory
2nd revised and enlarged edition
By F. J. Ynduniin
By R. Haag
Edgard Elbaz
Quantum
The Quantum Theory
of Particles,
Fields, and Cosmology
With 23 Figures
, Springer
Professor Dr. Edgard Elbaz
Institut de Physique Nucleaire de Lyon IN2P3-CNRS
Universite Claude Bernard Lyon - I
43, bd. du II Novembre 1918
F-69622 Villeurbanne Cedex, France
Editors
Roger Balian Nicolai Reshetikhin
CEA Department of Mathematics
Service de Physique TMorique de Saclay University of California
F-91191 Gif-sur-Yvette, France Berkeley, CA 94720-3840, USA
Wolf BeiglbOck Herbert Spohn
Institut fiir Angewandte Mathematik Theoretische Physik
Universitat Heidelberg Ludwig-Maximilians-Universitat Miinchen
1m Neuenheimer Feld 294 TheresienstraBe 37
D-69120 Heidelberg, Germany D-80333 Miinchen, Germany
Harald Grosse Walter Thirring
Institut fiir Theoretische Physik Institut fiir Theoretische Physik
Universitat Wien Universitat Wien
Boltzmanngasse 5 Boltzmanngasse 5
A-1090 Wien, Austria A-I090 Wien, Austria
Elliott H. Lieb
Jadwin Hall
Princeton University, P. O. Box 708
Princeton, NJ 08544-0708, USA
Title of the original French edition:
Quantique, published by Ellipses © Edition Marketing S. A. 1995
ISSN 0172-5998
ISBN-l3: 978-3-642-64327-9 e-ISBN-13:978-3-642-60266-5
DOl: 10.1007/978-3-642-60266-5
Library of Congress Cataloging-in-Publication Data
Elbaz, Edgard, 1937-[Quantique. English] Quantum: the quantum theory of particles, fields, and
cosmology 1 E. Elbaz. p. cm. - (Texts and monographs in physics) Includes bibliographical
references and index. ISBN-13: 978-3-642-64327-9 (Berlin: alk. paper) l. Quantum theory. I. Title.
II. Series. QC174.12.E4413 1998 530.12-dc21 97-33659
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To my wife
my children and gmnd-children
Ephmim, Levana, Shoshana,
Eliott, Yakov,
Rachel.
Preface
I consider teaching to have been one of my greatest privileges in life. I have al
ways attemped to make a matter, sometimes difficult, sometimes in constant
evolution, simpler and easier to understand. There comes a time then, when
one feels the necessity to write and publish a book. For this reason, I have
already published (in french) several lecture books on Classical Mechanics,
Quantum Mechanics, Matrix Optics, Electroweak Interaction, General Rel
ativity and Gravitation, Cosmology. Teaching quantum theory has been a
particular delight as its constant evolution and enlargement, embraces all
domains of physics. Even if the general layout remains relatively unchanged,
the evolution of our understanding of the physical world imposes the in
troduction of new approaches. It therefore seemed that a physics textbook,
even one with a graduate readership in mind, had to introduce the Dirac
Electron Theory and some rudimentary material on quantum field theory.
This holds for also for explanations of the spontaneous symmetry breaking
of Higgs scalar fields that gives mass to the bosons involved in short range
interactions. We have deliberately cut down on material on some "classical"
topics of quantum theory to make space for less known methods. Examples
of the latter include the Feynman path integral "3rd quantization" and the
interpretation of quantum mechanics in terms of phase focusing coherence.
After a review of the most important results of research in analytical
mechanics, it is shown that the description of a system of material parti
cles is possible either by using a "discrete" formalism, such as Poisson equa
tions, or a "continuous" formalism, such as Hamilton-Jacobi wave equations.
Next, drawing the parallel between mechanics and optics serves as the pre
text to introduce an equation for the propagation of mechanical waves and,
with the help of a further hypothesis, the Schrodinger equation for the wave
mechanical description of a classical system of particles with definite mass.
The next chapter develops wave-mechanical theory from the stand point of
the correspondence principle. It then goes on to briefly show the relation
ship between the Heisenberg uncertainty relation and the different interpre
tations of wave mechanics and wave function, interpretations stemming from
the Schrodinger equation: the pilot wave theory of Louis de Broglie and the
quantum potential theory of D. Bohm, the theory of universal Brownian mo
tion of E. Nelson, the probability theory of Max Born and Niels Bohr of the
VIII Preface
Copenhagen School. Mention is also made of the EPR (Einstein-Podolsky
Rosen) paradox as well as the Bell inequalities which where tested by Alain
Aspect and Philippe Grangier. Their experiments have definitely established
that quantum theory is a non-separable theory, that is without any hidden
variables.
Chapter 3 is a little more technical, describing classical stationary one
dimensional quantum systems, the potential well, the potential barrier and
the semi-classical approach (W.K.B.). The following three chapters return
to the Dirac formalism of nonrelativistic quantum mechanics and the two
representations it has generated: wave mechanics by projection on the con
figuration space and matrix mechanics by projection on the vector column
space.
After the study of quantum dynamics and the Schrodinger or Heisenberg
pictures of the evolution of a nonrelativistic quantum mechanical system, we
return to the interpretation of quantum mechanics in terms of the coherence
of the quantum states underlying all forms of observer-independent interpre
tation. For the Copenhagen School, measurement determines the history of
a quantum system with which a probability can be associated. When, as in
quantum cosmology, it is practically impossible to separate the observer from
the object being observed, the meaning of the probability postulate must be
stretched if the theory is to retain its internal coherence. In such a situa
tion, the Copenhagen School interpretation becomes an approximation of a
more general framework characteristic of measurement and the decoherence
resulting from the observer's measuring apparatus.
Chapter 7 develops the so-called 3rd quantization based on the Feynman
path integrals. This method has the advantage of being adapted to all forms
of interaction, including gravitation. It also emphasizes the importance of
the phase function in wave function. As with the Huyghens' principle, the
probability amplitude 'Ij;(x', t') is given by the superposition in the whole
space of the probability amplitude at an earlier instant. This underscores
the importance of the propagator developed with the Feynman conjecture
from classical data of the system. This approach to quantum physics sheds
new light on the role of action in quantum physics and its classical limit, the
Hamiltonian action.
Next, we examine the properties of symmetry and invariance in quan
tum physics generally: translation, rotation, parity, time reversal. Much space
has been alloted to non-local gauge transformations and to their associated
symmetry groups, the Abelian groups, just as in the gauge transformation
of the electromagnetic field gauge or the non-Abelian for Yang-Mills fields.
These transformations have, in the recent years, assumed great importance
in the description of electromagnetic and electroweak interactions in a unified
model. This point is also featured in the discussion of the standard theory of
electroweak interactions in Chapter 19.
Preface IX
Five chapters have been devoted to the angular momentum theory in
the standard representation, the theorems of addition of two or more an
gular momenta, rotation matrices and spherical harmonics and, finally, to
irreducible tensor operators. The innovation is in the relatively early intro
duction (Chapter 10) of the Clebsch-Gordan graphical representation and
"3jm" coefficients. This facilitates the visualization of calculations and the
understanding of the techniques of Racah algebra.
In the chapters on angular momentum (Chapters 9 through 13), the
graphical representation has been used to supplement analytical calculation.
It seemed to us an effective method for handling irreducible tensor opera
tors and the Wigner-Eckart theorem. It should also facilitate access to some
particularly difficult calculations, such as those for angular distributions and
particle polarization. We have also sought to extend this graphical approach
to classical vector calculus. This should simplify statements involving vec
tor products and vector inner products (these points are developed in the
appendix of Chapter 13).
Another innovation in the chapters on angular momentum is the intro
duction of the electroweak isospin of leptons and quarks, and of the magnetic
moment of atoms, particles and quarks, as applications of the standard rep
resentation and the Wigner-Eckart theorem.
We have brought together in Chapter 14 the various methods of approxi
mation in nonrelativistic quantum physics: stationary perturbations of degen
erate and non-degenerate states, the variational method, and time-dependent
variation. The Feynman diagram method is introduced to illustrate pertur
bative methods.
Feynman diagrams facilitate the understanding of the processes by which
relativistic or nonrelativistic particles are scattered.
The problems associated with scattering are studied in Chapter 15, using
the transition operator and its relationship with the transition amplitude and
the differential cross section of elastic scattering. The Lippmann-Schwinger
statement on the transition amplitude with respect to the interaction poten
tial inevitably brings us back to Feynman. Next, we seek to extend the scope
of the problem to the calculation of angular distribution and polarization in
direct nuclei interaction processes and it should be possible to expand the
results using G.S.A. graphical techniques without great difficulties.
In Chapter 16, "Second Quantization", we decided to treat together one
dimensional and n-dimensional harmonic oscillators and the second quanti
zation formalism for boson and fermion systems. Again, it seemed to us that
the formalism of creation and annihilation operators, when applied to the
harmonic oscillator, constituted an ideal introduction to the notion of second
quantization. We have also included a review of the present state of knowl
edge on the structure of matter with fermions that are emitters and receivers
of the bosons which mediate in fundamental interactions.
X Preface
Classical and quantum field theory has unfortunately been too often ne
glected in physics curricula. We have therefore devoted the next two chapters
to this subject.
In the chapter on boson fields, we have included a section on Higgs scalar
field and the process of spontaneous symmetry breaking which confers mass to
exchange bosons. Chapter 18 describes fermion fields. After a brief excursion
into the Dirac equation of a free particle, we return to the problem of gauge
invariance and, using the Noether theorem, try to show that a law of conser
vation of quadricurrent can be inferred from overall gauge invariance. Next,
we examine the interaction of a charged fermion with the electromagnetic
field before going on to redefine the conditions for local gauge invariance. It
is thus possible to introduce a little more naturally the Lagrangian electrody
namic quantum density. After describing the property of charge conjugation,
we apply the Dirac equation to a detailed study of the hydrogen atom, taking
into consideration both the relativistic effects and the electron spin.
The next chapter is entilted "Quantum X-dynamics" . It describes in very
brief terms quantum electrodynamic methods and the extensions quantum
chromo dynamics has brought to these theories. The standard model (G.S.W.)
of electroweak interaction with the process of spontaneous symmetry break
ing of Higgs fields which confers mass to the bosons carrying the weak in
teraction (the intermediate bosons W± and ZO) receives a more detailed
presentation. This chapter will no doubt appear a little more difficult than
the preceding ones. However, it should be remembered that the unification of
electromagnetic and electroweak interactions constituted such a stride in the
advancement of knowledge that one cannot do justice to the topics discussed
here without treating the method used and the spectacular results obtained
through the determination of the mass of intermediate bosons.
Finally, the last chapter is dedicated to the problems of quantum cosmo
logy. This is an introduction to a particularly difficult but thought-provoking
subject which conditions the very foundations of the fundamental notions
of space and time has been added at the suggestion of some colleagues. As
at the time of writing, there are no quantum physics textbooks that can
be considered accessible to cosmologists or cosmology textbooks that are
accessible to quantum physicists. Considering that we have had published a
textbook in cosmology which closes with a chapter on quantum cosmology,
it seemed natural that the present one too to close with a similar chapter.
We have therefore given a brief review of general relativity and the stan
dard models of Cosmology (the big-bang model) before going on to describe
the Hamiltonian formulation of Einstein's equations which, after quantiza
tion, lead to the fundamental equation of quantum cosmology: the Wheeler
De Witt equation. The use of this equation in well-chosen minisuperspaces
often leads to the (W.K.B.) expression of the Universal wave function. This
brief introduction is not intended as a substitute for specialized texts on the
question but rather to give an idea of the issues raised and, indeed, to incite
Preface XI
curiosity to find out more and perhaps also follow it up with a higher degree
and research in an almost virgin area.
This text is intended to be self-contained in itself without necessarily lay
ing claims to being exhaustive, to be clear without seeking to over-simplify.
Concerning the title: Quantum Mechanics? or Quantum Physics? or Quan
tum Theory? We finally settled for Quantum which contains all of them. To
facilitate the retention of the ideas introduced, each chapter closes with a ta
ble reproducing the main points where they are deemed to constitute an aid
in the understanding of the text. We can only hope then that this book will
be of real assistance to the target public: undergraduate students (Chapters
1 through 6 or, at best, through 8), masters students (Chapters 6 through 18)
and first-year doctoral students (the whole of the text). Because this text
book derives in part from an M.Sc. Physics course I run at Universite Claude
Bernard - Lyon 1, it has greatly benefitted from numerous observations by
the students I have had over the years.
I would like to thank Mr Z. Hernaus whole-heartedly for handling the
illustrations. My profound gratitude also goes to my colleagues in Lyon, and
especially to P. Desgrolard, for the patience and care with which he went
through the original manuscript and for the numerous suggestions that have
been instrumental in clarifying the ideas expressed in the text. I am grateful
too to Mrs S. Flores for keying in the original text in English, to M. Chartoire
for his technical assistance with the computer and to Uzoma Chukwu and
Swanny Prakash for translating the text from French.
Lyon, August 1997 E. Elbaz