QFT II

Overview

Quantum field theory becomes most powerful when it is no longer treated as a recipe for drawing diagrams, but as a language for scale, locality, symmetry, and topology. This course is the advanced sequel to QFT I. It begins with renormalization as a physical phenomenon: short-distance modes change the apparent couplings seen at long distances. From there it develops leading logarithms, Wilsonian effective actions, operator mixing, the operator product expansion, QED and Yang–Mills beta functions, Ward identities, Fermi-surface effective theory, two-dimensional field theory, bosonization, nonlinear sigma models, large-$N$ methods, theta terms, instantons, monopoles, confinement mechanisms, vacuum decay, Schwinger–Keldysh functionals, pair creation, and Rindler/Unruh physics.

The central theme is this: a quantum field theory is not a single Lagrangian at a single scale; it is a family of effective descriptions connected by locality and renormalization-group flow. Perturbative logarithms, screening clouds, dynamically generated scales, anomalous dimensions, topological sectors, and thermal horizons are different expressions of that same idea.

These notes are written for readers who have completed a first graduate course in QFT and want a coherent path into the tools used in modern high-energy theory, statistical field theory, and strongly coupled many-body physics. The pages emphasize derivations over slogans. When a calculation is convention-sensitive, the convention is stated explicitly; when a result is best understood physically, the physical interpretation is kept next to the formula rather than postponed to a comment at the end.

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.

Scope of QFT II

The course develops advanced field-theory methods through representative calculations rather than attempting an encyclopedic treatment. It concentrates on scale dependence, effective descriptions, exact structures in low dimensions, semiclassical nonperturbative physics, and real-time or accelerated-frame observables. Subjects such as higher-loop renormalization, lattice algorithms, supersymmetry, conformal bootstrap methods, and full curved-spacetime QFT are used only when they illuminate this path and are not developed systematically here.

The course deliberately keeps several complementary viewpoints in play: perturbative renormalization, Wilsonian coarse graining, background-field determinants, response functions, many-body scaling, two-dimensional exact methods, semiclassical saddles, and horizon thermality. No single viewpoint replaces the others; their agreement in overlapping regimes is one of the principal consistency checks.

Editorial principles

The goal is to make the course useful as both lecture notes and a reference. The guiding principles are:

• Renormalization is presented as physics, not bookkeeping. Counterterms remove cutoff dependence, but the deeper object is the scale dependence of effective descriptions.
• Effective field theory is treated as the default viewpoint. Relevant, marginal, and irrelevant operators are not labels for a table; they are instructions for what survives in the infrared and what must be measured or matched.
• Gauge invariance is kept visible. Vacuum polarization, Ward identities, background fields, Wilson lines, and beta functions are organized so that gauge-invariant content is separated from gauge-dependent language.
• Short-distance expansions and long-distance phases are connected. The same RG logic that organizes perturbative logarithms also controls critical phenomena, sigma models, Fermi surfaces, and confinement criteria.
• Topology is not ornamental. Theta terms, instantons, monopoles, winding numbers, Wilson loops, and vacuum sectors are treated as calculational structures with physical consequences.
• Figures explain structure. Diagrams, flow charts, contours, saddles, target spaces, and causal wedges are included when they clarify the argument rather than merely decorate it.

Course roadmap

A schematic path through QFT II. The course begins with renormalization and leading logarithms, passes through Wilsonian operator methods and gauge-field beta functions, then moves into many-body and two-dimensional field theory, sigma models, topology, confinement, vacuum transitions, and accelerated-frame QFT.

Prerequisites

The expected background is QFT I at the level of canonical quantization, path integrals, Feynman rules, one-loop diagrams, spinors, gauge fields, Ward identities in QED, and basic renormalization. The course assumes comfort with Gaussian functional integrals, Wick rotation, dimensional analysis, Feynman parameters, residues, and distributions.

A reader should also know the basic language of Lie groups and Lie algebras, especially $U(1)$, $SU(N)$, structure constants, generators, and representations. Differential geometry and topology enter later through target spaces, winding numbers, monopoles, instantons, and theta terms; the required ideas are introduced when needed, but prior exposure to forms, homotopy, and fiber-bundle language is helpful.

Several pages use examples from statistical mechanics and condensed matter physics: critical phenomena, Fermi surfaces, one-dimensional fermions, spin chains, and antiferromagnets. The notes do not assume a full many-body course, but they do assume the reader is willing to translate between Euclidean QFT language and statistical field theory language.

Convention lock

Unless a page says otherwise, the notes use natural units,

$$\hbar=c=1,$$

and the mostly-minus Minkowski metric

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

Plane waves and Fourier transforms use

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

and

$$f(x)=\int {d^dp\over (2\pi)^d}\,e^{-ip\cdot x}f(p), \qquad \int_p\equiv\int {d^dp\over (2\pi)^d}.$$

Dirac slash notation is written explicitly in a KaTeX-compatible form:

$$p\!\!\!/\equiv \gamma^\mu p_\mu, \qquad A\!\!\!/\equiv \gamma^\mu A_\mu, \qquad D\!\!\!/\equiv \gamma^\mu D_\mu.$$

The default Wick rotation is

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

For scalar fields,

$$\mathcal L_0={1\over2}\partial_\mu\phi\,\partial^\mu\phi-{1\over2}m^2\phi^2, \qquad S_{E,0}=\int d^dx\,\left[{1\over2}(\partial_\mu\phi)^2+{1\over2}m^2\phi^2\right].$$

Loop integrals are often written in Euclidean form. A logarithmic integral in four dimensions is normalized as

$$\int_{|k|<\Lambda}{d^4k\over(2\pi)^4}\,{1\over(k^2+m^2)^2} ={1\over16\pi^2}\log {\Lambda^2\over m^2}+\text{nonlogarithmic terms}.$$

The renormalization-group beta function is

$$\beta(g)=\mu {\partial g\over\partial \mu}\bigg|_{\text{bare}},$$

so an asymptotically free coupling has

$$\beta(g)=-{b_0\over16\pi^2}g^3+O(g^5), \qquad b_0>0,$$

and therefore becomes weak at short distances.

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

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

with

$$D_\mu=\partial_\mu-iqA_\mu, \qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.$$

For non-Abelian gauge theory, Hermitian generators obey

$$[T^a,T^b]=if^{abc}T^c,$$

and

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

with

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

A Wilson line along a path $\gamma$ is

$$U_\gamma=\mathcal P\exp\left[+ig\int_\gamma dx^\mu\,A_\mu^a(x)T^a\right].$$

In two Euclidean dimensions, the antisymmetric tensor is normalized by

$$\epsilon_{12}=+1.$$

For the $O(3)$ nonlinear sigma model, the Euclidean action and topological charge are written as

$$S_E={1\over2g}\int d^2x\,\partial_\mu\mathbf n\cdot\partial_\mu\mathbf n, \qquad \mathbf n^2=1,$$

and

$$Q={1\over8\pi}\int d^2x\,\epsilon_{\mu\nu}\,\mathbf n\cdot(\partial_\mu\mathbf n\times\partial_\nu\mathbf n)\in\mathbb Z.$$

Theta terms are included in Euclidean weights as

$$e^{-S_E+i\theta Q}.$$

Local pages may introduce additional conventions for finite density, real-time contours, spin-chain normalizations, or Rindler coordinates.

How to read these notes

The course is organized into forty lecture webpages in four parts. The first half builds the renormalization and gauge-theory tools; the second half applies the same ideas to many-body systems, two-dimensional models, topology, confinement, vacuum transitions, and accelerated observers.

The safest route is linear. Lectures 01–14 establish the renormalization viewpoint: leading logarithms, running couplings, effective actions, operator mixing, and QED screening. Lectures 15–22 develop background-field methods, Yang–Mills beta functions, Ward identities, and many-body/Fermi-surface applications. Lectures 23–33 move through two-dimensional gauge theory, bosonization, confinement and screening in two dimensions, nonlinear sigma models, large-$N$ methods, spin chains, theta terms, and instantons. Lectures 34–40 turn to monopoles, confinement in three-dimensional gauge theory, vacuum decay, real-time functionals, pair creation, and accelerated observers.

A high-energy-theory reading path can focus first on 01–20, 28–35, and 37. A statistical-field-theory path can focus on 01, 07–11, 21–22, and 27–33. A topology and nonperturbative-QFT path can begin with 27 and then read 30–40, returning to the RG pages as needed.

The exercises are part of the exposition. They are designed to check signs, normalizations, and physical interpretations that are easy to nod along with but hard to use correctly.

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 shows how a dimensionless coupling produces a physical mass scale through RG flow.
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 computes the perturbative RG of nonlinear sigma models and explains mass generation in two dimensions.
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.

Thematic map

Question	Where it appears	Core idea
Why do logarithms know about scale?	01–08	Logarithmic divergences are the perturbative shadow of RG flow.
What does Wilsonian locality buy us?	09–12	Short-distance physics becomes a controlled expansion in local operators and their mixing.
Why do QED and Yang–Mills run differently?	13–18	Screening and antiscreening come from the spin and self-interaction structure of charged fluctuations.
How do symmetries constrain correlation functions?	19–20	Ward identities and current correlators separate physical content from convention-dependent terms.
What changes at a Fermi surface?	21–22	Low-energy modes live near a surface in momentum space, changing scaling and instability structure.
Why are two-dimensional models so exact?	23–26	Gauge fields, fermions, and bosons have special infrared and topological structures in two dimensions.
How do sigma models generate scales?	27–29	Curved target spaces produce asymptotic freedom and nonperturbative mass gaps.
Why do theta terms matter?	30–33	Topological sectors contribute phases that cannot be removed by local perturbation theory.
How can confinement arise semiclassically?	34	Monopole gases and Wilson loops reveal confinement through long-distance disordering.
What does the vacuum mean out of equilibrium?	35–37	False vacua, in/out vacua, and real-time contours distinguish decay, production, and expectation values.
Why do accelerated observers see heat?	38–40	Horizon-adapted coordinates turn analyticity and Euclidean periodicity into thermal physics.

Diagnostic warm-ups

These short problems test background ideas that return throughout the course. They are not a barrier to entry, but they reveal which tools deserve a quick refresh before starting the lecture pages.

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 $\Lambda\gg m$.

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

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

This coefficient is the standard one-loop logarithmic normalization used repeatedly in the course.

Warm-up 2: relevant, marginal, or irrelevant

In $d$ spacetime dimensions, suppose a local operator $\mathcal O$ has scaling 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.

Quantum corrections can shift dimensions and turn marginal operators into marginally relevant or marginally irrelevant ones. That is the practical origin of many beta functions in the course.

Warm-up 3: transversality of vacuum polarization

Assume Lorentz invariance and current conservation imply

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

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

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

up to local convention-dependent contact terms already included 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.

Warm-up 4: Wilson loops and linear confinement

Suppose a rectangular Wilson loop with spatial size $R$ and Euclidean time size $T$ obeys

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

for large $T$. Use the relation

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

to find the static potential $V(R)$.

Solution

Compare the two large-$T$ exponents:

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

Therefore

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

An area law for large Wilson loops implies a linearly rising static potential. The coefficient $\sigma$ is the string tension.

References and further reading

The lecture pages cite focused references where needed. The following books are especially useful companions for the full course:

• Alexander M. Polyakov, Gauge Fields and Strings — a compact, original, and deeply physical source for gauge fields, sigma models, instantons, confinement, large-$N$ ideas, and stringy viewpoints.
• Sidney Coleman, Aspects of Symmetry and Lectures of Sidney Coleman on Quantum Field Theory — classic sources for renormalization, symmetry, effective actions, solitons, and semiclassical reasoning.
• Steven Weinberg, The Quantum Theory of Fields, Volumes I and II — a systematic reference for relativistic QFT, renormalization, gauge theory, RG methods, OPE, anomalies, and extended field configurations.
• Mark Srednicki, Quantum Field Theory — concise and useful for perturbation theory, renormalization, beta functions, gauge theory, instantons, and theta vacua.
• Matthew D. Schwartz, Quantum Field Theory and the Standard Model — clear on modern perturbation theory, Wilsonian thinking, effective actions, background fields, and the Standard Model.
• A. Zee, Quantum Field Theory in a Nutshell — physically intuitive on renormalization, sigma models, bosonization, topology, and condensed-matter connections.
• Jean Zinn-Justin, Quantum Field Theory and Critical Phenomena — a deep source for Euclidean field theory, RG, critical phenomena, large-$N$ methods, nonlinear sigma models, instantons, and resummation.
• Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory — a standard reference for perturbative gauge theory, renormalization, and non-Abelian beta functions.

A useful first cross-check by topic is:

Pages	Good companion references	Why they are useful
01–14	Srednicki; Schwartz; Zinn-Justin	Renormalization, effective field theory, one-loop logarithms, QED power counting, and critical scaling.
15–20	Schwartz; Weinberg I–II; Srednicki	Proper time, background fields, beta functions, Ward identities, current correlators, and anomalies.
21–26	Zee; Coleman; Zinn-Justin	Fermi surfaces, one-dimensional fermions, QED₂, bosonization, sine-Gordon/Thirring physics, and screening.
27–34	Polyakov; Zinn-Justin; Coleman; Srednicki	Nonlinear sigma models, large-$N$ methods, theta terms, instantons, monopoles, and confinement.
35–40	Coleman; Weinberg I; Birrell–Davies-style curved-spacetime QFT references	Semiclassical tunneling, vacuum decay, in–in functionals, pair creation, Rindler coordinates, and thermal periodicity.

Original landmarks that recur in the course include work by Gell-Mann and Low on charge renormalization, Wilson on the renormalization group and operator products, Gross–Wilczek and Politzer on asymptotic freedom, Schwinger on QED₂, Coleman on bosonization and false-vacuum decay, Polyakov on compact gauge theory and confinement, and Unruh on accelerated observers.

Where to begin

Start with Effective Actions, Dimensional Estimates, and IR Physics. The first page is deliberately modest: it asks what a low-energy observer can know without resolving short distances. That question quietly contains the whole course.