# GRQFT Lectures

I thank Geev Nahal Gelehpordsari for uploading the videos for this course to a YouTube channel allowing this course to be complete. .

I. Constructing GR through gauge invariance

Jan 20 – Constructing GR 1 Notes . Intro comments, Higgs theory for gravity?, Role of Equivalence Principle, Energy and Momentum, Gauge Symmetries – QED, Yang Mills notation, Preview of construction, Lorentz Invariance notation Video Constructing 1

Jan 22 – Constructing GR 2 Notes Review of philosophy, Local coordinate systems, Gauging Lorentz transformations, metric and tetrad, invariant volume, an invariant action for a scalar field, T_munu as source, equations of motion, covariant derivative, the connection. Video Constructing 2 (This was the class with the screen freeze on my computer. I put a summary of the material into the notes. The recording has the audio, which does not match the notes too well. But this is what survives…)

Jan 29 – Constructing GR 3 Notes Tensors, Aside: photon and lorentz invariance, metricity, derivation of connection, commutator of covariant derivatives and the Riemann curvature, Ricci scalar, properties of Riemann, Einstein Hilbert action, Einstein equation, FLRW exercise Video Constructing 3

Feb 3 – Constructing GR 4 Notes Review, Cosmological constant, First order or Palatini formalism, Boundary term, Alternate form of action, Canonical vs symmetric energy momentum tensor, Angular momentum and spin current, Belinfante construction, ‘Improved’ energy momentum for scalars. Video Constructing 4

II. The weak field limit and tree level Feynman rules

Feb 5 – Weak field 1 Notes Alternate form of Einstein Equation, weak field, expansion of the metric and curvatures, lack of a Green function, gauge invariance, harmonic gauge, gauge invariance for the scalar field, solution for a point mass, particle in a gravitational field, non-relativistic reduction, Einstein-Infield-Hoffman Hamiltonian and coordinate invariance, quantum effects in post-Newtonian limit. Video Weak field 1

Feb 10 – Weak field 2 Notes Energy-momentum for gravity, Constructing GR from self-interactions(Deser), Gravitational waves, Transverse-traceless condition, Polarization tensor from photon polarizations, gravitons from second quantization, weak field QFT, graviton propagator, Feynman rule for matter coupling, triple graviton vertex, Newton potential from graviton exchanges. Video Weak field 2

III. Some advanced field theory techniques

Feb 12 – Advanced 1 Notes Review of QED vacuum polarization, Vacuum polarization in gravity, Gaussian integrals for path integrals, Path integral for field, det O , exp Tr ln O, start background field method, Complex charged scalar, vacuum polarization via background field method. Video Advanced 1

Feb 19 – Advanced 2 Notes Background field method continued, renormalization in QED, the Effective Action, Effective Lagrangians, Generalization, full renormalization, a non-trivial example – chiral perturbation theory, definition of heat kernel. Video Advanced 2

Feb 24 – Advanced 3 Notes Background field exercises, Heat kernel in general, propagator from heat kernel, Example QED vacuum polarization, general heat kernel expansion and QFT divergences, Seeley DeWitt coefficients, Aside: Riemann normal coordinates, heat kernel coefficients for gravity, recall matter loop in gravity, reinterpret loop and divergence using heat kernel. Video Advanced 3

Feb 26 – Advanced 4 Notes Heat kernel expansion, Compare and Contrast methods, propagators in GR, heat kernel expansion of propagator, perturbative expansion. Ghost stories, constraining integrals Video Advanced 4

March 3 – Advanced 5 Notes Ghost day, Constrained integrals, the Jacobian, gauge fixing in gauge theory, QED, covariant gauges, Fadeev Popov trick and ghosts, review of Yang Mills, ghosts in QCD. Video Advanced 5

IV. Path integral quantization of General Relativity

March 5 – Path Integral 1 Notes Background field expansion of connection, gauge invariance w.r.t background field, curvatures, Lagrangian, harmonic gauge fixing, the ghost Lagrangian, full path integral. Video Path Integral 1

March 10 (part 1) – Path Integral 2 Notes One loop infinities – scalar field and perturbative matching, Gauss Bonnet identity, one loop divergences of gravity, pure gravity is one-loop finite, first indication of power counting, comments on quantizing gravity Path integral 2 (This video also contains some of the next lecture.

V. Effective field theory of General Relativity

March 10 (part 2) – GREFT 1 Notes Why do quantum calculations work, uncertainty principle and locality, example – vacuum polarization, Appelquist Carazzone theorem. Video GREFT 1 (This part starts at 49:30)

March 12- GREFT 2 Notes Organizing principles: #1 Locality, Effective Lagrangians, #2 energy expansion, integrating out a heavy field, #3 Loops renormalization, matching/measuring, the linear sigma model and its effective Lagrangian, renaming fields, equivalence of physical effects, Haag’s theorem Video GREFT 2

March 24 – GREFT 3 Notes Effective field theory as a full QFT, correspondence, tree-level matching to E^4, renormalize EFT, equivalence of finite parts of loops, matching the EFT to full theory, aside: imaginary parts of amplitudes, power counting, Weinberg theorem, matching vs measuring Video GREFT 3

March 26 – GREFT 4 Notes Rules for EFT, GR as EFT, the Lagrangian ordered in energy, bounding the R^2 terms, quantize and renormalize, power counting, the Newtonian potential, non-analytic terms, quantum correction to Newton, infrared photons and gravitons, soft IR divergence cancellation. Video GREFT 4

March 31 – GREFT 5 Notes Graviton-graviton scattering and IR effect, Reissner-Nordstrom metric and origin of classical terms, Bending of light and EP violation, (Lack of) running couplings in GR. Video GREFT 5

VI. Fermions, torsion, Holst etc.

April 2 – Fermions 1 Notes Non-local actions in GR, Limits of the EFT – UV and extreme IR, (End of EFT) Back to the foundations: Lorentz transformation for fermions, SO(1,3) algebra, gauged coordinate transformations and tetrad, gauged Lorentz/spin transformations and the spin connection. Video Fermions 1

April 7 – Fermions 2 Notes The two gauge symmetries with fermions, curvature from spin connection, relating spin connection and affine connection, equivalence of action formulations, introduction to torsion, normalization of torsion, properties, field strength for translation symmetry. Video Fermions 2

April 9 (part 1) – Fermions 3 Notes Two field strengths, intro torsion phenomenology, decomposition of torsion tensor, mass term, but normalization, (but)^2 Hayashi-Shirafuji analysis, (but)^3 field redefinitions, minimal coupling to fermions, Holst action Video Fermions 3 and Anomalies 1

VII. Anomalies

April 9 (part 2) – Anomalies 1 Notes Anomalies, 4 pathways, path integrals and current matrix elements, Noether current, path integral test for symmetries, Conditions: 1) invariance of Lagrangian 2) Invariance of P.I. measure (See previous lecture for video)

April 14 – Anomalies 2 Notes Scale invariance, the dilation current, virial current and the trace of the energy-momentum tensor, Fujikawa and the Jacobian of the scale transformation, heat kernel regularization, calculation of Jacobian, path integral test, the trace anomaly, application to QCD, sketch of Feynman diagram approach. Video Anomalies 2 Part 1 and Video Anomalies 2 Part 2 ( Recording of the second part of class, after the screen freeze).

April 16 – Anomalies 3 Notes IR derivations of anomalies, non-local actions, dispersion relations. Gravitational anomalies, scale, conformal symmetry, conformal scalar fields, Conformal anomaly, the Weyl tensor and conformal anomalies. The axial anomaly, heat kernel calculation, very brief phenomenology, the gravitational axial anomaly. Video Anomalies 3