QFT II

QFT II develops renormalization, symmetry and nonperturbative methods through forty lectures. Starting from one-loop integrals and running couplings, it moves through operator products, gauge fields, Fermi surfaces, two-dimensional models, sigma models and topology, then reaches vacuum decay, real-time observables and accelerated observers.

The course asks how to connect a calculation to a physical claim. A running coupling can generate an RG scale without establishing a mass gap; a semiclassical saddle contributes only in a regime where its fluctuations and measure are controlled; a thermal response depends on the state and observable as well as the coordinates. The lectures develop these distinctions through worked examples.

Start with lecture 01, choose a reading route, or use the four warm-ups to identify preparation you need.

Reading routes

For a first reading, follow the lecture list in order. Lectures 01–14 develop renormalization and operator methods; 15–22 add background fields, gauge identities and many-body applications; 23–33 develop two-dimensional theories and sigma models; 34–40 address monopoles, vacuum transitions and horizons.

For selective study, use these routes together with each lesson’s preparation links:

• High-energy theory: 01–20, then 27–35. Continue with 36–37 for real-time observables and particle production.
• Statistical field theory: 01–11, then 21–22 and 27–33. Readers already able to derive the running four-point vertex and renormalize operator insertions can enter at 07.
• Topology and nonperturbative physics: begin with constraints and sigma-model fields, then their running coupling and the large-N gap calculation before 30–40. The spin-chain and symmetry-restoration discussions use this preparation.

QFT III offers a complementary route through statistical models, conformal symmetry and fluctuating geometry. The two advanced courses can be consulted in parallel. Higher-loop calculations, lattice algorithms, supersymmetry and full curved-spacetime QFT are beyond this course’s systematic scope.

Preparation

The expected background is QFT I: canonical quantization, path integrals, Feynman rules, one-loop diagrams, spinors, gauge fields, Ward identities and basic renormalization. You should be comfortable with Gaussian integrals, Wick rotation, dimensional analysis, Feynman parameters, residues and distributions. The readiness guide can help locate a gap before beginning.

Use the following links to refresh a specific capability; completing every linked page is not a separate entry requirement.

Capability	Review	Used for
Normalize free-field modes from their equal-time algebra	Mode normalization	Propagators, oscillator traces and Bogoliubov modes
Complete a Gaussian with sources and continue correlators with their vacuum prescription	Gaussian integral; Euclidean continuation	Determinants and source correlators
Build a loop integrand and extract a regulated ultraviolet logarithm	Momentum-space rules; UV logarithm	Lectures 01–06 and warm-up 1
Differentiate at fixed bare coupling	Scalar beta function	Running couplings and RG equations
Keep spinor numerators and graded time ordering consistent	Fermion propagator	QED and two-dimensional fermions
Use covariant derivatives and representation-dependent contractions	Connections and curvature; Color algebra	Gauge-theory beta functions
Derive a Ward identity while retaining insertion contacts	Regulated Ward identity	Lectures 19–20 and warm-up 3

Lie groups and Lie algebras enter through gauge symmetry; familiarity with generators, representations and structure constants is useful. Later lectures introduce the geometry and topology needed for target spaces, winding numbers and theta terms. Statistical mechanics helps with the Fermi-surface and spin-chain examples, but a complete many-body course is not required.

Lecture list

Part I — Renormalization, leading logarithms, and OPE

1. Effective Actions, Dimensional Estimates, and IR Physics introduces effective actions, derivative expansions, dimensional estimates, and the distinction between ultraviolet and infrared sensitivity.
2. Contact Scattering and Renormalization in Quantum Mechanics uses a nonrelativistic contact interaction as the simplest laboratory for cutoff dependence, bubble sums, and renormalized couplings.
3. Euclidean Loop Integrals and Feynman Parameters develops Wick rotation, Schwinger parameters, Feynman parameters, and logarithmic loop integrals.
4. Scalar Propagators and One-Loop φ⁴ Theory computes the first local ultraviolet divergence in scalar four-point scattering.
5. Leading Logarithms and Nested Subgraphs explains why the highest powers of logarithms come from ordered momentum regions and nested divergent subgraphs.
6. RG Equation for the Four-Point Vertex derives the renormalization-group equation for the four-point vertex and resums leading logarithms.
7. Running Couplings and Critical Free Energy applies running couplings to critical thermodynamics and logarithmic corrections to scaling.
8. Mass Tuning and Relevant Deformations studies relevant perturbations, mass tuning, correlation lengths, and the emergence of physical scales.
9. Operator Product Expansion in Perturbation Theory introduces short-distance operator products, Wilson coefficients, and normal-product intuition.
10. Wilsonian RG and Operator Mixing develops coarse graining, operator bases, mixing matrices, and effective Lagrangians.
11. Callan–Symanzik Equations and Marginal Operators derives Callan–Symanzik equations and classifies relevant, marginal, and irrelevant directions near fixed points.
12. QED as an Effective Field Theory organizes QED interactions by operator dimension, including Pauli terms and four-fermion operators.
13. Vacuum Polarization and Gauge-Invariant Counterterms computes the structure of the vacuum polarization tensor and explains transversality and gauge-invariant counterterms.
14. Running Charge, Screening, and Antiscreening interprets charge renormalization through screening, dielectric response, and beta-function signs.

Part II — Background fields, determinants, beta functions, and Ward identities

15. Proper Time, Determinants, and Thermal Traces introduces functional determinants, proper-time representation, trace formulas, and finite-temperature trace logic.
16. Effective Actions in Background Fields derives one-loop effective actions in background fields and connects Landau levels to diamagnetic and paramagnetic effects.
17. QED and Yang–Mills Beta Functions compares spinor, scalar, and vector contributions to beta functions and explains the origin of non-Abelian antiscreening.
18. Dimensional Transmutation and Mass Gaps derives an RG-invariant scale and distinguishes dimensional transmutation from a demonstrated mass gap.
19. Ward Identities and Chiral Symmetries derives Ward identities from continuous symmetries and introduces chiral rotations and fermion bilinears.
20. Current Correlators and Polarization Tensors studies two-current functions, transverse tensor structures, contact terms, and local counterterms.
21. Fermi Surface and Nonrelativistic Many-Body Fields introduces finite-density Green functions, particles and holes, and low-energy modes near the Fermi surface.
22. Fermi-Surface Instabilities and One-Dimensional Fermions studies particle-hole singularities, nesting, Peierls-type instabilities, and one-dimensional fermion response.

Part III — Two-dimensional gauge fields, bosonization, and sigma models

23. Gauge Fields in Two Dimensions explains why gauge fields simplify in two dimensions and introduces light-cone variables and exact Green-function methods.
24. Schwinger Model and Gauge-Invariant Correlators develops QED₂, Wilson-line dressed correlators, anomaly-generated mass, and gauge-invariant fermion observables.
25. Bosonization and Sine-Gordon–Thirring Duality derives free-boson vertex-operator correlators and explains the sine-Gordon/massive-Thirring correspondence.
26. Confinement and Screening in Two-Dimensional QED compares linear potentials, screening by massless fermions, and confinement intuition in one spatial dimension.
27. Nonlinear Sigma Models and Constraints introduces fields valued on target spaces, constrained variables, Lagrange multipliers, and Goldstone coordinates.
28. Sigma-Model Beta Function and Asymptotic Freedom derives the curvature-dependent one-loop running and explains when the sphere model is asymptotically free.
29. Large-N Saddle Point in the O(N) Model derives the large-$N$ saddle, the gap equation, dynamical mass generation, and $1/N$ counting.
30. Antiferromagnets, Spin Chains, and Theta Terms connects antiferromagnetic spin chains to sigma models, Berry phases, and theta-angle physics.
31. Symmetry Restoration and Mermin–Wagner Physics explains how low-dimensional fluctuations restore continuous symmetries and modify order-parameter reasoning.
32. Sigma-Model Instantons and Topological Charge constructs $O(3)$ instantons, topological charge, CP¹ coordinates, and scale moduli.
33. Theta Angle and Instanton Corrections studies theta dependence, dilute instanton sums, and nonperturbative corrections.

Part IV — Monopoles, vacuum transitions, pair creation, and horizons

34. Monopoles and Confinement in Three-Dimensional Gauge Theory develops the monopole-plasma picture of confinement and the area-law intuition in compact gauge theory.
35. Instantons in Quantum Mechanics and Vacuum Decay derives imaginary-time tunneling, bounce solutions, vacuum persistence, and decay rates.
36. In–Out, In–In, and Schwinger–Keldysh Functionals distinguishes transition amplitudes from expectation values and introduces closed-time-path generating functionals.
37. Bogoliubov Coefficients and Pair Creation studies in/out mode bases, Bogoliubov transformations, vacuum persistence, and pair-creation probabilities.
38. Rindler Coordinates and Green Functions introduces Rindler wedges, accelerated coordinates, and Green functions adapted to horizons.
39. Unruh Temperature and Thermal Periodicity derives Euclidean periodicity, the KMS condition, detector response, and the Unruh temperature.
40. Spheres, de Sitter Continuation, and Outlook connects spheres, hyperboloids, analytic continuation, thermal interpretation, and broader field-theoretic outlooks.

Notation and conventions

The course inherits the site’s spacetime, Fourier, gauge and renormalization conventions, including the (+−−−) metric and natural units. The following local choices distinguish formulas that otherwise look similar.

Gauge and sigma-model couplings

For a four-dimensional gauge coupling $g$ in $D_\mu=\partial_\mu-igA_\mu^aT^a$, an asymptotically free one-loop example has

$$\beta_g=\mu\frac{\mathrm dg}{\mathrm d\mu}\bigg|_0 =-\frac{b_0}{16\pi^2}g^3+O(g^5), \qquad b_0>0.$$

The sign means that the coupling decreases as the energy scale increases. The power of the coupling depends on its definition; it is not a universal criterion for asymptotic freedom.

For the two-dimensional Euclidean $O(N)$ sigma model, use the dimensionless coupling $\alpha$ in

$$S_{\mathrm{kin}}=\frac{1}{2\alpha}\int \mathrm d^2x\, \partial_\mu\mathbf n\cdot\partial_\mu\mathbf n, \qquad \mathbf n^2=1.$$

With this normalization, lecture 28 gives

$$\beta_\alpha=-\frac{N-2}{2\pi}\alpha^2+O(\alpha^3).$$

The sphere target gives asymptotic freedom for $N>2$; the perturbative $O(2)$ case is different. A strong-coupling scale inferred from this running is not by itself a proof of a mass gap. Lecture 29 derives a gap in the controlled large-$N$ limit of this kinetic model without a topological term.

Topological charge and the theta sign

For $O(3)$, let $\mathbf n$ be a smooth map from a closed oriented two-dimensional surface to the unit sphere. On the plane, require boundary data that permit a smooth compactification to such a surface; in the usual construction the field approaches one fixed value at infinity. With $\epsilon_{12}=+1$, the degree is

$$Q=\frac{1}{8\pi}\int \mathrm d^2x\,\epsilon_{\mu\nu}\, \mathbf n\cdot(\partial_\mu\mathbf n\times\partial_\nu\mathbf n) \in\mathbb Z.$$

The integral represents the pullback of the sphere’s area form divided by $4\pi$. Its integer interpretation depends on the stated global conditions. On a surface with a physical boundary and unrestricted boundary data, the bulk integral need not be an integer; finite action alone does not supply the missing boundary condition. Lecture 32 develops the degree and instanton construction.

Here the theta convention is

$$S_E=S_{\mathrm{kin}}-i\theta Q, \qquad e^{-S_E}=e^{-S_{\mathrm{kin}}+i\theta Q}.$$

This agrees with lecture 32. Lecture 30 instead writes its action with $+i\theta_{30}Q$. For the same oriented charge, translate by $\theta_{30}=-\theta$; the Euclidean weight is then identical. Individual lessons also specify their finite-density state, real-time contour or accelerated-coordinate patch when needed.

Check your preparation

These problems test four tools used in the lectures: a loop integral, power counting, a Ward identity and a static-source observable. Work through each calculation before opening its solution; the linked lectures provide a longer derivation when needed.

Warm-up 1: a logarithmic Euclidean integral

Evaluate the leading large-$\Lambda$ behavior of

$$I(\Lambda,m)=\int_{|k|<\Lambda}{d^4k\over(2\pi)^4}\,{1\over(k^2+m^2)^2}$$

for $m>0$ and $\Lambda\gg m$. The positive mass regulates the infrared; the massless integral also needs an infrared regulator.

Solution

In four Euclidean dimensions,

$$d^4k=2\pi^2 k^3dk.$$

Therefore

$$I(\Lambda,m)={1\over(2\pi)^4}2\pi^2\int_0^\Lambda dk\,{k^3\over(k^2+m^2)^2}.$$

Set $u=k^2$, so $k^3dk={1\over2}u\,du$. Then

$$I(\Lambda,m)={1\over16\pi^2}\int_0^{\Lambda^2}du\,{u\over(u+m^2)^2}.$$

Since

$${u\over(u+m^2)^2}={1\over u+m^2}-{m^2\over(u+m^2)^2},$$

we get

$$I(\Lambda,m)={1\over16\pi^2}\left[\log{\Lambda^2+m^2\over m^2}-{\Lambda^2\over\Lambda^2+m^2}\right].$$

Thus

$$I(\Lambda,m)={1\over16\pi^2}\log{\Lambda^2\over m^2}+\text{nonlogarithmic terms}.$$

The coefficient is the one-loop logarithmic normalization used in lecture 03.

Warm-up 2: relevant, marginal, or irrelevant

Near the Gaussian fixed point in $d$ spacetime dimensions, suppose a local operator $\mathcal O$ has engineering mass dimension $\Delta$. The action contains

$$S\supset \int d^dx\,g_{\mathcal O}\mathcal O(x).$$

Find the engineering dimension of $g_{\mathcal O}$ and classify the perturbation by power counting.

Solution

The action is dimensionless and $[d^dx]=-d$. Hence

$$[g_{\mathcal O}]+\Delta-d=0,$$

so

$$[g_{\mathcal O}]=d-\Delta.$$

If $\Delta<d$, then $g_{\mathcal O}$ has positive mass dimension and the perturbation is relevant. If $\Delta=d$, the coupling is marginal by engineering dimension. If $\Delta>d$, the coupling has negative mass dimension and the perturbation is irrelevant.

At an interacting fixed point, diagonalize operator mixing and use a scaling operator’s full dimension, including its anomalous contribution, to determine its linear RG relevance. A linearly marginal direction still requires higher-order analysis to establish whether it remains marginal. Lecture 11 develops this distinction.

Warm-up 3: transversality of vacuum polarization

Work at generic nonzero momentum in a translation-invariant and Lorentz-invariant vacuum, using a regulator and renormalization prescription that preserve the gauge Ward identity. Let $\Pi_{\mu\nu}$ be the full renormalized vacuum-polarization tensor after the required contact terms have been included, so that

$$q^\mu\Pi_{\mu\nu}(q)=0.$$

Show that the parity-even vacuum polarization tensor in four dimensions can be written as

$$\Pi_{\mu\nu}(q)=\left(q_\mu q_\nu-q^2\eta_{\mu\nu}\right)\Pi(q^2)$$

including transverse local counterterms in $\Pi(q^2)$.

Solution

Lorentz invariance allows the parity-even rank-two tensor structure

$$\Pi_{\mu\nu}(q)=A(q^2)\eta_{\mu\nu}+B(q^2)q_\mu q_\nu.$$

Contracting with $q^\mu$ gives

$$q^\mu\Pi_{\mu\nu}(q)=\left[A(q^2)+q^2B(q^2)\right]q_\nu.$$

Transversality requires

$$A(q^2)=-q^2B(q^2).$$

Thus

$$\Pi_{\mu\nu}(q)=B(q^2)\left(q_\mu q_\nu-q^2\eta_{\mu\nu}\right).$$

Renaming $B(q^2)=\Pi(q^2)$ gives the desired form. Polynomial ambiguities in $\Pi(q^2)$ correspond to local gauge-invariant counterterms. See lecture 13 for the one-loop calculation.

Warm-up 4: Wilson loops and linear confinement

Consider a rectangular Wilson loop at zero temperature, with a fixed additive static-source energy subtraction. The two external sources have separation $R$ and Euclidean time extent $T$. Take large $T$ at fixed $R$ to extract the lowest energy with nonzero overlap in this source sector. In a regime with a stable confining flux tube, retain only the leading large-$R$ area contribution:

$$\langle W(R,T)\rangle\sim e^{-\sigma RT}$$

Use the spectral relation

$$\langle W(R,T)\rangle\sim e^{-V(R)T}$$

to find the leading large-$R$ term of the subtracted static potential $V(R)$.

Solution

Compare the two large-$T$ exponents:

$$e^{-V(R)T}\sim e^{-\sigma RT}.$$

Therefore

$$V(R)=\sigma R.$$

This is the leading large-distance term in the assumed area-law regime; $\sigma$ is the string tension. A different source-energy subtraction adds an $R$-independent constant, and subleading distance-dependent terms are not determined here. With dynamical screening charges, string breaking can invalidate this asymptotic area-law assumption. Lecture 34 explains the semiclassical setting used in this course.

Acknowledgement

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

Any errors in transcription, interpretation, exposition, convention choices, notation, or emphasis are the responsibility of the editor of these webpages.

Further reading

These companions offer complementary treatments. The individual lectures provide focused sources for their derivations.

• Altland, Alexander, and Ben Simons. Condensed Matter Field Theory. 3rd ed. Cambridge University Press, 2023. DOI. Many-body functional integrals, response and RG.
• Birrell, N. D., and P. C. W. Davies. Quantum Fields in Curved Space. Cambridge University Press, 1982. DOI. Particle creation and curved-spacetime examples for the final lectures.
• Coleman, Sidney. Aspects of Symmetry. Cambridge University Press, 1985. DOI. Symmetry and semiclassical reasoning.
• Polyakov, A. M. Gauge Fields and Strings. Routledge, 1987. DOI. Sigma models, instantons, large-N methods and confinement.
• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007. DOI. Perturbation theory and gauge fields.
• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. 5th ed. Oxford University Press, 2021. DOI. Euclidean field theory, RG and critical phenomena.