QFT I

QFT I develops the operator and functional methods of a first graduate course in quantum field theory. Starting from Fock space and scalar fields, it connects correlation functions to scattering amplitudes, introduces Euclidean and thermal methods, and ends with spinors, gauge fields, QED and renormalization. The expected starting point is quantum mechanics and special relativity.

Begin with Relativistic Spectrum and Oscillator Modes, choose a reading route, or try the three warm-ups. The lecture list gives the complete forty-lesson sequence; the notation summary collects the site’s global conventions and useful normalization translations.

Scope of QFT I

The course develops six connected groups of ideas:

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	Relate regulated real-scalar field integrals to statistical weights and derive imaginary-time thermal 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.

Non-Abelian gauge quantization beyond the first definitions, BRST symmetry, anomalies, instantons, conformal field theory and systematic effective field theory belong to later courses and the subject volumes. QFT II develops running couplings and nonperturbative applications; QFT III connects field theory with critical phenomena, conformal symmetry and geometry.

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.

Useful mathematical tools include distributions, contour integration, Gaussian integrals, linear algebra and symmetry arguments. The course builds the path integral from Gaussian integrals and time evolution. The readiness guide provides focused preparation, and the warm-ups below check oscillator normalization, engineering dimensions and propagator poles.

Suggested reading routes

For a first course, follow lectures 01–40 in order and work through the exercises as you go. The shorter routes below are for readers who already have the operator, Gaussian-integral and perturbation skills used at their entry points. Use a worked example to check a missing skill before continuing.

Reader goal	Suggested route	What to pay attention to
First complete pass through QFT	01–40	Do the 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. Assumes familiarity with stable-particle spectral poles, thresholds and scattering-state normalization; otherwise include 21–24.
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. Assumes the tree-level QED and Ward-identity skills developed in 37–38.

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 develops a coherent-state mean-field approximation at large occupation, 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 identifies the isolated real two-point pole of a stable particle with nonzero field overlap and relates its residue $Z$ to that overlap.
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 constructs finite regulated Gaussian measures with a positive-definite kernel.
28. Statistical Mechanics and Field Theory connects positive, normalizable weights for regulated real-scalar theories with a real confining Euclidean action to statistical partition functions and effective descriptions. Positivity alone does not establish a continuum limit.
29. Critical Behavior and Thermal Mass introduces critical temperature, correlation length, thermal masses, and infrared sensitivity.
30. Imaginary Time and Matsubara Propagators derives imaginary-time thermal boundary conditions from the thermal trace, then obtains 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.

Notation and conventions

These formulas summarize QFT.org’s global conventions. When comparing a calculation with a textbook, first translate the metric, Fourier phase, state normalization and gauge-sign convention. A diagram’s sign, factor of $i$, symmetry factor and external-line normalization depend on this same convention choice. 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,

$$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}, \qquad \overline\psi=\psi^\dagger\gamma^0, \qquad \gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3.$$

When a spatial alternating symbol is used, $\epsilon_{123}=+1$. It is a three-dimensional symbol, not the result of lowering indices on the four-dimensional $\epsilon^{0123}=+1$ convention.

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 the Dirac bilinear $j_{\mathrm{bil}}^\mu=\bar\psi\gamma^\mu\psi$, this convention gives $\mathcal L_{\mathrm{int}}=+qA_\mu j_{\mathrm{bil}}^\mu$. For an electron $q=-e$, so $D_\mu\psi_e=(\partial_\mu+ieA_\mu)\psi_e$, $\mathcal L_{\mathrm{int}}=-eA_\mu j_{\mathrm{bil}}^\mu$, and the photon vertex is $-ie\gamma^\mu$. If the Maxwell source is instead named through $\mathcal L_{\mathrm{source}}=-J_{\mathrm{source}}\!\cdot A$, then $J_{\mathrm{source}}^\mu=-qj_{\mathrm{bil}}^\mu$.

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

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

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

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

at fixed $E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}>0$, take $\epsilon\to0^+$ with $\epsilon/E_{\mathbf p}^2\ll1$. Write the denominator as a function of $p^0$ and identify the two poles to first order in $\epsilon$. The massless zero-momentum point is excluded: there the poles scale as $\sqrt\epsilon$, so expansion about nonzero $E_{\mathbf p}$ does not apply.

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.

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.

Further reading

These books provide complementary approaches to the course. The lecture pages give references for their individual calculations.

• Coleman, Sidney. Aspects of Symmetry. Cambridge University Press, 1985. DOI.
• Peskin, Michael E., and Daniel V. Schroeder. An Introduction to Quantum Field Theory. Addison-Wesley, 1995. Publisher catalogue.
• Polyakov, A. M. Gauge Fields and Strings. 1st ed. Routledge, 1987. DOI.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. DOI.
• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007. DOI.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, 1995. DOI.
• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. 5th ed. Oxford University Press, 2021. DOI.