Introduction to Quantum Field Theory - Arthur Jaffe

137 Pages·2005·0.65 MB·English

Introduction to Quantum Field Theory
Arthur Jaffe
Harvard University
Cambridge, MA 02138, USA
(cid:13)c by Arthur Jaffe. Reproduction only with permission of the author.
24 May, 2005 at 7:26
ii
Contents
I Life of a Single Particle 1
1 Introduction 3
2 Life of a Particle in Real Time 5
2.1 Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Poincar´e Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Special Features of a Single Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 The Configuration Space Representation . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5.1 The Momentum and Energy Operators . . . . . . . . . . . . . . . . . . . . . 9
2.6 The Momentum Space Representation . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.7 The Lorentz-Invariant Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.8 The Poincar´e Group on H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Life of a Particle at Imaginary Time 15
3.1 Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 The Euclidean Laplacian and its Green’s Function . . . . . . . . . . . . . . . . . . 18
3.3 Reflection Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Osterwalder-Schrader Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4.1 The Sobolev Space H (O) . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
−1
3.4.2 Why “Quantization”? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4.3 Quantization of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4.4 Some Examples of Quantized Operators . . . . . . . . . . . . . . . . . . . . 26
3.4.5 Unbounded Operators on H . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1
3.4.6 Quantization Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.7 Quantization of Space-Time Rotations . . . . . . . . . . . . . . . . . . . . . 30
3.5 Poincar´e Symmetry from Euclidean Symmetry . . . . . . . . . . . . . . . . . . . . . 30
3.6 Properties of Matrices and Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6.1 Operator Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.6.2 Two Monotonicity Preserving Functions . . . . . . . . . . . . . . . . . . . . 32
3.6.3 The Perron-Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 34
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3.7 Reflection Positivity Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.7.1 Mirror Charges and Classical Green’s Functions . . . . . . . . . . . . . . . . 35
3.7.2 Reflection Positivity & Operator Monotonicity . . . . . . . . . . . . . . . . . 37
3.7.3 Reflection Invariance Ensures Monotonicity . . . . . . . . . . . . . . . . . . 38
3.7.4 Monotonicity & Reflection Positivity . . . . . . . . . . . . . . . . . . . . . . 40
3.8 Space-Time Compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.8.1 Periodic Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.8.2 Periodic Time Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.8.3 Reflection Positivity on Td . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.8.4 Quantization on Td and the Role of S = ΘS . . . . . . . . . . . . . . . . . . 46
3.9 Mirror Space-Time Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.9.1 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.9.2 Time Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.9.3 Reflection Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
II Fock Space 51
4 Sums and Products 55
4.1 The Direct Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 The Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.1 Definition of K ⊗K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1 2
4.2.2 Tensor Products of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.3 The Pointwise Operator Product . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.4 Pointwise Products Preserve Positivity . . . . . . . . . . . . . . . . . . . . . 61
4.3 n-Fold Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Tensor Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.1 The Map Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Symmetric Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5.1 Bosonic Fock Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5.2 Bosonic Creation and Annihilation Operators . . . . . . . . . . . . . . . . . 68
4.6 Anti-Symmetric Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.7 Fermionic Fock Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7.1 Fermionic Creation and Annihilation Operators . . . . . . . . . . . . . . . . 71
5 Number Bounds 73
5.1 Estimates on m(f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Nice Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 The Weyl Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4 Some Additional Properties when H = H (Rd−1) . . . . . . . . . . . . . . . . . . 77
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CONTENTS v
III Quantum Fields 79
6 The Free Bosonic Field 83
6.1 The Local Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.1.1 The Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.1.2 Time-Zero Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 The Free Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2.1 Fields at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2.2 Momentum Space Representation . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2.3 Commutation Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3 Imaginary Time Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4 Compact Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.5 Forms and Number Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.6 Poincar´e Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.7 Locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.8 Wightman Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.9 Reeh-Schlieder Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7 The Fundamental Bound for Fields 91
7.1 The Fundamental Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.1.1 The Fundamental Bound and Field Operators . . . . . . . . . . . . . . . . . 93
7.1.2 The Fundamental Bound and Expectation Values . . . . . . . . . . . . . . . 103
IV Euclidean Fields 105
V Some Analytic Tools 109
8 Linear Transformations on Hilbert Space 111
8.1 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.3 Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.3.1 Analytic Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.4 Operators between Different Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . 118
8.5 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.5.1 The Graph of T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.6 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.7 Convergence of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.7.1 Convergence Based on Traces . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.7.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.7.3 Strong Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
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8.7.4 Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.7.5 Graph Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
9 Fourier Transformation 125
9.1 Fourier Transforms on L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
9.2 Schwartz Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Part I
Life of a Single Particle
1

Chapter 1
Introduction
Our goal is to present a brief and self-contained introduction to quantum field theory from the
constructive point of view. We try to motivate some basic results and relate them to interesting
open problems.
One should mention right at the start that one still does not understand whether quantum
mechanics and special relativity are compatible at a fundamental level in our Minkowski four-space
world. One generally assumes that this means finding a complete Yang-Mills gauge theory or the
interaction of gauge fields with fermionic matter fields, the simplest form being quantum chromo-
dynamics (QCD). Associated with this picture is the belief that the fundamental vector meson
excitations are massive (as opposed to photons, which arise in the limiting case of an abelian gauge
symmetry. The proof of the existence of a “mass gap” appears a necessary integral part of solving
the entire puzzle.
This question remains one of the deepest open issues in theoretical physics, as well as in math-
ematics. Basically the question remains: can one give a mathematical foundation to the theory of
fields in four-dimensions? In other words, can do quantum mechanics and special relativity lie on
the same footing as the classical physics of Newton, Maxwell, Einstein, or Schr¨odinger—all of which
fits into a mathematical framework that we describe as the language of physics. This glaring gap
in our fundamental knowledge even dwarfs questions of whether there are other more complicated
and sophisticated approaches to physics—those that incorporate gravity, strings, or branes—for
understanding their fundamental significance lies far in the future. In fact, one believes that stringy
proposals, if they can be fully implemented, have limiting cases that appear as relativistic quan-
tum fields, just as relativistic quantum fields describe non-relativistic quantum theory and classical
physics in various limiting cases.
We begin with the quantum mechanical treatment of a particle of a given mass. If we assume
that the symmetry of the quantum theory includes the transformations of special relativity, then
much of the structure follows naturally. We then develop the basic Euclidean point of view, that
arises from attempting to analytically continue Lorentz symmetry to Euclidean symmetry. This
provides also the natural connection with path integrals. We specialize the case of a single, free,
bosonic particle; this illustrates many of the main ideas.
3
4 CHAPTER 1. INTRODUCTION
Eachmethodgivesaroutetoquantization. Inthepathintegralframeworkweencounterclassical
fields defined on Euclidean space (with a positive metric and Euclidean symmetry). One encounters
a condition known as reflection (or Osterwalder-Schrader) positivity that allows one obtain a quan-
tum theory (on Hilbert space) from a path integral. The quantum theory that one finds agrees with
the usual picture of canonical quantization that one learns in standard field theory. The quantum
theory also comes with a representation of the inhomogeneous Lorentz group (the Poincar´e group)
that arises from an analytic continuation of the quantization of the Euclidean group.
Thus the two fundamental points of view mesh to one. We first investigate a special case
that relates to the Gaussian path integral and the free quantum field. We then give the general
construction that applies for bosonic non-linear fields.
IntroductiontoQuantumFieldTheory 24May,2005 at7:26