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QUANTUM FIELD THEORYa cyclist tour
Predrag Cvitanovi´c
What reviewers say:
N. Bohr:
“The most important work since that Schr¨ odinger killed thecat.”
···
R.P. Feynman: “Great doorstop!” Incomplete notes for aquantum ﬁeld theory course version 3.8 - Aug 29 2013
Contents
Things fall apart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Lattice ﬁeld theory 3
1.1 Wanderings of a drunken snail . . . . . . . . . . . . . . . . . . . 31.2 Lattice derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Periodic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Discrete Fourier transforms . . . . . . . . . . . . . . . . . . . . . 101.5 Continuum ﬁeld theory . . . . . . . . . . . . . . . . . . . . . . . 13
exercises 14
2 Path integral formulation of Quantum Mechanics 15
2.1 Quantum mechanics: a brief review . . . . . . . . . . . . . . . . 162.2 Matrix-valued functions . . . . . . . . . . . . . . . . . . . . . . . 182.3 Short time propagation . . . . . . . . . . . . . . . . . . . . . . . 202.4 Path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
exercises 22
3 Field theory path integrals 23
3.1 Field theory - setting up the notation . . . . . . . . . . . . . . . . 253.2 Saddle-point expansions . . . . . . . . . . . . . . . . . . . . . . 263.3 Saddle-point expansions are asymptotic . . . . . . . . . . . . . . 293.4 Free propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Free ﬁeld theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.6 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 32
commentary 33 exercises 34
4 WKB quantization 35
4.1 WKB ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Method of stationary phase . . . . . . . . . . . . . . . . . . . . . 374.3 WKB quantization . . . . . . . . . . . . . . . . . . . . . . . . . 384.4 Beyond the quadratic saddle point . . . . . . . . . . . . . . . . . 40
r´esum´e 41 commentary 42 exercises 43 references 43
A Group theory 45
A.1 Invariants and reducibility . . . . . . . . . . . . . . . . . . . . . 45
Index 48
1
2
CONTENTS
Preface
Ben Mottelson is the only physicist I personally know who thinks equally clearlyquantum-mechanically and classically, the rest of us are not so lucky. Still, Ihave never understood why my colleagues say that “while we understand classicalmechanics,” quantum mechanics is mysterious. I never got the memo: to me it isequally magical that both Newtonian and quantum mechanics follow variationalprinciples.On the other hand, almost every single thing we learn about quantum me-chanics and thus come to believe is quantum mechanics –operators, commuta-tors, complex amplitudes, unitary evolution operators, Green’s functions, Hilbertspaces, spectra, path integrals, spins, angular momenta– under a closer inspectionhas nothing speciﬁcally quantum mechanical to it. It is machinery equally suitedto classical, statistical and stochastic mechanics, which in
ChaosBook.org
arethought of together - in terms of evolution operators and their spectra. The com-mon theme of the three theories is that things fall apart, and inﬁnitely many frag-ments have to be pieced together to craft a theory. In the end it is only the
i
/
granularity of phase space that is the mystery of quantum mechanics; and why, acentury later, quantum mechanics is still a theory that refuses to fail?Over the years I have watched in amazement study group after study group of graduate students grovel in orgies of spacetime, spin and color indices, and havetried desperately todeprogram themthrough my
ChaosBook.org/FieldTheory
book [2], but all in vain: students
want
Quantum Field Theory to be mysteriousand accessed only by pages of index summations. Or two-forms. These notes areyet another attempt to demystify most of ﬁeld theory, inspired by young Feynmandriving yet younger Dyson across the continent to Los Alamos, hands o
ﬀ
thesteering wheel and gesticulating: “Path integrals are everything!” These lecturesare about “everything.” The theory is developed here at not quite the pedestrianlevel, perhaps a cyclist level. We start out on a ﬁnite lattice, without any functionalvoodoo; all we have to know is how to manipulate ﬁnite dimensional vectors andmatrices. Then we restart on a more familiar ground, by reformulating the oldfashioned Schr¨odinger quantum mechanics as Feynman path integral in chapter2.
More of such stu
ﬀ
can be found in ref. [2].
This version of ﬁeld theory presupposes prior exposure to the Ising model andthe Landau mean ﬁeld theory of critical phenomena on the level of ref. [1], or any
other decent introduction to critical phenomena.
Acknowledgments.
These notes owe its existence to the 1980’s Niels BohrInstitute’s and Nordita’s hospitable and nurturing environment, and the private,national and cross-national foundations that have supported this research over aspan of several decades. I am indebted to Benny Lautrup both for my ﬁrst intro-duction to lattice ﬁeld theory, and for the sect. 1.3 interpretation of the Fouriertransform as the spectrum of the stepping operator. And last but not least– pro-found thanks to all the unsung heroes–students and colleagues, too numerous tolist here–who have supported this project over many years in many ways, by sur-viving courses based on these notes, by providing invaluable insights, by teachingus, by inspiring us. I thank the Carlsberg Foundation and Glen P. Robinson forpartial support, and Dorte Glass, Tzatzilha Torres Guadarrama and Raenell Sollerfor typing parts of the manuscript.
CONTENTS
3Who is the 3-legged dog reappearing throughout the book? Long ago, whenI was innocent and knew not Borel measurable
α
to
Ω
sets, I asked V. Baladi aquestion about dynamical zeta functions, who then asked J.-P.Eckmann, who thenasked D. Ruelle. The answer was transmitted back: “The master says: ‘It is holo-morphic in a strip’.” Hence His Master’s Voice (H.M.V.) logo, and the 3-leggeddog is us, still eager to fetch the bone, or at least a missing ﬁgure, if a reader iskind enough to draw one for us. What is depicted on the cover? Roberto Ar-tuso found the smørrebrød at the Niels Bohr Institute indigestible, so he digestedH.M.V.’s wisdom on a strict diet of two Carlsbergs and two pieces of Danish pas-try for lunch every day. Frequent trips down to Milano’s ancestral home is whatkept him alive.
version 3.8 - Aug 29 2013 ackn.tex 9dec2013

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