Gauge theories and the standard model of elementary particle physics
|
|
- Janice Hudson
- 7 years ago
- Views:
Transcription
1 Gauge theories and the standard model of elementary particle physics Mark Hamilton 21st July / 35
2 Table of contents 1 The standard model / 35
3 The standard model The standard model is the most successful theory of elementary particle physics. Its predictions have been verified in numerous experiments at particle accelerators. The standard model describes all known interactions among elementary particles, except gravity. The mathematical foundation of the standard model is a gauge theory, based on the symmetry group U(1) SU(2) SU(3). The particles of the standard model consist of three generations of fermions (electron, neutrino, quarks, etc.), the gauge bosons (photon, gluon, etc.) that mediate the interactions between the fermions and the Higgs boson which is a part of the Higgs field that conveys mass to the fermions and some of the gauge bosons. 3 / 35
4 The particles of the standard model Figure: The elementary particles of the standard model (en.wikipedia.org) 4 / 35
5 History 1940s: Development of quantum electrodynamics. 1954: Yang and Mills develop non-abelian gauge theories. 1956: Lee, Wu and Yang discover chirality of the weak interaction. 1961: Glashow unites electromagnetic and weak interaction. 1964: Gell-Mann and Zweig postulate existence of quarks. 1964: Higgs and others develop the Higgs mechanism. 1967: Salam and Weinberg combine Glashow s model with the Higgs mechanism. 1972: t Hooft and Veltman prove renormalizability of electroweak interaction : Development of quantum chromodynamics. Several Nobel prizes for development of the standard model (in particular 1979 for Glashow, Salam and Weinberg, 1999 für t Hooft and Veltman as well as 2013 for Higgs and Englert). 5 / 35
6 Gauge theories as field theories Gauge theories are field theories that have two kinds of symmetries: Lorentz invariance Gauge invariance We will describe below two models for gauge theories: Geometrical model, in which the symmetries are implicit Physical model, in which the symmetries are explicit. We will also describe the transition from the geometrical to the physical model. We will only discuss the classical field theory, in particular the Lagrangian. In physics the field theory is quantized (quantum field theory) and the Feynman rules are deduced from the Lagrangian. Elementary particles correspond to quanta, i.e. minimal excitations of the fields. 6 / 35
7 Bundles Gauge theories are formulated geometrically in the language of bundles over manifolds, in particular principal bundles and vector bundles (spinor bundle, associated bundles). We have the following fundamental fact: Remark Every bundle over M = R 4 is trivial, i.e. a product M Σ consisting of base M and fiber Σ, since R 4 is contractible. However, such a bundle is not canonically trivial. In other words: There are trivializations, but in general none of them is preferred. In this regard bundles are similar to other mathematical objects. For example, every vector space has a basis, but none of them is preferred. In physics one often chooses trivializations and the independence of such a choice reflects itself in invariances and symmetries. 7 / 35
8 The geometrical model The geometrical model of a gauge theory consists of the following data: 1 Spacetime (M, η), a differentiable manifold with a metric of signature (+,,..., ), and the spinor bundle S over M. 2 The gauge group G, a compact, semisimple Lie group. 3 The gauge bundle P M, a G-principal bundle. 4 A unitary representation G V V on a complex vector space V. 5 The associated bundle E = P G V. 6 The fermion bundle or multiplet bundle F = S E. 7 The gauge boson field, a connection A on the principal bundle P with curvature F A. 8 / 35
9 Spinor bundle The spinor bundle is a certain complex vector bundle over the manifold M (it exists if M is spin). On the spinor bundle there exists a Clifford multiplication so that TM S S, (v, ψ) v ψ, v (v ψ) = 2η(v, v)ψ. If (M, η) = (R 4, η Mink ), then S = M C 4 after a choice of inertial frame. In this case, Clifford multiplication with a basis vector e µ in an inertial frame is given by multiplication of a spinor in C 4 with iγ µ, where γ µ are certain 4 4 Dirac matrices that satisfy {γ µ, γ ν } = γ µ γ ν + γ ν γ µ = 2η µν Id. 9 / 35
10 Principal bundle A G-principal bundle is a manifold P with a projection π : P M and an action P G P so that: 1 Every fibre of P is diffeomorphic to G and the action of G preserves the fibres and is simply transitive on them. 2 P is over small open sets U in M of the form U G. In other words, P is locally trivial. If (M, η) = (R 4, η Mink ), then P is globally trivial. In this case we have P = M G, together with the standard action of G. It is important that these trivializations are not canonical: A trivialization is given by a global section s : M P. We then get every element of P as s(x) g, for x M. 10 / 35
11 Gauge The standard model Definition (Gauge) We call a global section s of the principal bundle P a gauge. Every gauge defines a trivialization of P. The notion of gauge is of central importance for gauge theory (standard notion?). It plays a similar role to the notion of inertial frame in relativity. In both cases there is a manifold which is trivial in a certain sense, but that does not have a preferred trivialization. The change between two trivializations is described by a Lorentz and gauge transformation, respectively. The choice of gauge is the choice of a coordinate system. Gauge invariance will later mean: Invariance under the choice of gauge. 11 / 35
12 Associated bundle Let G V V be a representation. Then G acts on P V from the right via (p, v) g = (p g, g 1 v). The associated bundle is the quotient E = (P V )/G = P G V. It is a vector bundle over M with fibre isomorphic to V. If (M, η) = (R 4, η Mink ), then E is trivial, E = M V. A trivialization is not canonical, but given by a choice of gauge: If s : M P is a gauge, then sections Φ: M E correspond precisely to mappings φ: M V via Φ = [s, φ]. 12 / 35
13 Fermion bundle The fermion bundle F is given by F = S E. Let (M, η) = (R 4, η Mink ). We choose an inertial frame and a gauge s : M P. Let r denote the complex dimension of the representation space V. Then a section Ψ in F is given by ψ 1 Ψ =., ψ r where each component ψ i : M C 4 is a spinor. A fermion, i.e. a section of F, is thus described by a vector that has r components, each one of which is a spinor (multiplet). The representation of G mixes these components. 13 / 35
14 Connection and curvature I A connection A on the principal bundle P is a certain invariant 1-form on P with values in the Lie algebra g. The curvature of A is defined as F A = da + 1 [A, A]. 2 Here the commutator is to be taken in g. The curvature F A is a 2-form on P with values in g. One can think of the curvature as a 2-form on the basis M with values in the associated bundle Ad(P) = P G g, defined by the adjoint action. The difference between two connections is a 1-form on M with values in this bundle. One therefore says that gauge bosons transform under the adjoint action of the gauge group G. 14 / 35
15 Connection and curvature II Let (M, η) = (R 4, η Mink ) and choose a gauge s : M P. Then the differential of s defines the following forms on M with values in the Lie algebra g: With A = A ds F A = F(ds( ), ds( )). A µ = A(e µ ), F A µν = F A (e µ, e ν ) we have the fundamental equation Curvature F A µν = µ A ν ν A µ + [A µ, A ν ]. 15 / 35
16 Covariant derivative Every connection A on P defines a covariant derivative A on the associated bundle E: Let (M, η) = (R 4, η Mink ) and s : M P be a gauge. Then every section Φ in E is described by a map φ: M V, so that Φ = [s, φ]. In an inertial frame the corresponding covariant derivative is given by Covariant derivative A µφ = µ φ + A µ φ. On the right hand side the g-valued function A µ acts on the V -valued function φ via the representation of the group G. 16 / 35
17 Gauge fields Let n be the dimension of G. Choosing a basis T a of g, with a = 1,..., n, we can write A µ = A a µt a (Einstein summation convention). The connection A µ corresponds via the Lorentz metric to n vector fields A 1µ, A 2µ,..., A nµ on M. These vector fields describe the gauge bosons. There are thus precisely dim(g)-many gauge bosons in the gauge theory. The corresponding covariant derivative on F = S E describes the coupling of the gauge bosons to the fermions (the fermions interact with the gauge field and thus indirectly with each other, emission/absorption of gauge bosons). The term [A µ, A ν ] describes the interaction of the gauge bosons with each other in non-abelian gauge theories. 17 / 35
18 Dirac operator The covariant derivative A on E defines together with the spin connection S on the spinor bundle S a covariant derivative F on the fermion bundle F = S E. This defines with Clifford multiplication a twisted Dirac operator D A : C (S E) C (S E). We only need a formula for (M, η) = (R 4, η Mink ): Let s : M P be a gauge. Then we have in an inertial frame Dirac operator D A Ψ = iγ µ A µψ = iγ µ ( µ + A µ )Ψ, for Ψ = ψ 1. ψ r. The Dirac matrices γ µ here act on each 4-spinor component ψ i, while the gauge fields A µ act on the r components of Ψ. 18 / 35
19 Lagrangian I We can now write down the Lagrangian L of the gauge theory. The Lagrangian is a real valued function on M. It is given by where L = L fermion + L YM L fermion = Ψ, (D A m)ψ L YM = c 4g 2 FA F A. Here, is a hermitian scalar product on F = S E, m is the mass of the fermion, c a constant depending on the group G, g the coupling constant and a scalar product on the Ad(P)-valued 2-forms on M. 19 / 35
20 Lagrangian II We can also formulate the Lagrangian for (M, η) = (R 4, η Mink ). We choose a gauge s : M P and an inertial frame. Then we have L fermion = Ψ(iγ µ A µ m)ψ L YM = 1 4g 2 F Aaµν F Aa µν = 1 2g 2 tr(f Aµν F A µν). Here Ψ = Ψ γ 0, so that terms like ΨΨ and Ψγ µ A µψ transform as a scalar. In addition we choose a basis T a of the matrix algebra g, so that tr(t a T b ) = 1 2 δab. Then we write F A µν = F Aa µν T a. 20 / 35
21 The physical model The physical model is given precisely by this second Lagrangian. It is supposed to have the following symmetries: Definition (Lorentz invariance) The Lagrangian is independent of the choice of inertial frame. Definition (Gauge invariance) The Lagrangian is independent of the choice of gauge. It is clear that the Lagrangian is Lorentz invariant. We only have to check gauge invariance. 21 / 35
22 Gauge invariance I Let s, s : M P be two gauges. Then there is a gauge transformation U : M G so that s = s U. If Φ is a section of E, then Φ is described by φ, φ : M V with Φ = [s, φ] = [s, φ ]. We therefore have φ = U φ. The connection A is described in the gauges by 1-forms A, A on M with A = A ds, A = A ds. One can check (for a matrix group G): A µ = U A µ U 1 + U µ (U 1 ). 22 / 35
23 Gauge invariance II It follows that A µ φ = U A µ(u 1 φ ) F A µν = U F A µν U 1. These equations imply the gauge invariance of L fermion und L YM (for L fermion we use that the representation of G on V is unitary). Theorem The Lagrangian L = L fermion + L YM is gauge invariant. This was implicitly clear from the geometric formulation. 23 / 35
24 Normalized gauge fields In physics one often uses normalized gauge fields Then we have W µ = 1 ig A µ F W µν = 1 ig F A µν. F W µν = µ W ν ν W µ + ig[w µ, W ν ] W µ φ = µ φ + igw µ φ W µ = UW µ U 1 i g U µ(u 1 ) L YM = 1 2 tr(f W µν F W µν ). 24 / 35
25 In this section we describe some examples of gauge theories as well as the standard model of elementary particle physics. In every example we will indicate in particular the Lie group G and the vector space V. The representations of G on V depend on certain (rational) numbers that are called charges. Example (Charges) Quantum electrodynamics has gauge group U(1). The charge Q is called electric charge. The electroweak interaction has gauge group U(1) Y SU(2) L. The charges are called weak hypercharge Y and weak isospin T 3. We have Q = T 3 + Y / 35
26 QED The standard model The simplest example is quantum electrodynamics (QED). We have G = U(1) (abelian). The 1-form W µ has values in u(1) = R, hence W µ is after a choice of basis for R a standard 1-form. The gauge field W µ is called photon. The curvature F µν is thought of as the field strength. V = C. Therefore we have F = S E = S, i.e. fermions are described by 4-component spinors Ψ. We have g = e (elementary charge). One often writes A µ instead of W µ. We have L = L fermion + L YM = Ψ(iγ µ µ m)ψ 1 4 F µν F µν where µ = µ + iqa µ (charge q) and F µν = µ A ν ν A µ. 26 / 35
27 QCD I The standard model The next example is quantum chromodynamics (QCD). It describes the strong interaction between quarks. We have G = SU(3). The gauge field W µ has dim(su(3)) = 8 components (gluons). Since the group is non-abelian, there is an interaction between the gluons. V = C 3 with the standard representation. A quark (section in F = S E) is of the form q f = q r f q g f q b f, where f = u, d, c, s, t, b is one of six flavours, r, g, b one of three colours (red, green, blue) and q i f a 4-component spinor. The group SU(3) mixes the colours. 27 / 35
28 QCD II The notions of colour and hence quantum chromodynamics come from the triality of basic colours and because one only observes white combinations in nature (color confinement). One often writes G µ instead of W µ. We have L = L fermion + L YM = f q f (iγ µ µ m f )q f 1 2 tr(f µν F µν ) where µ = µ + igg µ F µν = µ G ν ν G µ + ig[g µ, G ν ]. Emission of a gluon can change the colour of a quark, different from the case of photons and electric charge. 28 / 35
29 Chirality Every 4-component spinor Ψ (Dirac spinor) over a 4-manifold M with a Lorentz metric decomposes into the direct sum of two 2-component spinors (Weyl spinors) Ψ R and Ψ L, Ψ = ( ΨR which are eigenvectors of the chirality operator γ 5 = iγ 0 γ 1 γ 2 γ 3 (orientation) with eigenvalues ±1 (right- and left-handed spinors). In the examples so far, right- and left-handed spinors transform in the same representation of the gauge group, which is why we can combine them into a 4-component spinor. The electroweak interaction on the other hand is a chiral gauge theory right- and left-handed spinors transform in different representations of the gauge group. Ψ L ), 29 / 35
30 Electroweak interaction I G = U(1) Y SU(2) L. One writes B µ for the gauge boson that belongs to U(1) Y (with coupling constant g ) and W µ for the gauge bosons that belong to SU(2) L (coupling constant g). The representations of G distinguish between right- and left-handed spinors. For left-handed spinors we have V = C 2 with the standard representation of SU(2). The fermions are of the form ( ) ( ) νel ul el, d L. Here ν e is the electron-neutrino, e the electron, u the up-quark and d the down-quark. Similar doublets exist for the other generations. Every component is a left-handed 2-component spinor. The isospin is T 3 = ± 1 2. The hypercharge is Y = 1 (leptons) and Y = 1 3 (quarks), respectively. 30 / 35
31 Electroweak interaction II For the left-handed quarks we have d L s L = V b L with the so-called CKM-matrix V. A quark of type u L can thus turn via weak interaction into different quarks of type d L, and vice versa (flavour change, β-decay d u + e + ν e ). For right-handed spinors we have V = C with the trivial representation of SU(2). The fermions are right-handed 2-component spinors of the form d L s L b L e R, u R, d R (with T 3 = 0 and Y = 2, 4 3, 2 3 ). One does not observe right-handed neutrinos (sterile). Similar singlets exist for the other generations. 31 / 35
32 The Higgs field I There are two problems: In chiral theories only mass terms with m = 0 in the Lagrangian are gauge invariant. But all fermions except the neutrinos have a mass different from zero. In gauge theories the gauge bosons have zero mass. However, one observes that the gauge bosons W ±, Z 0 of weak interaction have a non-zero mass. Solution: The fermions and gauge bosons have by themselves mass zero and acquire a mass only through interaction with a scalar field (Higgs field). This field is the only one which has a non-zero vacuum expectation value (the vacuum with Higgs field equal to zero is not stable). Since the field is non-zero in the vacuum, the vacuum is only invariant under a subgroup U(1) em of U(1) Y SU(2) L (spontaneous symmetry breaking). In addition a new particle arises, the Higgs boson, with non-zero mass. 32 / 35
33 The Higgs field II The Lagrangian of the Higgs field φ = ( φ1 φ 2 ) is: L = 1 2 ( µ φ) ( µ φ) V (φ), with V (φ) = 1 2 µφ φ λ(φ φ) 2. The minimum of the potential (vacuum) is at v = φ = µ 2λ. The mass of the fermions and weak gauge bosons is proportional to v. The mass of the Higgs boson is µ. Figure: Potential V (φ) of the Higgs field (en.wikipedia.org) 33 / 35
34 Further topics Field quantization, perturbation theory: free fields interacting fields (path integrals, Feynman diagrams, renormalization). Grand Unified Theories (SU(5) U(1) SU(2) SU(3), proton decay p e + + 2γ). Supersymmetry, Minimal Supersymmetric Standard Model (MSSM, superpartners: squark, slepton, gluino, etc.), candidates for dark matter (WIMPs, Weakly Interacting Massive Particles) in addition to sterile neutrinos. Quantum theory of gravity, superstrings. 34 / 35
35 References Helga Baum, Eichfeldtheorie, Springer-Verlag 2014 (in German). Ulrich Mosel, Fields, Symmetries, and Quarks, Springer-Verlag Thank you! 35 / 35
Theoretical Particle Physics FYTN04: Oral Exam Questions, version ht15
Theoretical Particle Physics FYTN04: Oral Exam Questions, version ht15 Examples of The questions are roughly ordered by chapter but are often connected across the different chapters. Ordering is as in
More informationSpontaneous symmetry breaking in particle physics: a case of cross fertilization
Spontaneous symmetry breaking in particle physics: a case of cross fertilization Yoichiro Nambu lecture presented by Giovanni Jona-Lasinio Nobel Lecture December 8, 2008 1 / 25 History repeats itself 1960
More informationAspects of Electroweak Symmetry Breaking in Physics Beyond the Standard Model
Aspects of Electroweak Symmetry Breaking in Physics Beyond the Standard Model Peter Athron Department of Physics and Astronomy University of Glasgow Presented as a thesis for the degree of Ph.D. in the
More informationStandard Model of Particle Physics
Standard Model of Particle Physics Chris Sachrajda School of Physics and Astronomy University of Southampton Southampton SO17 1BJ UK SUSSP61, St Andrews August 8th 3rd 006 Contents 1. Spontaneous Symmetry
More informationSTRING THEORY: Past, Present, and Future
STRING THEORY: Past, Present, and Future John H. Schwarz Simons Center March 25, 2014 1 OUTLINE I) Early History and Basic Concepts II) String Theory for Unification III) Superstring Revolutions IV) Remaining
More informationConcepts in Theoretical Physics
Concepts in Theoretical Physics Lecture 6: Particle Physics David Tong e 2 The Structure of Things 4πc 1 137 e d ν u Four fundamental particles Repeated twice! va, 9608085, 9902033 Four fundamental forces
More informationWeak Interactions: towards the Standard Model of Physics
Weak Interactions: towards the Standard Model of Physics Weak interactions From β-decay to Neutral currents Weak interactions: are very different world CP-violation: power of logics and audacity Some experimental
More informationQuantum Mechanics and Representation Theory
Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30
More informationIntroduction to SME and Scattering Theory. Don Colladay. New College of Florida Sarasota, FL, 34243, U.S.A.
June 2012 Introduction to SME and Scattering Theory Don Colladay New College of Florida Sarasota, FL, 34243, U.S.A. This lecture was given at the IUCSS summer school during June of 2012. It contains a
More informationQuantum Field Theory I
Quantum Field Theory I Wahlpflichtvorlesung Winter Semester 2009/2010 Fachbereich 08 Physik, Mathematik und Informatik DR. VLADIMIR PASCALUTSA Physik 05-326 Tel ext.: 27162 Email: vladipas@kph.uni-mainz.de
More informationFeynman diagrams. 1 Aim of the game 2
Feynman diagrams Contents 1 Aim of the game 2 2 Rules 2 2.1 Vertices................................ 3 2.2 Anti-particles............................. 3 2.3 Distinct diagrams...........................
More informationParticle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 7 : Symmetries and the Quark Model. Introduction/Aims
Particle Physics Michaelmas Term 2011 Prof Mark Thomson Handout 7 : Symmetries and the Quark Model Prof. M.A. Thomson Michaelmas 2011 206 Introduction/Aims Symmetries play a central role in particle physics;
More informationLecture Notes on the Standard Model of Elementary Particle Physics
Lecture Notes on the Standard Model of Elementary Particle Physics Preliminary, incomplete version, 15/11/2010 Jean-Pierre Derendinger Albert Einstein Center for Fundamental Physics Institute for Theoretical
More informationThroughout the twentieth century, physicists have been trying to unify. gravity with the Standard Model (SM). A vague statement about quantum
Elko Fields Andrew Lopez Throughout the twentieth century, physicists have been trying to unify gravity with the Standard Model (SM). A vague statement about quantum gravity is that it induces non-locality.
More informationA Theory for the Cosmological Constant and its Explanation of the Gravitational Constant
A Theory for the Cosmological Constant and its Explanation of the Gravitational Constant H.M.Mok Radiation Health Unit, 3/F., Saiwanho Health Centre, Hong Kong SAR Govt, 8 Tai Hong St., Saiwanho, Hong
More informationPhysics Department, Southampton University Highfield, Southampton, S09 5NH, U.K.
\ \ IFT Instituto de Física Teórica Universidade Estadual Paulista July/92 IFT-P.025/92 LEPTON MASSES IN AN SU(Z) L U(1) N GAUGE MODEL R. Foot a, O.F. Hernandez ", F. Pisano e, and V. Pleitez 0 Physics
More informationMathematicians look at particle physics. Matilde Marcolli
Mathematicians look at particle physics Matilde Marcolli Year of Mathematics talk July 2008 We do not do these things because they are easy. We do them because they are hard. (J.F.Kennedy Sept. 12, 1962)
More informationMonodromies, Fluxes, and Compact Three-Generation F-theory GUTs
arxiv:0906.4672 CALT-68-2733 Monodromies, Fluxes, and Compact Three-Generation F-theory GUTs arxiv:0906.4672v2 [hep-th] 1 Jul 2009 Joseph Marsano, Natalia Saulina, and Sakura Schäfer-Nameki California
More informationBeyond the Standard Model. A.N. Schellekens
Beyond the Standard Model A.N. Schellekens [Word cloud by www.worldle.net] Last modified 22 February 2016 1 Contents 1 Introduction 8 1.1 A Complete Theory?.............................. 8 1.2 Gravity
More informationAbout the Author. journals as Physics Letters, Nuclear Physics and The Physical Review.
About the Author Dr. John Hagelin is Professor of Physics and Director of the Doctoral Program in Physics at Maharishi International University. Dr. Hagelin received his A.B. Summa Cum Laude from Dartmouth
More informationParticle Physics. The Standard Model. A New Periodic Table
5 Particle Physics This lecture is about particle physics, the study of the fundamental building blocks of Nature and the forces between them. We call our best theory of particle physics the Standard Model
More informationGenerally Covariant Quantum Mechanics
Chapter 15 Generally Covariant Quantum Mechanics by Myron W. Evans, Alpha Foundation s Institutute for Advance Study (AIAS). (emyrone@oal.com, www.aias.us, www.atomicprecision.com) Dedicated to the Late
More informationIntroduction to Elementary Particle Physics. Note 01 Page 1 of 8. Natural Units
Introduction to Elementary Particle Physics. Note 01 Page 1 of 8 Natural Units There are 4 primary SI units: three kinematical (meter, second, kilogram) and one electrical (Ampere 1 ) It is common in the
More information0.33 d down 1 1. 0.33 c charm + 2 3. 0 0 1.5 s strange 1 3. 0 0 0.5 t top + 2 3. 0 0 172 b bottom 1 3
Chapter 16 Constituent Quark Model Quarks are fundamental spin- 1 particles from which all hadrons are made up. Baryons consist of three quarks, whereas mesons consist of a quark and an anti-quark. There
More informationThree Pictures of Quantum Mechanics. Thomas R. Shafer April 17, 2009
Three Pictures of Quantum Mechanics Thomas R. Shafer April 17, 2009 Outline of the Talk Brief review of (or introduction to) quantum mechanics. 3 different viewpoints on calculation. Schrödinger, Heisenberg,
More informationPhysics Beyond the Standard Model
2014 BUSSTEPP LECTURES Physics Beyond the Standard Model Ben Gripaios Cavendish Laboratory, JJ Thomson Avenue, Cambridge, CB3 0HE, United Kingdom. November 11, 2014 E-mail: gripaios@hep.phy.cam.ac.uk Contents
More informationChapter 22 The Hamiltonian and Lagrangian densities. from my book: Understanding Relativistic Quantum Field Theory. Hans de Vries
Chapter 22 The Hamiltonian and Lagrangian densities from my book: Understanding Relativistic Quantum Field Theory Hans de Vries January 2, 2009 2 Chapter Contents 22 The Hamiltonian and Lagrangian densities
More information3. Open Strings and D-Branes
3. Open Strings and D-Branes In this section we discuss the dynamics of open strings. Clearly their distinguishing feature is the existence of two end points. Our goal is to understand the effect of these
More informationConnections, Gauges and Field Theories
Connections, Gauges and Field Theories Lionel Brits August 3, 2005 Abstract The theory of gauges and connections in the principal bundle formalism is reviewed. The geometrical aspects of gauge potential,
More informationSpecial Theory of Relativity
June 1, 2010 1 1 J.D.Jackson, Classical Electrodynamics, 3rd Edition, Chapter 11 Introduction Einstein s theory of special relativity is based on the assumption (which might be a deep-rooted superstition
More informationarxiv:hep-ph/9902288v1 9 Feb 1999
A Quantum Field Theory Warm Inflation Model VAND-TH-98-01 Arjun Berera Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA arxiv:hep-ph/9902288v1 9 Feb 1999 Abstract A
More informationNon-Supersymmetric Seiberg Duality in orientifold QCD and Non-Critical Strings
Non-Supersymmetric Seiberg Duality in orientifold QCD and Non-Critical Strings, IAP Large N@Swansea, July 2009 A. Armoni, D.I., G. Moraitis and V. Niarchos, arxiv:0801.0762 Introduction IR dynamics of
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationThe Standard Model Higgs Boson
The Standard Model Higgs Boson Part of the Lecture Particle Physics II, UvA Particle Physics Master 011-01 Date: November 011 Lecturer: Ivo van Vulpen Assistant: Antonio Castelli Disclaimer These are private
More informationA tentative theory of large distance physics
hep-th/0204131 RUNHETC-2002-12 A tentative theory of large distance physics Daniel Friedan Department of Physics and Astronomy Rutgers, The State University of New Jersey Piscataway, New Jersey, USA and
More informationThe Standard Model of Particle Physics - II
The Standard Model of Particle Physics II Lecture 4 Gauge Theory and Symmetries Quantum Chromodynamics Neutrinos Eram Rizvi Royal Institution London 6 th March 2012 Outline A Century of Particle Scattering
More informationUnification - The Standard Model
Unification - The Standard Model Information on Physics level 4 Undergraduate Course PT.4.6 K.S. Stelle, Office Huxley 519 November 9, 2015 Rapid Feedback to be handed in to the UG office Level 3 (day
More informationOne of the primary goals of physics is to understand the wonderful variety of nature in a
A Unified Physics by by Steven Weinberg 2050? Experiments at CERN and elsewhere should let us complete the Standard Model of particle physics, but a unified theory of all forces will probably require radically
More informationPeriodic Table of Particles/Forces in the Standard Model. Three Generations of Fermions: Pattern of Masses
Introduction to Elementary Particle Physics. Note 01 Page 1 of 8 Periodic Table of Particles/Forces in the Standard Model Three Generations of Fermions: Pattern of Masses 1.0E+06 1.0E+05 1.0E+04 1.0E+03
More informationarxiv:0909.4541v1 [hep-ph] 25 Sep 2009 James D. Wells
CERN-PH-TH-009-154 MCTP-09-48 Lectures on Higgs Boson Physics in the Standard Model and Beyond arxiv:0909.4541v1 [hep-ph] 5 Sep 009 James D. Wells CERN, Theoretical Physics, CH-111 Geneva 3, Switzerland,
More informationThe Standard Model of Particle Physics. Tom W.B. Kibble Blackett Laboratory, Imperial College London
The Standard Model of Particle Physics Tom W.B. Kibble Blackett Laboratory, Imperial College London Abstract This is a historical account from my personal perspective of the development over the last few
More informationMASTER OF SCIENCE IN PHYSICS MASTER OF SCIENCES IN PHYSICS (MS PHYS) (LIST OF COURSES BY SEMESTER, THESIS OPTION)
MASTER OF SCIENCE IN PHYSICS Admission Requirements 1. Possession of a BS degree from a reputable institution or, for non-physics majors, a GPA of 2.5 or better in at least 15 units in the following advanced
More informationContents. Goldstone Bosons in 3He-A Soft Modes Dynamics and Lie Algebra of Group G:
... Vlll Contents 3. Textures and Supercurrents in Superfluid Phases of 3He 3.1. Textures, Gradient Energy and Rigidity 3.2. Why Superfuids are Superfluid 3.3. Superfluidity and Response to a Transverse
More informationPrecession of spin and Precession of a top
6. Classical Precession of the Angular Momentum Vector A classical bar magnet (Figure 11) may lie motionless at a certain orientation in a magnetic field. However, if the bar magnet possesses angular momentum,
More informationFlavour Physics. Tim Gershon University of Warwick. 31 March 2014
Flavour Physics Tim Gershon University of Warwick 31 March 2014 Outline Lecture 1 what is flavour physics? some history, some concepts, some theory charged lepton physics What is flavour physics? Parameters
More informationLecture 18 - Clifford Algebras and Spin groups
Lecture 18 - Clifford Algebras and Spin groups April 5, 2013 Reference: Lawson and Michelsohn, Spin Geometry. 1 Universal Property If V is a vector space over R or C, let q be any quadratic form, meaning
More informationInvariant Metrics with Nonnegative Curvature on Compact Lie Groups
Canad. Math. Bull. Vol. 50 (1), 2007 pp. 24 34 Invariant Metrics with Nonnegative Curvature on Compact Lie Groups Nathan Brown, Rachel Finck, Matthew Spencer, Kristopher Tapp and Zhongtao Wu Abstract.
More informationScientific Background on the Nobel Prize in Physics 2013
8 OCTOBER 2013 Scientific Background on the Nobel Prize in Physics 2013 The BEH-Mechanism, Interactions with Short Range Forces and Scalar Particles Compiled by the Class for Physics of the Royal Swedish
More informationElectromagnetic scattering of vector mesons in the Sakai-Sugimoto model.
Electromagnetic scattering of vector mesons in the Sakai-Sugimoto model Carlos Alfonso Ballon Bayona, Durham University In collaboration with H. Boschi-Filho, N. R. F. Braga, M. Ihl and M. Torres. arxiv:0911.0023,
More informationAllowed and observable phases in two-higgs-doublet Standard Models
Allowed and observable phases in two-higgs-doublet Standard Models G Sartori and G Valente Dipartimento di Fisica,Università di Padova and INFN, Sezione di Padova via Marzolo 8, I 35131 Padova, Italy (e-mail:
More informationTopologically Massive Gravity with a Cosmological Constant
Topologically Massive Gravity with a Cosmological Constant Derek K. Wise Joint work with S. Carlip, S. Deser, A. Waldron Details and references at arxiv:0803.3998 [hep-th] (or for the short story, 0807.0486,
More informationGravity and running coupling constants
Gravity and running coupling constants 1) Motivation and history 2) Brief review of running couplings 3) Gravity as an effective field theory 4) Running couplings in effective field theory 5) Summary 6)
More informationA CONFRONTATION WITH INFINITY
A CONFRONTATION WITH INFINITY Nobel lecture 1999. Gerard t Hooft Institute for Theoretical Physics University of Utrecht, Princetonplein 5 3584 CC Utrecht, the Netherlands e-mail: g.thooft@fys.ruu.nl 1.
More informationSearching for Solar Axions in the ev-mass Region with the CCD Detector at CAST
Searching for Solar Axions in the ev-mass Region with the CCD Detector at CAST Julia Katharina Vogel FAKULTÄT FÜR MATHEMATIK UND PHYSIK ALBERT-LUDWIGS-UNIVERSITÄT FREIBURG Searching for Solar Axions in
More informationINVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS
INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU Abstract. We classify the left-invariant metrics with nonnegative
More informationREALIZING EINSTEIN S DREAM Exploring Our Mysterious Universe
REALIZING EINSTEIN S DREAM Exploring Our Mysterious Universe The End of Physics Albert A. Michelson, at the dedication of Ryerson Physics Lab, U. of Chicago, 1894 The Miracle Year - 1905 Relativity Quantum
More informationVrije Universiteit Brussel. Faculteit Wetenschappen Departement Natuurkunde
Vrije Universiteit Brussel Faculteit Wetenschappen Departement Natuurkunde Measurement of the top quark pair production cross section at the LHC with the CMS experiment Michael Maes Promotor Prof. Dr.
More informationExtensions of the Standard Model (part 2)
Extensions of the Standard Model (part 2) Prof. Jorgen D Hondt Vrije Universiteit Brussel Inter-university Institute for High Energies Content: The Higgs sector of the Standard Model and extensions Theoretical
More informationarxiv:0805.3762v1 [hep-ph] 24 May 2008
6/008 arxiv:0805.376v1 [hep-ph] 4 May 008 Further Development of the Tetron Model Bodo Lampe e-mail: Lampe.Bodo@web.de Abstract After a prologue which clarifies some issues left open in my last paper,
More informationPhase Transitions in the Early Universe
Trick Phase Transitions in the Early Universe Electroweak and QCD Phase Transitions Master Program of Theoretical Physics Student Seminar in Cosmology Author: Doru STICLET Supervisors: Prof. Dr. Tomislav
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationFeynman Diagrams for Beginners
Feynman Diagrams for Beginners Krešimir Kumerički arxiv:1602.04182v1 [physics.ed-ph] 8 Feb 2016 Department of Physics, Faculty of Science, University of Zagreb, Croatia Abstract We give a short introduction
More informationA tentative theory of large distance physics
Published by Institute of Physics Publishing for SISSA/ISAS Received: May 3, 2002 Revised: October 27, 2003 Accepted: October 27, 2003 A tentative theory of large distance physics Daniel Friedan Department
More informationUniversity of Cambridge Part III Mathematical Tripos
Preprint typeset in JHEP style - HYPER VERSION Michaelmas Term, 2006 and 2007 Quantum Field Theory University of Cambridge Part III Mathematical Tripos Dr David Tong Department of Applied Mathematics and
More informationWhy the high lying glueball does not mix with the neighbouring f 0. Abstract
Why the high lying glueball does not mix with the neighbouring f 0. L. Ya. Glozman Institute for Theoretical Physics, University of Graz, Universitätsplatz 5, A-800 Graz, Austria Abstract Chiral symmetry
More informationElasticity Theory Basics
G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold
More information1 Symmetries of regular polyhedra
1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an
More informationLinear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
More informationLocalization of scalar fields on Branes with an Asymmetric geometries in the bulk
Localization of scalar fields on Branes with an Asymmetric geometries in the bulk Vladimir A. Andrianov in collaboration with Alexandr A. Andrianov V.A.Fock Department of Theoretical Physics Sankt-Petersburg
More information5 Homogeneous systems
5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m
More informationarxiv:hep-th/0404072v2 15 Apr 2004
hep-th/0404072 Tree Amplitudes in Gauge Theory as Scalar MHV Diagrams George Georgiou and Valentin V. Khoze arxiv:hep-th/0404072v2 15 Apr 2004 Centre for Particle Theory, Department of Physics and IPPP,
More informationLecture L3 - Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationMeasurement of the t -Channel Single Top-Quark Production Cross-Section with the ATLAS Detector at s = 7 TeV
FACHBEREICH MAHEMAIK UND NAURWISSENSCHAFEN FACHGRUPPE PHYSIK BERGISCHE UNIVERSIÄ WUPPERAL Measurement of the t -Channel Single op-quark Production Cross-Section with the ALAS Detector at s = 7 ev Dissertation
More informationarxiv:hep-ph/9812492v1 24 Dec 1998
MPI-PhT/96-14(extended version) July 1996 A Note on QCD Corrections to A b FB using Thrust to arxiv:hep-ph/9812492v1 24 Dec 1998 determine the b-quark Direction Bodo Lampe Max Planck Institut für Physik
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationFINDING SUPERSYMMETRY AT THE LHC
FINDING SUPERSYMMETRY AT THE LHC Tilman Plehn MPI München & University of Edinburgh TeV scale supersymmetry Signals at Tevatron and LHC Measurements at LHC SUSY parameters at LHC (and ILC) Tilman Plehn:
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationKINEMATIC ENDPOINT VARIABLES AND PHYSICS BEYOND THE STANDARD MODEL
KINEMATIC ENDPOINT VARIABLES AND PHYSICS BEYOND THE STANDARD MODEL A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the
More informationDirected by: Prof. Yuanning Gao, IHEP, Tsinghua University Prof. Aurelio Bay, LPHE, EPFL
Masters Thesis in High Energy Physics Directed by: Prof. Yuanning Gao, IHEP, Tsinghua University Prof. Aurelio Bay, LPHE, EPFL 1 Study for CP-violation in the ψ π + π J/ψ transition Vincent Fave July 18,
More informationElements of Abstract Group Theory
Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for
More informationFinite dimensional C -algebras
Finite dimensional C -algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for self-adjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n
More information1 Introduction. 1 There may, of course, in principle, exist other universes, but they are not accessible to our
1 1 Introduction Cosmology is the study of the universe as a whole, its structure, its origin, and its evolution. Cosmology is soundly based on observations, mostly astronomical, and laws of physics. These
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationold supersymmetry as new mathematics
old supersymmetry as new mathematics PILJIN YI Korea Institute for Advanced Study with help from Sungjay Lee Atiyah-Singer Index Theorem ~ 1963 Calabi-Yau ~ 1978 Calibrated Geometry ~ 1982 (Harvey & Lawson)
More informationUN PICCOLO BIG BANG IN LABORATORIO: L'ESPERIMENTO ALICE AD LHC
UN PICCOLO BIG BANG IN LABORATORIO: L'ESPERIMENTO ALICE AD LHC Parte 1: Carlos A. Salgado Universidade de Santiago de Compostela csalgado@usc.es http://cern.ch/csalgado LHC physics program Fundamental
More informationSearch for Third Generation Squarks in the Missing Transverse Energy plus Jet Sample at CDF Run II
Search for Third Generation Squarks in the Missing Transverse Energy plus Jet Sample at CDF Run II Búsquedas de squarks de la tercera familia en sucesos con jets y momento transverso neto en el experimento
More informationAxion/Saxion Cosmology Revisited
Axion/Saxion Cosmology Revisited Masahiro Yamaguchi (Tohoku University) Based on Nakamura, Okumura, MY, PRD77 ( 08) and Work in Progress 1. Introduction Fine Tuning Problems of Particle Physics Smallness
More informationarxiv:hep-th/9503040v1 7 Mar 1995
NBI-HE-95-06 hep-th/9503040 March, 1995 arxiv:hep-th/9503040v1 7 Mar 1995 Bosonization of World-Sheet Fermions in Minkowski Space-Time. Andrea Pasquinucci and Kaj Roland The Niels Bohr Institute, University
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More informationPreon Models in Particle Physics
Preon Models in Particle Physics Grunde Haraldsson Wesenberg Physics Submission date: June 2014 Supervisor: Jan Myrheim, IFY Norwegian University of Science and Technology Department of Physics NTNU Master
More informationRR charges of D2-branes in the WZW model
RR charges of 2-branes in the WZW model arxiv:hep-th/0007096v1 12 Jul 2000 Anton Alekseev Institute for Theoretical Physics, Uppsala University, Box 803, S 75108 Uppsala, Sweden Volker Schomerus MPI für
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationNot Even Wrong, ten years later: a view from mathematics February on prospects 3, 2016for fundamenta 1 / 32. physics without experiment
Not Even Wrong, ten years later: a view from mathematics on prospects for fundamental physics without experiment Peter Woit Columbia University Rutgers Physics Colloquium, February 3, 2016 Not Even Wrong,
More informationExploring physics beyond the Standard Electroweak Model in the light of supersymmetry
Exploring physics beyond the Standard Electroweak Model in the light of supersymmetry Thesis Submitted to The University of Calcutta for The Degree of Doctor of Philosophy (Science) By Pradipta Ghosh Department
More informationA SUSY SO(10) GUT with 2 Intermediate Scales
A SUSY SO(10) GUT with 2 Intermediate Scales Manuel Drees Bonn University & Bethe Center for Theoretical Physics SUSY SO(10) p. 1/25 Contents 1 Motivation: SO(10), intermediate scales SUSY SO(10) p. 2/25
More informationGrid Computing for LHC and Methods for W Boson Mass Measurement at CMS
. Forschungszentrum Karlsruhe in der Helmholtz-Gemeinschaft Wissenschaftliche Berichte FZKA 7402 Grid Computing for LHC and Methods for W Boson Mass Measurement at CMS C. Jung Steinbuch Centre for Computing
More informationThe Supersymmetric Standard Model. FB Physik, D-06099 Halle, Germany. Rutgers University. Piscataway, NJ 08855-0849, USA
hep-th/yymmnnn The Supersymmetric Standard Model Jan Louis a, Ilka Brunner b and Stephan J. Huber c a Martin{Luther{Universitat Halle{Wittenberg, FB Physik, D-06099 Halle, Germany email:j.louis@physik.uni-halle.de
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationJournal of Theoretics Journal Home Page
Journal of Theoretics Journal Home Page MASS BOOM VERSUS BIG BANG: THE ROLE OF PLANCK S CONSTANT by Antonio Alfonso-Faus E.U.I.T. Aeronáutica Plaza Cardenal Cisneros s/n 8040 Madrid, SPAIN e-mail: aalfonso@euita.upm.es
More information