Table Of ContentPHYSICS RESEARCH AND TECHNOLOGY
G T
AUGE HEORIES
AND DIFFERENTIAL GEOMETRY
No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or
by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no
expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No
liability is assumed for incidental or consequential damages in connection with or arising out of information
contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in
rendering legal, medical or any other professional services.
PHYSICS RESEARCH AND TECHNOLOGY
Additional books in this series can be found on Nova’s website
under the Series tab.
Additional e-books in this series can be found on Nova’s website
under the e-books tab.
MATHEMATICS RESEARCH DEVELOPMENTS
Additional books in this series can be found on Nova’s website
under the Series tab.
Additional e-books in this series can be found on Nova’s website
under the e-books tab.
PHYSICS RESEARCH AND TECHNOLOGY
G T
AUGE HEORIES
AND DIFFERENTIAL GEOMETRY
LANCE BAILEY
EDITOR
New York
Copyright © 2016 by Nova Science Publishers, Inc.
All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in
any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or
otherwise without the written permission of the Publisher.
We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse
content from this publication. Simply navigate to this publication’s page on Nova’s website and locate the
“Get Permission” button below the title description. This button is linked directly to the title’s permission
page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN.
For further questions about using the service on copyright.com, please contact:
Copyright Clearance Center
Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: [email protected].
NOTICE TO THE READER
The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied
warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for
incidental or consequential damages in connection with or arising out of information contained in this book.
The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or
in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government
reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of
such works.
Independent verification should be sought for any data, advice or recommendations contained in this book. In
addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property
arising from any methods, products, instructions, ideas or otherwise contained in this publication.
This publication is designed to provide accurate and authoritative information with regard to the subject
matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering
legal or any other professional services. If legal or any other expert assistance is required, the services of a
competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED
BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF
PUBLISHERS.
Additional color graphics may be available in the e-book version of this book.
Library of Congress Cataloging-in-Publication Data
Gauge theories and differential geometry / Lance Bailey, editor.
pages cm. -- (Physics research and technology) (Mathematics research developments)
Includes index.
ISBN:(cid:3)(cid:28)(cid:26)(cid:27)(cid:16)(cid:20)(cid:16)(cid:25)(cid:22)(cid:23)(cid:27)(cid:22)(cid:16)(cid:24)(cid:26)(cid:27)(cid:16)(cid:26) (eBook)
1. Gauge fields (Physics) 2. Geometry, Differential. I. Bailey, Lance, editor.
QC793.3.G38G36 2015
530.14'35--dc23
2015028225
Published by Nova Science Publishers, Inc. † New York
CONTENTS
Preface vii
Chapter 1 From Thermodynamics to Gauge Theory: 1
The Virial Theorem Revisited
J.-F. Pommaret
Chapter 2 Gravitational Gauge Theory of Spinorial and Vectorial Gravitating 45
Matter Fields
Jian Qi Shen
Chapter 3 Gravitational Gauge Theory as a Route to Gravity-Gauge 97
Unification
Jian Qi Shen
Chapter 4 Poincaré Gauge Theory of Gravity, Gravitational Interaction 179
and Regular Accelerating Universe
A. V. Minkevich
Chapter 5 Cutoff Regularization Method in Gauge Theories 199
G. Cynolter and E. Lendvai
Chapter 6 An Approach to Fractional Differential Geometry 219
and Fractal Space-Time via Fractional Calculus
for Non-Differentiable Functions
Guy Jumarie
Chapter 7 A Note on the Infinitesimal Baker-Campbell-Hausdorff Formula 271
Hirokazu Nishimura and Hirowaki Takamiya
Chapter 8 On the Curve Diffusion Flow: Invariant Functionals 297
and Gauge Transformations
Glen Wheeler
Index 303
PREFACE
This book revisits the mathematical foundations of thermodynamics and gauge theory by
using new differential geometric methods coming from the formal theory of systems of partial
differential equations and Lie pseudogroups. The gauge theory of gravity is also established,
in which spinorial and ventorial matter fields serve as gravitating sources. The potential
applications of the present gauge theory of gravity, including quantum-vacuum-energy
gravity, cosmological constant problem and gravity-gauge unification is also addressed. The
third chapter focuses on a gravitational gauge theory with spin connection and vierbein as
fundamental variables of gravity. Next, the place and physical significance of Poincaré gauge
theory of gravity (PGTG) in the framework of gauge approach to gravitation is discussed. A
cutoff regularization method in gauge theory is discussed in Chapter Five. The remaining
chapters in the book focus on differential geometry, in particular, the authors show how
fractional differential derived from fractional difference provides a basis to expand a theory
of fractional differential geometry which would apply to non-differentiable manifolds; a
review of the infinitesimal Baker-Campbell-Hausdorff formula is provided and the book
concludes with a short communication where the authors focus on local stability, and describe
how this leads naturally into the question of finite-time singularities and generalized soliton
solutions. In 1870, R. Clausius found the virial theorem which amounts to introduce the trace
of the stress tensor when studying the foundations of thermodynamics, as a way to relate the
absolute temperature of an ideal gas to the mean kinetic energy of its molecules.
In 1901, H. Poincaré introduced a duality principle in analytical mechanics in order to
study lagrangians invariant under the action of a Lie group of transformations. In 1909, the
brothers E. and F. Cosserat discovered another approach for studying the same problem
though using quite different equations. In 1916, H. Weyl considered again the same problem
for the conformal group of transformations, obtaining at the same time the Maxwell equations
and an additional specific equation also involving the trace of the impulsion-energy tensor.
Finally, having in mind the space-time formulation of electromagnetism and the Maurer-
Cartan equations for Lie groups, gauge theory has been created by C.N. Yang and R.L. Mills
in 1954 as a way to introduce in physics the differential geometric methods available at that
time, independently of any group action, contrary to all the previous approaches.
The main purpose of Chapter 1 is to revisit the mathematical foundations of
thermodynamics and gauge theory by using new differential geometric methods coming from
the formal theory of systems of partial differential equations and Lie pseudogroups, mostly
developped by D.C Spencer and coworkers around 1970. In particular, the author justifies and
viii Lance Bailey
extends the virial theorem, showing that the Clausius/Cosserat/Maxwell/Weyl equations are
nothing else but the formal adjoint of the Spencer operator appearing in the canonical
Spencer sequence for the conformal group of space-time and are thus totally dependent on the
group action. The duality principle also appeals to the formal adjoint of a linear differential
operator used in differential geometry and to the extension modules used in homological
algebra.
A gravitational gauge theory with spin connection (Lorentz-rotational gauge potential)
and vierbein (spacetime-translational gauge potential) as fundamental dynamical variables of
gravity is suggested in Chapter 2. In this theory, heavy field coupling, i.e., each gravitating
matter field is accompanied by a Planck-mass heavy partner, is introduced in order to account
for the dimensionful gravitational constant. The Einstein field equation appears as a first-
integral solution to the low-energy spin-connection gauge field equation of Yang-Mills type.
The most intriguing characteristics of the present scheme include: i) The gravitational
constant originates from the low-energy propagator of the Planck-mass heavy field that
mediates gravity between matter fields and spin-connection gauge field; ii) The large
cosmological constant resulting from quantum vacuum energy actually makes no
gravitational contribution since the spin-connection gauge field equation is a third-order
differential equation of metric, and an integral constant of its first-integral solution serves as
an effective cosmological constant that would cause the cosmic accelerated expansion; iii)
The rotational and translational gauge symmetries are unified in a five-dimensional de Sitter
spacetime, and a unified gravitational Lagrangian for rotational and translational gauge fields
will be constructed; iv) The prescription of gravitational gauge unification of fundamental
interactions can be suggested within the framework of the present gauge theory of gravity, in
which a higher-dimensional spin connection can serve as a Yang-Mills gauge potential, and a
higher-dimensional (spin-connection) curvature tensor can act as a Yang-Mills gauge field
strength (gauge field tensor). In order to realize gravity-gauge unification, the ordinary real-
manifold spacetime needs to be generalized to a complex-manifold spacetime, and therefore
the higher-dimensional spin current density of gravitating matter fields is a Yang-Mills charge
current density, and the higher-dimensional Lorentz rotational group (in higher-dimensional
complex-manifold internal space) is a Yang-Mills gauge symmetry group. The Dirac spinor
field theory and general/special theories of relativity (e.g., complex Lorentz group and
relevant topics) will be established in the complex-manifold spacetime. The present gauge
theory of gravity has some physical significance, e.g., the mechanism, in which the divergent
quantum vacuum energy term has been covariantly differentiated in the spin-connection
gravitational gauge field equation and an integral-constant cosmological constant term
emerges, provides a new insight into the cosmological constant problem. Since the present
framework of gravitational gauge theory within a higher-dimensional complex manifold
geometry can lead to a new route to gravity-gauge unification, the relevant topics such as
complex Lorentz groups in vector and spinor representations, Yang-Mills gauge symmetry
groups acting as higher-dimensional complex Lorentz groups, Yang-Mills action emerging
from the Lorentz-rotational symmetric gravitational action in complex manifolds have been
suggested and addressed in this chapter.
As explained in Chapter 3, in an attempt to explore the gauge theories of gravity
(particularly the topic of gravity-gauge unification), the author believes that the Einstein
gravity must be reformulated as a gauge theory of gravity with a new fundamental dynamical
variable (e.g., spin connection, the gauge potential related to local Lorentz-rotational gauge
Preface ix
symmetry). Though there have been a number of theories of gravity, where the spin
connection acts as one of the dynamical variables, yet the spin connection is merely a
supporting role, i.e., to obtain the gravitational field equation of Einstein, the most essential
dynamical variable is still the metric for functionally varying the gravitational Lagrangian
(the leading term is the Einstein-Hilbert Lagrangian). The author argues that the Einstein field
equation of gravity should be derived from a spin-connection gauge field theory (by varying a
curvature-squared Lagrangian with the spin connection). In such a theory, where the spin
connection (Lorentz-rotational gauge potential) serves as a dynamical variable, a Planck-mass
heavy coupling field (gravity-mediating field), which can mediate the interaction between
matter fields and spin-connection gauge field, can lead to dimensionless gravitational
coupling, and at lower energies, the lower-energy propagator of such a heavy field can
contribute the dimensionful Newtonian gravitational constant to the spin-connection
gravitational gauge field equation. The low-energy effective interaction Lagrangian density of
the matter field is obtained based on the functional integral approach, in which the Planck-
mass heavy field has been integrated out in the generating functional. The author will
establish the gauge theory of gravity, in which spinorial and vectorial matter fields serve as
gravitating sources. The low-energy effective interaction Lagrangian of vectorial and
spinorial gravitating matter fields will be constructed based on the scenario of heavy field
coupling for both Maxwell vector field and Dirac spinor field (as gravitating matter sources).
The functional variation of the effective Lagrangian of the spinning matter fields with respect
to the spin connection will lead to a spin-connection gravitational gauge field equation (a
third-order differential equation of metric). The potential applications of the present gauge
theory of gravity, including quantum-vacuum-energy gravity, cosmological constant problem
and gravity-gauge unification, will also be addressed.
The place and physical significance of Poincaré gauge theory of gravity (PGTG) in the
framework of gauge approach to gravitation is discussed in Chapter 4. Isotropic cosmology
built on the base of PGTG with general expression of gravitational Lagrangian with indefinite
parameters is considered. The most important physical consequences connected with the
change of gravitational interaction, with possible existence of limiting energy density and
gravitational repulsion at extreme conditions, and also with vacuum repulsion effect are
discussed. Regular inflationary Big Bang scenario with accelerating cosmological expansion
at present epoch proposed in the frame of PGTG is considered.
In quantum field theories divergences are inevitably turn up in loop calculations.
Renormalization is a part of the theory which can be performed only with a proper
regularization. In low energy effective theories there is a natural cutoff with well defined
physical meaning, but the naive cutoff regularization is unsatisfactory. In Chapter 5 a Lorentz
and gauge symmetry preserving regularization method is proposed in four dimension based
on momentum cutoff. First the authors give an overview of various regularization methods
then the new regularization is introduced. The author use the conditions of gauge invariance
or equivalently the freedom of shift of the loop momentum to define the evaluation of the
terms carrying even number of Lorentz indices, e.g., proportional to k k . The remaining
scalar integrals are calculated with a four dimensional momentum cutoff. The finite terms
(independent of the cutoff) are free of ambiguities coming from subtractions in non-trivial
cases. Finite parts of the result are equal with the results of dimensional regularization. The
proposed method can be applied to various physical processes where the use of dimensional
