Mathematical Methods of Many-Body Quantum Field Theory offers a comprehensive, mathematically rigorous treatment of many-body physics. It develops the mathematical tools for describing quantum many-body systems and applies them to the many-electron system. These tools include the formalism of second quantization, field theoretical perturbation theory, functional integral methods, bosonic and fermionic, and estimation and summation techniques for Feynman diagrams. Among the physical effects discussed in this context are BCS superconductivity, s-wave and higher l-wave, and the fractional quantum Hall effect. While the presentation is mathematically rigorous, the author does not focus solely on precise definitions and proofs, but also shows how to actually perform the computations.
Presenting many recent advances and clarifying difficult concepts, this book provides the background, results, and detail needed to further explore the issue of when the standard approximation schemes in this field actually work and when they break down. At the same time, its clear explanations and methodical, step-by-step calculations shed welcome light on the established physics literature.
SECOND QUANTIZATION
Coordinate and Momentum Space
The Many-Electron System
Annihilation and Creation Operators
PERTURBATION THEORY
The Perturbation Series for e(H0+lV)
The Perturbation Series for the Partition Function
The Perturbation Series for the Correlation Functions
GAUSSIAN INTEGRATION AND GRASSMANN INTEGRALS
Why Grassmann Integration? A Motivating Example
Grassmann Integral Representations
Ordinary Gaussian Integrals
Theory of Grassmann Integration
BOSONIC FUNCTIONAL INTEGRAL REPRESENTATION
The Hubbard Stratonovich Transformation
The Effective Potential
BCS THEORY AND SPONTANEOUS SYMMETRY BREAKING
The Quadratic Mean Field Model
The Quartic BCS Model
BCS with Higher l-Wave Interaction
THE MANY-ELECTRON SYSTEM IN A MAGNETIC FIELD
Solution of the Single Body Problem
Diagonalization of the Fractional Quantum Hall Hamiltonian
FEYNMAN DIAGRAMS
The Typical Behavior of Field Theoretical Perturbation Series
Connected Diagrams and the Linked Cluster Theorem
Estimates on Feynman Diagrams
Ladder Diagrams
RENORMALIZATION GROUP METHODS
Integrating Out Scales
A Single Scale Bound on the Sum of all Diagrams
A Multiscale Bound on the Sum of Convergent Diagrams
Elimination of Divergent Diagrams
The Feldman-Knorrer-Trubowitz Fermi Liquid Construction
RESUMMATION OF PERTURBATION SERIES
Starting Point and Typical Examples
Computing Inverse Matrix Elements
The Averaged Greens Function of the Anderson Model
The Many-Electron System with Attractive Delta-Interaction
Application to Bosonic Models
General Structure of the Integral Equations
THE 'MANY-ELECTRON MILLENNIUM PROBLEMS'
REFERENCES`
Biography
Lehmann, Detlef
"The book is clearly written, and all computations are performed in full detail."
—Mathematical Reviews"The presentation is mathematically rigorous, where possible. The author’s aim was to create a book containing enough motivation and enough mathematical details for those interested in this advanced and important field of contemporary mathematical physics."
—European Mathematical Society
