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Mathematically Rigorous Quantum Field Theories with a Nonlinear Normal Ordering of the Hamiltonian Operator

A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy

by Rachel 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 Hamiltonian 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 advisor V. Moncrief, in collaboration with M. Ryan, for quantum mechanical situations. Gauge, rotation, and translation invariance are directly verifiable for the nonlinear 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.

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c 2007 by Rachel Lash Maitra Copyright  All rights reserved.

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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

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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

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Acknowledgments My first thanks go to my advisor, Vincent Moncrief, for his generosity, encouragement, 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

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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 hospitality. 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 quantization. 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

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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,

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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.

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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], YangMills 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 approach. The gauge fields themselves can be taken as configuration variables, promoting gauge constraints to quantum operators whose commutation relations should reflect the classical Poisson brackets. Given a normalizable ground state wave functional 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 configurations). 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 generating spacetime diffeomorphisms). • Ground state should be normalizable with respect to some measure on configuration 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 YangMills 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   2 SCS (A) = T r A ∧ dA − A ∧ A ∧ A dx, 3 Σ 

where Σ denotes a spacelike slice of the spacetime manifold M ∼ = R × Σ. 3

(1.1)

The Kodama state exhibits many positive features, such as a semiclassical correspondence 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 undesirable 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 harmonic 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| + x, M x , 2

(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   x,T x Ω(x) = N exp − 2 4

(1.3)

(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| + V (x), 2

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)  2 1  ∂S − V (x) = 0. 2 i ∂xi 5

We can then order the quantum Hamiltonian operator as  ˆ =1 H 2 i



 ∂S − iˆ pi i ∂x



 ∂S i + iˆ p , ∂xi

admitting the zero-energy ground state

N exp (−S(x)) under usual assignment of canonical quantum operators xˆi : ψ(x) → xi ψ(x), pˆi : ψ(x) → −i ∂x∂ i ψ(x). This factorization can be illustrated with the anharmonic oscillator Hamiltonian 1 1 1 H = p2 + x2 + λx4 , 2 2 4

(1.4)

 2 1 2 1 4 in which case the Hamilton-Jacobi equation 12 dS = 2 x + 4 λx is easily integrated dx  3/2 2 2 to yield the solution S(x) = 3λ 1 + λ2 x2 − 3λ . While the resulting ground state is not the usual anharmonic oscillator ground state obtained from the factor ordering ˆ = H

1 2 mω2 2 1 pˆ + 2 xˆ + 4 λˆ x4 , 2m

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.

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1.4

Scope of this thesis

In this thesis, the nonlinear normal ordering will be extended from quantum mechanics to several quantum field theories. The technique is adaptable to any field theory of the form 

1 2 L (ϕ, ϕ) ˙ = ϕ˙ − V (ϕ, Dϕ) dx 2 

1 2 H (ϕ, π) = π + V (ϕ, Dϕ) dx, 2 ∂L π ≡ = 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 ˆ (x) : Ψ(ϕ) → ϕ(x)Ψ(ϕ) π ˆ (x) : Ψ(ϕ) → −i

δ Ψ(ϕ), δϕ(x)

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.

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If well-defined, this measure in turn yields a candidate for the Hilbert space of states:

L2 (Q, dµ).

Using the nonlinear normal ordering ˆ =1 H 2

   δS δS − iˆ π + iˆ π dx δϕ δϕ

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

 2 

δS

dx = V (ϕ, Dϕ) dx

δϕ

(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 



1 − ϕ˙ 2 − V (ϕ, Dϕ) dx 2     1 2 ˜ ˜ H = H(ϕ, π) dx = − π + V (ϕ, Dϕ) dx. 2 ˜ = L

˜ ϕ) L(ϕ, ˙ dx =

The full Hamilton-Jacobi equation is   ∂S δS ˜ + H ϕ, , t = 0; ∂t δϕ

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(1.6)

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 ∂S dS = + dt ∂t



δS ∂ϕt dx. δϕt ∂t

Subsituting from (1.6) we obtain    dS δS δS ∂ϕt ˜ ϕt , = −H + dx dt δϕt δϕt ∂t    δS ∂ϕt δS ˜ − H ϕt , dx. = δϕt ∂t δϕt For At the solution to

˜

∂ϕt δH =

∂t δπ

δS π= δϕ

t

with initial data At=0 = A, we get 

   δS ∂ϕt δS ˜ At , ˜ (ϕt , π t ) dx −H dx = π ϕ˙ t − H δϕt ∂t δϕt = L (ϕt , Dϕt )    ⇒ S ϕt0 − S (ϕ) =

t0

˜ (ϕt , ϕ˙ t ) dt. L

0

Taking S (ϕ) = −I˜ (ϕt ) = −





˜ (ϕt , ϕ˙ t ) dt L

0

  ∞ ˜ (ϕt , ϕ˙ t ) dt. clearly satisfies this relation, since we will then have S ϕt0 = − t0 L

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¯ + the set {x ∈ R : x ≥ 0}, we can alternatively write Denoting by R S (ϕ) = −



¯ + ×R3 R

L˜ (ϕt , ϕ˙ t ) .

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.

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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 variations, 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. Invariance under boosts remains to be investigated in future work.

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Chapter 2 Linear field theories 2.1

Free massive scalar field theory

Free massive scalar field theory, described by the classical Lagrangian 1 L= 2



R3

ϕ˙ 2 − |∇ϕ|2 − m2 ϕ2 dx

and Hamiltonian  1 H = π 2 + |∇ϕ|2 + m2 ϕ2 dx 2 R3   1 = π22 + ϕ, −ϕ + m2 ϕ 2 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)) ,

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where S (ϕ) is the solution to the iHJE 

R3



δS δϕ(x)

2

dx =



R3



 −ϕϕ + m2 ϕ2 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 + m2 ϕ. 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 H 1/2 . Taking   ϕ, H 1/2 ϕ 2 S (ϕ) = 2 as in (1.3), linearity and self-adjointness of H 1/2 yield 

R3



δS δϕ(x)

2

  dx = H 1/2 ϕ, H 1/2 ϕ 2 =



R3

δS δϕ(x)

= H 1/2 ϕ. Thus

  −ϕϕ + m2 ϕ2 dx,

verifying that S is nothing other than the solution of the Hamilton-Jacobi equation (2.1). The operator H 1/2 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). (α/2)

Proposition 1 The integral kernel Km

−α/2

(x) for the operator (Dm )−α/2 = (− + m2 )

is given by (α/2) Km

where

(x) =

m 2

1+α 2

3−α 2

π 3/2 Γ

α 2

|x|

3−α 2

K 3−α (m |x|) , 2

 z ν  1   ∞  ν− 12 Γ 2  Kν (z) = 2 e−zt t2 − 1 dt, 1 Γ ν+2 1

an integral representation of the modified Bessel function of the third kind [12]. (α/2)

Proof. The integral kernel Km (x) will be a function whose Fourier transform is  2 −α/2 m + 4π 2 |x|2 . To find such a function we need two facts: π|x|2

2

3

(i) F −1 (e−πδ|x| ) = e− δ δ − 2 (Fourier transform of a Gaussian)  ∞ −tδ a dδ 1 (ii) t−a = Γ(a) e δ δ (integral representation of Γ(x)) 0 The second fact (ii) implies

(4π)

α/2



2

2

m + 4π |x|

2 −α/2

=

Γ

1 α

14

2



0



e− 4π (m δ

2 +4π 2 |x|2

) δ α2 dδ . δ

So

F

−1



2

2

m + 4π |x|

2 −α/2



=F = = = = = =

−1





1



δ − 4π (m2 +4π2 |x|2 )

dδ δ δ α 2



e   (4π)α/2 Γ α2 0  ∞   δm2 α dδ 1 −1 −πδ|x|2 F e e− 4π δ 2   δ (4π)α/2 Γ α2 0  ∞ 2 2 α dδ 1 −3 − π|x| − δm δ δ 2 · e 4π δ 2 e (using (i))   δ (4π)α/2 Γ α2 0   −3+α  ∞ 2 |x|2 2 4πy dy m2 1 − πm4πy −y e e (y = δ)   m2 y 4π (4π)α/2 Γ α2 0  ∞ 2 |x|2 −3+α dy m3−α −y− m 4y e y 2   3/2 α y (4π) Γ 2 0 −3+α   2 m3−α m |x|   ·2 K 3−α (m |x|) 2 2 23 π 3/2 Γ α2 m

2

1+α 2

3−α 2

π 3/2 Γ

α 2

|x|

3−α 2

K 3−α (m |x|) , 2

where on the second to last line we have used another integral representation of Kν (z) from [12]. This result allows us to write     ϕ, H 1/2 ϕ 2 ϕ, H −1/2 Hϕ 2 S (ϕ) = = 2  2    1 (1/2) = ϕ(x) · Km (x − y) · −ϕ(y) + m2 ϕ(y) dx dy 2 R3 R3 (1/2)

where Km

(x) =

m K 2π 2 |x| 1

(m |x|) , or

      ϕ, H 1/2 ϕ 2 ϕ, HH −3/2 Hϕ 2 Hϕ, H −3/2 Hϕ 2 S (ϕ) = = = 2 2  2     1 2 (3/2) = −ϕ(x) + m ϕ(x) · Km (x − y) · −ϕ(y) + m2 ϕ(y) dx dy 2 R3 R3 (3/2)

where Km

(x) =

1 K 2π2 0

(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   ϕ, C −1 ϕ 2 dµ(ϕ)F = N exp − dϕ 2 where ϕ only runs over the finite-dimensional vector space F  , and hence for dϕ we can sensibly use Lebesgue measure on Rn , n = dim F . 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 ), dim F = 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 projections 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 measure for the free scalar case is obtained by taking

C −1 = 2H 1/2 .

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 Aµ dxµ , 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 = ∂µ Aν − ∂ν Aµ . 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 ∂j Ak . 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 = ∂i A0 − ∂0 Ai . To pass from a Lagrangian to a Hamiltonian formulation of Maxwell theory, one 1 As is customary, Greek indices are spacetime indices, while Latin indices run only over the spatial 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 E i = A˙ i (the negative of the electric field). In terms of these quantities the Hamiltonian is given as 1 H= 2



E 2 + B 2 dx; R3

using the L2 inner product ω, µ 2 =



R3

ω ∧ ∗µ

for 1-forms on R3 , we can re-express it as  1 E22 + B, B 2 2  1 = E22 + ∗dA, ∗dA 2 2  1 = E22 + A, ∗d ∗ dA 2 , 2

H =

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    A, ( ∗ d)−1/2 (∗d) A 2 Ω(A) = N exp − 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     1 (∇ × A(x)) · (∇ × A(y)) dx dy , Ω(A) = N exp − 2 4π R3 R3 |x − y|2

(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 Yµ T µ0 dx, where  µ να 1 µν  1 T µν = − 4π F α F − 4 η Fαβ F αβ is the stress-energy tensor and Y˜ denotes a Killing field in the direction of the infinitesimal transformation (translation, rotation, 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



  ijk x0 x0 δ µi + xi δ µ0 T µ0 dx as the sum of a translation generator 4π ε Ej Bk dx R3    1 plus the term 8π xi |E|2 + |B|2 dx. Translation invariance being already estabR3 R3



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

2 

δS

xi |B|2 dx, x dx = δA 3 3 R R



i

(2.3)

as shown by the following calculation:

          i δS δS x , = xi ∗d −1/2 ∗ dA , ∗d −1/2 ∗ dA 2 δA δA 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   i        −1/2     ∗ dx ∧ ∗d −1/2 ∗ dA , −1/2 ∗ dA 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 

i δS

δS x , δA 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

 i      x (∗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: i



x 

−1/2

B



  1 y Bj (y) 3 = x 2 2 d y 2π R3 |x − y|   1 y (xi Bj (y)) 3 = dy 2π 2 |x − y|2 R3   = −1/2  xi B i

This means that  i  −1/2         x   ∗ dA, −1/2 ∗ dA 2 = −1/2  xi B , −1/2 B 2   = xi B, B 2 by self-adjointness of −1/2 . The relation (2.3) allows the extra term

1 8π



R3

  xi |E|2 + |B|2 dx in the boost

generator to be ordered in the same way as the Hamiltonian, securing boost invariance.

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) ˙ = H (ϕ, π) =



3

R

R3

m2 2 λ 4 1 ∂µ ϕ∂ µ ϕ − ϕ − ϕ dx 2 2 2 2 1 2 1 m 2 λ 4 π + |∇ϕ|2 + ϕ + ϕ dx. 2 2 2 2

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 consider the zero-energy imaginary-time Hamilton-Jacobi equation

2 

δS

dx = |∇ϕ|2 + λϕ4 + m2 ϕ2 dx.

δϕ 3 3 R R



(3.1)

Constructing a solution by the method described in §1.3 requires us to find a min∞ ˜ (ϕt , ϕ˙ t ) dt satisfying the initial data ϕ| imizer ϕt for − 0 L 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) =



F (x, u(x), Du(x)) dx

U

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. lim F(uk ) = inf F . Clearly by definition of infimum, at least one minimizing k→∞

V

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) = inf F . If we have a function V

u = lim uk in some topology, the key to asserting F (u) = lim F(uk ) = inf F will be k→∞

k→∞

V

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 sequence {vk } ⊂ V , vk → v, F(v) ≤ lim inf F(vk ). k→∞

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 inf F ≤ F (u) ≤ lim inf F(uk ) = inf F , V

k→∞

V

or F (u) = inf F . V

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 

uk , u ∈ L1 (U, X), 1

 zk 2 z 

ν

zk , z ∈ L (U, R ), Then



U

F (x, u, z) dx ≤ lim inf k→∞



in L1loc (U );

(*)

F (x, uk , zk ) dx. U

A proof is given in [10], and will not be repeated here. To apply this theorem to functionals of the form F (u) = Rellich’s Theorem (Theorem 3.13 in [10]) can be employed:



U

F (x, u, Du) dx,

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∗ ≡

np . n−p

Then the immersion1

W 1,p (Λ) 5→ 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 1,p weakly to u in Wloc (U ) must also converge strongly to u in Lploc (U ). Weak conver1,p gence in Wloc (U ) is equivalent to weak W 1,p convergence in every open bounded set 1



Note that the immersion W 1,p (Λ) → Lp (Λ) is guarateed to exist by the Sobolev imbedding ∗ theorems, 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) = U F (x, u, Du) dx and 1,1 a sequence {uk } weakly convergent to u in Wloc (U, X). Such a sequence converges

strongly to u in L1loc (U, X), as discussed. Meanwhile {Duk } lies in L1loc (U), and any ∗

1,1 functional ω ∈ (L1loc (U )) and acting on Dv, v ∈ Wloc (U, X), can be interpreted to  1,1 ∗ 1,1 be a functional in Wloc (U, X) . Thus weak convergence of {uk } in Wloc (U, X)

implies weak convergence of {Duk } in L1loc (U ). Taking z in Theorem 3 to be Du

1,1 and zk to be Duk , (∗) holds for {uk } weakly convergent to u in Wloc (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) =



U

F (x, u, Du) dx,

one last observation simplifies the search for a minimizing sequence.

Holder’s in-

1,p 1,1 1,p equality ensures that Wloc (U, X) (p > 1) is a subset of Wloc (U, M) and that Wloc 1,1 1,p 1,1 convergence implies Wloc convergence and Wloc weak convergence implies Wloc weak 1,p 1,1 convergence. This is fortunate, because Wloc has the major advantage over Wloc of

being a reflexive space. By the Alaoglu Theorem, in a reflexive space every bounded sequence has a weakly convergent subsequence. 1,p Since a minimizing sequence uk 2 u in Wloc (U, X) will also converge weakly in 1,1 Wloc (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 1,1 respect to the weak topology of Wloc (U, X) implies lower semicontinuity of F(u) with 1,p 1,p respect to weak Wloc (U, X), also implying that uk 2 u in Wloc (U, X) is sufficient.

27

Moreover as remarked above, Alaoglu’s Theorem yields that in fact a minimizing 1,p sequence bounded in Wloc (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): 1,p Theorem 6 If F is lower semicontinuous in the weak topology of Wloc (U, X) and 1,p V is a weakly closed subset of Wloc (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 + m2 ϕ2 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 2 u implies Lploc convergence of of the traces ψ k → ψ on ∂U (subject to conditions on ∂U , which in our case is 28

¯ + × R3 , and therefore easily satisfies the requisite simply the t = 0 copy of R3 in R smoothness hypotheses). Therefore V is weakly closed, as required. The functional 1,2 F is coercive by inspection with respect to Wloc (R+ × R3 ), and equally easily seen

to be Caratheodory, positive, and convex, verifying that it is lower semicontinuous 1,2 in the weak topology of Wloc (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 + m2 ϕ2 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 + m2 ϕi + λϕ3i = 0. Use the parameter τ ∈ [0, 1] to interpolate linearly between ϕ1 and ϕ2 : ϕτ ≡ τ ϕ1 + (1 − τ )ϕ2 ,

29

so that

dϕτ dτ

= ϕ1 − ϕ2 . Starting from the Euler-Lagrange equation, we have

0 =



1

0 1

 d  (4) − ϕτ + m2 ϕτ + λϕ3τ dτ dτ

−(4)  (ϕ1 − ϕ2 ) + m2 (ϕ1 − ϕ2 ) + 3λϕ2τ (ϕ1 − ϕ2 ) dτ 0  1 (4) 2 = −  (ϕ1 − ϕ2 ) + m (ϕ1 − ϕ2 ) + 3λ (ϕ1 − ϕ2 ) ϕ2τ dτ . =

0

¯ + × R3 , we obtain Multiplying both sides by (ϕ1 − ϕ2 ) and integrating over R 0=



¯ + ×R3 R



(4)

2

(ϕ1 − ϕ2 ) −  (ϕ1 − ϕ2 ) + m (ϕ1 − ϕ2 ) + 3λ (ϕ1 − ϕ2 )



1 0

ϕ2τ





;

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=



¯ + ×R3 R

 1

(4)

2 2 2 2 2

∇ (ϕ1 − ϕ2 ) + m (ϕ1 − ϕ2 ) + 3λ (ϕ1 − ϕ2 ) ϕτ dτ . 0

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 ¯ + × R3 . For notational convenience, we now allow ϕ to denote the t = 0 slice of R ¯ + × R3 as above). Each field configuration field configurations on R3 (rather than R 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) =



¯ + ×R3 R

(4)

∇ϕt 2 + λϕ4t + m2 ϕ2t dx dt.

The corresponding zero-energy ground state for the nonlinear normal ordering is

Ω (ϕ) = N exp (−S (ϕ)) .

2

Note that for S (ϕ) to be finite, we are implicitly making the assumption that for every set of initial  ∞ data ϕ we are interested in, there exists at least one trajectory ϕs (ϕs=0 = ϕ), for which ˜ (ϕs , ϕ˙ s ) dt is finite. This constraint defines the set of physical fields, since for any ϕ on − 0 L 3 ¯ + × R3 can be found, allowing S (ϕ) to take an infinite value implies R for which no such ϕs on R 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 electromagnetism. 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 ∈ Λ1 P ⊗ g.

Given a local section σ α : Uα → P for some

neighborhood Uα ⊂ M , the connection 1-form A pulls back to a g-valued 1-form1 Aα = σ ∗α A on Uα ; the transformation of A on overlapping neighborhoods Uα and Uβ is given by a transition function τ αβ : Uα ∩ Uβ → G, defined by σ β (x) = σ α (x) τ αβ (x): Aα (x) = τ αβ (x)−1 dτ αβ (x) + τ αβ (x)−1 Aβ (x) τ αβ (x), x ∈ Uα ∩ Uβ . The important quantity for Yang-Mills theory is the curvature F ∈ Λ2 P ⊗ g of the connection A, given by F = dP A +

1 2

[A, A] where the bracket [·, ·] denotes

the graded commutator on forms, so that [A, A] = 2 (A ∧ A) . In terms of a local section σ α : Uα → P , F pulls back to a g-valued 2-form Fα = σ ∗α F on Uα , given by Fα = dM Aα + 12 [Aα , Aα ], transforming as Fα (x) = τ αβ (x)−1 Fβ (x) τ αβ (x) , x ∈ Uα ∩ Uβ for τ αβ as given above. In local coordinates, F reduces to

Fµν = ∂µ Aν − ∂ν Aµ + [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 . 1

For 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 1 I(A) = 2



1 tr (F ∧ ∗F ) = 4 M



M

√ trFµν F µν 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

dA h, F =

∂N



tr (dA h ∧ ∗F )  tr (h ∧ ∗F ) − tr (h ∧ dA ∗ F ) . N

N

It is evident that δ h (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 ¯ + × R3 . Since M is contractible, every bundle over M specialize to the case M = R 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 = A˙ Ii (this is the negative of the “electric field” variable). With respect to these variables we obtain the Hamiltonian 1 H= 2



R3

  tr E 2 + B 2 ,

(4.2)

∞ where B i = 21 εijk Fjk . Hamilton’s equations follow by writing the integral J = 0 H ∞ ∞ dt as 0 R3 H dx dt = −I + 0 R3 E A˙ dx dt. Varying both expressions with respect to a one-parameter family Aλ (where the variation has compact support in

¯ + × R3 and vanishes for t = 0), we arrive at the equality R dJ = dλ







δH δH δA + δE δE 0 R3 δA  ∞  dI ˙ = Eδ A˙ + AδE − dλ 3 0 ∞ R  dI ˙ ˙ = −EδA + AδE − , dλ 0 R3

using integration by parts. In order for equality to hold between the first and last

36

lines for all variations, Hamilton’s equations δH E˙ = − δA δH A˙ = δE must be equivalent to the vanishing of the variation

4.2

δI . δA

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  I    Ai (x) , AJj (y) = 0 = EIi (x) , EJj (y) ,

 j  EJ (y) , AIi (x) = δ 3 (x, y) δ ji δ IJ .

With the assignment of quantum operators AˆIi (x) : ψ(A) → AIi (x)ψ(A) δ EˆIi (x) : ψ(A) → −i I ψ(A), δAi (x)

(4.3)

the commutators of mirror the classical Poisson brackets, as required:   AˆIi (x), AˆJj (y) = 0 = EˆIi (x), EˆJj (y) ,

 EˆJj (y), AˆIi (x) = −iδ 3 (x, y) δ ji δ 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 nonlinear 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

Di E i = 0

must be dealt with separately, either by promoting to a quantum operator and verifying 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

ωp =



M

|ω|

p

 p1

=



p/2

(ω, ω)

M

 p1

=



M

[tr (ω ∧ ∗ω)]

p/2

 p1

.

(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 ω Ii1 ...ik (x) of ω to be Lp or Sobolev functions. In this notation, the Yang-Mills action is given by I(A) =

1 F 22 . 2

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   x ∈ ∂U (2) : x0 = 0 ,   ∂2 U = x ∈ ∂U (2) : |x| = 1 . ∂1 U =

¯ + × R3 with the Euclidean metIn our problem, the manifold of interest is R ric; however we solve the Yang-Mills Dirichlet problem for a general smooth 4dimensional Riemannian manifold with boundary, generalizing Marini’s procedure to the non-compact case (§4.4). Certain results used are also valid in general dimension 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 1,2 semicontinuous with respect to the weak topology on Wloc (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 2 A in W 1,2 (U), then I (A) ≤ lim inf i→∞ I (Ai ). Locally we can write Fi = dAi +

1 [Ai , Ai ] . 2

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 5→ 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 ] 2 [A, A], so that Fi 2 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 localizing 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 = ¯ + × R3 into a compact manifold with boundary, using inversion in the sphere (this R suggestion is due to T. Damour). In this approach, one considers the unit sphere ¯ + × R3 into R4 as the set {x : x0 ≥ 2}. The centered at the origin of R4 , imbedding R inversion mapping is yi = 2

2

2

xi , r2

2

where r2 = (x0 ) + (x1 ) + (x2 ) + (x3 ) . 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 ¯ + × R3 has been effectively mapped to a compact problem, problem of interest on R 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 Sobolevbounded 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 

 D = d + A : A ∈ W 1,p (U ), FA Ln/2 (U ) < K    A ∈ W 1,p (U) FA n/2 < K 1,p BK (U) = D = d+A : ,   Aτ ∈ W 1,p (∂1 U) FAτ Ln/2 (∂1 U ) < K A1,p K (U) =

    

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 1,p for connections belonging to A1,p K (U ) or BK (U). More precisely,

Theorem 9 For

n 2

≤ p < n, there exists K ≡ K(n) > 0 and c ≡ c(n) such that

1,p ˆ every connection D = d + A ∈ A1,p K (BK ) is gauge equivalent to a connection d + A,

43

A ∈ W 1,p (U ), satisfying 

 ˆ  d∗A=0  (a )   dτ ∗ Aτ = 0 on ∂1 U

(a) d ∗ Aˆ = 0 (b) Aˆν = 0 on ∂U

(b ) Aˆν = 0 on ∂2 U

* * * * (c = c ) *Aˆ* < c(n) FAˆ n/2 * *1,n/2 * * (d = d ) *Aˆ* < c(n) FAˆp 

1,p

1,p   (Unprimed conditions (a)-(d) refer to A1,p K (U); primed conditions (a )-(d ) to BK (U)).

Moreover, the gauge transformation s satisfying Aˆ = s−1 ds + s−1 As 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 (∂1 U ) picks up a factor of r under the dilation x = rx, so that the simultaneous conditions FA n/2 < K, FAτ Ln/2 (∂1 U ) < 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 (∂1 U ) 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 SY M ( Ai |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 }

|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

1 n

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   2 {A(i)} for which the corresponding subsequence of Bn (xn,1 ) |F (i)| 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-

a1 (1) a2 (1) 2 Diagonalizing over a list of sequences {aj (i)} such as a3 (1) .. .

a1 (3) . . . a2 (3) . . . a3 (3) . . . selects out .. .. . . the new sequence {ai (i)}. In the case important for this proof, each row represents a subsequence of the previous row, so that for any j, the diagonalized sequence {ak (k)}is a subsequence of {aj (i)} for k ≥ j.

45

a1 (2) a2 (2) a3 (2) .. .

quence {A(i)}I  ⊂ {A(i)}I such that for every ball in every cover, the sequence   2 |F (i)| dx converges. For each Cn , consider the collection of balls Bn (xn,m ) I   2 {Bn (yn,m )}, {yn,m } ⊂ {xn,m }, for which Bn (yn,m ) |F (i)| dx  converges to a value I

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 n1 for which Bn (yn,m ) |F (i)|2 dx ≥ ε : B ≥ Ni,n ε. B ε

limits the number of disjoint balls in the set {Bn (yn,m )} .  J Choose a maximal disjoint set Bn (yn,mj ) j=1 of balls in {Bn (yn,m )}, and consider  J the set Bn∗ (yn,mj ) j=1 of balls centered at the points yn,mj but having radius n3 . Then , , we have {yn,m } Bn (yn,m ) ⊂ Jj=1 Bn∗ (yn,mj ). This shows that if we discard the balls Thus the upper bound

{Bn (yn,m )} from the cover Cn , we will only have discarded a set which was contained balls of radius n3 . We can then safely discard the balls {Bn (yn,m )} from , each cover Cn , and form the union C = n∈N Cn \ {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. in J ≤

B 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 Aˆα (i) in the Hodge gauge, satisfying a uniform W 1,2 (U ) bound on Aˆα (i). Weak compactness of Sobolev   ˆ spaces now yields a further subsequence of Aα (i) , weakly convergent in W 1,2 to some Aˆα . It only remains to show that Aˆα retains the desired regularity properties   and boundary data, and that the set Aˆα 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 . from Theorem 9.

Let ε = K(4), where K(4) is the constant

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 σ α (i) : Uα → Pi (i ∈ I  ) and connections Aa on Uα such that (e)

σ ∗α (i) (Ai ) ≡ Aα (i) 2 Aα in W 1,2 (Uα )

(f)

F (Aα (i)) ≡ Fα (i) 2 Fα in L2 (Uα )

(g)

sαβ (i) 2 sαβ in W 2,2 (Uα ∩ Uβ )

(h)

∼ aτ |∂1 Uα by a smooth gauge transformation (Aα ) |  τ ∂1 Uα   d ∗ Aα = 0 on Uα    dτ ∗ (Aα )τ = 0 on ∂1 U

(i) (j)

−1 Aα ≡ s−1 αβ Aβ sαβ + sαβ dsαβ

Here sαβ (i) is the transition function Aβ (i) → Aα (i); i.e, −1 Aα (i) ≡ s−1 αβ (i) Aβ (i) sαβ (i) + sαβ (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 countable 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 ) ≡ min I (A) where A is the set of connections on G-bundles on M such A

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 (i)

Theorem 12 Let U (1) (U (2) ) be a neighborhood of type 1 (2); let U∗ = U (i) \ {0}. (i)

Let A be a connection in a bundle P over U∗ , FA L2 (U ) < B < ∞. Then (1)

(Type 1) If A is Yang-Mills on P | U∗ , there exists a C ∞ connection A0 (1)

defined on U (1) such that A0 is gauge-equivalent to A on U∗ . (2)

(Type 2) If A is Yang-Mills and C ∞ on P | U∗ , there exists a C ∞ connection A0 (2)

defined on U (2) such that A0 is gauge-equivalent to A on U∗ , by a gauge transformation in C ∞ (U∗ ). Proof. See [38], [24].

48

4.5

The Euclidean Yang-Mills Hamilton-Jacobi functional

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) =



¯ + ×R3 R

tr (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 existence 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 configuration 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 B 4 , but the solutions are topologically distinct, belonging to differing Chern classes. ¯ + × R3 , it seems likely that given initial data determines a On the domain M = R minimizer unique up to gauge transformation. In order to make the claim that S(A) solves the imaginary-time zero-energy YangMills 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 3 Again, we are implicitly assuming that for all initial data A of physical interest, there exists at least 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 ¯ + × R3 . Since extends to a smooth connection on a smooth bundle over all of M = R 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 = ¯ + × R3 ; part (b) extends the same principle to bound the growth of the connection R 1-form A. Lemma 13 Let D = d + A be a connection in a bundle P over an exterior region    ¯ + × R3 : |y| > N satisfying V = y∈R |F |2 < ∞. Then V (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|x2 ,   ¯ + × R3 : 0 < |x| ≤ 1 . By conformal invariance of the Yang-Mills where U∗ = x ∈ R action, we have



U∗



2

|f F | =



U∗



2

|F (f D)| =



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  −2 ≤ max |f ∗ F (x)| · N/ |x|2 x∈U

= C  N 2 |y|−4

(b) Define the gauge transformation s = σ ◦ f −1 : V → G.

Denoting As =

∗ A, we have f ∗ A ∗ A. ˜ = fs−1 ds + s−1 As by A˜ and (f ∗ A)σ = σ −1 dσ + σ −1 (f ∗ A) σ by f-

Thus again applying Theorem 11(b) and using the transformation law for 1-forms,



∗˜

˜ −1 f A(x) = A (y)

|df (x)|





−1

∗ A(x) · N/ |x|2 ≤ max fx∈U

= 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) = is functionally differentiable, and

δS δA



¯ + ×R3 R

tr (FAt ∧ ∗FAt ) dt

= E = A˙ t=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 S (Aλ ) − S(A0 ) [S (A0 + λh)]

= lim λ→0 dλ λ λ=0   ˜ 0,t ) −I˜ (Aλ,t ) − −I(A = lim λ→0 λ 1 ˜ ˜ 0,t ) = − lim I (Aλ,t ) − I(A λ→0 λ where for each Aλ = A0 + λh, Aλ,t denotes the absolute minimizer of −I˜ given initial ˜ 0,t ) can be expressed data Aλ . For any given value λ0 , the difference I˜ (Aλ0 ,t ) − I(A 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 λ ≡ Aλ0 ,t + 1 − A0,t , λ0 λ0

so that Xλ,0 = Aλ . The standard Taylor series expansion of I˜ (Xλ,t ) as a function of λ then gives ˜ 0,t ) = λ0 I˜ (Aλ0 ,t ) − I(A 1 λ0

Let ht =

˜ ∂ I

∂λ



∂ I˜ ∂λ



λ=0

  + O λ20 .

(4.4)

(Aλ0 ,t − A0,t ), so that Xλ,t = A0,t + λht . Then

λ=0

∂ = ∂λ  = 2

.

/   FXλ,t , FXλ,t

3

¯ + ×R R

λ=0

  dht + [A0,t , ht ] , FA0,t ¯ + ×R3 R      = 2 lim h, FA0 + ht , FA0,t − R→∞

∂1

∂2

0≤|x| 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 (|Aλ0 ,t | + |A0,t |) FA0,t λ0 1 ≤ (Kλ0 + K0 ) C0 · R−6 . λ0 ≤

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 = E = A˙ t=0 . δA

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 −1 dg + g −1 Ag = 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 =

¯ + ×R3 R

tr (FAt ∧ ∗FAt )

¯ + × R3 . The where ∗ denotes the Hodge star operator in the Euclidean metric on R ¯ + × R3 by taking gauge transformation g (x), x ∈ R3 can simply be extended to R ¯ + , and the cyclic property of the trace implies g (t, x) = g (x) constant over R 

−1

−1

S g dg + g Ag



= =



¯ + ×R3 R

tr (Fg·At ∧ ∗Fg·At )

¯ + ×R3 R

tr (FAt ∧ ∗FAt ) = S (A) .



Similarly, rotations and translations applied to R3 do not affect the value of ¯ + × R3 , and by a S (A), because we can extend them constantly through time over R 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



 x0 δµi + xi δ µ0 T µ0 dx,

54

(4.5)

 µ να 1 µν  1 where T µν = − 4π F α F − 4 η Fαβ F 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 directions in which the program can be extended.

Most important are its application

to coupled Yang-Mills-Higgs theory, and to general relativity. Another sideline concerns 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 1 I= 4



0

∞

R3

  2  tr Fµν F µν + (dA ϕ)µ (dA ϕ)µ + λ |ϕ|2 − 1 dx 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-MillsHiggs 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 {σ I }3I=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 q ij on Σ by det (q) qij = E iI EIj .

In terms of these variables, the Lagrangian action for general relativity is given by

I=



0

∞

Σ

  E iI A˙ iI − N H − N i Hi − w I GI dx dt.

Each term after the first is a constraint term:

GI = Di EIi = 0 Hi = EIj FijI = 0 IJK

H = ε

EIi EJj



FijK

Λ k − εijk EK 6



= 0,

where Λ is the cosmological constant and as before, F is the curvature of A, given in coordinates by FijI = ∂i AIj − ∂j AIi + [Ai , Aj ]I Just as in Yang-Mills theory, the Gauss law constraint w I GI generates an infinitesimal gauge transformation given by wI (x) σ I . The Hamiltonian constraint NH generates time evolution, and N i Hi generates infinitesimal diffeomorphisms. It is here that we encounter the largest difference between Yang-Mills theory and general relativity: 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 8π 2

  

  TT  · hik,l (y)

T hTik,l (x)

R3 R3

|x − y|2



dx dy  ,

in terms of the linearized metric tensor

hik = gik − η ik in vacuum gauge (denoted hTikT ).

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 formalism (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 satisfying the self-dual condition Fijc −

Λ εijk Eck = 0. 6

This condition leads to the Hamilton-Jacobi equation

Fijc +

Λ δS(A) εijk = 0, 6 δAck

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 −

¯ + ×R3 R



(4)

2 4 2 2

∇ϕt + λϕt + m ϕt 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



(4)

2 2 2

Ω0 (ϕ) = N0 exp − ∇ϕt + m ϕt dx dt ¯ + ×R3 R 0   1 ϕ, H 1/2 ϕ 2 = N0 exp − , 2 we can regard the interacting ground state Ω (ϕ) as a perturbation of Ω0 (ϕ) by the  term O(ϕ) = R¯ + ×R3 λϕ4t dxdt. 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 λϕ4 dx 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 standard, 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 differential 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) = ⊕k Λk (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 ∂f by df (v) = v (f ) = v µ ∂x µ.

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 Λk M ⊗g. Denote by ΛM ⊗g   the entire space ⊕k Λk M ⊗ 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 1

2

More accurately, the Laplace-de Rham operator extends the negative Laplacian −f = − ∂∂2 xf0 −

...−

∂2f ∂ 2 xn ,

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) ∈ Λk M ⊗ g and θ = (µ ⊗ T ) ∈ Λl M ⊗ 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 ∈ Λk M ⊗ g,   2 by using a basis T I of g to write η = I ηI ⊗ T I , where η I ∈ Λk M , and defining d : Λk M ⊗ g →

Λk+1 M ⊗ g 2 I  → I dη I ⊗ T ;

η

following immediately from this are generalizations of the codifferential and Laplacede Rham operator. Together with a connection A ∈ Λ1 P ⊗ g on a principal bundle π : P → M, the exterior derivative allows us to introduce the exterior covariant derivative determined by A: dA : Λk P ⊗ g → η

Λk+1 P ⊗ g

→ dη + [A, η] .

or locally on neighborhoods where A pulls back from Λ1 P ⊗ g to Λ1 M ⊗ g, dA : Λk M ⊗ g → Λk+1 M ⊗ 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) = αDv s Dv (s + t) = Dv s + Dv t Dv (fs) = v (f ) s + f Dv s Dv+w s = Dv s + Dw s Df v s = f Dv s.

The covariant derivative extends to D : Γ (⊗k T ∗ M ⊗ E) → Γ (⊗k+1 T ∗ M ⊗ E) via the definition

Dω (v0 , ..., vk ) = Dv0 (ω (v1 , ..., vk ))−ω (∇v0 v1 , v2 , ..., vk )−...−ω (v1 , ..., vk−1 , ∇v0 vk ) , 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

ωp =



M

66

|ω|

p

1/p

.

Define the W k,p Sobolev norm by

ωk,p =

 k  j=0

Dj ωpp

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 (⊗k T ∗ M ⊗ g) → Γ (T ∗ M ⊗ g) , DA0 η (v0 , ..., vk ) = DA0 (η (v1 , ..., vk )) (v0 )−η (∇v0 v1 , v2 , ..., vk )−...−η (v1 , ..., vk−1 , ∇v0 vk ) , 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

k,p to noncompact manifolds in this thesis, we work only over the spaces Wloc (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 ) = ⊕k Λk (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 ∂1 U, ∂2 U , 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

˜ (ϕ, π), H ˜ (ϕ, π), imaginary-time Hamiltonian density andHamiltonian for ϕ, see H page 8 I (ϕ) = I˜ (ϕ) =

∞ 0

∞ 0

L (ϕ, ϕ) ˙ dt, Lagrangian action for the field ϕ ˜ (ϕ, ϕ) L ˙ 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

˜ (ϕ, ϕ), L˜ (ϕ, ϕ), ˙ L ˙ 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} ¯ + = {x ∈ R : x ≥ 0} R S (Rn ), Schwartz space functions on Rn S  (Rn ), dual of Schwartz space on Rn τ αβ : 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 k,p Wloc (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 k,p Wloc (M ; E), space of locally Sobolev sections of a vector bundle, see page 67

71

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