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Physics 215C: Quantum Field Theory

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Physics 215C: Quantum Field Theory Physics 215C: Quantum Field Theory Lecturer: McGreevy Last updated: 2014/03/25, 15:49:16 0.1 Introductory remarks...............................4 0.2 Conventions....................................6 1 Ideas from quantum mechanics, I7 1.1 Broken scale invariance..............................7 1.2 Integrating out degrees of freedom........................ 13 1.2.1 Attempt to consolidate understanding................. 16 1.2.2 Wick rotation to real time......................... 18 2 Renormalization in QFT 22 2.1 Naive scale invariance in field theory...................... 22 2.2 Blob-ology: structure of diagrammatic perturbation theory.......... 24 2.3 Coleman-Weinberg(-Stone-Dasgupta-Ma-Halperin) potential......... 35 2.3.1 The one-loop effective potential..................... 37 2.3.2 Renormalization of the effective action................. 38 1 2.3.3 Useful properties of the effective action................. 41 2.4 The spectral density and consequences of unitarity............... 45 2.4.1 Cutting rules............................... 51 3 The Wilsonian perspective on renormalization 54 3.1 Where do field theories come from?....................... 54 3.1.1 A model with finitely many degrees of freedom per unit volume... 54 3.1.2 Landau and Ginzburg guess the answer................. 56 3.1.3 Coarse-graining by block spins...................... 58 3.2 The continuum version of blocking....................... 62 3.3 An extended example: a complex scalar field.................. 65 3.3.1 Important lessons............................. 72 3.3.2 Comparison with renormalization by counterterms........... 73 3.3.3 Comment on critical exponents..................... 74 3.3.4 Once more with feeling (and an arbitrary number of components).. 78 4 Effective field theory 84 4.1 Fermi theory of Weak Interactions........................ 87 4.2 Loops in EFT................................... 88 4.2.1 Comparison of schemes, case study................... 89 4.3 The SM as an EFT................................. 94 4.4 Quantum Rayleigh scattering.......................... 96 4.5 QFT of superconductors and superfluids.................... 98 4.5.1 Landau-Ginzburg description of superconductors............ 98 2 4.5.2 Lightning discussion of BCS........................ 100 4.5.3 Non-relativistic scalar fields....................... 103 4.5.4 Superfluids................................. 105 5 Roles of topology in QFT 107 5.1 Anomalies..................................... 107 5.1.1 Chiral anomaly.............................. 108 5.1.2 The physics of the anomaly....................... 113 5.2 Topological terms in QM and QFT....................... 116 5.2.1 Differential forms and some simple topological invariants of manifolds 116 5.2.2 Geometric quantization and coherent state quantization of spin systems118 5.2.3 Ferromagnets and antiferromagnets.................... 124 5.2.4 The beta function for non-linear sigma models............. 127 5.2.5 Coherent state quantization of bosons.................. 129 5.2.6 Where do topological terms come from?................ 130 6 Guide to unfulfilled promises 131 6.1 Linear response: nothing fancy, just QM.................... 131 6.1.1 Linear response, an example....................... 132 6.2 Next steps..................................... 132 3 0.1 Introductory remarks I will begin with some comments about my goals for this course. The main goal is to make a study of coarse- graining in quantum systems with extensive degrees of freedom. For silly historical rea- sons, this is called the renormalization group (RG) in QFT. By `extensive degrees of free- dom' I mean that we are going to study mod- els which, if we like, we can sprinkle over vast tracts of land, like sod (see Fig.1). And also like sod, each little patch of degrees of freedom only interacts with its neighboring patches: this property of sod and of QFT is Figure 1: Sod. called locality. By `coarse-graining' I mean ignoring things we don't care about, or rather only paying attention to them to the extent that they affect the things we do care about.1 In the course of doing this, I would like to try to convey the Wilsonian perspective on the RG, which provides an explanation of the totalitarian principle of physics that anything that can happen must happen. And I have a collection of subsidiary goals: • I would like to convince you that \non-renormalizable" does not mean \not worth your attention," and explain the incredibly useful notion of an Effective Field Theory. • There is more to QFT than perturbation theory about free fields in a Fock vacuum. In particular, we will spend some time thinking about non-perturbative physics, effects of topology, solitons. • I will try to resist making too many comments on the particle-physics-centric nature of the QFT curriculum. QFT is also quite central in many aspects of condensed matter physics, and we will learn about this. From the point of view of someone interested in QFT, high energy particle physics has the severe drawback that it offers only one 1To continue the sod example, a person laying the sod in the picture above cares that the sod doesn't fall apart, and rolls nicely onto the ground (as long as we don't do high-energy probes like bending it violently or trying to lay it down too quickly). These long-wavelength properties of rigidity and elasticity are collective, emergent properties of the microscopic constituents (sod molecules) { we can describe the dynamics involved in covering the Earth with sod (never mind whether this is a good idea in a desert climate) without knowing the microscopic theory of the sod molecules (I think they might be called `grass'). Our job is to think about the relationship between the microscopic model (grassodynamics) and its macroscopic counterpart (in this case, suburban landscaping). 4 example! (OK, for some purposes you can think about QCD and the electroweak theory separately...) • There is more to QFT than the S-matrix. In a particle-physics QFT course you learn that the purpose in life of correlation functions or green's functions or off-shell am- µ 2 plitudes is that they have poles (at p pµ − m = 0) whose residues are the S-matrix elements, which are what you measure (or better, are the distribution you sample) when you scatter the particles which are the quanta of the fields of the QFT. I want to make two extended points about this: 1. In many physical contexts where QFT is relevant, you can actually measure the off-shell stuff. This is yet another reason why including condensed matter in our field of view will deepen our understanding of QFT. 2. The Green's functions don't always have simple poles! There are lots of interesting field theories where the Green's functions instead have power-law singularities, like 1 G(p) ∼ p2∆ . If you fourier transform this, you don't get an exponentially-localized packet. The elementary excitations created by a field whose two point function does this are not particles. (Any conformal field theory (CFT) is an example of this.) The theory of particles (and their dance of creation and annihilation and so on) is a proper subset of QFT. Here is a confession, related to several of the points above: The following comment in the book Advanced Quantum Mechanics by Sakurai had a big effect on my education in physics: ... we see a number of sophisticated, yet uneducated, theoreticians who are conversant in the LSZ formalism of the Heisenberg field operators, but do not know why an excited atom radiates, or are ignorant of the quantum-theoretic derivation of Rayleigh's law that accounts for the blueness of the sky. I read this comment during my first year of graduate school and it could not have applied more aptly to me. I have been trying to correct the defects in my own education which this exemplifies ever since. I bet most of you know more about the color of the sky than I did when I was your age, but we will come back to this question. (If necessary, we will also come back to the radiation from excited atoms.) So I intend that there will be two themes of this course: coarse-graining and topology. Both of these concepts are important in both hep-th and in cond-mat. As for what these goals mean for what topics we will actually discuss, this depends some- what on the results of pset 00. Topics which I hope to discuss include: Theory of renor- malization, effective field theory, effects of topology in QFT, anomalies, solitons and defects, lattice models of QFT, CFT in d = 1 + 1 and in d > 1 + 1. We begin with some parables from quantum mechanics. 5 0.2 Conventions You will have noticed above that I already had to commit to a signature convention for the metric tensor. I will try to follow Zee and use + − −−. I am used to the other signature convention, where time is the weird one. We work in units where ~ and c are equal to one unless otherwise noted. The convention that repeated indices are summed is always in effect. I will try to be consistent about writing fourier transforms as Z d4p Z e−ipx=~f(x) ≡ ¯d4p e−ipx=~f(x) ≡ f~(p): (2π~)4 I reserve the right to add to this page as the notes evolve. Please tell me if you find typos or errors or violations of the rules above. 6 1 Ideas from quantum mechanics, I 1.1 Broken scale invariance Reading assignment: Zee chapter III. Here we will study a simple quantum mechanical example (that is: a finite number of degrees of freedom) which exhibits many interesting features that can happen strongly inter- acting quantum field theory { asymptotic freedom, dimensional transmutation. Because the model is simple, we can understand these phenomena without resort to perturbation theory. I learned this example from Marty Halpern. Consider the following (`bare') action: Z 1 Z 1 S[x] = dt ~x_ 2 + g δ(2)(~x) ≡ dt ~x_ 2 − V (~x) 2 0 2 where ~x = (x; y) are two coordinates of a quantum particle, and the potential involves δ(2)(~x) ≡ δ(x)δ(y), a Dirac delta function.
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