Physics 215A: Particles and Fields Fall 2016
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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 ........................... -
Pauli Crystals–Interplay of Symmetries
S S symmetry Article Pauli Crystals–Interplay of Symmetries Mariusz Gajda , Jan Mostowski , Maciej Pylak , Tomasz Sowi ´nski ∗ and Magdalena Załuska-Kotur Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland; [email protected] (M.G.); [email protected] (J.M.); [email protected] (M.P.); [email protected] (M.Z.-K.) * Correspondence: [email protected] Received: 20 October 2020; Accepted: 13 November 2020; Published: 16 November 2020 Abstract: Recently observed Pauli crystals are structures formed by trapped ultracold atoms with the Fermi statistics. Interactions between these atoms are switched off, so their relative positions are determined by joined action of the trapping potential and the Pauli exclusion principle. Numerical modeling is used in this paper to find the Pauli crystals in a two-dimensional isotropic harmonic trap, three-dimensional harmonic trap, and a two-dimensional square well trap. The Pauli crystals do not have the symmetry of the trap—the symmetry is broken by the measurement of positions and, in many cases, by the quantum state of atoms in the trap. Furthermore, the Pauli crystals are compared with the Coulomb crystals formed by electrically charged trapped particles. The structure of the Pauli crystals differs from that of the Coulomb crystals, this provides evidence that the exclusion principle cannot be replaced by a two-body repulsive interaction but rather has to be considered to be a specifically quantum mechanism leading to many-particle correlations. Keywords: pauli exclusion; ultracold fermions; quantum correlations 1. Introduction Recent advances of experimental capabilities reached such precision that simultaneous detection of many ultracold atoms in a trap is possible [1,2]. -
The Mechanics of the Fermionic and Bosonic Fields: an Introduction to the Standard Model and Particle Physics
The Mechanics of the Fermionic and Bosonic Fields: An Introduction to the Standard Model and Particle Physics Evan McCarthy Phys. 460: Seminar in Physics, Spring 2014 Aug. 27,! 2014 1.Introduction 2.The Standard Model of Particle Physics 2.1.The Standard Model Lagrangian 2.2.Gauge Invariance 3.Mechanics of the Fermionic Field 3.1.Fermi-Dirac Statistics 3.2.Fermion Spinor Field 4.Mechanics of the Bosonic Field 4.1.Spin-Statistics Theorem 4.2.Bose Einstein Statistics !5.Conclusion ! 1. Introduction While Quantum Field Theory (QFT) is a remarkably successful tool of quantum particle physics, it is not used as a strictly predictive model. Rather, it is used as a framework within which predictive models - such as the Standard Model of particle physics (SM) - may operate. The overarching success of QFT lends it the ability to mathematically unify three of the four forces of nature, namely, the strong and weak nuclear forces, and electromagnetism. Recently substantiated further by the prediction and discovery of the Higgs boson, the SM has proven to be an extraordinarily proficient predictive model for all the subatomic particles and forces. The question remains, what is to be done with gravity - the fourth force of nature? Within the framework of QFT theoreticians have predicted the existence of yet another boson called the graviton. For this reason QFT has a very attractive allure, despite its limitations. According to !1 QFT the gravitational force is attributed to the interaction between two gravitons, however when applying the equations of General Relativity (GR) the force between two gravitons becomes infinite! Results like this are nonsensical and must be resolved for the theory to stand. -
Charged Quantum Fields in Ads2
Charged Quantum Fields in AdS2 Dionysios Anninos,1 Diego M. Hofman,2 Jorrit Kruthoff,2 1 Department of Mathematics, King’s College London, the Strand, London WC2R 2LS, UK 2 Institute for Theoretical Physics and ∆ Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands Abstract We consider quantum field theory near the horizon of an extreme Kerr black hole. In this limit, the dynamics is well approximated by a tower of electrically charged fields propagating in an SL(2, R) invariant AdS2 geometry endowed with a constant, symmetry preserving back- ground electric field. At large charge the fields oscillate near the AdS2 boundary and no longer admit a standard Dirichlet treatment. From the Kerr black hole perspective, this phenomenon is related to the presence of an ergosphere. We discuss a definition for the quantum field theory whereby we ‘UV’ complete AdS2 by appending an asymptotically two dimensional Minkowski region. This allows the construction of a novel observable for the flux-carrying modes that resembles the standard flat space S-matrix. We relate various features displayed by the highly charged particles to the principal series representations of SL(2, R). These representations are unitary and also appear for massive quantum fields in dS2. Both fermionic and bosonic fields are studied. We find that the free charged massless fermion is exactly solvable for general background, providing an interesting arena for the problem at hand. arXiv:1906.00924v2 [hep-th] 7 Oct 2019 Contents 1 Introduction 2 2 Geometry near the extreme Kerr horizon 4 2.1 Unitary representations of SL(2, R).......................... -
Arxiv:1907.02341V1 [Gr-Qc]
Different types of torsion and their effect on the dynamics of fields Subhasish Chakrabarty1, ∗ and Amitabha Lahiri1, † 1S. N. Bose National Centre for Basic Sciences Block - JD, Sector - III, Salt Lake, Kolkata - 700106 One of the formalisms that introduces torsion conveniently in gravity is the vierbein-Einstein- Palatini (VEP) formalism. The independent variables are the vierbein (tetrads) and the components of the spin connection. The latter can be eliminated in favor of the tetrads using the equations of motion in the absence of fermions; otherwise there is an effect of torsion on the dynamics of fields. We find that the conformal transformation of off-shell spin connection is not uniquely determined unless additional assumptions are made. One possibility gives rise to Nieh-Yan theory, another one to conformally invariant torsion; a one-parameter family of conformal transformations interpolates between the two. We also find that for dynamically generated torsion the spin connection does not have well defined conformal properties. In particular, it affects fermions and the non-minimally coupled conformal scalar field. Keywords: Torsion, Conformal transformation, Palatini formulation, Conformal scalar, Fermion arXiv:1907.02341v1 [gr-qc] 4 Jul 2019 ∗ [email protected] † [email protected] 2 I. INTRODUCTION Conventionally, General Relativity (GR) is formulated purely from a metric point of view, in which the connection coefficients are given by the Christoffel symbols and torsion is set to zero a priori. Nevertheless, it is always interesting to consider a more general theory with non-zero torsion. The first attempt to formulate a theory of gravity that included torsion was made by Cartan [1]. -
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. -
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. -
Supersymmetry in Quantum Mechanics
supersy13u’rletry m intend to develop here some of the algebra pertinent to the example, the operation of changing a fermion to a boson and back basic concepts of supersymmetry. I will do this by showing an again results in changing the position of the fermion. I analogy between the quantum-mechanical harmonic os- If supersymmetry is an invariance of nature, then cillator and a bosonic field and a further analogy between the quantum-mechanical spin-% particle and a fermionic field. One [H, Q.]= O, (3) result of combining the two resulting fields will be to show that a “tower” ofdegeneracies between the states for bosons wtd fermions is that is, Q. commutes with the Hamiltonian H of the universe. Also, a natural feature of even the simplest of stspersymmetry theories. in this case, the vacuum is a supersymmetric singlet (Qalvac) = O). A supersymmetry operation changes bosons into fermions and Equations I through 3 are the basic defining equations of super- vice versa, which can be represented schematically with the operators symmetry. In the form given, however, the supcrsymmetry is solely Q: and Q, and the equations an external or space-time symmetry (a supersymmetry operation changes particle spin without altering any of the particle’s internal (?$lboson) = Ifermion)a symmetries). An exlended supersymmetry that connects external and internal symmetries can be constructed by expanding the number of and (1) operators of Eq. 2. However, for our purposes, we need not consider Q.lfermion) = Iboson)a. that complication. In the simplest version of supersymmetry, there are four such The Harmonic oscillator. -
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. -
7 Quantized Free Dirac Fields
7 Quantized Free Dirac Fields 7.1 The Dirac Equation and Quantum Field Theory The Dirac equation is a relativistic wave equation which describes the quantum dynamics of spinors. We will see in this section that a consistent description of this theory cannot be done outside the framework of (local) relativistic Quantum Field Theory. The Dirac Equation (i∂/ m)ψ =0 ψ¯(i∂/ + m) = 0 (1) − can be regarded as the equations of motion of a complex field ψ. Much as in the case of the scalar field, and also in close analogy to the theory of non-relativistic many particle systems discussed in the last chapter, the Dirac field is an operator which acts on a Fock space. We have already discussed that the Dirac equation also follows from a least-action-principle. Indeed the Lagrangian i µ = [ψ¯∂ψ/ (∂µψ¯)γ ψ] mψψ¯ ψ¯(i∂/ m)ψ (2) L 2 − − ≡ − has the Dirac equation for its equation of motion. Also, the momentum Πα(x) canonically conjugate to ψα(x) is ψ δ † Πα(x)= L = iψα (3) δ∂0ψα(x) Thus, they obey the equal-time Poisson Brackets ψ 3 ψα(~x), Π (~y) P B = iδαβδ (~x ~y) (4) { β } − Thus † 3 ψα(~x), ψ (~y) P B = δαβδ (~x ~y) (5) { β } − † In other words the field ψα and its adjoint ψα are a canonical pair. This result follows from the fact that the Dirac Lagrangian is first order in time derivatives. Notice that the field theory of non-relativistic many-particle systems (for both fermions on bosons) also has a Lagrangian which is first order in time derivatives. -
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. -
Physics 234C Lecture Notes
Physics 234C Lecture Notes Jordan Smolinsky [email protected] Department of Physics & Astronomy, University of California, Irvine, ca 92697 Abstract These are lecture notes for Physics 234C: Advanced Elementary Particle Physics as taught by Tim M.P. Tait during the spring quarter of 2015. This is a work in progress, I will try to update it as frequently as possible. Corrections or comments are always welcome at the above email address. 1 Goldstone Bosons Goldstone's Theorem states that there is a massless scalar field for each spontaneously broken generator of a global symmetry. Rather than prove this, we will take it as an axiom but provide a supporting example. Take the Lagrangian given below: N X 1 1 λ 2 L = (@ φ )(@µφ ) + µ2φ φ − (φ φ ) (1) 2 µ i i 2 i i 4 i i i=1 This is a theory of N real scalar fields, each of which interacts with itself by a φ4 coupling. Note that the mass term for each of these fields is tachyonic, it comes with an additional − sign as compared to the usual real scalar field theory we know and love. This is not the most general such theory we could have: we can imagine instead a theory which includes mixing between the φi or has a nondegenerate mass spectrum. These can be accommodated by making the more general replacements X 2 X µ φiφi ! Mijφiφj i i;j (2) X 2 X λ (φiφi) ! Λijklφiφjφkφl i i;j;k;l but this restriction on the potential terms of the Lagrangian endows the theory with a rich structure.