Unformatted text preview: Graduate Texts in Physics Edouard B. Manoukian Quantum
Field Theory II
Introductions to Quantum Gravity,
Supersymmetry and String Theory Graduate Texts in Physics Series editors
Kurt H. Becker, Polytechnic School of Engineering, Brooklyn, USA
Sadri Hassani, Illinois State University, Normal, USA
Bill Munro, NTT Basic Research Laboratories, Atsugi, Japan
Richard Needs, University of Cambridge, Cambridge, UK
Jean-Marc Di Meglio, Université Paris Diderot, Paris, France
William T. Rhodes, Florida Atlantic University, Boca Raton, USA
Susan Scott, Australian National University, Acton, Australia
H. Eugene Stanley, Boston University, Boston, USA
Martin Stutzmann, TU München, Garching, Germany
Andreas Wipf, Friedrich-Schiller-Univ Jena, Jena, Germany Graduate Texts in Physics
Graduate Texts in Physics publishes core learning/teaching material for graduateand advanced-level undergraduate courses on topics of current and emerging fields
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suitable as introductions or references for scientists entering, or requiring timely
knowledge of, a research field. More information about this series at Edouard B. Manoukian Quantum Field Theory II
Introductions to Quantum Gravity,
Supersymmetry and String Theory 123 Edouard B. Manoukian
The Institute for Fundamental Study
Naresuan University
Phitsanulok, Thailand ISSN 1868-4513
Graduate Texts in Physics
ISBN 978-3-319-33851-4
DOI 10.1007/978-3-319-33852-1 ISSN 1868-4521 (electronic)
ISBN 978-3-319-33852-1 (eBook) Library of Congress Control Number: 2016935720
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The registered company is Springer International Publishing AG Switzerland Preface to Volume II My motivation in writing this second volume was to have a rather introductory
book on quantum gravity,1 supersymmetry,2 and string theory3 for a reader who
has had some training in conventional quantum field theory (QFT) dealing with
its foundations, with abelian and non-abelian gauge theories including grand
unification, and with the basics of renormalization theory as already covered in
Vol. I Quantum Field Theory I: Foundations and Abelian and Non-Abelian Gauge
Theories. This volume is partly based on lectures given to graduate students in
theoretical and experimental physics, at an introductory level, emphasizing those
parts which are reasonably well understood and for which satisfactory theoretical
descriptions have been given.
Quantum gravity is a vast subject,4 and I obviously have to make a choice in this
introductory treatment of the subject. As an introduction, I restrict the study to two
different approaches to quantum gravity: the perturbative quantum general relativity
approach as the main focus and a non-perturbative background-independent one
referred to as “loop quantum gravity” (LQG), where space emerges from the theory
itself and is quantized. In LQG we encounter a QFT in a three-dimensional space. 1
For more advanced books on quantum gravity that I am familiar with, see the following: C. Kiefer
(2012): Quantum Gravity, by Oxford University Press, T. Thiemann (2007): Modern Canonical
Quantum Gravity, C. Rovelli (2007): Quantum Gravity, as well as of the collection of research
investigations in D. Oriti (2009): Approaches to Quantum Gravity, by Cambrige University Press.
2
For more advanced books on supersymmetry that I am familiar with, see the following books:
H. Baer & X. Tata (2006): Weak scale supersymmetry: from superfields to scattering events,
M. Dine (2007): Supersymmetry and string theory - beyond the stadard model, S. Weinberg (2000):
The Quantum theory of fields III: Supersymmetry, by Cambridge University Press, and P. Binetruy
(2006): Supersymmetry, experiments and cosmology by Oxford University Press.
3
For more advanced books on string theory that I am familiar with, see the following books:
K. Becker, M. Becker & J. H. Schwarz (2006): String theory and M-theory - a modern approach,
M. Dine (2007): Supersymmetry and string theory - beyond the standard model, and J. Polchinski
(2005) : Superstring theory I & II by Cambridge University Press.
4
See the references given above on quantum gravity. v vi Preface to Volume II Some unique features of the treatment given are:
• No previous knowledge of general relativity is required, and the necessary
geometrical aspects needed are derived afresh.
• The derivation of field equations and of the expression for the propagator of
the graviton in the linearized theory is solved with a gauge constraint, and a
constraint necessarily implies that not all the components of the gravitational
field may be varied independently—a point which is most often neglected in the
literature.
• An elementary treatment is given of the so-called Schwinger-DeWitt technique.
• Non-renormalizability aspects of quantum general relativity are discussed as well
as of the renormalizability of some higher-order derivative gravitational theories.
• A proof is given of the Euler-Poincaré Characteristic Theorem which is most
often omitted in textbooks.
• A uniqueness property of the invariant product of three Riemann tensors is
proved which is also most often omitted in textbooks.
• An introductory treatment is provided of “loop quantum gravity” with sufficient
details to get the main ideas across and prepare the reader for more advanced
studies.
Supersymmetry is admittedly a theory with mathematical beauty. It unites
particles of integer and half-integer spins, i.e., with different spins, but with equal
masses in symmetry multiplets. Some important aspects in the treatment of the
subject are the following:
• A fundamental property of supersymmetric theories is that the supersymmetry charge (supercharge) operator responsible for interchanging bosonic
and fermionic degrees of freedom obviously does not commute with angular
momentum (spin) due to different spins arising in a given supermultiplet.
This commutation relation is explicitly derived which is most often omitted in
textbooks.
• The concept of superspace is introduced, as a direct generalization of the
Minkowski one, and the basic theory of integration and differentiation in
superspace is developed.
• A derivation is given of the so-called Super-Poincaré algebra satisfied by the
generators of supersymmetry and spacetime transformations, which involves
commutators and anti-commutators5 and generalizes the Poincaré algebra of
spacetime transformations derived in Vol. I.
• The subject of supersymmetric invariance of integration theory in superspace is
developed as it is a key ingredient in defining supersymmetric actions and in
constructing supersymmetric extensions of various field theories.
• A panorama of superfields is given including that of the pure vector superfield,
and complete derivations are provided. 5 Such an algebra is referred to as a graded algebra. Preface to Volume II vii • Once the theory of supersymmetric invariant integration is developed, and
superfields are introduced, supersymmetric extensions of basic field theories are
constructed, such as that of Maxwell’s theory of electrodynamics; a spin 0–spin
1/2 field theory, referred to as the Wess-Zumino supersymmetric theory with
interactions; the Yang-Mill field theory; and the standard model.
• There are several advantages of a supersymmetric version of a theory over
its non-supersymmetric one. For one thing, the ultraviolet divergence problem
is much improved in the former in the sense that divergences originating
from fermions loops tend, generally, to cancel those divergent contributions
originating from bosons due to their different statistics. The couplings in the
supersymmetric version of the standard model merge together more precisely
at a high energy. Moreover, this occurs at a higher energy than in the nonsupersymmetric theory, getting closer to the Planck one at which gravity is
expected to be significant. This gives the hope of unifying gravity with the rest
of interactions in a quantum setting.
• Spontaneous symmetry breaking is discussed to account for the mass differences
observed in nature of particles of bosonic and fermionic types.
• The underlying geometry necessary for incorporating spinors in general relativity
is developed to finally and explicitly derive the expression of the action of the full
supergravity theory.
In string theory, one encounters a QFT on two-dimensional surfaces traced by
strings in spacetime, referred to as their worldsheets, with remarkable consequences
in spacetime itself, albeit in higher dimensions. If conventional field theories are
low-energy effective theories of string theory, then this alone justifies introducing
this subject to the student. Some important aspects of the treatment of the subject
are the following:
• In string theory, particles that are needed in elementary particle physics arise
naturally in the mass spectra of oscillating strings and are not, a priori, assumed
to exist or put in by hand in the underlying theory. One of such particles emerging
from closed strings is the evasive graviton.
• With the strings being of finite extensions, string theory may, perhaps, provide a
better approach than conventional field theory since the latter involves products
of distributions at the same spacetime points which are generally ill defined.
• Details are given of all the massless fields in bosonic and superstring theories,
including the determination of their inherited degrees of freedom.
• The derived degrees of freedom associated with a massless field in Ddimensional spacetime, together with the eigenvalue equation associated with
the mass squared operator associated with such a given massless field, are
consistently used to determine the underlying spacetime dimensions D of the
bosonic and superstring theories.
• Elements of space compactifications are introduced.
• The basics of the underlying theory of vertices, interactions, and scattering of
strings are developed.
• Einstein’s theory of gravitation is readily obtained from string theory.
• The Yang-Mills field theory is readily obtained from string theory. viii Preface to Volume II This volume is organized as follows. In Chap. 1, the reader is introduced to quantum gravity, where no previous knowledge of general relativity (GR) is required.
All the necessary geometrical aspects are derived afresh leading to explicit general
Lagrangians for gravity, including that of GR. The quantum aspect of gravitation, as
described by the graviton, is introduced, and perturbative quantum GR is discussed.
The so-called Schwinger-DeWitt formalism is developed to compute the oneloop contribution to the theory, and renormalizability aspects of the perturbative
theory are also discussed. This follows by introducing the very basics of a nonperturbative, background-independent formulation of quantum gravity, referred to
as “loop quantum gravity” which gives rise to a quantization of space and should
be interesting to the reader. In Chap. 2, we introduce the reader to supersymmetry
and its consequences. In particular, quite a detailed representation is given for the
generation of superfields, and the underlying section should provide a useful source
of information on superfields. Supersymmetric extensions of Maxwell’s theory, as
well as of Yang-Mills field theory, and of the standard model are worked out,
as mentioned earlier. Spontaneous symmetry breaking, and improvement of the
divergence problem in supersymmetric field theory are also covered. The unification
of the fundamental couplings in a supersymmetric version of the standard model 6
is then studied. Geometrical aspects necessary to study supergravity are established
culminating in the derivation of the full action of the theory. In the final chapter,
the reader is introduced to string theory, involving both bosonic and superstrings,
and to the analysis of the spectra of the mass (squared) operator associated with the
oscillating strings. The properties of the underlying fields, associated with massless
particles, encountered in string theory are studied in some detail. Elements of
compactification, duality, and D-branes are given, as well as of the generation of
vertices and interactions of strings. In the final sections on string theory, we will see
how one may recover general relativity and the Yang-Mills field theory from string
theory. We have also included two appendices at the end of this volume containing
useful information relevant to the rest of this volume and should be consulted by
the reader. The problems given at the end of the chapters form an integral part of
the books, and many developments in the text depend on the problems and may
include, in turn, additional material. They should be attempted by every serious
student. Solutions to all the problems are given right at the end of the book for
the convenience of the reader. We make it a point pedagogically to derive things in
detail, and some of such details are sometimes relegated to appendices at the end of
the respective chapters, or worked out in the problems, with the main results given
in the chapters in question. The very detailed introduction to QFT since its birth in
1926 in Vol. I,7 as well as the introductions to the chapters, provide the motivations 6 The standard model consists of the electroweak and QCD theories combined, with a priori
underlying symmetry represented by the group products SU.2/ U.1/ SU.3/.
7
Quantum Field Theory I: Foundations and Abelian and Non-Abelian Gauge Theories. I strongly
suggest that the reader goes through the introductory chapter of Vol. I to obtain an overall view of
QFT. Preface to Volume II ix and the pedagogical means to handle the technicalities that follow them in these
studies.
This volume is suitable as a textbook. Its content may be covered in a 1 year
(two semesters) course. Short introductory seminar courses may be also given on
quantum gravity, supersymmetry, and string theory.
I often meet students who have a background in conventional quantum field
theory mentioned earlier and want to learn about quantum gravity, supersymmetry
and string theory but have difficulty in reading more advanced books on these
subjects. I thus felt a pedagogical book is needed which puts these topics together
and develops them in a coherent introductory and unified manner with a consistent
notation which should be useful for the student who wants to learn the underlying
different approaches in a more efficient way. He or she may then consult more
advanced specialized books, also mentioned earlier, for additional details and further
developments, hopefully, with not much difficulty.
I firmly believe that different approaches taken in describing fundamental physics
at very high energies or at very small distances should be encouraged and considered
as future experiments may confirm directly, or even indirectly, their relevance to the
real world.
I hope this book will be useful for a wide range of readers. In particular, I
hope that physics graduate students, not only in quantum field theory and highenergy physics but also in other areas of specializations, will also benefit from it
as, according to my experience, they seem to have been left out of this fundamental
area of physics, as well as instructors and researchers in theoretical physics.
Edouard B. Manoukian Contents 1 Introduction to Quantum Gravity . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.1 Geometrical Aspects, Structure of Spacetime
and Development of the General Theory of Relativity . . . . . . . . . . . . . .
1.2 Lagrangians for Gravitation: The Einstein-Hilbert
Action, Einstein’s Equation of GR, Energy-Momentum
Tensor, Higher-Order Derivatives Lagrangians . .. . . . . . . . . . . . . . . . . . . .
1.3 Quantum Particle Aspect of Gravitation: The Graviton
and Polarization Aspects . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.3.1 Second Order Covariant Formalism . . . . . .. . . . . . . . . . . . . . . . . . . .
1.3.2 First Order Formalism.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.3.3 The Quanta of Gravitation in Evidence: Graviton
Emission and Gravitational Radiation . . . .. . . . . . . . . . . . . . . . . . . .
1.4 Quantum Fluctuation About a Background Metric . . . . . . . . . . . . . . . . . .
1.5 The Schwinger-DeWitt Technique .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.6 Loop Expansion and One-Loop Contribution
to Quantum General Relativity. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.7 Dimensional Regularization of the One Loop
Contribution to Quantum General Relativity . . . . .. . . . . . . . . . . . . . . . . . . .
1.8 Renormalization Aspects of Quantum Gravity: Explicit
Structures of One- and Two-Loop Divergences
of Quantum GR, The Full Theory of GR Versus Higher
Derivatives Theories: The Low Energy Regime .. . . . . . . . . . . . . . . . . . . .
1.8.1 Two and Multi Loops . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.8.2 Higher Order Derivatives Corrections .. . .. . . . . . . . . . . . . . . . . . . .
1.8.3 The Low Energy Regime: Quantum GR
as an Effective Field Theory and Modification
of Newton’s Gravitational Potential . . . . . .. . . . . . . . . . . . . . . . . . . .
1.9 Introduction to Loop Quantum Gravity.. . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.9.1 The ADM Formalism and Intricacies of the
Underlying Geometry . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1
6 23
27
29
33
36
40
45
49
52 54
57
60 62
66
67 xi xii Contents 1.9.2 Gravitational “Electric” Flux Across a Surface
in 3D Space .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.9.3 Concept of a Holonomy and Some of its Properties .. . . . . . . .
1.9.4 Definition of Spin Networks, Spin Network
States, States of Geometry . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.9.5 Quanta of Geometry .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Appendix A: Variation of a Determinant .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Appendix B: Parametric Integral Representation of the
Logarithm of a...
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