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Proseminar Theoretical Physics PHY391 Proseminar Theoretical Physics PHY391 Massimiliano Grazzini University of Zurich FS21, February 22, 2021 Introduction A selection of topics in theoretical physics relevant for high-energy and condensed matter physics Each student is supposed to give one presentation and to attend at least 80% of the presentations by the other students Active participation is required Assistants: Luca Buonocore Bastien Lapierre Chiara Savoini Ben Stefanek Luca Rottoli 1) Lorentz and Poincare groups Quantum mechanics is an intrinsically nonrelativistic theory. A change of viewpoint — moving from wave equations to quantum field theory — is necessary in order to make it consistent with special relativity. For this reason, it is therefore paramount to understand how Lorentz symmetry is realised in a quantum setting. Besides invariance under Lorentz transformation, invariance under space-time translations is another necessary requirement in the construction of quantum field theory. Translations plus Lorentz transformation form the inhomogeneous Lorentz group, or the Poincaré group. The study of the Poincaré group and its representation allows one to understand how the concept of particle emerges. References: M. Maggiore, A Modern Introduction to Quantum Field Theory, Ch. 2. Luca R 2) Noether theorem You have already seen that in physics we have a deep relation between symmetries and conserved quantities (translation invariance -> momentum conservation; space isotropy -> angular momentum conservation…) The Noether theorem states that every continuous symmetry of the action functional leads to a conservation law derivation of the theorem starting from a classical field Lagrangian (Goldstein chap. 12.7) application of the theorem in the case of invariance under translations and Lorentz transformations: energy-momentum and angular momentum conservation (Itzykson-Zuber chap. 1-2-2) The case of internal symmetries (Itzykson-Zuber chap. 1-2-3) Chiara 3) Phase transitions, the Ising model and spontaneous symmetry breaking A phase transition is a point in parameter space, where the physical properties of a many-particle system are subject to a sudden change. A typical example is the paramagnet to ferromagnet transition in Fe. The Ising Model describes the behaviour of ferromagnetic systems through a simple interaction of each spin with its neighbours and is one the most famous model of statistical mechanics. The one dimensional case can be solved exactly (no phase transition) The solution of the two dimensional case is considerably more cumbersome, but one can show that the model has a finite temperature phase transition between a paramagnetic and a ferromagnetic phase. References: Luca Peliti, Statistical Mechanics in a Nutshell, Ch. 5; Silvio R. A. Salinas, Introduction to Statistical Physics, Ch. 12 and 13 Luca R 4) Renormalisation group By renormalisation group we refer to a technique allowing the systematic investigation of the changes of a physical system as viewed at different scales The introduction of the renormalisation group has been an immense conceptual step forward in theoretical physics, both in particle physics as well as statistical physics In the context of statistical physics, it enables to obtain analytically the set of critical exponents of a given phase transition. It also led to the discovery and understanding of different problems such as the Kondo effect, the BKT phase transition as well as the Luttinger liquid Goal of the proseminar: introduce the renormalisation group method in statistical physics, focussing on simple examples References: Zinn-Justin, “Phase transitions and Renormalisation group”, Oxford Bastien 5) Brehmstrahlung and soft singularities The emission of soft photons by accelerated charges is a phenomenon that can be described at purely classical level p3 Soft (large wavelength) photons do not p1 resolve the details of a scattering p4 process but are just sensitive to the electric charges of the initial and final p2 state particles pn q classical computation and infrared catastrophe (Peskin chap. 6.1 & Itzykson-Zuber chap. 1-3-2) quantum computation: soft eikonal factor, example of Coulomb scattering off a fixed nucleus (Peskin chap. 6.1 & Itzykson-Zuber chap. 5-2-4) Chiara 6) Kinoshita-Lee-Nauenberg theorem Transition probabilities between on-shell states in Quantum Field Theories with massless particles are plagued by infrared divergences Example: electron splitting in electron+photon It is not possible to have the three particles on-shell e− − 2 2 2 2 e (p = (p′) = me and q = 0) and simultaneously conserve p′ energy and momentum unless p q → 0: soft limit q q||p′: collinear limit (when me = 0) These configurations lead to infrared divergences The problem can be entirely cast in the language of Quantum Mechanics The solution is provided by the so-called Kinoshita-Lee- Nauenberg theorem which states that all infrared singularities are canceled provided that summation over all degenerate (in energy) states is performed. Luca B 7) Antiparticles Quantum Mechanics is based on the Schrödinger wave equation, which is intrinsically non relativistic Can we find a relativistic wave equation ? Yes, but relativistic wave equations lead to solutions with negative energy The interpretation of these solutions leads to the concept of antiparticles Particles and antiparticles can be created in pairs out of the vacuum: their number is not any more fixed (second quantisation) Halzen & Martin, Quark and Leptons: an introductory course in modern particle physics, Chapter 3 Ben 8) Beta decay In nuclear physics β-decay is a type of radioactive decay in which a beta particle (energetic electron or positron) is emitted from a nucleus In 1930 Pauli proposed in his famous letter: “Liebe Radioactive Damen und Herren,…..” the existence of neutrino to explain the continuous energy spectrum of the electron 3-body decay and Kurie plot (Krane 9.1-9.3 & Perkins 7.3) Fermi theory (Krane chap. 9.2 & Halzen-Martin chap. 12.1-12.3) Parity violation and Madame Wu experiment (Krane chap. 9.9 & Perkins chap. 7.5) Chiara 9) Static quark model In the 1950-60s the situation in particle physics was particularly confusing Indeed, many strongly interacting particles (hadrons) were known and believed to be distinct elementary particles in their own right. The solution to this problem came by exploiting the notion of Symmetry Group theory and its techniques has proved themselves invaluable tools to get quantitative results and predictions. Group representations and Yang Tableaux Meson and barions as multiplets in the decomposition of products of SU(3) flavour representations “Gauge Theory of Elementary Particle Physics", Ta-Pei Cheng and Ling-Fong Li, Oxford University Press, 1984, Chap.4 Luca B 10) Quarkonia Quarkonium is a bound state of a heavy quark and its own antiquark, which mutually interact (mainly) through the strong interaction Examples: J/psi (charmonium) and Y (bottonium) Historically they represent the first evidence of the existence of charm and bottom quarks From a theoretical point of view, quarkonia represent an interesting problem since perturbative methods cannot be applied. On the other hand, the speed of the charm and the bottom quarks in their respective quarkonia is sufficiently small for relativistic effects in these states to be much reduced An effective phenomenological description is possible in terms of non-relativistic QM Quantum Mechanics with Application to Quarkonium“, C. Quigg and J. L. Rosner, Phys.Rept. 56 (1979) 167-235 A modern introduction to Particle Physics, Third Edition”, Fayyazuddin and Riazuddin, World Scientific Publishing, Singapore, 2012, Secs. 8.5, 8.8, 8.9. Luca B 11) Kaon oscillations Strangeness was introduced to explain the fact that certain particles, such as the kaons or the hyperons are created easily in pairs in particle collisions, but decay much more slowly A new quantum number “strangeness” preserved during their creation but violated in decay Modern picture: strangeness conserved in strong and electromagnetic interactions but not in weak interactions Neutral kaons: K¯0(d¯s) and K0(ds¯) transform into each other through strangeness oscillations non-leptonic decays of neutral kaons: CP eigenstates (Okun chap. 10.1 & Perkins chap. 7.15) Strangeness oscillations in vacuum (Okun chap. 11.3 & Perkins chap. 7.15) Glashow-Iliopoulos-Maiani (GIM) mechanism Chiara 12) Neutrinos Neutrinos are special among the other elementary particles: they are electrically neutral, do not feel the strong interactions, and interact only weakly They are several order of magnitude lighter than all other fermions, and in fact no direct measurement has found evidence for a non zero neutrino mass Measurements of neutrinos coming from the sun or produced by cosmic rays in the Earth’s atmosphere have revealed that neutrinos “oscillate”, that is they change their flavour periodically with time This phenomenon has a characteristic Quantum Mechanics origin By restricting ourselves to the case of 2 flavours, one can describe neutrino oscillations through a two-state system with a time-independent Hamiltonian Neutrino physics“, E. Kh. Akhmedov, Contribution to: ICTP Summer School in Particle, Physics, hep-ph/0001264 [hep-ph] Luca B 13) Higgs mechanism 1 In particle physics the Higgs mechanism is essential to explain the generation of masses for elementary particles The simplest description requires a quantum scalar field (Higgs field) whose vacuum expectation value spontaneously breaks the symmetry of the theory (SSB) concept of vacuum and SSB (example
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