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Physics - Classical Continuum Physics | Quantum Physics - A Functional Integral Point of View

Quantum Physics

A Functional Integral Point of View

Glimm, J., Jaffe, A.

1981, 417p.

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  • About this textbook

This book is addressed to one problem and to three audiences. The problem is the mathematical structure of modem physics: statistical physics, quantum mechanics, and quantum fields. The unity of mathemati­ cal structure for problems of diverse origin in physics should be no surprise. For classical physics it is provided, for example, by a common mathematical formalism based on the wave equation and Laplace's equation. The unity transcends mathematical structure and encompasses basic phenomena as well. Thus particle physicists, nuclear physicists, and con­ densed matter physicists have considered similar scientific problems from complementary points of view. The mathematical structure presented here can be described in various terms: partial differential equations in an infinite number of independent variables, linear operators on infinite dimensional spaces, or probability theory and analysis over function spaces. This mathematical structure of quantization is a generalization of the theory of partial differential equa­ tions, very much as the latter generalizes the theory of ordinary differential equations. Our central theme is the quantization of a nonlinear partial differential equation and the physics of systems with an infinite number of degrees of freedom. Mathematicians, theoretical physicists, and specialists in mathematical physics are the three audiences to which the book is addressed. Each of the three parts is written with a different scientific perspective.

Content Level » Research

Keywords » Finite - Identity - Phase - Physics - Wave - calculus - equation - function - proof - scattering theory - theorem

Related subjects » Applied & Technical Physics - Classical Continuum Physics - Quantum Physics

Table of contents 

I An Introduction to Modem Physics.- 1 Quantum Theory.- 1.1 Overview.- 1.2 Classical Mechanics.- 1.3 Quantum Mechanics.- 1.4 Interpretation.- 1.5 The Simple Harmonic Oscillator.- 1.6 Coulomb Potentials.- 1.7 The Hydrogen Atom.- 1.8 The Need for Quantum Fields.- 2 Classical Statistical Physics.- 2.1 Introduction.- 2.2 The Classical Ensembles.- 2.3 The Ising Model and Lattice Fields.- 2.4 Series Expansion Methods.- 3 The Feynman-Kac Formula.- 3.1 Wiener Measure.- 3.2 The Feynman-Kac Formula.- 3.3 Uniqueness of the Ground State.- 3.4 The Renormalized Feynman-Kac Formula.- 4 Correlation Inequalities and the Lee-Yang Theorem.- 4.1 Griffiths Inequalities.- 4.2 The Infinite Volume Limit.- 4.3 ?4 Inequalities.- 4.4 The FKG Inequality.- 4.5 The Lee-Yang Theorem.- 4.6 Analyticity of the Free Energy.- 4.7 Two Component Spins.- 5 Phase Transitions and Critical Points.- 5.1 Pure and Mixed Phases.- 5.2 The Mean Field Picture.- 5.3 Symmetry Breaking.- 5.4 The Droplet Model and Peierls’ Argument.- 5.5 An Example.- 6 Field Theory.- 6.1 Axioms.- (i) Euclidean Axioms.- (ii) Minkowski Space Axioms.- 6.2 The Free Field.- 6.3 Fock Space and Wick Ordering.- 6.4 Canonical Quantization.- 6.5 Fermions.- 6.6 Interacting Fields.- II Function Space Integrals.- 7 Covariance Operator = Green’s Function = Resolvent Kernel = Euclidean Propagator = Fundamental Solution.- 7.1 Introduction.- 7.2 The Free Covariance.- 7.3 Periodic Boundary Conditions.- 7.4 Neumann Boundary Conditions.- 7.5 Dirichlet Boundary Conditions.- 7.6 Change of Boundary Conditions.- 7.7 Covariance Operator Inequalities.- 7.8 More General Dirichlet Data.- 7.9 Regularity of CB.- 7.10 Reflection Positivity.- 8 Quantization = Integration over Function Space.- 8.1 Introduction.- 8.2 Feynman Graphs.- 8.3 Wick Products.- 8.4 Formal Perturbation Theory.- 8.5 Estimates on Gaussian Integrals.- 8.6 Non-Gaussian Integrals, d = 2.- 8.7 Finite Dimensional Approximations.- 9 Calculus and Renormalization on Function Space.- 9.1 A Compilation of Useful Formulas.- (i) Wick Product Identities.- (ii) Gaussian Integrals.- (iii) Integration by Parts.- (iv) Limits of Measures.- 9.2 Infinitesimal Change of Covariance.- 9.3 Quadratic Perturbations.- 9.4 Perturbative Renormalization.- 9.5 Lattice Laplace and Covariance Operators.- 9.6 Lattice Approximation of P(?)2 Measures.- 10 Estimates Independent of Dimension.- 10.1 Introduction.- 10.2 Correlation Inequalities for P(?)2 Fields.- 10.3 Dirichlet or Neumann Monotonicity and Decoupling.- 10.4 Reflection Positivity.- 10.5 Multiple Reflections.- 10.6 Nonsymmetric Reflections.- 11 Fields Without Cutoffs.- 11.1 Introduction.- 11.2 Monotone Convergence.- 11.3 Upper Bounds.- 12 Regularity and Axioms.- 12.1 Introduction.- 12.2 Integration by Parts.- 12.3 Nonlocal ?j Bounds.- 12.4 Uniformity in the Volume.- 12.5 Regularity of the P(?)2 Field.- III The Physics of Quantum Fields.- 13 Scattering Theory: Time-Dependent Methods.- 13.1 Introduction.- 13.2 Multiparticle Potential Scattering.- 13.3 The Wave Operator for Quantum Fields.- 13.4 Wave Packets for Free Particles.- 13.5 The Haag-Ruelle Theory.- 14 Scattering Theory: Time-Independent Methods.- 14.1 Time-Ordered Correlation Functions.- 14.2 The S Matrix.- 14.3 Renormalization.- 14.4 The Bethe-Salpeter Kernel.- 15 The Magnetic Moment of the Electron.- 15.1 Classical Magnetic Moments.- 15.2 The Fine Structure of the Hydrogen Atom and the Dirac Equation.- 15.3 The Dirac Theory.- 15.4 The Anomalous Moment.- 15.5 The Hyperfine Structure and the Lamb Shift of the Hydrogen Atom.- 16 Phase Transitions.- 16.1 Introduction.- 16.2 The Two Phase Region.- 16.3 Symmetry Unbroken, d = 2.- 16.4 Symmetry Broken, 3 ? d.- 17 The ?4 Critical Point.- 17.1 Elementary Considerations.- 17.2 The Absence of Even Bound States.- 17.3 A Bound on the Coupling Constant ?phys.- 17.4 Existence of Particles and a Bound on dm2/d?.- 17.5 Existence of the ?4 Critical Point.- 17.6 Continuity of d? at the Critical Point.- 17.7 Critical Exponents.- 17.8 ? ? 1.- 17.9 The Scaling Limit.- 17.10 The Conjecture ?(6) ? 0.- 18 The Cluster Expansion.- 18.1 Introduction.- 18.2 The Cluster Expansion.- 18.3 Clustering and Analyticity.- 18.4 Convergence: The Main Ideas.- 18.5 An Equation of Kirkwood-Salsburg Type.- 18.6 Covariance Operators.- 18.7 Convergence: The Proof Completed.- 19 From Path Integrals to Quantum Mechanics.- 19.1 Reconstruction of Quantum Fields.- 19.2 The Feynman-Kac Formula.- 19.3 Self-Adjoint Fields.- 19.4 Commutators.- 19.5 Lorentz Covariance.- 19.6 Locality.- 19.7 Uniqueness of the Vacuum.- 20 Further Directions.- 20.1 The $$\phi _3^4 $$ Model.- 20.2 Borel Summability.- 20.3 Euclidean Fermi Fields.- 20.4 Yukawa Interactions.- 20.5 Low Temperature Expansions and Phase Transitions.- 20.6 Debye Screening and the Sine-Gordon Transformation.- 20.7 Dipoles Don’t Screen.- 20.8 Solitons.- 20.9 Gauge Theories.- 20.10 The Higgs Model and Superconductivity.

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