Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

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Special Relativity & Classical Fields

Leonard Susskind's intuitive introduction to relativistic field theory

Stanford Continuing Studies β€’ Spring 2012 β€’ 10 lectures

Course Overview

This course builds the bridge from special relativity to classical field theory - the essential foundation for understanding quantum fields.

Susskind emphasizes physical intuition over mathematical formalism. You'll learn why fields are necessary, how they transform under Lorentz boosts, and what the Lagrangian formulation really means physically.

🎯 What You'll Learn

  • Spacetime structure: Why space and time mix under boosts
  • Lorentz transformations: The geometry of special relativity
  • Four-vectors: How to write physics in covariant form
  • Classical fields: From particles to continuous systems
  • Lagrangian formalism: The action principle for fields
  • Maxwell's equations: Electromagnetism as a field theory
  • Energy-momentum: Conserved quantities and Noether's theorem

Prerequisites: Basic calculus, classical mechanics (Lagrangian formulation helpful but not required). Susskind builds everything from first principles!

Lecture 1: Introduction to Special Relativity

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Video Lecture

Special Relativity and Classical Field Theory - Lecture 1

Introduction to special relativity, spacetime diagrams, and the principle of relativity (Stanford)

πŸ’‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Key Concepts

The Principle of Relativity

Physics looks the same in all inertial reference frames. No experiment can detect absolute motion!

This seemingly simple statement leads to time dilation, length contraction, and the unification of space and time into spacetime.

Spacetime Interval (Invariant)

$$\Delta s^2 = -c^2\Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$

This quantity is the same in all reference frames - the fundamental invariant of special relativity.

πŸ’‘ Susskind's Insight

"Think of spacetime as a four-dimensional geometry. Lorentz transformations are just rotations in this geometry - mixing time and space the way ordinary rotations mix x and y."

Topics covered: Light cones, timelike vs spacelike intervals, simultaneity is relative, spacetime diagrams, worldlines.

Lectures 4-5: Classical Fields & Lagrangians

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Video Lecture

Special Relativity and Classical Field Theory - Lecture 4

Introduction to fields and Lagrangian formulation (Stanford)

πŸ’‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

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Video Lecture

Special Relativity and Classical Field Theory - Lecture 5

Lagrangian field theory and equations of motion (Stanford)

πŸ’‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Key Concepts

Why Fields?

Particles have a problem with relativity: action at a distance is instantaneous, but nothing can travel faster than light!

Solution: Replace particles with fields - continuous functions Ο†(x,t) that fill all of space. Disturbances propagate through the field at finite speed (like ripples on a pond).

Lagrangian Density

For fields, we use Lagrangian density β„’ instead of Lagrangian L:

$$\mathcal{L} = \mathcal{L}(\phi, \partial_\mu\phi)$$
$$S = \int d^4x \, \mathcal{L}(\phi, \partial_\mu\phi)$$

The action S is an integral over all spacetime. Principle of least action β†’ Euler-Lagrange equations for fields!

Euler-Lagrange Equations (Fields)

$$\frac{\partial \mathcal{L}}{\partial \phi} - \partial_\mu\left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu\phi)}\right) = 0$$

This generalizes Newton's equations to fields. It's the fundamental equation of classical field theory!

πŸ’‘ Susskind's Insight

"A field is just an infinite collection of coupled oscillators - one at each point in space. The Lagrangian formulation makes this coupling automatic through derivatives βˆ‚Ο†."

Topics covered: Transition from particles to fields, action functional, functional derivatives, Klein-Gordon equation from Lagrangian, relativistic wave equations.

Lectures 9-10: Electromagnetism & Field Theory

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Video Lecture

Special Relativity and Classical Field Theory - Lecture 9

Electromagnetic field tensor and Maxwell's equations (Stanford)

πŸ’‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

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Video Lecture

Special Relativity and Classical Field Theory - Lecture 10

Gauge invariance and conservation laws (Stanford)

πŸ’‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Key Concepts

The Electromagnetic Field Tensor

Electric field E⃗ and magnetic field B⃗ are not separate! They're components of a single relativistic object: Fμν

$$F^{\mu\nu} = \begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{pmatrix}$$

What looks like electric field in one frame looks partly magnetic in another! E⃗ and B⃗ mix under Lorentz boosts.

Maxwell's Equations (Covariant Form)

Gauss & Faraday (homogeneous):

$$\partial_\mu F^{\mu\nu} = j^\nu$$

Bianchi identity:

$$\partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0$$

Four messy vector equations collapse into two beautiful tensor equations!

Gauge Invariance

The electromagnetic potential AΞΌ has redundancy:

$$A_\mu \to A_\mu + \partial_\mu\Lambda$$

Physics (FΞΌΞ½) unchanged! This "gauge freedom" is not a bug - it's a feature that leads to charge conservation and eventually to quantum field theory's deepest structure.

πŸ’‘ Susskind's Insight

"Gauge invariance tells us there's no unique way to split the photon field into 'this part' and 'that part'. Only the total field strength FΞΌΞ½ is physical."

"This redundancy isn't wasteful - it's protecting a deep symmetry. And symmetries, by Noether's theorem, give us conservation laws!"

Topics covered: Four-potential AΞΌ, field strength tensor, Lorentz force law, gauge transformations, electromagnetic Lagrangian, energy-momentum tensor, Noether's theorem for fields.

πŸ”— Connection to MIT QFT Course

Susskind's lectures provide the physical intuition for concepts formalized in:

MIT Part I: Lagrangian Formalism

Rigorous treatment of field theory fundamentals

β†’

MIT Part I: Scalar Fields

Klein-Gordon equation and relativistic fields

β†’

MIT Part I: Symmetries

Gauge invariance and internal symmetries

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πŸ“š Recommended Study Path

  1. Watch Susskind lectures β†’ build physical intuition
  2. Read MIT notes β†’ understand mathematical structure
  3. Work MIT problem sets β†’ develop calculation skills
  4. Return to Susskind when concepts feel too abstract!

πŸ“– Additional Resources

Book

Special Relativity and Classical Field Theory: The Theoretical Minimum
Leonard Susskind & Art Friedman (2017)
Perfect companion to these lectures - same conversational style!

Full Lecture Playlist

All 10 lectures available free on YouTube
Search: "Leonard Susskind Special Relativity and Classical Field Theory 2012"

Prerequisites

  • Basic calculus (derivatives, integrals)
  • Classical mechanics (F = ma level)
  • Lagrangian mechanics helpful but not required
← Back to Theoretical MinimumNext: Advanced Quantum Mechanics β†’