mathematically rigorous quantum field theories with a...
Post on 13-Apr-2018
224 Views
Preview:
TRANSCRIPT
Mathematically Rigorous Quantum Field
Theories with a Nonlinear Normal
Ordering of the Hamiltonian Operator
A DissertationPresented to the Faculty of the Graduate School
ofYale University
in Candidacy for the Degree ofDoctor of Philosophy
byRachel Lash Maitra
Dissertation Director: Dr. Vincent Moncrief
December 2007
Abstract
Mathematically Rigorous Quantum Field Theories
with a Nonlinear Normal Ordering of the
Hamiltonian Operator
Rachel Lash Maitra
2007
The ongoing quantization of the four fundamental forces of nature represents one
of the most fruitful grounds for cross-pollination between physics and mathematics.
While remaining vastly open, substantial progress has been made in the last decades:
the expression of all basic physical theories in terms of geometry, specifically as
gauge theories. This is accomplished by the recognition of the strong, weak, and
electromagnetic fields as Yang-Mills (gauge) fields, and by the re-writing of general
relativity in terms of gauge connection variables.
The method of canonical quantization offers several advantages in treating gauge
theories: the gauge fields themselves are the basic variables, while gauge constraints
promote to quantum operators whose commutation relations reflect the classical
Poisson brackets.
In this thesis I construct a zero-energy ground state for canonically quantized
Yang-Mills theory, for a particular (“nonlinear normal”) factor ordering of the Hamil-
tonian operator. The inspiration for this project is to find an alternative to the
Chern-Simons and Kodama states. These are closely related ground state solutions
for (respectively) quantum Yang-Mills theory and quantum gravity with a positive
cosmological constant. Objections to the Chern-Simons and Kodama states come
from, among other arguments, their apparent lack of well-defined decay “at infinity.”
The ground state I have constructed, as the exponentiation of a strictly non-positive
functional, manifestly enjoys good decay properties. In addition, I have constructed
a similar ground state for scalar ϕ4 theory. The construction of these ground states
represents a generalization to quantum field theories of work done by my thesis advi-
sor V. Moncrief, in collaboration with M. Ryan, for quantum mechanical situations.
Gauge, rotation, and translation invariance are directly verifiable for the nonlin-
ear normal ordered Yang-Mills ground state; invariance under boosts remains as a
question for future work. The analogous state for the abelian case (free Maxwell
theory) enjoys full Poincare invariance.
ii
Contents
1 Introduction 1
1.1 Quantized gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Chern-Simons and Kodama states . . . . . . . . . . . . . . . . . 3
1.3 Nonlinear normal ordering . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Scope of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Linear field theories 12
2.1 Free massive scalar field theory . . . . . . . . . . . . . . . . . . . . . 12
2.2 Gaussian measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Free Maxwell theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Scalar ϕ4 theory 23
3.1 Direct method: General results and techniques . . . . . . . . . . . . 24
3.2 Application to ϕ4 theory . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Yang-Mills theory 32
4.1 Mathematical formalism . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Canonical quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Preliminaries for minimizing procedure . . . . . . . . . . . . . . . . . 38
ii
4.4 Solving the Yang-Mills Dirichlet problem . . . . . . . . . . . . . . . . 40
4.5 The Euclidean Yang-Mills Hamilton-Jacobi functional . . . . . . . . . 49
4.6 Gauge and Poincare invariance . . . . . . . . . . . . . . . . . . . . . 53
5 Future work 56
5.1 Yang-Mills-Higgs theory . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 Scalar ϕ4 non-Gaussian measure . . . . . . . . . . . . . . . . . . . . . 60
Appendix 62
Index of Notation 69
iii
Acknowledgments
My first thanks go to my advisor, Vincent Moncrief, for his generosity, encourage-
ment, and support throughout my graduate studies. He always made time to meet
with me, as long as we were in the same country, and when not, he never failed to
answer my many emailed questions with prompt and detailed replies. From these
meetings (or emails) I always came away inspired by new ideas, not only restricted to
my thesis project but extending to the larger concepts underpinning mathematics,
physics, and the relation of these two fields. I hope the completion of my thesis
under Vince marks only the beginning of a much longer collaboration on the many
avenues of inquiry left open by this project.
My gratitude extends to the whole Yale mathematics department, in whose lounge
I spent five happy years working, chatting, and eating cookies. Among our faculty, I
would especially like to thank Gregg Zuckerman, Howard Garland, Andrew Casson,
Roger Howe, and Igor Frenkel, for useful conversations, interesting classes, and good
advice. As for my fellow students, everyone deserves thanks for helping to create such
a friendly and collegial atmosphere, but I owe a special debt to Masood Aryapoor,
Ignacio Uriarte-Tuero, and Luke Rogers, for teaching me all the analysis I know; to
Helen Wong, Jon Hibbard, Arthur Szlam, and Raanan Schul, for their friendship both
mathematical and personal, and especially for their support during my early years of
qualifying exams; and to Josh Sussan and Manish Patnaik, for so many great games
iv
of Taboo. I would like to thank all of our wonderful staff, in particular Bernadette
Alston-Facey, for being omniscient, omnipotent, and beneficent; Karen Fitzgerald
and Mel del Vecchio, for helping me with everything from faxing my applications
to re-gluing my broken flip-flop; Mary Belton, Donna Younger, and Jo-Ann Ahearn,
for assistance on innumerable occasions with mailing and copying; Ella Sandor, for
sparing me any worry about financial arrangements; and Paul Lukasiewicz, for not
only knowing the appearance and whereabouts of every book in the library, but for
recommending to me many useful volumes as well.
During my final months of research and writing, I am very grateful to have been
generously hosted by the University of Amsterdam’s Korteweg-de Vries Institute for
Mathematics. Special thanks go to Eric Opdam, Jan Wiegerinck, Thomas Quella,
and Robbert Dijkgraaf, for kind hospitality, valuable discussions, and much help in
navigating the NWO. Thanks to Evelien Wallet for efficiently managing all official
matters.
During my graduate studies I have been fortunate to visit the Perimeter Institute
on several occasions, and wish to thank everyone at this wonderful place for hospi-
tality. In particular, many thanks to Lee Smolin, for hosting me, for guiding me as
I learned the Ashtekar/loop variables approach to quantum gravity, and for sharing
insights which helped inspire this project. To John Baez at University of California
Riverside, I am grateful for email correspondence which also inspired the direction
of my thesis research, and greatly contributed to my understanding of field quanti-
zation. Additional thanks are due to John Baez as well as to Gregg Zuckerman, for
acting as readers of this thesis. To Hans Lindblad at UC San Diego, I am grateful
for providing me his notes on the Euclidean Dirichlet problem for scalar ϕ4 theory.
Over the years many wonderful teachers have shared their knowledge and ideas
with me. My first teachers, and still my most admired and beloved mentors, are my
v
own parents Martha and Barry Lash, who educated me at home. As well as guiding
my early studies, they gave me the priceless gift of helping me learn how to think
for myself. They gave unstintingly of their time and energy to discuss and explore
ideas with me, and to this day they make time for my cell-phone calls, even if they
are in the mouthwash aisle in Wegmans when the phone rings.
For helping me first discover the beauties of mathematics and physics, I am
grateful to my teachers Dean Hoover, Paul Manikowski, Elaine Nye, Henry Nebel,
John Stull, and Dave Toot. Ten or fifteen years later, I still can recall moments of
understanding each helped me to achieve.
I am very lucky to have first learned differential geometry and general relativity as
an undergraduate at Alfred University from Stuart Boersma. Since the curriculum
had no formal course in these subjects, he taught them to me as independent studies.
I wish to thank him not only for his lucid explanations of tensors and abstract
indices, but for his insights into the meaning of general relativity and what it says
about space, time, and matter. I remember in particular his explanation to me of
Einstein’s equation: matter equals curvature; physics equals mathematics.
In my undergraduate advisor, Rob Williams, I found a generous mentor from
whom I learned uncountably many valuable pieces of mathematics, from the meaning
of De Moivre’s Theorem, to the methods of clear and succinct proof-writing. With
Debra Waugh at Alfred University and Juha Pohjanpelto at Oregon State University,
he guided me through the completion of my senior thesis project on topology and
knot theory of surfaces. Also while at Alfred, I am grateful to have learned much
topology and analysis from Roger Douglass. For many wonderful discussions of
philosophy and writing, I wish to thank Emrys and Vicky Westacott. From The
Pennsylvania State University’s Mathematics Advanced Study Semester Program,
thanks to Svetlana Katok, Adrian Ocneanu, Victor Nistor, and Sergei Tabachnikov,
vi
for a semester of fascinating mathematics as well as much good advice.
For making the non-academic side of my life such fun, I am indebted to many
friends far and near. Thank you to Rebecca Sawyer, Melody Lo, and LT Palmer,
for all the laughter, cooking sprees, and late-night conversations in Rosecliff and
Balmoral; to Nawreen Sattar, Sharon Ma, and Kee Chan, for Sunday Thai brunches,
Boggle, kettle corn, and cherry blossoms; and to Oana Catu, for gelato, movie nights,
and Romanian poetry. Thank you to Adam Poswolsky, for solidarity as a fellow Yalie
transplant to Northern Europe; and to Rashad Ullah, for afternoon teas and Bengali
conversation practice, as well as special thanks for printing out this thesis for me.
Finally, thank you to Missy Pritchard and Leah Sarat, for keeping in touch through
ten years all around the globe; and to my own sister Hannah Lash, my oldest and
dearest friend.
Along with my parents and sister, my heartfelt thanks and love go to my brother
Rob Lash, sister-in-law Amy Rees, and niece and nephew Michelle and John Sawyer.
My large and warm extended family are too numerous to be named individually, but
each one has my gratitude and love.
I am doubly lucky to belong to two wonderful families, once by birth and once
by marriage. I would like to thank the Maitras in Seattle and in Kolkata for the
warm and loving welcome they have shown me — pronam ar amar onek bhalobasa
neben. This brings me to my final and very great thanks, to my husband and friend,
Dipankar, for unwavering support and encouragement, for keeping me sane, and best
of all, for making life a joy.
Many more people have contributed to this thesis than I can acknowledge. Not all
contributions are neatly packaged for inclusion under this heading, but my gratitude
to those who have shared with me advice, friendship, and kindness is undiminished,
although not formally expressed. Any errors in this work are mine, and mine alone.
vii
Chapter 1
Introduction
1.1 Quantized gauge theories
The framing of our basic physical theories in terms of geometry is surely one of the
greatest advances of mathematical physics during the 20th century. Independently
of the concurrently developing mathematics of fiber bundle theory, Yang and Mills
developed a nonabelian gauge theory of physical fields, successfully describing the
strong and weak fundamental forces of nature by generalizing Maxwell’s theory of
electromagnetic fields. Subsequently, the remarkable convergence of ideas from the
physical theory of nonabelian gauge fields with the mathematics of connections on
principal bundles became a well-spring of new ideas and developments for both. In
its original Einsteinian form, general relativity encodes the gravitational field in a
space-time metric rather than a gauge connection, but over the last decades Sen,
Ashtekar, and Barbero have recast general relativity as a gauge theory ([2], and
references therein), adding gravity to complete the list of fundamental forces thus
describable. Gauge theories such as Yang-Mills theory and the theory of general
relativity identify fundamental physical fields with sections of principal bundles, and
1
describe physical interactions as dictated by the action of a structure group. These
theories represent a true reduction of natural phenomena to more basic geometrical
principles. In the words of J.A. Wheeler, we can begin to explain mass without
mass, charge without charge, and field without field.
Hand in hand with the conceptual simplification comes much greater challenge
in quantizing these theories. To borrow terminology from Ashtekar et al. [3], Yang-
Mills theory and general relativity exhibit two types of nonlinearity, dynamical and
kinematical. Dynamically, the field equations are nonlinear; kinematically, the space
of physically allowable connections is itself not a linear space, since it is composed
of connections modulo gauge transformations (A/G). Because of these subtleties as
well as the inherently geometrical character of the basic quantities, a mathematically
rigorous approach is fundamental to the development of a sound theory of quantized
gauge fields. Conversely, the search for rigorous quantizations of gauge theories
continues to generate new mathematics of independent interest, as in the search for
a well-defined measure on the space A/G ([3], [6]), and the correspondence of such
generalized measures on A/G with link invariants ([4], and references therein).
In quantizing gauge theories, there are several reasons to adopt a canonical ap-
proach. The gauge fields themselves can be taken as configuration variables, pro-
moting gauge constraints to quantum operators whose commutation relations should
reflect the classical Poisson brackets. Given a normalizable ground state wave func-
tional Ω(ϕ) well-behaved within the canonical approach, one hopes to define from it
a (non-)Gaussian measure heuristically given as dµ(ϕ) = Ω2(ϕ)dϕ, and from this in
turn a candidate state space L2(Q, dµ) (where Q denotes the space of field configu-
rations).
Consistency, Poincare/diffeomorphism invariance, and normalizability dictate the
following requirements for a good operator ordering and ground state wave functional
2
in a quantized gauge theory:
• Commutators of quantum constraint operators should be equal to Poisson
brackets of classical constraints, promoted to quantum operators using the
operator ordering.
• Quantum Hamiltonian operator should admit a zero-energy ground state (to
ensure time translation invariance).
• Ground state should be invariant under quantum constraints generating Poincare
transformations (in the case of quantum general relativity, constraints gener-
ating spacetime diffeomorphisms).
• Ground state should be normalizable with respect to some measure on config-
uration space.
1.2 The Chern-Simons and Kodama states
General relativity’s description as a gauge theory bypasses many of the operator
ordering problems which hindered its quantization as a theory of metrics. Moreover
the Ashtekar variables make available the use of results and techniques from Yang-
Mills theory, a notable instance being the construction of the Kodama state. At
present the only known candidate ground state for background-independent quantum
gravity, it is analogous to the Chern-Simons state in Yang-Mills theory ([19], [20]),
the exponential of the famous Chern-Simons functional
SCS(A) =
∫
Σ
Tr
(A ∧ dA− 2
3A ∧ A ∧ A
)dx, (1.1)
where Σ denotes a spacelike slice of the spacetime manifold M ∼= R× Σ.
3
The Kodama state exhibits many positive features, such as a semiclassical cor-
respondence to de Sitter space-time ([19], [20], [34]) and correspondence under the
loop transform to the Kauffman bracket link invariant ([4], [43]), as well as having
excitations resembling graviton states [34]. However the usual construction of the
Kodama state necessitates complexifying the tangent bundle, a rectifiable but unde-
sirable recourse. More seriously, as discussed in [4] and [42] both the Kodama and
Chern-Simons states lack CPT invariance, and are nonnormalizable due to the fact
that the Chern-Simons functional (1.1) is indefinite: under space inversion x → −x,
the sign of SCS(A) is reversed. In [42], Witten analyzes the Chern-Simons and
Kodama states by constructing analogous nonnormalizable ground states in linear
theories for which the correct normalizable ground state is known. States consisting
of a Chern-Simons-like indefinite Gaussian are easily constructible for the simple har-
monic oscillator as well as the free Maxwell theory of electromagnetism (an abelian
(U(1)) gauge theory).
The most important point which emerges from Witten’s analysis is the relative
ease of identifying a normalizable ground state for a linear theory, using methods
which immediately fail at the introduction of any nonlinearities as in nonabelian
gauge theory. In a linear theory, the classical Hamiltonian takes the form
H =1
2
(|p|2 + 〈x,Mx〉
), (1.2)
where x and p are the canonical position and momentum variables and M is a
positive self-adjoint operator on configuration space. Thus M has a unique positive
(self-adjoint) square root T , yielding a ground state for the quantum Hamiltonian
Ω(x) = N exp
(−〈x,Tx〉
2
)(1.3)
4
(N a normalization constant) with energy
Tr T
2.
Normalization of this ground state is made possible precisely by the positivity of T ,
which ensures rapid decay of the ground state for x sufficiently large. However for
a nonlinear theory, such a factorization clearly breaks down, leaving us without a
simple recipe for constructing a normalizable ground state. At this level, kinematical
nonlinearity is not even yet relevant; the difficulties in finding a positive functional
to exponentiate are caused solely by dynamical nonlinearity. This, then, must be
the first matter to address, in a manner which extends as smoothly as possible the
techniques successful in linear cases. We will find that the approach we develop in
fact lends itself especially well to gauge theories.
1.3 Nonlinear normal ordering
For nonlinear quantum mechanical situations, Moncrief [25] and Ryan [30] present
a “normal” ordering scheme for the Hamiltonian operator, yielding a well-behaved
associated ground state. Consider a nonlinear quantum mechanical Hamiltonian of
the form
H =1
2|p|2 + V (x),
where the generic function V (x) ≥ 0 has replaced the quadratic form 〈x,Mx〉 in
the Hamiltonian (1.2). The idea is to factorize V (x) by solving the imaginary-time
zero-energy Hamilton-Jacobi equation (iHJE)
1
2
∑
i
(∂S
∂xi
)2− V (x) = 0.
5
We can then order the quantum Hamiltonian operator as
H =1
2
∑
i
(∂S
∂xi− ipi
)(∂S
∂xi+ ipi
),
admitting the zero-energy ground state
N exp (−S(x))
under usual assignment of canonical quantum operators xi : ψ(x) → xiψ(x), pi :
ψ(x) → −i ∂∂xi
ψ(x).
This factorization can be illustrated with the anharmonic oscillator Hamiltonian
H =1
2p2 +
1
2x2 +
1
4λx4, (1.4)
in which case the Hamilton-Jacobi equation 12
(dSdx
)2= 1
2x2+ 1
4λx4 is easily integrated
to yield the solution S(x) = 23λ
(1 + λ
2x2
)3/2− 23λ. While the resulting ground state is
not the usual anharmonic oscillator ground state obtained from the factor ordering
H = 12m
p2+mω2
2x2+ 1
4λx4, it is the correct ground state for nonlinear normal ordering,
and moreover is a zero-energy ground state.
Finding a ground state with zero energy is not a priority for an ordinary quantum
mechanical system like the anharmonic oscillator, but in the realm of relativistic
field theories, a quantum ground state must have zero energy to exhibit Poincare
invariance. Moncrief and Ryan [26] show that for cosmologies such as vacuum
Bianchi IX having enough symmetry to reduce to finitely many degrees of freedom,
a nonlinear normal ordered ground state does exist for general relativity, suggesting
the possibility of extending this technique to deal with full quantum field theories,
in particular quantum gravity.
6
1.4 Scope of this thesis
In this thesis, the nonlinear normal ordering will be extended from quantum me-
chanics to several quantum field theories. The technique is adaptable to any field
theory of the form
L (ϕ, ϕ) =
∫ 1
2ϕ2 − V (ϕ,Dϕ)
dx
H (ϕ, π) =
∫ 1
2π2 + V (ϕ,Dϕ)
dx,
π ≡ ∂L∂ϕ
= ϕ,
where the field ϕ may be a scalar field, or may take values in more general vector
bundles (as in the case of Yang-Mills theory, Chapter 4).
The first step is to introduce the Schrödinger representation of a canonically
quantized field theory. Taking as a “position” variable the field configuration ϕ,
with “momentum” given by π = ϕ, these canonical variables can be promoted to
quantum operators as
ϕ(x) : Ψ(ϕ) → ϕ(x)Ψ(ϕ)
π(x) : Ψ(ϕ) → −iδ
δϕ(x)Ψ(ϕ),
acting on the wave functional Ψ(ϕ) defined on the space Q of field configurations.
To define the state space of physically meaningful wave functionals, we need some
measure on the space Q; formally a ground state wave functional Ω (ϕ) offers such a
measure of the form
dµ(ϕ) = Ω2(ϕ)dϕ.
7
If well-defined, this measure in turn yields a candidate for the Hilbert space of states:
L2(Q, dµ).
Using the nonlinear normal ordering
H =1
2
∫ (δS
δϕ− iπ
)(δS
δϕ+ iπ
)dx
of the Hamiltonian operator, we can find a ground state wave functional Ω(ϕ) =
N exp (−S(ϕ)) by solving the zero-energy imaginary time Hamilton-Jacobi equation
1
2
∫ ∣∣∣∣δS
δϕ
∣∣∣∣2
dx =
∫V (ϕ,Dϕ) dx (1.5)
for the functional S (ϕ) . While (1.5) is not as easily solved as in the anharmonic
oscillator problem, classical Hamilton-Jacobi theory fortunately provides us with a
means of constructing the solution, essentially as Hamilton’s principal function for
the imaginary-time problem. With the transformation to imaginary time t → it,
the chain rule yields ϕ → ϕ, ϕ → −iϕ, and π → −iπ, so that the imaginary-time
Lagrangian and Hamiltonian are
L =
∫L(ϕ, ϕ) dx =
∫−1
2ϕ2 − V (ϕ,Dϕ) dx
H =
∫H(ϕ, π) dx =
∫ (−1
2π2 + V (ϕ,Dϕ)
)dx.
The full Hamilton-Jacobi equation is
∂S
∂t+ H
(ϕ,
δS
δϕ, t
)= 0; (1.6)
8
a time-independent solution S(ϕ) to this equation will be the solution we seek for
(1.5). In fact, the solution we construct will be time-independent, but for the
moment we assume that an explicit time dependence is possible, using the definition
of functional derivatives to write
dS
dt=
∂S
∂t+
∫δS
δϕt
∂ϕt
∂tdx.
Subsituting from (1.6) we obtain
dS
dt= −H
(ϕt,
δS
δϕt
)+
∫δS
δϕt
∂ϕt
∂tdx
=
∫δS
δϕt
∂ϕt
∂t− H
(ϕt,
δS
δϕt
)dx.
For At the solution to
∂ϕt
∂t=
δH
δπ
∣∣∣∣∣π= δS
δϕt
with initial data At=0 = A, we get
∫δS
δϕt
∂ϕt
∂t− H
(At,
δS
δϕt
)dx =
∫πϕt − H (ϕt, πt) dx
= L (ϕt, Dϕt)
⇒ S(ϕt0
)− S (ϕ) =
∫ t0
0
L (ϕt, ϕt) dt.
Taking
S (ϕ) = −I (ϕt) = −∫ ∞
0
L (ϕt, ϕt) dt
clearly satisfies this relation, since we will then have S(ϕt0
)= −
∫∞t0
L (ϕt, ϕt) dt.
9
Denoting by R+ the set x ∈ R : x ≥ 0, we can alternatively write
S (ϕ) = −∫
R+×R3L (ϕt, ϕt) .
The exponential exp (−S(ϕ)) will peak about the field configuration ϕ = 0. To
prove that the functional S(ϕ) exists, we need only prove the existence of a solution
ϕt of the Euclidean Euler-Lagrange equations, with initial data ϕ. We can refer to
this problem as the Euclidean Dirichlet problem, for whatever field theory we are
currently handling.
The aim of this thesis is to develop a means for incorporating nonlinearities
directly into quantization of field theories, in particular gauge field theories. Because
of operator ordering, the quantizations obtained are not expected to agree with more
usual quantizations, except in the free case. Different operator orderings correspond
to different Dyson-Wick expansions and Feynman rules (see [36]), so equivalence with
conventional perturbative results is not in general expected.
As evidenced by the preceding discussion, the gathering of insights valuable to the
search for a well-behaved quantum ground state of the gravitational field is central
to the motivation of this program. A direct approach to nonlinearity is essential to
any complete theory of quantum gravity, since in general relativity even spacetime
itself is not a fixed linear background structure, but a dynamically changing curved
manifold.
10
1.5 Layout
Chapter 2 begins by dealing with the properties of normal-ordered ground states
for free field theories, specifically free massive scalar field theory and free Maxwell
theory. Here the absense of nonlinearities allows us easily to carry out constructions
we hope to extend to nonlinear normal ordered field theories. We solve explicitly
for the ground state in free scalar field theory and construct a canonical Gaussian
measure on the space of field configurations S ′ (R3). For free Maxwell theory, a
ground state in closed form has been obtained by Wheeler using Fourier analysis;
this is the same as that corresponding to the normal ordering. For this ground state
we verify gauge and full Poincare invariance.
Advancing to dynamically nonlinear (though kinematically linear) field theories,
we consider scalar ϕ4 theory in Chapter 3, proving the existence of a zero-energy
ground state for the nonlinear normal ordered scalar ϕ4 Hamiltonian.
The methods of Chapter 3, particularly the direct method in the calculus of vari-
ations, serve as a model for the analogous problem in Yang-Mills theory (Chapter
4). To find a zero-energy ground state for the nonlinear normal ordered Yang-Mills
Hamiltonian, we use results of Uhlenbeck, Sedlacek, and Marini ([37], [32], [24]) in
solving the Euclidean Dirichlet problem. Gauge invariance, as well as invariance
under spatial rotations and translations, is automatic from the construction. Invari-
ance under boosts remains to be investigated in future work.
11
Chapter 2
Linear field theories
2.1 Free massive scalar field theory
Free massive scalar field theory, described by the classical Lagrangian
L =1
2
∫
R3
ϕ2 − |∇ϕ|2 −m2ϕ2 dx
and Hamiltonian
H =1
2
∫
R3
π2 + |∇ϕ|2 + m2ϕ2 dx
=1
2
(‖π‖22 +
⟨ϕ,−ϕ + m2ϕ
⟩2
)
(π = ϕ),
furnishes a good testing ground to show that in a linear situation, the ground state
obtained through the usual normal ordering coincides with that given by
Ω (ϕ) = N exp (−S (ϕ)) ,
12
where S (ϕ) is the solution to the iHJE
∫
R3
(δS
δϕ(x)
)2dx =
∫
R3
(−ϕϕ + m2ϕ2
)dx. (2.1)
To find the normal ordered ground state as described at the end of §1.2, we must
establish the existence of a unique positive square root for the operator
Hϕ = −ϕ + m2ϕ.
By Theorem 5.4, Chapter V, in [18], this operator H with domain
D(H) =ϕ ∈ L2(R3) : ϕ ∈ L2(R3)
is self-adjoint with respect to the L2 inner product, and bounded from below. This
result combined with Theorem 3.35, Chapter V, [18] implies that H has a unique
positive (self-adjoint) square root H1/2. Taking
S (ϕ) =
⟨ϕ,H1/2ϕ
⟩2
2
as in (1.3), linearity and self-adjointness of H1/2 yield δSδϕ(x)
= H1/2ϕ. Thus
∫
R3
(δS
δϕ(x)
)2dx =
⟨H1/2ϕ,H1/2ϕ
⟩2
=
∫
R3
(−ϕϕ + m2ϕ2
)dx,
verifying that S is nothing other than the solution of the Hamilton-Jacobi equation
(2.1).
The operator H1/2 can be written more explicitly by combining powers of H with
the integral kernel of the pseudodifferential operator H−α/2. Following Stein’s [35]
13
analysis for the integral kernel of (1 −)−α/2 we obtain the an integral kernel for
H−α/2 (expressible in terms of Bessel functions).
Proposition 1 The integral kernel K(α/2)m (x) for the operator (Dm)−α/2 = (− + m2)
−α/2
is given by
K(α/2)m (x) =
m3−α2
21+α2 π3/2Γ
(α2
)|x| 3−α2
K 3−α2
(m |x|) ,
where
Kν (z) =
(z2
)νΓ(12
)
Γ(ν + 1
2
)∫ ∞
1
e−zt(t2 − 1
)ν− 1
2 dt,
an integral representation of the modified Bessel function of the third kind [12].
Proof. The integral kernel K(α/2)m (x) will be a function whose Fourier transform is
(m2 + 4π2 |x|2
)−α/2. To find such a function we need two facts:
(i) F−1(e−πδ|x|2
) = e−π|x|2
δ δ−3
2 (Fourier transform of a Gaussian)
(ii) t−a = 1Γ(a)
∫∞0
e−tδδa dδδ
(integral representation of Γ(x))
The second fact (ii) implies
(4π)α/2(m2 + 4π2 |x|2
)−α/2=
1
Γ(α2
)∫ ∞
0
e−δ4π (m2+4π2|x|2)δ
α2dδ
δ.
14
So
F−1((
m2 + 4π2 |x|2)−α/2)
= F−1
(1
(4π)α/2 Γ(α2
)∫ ∞
0
e−δ4π (m2+4π2|x|2)δ
α2
dδ
δ
)
=1
(4π)α/2 Γ(α2
)∫ ∞
0
F−1(e−πδ|x|
2)e−
δm2
4π δα2
dδ
δ
=1
(4π)α/2 Γ(α2
)∫ ∞
0
e−π|x|2
δ δ−3
2 · e− δm2
4π δα2dδ
δ(using (i))
=1
(4π)α/2 Γ(α2
)∫ ∞
0
e−πm2|x|2
4πy e−y(
4πy
m2
)−3+α2 dy
y(y =
m2
4πδ)
=m3−α
(4π)3/2 Γ(α2
)∫ ∞
0
e−y−m2|x|2
4y y−3+α2
dy
y
=m3−α
23π3/2Γ(α2
) · 2
(m |x|
2
)−3+α2
K 3−α2
(m |x|)
=m
3−α2
21+α2 π3/2Γ
(α2
)|x| 3−α2
K 3−α2
(m |x|) ,
where on the second to last line we have used another integral representation of
Kν (z) from [12].
This result allows us to write
S (ϕ) =
⟨ϕ,H1/2ϕ
⟩2
2=
⟨ϕ,H−1/2Hϕ
⟩2
2
=1
2
∫
R3
∫
R3
[ϕ(x) ·K(1/2)
m (x− y) ·(−ϕ(y) + m2ϕ(y)
)]dx dy
where K(1/2)m (x) = m
2π2|x|K1 (m |x|) , or
S (ϕ) =
⟨ϕ,H1/2ϕ
⟩2
2=
⟨ϕ,HH−3/2Hϕ
⟩2
2=
⟨Hϕ,H−3/2Hϕ
⟩2
2
=1
2
∫
R3
∫
R3
[(−ϕ(x) + m2ϕ(x)
)·K(3/2)
m (x− y) ·(−ϕ(y) + m2ϕ(y)
)]dx dy
where K(3/2)m (x) = 1
2π2K0 (m |x|).
15
2.2 Gaussian measure
Having obtained a ground state for a quantum theory, one would also like to define
a probability measure dµ on its configuration space Q of field configurations, leading
to a Hilbert space H = L2(Q, dµ) of quantum states. Formally, one would like
to use the ground state wave functional as a “damping factor” multiplying a naive
“Lebesgue measure” “dϕ” on the space Q of field configurations. The heuristic for
such a probability measure dµ(ϕ) on Q is
“dµ(ϕ) = N [exp (−S(ϕ))]2 dϕ.”
Glimm and Jaffe ([11], and references therein) have brought ideas of Kolmogorov
to bear on the problem of defining a rigorous measure in this spirit. In scalar
field theory (linear or nonlinear), the configuration space Q should be (a subset
of) the space of tempered distributions S ′(R3) (the dual of Schwartz space S(R3));
this is the case best suited to the construction, so we specialize to Q = S ′(R3).
The construction starts with the observation that if we already had a measure dµ
on S ′(R3), we could define from it a bilinear form known as the “covariance” on
S(R3) × S(R3)
(f, g) →∫
f(ϕ)g(ϕ) dµ(ϕ),
where ϕ ∈ S ′(R3) and f(ϕ), g(ϕ) denote the usual linear pairing between vectors
and dual vectors.
This definition of the covariance in terms of the measure can in fact be reversed:
one starts with a continuous non-degenerate bilinear form 〈· , C·〉2 on S(R3)×S(R3),
and from it recovers the measure dµ on S ′(R3) for which 〈f, Cg〉2 =∫f(ϕ)g(ϕ)dµ(ϕ).
For any finite-dimensional subspace F ⊂ S(R3), the restriction of the (still to be
16
defined) measure dµ to F ′ ⊂ S ′(R3) is the Gaussian measure
dµ(ϕ)F = N exp
(−〈ϕ,C−1ϕ〉2
2
)dϕ
where ϕ only runs over the finite-dimensional vector space F ′, and hence for dϕ we
can sensibly use Lebesgue measure on Rn, n = dimF . Since the two restrictions
dµ(ϕ)E and dµ(ϕ)F agree for two finite-dimensional subspaces E and F , E ⊂ F ⊂
S(R3), we can view the set of measures
Ξ =dµ(ϕ)F : F ⊂ S(R3), dimF = n
n∈N
as allowing us to integrate any “Borel F -cylinder function” on S ′(R3) defined by
f(ϕ) = F (f1(ϕ), ..., fn(ϕ)), where f1, ..., fn ∈ F , F is a Borel function on Rn. The
measurable sets for the Borel F-cylinder functions are the Borel F -cylinder sets
ϕ ∈ S ′(R3) : (f1(ϕ), ..., fn(ϕ)) ∈ B, where B is a Borel set in Rn (inverse projec-
tions of S ′(R3) onto n-dimensional Borel sets). Thus by taking M to be the Borel
σ-algebra generated by all F-cylinder sets for F any finite-dimensional subspace of
S(R3), we can extend from the set of measures Ξ above to a well-defined measure
(dµ,M,S ′(R3)).
Comparing with the results of the previous section, we see that a Gaussian mea-
sure for the free scalar case is obtained by taking
C−1 = 2H1/2.
This proves the existence of a well-defined Gaussian measure on S ′(R3) corresponding
to the heuristic “dµ(ϕ) = N [exp (−S(ϕ))]2 dϕ.”
17
2.3 Free Maxwell theory
Another linear test-case is the abelian gauge theory having structure group U(1);
namely, Maxwell’s theory of the free electromagnetic field. The basic quantity
is the potential, a differential 1-form A given in coordinates by Aµdxµ, where µ
is a spacetime index1 running from 0 to 3. However, the physically measurable
quantity is the field strength, given by the 2-form F = dA, or in coordinates Fµν =
∂µAν − ∂νAµ. From this one defines the electric field E, a 1-form on space, and the
magnetic field B, a 2-form on space, by
F = B + E ∧ dx0.
In components, B is usually given in terms of the 1-form which is its Hodge dual
with respect to R3:
Bi = εijk∂jAk.
For the remainder of the present discussion, let ∗ be understood to denote the spatial
Hodge dual on differential forms over R3, and similarly let d refer to the spatial
exterior derivative (for details on Hodge theory used throughout this section, see the
Appendix). Then
B = ∗dA,
where only the spatial components of A are used. The electric field E can be written
in components as
Ei = ∂iA0 − ∂0Ai.
To pass from a Lagrangian to a Hamiltonian formulation of Maxwell theory, one
1As is customary, Greek indices are spacetime indices, while Latin indices run only over thespatial coordinates 1,2,3.
18
must first specialize to the Weyl gauge A0 = 0, since the Lagrangian is independent
of A0 and therefore the Legendre transformation breaks down for an arbitrary gauge
(see e.g. [15]). Thus our canonical position variable is Ai, where i runs over the
three spatial parameters, and the canonical momentum is Ei = Ai (the negative of
the electric field). In terms of these quantities the Hamiltonian is given as
H =1
2
∫
R3
E2 + B2 dx;
using the L2 inner product
〈ω, µ〉2 =
∫
R3
ω ∧ ∗µ
for 1-forms on R3, we can re-express it as
H =1
2
(‖E‖22 + 〈B,B〉2
)
=1
2
(‖E‖22 + 〈∗dA, ∗dA〉2
)
=1
2
(‖E‖22 + 〈A, ∗d ∗ dA〉2
),
where in the last line we have used the fact that for 1-forms on R3, (∗d)∗ = ∗d, a
consequence of Stokes’ Theorem.
The similarity to the linear template described in §1.2 is now apparent, as the
operator ∗d ∗ d is equal to δd = d∗d, and therefore can be shown to have a unique
positive square root. Taking the domain of d to be W 1,2 1-forms on R3 (see Appendix
for detailed definition), d is a closed operator from Λ1 (R3) to Λ2 (R3), and therefore
we can apply Theorem 3.24, Chapter V, in [18] to assert that δd is self-adjoint. This
allows us then to deduce from Theorem 3.35, Chapter V [18] that δd possesses a
unique nonnegative self-adjoint square root T .
To write the square root T of δd explicitly, note that δd is closely related to the
19
Laplace-de Rham operator
= δd + dδ = ∗d ∗ d− d ∗ d∗
defined on 1-forms over R3. Applied to B = ∗dA, the two operators have the
same effect, since (∗)2 = ±I and d2 = 0. Thus T can be written in the form
( ∗ d)−1/2 (∗d), yielding a ground state
Ω(A) = N exp
(−⟨A, ( ∗ d)−1/2 (∗d)A
⟩2
2
)
for the normal ordering of H. Converting to coordinates and writing −1/2 in its
integral kernel representation (see e.g. [35]), we obtain
Ω(A) = N exp
(− 1
4π2
∫
R3
∫
R3
(∇×A(x)) · (∇×A(y))
|x− y|2dx dy
), (2.2)
which agrees exactly with the usual closed-form Maxwell theory ground state, as
written by Wheeler in [41]. According to the naive factor ordering of the Hamiltonian
operator, this ground state has an infinite constant as its ground state energy; a
pleasant effect of the normal ordering is to assign zero energy to the same ground
state.
Free Maxwell theory also serves as a testing ground for the implementation
of Poincare invariance in a quantized gauge theory. The conserved quantities
generating infinitesimal transformations can be written CY =∫R3
YµTµ0dx, where
T µν = − 14π
F µ
αFνα − 1
4ηµνFαβF
αβ
is the stress-energy tensor and Y denotes a
Killing field in the direction of the infinitesimal transformation (translation, ro-
tation, or boost). Rotation and translation invariance may be directly verified;
for boosts, it is not difficult to write the generator in the xi direction CB(i) =
20
∫R3
(x0δµi + xiδµ0
)T µ0dx as the sum of a translation generator x0
4π
∫R3
εijkEjBkdx
plus the term 18π
∫R3
xi(|E|2 + |B|2
)dx. Translation invariance being already estab-
lished, it only remains to verify that (2.2) is annihilated by the remaining term under
our ordering. Indeed, the functional S(A) in the exponent of (2.2) satisfies
∫
R3
xi
∣∣∣∣δS
δA
∣∣∣∣2
dx =
∫
R3
xi |B|2 dx, (2.3)
as shown by the following calculation:
⟨xi δS
δA,δS
δA
⟩
2
=⟨xi
(∗d
(−1/2
)∗ dA
), ∗d
(−1/2
)∗ dA
⟩2
=⟨∗d
(xi
(∗d
(−1/2
)∗ dA
)),(−1/2
)∗ dA
⟩2
=⟨∗(dxi ∧ ∗d
(−1/2
)∗ dA
),(−1/2
)∗ dA
⟩2
+⟨xi (∗d ∗ d)
(−1/2
)∗ dA,
(−1/2
)∗ dA
⟩2
.
The first term of the last equality can be shown to vanish, by recognizing it as
⟨∗(dxi ∧ ∗d
(−1/2
)∗ dA
),(−1/2
)∗ dA
⟩2
=⟨(−1/2 ∗ dA
)i, − ∗ d ∗
(−1/2
)∗ dA
⟩2
−⟨∂i
(−1/2 ∗ dA
),(−1/2
)∗ dA
⟩2
.
On the right hand side, the first term is 0 since −1/2 and ∗d commute, (∗)2 = 1,
and d2 = 0. The second term is one-half of a boundary, and therefore also 0.
We are then left with
⟨xi δS
δA,δS
δA
⟩
2
=⟨xi (∗d ∗ d)
(−1/2
)∗ dA,
(−1/2
)∗ dA
⟩2
;
21
using the fact that −d ∗ d ∗(−1/2
)∗ dA = 0, we can write
⟨xi (∗d ∗ d)
(−1/2
)∗ dA,
(−1/2
)∗ dA
⟩2
=⟨xi (∗d ∗ d− d ∗ d∗)
(−1/2
)∗ dA,
(−1/2
)∗ dA
⟩2
=⟨xi
(−1/2
)∗ dA,
(−1/2
)∗ dA
⟩2
=⟨xi
(−1/2
)∗ dA,
(−1/2
)∗ dA
⟩2
Using the integral kernel representation of −1/2, and writing Bj for the coordinate
representation of B = ∗dA, we can rewrite the first slot in the inner product:
xi(−1/2B
)= xi 1
2π2
(∫
R3
yBj (y)
|x− y|2d3y
)
=1
2π2
(∫
R3
y (xiBj (y))
|x− y|2d3y
)
= −1/2(xiB
)
This means that
⟨xi
(−1/2
)∗ dA,
(−1/2
)∗ dA
⟩2
=⟨−1/2
(xiB
), −1/2B
⟩2
=⟨xiB, B
⟩2
by self-adjointness of −1/2.
The relation (2.3) allows the extra term 18π
∫R3
xi(|E|2 + |B|2
)dx in the boost
generator to be ordered in the same way as the Hamiltonian, securing boost invari-
ance.
22
Chapter 3
Scalar ϕ4 theory
Adding a ϕ4 interaction term to the free massive scalar field theory considered in
§2.1 results in the classical Lagrangian and Hamiltonian
L (ϕ, ϕ) =
∫
R3
1
2∂µϕ∂
µϕ− m2
2ϕ2 − λ
2ϕ4 dx
H (ϕ, π) =
∫
R3
1
2π2 +
1
2|∇ϕ|2 +
m2
2ϕ2 +
λ
2ϕ4 dx.
In the context of more conventional perturbative quantizations, it is widely expected
that quantized scalar ϕ4 theory will in fact turn out to be free, its interactions being
rendered trivial by a vanishing renormalized coupling constant. In five or more
spacetime dimensions, this has been proven to be the case; in four dimensions there
is no complete proof, but evidence exists [9].
By contrast, a nonlinear normal ordered canonical quantization of scalar ϕ4 theory
results in a ground state distinct from that of the free theory. This is a visible
demonstration of inequivalence between our quantization and the more usual schemes
(recall the discussion of §1.4). Since no fundamental scalar fields have been observed
in nature, we have no basis for any evaluative judgement, and for this reason primarily
23
regard the nonlinear normal ordered quantization of scalar ϕ4 theory as a model
problem for techniques to be used for Yang-Mills theory (Chapter 4), and eventually
in future work general relativity (Chapter 5).
To find the nonlinear normal ordered ground state for scalar ϕ4 theory, we con-
sider the zero-energy imaginary-time Hamilton-Jacobi equation
∫
R3
∣∣∣∣δS
δϕ
∣∣∣∣2
dx =
∫
R3
|∇ϕ|2 + λϕ4 + m2ϕ2 dx. (3.1)
Constructing a solution by the method described in §1.3 requires us to find a min-
imizer ϕt for −∫∞0
L (ϕt, ϕt) dt satisfying the initial data ϕ|t=0 = ϕ. We proceed
to do this using notation and results from Giusti [10], within the “direct method”
for calculus of variations. Another possible approach to the Euclidean Dirichlet
problem for scalar ϕ4 theory has been investigated by H. Lindblad, using existence
results for solutions of partial differential equations given restrictions on the initial
data [22].
3.1 Direct method: General results and tech-
niques
The argument for existence of a minimizer of the functional
F(u) =
∫
U
F (x, u(x), Du(x)) dx
depends on the concept of “minimizing sequences.” For a functional F(u) considered
over functions u belonging to some set V , a minimizing sequence is a sequence uk ⊂
V s.t. limk→∞
F(uk) = infV
F . Clearly by definition of infimum, at least one minimizing
24
sequence must exist. The method of finding a minimizer will be to show that under
suitable conditions, there exists a minimizing sequence which converges (in some
topology) to a limiting function u in V , such that F(u) = infV
F . If we have a function
u = limk→∞
uk in some topology, the key to asserting F(u) = limk→∞
F(uk) = infV
F will be
the property of “lower semicontinuity,” as follows. For the present we will assume
without specification an appropriate topology in which to work; later this issue will
be settled.
Definition 2 A functional is called “lower semicontinuous” on V if for every se-
quence vk ⊂ V , vk → v,
F(v) ≤ limk→∞
inf F(vk).
Thus for a lower semicontinuous functional F on V , a convergent minimizing
sequence uk ⊂ V , uk → u ∈ V in fact yields a minimizer u ∈ V of F , since
infV
F ≤ F(u) ≤ limk→∞
inf F(uk) = infV
F ,
or
F(u) = infV
F .
It only remains, then, to show that in some suitable topology, our functional F is
lower semicontinuous and a convergent minimizing sequence exists. The following
theorem from [10] (Theorem 4.5) gives conditions guaranteeing lower semicontinuity
of F :
Theorem 3 Let U be an open set in Rn, X a closed set in RN , and F (x, u, z) a
25
function defined in U ×X × Rν such that
(i) F is measurable in x ∀ (u, z) ∈ X × Rν; continuous in (u, z)
for almost every x ∈ U
(ii) F (x, u, z) is convex in z for almost every x ∈ U , ∀ u ∈ X
(iii) F ≥ 0.
Suppose
uk, u ∈ L1(U,X), uk → u
zk, z ∈ L1(U,Rν), zk z
in L1loc(U); (*)
Then ∫
U
F (x, u, z) dx ≤ limk→∞
inf
∫
U
F (x, uk, zk) dx.
A proof is given in [10], and will not be repeated here.
To apply this theorem to functionals of the form F(u) =∫UF (x, u,Du) dx,
Rellich’s Theorem (Theorem 3.13 in [10]) can be employed:
Theorem 4 Let Λ be a bounded open set in Rn, with Lipschitz-continuous boundary
∂Λ. Let p, q be such that 1 ≤ p < n, 1 ≤ q < p∗ ≡ npn−p
. Then the immersion1
W 1,p(Λ) → Lq(Λ)
is compact (i.e. continuous and maps bounded sets to relatively compact sets).
The important point is that by Rellich’s Theorem, a sequence uk converging
weakly to u in W 1,ploc (U) must also converge strongly to u in Lp
loc(U). Weak conver-
gence in W 1,ploc (U) is equivalent to weak W 1,p convergence in every open bounded set
1Note that the immersion W 1,p(Λ) → Lp∗
(Λ) is guarateed to exist by the Sobolev imbeddingtheorems, e.g. Thm 3.11 in [10]. Since Λ is bounded, Holder’s inequality shows that Lp
∗
(Λ) ⊂L1(Λ), and from here we can again use Holder’s inequality to show that ∀ r ∈ (1, p∗), Lp∗(Λ) ⊂Lr(Λ) (this is a standard result; see for example [8]).
26
Λ ⊂ U . By the Principle of Uniform Boundedness,‖uk‖1,p,Λ
is bounded, which
implies by Rellich’s Theorem that uk is relatively compact in Lp(Λ) (since p < p∗).
Thus every subsequence of uk has a convergent subsequence, and since uk con-
verges weakly to u in Lp(Λ), all convergent subsequences of uk must converge
strongly to u in Lp(Λ). Therefore uk converges strongly to u in Lploc(U).
This analysis provides the necessary information to verify that the hypotheses of
Theorem 3 are satisfied by a functional of the form F(u) =∫UF (x, u,Du) dx and
a sequence uk weakly convergent to u in W 1,1loc (U,X). Such a sequence converges
strongly to u in L1loc(U,X), as discussed. Meanwhile Duk lies in L1loc(U), and any
functional ω ∈ (L1loc(U))∗
and acting on Dv, v ∈ W 1,1loc (U,X), can be interpreted to
be a functional in(W 1,1
loc (U,X))∗
. Thus weak convergence of uk in W 1,1loc (U,X)
implies weak convergence of Duk in L1loc(U). Taking z in Theorem 3 to be Du
and zk to be Duk, (∗) holds for uk weakly convergent to u in W 1,1loc (U,X). It would
only remain to verify conditions (i), (ii), and (iii) for F .
Before finally applying Theorem 3 to the functional F(u) =∫UF (x, u,Du) dx,
one last observation simplifies the search for a minimizing sequence. Holder’s in-
equality ensures that W 1,ploc (U,X) (p > 1) is a subset of W 1,1
loc (U,M) and that W 1,ploc
convergence implies W 1,1loc convergence and W 1,p
loc weak convergence implies W 1,1loc weak
convergence. This is fortunate, because W 1,ploc has the major advantage over W 1,1
loc of
being a reflexive space. By the Alaoglu Theorem, in a reflexive space every bounded
sequence has a weakly convergent subsequence.
Since a minimizing sequence uk u in W 1,ploc (U,X) will also converge weakly in
W 1,1loc (U,X), such a uk would be sufficient to the matter at hand. On the other
side of the coin, we could also say that lower semicontinuity of a functional F(u) with
respect to the weak topology of W 1,1loc (U,X) implies lower semicontinuity of F(u) with
respect to weak W 1,ploc (U,X), also implying that uk u in W 1,p
loc (U,X) is sufficient.
27
Moreover as remarked above, Alaoglu’s Theorem yields that in fact a minimizing
sequence bounded in W 1,ploc (U,X) would be enough. A simple condition on the
functional F(u) to be “coercive” guarantees that in fact all minimizing sequences
will be bounded:
Definition 5 A functional F(u) on W 1,p(Λ) is called “coercive” if
lim‖u‖1,p→∞
F(u) = +∞.
The result of this analysis is an existence theorem (Theorem 4.6 in [10]) for a
minimizer of F(u):
Theorem 6 If F is lower semicontinuous in the weak topology of W 1,ploc (U,X) and
V is a weakly closed subset of W 1,ploc (U,X) on which F is coercive, then F takes a
minimum in V .
3.2 Application to ϕ4 theory
At last Theorem 6 can be applied to the functional
F (ϕ) = −I (ϕ) =
∫
R+×R3
∣∣(4)∇ϕ∣∣2 + λϕ4 + m2ϕ2 dx dt,
allowing us to find the imaginary-time zero-energy Hamilton-Jacobi functional for
scalar ϕ4 theory. In this ansatz, U is R+ × R3 and X is R. For given initial
data ϕ0, the set V in Theorem 6 is ϕ : ϕ|t=0 = ϕ0. As discussed in §3.7 of [10],
the concept of boundary values (“traces”) for W 1,p(U) functions is meaningful when
suitably defined, and weak W 1,p(U) convergence uk u implies Lploc convergence
of of the traces ψk → ψ on ∂U (subject to conditions on ∂U , which in our case is
28
simply the t = 0 copy of R3 in R+ × R3, and therefore easily satisfies the requisite
smoothness hypotheses). Therefore V is weakly closed, as required. The functional
F is coercive by inspection with respect to W 1,2loc (R+ × R3), and equally easily seen
to be Caratheodory, positive, and convex, verifying that it is lower semicontinuous
in the weak topology of W 1,2loc (R+ × R3). Therefore the desired minimizer exists in
V .
The minimizer we have just found can also be shown to be unique. The following
proof is due to V. Moncrief.
Proposition 7 The minimizer found above for the functional
F (ϕ) =
∫
R+×R3
∣∣(4)∇ϕ∣∣2 + λϕ4 + m2ϕ2 dx dt,
for given initial data ϕ|t=0 = ϕ0, is unique.
Proof. Suppose ϕ1 and ϕ2 are two distinct minimizers for F , for given initial data
ϕ|t=0 = ϕ0. Then both are solutions of the corresponding Euler-Lagrange equation:
−(4)ϕi + m2ϕi + λϕ3i = 0.
Use the parameter τ ∈ [0, 1] to interpolate linearly between ϕ1 and ϕ2:
ϕτ ≡ τϕ1 + (1 − τ)ϕ2,
29
so that dϕτdτ
= ϕ1 − ϕ2. Starting from the Euler-Lagrange equation, we have
0 =
∫ 1
0
d
dτ
[−(4)ϕτ + m2ϕτ + λϕ3τ
]dτ
=
∫ 1
0
−(4) (ϕ1 − ϕ2) + m2 (ϕ1 − ϕ2) + 3λϕ2τ (ϕ1 − ϕ2) dτ
= −(4) (ϕ1 − ϕ2) + m2 (ϕ1 − ϕ2) + 3λ (ϕ1 − ϕ2)
∫ 1
0
ϕ2τ dτ .
Multiplying both sides by (ϕ1 − ϕ2) and integrating over R+ ×R3, we obtain
0 =
∫
R+×R3(ϕ1 − ϕ2)
−(4) (ϕ1 − ϕ2) + m2 (ϕ1 − ϕ2) + 3λ (ϕ1 − ϕ2)
∫ 1
0
ϕ2τ dτ
;
at this point, equality of ϕ1 and ϕ2 at t = 0 and decay of both fields as |x| → ∞
allow us to integrate by parts in the first term of the above integral, yielding
0 =
∫
R+×R3
∣∣(4)∇ (ϕ1 − ϕ2)∣∣2 + m2 (ϕ1 − ϕ2)
2 + 3λ (ϕ1 − ϕ2)2
∫ 1
0
ϕ2τ dτ
.
Since all terms inside the integral are greater than or equal to 0, we must conclude
ϕ1 − ϕ2 = 0.
We can now define our Hamilton-Jacobi functional. This has for its domain
the set of scalar field configurations on R3, interpreted in the Dirichlet problem as
the t = 0 slice of R+ × R3. For notational convenience, we now allow ϕ to denote
field configurations on R3 (rather than R+×R3 as above). Each field configuration
ϕ is the initial value for a guaranteed minimizer ϕt of the functional F , so that
ϕt=0 = ϕ. The imaginary-time zero-energy Hamilton-Jacobi functional S (ϕ) can
30
then be written2
S (ϕ) =
∫
R+×R3
∣∣(4)∇ϕt
∣∣2 + λϕ4t + m2ϕ2t dx dt.
The corresponding zero-energy ground state for the nonlinear normal ordering is
Ω (ϕ) = N exp (−S (ϕ)) .
2Note that for S (ϕ) to be finite, we are implicitly making the assumption that for every set ofinitial data ϕ we are interested in, there exists at least one trajectory ϕs (ϕs=0 = ϕ), for which−∫∞
0L (ϕs, ϕs) dt is finite. This constraint defines the set of physical fields, since for any ϕ on
R3 for which no such ϕs on R+ ×R3 can be found, allowing S (ϕ) to take an infinite value implies
that evaluated on this ϕ, the ground state Ω(ϕ) is zero.
31
Chapter 4
Yang-Mills theory
Yang-Mills theory arose as a generalization of the Maxwell theory of electromag-
netism. While the concept of symmetries constraining the laws of physics has been
central since Galilean relativity, the new and powerful idea in Maxwell theory is
that symmetries can act not only on space and time but on internal spaces in which
the physical fields take their values. Solutions to a physical system are no longer
represented by a single function over spacetime, but by an equivalence class of such
functions related by smoothly varying local symmetries. More physically stated,
a field has local internal degrees of freedom (gauge freedom) on which a structure
group can act, without changing the measurable properties of the field.
In electromagnetism, the structure group is U (1), meaning that gauge symmetry
acts on the potential A by means of a function valued in i ·u(1) (that is, a real-valued
function f). Because U (1) is abelian, the action of the gauge group on A is given
by
A → A + df,
and the field strength F (and electric and magnetic fields E and B) are left unchanged
by gauge transformations. The success of electrodynamics prompted the question
32
of whether other — nonabelian — structure groups could be used to construct gauge
theories representing the remaining fundamental forces of nature. Yang and Mills’s
discovery of nonabelian gauge theories led to the description of the electroweak force
as an SU (2)×U (1) gauge theory (modified by the Higgs mechanism to account for
the mass of weak gauge bosons), and of the strong nuclear force as an SU (3) gauge
theory. However, it is still an open question whether rigorous quantum Yang-Mills
theories exist. In fact, demonstrating a rigorous quantization of pure Yang-Mills
theory in four dimensions for a compact simple gauge group G, and proving that
this theory exhibits a “mass gap” > 0 (i.e., that the Hamiltonian operator has no
eigenvalues in the interval (0,)), is a Clay Prize Millenium Problem (see [17]).
Our primary aim here lies not in the direction of the mass-gap problem (though
of course any progress on rigorously quantized Yang-Mills theory is interesting), but
more generally investigating promising strategies for overcoming hurdles inherent in
quantization of theories with gauge degrees of freedom. In particular, nonabelian
gauge theory is a reasonable testing ground for ideas to be applied to quantum
gravity. After all, the Kodama state arose as a generalization of the Chern-Simons
state from Yang-Mills theory; a first place to look for alternatives to the Kodama
state, then, might be in more well-behaved candidate ground states for Yang-Mills.
In the context of general relativity, no mass gap is expected, and therefore in this
quantization of Yang-Mills theory we do not seek a mass gap, since our goal is a
quantization generalizable to quantum gravity.
In the following sections, we present a nonlinear normal ordered ground state
for Yang-Mills theory with a compact gauge group G ⊂ SO (l). We prove that this
ground state is gauge invariant and invariant under spatial translations and rotations.
Completing the Poincare group, invariance under boosts is conjectured, and remains
as a question for future work.
33
4.1 Mathematical formalism
The main ingredient in a Yang-Mills theory is the structure group; this is a compact
Lie group G with Lie algebra g. For P a principal G-bundle over M , the Yang-Mills
field is a connection A ∈ Λ1P ⊗ g. Given a local section σα : Uα → P for some
neighborhood Uα ⊂ M , the connection 1-form A pulls back to a g-valued 1-form1
Aα = σ∗αA on Uα; the transformation of A on overlapping neighborhoods Uα and Uβ is
given by a transition function ταβ : Uα ∩Uβ → G, defined by σβ (x) = σα (x) ταβ(x):
Aα(x) = ταβ(x)−1dταβ(x) + ταβ(x)−1Aβ (x) ταβ(x), x ∈ Uα ∩ Uβ.
The important quantity for Yang-Mills theory is the curvature F ∈ Λ2P ⊗ g of
the connection A, given by F = dPA + 12
[A,A] where the bracket [·, ·] denotes
the graded commutator on forms, so that [A,A] = 2 (A ∧A) . In terms of a local
section σα : Uα → P , F pulls back to a g-valued 2-form Fα = σ∗αF on Uα, given by
Fα = dMAα + 12
[Aα, Aα], transforming as
Fα(x) = ταβ(x)−1Fβ (x) ταβ (x) , x ∈ Uα ∩ Uβ
for ταβ as given above. In local coordinates, F reduces to
Fµν = ∂µAν − ∂νAµ + [Aµ, Aν]
(here [·, ·] is the ordinary commutator in g).
In order to describe the Yang-Mills action, we use the local expressions of F as a
g-valued 2-form on neighborhoods of M ; however all definitions are gauge-invariant
and therefore do not depend on the particular section used to pull back F .
1For details on Hodge theory of Lie-algebra-valued forms used in this section, see the Appendix.
34
The Yang-Mills action can be conveniently couched in terms of the inner product
for g-valued k-forms on the manifold M :
〈η, θ〉2 =
∫
M
tr (η ∧ ∗θ) , (4.1)
where ∗ denotes the Hodge dual with respect to the metric g on M . We occasionally
also write 〈η, θ〉 for the pointwise inner product 〈η, θ〉 = tr (η ∧ ∗θ). In this notation,
the action is given as
I(A) =1
2
∫
M
tr (F ∧ ∗F ) =1
4
∫
M
trFµνFµν√gdx1 · · · dxn.
Gauge invariance of the form tr (F ∧ ∗F ) negates any ambiguity due to choice of
local trivialization.
The Yang-Mills field equations are
dA ∗ F = 0
where dA = d + [A, ·] is the exterior covariant derivative; solutions of this system
correspond exactly to critical points of the Yang-Mills action I(A). To see this, vary
I(A) by varying A as A + λh, where h vanishes at t = 0 and is supported on some
compact subset N (dependent on h) of M :
δh (I) (A) =
∫
N
〈dAh, F 〉 =
∫
N
tr (dAh ∧ ∗F )
=
∫
∂N
tr (h ∧ ∗F ) −∫
N
tr (h ∧ dA ∗ F ) .
It is evident that δh (I) (A) vanishes for all variations h precisely when dA ∗ F is
identically 0.
35
To make the transformation from a Lagrangian to a Hamiltonian formulation, we
specialize to the case M = R+ × R3. Since M is contractible, every bundle over M
is trivial and therefore admits a global section. We can then drop the distinction
between A and its local coordinate representation.
As in the case of Maxwell theory (§2.3), working in Weyl gauge is necessary for
the Legendre transform to be well-defined. Thus our canonical position variable
is AIi , where i runs over the three spatial parameters and I over the basis of the
Lie algebra g, and the canonical momentum is EiI = AI
i (this is the negative of the
“electric field” variable).
With respect to these variables we obtain the Hamiltonian
H =1
2
∫
R3
tr(E2 + B2
), (4.2)
where Bi = 12εijkFjk. Hamilton’s equations follow by writing the integral J =
∫∞0
H
dt as∫∞0
∫R3
H dx dt = −I +∫∞0
∫R3
EA dx dt. Varying both expressions with
respect to a one-parameter family Aλ (where the variation has compact support in
R+ × R3 and vanishes for t = 0), we arrive at the equality
dJ
dλ=
∫ ∞
0
∫
R3
δH
δAδA +
δH
δEδE
=
∫ ∞
0
∫
R3
[EδA + AδE
]− dI
dλ
=
∫ ∞
0
∫
R3
[−EδA + AδE
]− dI
dλ,
using integration by parts. In order for equality to hold between the first and last
36
lines for all variations, Hamilton’s equations
E = −δH
δA
A =δH
δE
must be equivalent to the vanishing of the variation δIδA
.
4.2 Canonical quantization
The position and momentum variables described in §4.1 are promoted to operators
in a canonically quantized gauge theory. Classically, the Poisson bracket relations
are
AI
i (x) , AJj (y)
= 0 =
Ei
I (x) , EjJ (y)
,
Ej
J (y) , AIi (x)
= δ3 (x, y) δjiδ
IJ .
With the assignment of quantum operators
AIi (x) : ψ(A) → AI
i (x)ψ(A)
EiI(x) : ψ(A) → −i
δ
δAIi (x)
ψ(A),(4.3)
the commutators of mirror the classical Poisson brackets, as required:
[AI
i (x), AJj (y)
]= 0 =
[Ei
I(x), EjJ(y)
],
[Ej
J(y), AIi (x)
]= −iδ3 (x, y) δjiδ
IJ .
where we have set Planck’s constant equal to 1.
In terms of these operators, we search for a quantum ground state for the non-
linear normal ordered Hamiltonian operator. Notice that in taking the Weyl gauge
A0 = 0 in §4.1, we have lost the field equations describing gauge transformations,
37
and therefore in the quantized theory, the Gauss law constraint
DiEi = 0
must be dealt with separately, either by promoting to a quantum operator and veri-
fying that it annihilates the ground state, or by other means. For the ground state
we construct here, gauge invariance in fact turns out to be directly verifiable (see
§4.6).
4.3 Preliminaries for minimizing procedure
Since finding a ground state for Yang-Mills theory in the nonlinear normal ordering
demands that we solve the Euclidean Yang-Mills Dirichlet problem, as explained in
§1.3 for a general field theory, we now consider the problem of Yang-Mills theory on a
Riemannian manifold (M, g). Let M be a smooth n-dimensional manifold equipped
with a Riemannian metric g, and let ∂M be its boundary.
The inner product on g-valued forms allows us to define Lp and Sobolev spaces
of such forms, using the norm
‖ω‖p =
(∫
M
|ω|p) 1
p
=
(∫
M
(ω, ω)p/2) 1
p
=
(∫
M
[tr (ω ∧ ∗ω)]p/2) 1
p
.
(for details, see Appendix). Note that with respect to a choice of coordinates xi
on M and a basis T I of the Lie algebra g, this is the same as requiring all components
ωIi1...ik
(x) of ω to be Lp or Sobolev functions. In this notation, the Yang-Mills action
is given by
I(A) =1
2‖F‖22 .
38
To solve the Yang-Mills Dirichlet problem for a compact manifold M , Marini [24]
introduces a terminology for coverings of M by geodesic balls and half-balls; these are
described respectively as neighborhoods of type 1 and type 2. Thus neighborhoods
of type 1, in the manifold’s interior, are denoted
U (1) ≡x =
(x0, ..., xn−1
): |x| < 1
while neighborhoods of type 2, centered around points x ∈ ∂M , are of the form
U (2) ≡x =
(x0, ..., xn−1
): |x| < 1, x0 ≥ 0
,
where the coordinate x0 parametrizes unit-speed geodesics orthogonal to ∂M =
x0 = 0. The boundary of a type 2 neighborhood divides into
∂1U =x ∈ ∂U (2) : x0 = 0
,
∂2U =x ∈ ∂U (2) : |x| = 1
.
In our problem, the manifold of interest is R+ × R3 with the Euclidean met-
ric; however we solve the Yang-Mills Dirichlet problem for a general smooth 4-
dimensional Riemannian manifold with boundary, generalizing Marini’s procedure
to the non-compact case (§4.4). Certain results used are also valid in general dimen-
sion n; such distinctions are clearly noted in the statements. We return to consider
the importance of dimension more thoroughly in §4.4.
To use the direct method for minimizing the Yang-Mills action, we need lower
semicontinuity, as in the Dirichlet problem for scalar ϕ4 theory (Chapter 3).
Theorem 8 The Yang-Mills functional on a manifold M of dimension 4 is lower
semicontinuous with respect to the weak topology on W 1,2loc (M) .
39
Proof. Recalling the definition of lower semicontinuity (Definition 2, Chapter 3),
it is good enough to prove that on any open bounded set U ⊂ M , if Ai A in
W 1,2 (U), then I (A) ≤ lim infi→∞ I (Ai). Locally we can write
Fi = dAi +1
2[Ai, Ai] .
Using the same reasoning as Sedlacek’s in Lemma 3.6 of [32], weak convergence
of Ai to A in W 1,2 (U) implies weak convergence of dAi to dA in L2 (U) . The
continuity of the imbedding W 1,2 → L4 and of the multiplication L4×L4 → L2 along
with boundedness of||Ai||2,1
implies that || [Ai, Ai] ||2 is bounded. This together
with a.e. pointwise convergence yield [Ai, Ai] [A,A], so that Fi F in L2 (U ;P ).
Finally, lower semicontinuity of the ||·||2 norm concludes lower semicontinuity of the
Yang-Mills functional.
4.4 Solving the Yang-Mills Dirichlet problem
For M a compact manifold with boundary ∂M , the Yang-Mills Dirichlet problem
has been collectively solved by Uhlenbeck [37], Sedlacek [32], and Marini [24], using
the direct method in the calculus of variations. The argument begins with a local-
izing theorem, proving that given a sequence of connections with a uniform global
bound on the Yang-Mills action, there exists a cover for M (possibly missing a finite
collection of points) such that on neighborhoods of the cover, the Yang-Mills action
for connections in the sequence eventually becomes lower than an arbitrary pre-set
bound ε. This result depends on compactness as proved in [32] (see Proposition 3.3,
or in [24] Theorem 3.1). We reprove the result here in a manner independent of
compactness (Theorem 10), so that the overall argument now applies to noncompact
40
manifolds with boundary as well.
Note that another possible solution to the problem would be to transform M =
R+×R3 into a compact manifold with boundary, using inversion in the sphere (this
suggestion is due to T. Damour). In this approach, one considers the unit sphere
centered at the origin of R4, imbedding R+×R3 into R4 as the set x : x0 ≥ 2. The
inversion mapping is
yi =xi
r2,
where r2 = (x0)2
+ (x1)2
+ (x2)2
+ (x3)2. Under this transformation, the hyperplane
x0 = 2 maps to a sphere S1/4 of radius 14, with its south pole (the image of all points
at infinity) at the origin. The half-space x : x0 > 2 maps to the interior of S1/4.
Since the mapping is conformal, the Yang-Mills action remains invariant, and the
problem of interest on R+ × R3 has been effectively mapped to a compact problem,
to which the arguments of [37], [32], and [24] should apply directly. We do not
pursue this approach here, since the crucial result using compactness can be shown
to generalize (Theorem 10); however we note its potential usefulness to future work,
such as the investigation of uniqueness of solution to the Yang-Mills Dirichlet problem
(see §4.5). An issue pertinent to the conformal mapping approach is the behavior
of initial data at the south pole of S1/4, the image of points at infinity.
Returning to our sketch of the Yang-Mills Dirichlet problem’s solution, local
control over the Yang-Mills action is used to prove existence and regularity of a
minimizer. From this point on, the proofs in [37], [32], and [24] are purely local and
hold unchanged in the noncompact case; proofs are thus not repeated here. Locally,
the argument for existence of a Yang-Mills minimizer consists in finding a Sobolev-
bounded minimizing sequence satisfying the boundary conditions; this sequence then
has a weakly convergent subsequence, which proves to be a solution to the original
41
Dirichlet problem. Local solutions are related by transition functions on overlapping
neighborhoods.
Gauge freedom turns out to be a help as well as a hindrance. Of course it
forces the necessity of working locally and proving compatibility on overlaps, but at
the same time gauge freedom offers an elegant solution to the regularity problem.
A judicious choice of gauge — the “Hodge gauge” — complements the Yang-Mills
equation in such a way as to yield an elliptic system. In the Yang-Mills equation
d ∗ dA + [A, dA] + [A, [A,A]] ,
the highest order term is related to the first term of the Laplace-de Rham operator
∆ = δd+dδ, where δ is the codifferential δ = (−1)n(k+1)+1 ∗d∗ (k being the degree of
the differential form operated upon). Choosing the Hodge gauge, in which d∗A = 0,
ensures that every solution of our system in this gauge is also a solution of the
elliptic system ∆A+ ∗ ([A, dA] + [A, [A,A]]) = 0, and therefore enjoys the regularity
properties of such solutions. (Additional work is needed to establish boundary
regularity; Marini accomplishes this in [24] using the technique of local doubling.)
In the physical problem, we are interested in Yang-Mills theory over a 4-dimensional
manifold with boundary. However many theorems which follow are also valid over
any smooth n-dimensional Riemannian manifold with boundary, and we retain this
level of generality in stating and proving results. The caveat lies in stringing together
the individual theorems into a complete argument for existence and regularity of a
solution to the Yang-Mills Dirichlet problem; to accomplish this, the dimension must
be 4 (see the remarks in [24] following Theorem 3.1). The “good cover” theorem
(Theorem 10 here, or Theorem 3.1 in [24]) guarantees a cover of M\ x1, ..., xk on
whose neighborhoods the local Yang-Mills action for the connections in the sequence
42
is eventually bounded by an arbitrary pre-set bound ε. However, the condition for
existence of a Hodge gauge solution to the Yang-Mills Dirichlet problem is a bound
on the local Ln/2 norm of the Yang-Mills field strength F , which except in dimension
4 is not the same as a local bound on the Yang-Mills action.
Without further ado, we give the precise statements of all theorems needed for the
existence and regularity of a Yang-Mills minimizer on a 4-dimensional manifold with
boundary. On a neighborhood U of type 1 or 2, the condition for local existence of
a gauge satisfying d∗A = 0 is an Ln/2 bound on Yang-Mills field strength. Consider
the sets
A1,pK (U) =
D = d + A : A ∈ W 1,p(U), ‖FA‖Ln/2(U) < K
B1,pK (U) =
D = d + A :
A ∈ W 1,p(U)
Aτ ∈ W 1,p(∂1U),
‖FA‖n/2 < K
‖FAτ‖Ln/2(∂1U) < K
describing connections with field strength locally Ln/2-bounded on a neighborhood
U of type 1 and type 2, respectively. (All norms are defined on U , unless otherwise
specified.) As proven in [37] (Theorem 2.1) for interior neighborhoods and in [24]
(Theorems 3.2 and 3.3) for boundary neighborhoods, a good choice of gauge exists
for connections belonging to A1,pK (U) or B1,pK (U). More precisely,
Theorem 9 For n2≤ p < n, there exists K ≡ K(n) > 0 and c ≡ c(n) such that
every connection D = d+A ∈ A1,pK (B1,pK ) is gauge equivalent to a connection d+ A,
43
A ∈ W 1,p(U), satisfying
(a) d ∗ A = 0 (a′)
d ∗ A = 0
dτ ∗ Aτ = 0 on ∂1U
(b) Aν = 0 on ∂U (b′) Aν = 0 on ∂2U
(c = c′)∥∥∥A
∥∥∥1,n/2
< c(n) ‖FA‖n/2
(d = d′)∥∥∥A
∥∥∥1,p
< c(n) ‖FA‖p
(Unprimed conditions (a)-(d) refer to A1,pK (U); primed conditions (a′)-(d′) toB1,pK (U)).
Moreover, the gauge transformation s satisfying A = s−1ds + s−1As can be taken in
W 2,n/2(U) (s will in fact always be one degree smoother than A; see Lemma 1.2 in
[37]).
Proof. See [37],[24].
As noted in [24], the condition ‖FA‖n/2 < K is conformally invariant, while by
contrast the norm ‖FAτ‖Ln/2(∂1U) picks up a factor of r under the dilation x′ =
rx, so that the simultaneous conditions ‖FA‖n/2 < K, ‖FAτ‖Ln/2(∂1U) < K on a
neighborhood U of type 2 can always be achieved by applying an appropriate dilation
(the Dirichlet boundary data is prescribed to be smooth, so ‖FAτ‖Ln/2(∂1U) already
satisfies some bound).
To find a regular minimizer of the Yang-Mills action on a 4-dimensional manifold
M , we must find a cover Uα of M and a minimizing sequence Ai whose members
satisfy
SYM(Ai|Uα) =
∫
Uα
|FAi|2 dx < K ∀ α, i,
where K ≡ K(4) is as given in Theorem 9. For a compact manifold this is proved
in [32] using a counting argument. Here we use dilations of the neighborhoods in a
44
cover to construct a proof valid for any smooth Riemannian manifold.
Theorem 10 Let A(i) be a sequence of connections in G-bundles Pi over M ,
with uniformly bounded action∫M|F (i)|2 dx < B ∀ i. For any ε > 0, there
exists a countable collection Uα of neighborhoods of type 1 and 2, a collection of
indices Iα, a subsequence A(i)I′ ⊂ A(i)I and at most a finite number of points
x1, ..., xk ∈ M such that
⋃Uα ⊃ M\ x1, ..., xk
∫
Uα
|F (i)|2 dx < ε ∀ i ∈ I ′, i > Iα.
Proof. For each n ∈ N, consider the cover Bn(x) : x ∈ M of M given by geodesic
balls of radius 1n
centered at each point x ∈ M (for x ∈ ∂M , the geodesic “ball”
Bn(x) will actually be a half-ball, a fact which makes no difference in the proof). By
separability, each such cover has a countable subcover Cn = Bn(xn,m) : m ∈ N.
On any ball Bn(xn,m), we have the uniform bound∫Bn(xn,m)
|F (i)|2 dx < B ∀
i. Therefore for the ball Bn(xn,1) in a given cover Cn, there exists a subsequence of
A(i) for which the corresponding subsequence of∫
Bn(xn,1)|F (i)|2 dx
converges.
Of this subsequence, there exists a further subsequence such that the corresponding
subsequence of∫
Bn(xn,2)|F (i)|2 dx
converges, and so on, for every m. Diagonal-
izing2 over these nested subsequences, we obtain a subsequence of A(i) such that
the corresponding subsequence of∫
Bn(xn,m)|F (i)|2 dx
converges for every m ∈ N.
Performing a similar diagonalization over all covers Cn, there exists a subse-
2Diagonalizing over a list of sequences aj (i) such as
a1 (1) a1 (2) a1 (3) . . .
a2 (1) a2 (2) a2 (3) . . .
a3 (1) a3 (2) a3 (3) . . ....
......
. . .
selects out
the new sequence ai (i). In the case important for this proof, each row represents a subsequenceof the previous row, so that for any j, the diagonalized sequence ak (k)is a subsequence of aj (i)for k ≥ j.
45
quence A(i)I′ ⊂ A(i)I such that for every ball in every cover, the sequence∫
Bn(xn,m)|F (i)|2 dx
I′
converges. For each Cn, consider the collection of balls
Bn(yn,m), yn,m ⊂ xn,m, for which∫
Bn(yn,m)|F (i)|2 dx
I′
converges to a value
greater than or equal to ε. Note that for any i ∈ I, there is an upper bound on the
number Ni,n of disjoint balls of radius 1n
for which∫Bn(yn,m)
|F (i)|2 dx ≥ ε :
B ≥ Ni,nε.
Thus the upper bound Bε
limits the number of disjoint balls in the set Bn(yn,m) .
Choose a maximal disjoint setBn(yn,mj
)J
j=1of balls in Bn(yn,m), and consider
the setB∗
n(yn,mj)J
j=1of balls centered at the points yn,mj
but having radius 3n. Then
we have⋃yn,m
Bn(yn,m) ⊂ ⋃Jj=1B
∗n(yn,mj
). This shows that if we discard the balls
Bn(yn,m) from the cover Cn, we will only have discarded a set which was contained
in J ≤ Bε
balls of radius 3n. We can then safely discard the balls Bn(yn,m) from
each cover Cn, and form the union C =⋃
n∈NCn\ Bn(yn,m) to obtain a cover C of
M\ x1, ..., xk, where k ≤ Bε, and each ball Bn (xn,m) ∈ C satisfies
∫Bn(xn,m)
|F |2 < ε.
Since a minimizing sequence A(i)i∈I by definition admits a uniform bound
on the action, we can use Theorem 10 to select a subsequence A(i)i∈I′ of the
minimizing sequence and a cover Uα satisfying∫Uα
|F (i)|2 dx < K(4) ∀ i ∈ I ′,
i > Iα. On any neighborhood Uα in the cover, Theorem 9 implies that each member
Aα(i) of the subsequence is gauge-equivalent to a connection Aα(i) in the Hodge
gauge, satisfying a uniform W 1,2(U) bound on Aα(i). Weak compactness of Sobolev
spaces now yields a further subsequence ofAα(i)
, weakly convergent in W 1,2 to
some Aα. It only remains to show that Aα retains the desired regularity properties
and boundary data, and that the setAα
can be patched to a global connection on
46
M. These objectives are accomplished by Theorem 3.4 in [24] (generalizing Theorem
3.6 in [37] and Theorem 3.1 in [32]). Their results are paraphrased below; the proof
is by weak compactness of Sobolev spaces:
Theorem 11 Let A(i)i∈I be a sequence of G-connections with uniformly bounded
action as described in Theorem 10, and with prescribed smooth tangential boundary
components (A (i))τ = aτ on ∂M . Let ε = K(4), where K(4) is the constant
from Theorem 9. Then, for the subsequence A (i)i∈I′ found in Theorem 10 and
cover Uα, there exists a further subsequence A (i)i∈I′′, sections σα (i) : Uα → Pi
(i ∈ I ′′) and connections Aa on Uα such that
(e) σ∗α (i) (Ai) ≡ Aα (i) Aα in W 1,2 (Uα)
(f) F (Aα (i)) ≡ Fα (i) Fα in L2 (Uα)
(g) sαβ (i) sαβ in W 2,2 (Uα ∩ Uβ)
(h) (Aα)τ |∂1Uα ∼ aτ |∂1Uα by a smooth gauge transformation
(i)
d ∗ Aα = 0 on Uα
dτ ∗ (Aα)τ = 0 on ∂1U
(j) Aα ≡ s−1αβAβsαβ + s−1αβdsαβ
Here sαβ(i) is the transition function Aβ (i) → Aα (i); i.e,
Aα (i) ≡ s−1αβ (i)Aβ (i) sαβ (i) + s−1αβ (i) dsαβ (i) .
Proof. See [32], [24]. Note that the result follows by weak compactness of Sobolev
spaces, after applying diagonalization (as in the proof of Theorem 10) over the count-
able cover Uα.
Lower semicontinuity of the Yang-Mills functional now implies that the value of
the action on the limiting connection A of the sequence described in Theorem 11 is
47
in fact m (aτ ) ≡ minA
I (A) where A is the set of connections on G-bundles on M such
that Aτ |∂M = aτ . In Theorem 3.5 in [24] and 4.1 in [32], it is proved by contradiction
that A in fact satisfies the Yang-Mills equations. These proofs are completely local,
and hold unchanged in our case.
The proofs of regularity of the connection in Hodge gauge are also local and hold
unchanged. Regularity except for at the points x1, ..., xk from Theorem 10 is a
consequence of the ellipticity of the Yang-Mills equations in Hodge gauge (Theorem
4.5, [24]). At the points x1, ..., xk, the limiting connection may not be defined, so
removable singularity theorems are needed to extend A to these points. The case of
interior points is covered by Theorem 4.1 in [38], and by Theorem 4.6 in [24], so that
the connection A extends to a smooth connection (provided the Dirichlet boundary
data is smooth). More precisely, we have
Theorem 12 Let U (1) (U (2)) be a neighborhood of type 1 (2); let U(i)∗ = U (i)\ 0.
Let A be a connection in a bundle P over U(i)∗ , ‖FA‖L2(U) < B < ∞. Then
(Type 1) If A is Yang-Mills on P |U (1)∗ , there exists a C∞ connection A0
defined on U (1) such that A0 is gauge-equivalent to A on U(1)∗ .
(Type 2) If A is Yang-Mills and C∞ on P |U (2)∗ , there exists a C∞ connection A0
defined on U (2) such that A0 is gauge-equivalent to A on U(2)∗ ,
by a gauge transformation in C∞ (U∗).
Proof. See [38], [24].
48
4.5 The Euclidean Yang-Mills Hamilton-Jacobi func-
tional
Having shown the existence of an absolute minimizer At for the Euclidean Yang-Mills
action given prescribed smooth initial tangential components A = aτ , we can now
define the Hamilton-Jacobi functional3
S (A) =
∫
R+×R3tr (FAt ∧ ∗FAt) dt,
where ∗ indicates the Hodge star operator in the Euclidean metric.
The values of this functional are well-defined even allowing for the possible ex-
istence of more than one gauge-equivalence class of minimizers for the given initial
data; in principle we can simply choose a minimizer starting from the field config-
uration A = aτ . However, while still an open question, there exist partial results
toward establishing uniqueness of a minimizer for given initial data in the compact
case. In [13], Isobe has shown that for flat boundary values, the Dirichlet problem on
a star-shaped bounded domain in Rn can only have a flat solution. Non-uniqueness
results are proven by Isobe and Marini [14] for Yang-Mills connections in bundles over
B4, but the solutions are topologically distinct, belonging to differing Chern classes.
On the domain M = R+ × R3, it seems likely that given initial data determines a
minimizer unique up to gauge transformation.
In order to make the claim that S(A) solves the imaginary-time zero-energy Yang-
Mills Hamilton-Jacobi equation, we must also verify its functional differentiability.
This can be done using the same integration by parts argument as in the derivation
of the Euler-Lagrange equation. However, we must first write the solution to the
3Again, we are implicitly assuming that for all initial data A of physical interest, there exists atleast one trajectory As (As=0 = A) such that −I (As) <∞. See the footnote at the end of §3.2.
49
Euclidean Dirichlet problem in a global gauge which is smooth and decays sufficiently
rapidly at spatial and temporal infinity.
First, Theorem 12 implies that the solution A to the Yang-Mills Dirichlet problem
extends to a smooth connection on a smooth bundle over all of M = R+×R3. Since
the only bundle over a contractible base manifold is the trivial one (see e.g. [27]), A
is also a connection on the trivial bundle P ∼= M ×G. Therefore we can write A in
terms of a smooth global section σ : M → G. Using this trivialization, D = d + A
is smoothly defined over all of M .
The following lemma controls the growth of A and F, for a good choice of gauge.
Part (a) is a version of Uhlenbeck’s Corollary 4.2 [38] for our base manifold M =
R+×R3; part (b) extends the same principle to bound the growth of the connection
1-form A.
Lemma 13 Let D = d + A be a connection in a bundle P over an exterior region
V =y ∈ R+ ×R3 : |y| > N
satisfying
∫V|F |2 < ∞. Then
(a) |F | ≤ C |y|−4 for some constant C (not uniform);
(b) There exists a gauge in which D = d + A satisfies∣∣∣A
∣∣∣ ≤ K |y|−2.
Proof. (a) Following the reasoning in [38], we define the conformal mapping
f : U∗ → V
y = f (x) = N x|x|2
,
where U∗ =x ∈ R+ × R3 : 0 < |x| ≤ 1
. By conformal invariance of the Yang-Mills
action, we have ∫
U∗
|f∗F |2 =
∫
U∗
|F (f ∗D)|2 =
∫
V
|F |2 .
Applying part (b) of Theorem 12 to the pullback f∗D of D under f , there exists a
50
gauge transformation σ : U∗ → G in which f ∗D extends smoothly to U . Thus using
the transformation law for 2-forms, we have the following
|F (y)| = |f ∗F (x)| |df (x)|−2
≤ maxx∈U
|f ∗F (x)| ·(N/ |x|2
)−2
= C ′N2 |y|−4
(b) Define the gauge transformation s = σ f−1 : V → G. Denoting As =
s−1ds+s−1As by A and (f ∗A)σ = σ−1dσ+σ−1 (f ∗A)σ by f ∗A, we have f ∗A = f∗A.
Thus again applying Theorem 11(b) and using the transformation law for 1-forms,
∣∣∣A (y)∣∣∣ =
∣∣∣f ∗A(x)∣∣∣ |df (x)|−1
≤ maxx∈U
∣∣∣f ∗A(x)∣∣∣ ·
(N/ |x|2
)−1
= C ′′N |y|−2 .
We are now ready to prove differentiability of our Hamilton-Jacobi functional.
Thanks are due to V. Moncrief for suggesting the form of this argument.
Theorem 14 The functional
S (A) = −I (A) =
∫
R+×R3tr (FAt ∧ ∗FAt) dt
is functionally differentiable, and δSδA
= E = At=0.
Proof. To find the functional derivative of S(A) = −I (At) at a given connection
51
A0 on the slice x0 = 0, consider the 1-parameter family A0 + λh, constructing
d
dλ[S (A0 + λh)]
∣∣∣∣λ=0
= limλ→0
S (Aλ) − S(A0)
λ
= limλ→0
−I (Aλ,t) −(−I(A0,t)
)
λ
= − limλ→0
1
λ
[I (Aλ,t) − I(A0,t)
]
where for each Aλ = A0+λh, Aλ,t denotes the absolute minimizer of −I given initial
data Aλ. For any given value λ0, the difference I (Aλ0,t) − I(A0,t) can be expressed
in terms of a Taylor series, as follows. First, use the parameter λ to interpolate
between A0,t and Aλ0,t, describing a 1-parameter family Xλ,t,
Xλ,t ≡λ
λ0Aλ0,t +
(1 − λ
λ0
)A0,t,
so that Xλ,0 = Aλ. The standard Taylor series expansion of I (Xλ,t) as a function
of λ then gives
I (Aλ0,t) − I(A0,t) = λ0
(∂I
∂λ
)
λ=0
+ O(λ20
). (4.4)
Let ht = 1λ0
(Aλ0,t − A0,t), so that Xλ,t = A0,t + λht. Then
∂I
∂λ
∣∣∣∣∣λ=0
=∂
∂λ
[∫
R+×R3
⟨FXλ,t
, FXλ,t
⟩]∣∣∣∣λ=0
= 2
∫
R+×R3
⟨dht + [A0,t, ht] , FA0,t
⟩
= 2 limR→∞
(∫
∂1
〈h, FA0〉 +
∫
∂2
⟨ht, FA0,t
⟩−
∫
0≤|x|<R
⟨ht, D
∗FA0,t
⟩)
where ∂1 = |x| < R,x0 = 0 , ∂2 = |x| = R, x0 > 0.
The last term on the right-hand side vanishes due to the fact that FA0,t is a
solution to the Yang-Mills equations. Working with Aλ0,t and A0,t both in the
52
gauge guaranteed by Lemma 13 (for some fixed N which R eventually surpasses),
the middle term also approaches zero as R approaches infinity, since
⟨ht, FA0,t
⟩≤ |ht|
∣∣FA0,t
∣∣
≤ 1
λ0(|Aλ0,t| + |A0,t|)
∣∣FA0,t
∣∣
≤ 1
λ0(Kλ0 + K0)C0 ·R−6.
Since the area element on ∂2 contributes only a factor of R2, the middle term is easily
seen to vanish. Thus we are left with only the first term, so that
∂I
∂λ
∣∣∣∣∣λ=0
=
∫
R3
〈h, FA0〉 ,
and the definition of functional derivative implies that
δS
δA= E = At=0.
4.6 Gauge and Poincare invariance
In order for the candidate ground state wave functional
Ω(A) = N exp (−S (A))
53
to be physical, it must remain invariant under the action of gauge transformations
g (x), x ∈ R3, on the connection A(x) :
S(g−1dg + g−1Ag
)= S (A) ,
so that S is in fact a functional on the physical configuration space A/G of connections
modulo gauge transformations, rather than the kinematical configuration space A.
Gauge invariance of S follows immediately from its form
S(A) = −∫ ∞
0
L(At, At
)dt =
∫
R+×R3tr (FAt ∧ ∗FAt)
where ∗ denotes the Hodge star operator in the Euclidean metric on R+ ×R3. The
gauge transformation g (x), x ∈ R3 can simply be extended to R+ × R3 by taking
g (t, x) = g (x) constant over R+, and the cyclic property of the trace implies
S(g−1dg + g−1Ag
)=
∫
R+×R3tr (Fg·At ∧ ∗Fg·At)
=
∫
R+×R3tr (FAt ∧ ∗FAt) = S (A) .
Similarly, rotations and translations applied to R3 do not affect the value of
S (A), because we can extend them constantly through time over R+×R3, and by a
change of coordinates the value of the integral defining S (A) is unchanged. The only
remaining Poincare transformations are boosts, which cannot be verified directly in
our canonical framework. The conserved quantity generating an infinitesimal boost
in the xi direction is
CB(i) =
∫
R3
(x0δµi + xiδµ0
)T µ0 dx, (4.5)
54
where T µν = − 14π
F µ
αFνα − 1
4ηµνFαβF
αβ
is the stress-energy tensor of Yang-Mills
theory. This constraint must be promoted to a quantum operator which annihilates
our candidate ground state. A test case in which this can be done is the abelian
case of U (1) gauge theory (see §2.3) Using the abelian case as a model, we hope to
extend invariance under boosts to the nonabelian case in future work.
55
Chapter 5
Future work
Immediate plans for future work center around the question of invariance under
boosts of the nonlinear normal ordered ground state for Yang-Mills theory. In
the abelian case, we have verified invariance under boosts as described in §2.3; the
nonabelian case remains.
Additionally, uniqueness of the solution to the Euclidean Yang-Mills Dirichlet
problem is still an open question, and one which it would be desirable to settle.
Gauge freedom renders this problem more difficult to address than, for example, in
the case of scalar ϕ4 theory; however in the compact case, partial uniqueness results
have been established by Isobe and Marini [14], and Isobe [13]. Using a conformal
transformation to map the problem to the compact case, as described in §4.4, is a
possible means of approach.
Beyond these questions filling out the current framework, there are several direc-
tions in which the program can be extended. Most important are its application
to coupled Yang-Mills-Higgs theory, and to general relativity. Another sideline con-
cerns the possibility of defining a non-Gaussian measure from the ground state found
in Chapter 3 for scalar ϕ4 theory.
56
5.1 Yang-Mills-Higgs theory
In the nonabelian Higgs model, the Yang-Mills connection A is coupled to a g-valued
scalar field ϕ. We follow notation of Jaffe and Taubes in [16]. The Lagrangian
action is given by
I =1
4
∫ ∞
0
∫
R3
trFµνF
µν + (dAϕ)µ (dAϕ)µ + λ(|ϕ|2 − 1
)2dx dt.
Canonically, the “position” variables are ϕ,A.
In performing a Legendre transformation, we will again encounter the problem
of indefiniteness due to gauge freedom. In [31], Salmela discusses resolutions of this
issue by gauge-fixing, examining various methods of subsequently implementing the
Gauss constraint. This work will likely prove valuable to our project.
In order to construct a nonlinear normal ordered ground state for Yang-Mills-
Higgs theory, we need to solve the Euclidean Dirichlet problem. Work on regularity
properties of solutions to the Yang-Mills-Higgs system has been done by Parker [29]
and Otway [28].
5.2 General relativity
In presenting the action for general relativity, we follow Smolin’s notation in [34].
The structure group for general relativity is SU (2); we denote by σI3I=1 a basis
of the Lie algebra su(2). Capital Latin indices run over this basis, while lowercase
letters index the three spatial parameters on each spacelike slice Σ of the spacetime
manifold M , assumed to be of the form R × Σ. The “position” in the Ashtekar-
variables framework is an SU (2) connection AIi on Σ; its conjugate momentum is a
57
densitized inverse triad EiI , related to the spacelike metric qij on Σ by
det (q) qij = EiIEjI .
In terms of these variables, the Lagrangian action for general relativity is given by
I =
∫ ∞
0
∫
Σ
EiIAiI −NH−N iHi − wIGI
dx dt.
Each term after the first is a constraint term:
GI = DiEiI = 0
Hi = EjIF
Iij = 0
H = εIJKEiIE
jJ
(FijK − Λ
6εijkE
kK
)= 0,
where Λ is the cosmological constant and as before, F is the curvature of A, given
in coordinates by
F Iij = ∂iA
Ij − ∂jA
Ii + [Ai, Aj ]
I
Just as in Yang-Mills theory, the Gauss law constraint wIGI generates an infinites-
imal gauge transformation given by wI (x) σI . The Hamiltonian constraint NH gen-
erates time evolution, and N iHi generates infinitesimal diffeomorphisms. It is here
that we encounter the largest difference between Yang-Mills theory and general rela-
tivity: where Yang-Mills theory only requires invariance under the Poincare group,
a finite-dimensional subgroup of the diffeomorphism group, general relativity must
respect full diffeomorphism invariance. We thus have infinitely many constraints
to promote to quantum operators, which will undoubtedly render the problem of
factor-ordering the quantum constraints much more difficult.
58
On the other hand, encouragement for the goal of constructing a nonlinear normal
ordered ground state for general relativity comes from Moncrief and Ryan’s explicit
solution in the case of vacuum Bianchi IX cosmology (see [26]). Another starting
point for investigations on a ground state for quantum gravity is Kuchar’s ground
state for linearized gravity [21]
Ω (h) = N exp
− 1
8π2
∫
R3
∫
R3
(hTTik,l(x)
)·(hTTik,l(y)
)
|x− y|2dx dy
,
in terms of the linearized metric tensor
hik = gik − ηik
in vacuum gauge (denoted hTTik ). Strong analogy between this functional and
Wheeler’s ground state for free Maxwell theory (cf. §2.3) is manifest.
A caveat for the Ashtekar variables is the fact that certain versions of the for-
malism (e.g. the original presentation of Sen [33], Ashtekar [1]) require the basic
variables to take complex values. In fact, the usual construction of the Kodama
state necessitates complexifying in order to obtain nontrivial connections A satisfy-
ing the self-dual condition
Fijc −Λ
6εijkE
kc = 0.
This condition leads to the Hamilton-Jacobi equation
Fijc +Λ
6εijk
δS(A)
δAck
= 0,
satisfied by the Chern-Simons functional SCS as given in (1.1); thus exp (SCS (A))
is annihilated by the self-dual condition when promoted to a quantum operator.
59
Working with complex fields, however, is physically undesirable.
5.3 Scalar ϕ4 non-Gaussian measure
In Chapter 3 we obtained the ground state
Ω (ϕ) = N exp
−∫
R+×R3
∣∣(4)∇ϕt
∣∣2 + λϕ4t + m2ϕ2t dx dt
for scalar ϕ4 theory (where as before, ϕt is a minimizer of −I(ϕ) for the initial data
ϕt=0 = ϕ(x)). The natural next question is whether we can use Ω (ϕ) to define a
non-Gaussian measure of the form
dµ(ϕ) = Ω2 (ϕ) dϕ.
Denoting the free scalar field ground state by
Ω0 (ϕ) = N0 exp
−∫
R+×R3
∣∣(4)∇ϕt
∣∣2 + m2ϕ2t dx dt
= N0 exp
−⟨ϕ,H1/2ϕ
⟩2
2
,
we can regard the interacting ground state Ω (ϕ) as a perturbation of Ω0 (ϕ) by the
term O(ϕ) =∫R+×R3
λϕ4tdxdt. From §2.2 we know that Ω0 (ϕ) indeed determines a
Gaussian measure on S ′ (R3).
Since ϕt is determined by ϕ, and moreover its decay is controlled by the fact that it
must minimize −I(ϕ), the time dependence on the right-hand side of O(ϕ) integrates
out. It thus seems possible that this perturbation might be usefully estimated in
terms of the perturbation∫R3
λϕ4dx from the 3-dimensional Euclidean path-integral
approach. Techniques such as those used in [7] might then be employed to define a
60
non-Gaussian measure for our situation.
In the path-integral formalism, functional measures for scalar ϕ4 theory have only
been shown to exist for three spacetime dimensions, as in [7]; a rigorous measure for
scalar ϕ4 path-integrals in four spacetime dimensions has so far eluded definition.
Canonically, however, the full four-dimensional spacetime theory is described by
integration over field configurations on the three-dimensional spatial slice; hence one
fewer dimension is necessary to recover the full theory than in the path-integral
approach. Thus existing results from ϕ43 path integral theory may possibly be
relevant to the full theory here.
61
Appendix
Collected here are a few results and definitions used in this thesis. Throughout, we
follow notation of Baez and Muniain [5], Nakahara [27], and Westenholz [40].
Hodge theory
Real-valued differential forms: The following material is completely stan-
dard, and is recorded here for reference within the thesis, as well as for comparison
with the definitions given below of similar objects in the theory of g-valued differ-
ential forms. For full details, see [5] or [40]. We follow a similar notation to these
authors.
We denote the space of differential k-forms on the manifold M as Λk (M). The
set of all differential forms Λ (M) = ⊕kΛk (M) can be thought of as the algebra
generated by Λ1 (M) with the relation ω ∧ µ = −µ ∧ ω. For higher-order forms
ω ∈ Λk (M) and µ ∈ Λl (M), we have
ω ∧ µ = (−1)kl µ ∧ ω.
A Riemannian metric g on the manifold M allows us to define the Hodge star operator
∗ : Λk (M) → Λn−k (M) . For a positively oriented orthonormal basis of 1-forms
62
e1, ..., en,
∗(ei1 ∧ · · · ∧ eik
)= εi1,...,in
(eik+1 ∧ · · · ∧ ein
);
extend linearly from this definition to the whole of Λk (M). The Hodge star operator
satisfies (∗)2 = (−1)k(n−k).
From the Hodge star operator, we can in turn define an L2 inner product on the
space of k-forms:
〈ω, µ〉2 =
∫
M
ω ∧ ∗µ. (5.1)
The other basic operator on Λ (M) is the exterior derivative d : Λk (M) →
Λk+1 (M), defined as the set of linear maps satisfying
(i) d (ω ∧ µ) = dω ∧ µ + (−1)p ω ∧ dµ, for ω ∈ Λp (M) and µ ∈ Λ (M) .
(ii) d (dω) = 0 for all ω ∈ Λ (M) ,
and such that d : Λ0 (M) → Λ1 (M) on real-valued functions is the 1-form defined
by df (v) = v (f) = vµ ∂f∂xµ
.
With respect to the inner product (5.1), the exterior derivative has an adjoint,
the codifferential, given as
δ ≡ d∗ = (−1)n(k+1)+1 ∗ d∗ (5.2)
when applied to k-forms. The expression in terms of d follows from Stokes’ Theorem.
Just as the exterior derivative extends the gradient operator on functions to all of
Λ (M), the Laplace-de Rham operator, defined as
= dδ + δd,
63
extends the ordinary Laplacian1. In terms of (5.2), we can write
= (−1)nk+1 ((−1)n d ∗ d ∗ + ∗ d ∗ d)
for the Laplace-de Rham operator on k-forms.
g-valued differential forms: Most of the preceding section’s definitions can
be extended to apply not only to real-valued differential forms, but to forms taking
values in the Lie algebra g. Again we follow the presentation of [5], [27], and [39].
For a manifold M , the g-valued k-forms are members of ΛkM⊗g. Denote by ΛM⊗g
the entire space ⊕k
(ΛkM ⊗ g
). Every g-valued k-form can be written as the sum
of terms of the form ω ⊗ S, for ω ∈ Λk (M), S ∈ g. The Hodge dual generalizes to
∗ (ω ⊗ S) = ∗ω ⊗ S,
and we can similarly define the wedge product of g-valued differential forms by
extending linearly from the basic definition
(ω ⊗ S) ∧ (µ⊗ T ) = (ω ∧ µ) ⊗ ST.
These two notions give us the necessary background for introducing the inner product
of g-valued k-forms η and θ:
〈η, θ〉2 =
∫
M
tr (η ∧ ∗θ) .
Note that in the wedge product above, the values from g are combined using the
1More accurately, the Laplace-de Rham operator extends the negative Laplacian −f = − ∂2f∂2x0
−. . .− ∂2f
∂2xn, a convention adopted because − has nonnegative eigenvalues.
64
product operation, but also important is their Lie bracket. For two g-valued forms
η = (ω ⊗ S) ∈ ΛkM ⊗ g and θ = (µ⊗ T ) ∈ ΛlM ⊗ g, we can use their Lie bracket
to form the graded commutator
[η, θ] ≡ η ∧ θ − (−1)kl θ ∧ η
= (ω ∧ µ) ⊗ [S, T ]
where in the last equality, [·, ·] denotes the Lie bracket and the wedge product is
taken in ΛM (using the rule for interchanging ω and µ).
We can apply the exterior derivative on M to the g-valued form η ∈ ΛkM ⊗ g,
by using a basisT I
of g to write η =
∑I ηI ⊗ T I , where ηI ∈ ΛkM , and defining
d : ΛkM ⊗ g → Λk+1M ⊗ g
η → ∑I dηI ⊗ T I ;
following immediately from this are generalizations of the codifferential and Laplace-
de Rham operator.
Together with a connection A ∈ Λ1P ⊗ g on a principal bundle π : P → M , the
exterior derivative allows us to introduce the exterior covariant derivative determined
by A:
dA : ΛkP ⊗ g → Λk+1P ⊗ g
η → dη + [A, η] .
or locally on neighborhoods where A pulls back from Λ1P ⊗ g to Λ1M ⊗ g,
dA : ΛkM ⊗ g → Λk+1M ⊗ g
η → dη + [A, η] .
65
Sobolev spaces on vector bundles
Following Wehrheim [39], we define Sobolev spaces of sections of vector bundles.
Let (M, g) be a Riemannian manifold forming the base space for a vector bundle
π : E → M with an inner product on the fibers and a covariant derivative D; i.e., a
map D : Γ (E) → Γ (T ∗M ⊗ E) satisfying the following relations, for all vector fields
v,w ∈ V ect (M), s, t ∈ Γ (E), f ∈ C∞ (M) , and all scalars α:
Dv (αs) = αDvs
Dv (s + t) = Dvs + Dvt
Dv (fs) = v (f) s + fDvs
Dv+ws = Dvs + Dws
Dfvs = fDvs.
The covariant derivative extends to D : Γ (⊗kT∗M ⊗ E) → Γ (⊗k+1T
∗M ⊗ E) via
the definition
Dω (v0, ..., vk) = Dv0 (ω (v1, ..., vk))−ω (∇v0v1, v2, ..., vk)−...−ω (v1, ..., vk−1, ∇v0vk) ,
for v0, ..., vk ∈ V ect (M) and ∇ the Levi-Civita connection on M induced by the
metric g.
Denote by Γ0 (E) the space of smooth sections of E having compact support on
M , and define the Lp norm on Γ0 (E) by
‖ω‖p =
(∫
M
|ω|p)1/p
.
66
Define the W k,p Sobolev norm by
‖ω‖k,p =
(k∑
j=0
‖Djω‖pp
)1/p
.
Definition 15 The Sobolev space W k,p (M ;E) of sections of the vector bundle π :
E → M is the completion of Γ0 (E) with respect to the norm ‖·‖k,p.
In the case of a principal G-bundle π : P → M , we can define the Sobolev space
of connections using the inner product
〈η, θ〉2 =
∫
M
tr (η ∧ ∗θ) .
To apply the above definition, we must also choose a base connection A0 ∈ A (P )
in the affine space of connections, so that every other connection P can be written
as the sum of A0 and a section of T ∗M ⊗ g. The base connection A0 determines a
covariant derivative DA0 : Γ (⊗kT∗M ⊗ g) → Γ (T ∗M ⊗ g) ,
DA0η (v0, ..., vk) = DA0 (η (v1, ..., vk)) (v0)−η (∇v0v1, v2, ..., vk)−...−η (v1, ..., vk−1, ∇v0vk) ,
where as before ∇ is the Levi-Civita connection on M induced by g, and the k = 0
case of DA0 is given by s → ds + [A, s]. The Sobolev space of connections is then
taken to be
A0 + W k,p (M, T ∗M ⊗ g) .
As noted in [39], if the base manifold M is compact, the space does not depend
upon these choices, although the value of the norm will. Although we extend
to noncompact manifolds in this thesis, we work only over the spaces W k,ploc (M ;E)
comprising sections of E with finite W k,p-norm over any Λ M . Thus choice of
67
base connection makes no difference for us. In fact, since the local trivializations
important for us are over geodesic balls or half-balls, we are free to use a flat base
connection locally.
68
Index of Notation
, Λ M is an open subset of M contained in a compact subset Ω ⊂ M ; Λ ⊂ Ω ⊂
M .
〈·, ·〉2, L2 inner product on functions, L2 inner product on forms, see page 63, 64
‖·‖2, norm corresponding to L2 inner product
(4)∇ =(
∂∂x0
, ∂∂x1
, ∂∂x2
, ∂∂x3
), gradient operator in four Euclidean dimensions
, Laplacian on real-valued functions, Laplace-de Rham operator on differential
forms, see page 63
(4), Laplacian on real-valued functions over four Euclidean dimensions
Λ (M) = ⊕kΛk (M), real-valued differential forms on M , see page 62
ΛM ⊗ g = ⊕k
(Λk (M) ⊗ g
), g-valued differential forms on M , see page 64
Ω (ϕ), ground state wave functional on field configuration ϕ
A, space of G-connections, see page 2, 34
A/G, space of G-connections modulo gauge transformations, see page 2
D, covariant derivative, see page 66
d, exterior derivative on differential forms, see page 63, 65
69
dM , exterior derivative on the manifold M (for disambiguation)
dA, exterior covariant derivative induced by connection A, see page 65
δ, codifferential on differential forms, see page 63
∂1U , ∂2U , boundary of a neighborhood of type 2, see page 39
εi1...ip, completely antisymmetric symbol on i1, ..., ip
G, compact Lie group
G, group of gauge transformations
H (ϕ, π), Hamiltonian density for the field ϕ
H (ϕ, π) =∫R3
H (ϕ, π) dx, Hamiltonian for the field ϕ
H (ϕ, π), H (ϕ, π), imaginary-time Hamiltonian density andHamiltonian for ϕ, see
page 8
I (ϕ) =∫∞0
L (ϕ, ϕ) dt, Lagrangian action for the field ϕ
I (ϕ) =∫∞0
L (ϕ, ϕ) dt, imaginary-time Lagrangian action for the field ϕ, see page 9
L (ϕ, ϕ), Lagrangian density for the field ϕ
L (ϕ, ϕ) =∫R3
L (ϕ, ϕ) dx, Lagrangian for the field ϕ
L (ϕ, ϕ), L (ϕ, ϕ), imaginary-time Lagrangian density and Lagrangian for ϕ, see
page 8
π : E → M , vector bundle over M
π : P → M , principal G-bundle over M
70
R+ = x ∈ R : x > 0
R+ = x ∈ R : x ≥ 0
S (Rn), Schwartz space functions on Rn
S ′ (Rn), dual of Schwartz space on Rn
ταβ : Uα ∩ Uβ → G, transition function on G-bundle, see page 34
U (1), U (2), neighborhoods of type 1 and 2, see page 39
W k,p (M), Sobolev space of functions on M
W k,ploc (M), space of locally Sobolev functions on M
W k,p (M,N), space of Sobolev functions on M taking values in the domain N ⊂ Rn
W k,p (M ;E), space of Sobolev sections of a vector bundle, see page 67
W k,ploc (M ;E), space of locally Sobolev sections of a vector bundle, see page 67
71
Bibliography
[1] A. Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett.
57 (1986), 2244-2247.
[2] A. Ashtekar and J. Lewandowski, Background independent quantum gravity:
A status report, Class. Quant. Grav. 21 (2004), R53-R152.
[3] A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao, T. Thiemann, A manifestly
gauge invariant approach to quantum theories of gauge fields, in Geometry of
constrained dynamical systems, ed. J.M. Charap, Cambridge University Press,
Cambridge, 1995.
[4] J. Baez, Knots and quantum gravity: Progress and prospects, arXiv:gr-
qc/9410018v1 13 Oct 1994.
[5] J. Baez and J.P. de Muniain, Gauge Fields, Knots, and Gravity, World Scien-
tific Press, 1994.
[6] J. Baez and S. Sawin, Functional integration on spaces of connections, arXiv:q-
alg/9507023 v1 20 Jul 1995.
[7] J. Feldman and K. Osterwalder, The Wightman axioms and the mass gap for
weakly coupled (ϕ4)3 quantum field theories, Ann. Phys. 97 (1976), 80-135.
72
[8] G. Folland, Real Analysis: Modern Techniques and Their Applications, Wiley-
Interscience, 1999.
[9] J. Frohlich, On the triviality of λϕ4d theories and the approach to the critical
point in d ≥ 4 dimensions, Nucl. Phys. B 200 (1982), 281-296.
[10] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003.
[11] J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of
View, Springer-Verlag, 1987.
[12] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products,
Academic Press, 1994.
[13] T. Isobe, Non-existence and uniqueness results for boundary value problems
for Yang-Mills connections, Proc. Am. Math. Soc. 125 (1997), 1737-1744.
[14] T. Isobe and A. Marini, On topologically distinct solutions of the Dirichlet
problem for Yang-Mills connections, Calc. Var. 5 (1997), 345-358.
[15] R. Jackiw, Topological investigations of quantized gauge theories, in Current
Algebra and Anoma lies, ed. S.B. Treiman et al, Princeton University Press,
Princeton, 1985.
[16] A. Jaffe and C. Taubes, Vortices and Monopoles: Structure of Static Gauge
Theories, Birkhäuser, 1980.
[17] A. Jaffe and E. Witten, Quantum Yang-Mills theory,
www.claymath.org/millennium/Yang-Mills_Theory/yangmills.pdf
[18] T. Kato, Perturbation Theory for Linear Operators, Springer, 1966.
73
[19] H. Kodama, Holomorphic wavefunction of the universe, Phys. Rev. D 42 (1990),
2548-2565.
[20] H. Kodama, Specialization of Ashtekar’s formalism to Bianchi cosmology, Prog.
Theor. Phys. 80 (1988), 1024-1040.
[21] K. Kuchar, Ground state functional of the linearized gravitational field, Jour.
Math. Phys. 11 (1970), 3322-3334.
[22] H. Lindblad, unpublished notes.
[23] J. Magnen, V. Rivasseau, and R. Sénéor, Rigorous results on the ultraviolet
limit of non-Abelian gauge theories, Commun. Math. Phys. 103 (1986), 67-103.
[24] A. Marini, Dirichlet and Neumann boundary value problems for Yang-Mills
connections, Comm. Pure Appl. Math. 45 (1992), 1015-1050.
[25] V. Moncrief, Can one ADM quantize relativistic bosonic strings and mem-
branes? Gen. Rel. Grav. 38 (2006), 561-575.
[26] V. Moncrief and M. Ryan, Amplitude-real-phase exact solutions for quantum
mixmaster universes, Phys. Rev. D. 44 (1991), 2375-2379.
[27] M. Nakahara, Geometry, Topology, and Physics, IOP Publishing, 2003.
[28] T. Otway, Higher-order singularities in coupled Yang-Mills-Higgs fields, Nonlin.
An. Th., Meth. & Appl. 15 (1990), 239-244.
[29] T. Parker, Gauge theories on four dimensional Riemannian manifolds, Comm.
Math. Phys. 85 (1982), 563-602.
[30] M. Ryan, Cosmological “ground state” wave functions in gravity and electro-
magnetism, arXiv:gr-qc/9312024v1 15 Dec 1993.
74
[31] A. Salmela, Gauss’s Law in Yang-Mills Theory, University of Helsinki doctoral
thesis, 2005.
[32] S. Sedlacek, A direct method for minimizing the Yang-Mills functional over
4-manifolds, Comm. Math. Phys. 86 (1982), 515-527.
[33] A. Sen, Gravity as a spin system, Phys. Lett. B 119 (1982), 89-91.
[34] L. Smolin, Quantum gravity with a positive cosmological constant, arXiv:hep-
th/0209079v1 9 Sep 2002.
[35] E. Stein, Singular Integrals and Differentiability Properties of Functions,
Princeton University Press, 1970.
[36] T. Suzuki, A.C. Hirshfeld, and H. Leschke, The role of operator ordering in
quantum field theory, Prog. Theor. Phys. 63 (1980), 287-302.
[37] K. Uhlenbeck, Connections with Lp bounds on curvature, Comm. Math. Phys.
83 (1982), 31-42.
[38] K. Uhlenbeck, Removable singularities in Yang-Mills fields, Comm. Math.
Phys. 83 (1982), 11-29.
[39] K. Wehrheim, Uhlenbeck Compactness, European Mathematical Society, 2004.
[40] C. von Westenholz, Differential Forms in Mathematical Physics, North-Holland
Publishing Co., 1978.
[41] J.A. Wheeler, Geometrodynamics, Academic Press, 1962.
[42] E. Witten, A note on the Chern-Simons and Kodama wavefunctions, arXiv:gr-
qc/0306083 v2 19 Jun 2003.
75
top related