Quantum Field Theory Introduction

Jul 17, 2024

Introduction to Quantum Field Theory

Instructor

  • Instructor: Hong
  • Theoretical physicist specializing in high-energy theory, string theory, quantum gravity, statistical physics, etc.

Quantum Field Theory (QFT) Overview

  • Objective: Develop concepts for momentum analytics for quantum dynamics of fields.
  • Classical dynamics (e.g., Maxwell's equations) describe electric and magnetic fields classically. However, quantum mechanics provides a different picture.
  • Quantum Electrodynamics (QED): Leads to the concept of photons as mediators of electromagnetic interactions.
  • Quantum field theory offers new physical insights different from classical theory.

Importance of Quantum Field Theory

  • Fundamental Interactions: Three out of four fundamental interactions in nature are described by QFT.
  • Quantum Gravity: While not complete, QFT techniques help understand certain quantum gravity questions.
  • Applications: Over the years, QFT methods have become important in other areas like condensed matter and statistical physics.
  • Universal Language: Essential for theoretical physics across various fields. Important for both theorists and experimentalists.

Course Structure

  • Course Goal: To appreciate and understand quantum electrodynamics.
  • Sequence: First in a series of master-level courses. This semester focuses on fundamental concepts; subsequent courses will delve into technical development.
  • Outline: Provided on the course website; serves as a rough roadmap subject to change.
  • Difficulty: QFT is often seen as difficult mainly due to conceptual challenges rather than the complexity of calculations.
  • Learning Approach: Emphasizes developing intuition through multiple examples.

Instructions for Students

  • Philosophy: Fundamental understanding and formalism should be tied to solving concrete physical problems.
  • Examples and Problems: Essential for developing intuition and reinforcing learning. Important to reflect on problems after solving them.
  • Learning Outside Class: Most learning will happen outside the classroom through problem sets (Psets) and examples.
  • Rotations: Notations in lectures might differ from textbooks (Peskin and Weinberg). Students need to adapt between different conventions.

Concepts and Principles

Principle of Locality

  • Action at a Distance: Classical Newtonian mechanics involves action at a distance (e.g., gravity, Coulomb interactions).
  • Locality: Principle stating that physical points in space participate in the physical process, as formulated by Faraday.
  • Fields as Mathematical Devices: Fields are tools to realize the principle of locality.
  • Field Examples: Scalar fields, vector fields (electric and magnetic fields), tensor fields, and spinor fields.

Formalism

  • Action Principle for Classical Fields: Derived from the principle of locality.
  • Field Lagrangian: Must only involve single integrals over space-time and should depend on value and first derivatives of fields.
  • Canonical Momentum and Hamiltonian: Can be defined for field theories similar to particle mechanics.

Classical Field Theory Examples

  1. Maxwell's Theory
    • Dynamical Variable: Four-vector potential, $A_\mu$.
    • Action: Integral involving electromagnetic field tensor, $F_{\mu\nu}$.
  2. Einstein Gravity
    • Variable: Space-time metric $G$.
    • Action: Involving Ricci scalar $R$ and Newton's constant.
  3. Scalar Field Theory
    • Simple Case: Real-valued scalar field $ abla$ with an action involving only $ abla$ and its first derivatives.
    • Potential $V( abla)$: Should be constructed to maintain locality and Lorentz invariance (e.g., Klein-Gordon equation).