← Back to All Courses

Quantum Field Theory

Quantum field theory is the theoretical framework that combines quantum mechanics with special relativity, providing the language in which the laws of particle physics are written. This course takes you from classical field theory through canonical quantization and path integrals, to the full machinery of the Standard Model and beyond — including renormalization, gauge theories, spontaneous symmetry breaking, and modern topics such as supersymmetry and effective field theory.

Course Overview

What You'll Learn

  • • Classical fields and the principle of least action
  • • Canonical quantization of scalar, spinor, and vector fields
  • • Path integral formulation and generating functionals
  • • Feynman rules and scattering cross sections
  • • QED: the quantum theory of light and matter
  • • Renormalization and the renormalization group
  • • Non-abelian gauge theories and the Standard Model

Key Topics

  • • Lorentz invariance and the Poincaré group
  • • Noether's theorem and conserved currents
  • • Spontaneous symmetry breaking and Goldstone bosons
  • • The Higgs mechanism and mass generation
  • • Asymptotic freedom and confinement in QCD
  • • Anomalies and their physical consequences
  • • Effective field theory and the Wilsonian perspective

Course Structure

This course is organized into 4 comprehensive parts, progressing from foundational concepts to cutting-edge topics in modern theoretical physics. Each part includes rigorous derivations, worked examples, and connections to experimental results.

Part I
Foundations
Part II
QED & Renormalization
Part III
Gauge Theories
Part IV
Beyond SM

Key Equations of QFT

Klein-Gordon Equation

The relativistic wave equation for a free scalar field of mass $m$:

$$(\partial_\mu \partial^\mu + m^2)\,\phi(x) = 0$$

In covariant form: $(\Box + m^2)\phi = 0$, where $\Box = \partial_\mu \partial^\mu$ is the d'Alembertian operator.

Dirac Equation

The relativistic equation for spin-$\tfrac{1}{2}$ fermions:

$$(i\gamma^\mu \partial_\mu - m)\,\psi(x) = 0$$

The $\gamma^\mu$ are $4 \times 4$ matrices satisfying the Clifford algebra $\{\gamma^\mu, \gamma^\nu\} = 2g^{\mu\nu}$.

Path Integral

The generating functional for a scalar field theory with source $J$:

$$Z[J] = \int \mathcal{D}\phi \; \exp\!\Bigl(i\!\int\! d^4x \;\bigl[\mathcal{L}(\phi) + J\phi\bigr]\Bigr)$$

All correlation functions are obtained via functional derivatives: $\langle \phi(x_1)\cdots\phi(x_n)\rangle = \frac{1}{Z[0]}\frac{\delta^n Z}{\delta J(x_1)\cdots\delta J(x_n)}\Big|_{J=0}$.

Feynman Propagator

The time-ordered two-point function for a free scalar field:

$$D_F(x - y) = \int \frac{d^4p}{(2\pi)^4} \; \frac{i}{p^2 - m^2 + i\epsilon} \; e^{-ip\cdot(x-y)}$$

The $i\epsilon$ prescription enforces the correct causal (Feynman) boundary conditions.

Yang-Mills Lagrangian

The gauge-invariant kinetic term for non-abelian gauge fields:

$$\mathcal{L}_{\text{YM}} = -\frac{1}{4}\,F^{a}_{\mu\nu}\,F^{a\mu\nu}, \quad F^{a}_{\mu\nu} = \partial_\mu A^{a}_\nu - \partial_\nu A^{a}_\mu + g\,f^{abc}\,A^{b}_\mu\,A^{c}_\nu$$

Here $f^{abc}$ are the structure constants of the gauge group and $g$ is the coupling constant.

Higgs Potential

The Mexican-hat potential responsible for electroweak symmetry breaking:

$$V(\Phi) = -\mu^2\,\Phi^\dagger\Phi + \lambda\,(\Phi^\dagger\Phi)^2$$

For $\mu^2 > 0$ the minimum is at $\langle\Phi\rangle = v/\sqrt{2}$ with $v = \mu/\sqrt{\lambda}$, giving masses to the $W^\pm$ and $Z$ bosons.

Course Parts

I

Foundations

Classical field theory, Noether's theorem, canonical quantization of scalar and spinor fields, the path integral formalism, and Feynman diagram techniques.

Classical FieldsCanonical QuantizationPath IntegralsFeynman Diagrams
II

QED & Renormalization

Quantum electrodynamics, loop calculations, ultraviolet divergences, regularization and renormalization, running coupling constants, and quantum anomalies.

QED BasicsRenormalizationRunning CouplingsAnomalies
III

Gauge Theories

Non-abelian gauge invariance, Yang-Mills theory, quantum chromodynamics, electroweak unification, spontaneous symmetry breaking, and the Higgs mechanism.

Non-Abelian GaugeQCDElectroweak TheoryHiggs Mechanism
IV

Beyond the Standard Model

Grand unification, supersymmetry and superfields, the Wilsonian approach to effective field theory, and non-perturbative methods including lattice QFT.

Beyond SMSupersymmetryEffective Field TheoryLattice QFT

References & Bibliography

Peskin & Schroeder

An Introduction to Quantum Field Theory

M. E. Peskin and D. V. Schroeder, Westview Press (1995). The standard graduate-level textbook, covering Feynman diagrams, QED, renormalization, and non-abelian gauge theories with detailed calculations.

Weinberg

The Quantum Theory of Fields (Vols. I-III)

S. Weinberg, Cambridge University Press (1995-2000). A comprehensive and rigorous treatment from first principles, emphasizing symmetries, effective field theory, and supersymmetry.

Schwartz

Quantum Field Theory and the Standard Model

M. D. Schwartz, Cambridge University Press (2014). A modern textbook balancing conceptual clarity with calculational detail, with up-to-date coverage of the Higgs boson and LHC physics.

Srednicki

Quantum Field Theory

M. Srednicki, Cambridge University Press (2007). An accessible and well-organized textbook using the path integral approach from the start, with a free draft available online.