Methods of Quantum Field Theory in Statistical Physics (with L.P. Gorkov and I.E. Dzyaloshinskii)

Summary

Alexei Abrikosov's "Methods of Quantum Field Theory in Statistical Physics" (co-authored with L.P. Gorkov and I.E. Dzyaloshinskii) establishes that quantum field theory, developed for particle physics, offers powerful, rigorous techniques for solving complex problems in condensed matter physics. The central thesis is that the mathematical formalism of Green's functions and Feynman diagrams, when adapted, provides a unified approach to understanding collective phenomena in many-body systems. The book demonstrates how these methods can accurately describe phase transitions, superconductivity, and other emergent properties by treating particles as excitations of quantum fields.

Readers gain proficiency in applying these sophisticated theoretical tools to analyze strongly correlated electron systems and understand phenomena like critical exponents and thermodynamic properties. Key takeaways include the ability to perform diagrammatic expansions, calculate correlation functions, and utilize techniques like the renormalization group to tackle divergences and describe universal behaviors. The work is essential for researchers and advanced students seeking a deep, quantitative understanding of condensed matter physics through the lens of quantum field theory.

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Key concepts

• Green's Functions — Mathematical objects that describe the propagation of particles and excitations in a quantum system, crucial for calculating response functions and correlation functions.
• Feynman Diagrams — Graphical representations of perturbation theory calculations in quantum field theory, allowing for systematic summation of complex interactions.
• Perturbation Theory — A mathematical method used to find approximate solutions to complex problems by starting with a simpler, solvable problem and adding corrections.
• Superconductivity — A quantum mechanical phenomenon where a material exhibits zero electrical resistance below a critical temperature, explained here using field theoretic methods.
• Renormalization Group — A theoretical framework used to understand how physical properties of a system change with scale, particularly important for critical phenomena and divergences.
• Collective Excitations — Phenomena in many-body systems that arise from the cooperative behavior of many particles, described as quantized waves or quasiparticles.