Professor sachin is a faculty member at the Center for High Energy Physics, Indian Institute of Science (IISc) in Bangalore, India. He specializes in theoretical physics. For more information please visit the link.

The handwritten notes of the lectures are shared on the google drive link.

This transcription was done with Claude Opus 5, with every diagram redrawn in TikZ.

1. Lecture 1 -- goals of the course, textbooks, exams, notations and the Minkowski metric, QFT-1 recap (particle \(\leftrightarrow\) field dictionary), scalar mode functions and the K.G. equation, the Dirac equation and the spinor-field expansion, functional derivatives and the space \(\mathcal{F}\) of field configurations, and canonical quantisation of the scalar and Dirac fields.
2. Lecture 2 -- propagators for free fields and \(\Delta(x-y)\), Green's functions and the choice of \(k^0\) contour (\(C_R\), \(C_A\), \(C_F\)), the Feynman propagator, the generating functional \(Z[J]\) and \(Z_0[J]\) for free fields (with the long marginal derivation), interactions in \(\lambda\varphi^4\), the LSZ formulation and the scattering amplitude, and the electromagnetic field -- gauge symmetry, radiation gauge, the transverse/longitudinal split and polarisation sums.
3. Lecture 3 -- quantisation of QED and the photon propagator, the S-matrix functional, scalar QED and spinor QED, the functional-integral representation (metric on field space, \(dV\), Gaussian functional integrals and \(\det M\)), Grassmann variables and the functional integral for fermions, and a working block on lattice discretisation of the Klein--Gordon operator (fermion doubling as homework).
4. Lecture 4 -- Assignment 1, functional integrals for fermions (the \(2\times 2\) Grassmann example giving \(\det A\)), the SHO partition function rebuilt as a position-basis path integral with Euclidean action \(S_E\), the fermionic oscillator (anticommutators, the two-state space, \(H = \frac{\hbar\omega}{2}\sigma_3\) and \(Z = 2\cosh\frac{\beta\hbar\omega}{2}\)), the axioms of Grassmann integration, and coherent states \(|c\rangle = e^{-ca^{\dagger}}|0\rangle\) with \(a|c\rangle = c|c\rangle\).
5. Lecture 5 -- the bra \(\langle c|\) and the overlap \(\langle c'|c\rangle = e^{c'^{*}c}\), the resolution of identity and the trace formula over Grassmann coherent states, \(Z = \mathrm{Tr}\,e^{-\beta H}\) as a fermionic path integral with antiperiodic boundary conditions in imaginary time, a problem on the forced oscillator, and the start of regularization \& renormalization -- the infinite line charge in electrostatics and dimensional regularization.
6. Lecture 6 -- poles of the Gamma function and the \(\epsilon = 1-d\) expansion, the renormalized potential \(\phi^{R}\) (only potential differences are finite), the 2d charge configuration, the generating functional for the interacting scalar field, Gaussian functional integrals and the Euclidean action, why renormalisation is needed in QFT (loop integrals, the cut off \(\Lambda\), bare vs measured couplings), and mass renormalisation through the two-point vertex function \(\Gamma^{(2)}(k)\).
7. Lecture 7 -- mass renormalisation continued (the \(d=4\) cut-off integral and \(\mu^2 = \mu^2(\Lambda; m^2)\)), the two-loop contributions to \(\Gamma^{(2)}\) and \(\Delta A(k)\), defining \(a^2\) from \(d\Gamma^{(2)}/dk^2\), and renormalisation of \(\lambda\) in favour of \(g \equiv \Gamma^{(4)}(0,0,0)\) -- the one-loop four-point diagrams and the cancellation of disconnected diagrams.
8. Lecture 8 -- infrared vs ultraviolet cut off, the perturbative expansion of \(Z(m,\lambda)\) and the Gaussian integrals \(I(a)\), \(I(a,\lambda)\) (why the series is asymptotic and not analytic at \(\lambda=0\)), renormalisation of \(\lambda\) (the four one-loop \(\Gamma^{(4)}\) graphs, the renormalisation condition and the symmetric point), the three \(\Lambda\)-sensitive integrals and the one-parameter family of lagrangians, the counting of internal lines, loop momenta \(L = V - \frac{n}{2} + 1\) and the naive \(q^{4-n}\) behaviour, and 1PI / primitively divergent graphs.
9. Lecture 9 -- primitively divergent graphs in \(\lambda\phi^4\) (tadpole, sunrise/saturn/sunset, the four-point bubble), \underline{dimensional regularisation}: extending \(\int d^d q\) to complex \(w\) by analytic continuation, the split into radial and angular pieces, the derivation of \(V(S^{N-1}) = 2\pi^{N/2}/\Gamma(N/2)\) (checked for \(N=3\)), successive integrations by parts that widen the domain to \(0<w<1\), and the continuation to \(w=2\) with the \(\Gamma(-n+\epsilon)\), \(\psi\) and \(\psi'\) expansions.
10. Lecture 10--11 -- \(\phi^4\) renormalisation in \(2w\) dimensions (\(\mu\) introduced to keep the action dimensionless, \(n=2-w\)), the modified Feynman rules, the tadpole \(T\) and its simple pole in \(\epsilon\), a digression on integrals over arbitrary dimensions (the \(\Gamma\)/Beta-function formulae and the \(\ell_\mu\ell_\nu\) integral), \(\Gamma^{(4)}\) via Feynman parameterization, the two 2-loop diagrams and \(\Sigma(p)\) with \(K(p)\) and \(K_\mu(p)\), and renormalisation by counterterms -- the \(\varphi^2\) counterterm with \(F_1\), the \(\varphi^4\) counterterm with \(G_1\), and the finite \(\Gamma^{(2)}_{\text{new}}\), \(\Gamma^{(4)}_{\text{new}}\).
11. Lecture 12 -- \(\Gamma^{(2)}\) to two loops (the three new diagrams and their \(1/\epsilon^2\) and \(1/\epsilon\) poles, and the disappearance of \(\ln\hat{m}^2\)), the new mass counterterm Feynman rule and the counterterm in \(\mathcal{L}\), the extra \(\frac{1}{2}(\partial\varphi)^2\) counterterm carrying \(H_2\), \(\mathcal{L}^{\text{ren}} = \mathcal{L} + \mathcal{L}_{ct}\) with \(A, B, C\) (i.e. \(Z_\varphi, Z_m, Z_\lambda\)), the bare field / mass / coupling redefinitions, and the derivation of the \underline{Renormalization Group (Gell-Mann -- Low) equation} from the \(\mu\)-independence of the bare vertex functions.
12. Lecture 13 -- bare parameters as a Laurent series in the renormalized ones (\(a_k, b_k, C_k\)), \(\phi^6\) theory in 4d and why it needs infinitely many couplings (renormalizable vs non-renormalizable theories), the Callen -- Symanzik equation, and the three renormalisation schemes -- A (zero-momentum subtraction), B (space-like subtraction point \(M^2\)) and C (minimal, \underline{mass-independent} renormalization) -- ending with \(\mu\,\partial\lambda/\partial\mu = 3\lambda^2/16\pi^2\) and the running coupling \(\lambda(\mu)\).
13. A Short Break from Renormalisation -- an introduction to the infrared problem in QED through the forced harmonic oscillator: time evolution in QM and the Schr\"odinger / Heisenberg / Dirac pictures, the SHO with a forcing function \(f(t)\) switched on between \(T_1\) and \(T_2\), retarded and advanced Green's functions and the in / out operators, the time-ordered exponential for \(\bar{T}\) and its BCH resummation, the scattering operator \(S\), the Poisson transition probability \(|\langle n|S|0\rangle|^2\), and the IR divergence that appears when \(f(t)\) is not turned off fast enough.
14. Lecture 15 -- the running coupling in the mass-independent (minimal) scheme: the \(\mu\)-independence of \(\lambda_0\) giving \(\beta(\lambda) = -2(1-\lambda\partial_\lambda)a_1\) and the recursion for the higher \(a_k\), the one-loop running \(\lambda(\mu)\) and the \underline{Landau pole}, the four scenarios for \(\beta(\lambda)\) (UV and IR fixed points, asymptotic freedom, Yang--Mills), the running mass and \(\gamma_m(\lambda)\), the anomalous dimension \(\gamma_d(\lambda) = -\lambda\,dC_1/d\lambda\), the integrated RG equation with the scale-dependent \(\bar{\lambda}(s)\), \(\bar{m}(s)\), and an aside on integrable models in \(1+1\) dimensions.
15. Lecture 16 -- \underline{gauge theories}: minimal coupling as the requirement of gauge invariance, \(U(1)\) written group-theoretically as a map \(g: M^{1,3}\to U(1)\), the generalisation to \(SU(N)\) (fundamental and higher representations \(D^{(R)}\)), \(SU(2)\) vs \(SO(3)\) and the \(4\pi\) rotation, the covariant derivative \(D_\mu = \partial_\mu + A_\mu\) and the transformation \(A^{g}_{\mu} = gA_\mu g^{-1} - (\partial_\mu g)g^{-1}\), why \(A_\mu\) is Lie-algebra valued, and the infinitesimal gauge transformation \(A^{g}_\mu - A_\mu = i t^a (D_\mu\theta)^a\).
16. Lecture 17 -- the covariant derivative revisited (global vs local), the explicit \(SU(2)\) computation of \((\partial_\mu g)g^{-1}\) showing it lies in the Lie algebra, the infinitesimal transformation worked out term by term, the \underline{adjoint representation} \((T^a)_{bc} = -if_{abc}\) (with \(T^1,T^2,T^3\) for \(SU(2)\)), and the dynamics -- the field strength \(F_{\mu\nu} = [D_\mu,D_\nu] = -it^aF^{a}_{\mu\nu}\), its covariant (adjoint) transformation, and the convention \(\mathrm{Tr}(t^at^b) = \frac12\delta^{ab}\).
17. Lecture 18 -- dynamics for \(A_\mu\): the Yang--Mills lagrangian \(-\frac{1}{4g^2}F^a_{\mu\nu}F^{\mu\nu a}\), the Chern--Simons term in \(2+1\) dimensions, the equations of motion \((D_\mu F^{\mu}_{\ \nu})^a + g^2\delta S_m/\delta A^a_\nu = 0\), the \underline{Bianchi identity} as the Jacobi identity for covariant derivatives, charges as representations, matter fields, and \underline{parallel transport} -- the path-ordered exponential \(U(x,y;C) = P\exp(-\int A_\mu dx^\mu)\).
18. Lecture 19 -- integrability (\(D_\mu U = 0 \Leftrightarrow F_{\mu\nu}=0\)) and \(A_\mu = -(\partial_\mu U)U^{-1}\), the \underline{Wilson loop} \(W(C) = P\exp\oint_C A_\mu dx^\mu \simeq 1 - F_{\mu\nu}\sigma^{\mu\nu}\) and the Wilson line, the Yang--Mills action and the derivation of \(D^\mu F_{\mu\nu} = 0\) by varying \(S\), the Hodge dual \(\tilde{F}_{\mu\nu}\) and the Bianchi identity, Euclidean solutions and the \underline{Bogomolny bound} \(\mathrm{Tr}(FF) \geq \mp\mathrm{Tr}(F\tilde{F})\) with (anti) self dual solutions, \(\int\mathrm{Tr}(F\tilde{F})\) as a total derivative of the Chern--Simons current \(W^\varsigma\), pure gauge at Euclidean infinity, and \underline{homotopy classes} -- maps \(S^3_\infty \to SU(2)\) and the winding number illustrated with \(S^1\to U(1)\).
19. Lecture 20 -- \underline{path integral formulation of gauge theories}: recap of the QM propagator and time slicing, the Hamiltonian formulation of QED, canonical momenta \(\Pi_\mu = F_{0\mu}\) and the Poisson bracket \(\{A_\mu,\Pi_\nu\} = -g_{\mu\nu}\delta(\vec{x}-\vec{y})\), the difficulty \(\Pi_0 = 0\) (4 velocities but 3 momenta -- a singular Legendre transform), the non-uniqueness \(H = H_0 + \int C\Pi_0\) and \(\dot{A}_0 = C\) as a gauge transformation, the secondary constraint \(\partial_i\Pi^i = 0\), the \underline{Dirac theory of constraints} with \(H_{\text{extra}}\) generating gauge transformations on \(A_i\), and \underline{Gauss' law} \(G(A_i,\Pi^j)=0\) as the physical surface in phase space.
20. Lecture 21 -- gauge fixing as a non-singular change of variables and the Jacobian \(\det|\{G,\partial_i\Pi^i\}_{PB}| \neq 0\), \underline{Coulomb gauge} \(G = \partial_iA^i\) with the zero modes of the Laplacian and the longitudinal/transverse split giving \(H_{\text{EM rad}} = \frac12\int(\Pi^T_i\Pi^T_i + B_iB_i)\), \underline{axial (Arnowitt--Fickler) gauge} \(A_3 = 0\) where \(\Pi^3\) is a non-local functional of \(\Pi_1,\Pi_2\), the magnetic field in that gauge, the \underline{phase space path integral} with \(\delta(\mathcal{G})\det|\{\partial_i\Pi^i,\mathcal{G}\}|\) and \(A_0\) introduced to exponentiate \(\delta(\partial_i\Pi^i)\), and the homework.

The same material, typeset in LaTeX as a single document with all diagrams redrawn:

• Quantum Field Theory -- 2 -- PDF (Lec 1-13 + Short Break + Lec 15-21)

• Handwritten scans -- Lectures 15--18
• Handwritten scans -- Lectures 19--21
• Handwritten scans -- Lectures 1--18
• Handwritten scans -- Lectures 1--21 (complete)

• Quantum Field Theory - 2
• QFT2 -- Sunil Mukhi (2024)
• QFT2 -- Ashoke Sen (2022)

Acknowledgements:

All these tex files are created by use of Anthropic's Claude AI using opus 5 with high usage.

For any discrepancies, please refer to the original lectures or my handwritten notes.