Physics 234C Lecture Notes
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An Introduction to Quantum Field Theory
AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ........................... -
3. Models of EWSB
The Higgs Boson and Electroweak Symmetry Breaking 3. Models of EWSB M. E. Peskin Chiemsee School September 2014 In this last lecture, I will take up a different topic in the physics of the Higgs field. In the first lecture, I emphasized that most of the parameters of the Standard Model are associated with the couplings of the Higgs field. These parameters determine the Higgs potential, the spectrum of quark and lepton masses, and the structure of flavor and CP violation in the weak interactions. These parameters are not computable within the SM. They are inputs. If we want to compute these parameters, we need to build a deeper and more predictive theory. In particular, a basic question about the SM is: Why is the SU(2)xU(1) gauge symmetry spontaneously broken ? The SM cannot answer this question. I will discuss: In what kind of models can we answer this question ? For orientation, I will present the explanation for spontaneous symmetry breaking in the SM. We have a single Higgs doublet field ' . It has some potential V ( ' ) . The potential is unknown, except that it is invariant under SU(2)xU(1). However, if the theory is renormalizable, the potential must be of the form V (')=µ2 ' 2 + λ ' 4 | | | | Now everything depends on the sign of µ 2 . If µ2 > 0 the minimum of the potential is at ' =0 and there is no symmetry breaking. If µ 2 < 0 , the potential has the form: and there is a minimum away from 0. That’s it. Don’t ask any more questions. -
Goldstone Theorem, Higgs Mechanism
Introduction to the Standard Model Lecture 12 Classical Goldstone Theorem and Higgs Mechanism The Classical Goldstone Theorem: To each broken generator corresponds a massless field (Goldstone boson). Proof: 2 ∂ V Mij = ∂φi∂φj φ~= φ~ h i V (φ~)=V (φ~ + iεaT aφ~) ∂V =V (φ~)+ iεaT aφ~ + ( ε 2) ∂φj O | | ∂ ∂V a Tjlφl =0 ⇒∂φk ∂φj 2 ∂ ∂V a ∂ V a ∂V a Til φl = Tjlφl + Tik ∂φk ∂φi ∂φi∂φj ∂φi φ~= φ~ h i 0= M T a φ +0 ki il h li =~0i | 6 {z } If T a is a broken generator one has T a φ~ = ~0 h i 6 ⇒ Mik has a null eigenvector null eigenvalues massless particle since the eigenvalues of the mass matrix are the particle⇒ masses. ⇒ We now combine the concept of a spontaneously broken symmetry to a gauge theory. The Higgs Mechanism for U(1) gauge theory Consider µ 2 2 1 µν = D φ∗ D φ µ φ∗φ λ φ∗φ F F L µ − − − 4 µν µν µ ν ν µ with Dµ = ∂µ + iQAµ and F = ∂ A ∂ A . Gauge symmetry here means invariance under Aµ Aµ ∂µΛ. − → − 2 2 2 case a) unbroken case, µ > 0 : V (φ)= µ φ∗φ + λ φ∗φ with a minimum at φ = 0. The ground state or vaccuum is U(1) symmetric. The corresponding theory is known as Scalar Electrodynamics of a massive spin-0 boson with mass µ and charge Q. 1 case b) nontrivial vaccuum case 2 µ 2 v iα V (φ) has a minimum for 2 φ∗φ = − = v which gives φ = e . -
Doi:10.5281/Zenodo.2566644
Higgs, dark sector and the vacuum: From Nambu-Goldstone bosons to massive particles via the hydrodynamics of a doped vacuum. Marco Fedi * v2, February 16, 2019 Abstract is the energy density of the vacuum, whose units corre- spond to pressure (J=m3 = Pa), hence justifying the re- Here the physical vacuum is treated as a superfluid, fun- pulsive action of dark energy. One can describe the damental quantum scalar field, coinciding with dark en- virtual pairs forming and annihilating in quantum vac- ergy and doped with particle dark matter, able to pro- uum – considered as a fundamental, scalar, quantum duce massive particles and interactions via a hydrody- field – as vortex-antivortex pairs of vacuum’s quanta, namic reinterpretation of the Higgs mechanism. Here via a mechanism analogous to the Higgs mechanism, the Nambu-Goldstone bosons are circularly polarized where phonons in the superfluid vacuum are the Nambu- phonons around the edge of the Brillouin zone of vac- Goldstone bosons, which here trigger quantized vortices uum’s quasi-lattice and they give mass to particles by trig- and the mass-acquisition process, due to the interaction gering quantized vortices, whose dynamics reproduces with diffused particle dark matter [2], which acts as a any possible spin. Doped vortices also exert hydrody- dopant of the superfluid vacuum and that could be the rea- namic forces which may correspond to fundamental in- son for vacuum dilatancy, described and proven in [22]. teractions. Hence, is the Higgs field really something different or along with the dark sector and quantum vacuum we are Keywords— quantum vacuum; dilatant vacuum; dark en- using different names to refer to the same thing? Dilatant ergy; dark matter; Higgs mechanism; spin; fundamental vacuum [22] could refer to the possible apparent viscosity interactions. -
Finite Quantum Field Theory and Renormalization Group
Finite Quantum Field Theory and Renormalization Group M. A. Greena and J. W. Moffata;b aPerimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada bDepartment of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada September 15, 2021 Abstract Renormalization group methods are applied to a scalar field within a finite, nonlocal quantum field theory formulated perturbatively in Euclidean momentum space. It is demonstrated that the triviality problem in scalar field theory, the Higgs boson mass hierarchy problem and the stability of the vacuum do not arise as issues in the theory. The scalar Higgs field has no Landau pole. 1 Introduction An alternative version of the Standard Model (SM), constructed using an ultraviolet finite quantum field theory with nonlocal field operators, was investigated in previous work [1]. In place of Dirac delta-functions, δ(x), the theory uses distributions (x) based on finite width Gaussians. The Poincar´eand gauge invariant model adapts perturbative quantumE field theory (QFT), with a finite renormalization, to yield finite quantum loops. For the weak interactions, SU(2) U(1) is treated as an ab initio broken symmetry group with non- zero masses for the W and Z intermediate× vector bosons and for left and right quarks and leptons. The model guarantees the stability of the vacuum. Two energy scales, ΛM and ΛH , were introduced; the rate of asymptotic vanishing of all coupling strengths at vertices not involving the Higgs boson is controlled by ΛM , while ΛH controls the vanishing of couplings to the Higgs. Experimental tests of the model, using future linear or circular colliders, were proposed. -
Exotic Goldstone Particles: Pseudo-Goldstone Boson and Goldstone Fermion
Exotic Goldstone Particles: Pseudo-Goldstone Boson and Goldstone Fermion Guang Bian December 11, 2007 Abstract This essay describes two exotic Goldstone particles. One is the pseudo- Goldstone boson which is related to spontaneous breaking of an approximate symmetry. The other is the Goldstone fermion which is a natural result of spontaneously broken global supersymmetry. Their realization and implication in high energy physics are examined. 1 1 Introduction In modern physics, the idea of spontaneous symmetry breaking plays a crucial role in understanding various phenomena such as ferromagnetism, superconductivity, low- energy interactions of pions, and electroweak unification of the Standard Model. Nowadays, broken symmetry and order parameters emerged as unifying theoretical concepts are so universal that they have become the framework for constructing new theoretical models in nearly all branches of physics. For example, in particle physics there exist a number of new physics models based on supersymmetry. In order to explain the absence of superparticle in current high energy physics experiment, most of these models assume the supersymmetry is broken spontaneously by some underlying subtle mechanism. Application of spontaneous broken symmetry is also a common case in condensed matter physics [1]. Some recent research on high Tc superconductor [2] proposed an approximate SO(5) symmetry at least over part of the theory’s parameter space and the detection of goldstone bosons resulting from spontaneous symmetry breaking would be a ’smoking gun’ for the existence of this SO(5) symmetry. From the Goldstone’s Theorem [3], we know that there are two explicit common features among Goldstone’s particles: (1) they are massless; (2) they obey Bose-Einstein statistics i.e. -
U(1) Symmetry of the Complex Scalar and Scalar Electrodynamics
Fall 2019: Classical Field Theory (PH6297) U(1) Symmetry of the complex scalar and scalar electrodynamics August 27, 2019 1 Global U(1) symmetry of the complex field theory & associated Noether charge Consider the complex scalar field theory, defined by the action, h i h i I Φ(x); Φy(x) = d4x (@ Φ)y @µΦ − V ΦyΦ : (1) ˆ µ As we have noted earlier complex scalar field theory action Eq. (1) is invariant under multiplication by a constant complex phase factor ei α, Φ ! Φ0 = e−i αΦ; Φy ! Φ0y = ei αΦy: (2) The phase,α is necessarily a real number. Since a complex phase is unitary 1 × 1 matrix i.e. the complex conjugation is also the inverse, y −1 e−i α = e−i α ; such phases are also called U(1) factors (U stands for Unitary matrix and since a number is a 1×1 matrix, U(1) is unitary matrix of size 1 × 1). Since this symmetry transformation does not touch spacetime but only changes the fields, such a symmetry is called an internal symmetry. Also note that since α is a constant i.e. not a function of spacetime, it is a global symmetry (global = same everywhere = independent of spacetime location). Check: Under the U(1) symmetry Eq. (2), the combination ΦyΦ is obviously invariant, 0 Φ0yΦ = ei αΦy e−i αΦ = ΦyΦ: This implies any function of the product ΦyΦ is also invariant. 0 V Φ0yΦ = V ΦyΦ : Note that this is true whether α is a constant or a function of spacetime i.e. -
Vacuum Energy
Vacuum Energy Mark D. Roberts, 117 Queen’s Road, Wimbledon, London SW19 8NS, Email:[email protected] http://cosmology.mth.uct.ac.za/ roberts ∼ February 1, 2008 Eprint: hep-th/0012062 Comments: A comprehensive review of Vacuum Energy, which is an extended version of a poster presented at L¨uderitz (2000). This is not a review of the cosmolog- ical constant per se, but rather vacuum energy in general, my approach to the cosmological constant is not standard. Lots of very small changes and several additions for the second and third versions: constructive feedback still welcome, but the next version will be sometime in coming due to my sporadiac internet access. First Version 153 pages, 368 references. Second Version 161 pages, 399 references. arXiv:hep-th/0012062v3 22 Jul 2001 Third Version 167 pages, 412 references. The 1999 PACS Physics and Astronomy Classification Scheme: http://publish.aps.org/eprint/gateway/pacslist 11.10.+x, 04.62.+v, 98.80.-k, 03.70.+k; The 2000 Mathematical Classification Scheme: http://www.ams.org/msc 81T20, 83E99, 81Q99, 83F05. 3 KEYPHRASES: Vacuum Energy, Inertial Mass, Principle of Equivalence. 1 Abstract There appears to be three, perhaps related, ways of approaching the nature of vacuum energy. The first is to say that it is just the lowest energy state of a given, usually quantum, system. The second is to equate vacuum energy with the Casimir energy. The third is to note that an energy difference from a complete vacuum might have some long range effect, typically this energy difference is interpreted as the cosmological constant. -
Goldstone Bosons in a Crystalline Chiral Phase
Goldstone Bosons in a Crystalline Chiral Phase Goldstone Bosonen in einer Kristallinen Chiralen Phase Zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von M.Sc. Marco Schramm, Tag der Einreichung: 29.06.2017, Tag der Prüfung: 24.07.2017 Darmstadt 2017 — D 17 1. Gutachten: PD Dr. Michael Buballa 2. Gutachten: Prof. Dr. Jens Braun Fachbereich Physik Institut für Kernphysik NHQ Goldstone Bosons in a Crystalline Chiral Phase Goldstone Bosonen in einer Kristallinen Chiralen Phase Genehmigte Dissertation von M.Sc. Marco Schramm, 1. Gutachten: PD Dr. Michael Buballa 2. Gutachten: Prof. Dr. Jens Braun Tag der Einreichung: 29.06.2017 Tag der Prüfung: 24.07.2017 Darmstadt 2017 — D 17 Bitte zitieren Sie dieses Dokument als: URN: urn:nbn:de:tuda-tuprints-66977 URL: http://tuprints.ulb.tu-darmstadt.de/6697 Dieses Dokument wird bereitgestellt von tuprints, E-Publishing-Service der TU Darmstadt http://tuprints.ulb.tu-darmstadt.de [email protected] Die Veröffentlichung steht unter folgender Creative Commons Lizenz: Namensnennung – Keine kommerzielle Nutzung – Keine Bearbeitung 4.0 International https://creativecommons.org/licenses/by-nc-nd/4.0/ Abstract The phase diagram of strong interaction matter is expected to exhibit a rich structure. Different models have shown, that crystalline phases with a spatially varying chiral condensate can occur in the regime of low temperatures and moderate densities, where they replace the first-order phase transition found for spatially constant order parameters. We investigate this inhomogeneous phase, where in addition to the chiral symmetry, transla- tional and rotational symmetry are broken as well, in a two flavor Nambu–Jona-Lasinio model. -
Ft». 421* Clffis-Lh'- Fjf.F.Ol
ClffiS-lH'- fjf.f.Ol THE ABELIAN HIGGS KIEBLE MODEL. UNITARITY OF THE S- OPERATOR. C. BECCHI* by A. ROUET '•"• R. STORA Centre de Physique Théorique CNRS Marseille. : Results concerning the renormali nation of the abelian Higgs Kibble model in the 't Hooft gauges are presented. A direct combinatorial proof of the unitarity of the physical S -operator is described. ft». 421* x On leave of absence from the University of Genova xx. Boursier de thèse CEA Postal Address i Centre de Physique Théorique 31. chemin J. Aiguier 13274 MARSEILLE CEDEX 2 The Higgs-Kibble model [IJ provides the best known exception to the Goldstone theorem I 2 j concerning the phenomenon of spontaneous symmetry breaking in local field theory. If zero mass vector particles are present in the unbroken theory the degrees of freedom corresponding to the Goldstone boson may be lost, the vector particles becoming massive. In order to discuss this phenomenon in the framework of renormalized quantum field theory [3,4,5j it is necessary to introduce a certain number of non physical degrees of freedom, some of which are associated with negative norm one-particle states. The number of non physical fields and their properties depend on the choice of the gauge [3,4,5,6] , It is however convenient to exclude gauges such as the Stueckelberg gauge which lead [sj to massless particles, in order to avoid unnecessary infrared problems. A choice of gauge satisfying this requirement has been proposed by G. t'Hoofc 14 . In this gauge, according to the Faddeev-Popov fTJ prescription, a system of anticommuting scalar ghost fields must be introduced, which are coupled to the remaining fields. -
8. Quantum Field Theory on the Plane
8. Quantum Field Theory on the Plane In this section, we step up a dimension. We will discuss quantum field theories in d =2+1dimensions.Liketheird =1+1dimensionalcounterparts,thesetheories have application in various condensed matter systems. However, they also give us further insight into the kinds of phases that can arise in quantum field theory. 8.1 Electromagnetism in Three Dimensions We start with Maxwell theory in d =2=1.ThegaugefieldisAµ,withµ =0, 1, 2. The corresponding field strength describes a single magnetic field B = F12,andtwo electric fields Ei = F0i.Weworkwiththeusualaction, 1 S = d3x F F µ⌫ + A jµ (8.1) Maxwell − 4e2 µ⌫ µ Z The gauge coupling has dimension [e2]=1.Thisisimportant.ItmeansthatU(1) gauge theories in d =2+1dimensionscoupledtomatterarestronglycoupledinthe infra-red. In this regard, these theories di↵er from electromagnetism in d =3+1. We can start by thinking classically. The Maxwell equations are 1 @ F µ⌫ = j⌫ e2 µ Suppose that we put a test charge Q at the origin. The Maxwell equations reduce to 2A = Qδ2(x) r 0 which has the solution Q r A = log +constant 0 2⇡ r ✓ 0 ◆ for some arbitrary r0. We learn that the potential energy V (r)betweentwocharges, Q and Q,separatedbyadistancer, increases logarithmically − Q2 r V (r)= log +constant (8.2) 2⇡ r ✓ 0 ◆ This is a form of confinement, but it’s an extremely mild form of confinement as the log function grows very slowly. For obvious reasons, it’s usually referred to as log confinement. –384– In the absence of matter, we can look for propagating degrees of freedom of the gauge field itself. -
Unitary Gauge, Stueckelberg Formalism and Gauge Invariant
Unitary Gauge, St¨uckelberg Formalism and Gauge Invariant Models for Effective Lagrangians Carsten Grosse-Knetter∗† and Reinhart K¨ogerler Universit¨at Bielefeld Fakult¨at f¨ur Physik D-4800 Bielefeld 1 Germany BI-TP 92/56 December 1992 Abstract arXiv:hep-ph/9212268v1 16 Dec 1992 Within the framework of the path-integral formalism we reinvestigate the different methods of removing the unphysical degrees of freedom from spontanously broken gauge theories. These are: construction of the unitary gauge by gauge fixing; Rξ-limiting procedure; decoupling of the unphysi- cal fields by point transformations. In the unitary gauge there exists an extra quartic divergent Higgs self-interaction term, which cannot be ne- glected if perturbative calculations are performed in this gauge. Using the St¨uckelberg formalism this procedure can be reversed, i. e., a gauge theory can be reconstructed from its unitary gauge. We also discuss the equiv- alence of effective-Lagrangian theories, containing arbitrary interactions, to (nonlinearly realized) spontanously broken gauge theories and we show how they can be extended to Higgs models. ∗Supported in part by Deutsche Forschungsgemeinschaft, Project No.: Ko 1062/1-2 †E-Mail: [email protected] 0 1 Introduction The purpose of the present paper is primarily to reinvestigate the various ap- proaches to the unitary gauge within quantized spontanously broken gauge the- ories (SBGTs), thereby putting the emphasis on the connections between the different methods and their common basis. Although most of the described tech- niques are known (at least to several groups of experts) we find it worthwile to clarify these different approaches and, especially, to analyze the powerful method of St¨uckelberg transformations.