Gauge Field Theories: Spin One and Spin Two: 100 Years After General Relativity

478 Pages·2016·32.95 MB·English

by  Gunter Scharf

GAUGE FIELD THEORIES: SPIN
ONE AND SPIN TWO
100 Years after General Relativity
Günter Scharf
Physics Institute, University of Zürich
DOVER PUBLICATIONS, INC.
Mineola, New York
Copyright
Copyright © 2016 by Günter Scharf All rights reserved.
Bibliographical Note
Gauge Field Theories: Spin One and Spin Two: 100 Years after General Relativity, first published by
Dover Publications, Inc., in 2016, is a revised edition of Quantum Gauge Theories: A True Ghost Story,
originally published in 2001 by John Wiley & Sons, Inc., New York.
International Standard Book Number ISBN-13: 978-0-486-80524-5
ISBN-10: 0-486-80524-7
Manufactured in the United States by RR Donnelley 80524701 2106
www.doverpublications.com
Preface to the Dover Edition
In 1916, Albert Einstein published his general theory of relativity in the
“Annalen der Physik 49.” A hundred years later, this milestone event in the
history of science will rightly be celebrated. However, the celebrations may be
disturbed by physicists deep under the mountains of southern China who are
trying to detect dark matter particles.1 This hypothetical matter is necessary for
the analysis of rotation curves in galaxies on the basis of standard general
relativity. The Milky Way, too, must be full of dark matter. So far, however, the
hunt for these new undiscovered particles has met with little success. If dark
matter cannot be found, then the standard theory of general relativity is in
serious trouble and the 100th anniversary celebrations are ruined. One should
frankly admit that general relativity is not established on the scale of galaxies.
Standard general relativity is a geometric theory. Einstein postulated a fusion
between geometry and gravity. In his theory, the metric tensor g describes both
μν
the gravitational field and the geometry of space and time. Here is the point
where we deviate. Following H. Poincaré (in Science and Hypothesis) we
consider geometry as a convention and the g (x) on the other hand describes the
μv
gravitational field only. The simplest geometry is the Minkowski space. So all
the fields in this book will be defined on Minkowski space, g (x), as well.
μv
Otherwise, Einstein’s equations remain unchanged. But now more solutions are
physically possible, and there exist solutions with general form of the
corresponding rotation curves. Consequently, in non-geometric general relativity
no dark matter is needed for galactic dynamics and the same is true for
cosmology. In this way the 2016 Einstein celebrations can be saved.
This is the third edition of my book Quantum Gauge Theories: A True Ghost
Story, which was published in 2001 by John Wiley & Sons, Inc. The title of the
second edition was changed to Quantum Gauge Theories— Spin 1 and 2. This
already indicates the increasing importance of gravity (spin 2) in the framework
of gauge theories. In this third edition, a further small variation of the title was
necessary. By choosing the title Gauge Field Theories instead of quantum gauge
theories, we take into consideration that gravity mainly acts as a classical field
(although we shall derive it as a quantum gauge field). So the last chapter “Non-
geometric General Relativity” is 100% classical field theory. In fact,
astrophysicists may read only this chapter to study the nonstandard approach to
gravity and cosmology.
One may ask why I did not write one book for spin 1 concerning the
microcosmos and a second one for spin 2 which is the macrocosmos. Maybe this
has to be done in the future; at present it is my main aim to treat the electroweak
and strong interactions and gravity on the same footing. In fact, it had quite often
been said that the unification of gravity with quantum theory is the main open
problem in theoretical physics. Our basic principle to achieve this unification is
the appropriate formulation of quantum gauge invariance. In the case of the
massless spin 2 theory, this gauge-theoretical foundation leads directly to the
non-geometric interpretation of general relativity. So besides the dark matter
problem, there is a strong theoretical reason to favor this approach above the
standard geometrical theory. To see the microcosmos (i.e., particle physics) and
the Universe in the large be governed by the same basic principle gives great
intellectual satisfaction—or even more.
Zürich, April 2015
Günter Scharf

1
Another dark matter search is being performed by the XENON Collaboration under the Gran Sasso
mountain in Italy.
Preface (2011)
It has quite often been said that the unification of gravity with quantum theory is
the main open problem in theoretical physics. The subtitle “spin one and two”
means that the book is concerned with this problem, because spin-1 gauge
theories include the successful standard theory of electroweak and strong
interactions and spin-2 is gravitation. In fact we shall see that the notion of
quantum gauge theory gives the natural framework for both spin-1 (non-abelian)
gauge theories and gravity.
Quantum gauge theories are mostly treated with the functional method. For
writing down the basic functional integral a classical Lagrangian must be given.
This is a disadvantage because the correct choice of the classical Lagrangian
(including ghost, Higgs and auxiliary fields) requires great skill. The resulting
final theory has some artificial ad-hoc character so that it lacks strong evidence
of being truly fundamental. Indeed, it is a common belief today that quantum
gauge theory might only be the low energy limit of a more fundamental theory
like string theory for example. A strong support for this belief comes from the
necessity to include gravity.
In this monograph we use an alternative method which does not require any
classical Lagrangian. This gives us the chance to discover new physics, and we
shall see this in Chapter 5 when we consider massive gravity. In the spin-1 case
we shall recover the results of the standard theory in all details. This now gives
us stronger evidence that quantum gauge theories may be really fundamental. In
the spin-2 case with mass zero we recover the coupling given by general
relativity, of course, but we also find a very interesting modification of it when
we consider the massless limit of massive gravity.
The reader now certainly asks: What is the basis of this alternative theory if
we do not use a classical Lagrangian ? Answer: It is the proper definition of
gauge invariance for the S-matrix. The S-matrix is defined on the space of free
asymptotic fields which is the only thing that must be given. On these free
quantum fields one introduces a gauge structure which involves ghost fields and
further auxiliary fields if some gauge fields are massive. The gauge structure is
given by the gauge variations of the asymptotic fields. These gauge variations
are the quantum counterpart of classical gauge transformations and general
coordinate transformations in general relativity. There is a formal similarity with
the BRS transformation in the functional method. However, the BRS
transformation operates on the interacting fields, and these can only be
constructed if the coupling, i.e. the classical Lagrangian is given. We will not
introduce interacting fields in this monograph.
The gauge variation will be denoted by d where the suffix Q indicates that
Q
it is the (super)commutator with a nilpotent gauge charge Q (Q2 = 0). The gauge
charge Q serves for two purposes: (i) It defines the physical subspace of the big
Fock space generated by the asymptotic fields. (ii) d allows to calculate the
Q
gauge variation of the S-matrix. By S-matrix we always mean the scattering
operator S(g) where the coupling is smeared with a (Schwartz) test function g:
∫T(x)g(x)dx. This g(x) is a natural infrared regulator, so that S(g) also exists if
massless fields are present in the theory. The S-matrix is defined perturbatively
by means of the time-ordered products T (x ,…, x ). The use of perturbation
n 1 n
theory might look old-fashioned and restricts the possibility to calculate strong
coupling phenomena. But it is good enough to gain the theory from scratch and
to analyze its properties. Non-perturbative methods can be introduced at a later
stage.
The time-ordered products T are constructed recursively from T = T by the
n 1
causal method of Epstein and Glaser. In this way the nasty ultraviolet
divergences which appear if one uses Feynman rules are avoided. No cutoff and
no regularization is needed, everything is finite. This is of great advantage in the
gravitational case. Gauge invariance of the S-matrix is now expressed by a
condition on the time-ordered products. The basic condition called “causal gauge
invariance” reads
Note that the gauge variation of T does not vanish, but it must be a divergence;
Τμ is called Q-vertex in distinction to the ordinary vertex T. The solutions of
equation (1) form the so-called relative cohomology group of d . It is a crucial
Q
fact that this group contains very few non-trivial elements only. These non-
trivial couplings T give the physical theories. It is a nice feature of this approach
that T automatically contains all couplings of the quantum gauge theory
including ghost, Higgs and auxiliary field couplings.
After this overview the plan of the book is clear. In Chapter 1 we introduce
the free quantum fields and we prepare the ground for the gauge structure. In
Chapter 2 we describe the inductive construction of the time-ordered products T
n
starting from T = T by causal perturbation theory. The analysis of causal gauge
1
invariance (1) begins with Chapter 3 where massless spin-1 gauge fields are
considered. The solution is given by Yang-Mills theories (up to trivial
modifications). In Chapter 4 the method is applied to massive gauge fields.
Causal gauge invariance forces us to introduce un-physical and physical (Higgs)
scalar fields and determines their couplings. Spontaneous symmetry breaking
and the Higgs mechanism are not needed.
The construction of spin-2 gauge theories starts in Chapter 5 with the mass
zero case. Needless to say that we use no input from general relativity. A
stronger formulation of causal gauge invariance (in the form of descent
equations) allows to derive the coupling T in an elegant way. The pure
gravitational terms in T and T agree precisely with the expansion of the
1 2
Einstein-Hilbert Lagrangian. We also consider the coupling to spin-1 gauge
fields. This is a mixed spin-1 and spin-2 gauge theory, and we obtain some
interesting new results in Section 5.11. There exists the technical problem of
non-renormalizability of gravity. This problem gets simplified by using the
cohomological definition of gauge invariance. But we do not discuss this issue
because the complete solution is still not known.
We treat the massive spin-2 case parallel to ordinary massless gravity. The
gauge structure, i.e. the nilpotency of d , forces us to introduce a vector field υμ
Q
with the same mass m as the graviton. To have a smooth limit for m → 0 the
physical modes of the massive graviton must be chosen as follows: two modes
agree with the two transversal polarizations of the massless graviton, the
remaining four modes are given by υμ. (The physical subspace contains one
mode more than a pure spin-2 field, 5+1=6.) It is a very interesting observation
that the υ-field does not decouple from the other degrees of freedom in the limit
m → 0. That means the massless limit of massive gravity does not agree with
massless gravity, because the υ-field survives.
Zürich, April 2011
Günter Scharf
Contents
1. Free fields
1.1 Bosonic scalar fields
1.2 Fermionic scalar (ghost) fields
1.3 Massless vector fields
1.4 Operator gauge transformations
1.5 Massive vector fields
1.6 Fermionic vector (ghost) fields
1.7 Tensor fields
1.8 Spinor fields
1.9 Normally ordered products in free fields
1.10 Problems
2. Causal perturbation theory
2.1 The S-matrix in quantum mechanics
2.2 The method of Epstein and Glaser
2.3 Splitting of causal distributions in x-space
2.4 Splitting in momentum space
2.5 Calculation of tree graphs
2.6 Calculation of loop graphs
2.7 Normalizability
2.8 Problems
3. Spin-1 gauge theories: massless gauge fields
3.1 Causal gauge invariance
3.2 Self-coupled gauge fields to first order
3.3 Divergence-and co-boundary-couplings
3.4 Yang-Mills theory to second order
3.5 Reductive Lie algebras
3.6 Coupling to matter fields