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The 1996 NATO Advanced Study Institute (ASI) followed the
international tradi tion of the schools held in Cargese in 1976,
1979, 1983, 1987 and 1991. Impressive progress in quantum field
theory had been made since the last school in 1991. Much of it is
connected with the interplay of quantum theory and the structure of
space time, including canonical gravity, black holes, string
theory, application of noncommutative differential geometry, and
quantum symmetries. In addition there had recently been important
advances in quantum field theory which exploited the
electromagnetic duality in certain supersymmetric gauge theories.
The school reviewed these developments. Lectures were included to
explain how the "monopole equations" of Seiberg and Witten can be
exploited. They were presented by E. Rabinovici, and supplemented
by an extra 2 hours of lectures by A. Bilal. Both the N = 1 and N =
2 supersymmetric Yang Mills theory and resulting equivalences
between field theories with different gauge group were discussed in
detail. There are several roads to quantum space time and a
unification of quantum theory and gravity. There is increasing
evidence that canonical gravity might be a consistent theory after
all when treated in. a nonperturbative fashion. H. Nicolai
presented a series of introductory lectures. He dealt in detail
with an integrable model which is obtained by dimensional reduction
in the presence of a symmetry."
Soon after the discovery of quantum mechanics, group theoretical
methods were used extensively in order to exploit rotational
symmetry and classify atomic spectra. And until recently it was
thought that symmetries in quantum mechanics should be groups. But
it is not so. There are more general algebras, equipped with
suitable structure, which admit a perfectly conventional
interpretation as a symmetry of a quantum mechanical system. In any
case, a "trivial representation" of the algebra is defined, and a
tensor product of representations. But in contrast with groups,
this tensor product needs to be neither commutative nor
associative. Quantum groups are special cases, in which
associativity is preserved. The exploitation of such "Quantum
Symmetries" was a central theme at the Ad vanced Study Institute.
Introductory lectures were presented to familiarize the
participants with the al gebras which can appear as symmetries and
with their properties. Some models of local field theories were
discussed in detail which have some such symmetries, in par ticular
conformal field theories and their perturbations. Lattice models
provide many examples of quantum theories with quantum symmetries.
They were also covered at the school. Finally, the symmetries which
are the cause of the solubility of inte grable models are also
quantum symmetries of this kind. Some such models and their
nonlocal conserved currents were discussed.
Bibliograpby . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 325 Critical
point dominance in quantum field models . . . . . . . . . . . . . .
. . . . . . 326 lp, ' quantum fieId model in the single-phase
regioni: Differentiability of the mass and bounds on critical
exponents . . . . 341 Remark on the existence of lp: . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 On
the approach to the critical point . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 348 Critical exponents and elementary
partic1es . . . . . . . . . . . . . . . . . . . . . . . . . . 362 V
Particle Structure Introduction. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 371 Bibliography . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 The
entropy principle for vertex funetions in quantum fieId models . .
. . . 372 Three-partic1e structure of lp' interactions and the
sealing limit . . . . . . . . . 397 Two and three body equations in
quantum field models . . . . . . . . . . . . . . . 409 Partic1es
and scaling for lattice fields and Ising models . . . . . . . . . .
. . . . . . 437 The resununation of one particIe lines. . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 450 VI Bounds on
Coupling Constants Introduction. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 479 Bibliography . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
Absolute bounds on vertices and couplings . . . . . . . . . . . . .
. . . . . . . . . . . . . 480 The coupling constant in a lp' field
theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
VII Confinement and Instantons Introduction. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 497 Bibliography . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
497 Instantons in a U(I) lattice gauge theory: A coulomb dipole gas
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 498 Charges, vortiees and confinement. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 516 vi VIII
ReOectioD Positivity Introduction. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 531 Bibliography . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 A
note on reflection positivity . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 532 vii Collected Papers -
Volume 1 Introduction. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 5 I Infinite
Renormalization of the Hamiltonian Is Necessary 9 II Quantum Field
Theory Models: Parti. The ep;" Model 13 Introduction. . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 13 Fock space. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 17 Q space. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
The Hamiltonian H(g). . . . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 325 Critical
point dominance in quantum field models. . . . . . . . . . . . . .
. . . . . . 326 q>,' quantum field model in the single-phase
regions: Differentiability of the mass and bounds on critical
exponents. . . . 341 Remark on the existence of q>:. . . * . . .
. * . . . . * . . . . . . . . * . * . . . . . . . . . . * . 345 On
the approach to the critical point . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 348 Critical exponents and elementary
particles. . . . . . . . . . . . . . . . . . . . . . . . . . 362 V
Particle Structure Introduction. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 371 Bibliography . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 The
entropy principle for vertex functions in quantum field models. . .
. . 372 Three-particle structure of q>4 interactions and the
scaling limit . . . . . . . . . 397 Two and three body equations in
quantum field models. . . . . . . . . . . . . . . 409 Particles and
scaling for lattice fields and Ising models. . . . . . . . . . . .
. . . . 437 The resummation of one particle lines. . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 450 VI Bounds on
Coupling Constants Introduction. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 479 Bibliography . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
Absolute bounds on vertices and couplings. . . . . . . . . . . . .
. . . . . . . . . . . . . 480 The coupling constant in a q>4
field theory. . . . . . . . . . . . . . . . . . . . . . . . . . .
491 VII Confinement and Instantons Introduction. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 497 Bibliography . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 497 Instantons in a U(I) lattice gauge theory: A coulomb dipole
gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 498 Charges, vortices and confinement. . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 516 ix VIII
Reflection Positivity Introduction. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 531 Bibliography . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 A
note on reflection positivity . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 532 x Introduction This volume
contains a selection of expository articles on quantum field theory
and statistical mechanics by James Glimm and Arthur Jaffe. They
include a solution of the original interacting quantum field
equations and a description of the physics which these equations
contain. Quantum fields were proposed in the late 1920s as the
natural framework which combines quantum theory with relativ ity.
They have survived ever since.
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 325 Critical
point dominance in quantum field models. . . . . . . . . . . . . .
. . . . . . 326 q>/ quantum field model in the single-phase
regions: Differentiability of the mass and bounds on critical
exponents. . . . 341 Remark on the existence of q>. ' . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
On the approach to the critical point . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 348 Critical exponents and
elementary particles. . . . . . . . . . . . . . . . . . . . . . . .
. . 362 V Particle Structure Introduction. . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 371 Bibliography . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
371 The entropy principle for vertex functions in quantum field
models . . . . . 372 Three-particle structure of q>4
interactions and the scaling limit . . . . . . . . . 397 Two and
three body equations in quantum field models . . . . . . . . . . .
. . . . 409 Particles and scaling for lattice fields and Ising
models. . . . . . . . . . . . . . . . 437 The resummation of one
particle lines. . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 450 VI Bounds on Coupling Constants Introduction. . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 479 Bibliography . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 479 Absolute bounds on vertices and couplings . . . . .
. . . . . . . . . . . . . . . . . . . . . 480 The coupling constant
in a q>4 field theory. . . . . . . . . . . . . . . . . . . . . .
. . . . . 491 VII Confinement and Instantons Introduction. . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 497 Bibliography . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 497 Instantons in a U(I) lattice gauge theory: A
coulomb dipole gas. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 498 Charges, vortices and confinement.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
516 vi VIII Reflection Positivity Introduction. . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 531 Bibliography . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 531 A note on reflection positivity . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 532 vii Collected
Papers - Volume 1 Introduction. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 5 I Inimite
Reoormalization of the Hamiltonian Is Necessary 9 II Quantum Field
Theory Models: Part I. The cp~ Model 13 Introduction. . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 13 Fock space. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 17 Qspace. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
The Hamiltonian H(g). . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 39 Removing the space
cutoff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 50 Lorentz covariance and the Haag-Kastler
axioms. . . . . . . . . . . . . . . . . . . . . . 61 Part II. The
Yukawa Model 71 Preliminaries . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72 First and second order estimates. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 86 Resolvent convergence
and self adjointness . . . . . . . . . . . . . . . . . . . . . . .
. . . . 98 The Heisenberg picture. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
The 1996 NATO Advanced Study Institute (ASI) followed the
international tradi tion of the schools held in Cargese in 1976,
1979, 1983, 1987 and 1991. Impressive progress in quantum field
theory had been made since the last school in 1991. Much of it is
connected with the interplay of quantum theory and the structure of
space time, including canonical gravity, black holes, string
theory, application of noncommutative differential geometry, and
quantum symmetries. In addition there had recently been important
advances in quantum field theory which exploited the
electromagnetic duality in certain supersymmetric gauge theories.
The school reviewed these developments. Lectures were included to
explain how the "monopole equations" of Seiberg and Witten can be
exploited. They were presented by E. Rabinovici, and supplemented
by an extra 2 hours of lectures by A. Bilal. Both the N = 1 and N =
2 supersymmetric Yang Mills theory and resulting equivalences
between field theories with different gauge group were discussed in
detail. There are several roads to quantum space time and a
unification of quantum theory and gravity. There is increasing
evidence that canonical gravity might be a consistent theory after
all when treated in. a nonperturbative fashion. H. Nicolai
presented a series of introductory lectures. He dealt in detail
with an integrable model which is obtained by dimensional reduction
in the presence of a symmetry."
Describes fifteen years' work which has led to the construc- tion
of solutions to non-linear relativistic local field e- quations in
2 and 3 space-time dimensions. Gives proof of the existence theorem
in 2 dimensions and describes many properties of the solutions.
Kurt Symanzik was certainly one of the most outstanding theoretical
physicists of our time. For thirty years, until his untimely death
in 1983, he helped to shape the present form of quantum field
theory and its application to elementary particle physics. In
memoriam of Kurt" Symanzik leading scientists present their most
recent results, giving, at the same time, an overview of the state
of the art. This collection was originally published in Vol. 97,
1/2 (1985) of Communications in Mathematical Physics. They range
over various inter related topics of interest to Kurt Symanzik. We
hope that making this collection available in an accessible and
inexpensive way will benefit the physics community. The Publisher
Contents To the Memory of Kurt Symanzik 1 By A. Jaffe, H. Lehmann,
and G. Mack Monte Carlo Simulations for Quantum Field Theories
Involving Fermions. By M. Karowski, R. Schrader, and H. J. Thun
(With 8 Figures) . . . . . . . . . . . . . . . . . . . 5 SU(2)
Lattice Gauge Theory: Standard Action Versus Symanzik's
Tree-Improved Action. By B. Berg, A. Billoire, S. Meyer, and C.
Panagiotakopoulos (With 13 Figures). . . . . . . . . . 31 .
On-shell Improved Lattice Gauge Theories By M. Luscher and P. Weisz
(With 3 Figures) . . . . . 59 On the Modular Structure of Local
Algebras of Observables By K. Fredenhagen . . . . . . . . . . . . .
. . . . 79 . . . The Intersection of Brownian Paths as a Case Study
of a Renormalization Group Method for Quantum Field Theory By M.
Aizenman (With 3 Figures). . . . . . . . . . . . 91 Intersection
Properties of Simple Random Walks: A Renormalization Group
Approach. By G. Felder and J. Frohlich. . . . . . . 111 ."
This volume contains a selection of expository articles on quantum
field theory and statistical mechanics by James Glimm and Arthur
Jaffe. They include a solution of the original interacting quantum
field equations and a description of the physics which these
equations contain. Quantum fields were proposed in the late 1920s
as the natural framework which combines quantum theory with relativ
ity. They have survived ever since. The mathematical description
for quantum theory starts with a Hilbert space H of state vectors.
Quantum fields are linear operators on this space, which satisfy
nonlinear wave equations of fundamental physics, including coupled
Dirac, Max well and Yang-Mills equations. The field operators are
restricted to satisfy a "locality" requirement that they commute
(or anti-commute in the case of fer mions) at space-like separated
points. This condition is compatible with finite propagation speed,
and hence with special relativity. Asymptotically, these fields
converge for large time to linear fields describing free particles.
Using these ideas a scattering theory had been developed, based on
the existence of local quantum fields."
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 325 Critical
point dominance in quantum field models. . . . . . . . . . . . . .
. . . . . . 326 q>/ quantum field model in the single-phase
regions: Differentiability of the mass and bounds on critical
exponents. . . . 341 Remark on the existence of q>. ' . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
On the approach to the critical point . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 348 Critical exponents and
elementary particles. . . . . . . . . . . . . . . . . . . . . . . .
. . 362 V Particle Structure Introduction. . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 371 Bibliography . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
371 The entropy principle for vertex functions in quantum field
models . . . . . 372 Three-particle structure of q>4
interactions and the scaling limit . . . . . . . . . 397 Two and
three body equations in quantum field models . . . . . . . . . . .
. . . . 409 Particles and scaling for lattice fields and Ising
models. . . . . . . . . . . . . . . . 437 The resummation of one
particle lines. . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 450 VI Bounds on Coupling Constants Introduction. . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 479 Bibliography . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 479 Absolute bounds on vertices and couplings . . . . .
. . . . . . . . . . . . . . . . . . . . . 480 The coupling constant
in a q>4 field theory. . . . . . . . . . . . . . . . . . . . . .
. . . . . 491 VII Confinement and Instantons Introduction. . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 497 Bibliography . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 497 Instantons in a U(I) lattice gauge theory: A
coulomb dipole gas. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 498 Charges, vortices and confinement.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
516 vi VIII Reflection Positivity Introduction. . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 531 Bibliography . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 531 A note on reflection positivity . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 532 vii Collected
Papers - Volume 1 Introduction. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 5 I Inimite
Reoormalization of the Hamiltonian Is Necessary 9 II Quantum Field
Theory Models: Part I. The cp~ Model 13 Introduction. . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 13 Fock space. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 17 Qspace. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
The Hamiltonian H(g). . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 39 Removing the space
cutoff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 50 Lorentz covariance and the Haag-Kastler
axioms. . . . . . . . . . . . . . . . . . . . . . 61 Part II. The
Yukawa Model 71 Preliminaries . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72 First and second order estimates. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 86 Resolvent convergence
and self adjointness . . . . . . . . . . . . . . . . . . . . . . .
. . . . 98 The Heisenberg picture. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliograpby . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 325 Critical
point dominance in quantum field models . . . . . . . . . . . . . .
. . . . . . 326 lp, ' quantum fieId model in the single-phase
regioni: Differentiability of the mass and bounds on critical
exponents . . . . 341 Remark on the existence of lp: . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 On
the approach to the critical point . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 348 Critical exponents and elementary
partic1es . . . . . . . . . . . . . . . . . . . . . . . . . . 362 V
Particle Structure Introduction. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 371 Bibliography . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 The
entropy principle for vertex funetions in quantum fieId models . .
. . . 372 Three-partic1e structure of lp' interactions and the
sealing limit . . . . . . . . . 397 Two and three body equations in
quantum field models . . . . . . . . . . . . . . . 409 Partic1es
and scaling for lattice fields and Ising models . . . . . . . . . .
. . . . . . 437 The resununation of one particIe lines. . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 450 VI Bounds on
Coupling Constants Introduction. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 479 Bibliography . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
Absolute bounds on vertices and couplings . . . . . . . . . . . . .
. . . . . . . . . . . . . 480 The coupling constant in a lp' field
theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
VII Confinement and Instantons Introduction. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 497 Bibliography . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
497 Instantons in a U(I) lattice gauge theory: A coulomb dipole gas
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 498 Charges, vortiees and confinement. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 516 vi VIII
ReOectioD Positivity Introduction. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 531 Bibliography . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 A
note on reflection positivity . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 532 vii Collected Papers -
Volume 1 Introduction. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 5 I Infinite
Renormalization of the Hamiltonian Is Necessary 9 II Quantum Field
Theory Models: Parti. The ep;" Model 13 Introduction. . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 13 Fock space. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 17 Q space. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
The Hamiltonian H(g). . . . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 325 Critical
point dominance in quantum field models. . . . . . . . . . . . . .
. . . . . . 326 q>,' quantum field model in the single-phase
regions: Differentiability of the mass and bounds on critical
exponents. . . . 341 Remark on the existence of q>:. . . * . . .
. * . . . . * . . . . . . . . * . * . . . . . . . . . . * . 345 On
the approach to the critical point . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 348 Critical exponents and elementary
particles. . . . . . . . . . . . . . . . . . . . . . . . . . 362 V
Particle Structure Introduction. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 371 Bibliography . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 The
entropy principle for vertex functions in quantum field models. . .
. . 372 Three-particle structure of q>4 interactions and the
scaling limit . . . . . . . . . 397 Two and three body equations in
quantum field models. . . . . . . . . . . . . . . 409 Particles and
scaling for lattice fields and Ising models. . . . . . . . . . . .
. . . . 437 The resummation of one particle lines. . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 450 VI Bounds on
Coupling Constants Introduction. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 479 Bibliography . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
Absolute bounds on vertices and couplings. . . . . . . . . . . . .
. . . . . . . . . . . . . 480 The coupling constant in a q>4
field theory. . . . . . . . . . . . . . . . . . . . . . . . . . .
491 VII Confinement and Instantons Introduction. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 497 Bibliography . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 497 Instantons in a U(I) lattice gauge theory: A coulomb dipole
gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 498 Charges, vortices and confinement. . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 516 ix VIII
Reflection Positivity Introduction. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 531 Bibliography . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 A
note on reflection positivity . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 532 x Introduction This volume
contains a selection of expository articles on quantum field theory
and statistical mechanics by James Glimm and Arthur Jaffe. They
include a solution of the original interacting quantum field
equations and a description of the physics which these equations
contain. Quantum fields were proposed in the late 1920s as the
natural framework which combines quantum theory with relativ ity.
They have survived ever since.
A survey of mathematical developments during 1998. Topics covered
include mirror principle, symplectic topology, o-minimal
structures, and asymptotic solutions to dynamics of many-body
systems and classical continuum equations.
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