QFT I

Overview

Quantum field theory is the language in which particles, waves, and symmetries become different faces of the same object. This course begins with the most concrete viewpoint: a relativistic many-particle Hilbert space. From there it introduces creation and annihilation operators, field operators, relativistic wave equations, perturbation theory, Feynman propagators, path integrals, scattering amplitudes, Euclidean continuation, finite-temperature field theory, spinors, gauge fields, QED, and the first ideas of renormalization.

The organizing principle is simple but powerful: a field is the operator that creates and destroys the quanta whose motion and interactions we observe. The rest of the course develops the calculational machinery needed to make that principle precise.

These webpages are written for readers who already know quantum mechanics and special relativity, but who want a careful bridge from those subjects to modern QFT. The goal is not to hide the formalism behind slogans. Global qft.org conventions are fixed here, and local convention notes appear only when a page uses a temporary normalization or sign choice that deserves extra warning. Each lecture page derives its main formulas, explains the physical meaning, and includes exercises with solutions.

Acknowledgement

These webpages are based on handwritten notes taken by Jie Ren from Alexander M. Polyakov’s one-semester course Introduction to Quantum Field Theory. The notes have been edited, expanded, typeset, supplemented with derivations and figures, and adapted for QFT.org. Any errors are the responsibility of the editor of these webpages.

Scope of QFT I

This is a first graduate course in quantum field theory, not a survey of every technique used in modern field theory. It focuses on the conceptual and computational core that later courses rely on:

Emphasis	Appears in	Purpose
Fock space and field operators	01–08	Build the particle interpretation of free fields and the operator language for many-body systems.
Green functions and perturbation theory	09–18	Turn time evolution, Wick contractions, and source functionals into calculable diagrams.
Poles, cuts, and scattering	19–26	Connect exact correlators to physical particles, spectral densities, amplitudes, and cross sections.
Euclidean and thermal methods	27–30	Reinterpret QFT as statistical field theory and introduce imaginary-time correlators.
Spinors, gauge fields, and QED	31–38	Add Lorentz representations, Dirac fields, gauge redundancy, and tree-level QED amplitudes.
Ultraviolet structure	39–40	Explain one-loop divergences, local counterterms, and running couplings.

Several important topics are intentionally left for later courses or specialized notes: spontaneous symmetry breaking, non-Abelian gauge quantization beyond the first structural definitions, ghosts and BRST symmetry, anomalies, instantons, solitons, conformal field theory, the full Standard Model, and modern effective field theory. Those topics make much more sense after the machinery developed here is secure.

Editorial principles

The course is designed to be useful both as a first serious pass through QFT and as a reference for later study. The guiding principles are:

• Conventions are locked globally and flagged locally. Factors of $i$, signs in the metric, Fourier-transform conventions, and normalizations of states follow the qft.org convention page. Local notes appear only when a page temporarily uses a special normalization, such as finite-volume states or oscillator-normalized modes.
• Important identities are derived, not merely quoted. Wick’s theorem, the Feynman propagator, Schwinger–Dyson identities, pole residues, and LSZ reduction are treated as calculational tools whose assumptions should be visible.
• Physics and analysis are kept together. Poles, cuts, thresholds, spectral densities, Euclidean continuation, and renormalization are presented as different ways of reading the same quantum dynamics.
• Diagrams are explanatory devices. Feynman diagrams are not decorative pictures; they encode algebra, combinatorics, singularities, and approximations.

Course roadmap

A schematic path through QFT I. The course begins with many-particle quantum mechanics and ends with the ultraviolet logic of renormalization. The middle of the course develops the main computational bridge: Green functions, path integrals, diagrams, and scattering amplitudes.

Prerequisites

The expected background is a standard graduate-level command of quantum mechanics, including harmonic oscillators, Hilbert spaces, perturbation theory, and identical particles. Familiarity with special relativity is essential: four-vectors, Lorentz transformations, mass shells, and invariant phase space appear throughout the course. Classical field theory, complex analysis, and statistical mechanics are used repeatedly, but the relevant tools are reviewed when they become central.

The most useful mathematical habits are comfort with distributions, contour integration, Gaussian integrals, linear algebra in infinite-dimensional notation, and symmetry arguments. A reader does not need to know the path integral in advance; it is built from Gaussian integrals and time evolution.

Convention lock

These pages follow the qft.org convention page. The formulas below are the convention lock most often needed inside QFT I; later lecture pages do not repeat them unless a local normalization or sign choice would otherwise be easy to miss. The default spacetime metric is mostly-minus,

$$\eta_{\mu\nu}=\operatorname{diag}(+,-,-,-), \qquad p^2=(p^0)^2-\mathbf p^2,$$

with $c=\hbar=1$. Plane waves use

$$e^{-ip\cdot x}=e^{-ip^0t+i\mathbf p\cdot\mathbf x}, \qquad p\cdot x=p^0t-\mathbf p\cdot\mathbf x.$$

For a free real scalar field,

$$\mathcal L_0={1\over2}\partial_\mu\phi\,\partial^\mu\phi-{1\over2}m^2\phi^2, \qquad (\Box+m^2)\phi=0,$$

and the scalar Feynman propagator is

$$D_F(x-y)=\langle0|T\phi(x)\phi(y)|0\rangle =\int {d^4p\over(2\pi)^4}\,{i\,e^{-ip\cdot(x-y)}\over p^2-m^2+i\epsilon},$$

so

$$(\Box_x+m^2)D_F(x-y)=-i\delta^{(4)}(x-y).$$

Relativistic one-particle states are covariantly normalized by default:

$$\langle \mathbf p',s'|\mathbf p,s\rangle =(2\pi)^3 2E_{\mathbf p}\,\delta^{(3)}(\mathbf p-\mathbf p')\delta_{ss'}, \qquad E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}.$$

Accordingly,

$$d\Pi_p={d^3\mathbf p\over(2\pi)^3 2E_{\mathbf p}}$$

is the Lorentz-invariant on-shell measure. Several early operator pages use the equivalent oscillator-normalized scalar expansion

$$\phi(x)=\int {d^3\mathbf p\over(2\pi)^3}{1\over\sqrt{2E_{\mathbf p}}} \left[\alpha(\mathbf p)e^{-ip\cdot x}+\alpha^\dagger(\mathbf p)e^{ip\cdot x}\right], \qquad [\alpha(\mathbf p),\alpha^\dagger(\mathbf q)]=(2\pi)^3\delta^{(3)}(\mathbf p-\mathbf q).$$

This is the same convention as the covariant expansion after the rescaling

$$a(\mathbf p)=\sqrt{2E_{\mathbf p}}\,\alpha(\mathbf p),$$

which gives

$$[ a(\mathbf p),a^\dagger(\mathbf q)] =(2\pi)^3 2E_{\mathbf p}\delta^{(3)}(\mathbf p-\mathbf q).$$

For spinors,

$$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}, \qquad \overline\psi=\psi^\dagger\gamma^0, \qquad \gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3.$$

For Abelian gauge theory, a field of charge $q$ transforms as

$$\psi(x)\mapsto e^{iq\alpha(x)}\psi(x), \qquad A_\mu(x)\mapsto A_\mu(x)+\partial_\mu\alpha(x),$$

and

$$D_\mu=\partial_\mu-iqA_\mu.$$

For an electron $q=-e$, so $D_\mu\psi_e=(\partial_\mu+ieA_\mu)\psi_e$. For non-Abelian gauge theory, Hermitian generators obey $[T^a,T^b]=if^{abc}T^c$, and the default convention is

$$D_\mu=\partial_\mu-igA_\mu^aT^a, \qquad [D_\mu,D_\nu]=-igF_{\mu\nu}^aT^a,$$

with

$$F_{\mu\nu}^a=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a+gf^{abc}A_\mu^bA_\nu^c.$$

The default Wick rotation is

$$t=-i\tau, \qquad p^0=ip_E^0, \qquad e^{iS}\mapsto e^{-S_E},$$

and for the real scalar field

$$S_E=\int d\tau\,d^3\mathbf x\, \left[{1\over2}(\partial_\tau\phi)^2+{1\over2}(\nabla\phi)^2+{1\over2}m^2\phi^2\right].$$

How to read these notes

The safest route is linear. Lectures 01–08 build the operator language of fields and particles. Lectures 09–18 develop time ordering, Green functions, Wick’s theorem, path integrals, and Feynman diagrams. Lectures 19–26 explain how interacting particles are recognized through poles, residues, spectral representations, and scattering amplitudes. Lectures 27–30 translate the theory into Euclidean and thermal language. Lectures 31–38 add Lorentz representations, spinors, vector fields, gauge redundancy, and QED. Lectures 39–40 introduce ultraviolet divergences, power counting, local counterterms, and running couplings.

A first reading can focus on the main text, worked examples, and conceptual checks. A second reading should include the exercises, because the exercises are where normalization conventions, signs, and diagrammatic factors become real.

Suggested reading routes

Different readers can use the same material at different depths.

Reader goal	Suggested route	What to pay attention to
First complete pass through QFT	01–18, then 21, 25–27, 31–38, 39–40	Do the basic exercises and track every normalization convention.
Fast perturbation-theory bootstrapping	09–20, then 25–26	Focus on time ordering, Wick contractions, source functionals, symmetry factors, and amputation.
Scattering and analytic structure	19–26	Treat poles, residues, spectral densities, thresholds, and LSZ as one connected story.
Euclidean and thermal field theory	27–30, with 14–17 as background	Watch the Wick-rotation signs and the difference between Euclidean, retarded, and time-ordered correlators.
Spinors and gauge theory	31–38	Keep Lorentz representations, field components, particle helicities, and gauge redundancy separate.
Renormalization preview	18–20 and 39–40	Separate regularization, counterterms, physical renormalization conditions, and running couplings.

Lecture list

Part I — Fock space, particles, and field operators

1. Relativistic Spectrum and Oscillator Modes introduces relativistic energy, additive spectra, and the harmonic oscillator as the prototype of a quantum field mode.
2. Identical Particles and Fock Space builds many-particle Hilbert spaces, Bose symmetrization, occupation numbers, and normalized Fock states.
3. Nonrelativistic Field Operators defines field operators, density operators, and the second-quantized form of many-body Hamiltonians.
4. Interacting Bose Fields and Bogoliubov Sound derives the classical large-occupation limit, linearized collective modes, the Bogoliubov dispersion, and the Landau critical-velocity idea.
5. Statistics, Occupation Algebra, and Anyons compares bosonic and fermionic ladder algebras and explains why two spatial dimensions permit anyonic exchange phases.
6. Free Nonrelativistic Fields and Mode Hamiltonians diagonalizes free nonrelativistic fields in momentum space and relates the Hamiltonian to number operators.
7. Klein–Gordon Equation and Scalar Modes introduces relativistic scalar waves, positive and negative frequencies, and the mass-shell condition.
8. Relativistic Scalar Action and Complex Fields develops real and complex scalar actions, conserved charge, the nonrelativistic limit, and quartic interactions.

Part II — Green functions, Wick theorem, and diagrams

9. Interaction Picture and Dyson Expansion derives the interaction-picture evolution operator and the time-ordered Dyson series.
10. Time Ordering, Contact Terms, and Green Functions explains time-ordered products, derivative contact terms, and equations obeyed by Green functions.
11. Feynman Propagator and the iε Prescription derives the Feynman propagator from oscillator spectral sums and explains its pole prescription.
12. Wick Theorem and n-Point Functions proves the free-field contraction rule and computes two-point and four-point functions.
13. Wick Contractions and Diagrammatic Expansion turns contractions into diagrams and organizes Gaussian moments for real and complex fields.
14. Gaussian Integrals and Classical Paths reviews finite-dimensional Gaussian integration, determinants, saddle points, and the classical path in the path integral.
15. Source Functionals in Quantum Mechanics and QFT introduces sources, generating functionals, and functional derivatives.
16. Perturbation Theory from Functional Derivatives derives interaction expansions by replacing fields with functional derivatives with respect to sources.
17. Schwinger–Dyson Identities and the Classical Limit derives field equations inside correlation functions and clarifies the relation between quantum identities and classical equations of motion.
18. φ⁴ Theory and First Diagrams introduces quartic interactions, vacuum bubbles, tadpoles, two-point corrections, and symmetry factors.

Part III — Self-energy, spectral structure, and scattering

19. Self-Energy and the Dyson Equation defines one-particle-irreducible self-energy insertions and sums them into the full propagator.
20. Pole Shifts, Mass Renormalization, and 1PI Diagrams explains how interactions move particle poles and how counterterms enter perturbatively.
21. Causality, Commutators, and Support compares Feynman, retarded, and commutator functions and shows how relativistic causality appears through light-cone support.
22. Spectral Representation and Dispersion Integrals develops spectral densities, analytic continuation, branch cuts, positivity, and dispersion relations.
23. Thresholds, Cuts, and Imaginary Parts studies multiparticle thresholds, imaginary parts of amplitudes, and the optical-theorem intuition.
24. Scattering Amplitudes and Dyson Resummation organizes reducible and irreducible contributions to scattering and relates them to resummed propagators.
25. Pole Residue and Wavefunction Renormalization defines the residue $Z$, physical one-particle states, and quasiparticle normalization.
26. LSZ Reduction and Cross Sections derives external-leg amputation, LSZ reduction, invariant phase space, and cross-section normalization.

Part IV — Euclidean and thermal field theory

27. Euclidean Continuation and Gaussian Path Integrals performs Wick rotation and explains why Euclidean Gaussian measures are well behaved.
28. Statistical Mechanics and Field Theory connects partition functions, spin systems, Ising intuition, and field-theoretic effective descriptions.
29. Critical Behavior and Thermal Mass introduces critical temperature, correlation length, thermal masses, and infrared sensitivity.
30. Imaginary Time and Matsubara Propagators derives the thermal trace, imaginary-time periodicity, Matsubara frequencies, and finite-temperature propagators.

Part V — Lorentz representations, spinors, and gauge fields

31. Lorentz Group and Complex Coordinates reviews Lorentz transformations, light cones, invariant intervals, and the spinorial $SL(2,\mathbb C)$ viewpoint.
32. Scalar and Vector Representations on the Mass Shell treats scalar and vector representations, massive and massless momenta, and little-group intuition.
33. Spinor Representations and Gamma Matrices introduces the Clifford algebra, gamma matrices, and Lorentz generators acting on spinors.
34. Dirac Equation and Spinor Solutions derives the Dirac equation, plane-wave spinors, covariance, and spin-sum structures.
35. Dirac Field Quantization and Propagators quantizes fermion fields, derives anticommutators, and computes the Dirac propagator.
36. Grassmann Integrals and Fermionic Wick Theorem develops Berezin integration, fermionic Gaussian determinants, and fermionic Wick contractions.
37. Vector Fields and Gauge Redundancy introduces polarization vectors, Maxwell theory, gauge transformations, constraints, and physical degrees of freedom.
38. QED Vertices and Tree Amplitudes derives minimal coupling, QED Feynman rules, photon exchange, and the first Ward-identity checks.

Part VI — Divergences and renormalization

39. One-Loop Divergences and Power Counting studies loop integrals, superficial degree of divergence, and the locality of ultraviolet counterterms.
40. Renormalization and Running Couplings explains logarithmic divergences, coupling renormalization, running couplings, and beta-function intuition.

Thematic map

Question	Where it appears	Core idea
How do particles become field quanta?	01–08	A field decomposes into oscillator modes, and Fock space organizes arbitrary particle number.
Why do time-ordered correlators calculate amplitudes?	09–13	The Dyson expansion and Wick theorem convert quantum time evolution into propagators and contractions.
What does a Feynman diagram really encode?	13–20	A diagram records an algebraic term, its propagators, vertices, integrations, and symmetry factor.
How are physical particles identified in an interacting theory?	19–26	Particles correspond to poles of exact Green functions, with residues giving state normalization.
Why is Euclidean field theory useful?	27–30	Wick rotation turns oscillatory path integrals into statistical weights and reveals thermal periodicity.
Where do spin and gauge redundancy enter?	31–38	Lorentz representation theory fixes possible fields, while gauge redundancy removes unphysical vector modes.
Why must couplings run?	39–40	Ultraviolet divergences are local and can be absorbed into scale-dependent parameters.

Diagnostic warm-ups

These short problems are meant to test the background assumptions for the course. They are not prerequisites for reading the first page, but they flag ideas that will return constantly.

Warm-up 1: oscillator normalization

Let $a$ and $a^\dagger$ obey $[a,a^\dagger]=1$, and define $|n\rangle=C_n(a^\dagger)^n|0\rangle$ with $a|0\rangle=0$ and $\langle 0|0\rangle=1$. Find $C_n$ such that $\langle n|n\rangle=1$.

Solution

Using $[a,a^\dagger]=1$, one obtains

$$a(a^\dagger)^n=(a^\dagger)^n a+n(a^\dagger)^{n-1}.$$

Since $a|0\rangle=0$,

$$\langle 0|a^n(a^\dagger)^n|0\rangle=n!.$$

Thus

$$1=\langle n|n\rangle=|C_n|^2 n!,$$

so a standard real positive choice is

$$C_n=\frac{1}{\sqrt{n!}}.$$

This normalization is the seed of the Fock-space normalization used for field modes.

Warm-up 2: mass dimension of a scalar field

In four spacetime dimensions, use the action

$$S=\int d^4x\,\frac12\partial_\mu\phi\,\partial^\mu\phi$$

to find the mass dimension of $\phi$. Then find the dimension of the coupling $\lambda$ in the interaction $\lambda\phi^4/4!$.

Solution

In natural units the action is dimensionless, so the Lagrangian density has mass dimension $4$. The derivative has dimension $1$, hence

$$[\partial_\mu\phi\,\partial^\mu\phi]=2+2[\phi]=4.$$

Therefore

$$[\phi]=1.$$

The interaction term must also have dimension $4$:

$$[\lambda]+4[\phi]=4.$$

Since $[\phi]=1$,

$$[\lambda]=0.$$

This is the first hint that $\phi^4$ theory in four dimensions is marginal by power counting.

Warm-up 3: locating the scalar propagator pole

For the free scalar Feynman propagator

$$\frac{i}{p^2-m^2+i\epsilon},$$

write the denominator as a function of $p^0$ and identify the two poles to first order in $\epsilon$.

Solution

Since $p^2=(p^0)^2-\mathbf p^2$,

$$p^2-m^2+i\epsilon=(p^0)^2-E_{\mathbf p}^2+i\epsilon, \qquad E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}.$$

The poles solve

$$(p^0)^2=E_{\mathbf p}^2-i\epsilon.$$

To first order in $\epsilon$,

$$p^0\approx +E_{\mathbf p}-i\frac{\epsilon}{2E_{\mathbf p}}, \qquad p^0\approx -E_{\mathbf p}+i\frac{\epsilon}{2E_{\mathbf p}}.$$

Thus the positive-energy pole lies just below the real axis and the negative-energy pole lies just above it. This displacement is the Feynman boundary condition in momentum space.

Reliability notes

The pages have been edited for consistency with the qft.org convention lock, especially the mostly-minus metric, $e^{-ip\cdot x}$ plane waves, covariant relativistic state normalization, scalar and Dirac propagators, Abelian gauge transformations, and Wick rotation. When comparing to a textbook, first translate the metric, Fourier phase, state normalization, and gauge-sign convention before comparing individual formulas.

Diagrams in these notes use textbook-style symbolic normalization. A diagram is not a standalone observable: its sign, factor of $i$, symmetry factor, and external-line normalization are meaningful only together with the convention package stated on the relevant page.

References and further reading

The lecture pages cite a small number of references as needed. The following books are especially useful companions for the full course:

• Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory — a standard reference for perturbation theory, Feynman rules, scattering, and renormalization.
• Steven Weinberg, The Quantum Theory of Fields, Volume I — especially valuable for Lorentz symmetry, particles, fields, and the logic of relativistic quantum theory.
• Mark Srednicki, Quantum Field Theory — concise and systematic, with a useful path-integral-first organization.
• Matthew D. Schwartz, Quantum Field Theory and the Standard Model — very clear on modern perturbative methods, spinors, gauge theory, and renormalization.
• Jean Zinn-Justin, Quantum Field Theory and Critical Phenomena — a deep source for Euclidean methods, statistical field theory, critical behavior, and renormalization.
• Sidney Coleman, Aspects of Symmetry — classic lectures on physical reasoning in QFT, symmetry, solitons, and effective field theory.
• Alexander M. Polyakov, Gauge Fields and Strings — a compact and profound view of gauge fields, path integrals, topology, and field-theoretic structures beyond this introductory course.

Where to begin

Start with Relativistic Spectrum and Oscillator Modes. The harmonic oscillator may look elementary, but in QFT it becomes the local grammar of particles: every momentum mode of a free field is an oscillator, and interactions couple these oscillators together.