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Notes from Sidney Coleman's Physics 253A Notes from Sidney Coleman’s Physics 253a Sidney Coleman∗ Harvard, Fall 1985-1986 semester† ∗typeset by Bryan Gin-ge Chen and Ting Yuan Sen from scans of the handwritten notes of Brian Hill † LATEX version of March 14, 2011 1 0.Preface NotesfromSidneyColeman’sPhysics253a 2 0 Preface It’s unexpected and heart-warming to be asked by Bryan Chen to write something about these notes, 25 years after taking them. I was the teaching assistant for Sidney’s quantum field theory course for three years. In the first year, I sat in, because frankly, I hadn’t learned quantum field theory well enough the first time that I took it. When I have the good fortune to hear a really good lecturer, I often re-copy my notes, preferably the evening on the day that I took them. Once in a while, students would miss a class, and then ask me if they could look at my notes. At some point, the requests started happening enough that it was suggested that a copy be put on reserve in the Harvard physics library. From there, copies of the notes just kept spreading. Sidney once expressed disappointment about the spread of the notes. For one thing, I even wrote down some of his anecdotes and jokes, and that made it less fun for him to re-tell them. For another, he wrote Aspects of Symmetry which shared a lot of material with what he taught in Physics 253b. He may have had in mind that he would write a field theory book as a companion volume. Of course, he never did write a field theory book, or you’d be reading that, and he never tried to rein the copies in. Now that he is gone, we are lucky that his clarity lives on. Thanks to Bryan Chen and Ting Yuan Sen for creating this lovely LATEX version. Some- times transcription can seem tedious, but I hope it was as valuable for them as the first re-copying was for me, and that for you – fellow student of quantum field theory – the existence of these notes is similarly valuable. –Brian Hill, www.lingerhere.org, March 10, 2011 Typesetters’ notes The great field theorist Sidney Coleman for many years taught the course Physics 253 at Harvard on Quantum Field Theory. The notes you are reading were typeset from a scanned version of handwritten lecture notes by Brian Hill from the Fall 1985-1986 semester of the first half of the course: Physics 253a. The Harvard Physics Department has made films of the lectures from the 1975-1976 version of the course available on their website as well. The typesetting for lectures 1-11 was done by Bryan Gin-ge Chen and for lectures 12-28 by Ting Yuan Sen, who also recreated most of the figures. We have attempted to stay as faithful as possible to the scanned notes, correcting only obvious typos and changing some of the in-text references. We have noted some of these changes in comments in the LATEX source, though this was not done systematically. 0.Preface NotesfromSidneyColeman’sPhysics253a 3 Contents 0 Preface 2 1 September 23 7 Units; Translational invariance, Rotational invariance and Lorentz invariance for a singlefreeparticle. ................................ 7 2 September 25 14 Position operator; Violation of Causality; Pair Production; Fock Space; Occupation number representation; SHO; Oscillator-like formalism for Fockspace. 14 3 September 30 21 Causality, observables and quantum fields; Constructing the quantum field from Fock space operators axiomatically; Translational invariance; Lorentz invari- ance and Relativistic normalization; Field constructed satisfies Klein-Gordon equation....................................... 21 4 October 2 28 Constructing Fock space from the quantum field axiomatically; The Method of the Missing Box, Classical Particle Mechanics, Quantum Particle Mechanics, Classical Field Theory, Quantum Field Theory; Quantum field from free scalar theory........................................ 28 5 October 7 38 Hamiltonian recovered in free scalar theory up to infinite constant; Normal ordering; Symmetries and conservation laws, Noether’s theorem; Noether’s theorem in field theory, conserved currents; ambiguity in currents; Energy-momentum tensor........................................ 38 6 October 9 49 Lorentz transformations; Angular momentum conservation; Internal symmetries; SO(2) internal symmetry; Charged field; SO(n)internalsymmetry. 49 7 October 14 58 Lorentz transformation, properties of conserved quantities; Discrete symmetries; φ φ; Charge conjugation; Parity; Ambiguity of choice of parity; Time reversal;→ − Unitary and anti-unitary operators, angular momentum; Dilatations. 58 8 October 16 70 0.Preface NotesfromSidneyColeman’sPhysics253a 4 Scattering theory overview; Low budget scattering theory; Turning on and off func- tion; Schr¨odinger picture, Heisenberg picture and interaction picture; Evolu- tion operator; Time ordered product; Three models; Wick’s theorem. 70 9 October 21 83 Diagrammatic perturbation theory in Model 3; Vertex in model 1; Connected di- agrams; Thm: all Wick diagrams = : e connected Wick diagrams :; Model 1 solved;Model2begun. ..............................P 83 P 10 October 23 93 Model 2 finished; Vacuum energy c.t.; S matrix is 1; Ground state energy; Yukawa potential; Ground state wave function; Model 3 and Mass renormalization; Renormalizedperturbationtheory. ... 93 11 October 28 103 Feynman diagrams in Model 3; Feynman rules in model 3; A catalog of all Feynman diagrams in model 3 to (g2); Scattering amplitude at (g2); Direct and exchangeYukawapotentials.O . .O . 103 12 October 30 112 “Nucleon”-anti“nucleon” scattering at (g2); Energy eigenstate probe; Meson- “nucleon” scattering; “Nucleon”-anti“nucleon”O annihilation; Assembling the amplitudes for various processes into one amplitude; Mandelstam variables; Mandelstam-Kibble plot; crossing symmetry; CPT; Phase space and the S Differential tran. prob. matrix; unit time . ............................ 112 13 November 4 131 Differential tran. prob. Applications of unit time ; Decay; Cross sections, flux; Final state phase dσ space simplified for two bodies; dΩ ; Optical theorem; Final state phase space for three bodies; Feynman diagrams with external lines off the mass shell; they could be an internal part of a larger diagram. ...... 131 14 November 6 144 Fourier transform of the new blob; A second interpretation of Feynman diagrams n with lines off the mass shell; they are the coefficients of ρ in 0 S 0 ρ;A third interpretation of the blob; they are the Fourier transform ofh | the| i VEV of a string of Heisenberg fields; Reformulation of scattering theory; S matrix elements without the turning on and off function; LSZ formula stated. 144 15 November 13 164 0.Preface NotesfromSidneyColeman’sPhysics253a 5 LSZ formula proved; A second look at Model 3 and its renormalization; Renormal- izationconditions.................................. 164 16 November 18 172 Perturbative determination of a c.t.; Problems with derivative couplings; Rephras- ing renormalization conditions in terms of Green’s functions; Lehmann-Kallen spectral representation for the propagator; Rephrasing renormalization con- ditionsintermsof1PIfunctions. 172 17 November 20 183 Perturbative determination of c.t.; Corrections to external lines in the computation of S matrix elements; One loop correction to meson self energy; Feynman’s trick for combining 2 denominators; Shift to make denominator SO(3, 1) in- variant; Wick notation to make denominator O(4) invariant; Integral tables for convergent combinations; Self-energy at one loop studied; Combining lots of denominators; The shift in the general case to reduce any multi-loop inte- gral to an integral over Feynman parameters.. ..... 183 18 November 25 197 Rephrasing coupling constant renormalization in terms of a 1PI function; Exper- imental significance of the definition; Renormalization versus the infinities; Renormalizable Lagrangians; Unstable particles, Decay products. 197 19 December 2 208 Unstable particles, lifetime, method of stationary phase; Where it begins again; Lorentz transformation laws of fields; Equivalent representations; Reducible reps; The finite dimensional inequivalent irreducible representations of SO(3); Unitarity; Complex conjugation; Direct product; Projection operators and reducibility. .................................... 208 20 December 4 227 Parametrizing the Lorentz group; Commutation relations for the generators; de- composition into two sets obeying SO(3) commutation relations; The cata- log; Complex conjugation properties; Tensor product properties; Restriction to SO(3);Thevector;Rank2tensors;Spinors.. 227 21 December 9 241 Lagrangian made of two component spinors; Solution of Weyl equations of motion; Weyl particles; Dirac Lagrangian; Four-component spinors; Weyl basis, Dirac basis; Plane wave solutions of the Dirac equation. ....... 241 0.Preface NotesfromSidneyColeman’sPhysics253a 6 22 December 11 250 Plane wave solutions of the Dirac equation; Pauli’s theorem; Dirac adjoint; Pauli- Feynman notation; Parity; Bilinears; Orthogonality; Completeness; Summary. 250 23 December 16 268 Canonical quantization of the Dirac Lagrangian. .......... 268 24 December 18 274 Perturbation theory for spinors; Time ordered product; Wick’s theorem; Calcu- lation of the contraction (propagator); Wick diagrams; Feynman diagrams; Matrix multiplication; Spin averages and spin sums. ........ 274 25 January 6 290 Parity for spinors; Fermion and antifermion have opposite intrinsic parity; Charge conjugation; Majorana basis; Charge conjugation properties of fermion bilin- ears; Decay of ortho and para positronium; U U = U U ( 1)NF ;PT. 290 P C C P − 26 January 8 302 Effect of PT on states;
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