Modern Functional Quantum Field Theory: Summing Feynman Graphs

281 Pages·2013·3.83 MB·English

by  Herbert M. Fried

Modern Functional
Quantum Field Theory
Summing Feynman Graphs
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Modern Functional
Quantum Field Theory
Summing Feynman Graphs
QED
QCD
QTD
Herbert M. Fried
Brown University, USA
World Scientific
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MODERN FUNCTIONAL QUANTUM FIELD THEORY
Summing Feynman Graphs
Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd.
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ISBN 978-981-4415-87-3
In-house Editor: Rhaimie Wahap
Printed in Singapore
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This book is dedicated to the memory of two extraordinary theoretical physicists,
both for personal reasons but also in the literal sense that it could not have been
written without their prior efforts: Julian Schwinger and Efimov Fradkin.
v
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Preface
This is the author’s fourth book, in a series [Fried (1972)]; [Fried (1990)];
[Fried (2002)] which has stretched over four decades, during which the
applicability of Schwinger’s format, functional solutions for the Generating
Functional of a relativistic Quantum Field Theory have expanded from
the perturbative estimates of QED radiative corrections, given by the first
few orders of perturbation theory, to the non-perturbative inclusion of all
relevant radiative corrections in the strong-coupling sectors of QCD.
During this period, many “partial models” have been invented, models
which sum up a certain class of Feynman graphs to produce a result con-
tainingallpowersofthecoupling,butwhichleaveuntouchedhugenumbers
ofothergraphscontributingtothesameresult;andtherefore,such”partial
models” can only be convincing in certain limiting situations. An imme-
diate example are the eikonal “multiperipheral” or “tower-graph” models
of particle scattering, which in the early ’70s produced the first theoretical
examples of total cross sections that satisfy the unitary requirement of the
Froissart bound. As energies continue to increase, however, other unitary
cancellations within the interacting structure of the omitted radiative cor-
rections come into play, and it may be argued that1 - within a given QFT
- the total cross sections must drop below their upper, Froissart bound.
How can a sum over ALL relevant Feynman graphs describing a spe-
cific physical process be achieved? The difficulty with summing subsets of
Feynman graphs is that for every increase of the order of the perturbation
expansion for the desired amplitude, a new class of graphs appears and as
the perturbation order increases, that new class contains an infinite num-
ber of graphs. Simply stated, summing Feynman graphs when all orders of
perturbation theory are needed is a mental and physical impossibility; and
1See,forexample,Chapter8of[Fried (2002)]
vii
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viii Modern Functional Quantum Field Theory
one must turn to a truly non-perturbative approach, one which is simple
touseandpowerfulenoughtocorrectlydescribethecomplexnatureofthe
amplitudes desired.
Happily, this approach is provided by employing (a simple variant of)
the Fradkin representations for the potential-theory Green’s functions and
related functional, which form a basic element of (a simple variant of) the
Schwinger functional solutions for every n-point function of any Quantum
Field Theory (QFT). These representations allow one to perform the sum
over all relevant Feynman graphs, and express the result in terms of a set
of functionalintegrals that describethe internal, space-timefluctuations of
those potential-theory Green’s functions, and which can be approximated
in a variety of ways, as described in the following Chapters.
For example: All previous eikonal models ever written appear trivially
in the context of a Bloch-Nordsieck approximation to the exact Fradkin
representation for a scattering particle.
For example: One can easily see the origins, and perform an approx-
imate extraction, and obtain the sum of all leading UV divergences (of
the inverse of the wave-function renormalization constant) of the dressed
photon propagator in four-dimensional “text-book” QED; and find, upon
summingtheeffectsofaninfinitesetofgraphsneverbeforecalculated,that
theresultappearstobefinite. Further,inthecontextofthis(rathergood)
approximation, one finds that the renormalized charge is within an order
of magnitude of 1/137; and that all charged fermions obeying pure QED
must have the same renormalized charge, independently of their mass.
For example: One can display a manifestly gauge — and Lorentz-
invariant summation over all gluons — including cubic and quartic gluon
interactions — exchanged between quarks and/or antiquarks for the cal-
culation of all QCD n-point functions, and find that the result has the
unexpected property of “Effective Locality” (EL). This property converts
amainfunctionalintegralintoasetofordinaryintegrals; andonefindsex-
plicitdemonstrationsofasymptoticfreedom,ananalyticpropertynolonger
dependent upon any perturbative approximation.
Further, one finds that EL requires a distinction to be made between
quanta of Abelian fields whose asymptotic positions or momenta can, in
principle, be measured, and those of quarks which are, asymptotically, al-
ways bound, and whose transverse coordinates are, in principle, not mea-
surable. Once this distinction between “Ideal” and “Realistic” QCD is
enforced, non-perturbative absurdities of the “Ideal” theory are removed,
and one can explicitly and easily obtain quark binding potentials. One
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Preface ix
small step past this realization leads, for probably the first time, to an
analytic method of obtaining effective scattering and binding potentials
between nucleons; and this, in turn, suggests a new way of understanding
how nucleons may bind together to form nuclei.
This methodology effectively opens a new door to an understand-
ing of the full, non-perturbative, non-linearities of QFT. It is the aim
and intention of this book to state in simple, functional language this
Schwinger/Fradkinapproach,whichhasleadtotheexamplesquotedabove
and described in subsequent Chapters; and to provide as much informa-
tion as is presently known about sensible approximations that have been
explored in this context.
For these purposes, the subject matter has been divided into four main
Parts, the first of which consists of a preliminary Functional description of
the techniques used; most of this preliminary material has appeared in the
author’s previous books, noted below, especially [Fried (1990)]. Parts II
and III are devoted to QED and QCD, respectively; portions of the first
Chapters of each have also been discussed, from a functional point of view,
in[Fried (1972)];and[Fried (2002)];thenewmaterialliesintheremaining
ChaptersofthoseParts. Itshouldbeemphasizedthatthesetechniquescan
beappliedtoanylocalQFT;and, indeed, toavarietyofcausal, non-linear
problems in Physics.2
Part IV of this book, entitled “Astrophysical Speculations”, is devel-
oped in terms of basic QFT principles, and represents the author’s sci-
entific musing on current questions of high astrophysical interest. Two
basic, field-theoretic suggestions are posed, the first as the adoption of a
new QED-based Model of Dark Energy; and the second as an obvious, if
somewhat unusual method of extending that Model to include Inflation,
which in passing provides a perfect candidate for Dark Matter and for the
understanding of galactic Gamma-Ray Bursts of GeV magnitude, and of
Ultra-High-EnergyCosmicRays. Onefurtherandseeminglyreasonableas-
sumption leads to a possible description of the Big Bang, and of the Birth
and Death of a Universe.
Ofcourse,atthistimethereisnoobservationalassuranceforthephysi-
calcorrectnessoftheseexplanations,norisanysuchclaimintended;rather
these last Chapters are meant to show how a first, and very simple field-
theoretic idea can lead to another, rather revolutionary idea, which is sud-
denly applicable to a wide variety of astrophysical observations. The pur-
2See, for example, Chapter 15 of [Fried (1990)], and Chapters 6, 7, and 9 of [Fried
(2002)]