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 of the double-well potential in MQ)

Goldstone theorem: SSB of global and gauge symmetries

Higgs mechanism in the case of a complex scalar ﬁeld Lagrangian

Ryder, Quantum Field Theory, Cambridge, 8.1-8.3 Chiara 14) Higgs mechanism 2

The Higgs mechanism turns out to also be relevant in the context of nonrelativistic physics, and in particular in explaining superconductivity Superconductivity occurs through the formation of Cooper pairs: electron- electron pairs whose repulsion is screened A signiﬁcant fraction of such Cooper pairs occupy a single quantum state (Bose condensation)

The magnetic ﬁeld is expelled (Meissner effect), or, more precisely, exponentially suppressed beyond the

penetration length λ ⟶ photon mass Mγ = ℏ/λc

Goal: understand the formulation of the Higgs-Anderson mechanism for superconductors, and how it can explain the Meissner effect

L.Dixon, “From superconductors to supercolliders”, 1996 Bastien R.Shimano, N.Tsuji, “Higgs mode in superconductors”, arXiv 1906.09401 15) Inﬂation

Inflation is a period of exponential expansion of space in the early universe. The inflationary epoch lasted from 10−36 seconds after the conjectured Big Bang singularity to some time between 10−33 and 10−32 seconds after the singularity. Following the inflationary period, the universe continued to expand, but at a slower rate.

Inﬂation is considered responsible for the large-scale homogeneity of the universe and for the small ﬂuctuations that were the seeds for the formation of structures like galaxies

Inﬂation as a solution of the ﬂatness and horizon problems of Big Bang cosmology

TASI lectures on inﬂation, Daniel Baumann, arXiv:0907.5424 Ben 16) Black holes

A black hole is a region of spacetime where gravity is so strong that nothing, not even electromagnetic radiation can escape from it The theory of general relativity predicts that a sufﬁciently compact mass can deform space time to form a black hole

Find the Schwarzschild black hole solution in general relativity

Hawking radiation and black hole evaporation

J.B.Hartle, An introduction to Einstein’s general relativity, Addison-Wesley (2003)

Ben 17) Majorana fermions

Majorana fermions are hypothetical particles postulated to be their own antiparticle. In relativistic particle physics, such particles have never been observed, and all known fundamental fermions are of Dirac type. In the context of condensed matter physics, Majorana fermions can arise as quasiparticles, which are collective excitations of several particles in a many-body system.

Goal: study the one dimensional Kitaev wire, which displays Majorana zero modes at its boundary. It will also be interesting to discuss potential applications of this model, which is a building block for topological quantum computation.

C.W.J Beenakker, “Search for Majorana fermions in Superconductors”, Annual Review of Condensed Matter Physics, Vol. 4:113-13

A. Yu. Kitaev, "Fault-tolerant quantum computation by anyons”, quant-ph/9707021

Bastien 18) Quantum Hall effect

The quantum Hall effect appears in particular two dimensional systems subject to an external magnetic ﬁeld, where the conductivity displays integer as well as fractional quantised values. Because of its topological origin, this phenomena is robust to the presence of perturbations and disorder

2 σH = νe /h Goal of the project: understand the phenomenology of the integer quantum Hall effect, using the notion of Landau levels

25 Years of Quantum Hall Effect (QHE) A Personal View on the Discovery, Physics and Applications of this Quantum Effect, Klaus von Klitzing, Seminar Poincaré, 2004 Bastien 19) Berry phase

The concept of Berry phase emerges in the context of Quantum Mechanics as a geometric phase obtained by adiabatically performing a closed cycle in the parameter space of the Hamiltonian Elementary example: parallel transport of a vector on a curved surface

Since its discovery (Berry, 1984), it became an important tool in the ﬁeld of condensed matter physics to study topological phenomena such as the quantum Hall effect, or to formulate the modern theory of electric polarisation.

Goal: introduce the Berry phase in a quantum mechanical system, and discuss its applications to condensed matter physics. Di Xiao, Ming-Che Chang, Qian Niu, “Berry phase effects on electronic properties”, Reviews of Modern Physics, 82, 2010 Bastien 20) Bell Inequalities and Hidden variables

In Quantum Mechanics one has to abandon the classical concept of local realism For example, in the two-slit experiment it does not make sense to ask if one particle went through one or the other slit

The existence of ‘Hidden Variables’ in Quantum Mechanics is a possible way to reconcile local realism and Quantum Mechanics.

John Bell showed that it is possible to experimentally verify the hypothesis of local realism, and to test whether Quantum Mechanics and local realism are compatible.

References: Gottfried & Yang, Quantum Mechanics: Fundamentals, Ch. 12, Sakurai, Modern Quantum Mechanics, Sect. 3.10

Luca R 21) Extra dimensions

Extra dimensions: proposed additional dimensions beyond the 3+1 dimensions of space and time First proposals date back to the work of Kaluza and Klein in 1920 in an attempt to unify the forces of Nature

Study the Kaluza-Klein model with a ﬂat, periodic 5th dimension

Understand how a U(1) gauge theory (EM) emerges from 5D gravity after integrating out the extra dimension

PDG review on Extra Dimensions; Eduardo Ponton, TASI 2011: Four Lectures on TeV Scale Extra Dimensions, 1207.3827 Ben 22) Randall-Sundrum model

The RS1 model attempts to solve the hierarchy problem (huge difference between the electroweak scale and the Planck scale) assuming that all the elementary particles are localised on a 3+1 dimensional brane and that the warped geometry generates an exponential hierarchy between the UV (Planck) and IR (EW) branes

Study the model (5d with two endpoint branes, only gravity in the bulk. SM ﬁelds all in the IR brane) Solve the 5D Einstein equations and show how the warped geometry generates an exponential hierarchy between the UV (Planck) and IR (EW) branes

L.Randall,R.Sundrum, “A Large mass hierarchy from a small extra dimension”, Phys.Rev.Lett. 83 (1999) 3370, hep-ph/9905221 Ben 23) Random systems and Markov processes

Many phenomena in nature are characterised by quantities which vary in a random way

One example is brownian motion: the motion of a particle suspended in a ﬂuid, observed under a microscope, looks random. Although one cannot compute the position of the particle in detail, certain average features display simple laws.

One of the most important stochastic processes are so-called Markov processes: they are random processes without memory. Random walk and Brownian motion are two typical example of Markov processes. These processes can be easily simulated on a computer.

Luca Peliti, Statistical Mechanics in a Nutshell, Ch. 5; Silvio R. A. Salinas, Introduction to Statistical Physics, Ch. 12 and 13 Luca R 24) Solitons Very general kind of waves characterised by unusual stability due to the cancellation of non-linear and dispersive effects

They have fascinated scientists since their ﬁrst observation by J. Scott- Russel in 1834

They occur in many areas of physics, from ﬁbre optics to solid state physics and biology Peregrine soliton Goal: understand the main ideas that underline the soliton concept and to present some applications.

M. Peyrard, Introduction to solitons and their applications in physics and biology; Michel Peyrard et Thierry Dauxois, Physique des solitons (French), Physics of Solitons (English), Ch. 1. Luca R 25) Chaos theory

In modern Physics, probability and randomness are usually connected with Statistical Physics or with the principles of Quantum Mechanics, as opposed to the deterministic equation of motions of Classical Mechanics. However, even "simple" non-linear dynamical systems can exhibit surprising dynamical behaviours Small differences in initial conditions can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behaviour impossible in general.

Goal: introduce the logistic map as a model for how bug populations achieve dynamic equilibrium. It is an example of a very simple, but nonlinear, equation producing surprising complex behaviour. The main concepts of chaos theory, ﬁxed points, cycles bifurcations and Lyapunov exponents, will be introduced (if possible with computer program examples)

“Computational Physics, Problem Solving with Computers”, R. H. Landau, M, J. Paez and C. C. Bordeianu, 2nd Edition, WILEY-VCH, 2007. Chapter 18 (and 19) “The Logistic Map”, https://young.physics.ucsc.edu/115/logistic.pdf, (with Mathematica) Luca B