Canonical Quantization C6, HT 2016
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Quantum Field Theory with No Zero-Point Energy
Published for SISSA by Springer Received: March 16, 2018 Revised: May 20, 2018 Accepted: May 27, 2018 Published: May 30, 2018 JHEP05(2018)191 Quantum field theory with no zero-point energy John R. Klauder Department of Physics, University of Florida, Gainesville, FL 32611-8440, U.S.A. Department of Mathematics, University of Florida, Gainesville, FL 32611-8440, U.S.A. E-mail: [email protected] Abstract: Traditional quantum field theory can lead to enormous zero-point energy, which markedly disagrees with experiment. Unfortunately, this situation is built into con- ventional canonical quantization procedures. For identical classical theories, an alternative quantization procedure, called affine field quantization, leads to the desirable feature of having a vanishing zero-point energy. This procedure has been applied to renormalizable and nonrenormalizable covariant scalar fields, fermion fields, as well as general relativity. Simpler models are offered as an introduction to affine field quantization. Keywords: Nonperturbative Effects, Renormalization Regularization and Renormalons, Models of Quantum Gravity ArXiv ePrint: 1803.05823 Open Access, c The Authors. 3 https://doi.org/10.1007/JHEP05(2018)191 Article funded by SCOAP . Contents 1 Introduction 1 2 Nonrenormalizable models exposed 2 3 Ultralocal scalar model: a nonrenormalizable example 3 4 Additional studies using affine variables 5 JHEP05(2018)191 1 Introduction The source of zero-point energy for a free scalar field is readily canceled by normal order- ing of the Hamiltonian operator, which leaves every term with an annihilation operator. Unfortunately, for a non-free scalar field, say with a quartic potential term, normal order- ing cannot eliminate the zero-point energy since there is always a term composed only of creation operators. -
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 ........................... -
Arxiv:Cond-Mat/0203258V1 [Cond-Mat.Str-El] 12 Mar 2002 AS 71.10.-W,71.27.+A PACS: Pnpolmi Oi Tt Hsc.Iscoeconnection Close Its High- of Physics
Large-N expansion based on the Hubbard-operator path integral representation and its application to the t J model − Adriana Foussats and Andr´es Greco Facultad de Ciencias Exactas Ingenier´ıa y Agrimensura and Instituto de F´ısica Rosario (UNR-CONICET). Av.Pellegrini 250-2000 Rosario-Argentina. (October 29, 2018) In the present work we have developed a large-N expansion for the t − J model based on the path integral formulation for Hubbard-operators. Our large-N expansion formulation contains diagram- matic rules, in which the propagators and vertex are written in term of Hubbard operators. Using our large-N formulation we have calculated, for J = 0, the renormalized O(1/N) boson propagator. We also have calculated the spin-spin and charge-charge correlation functions to leading order 1/N. We have compared our diagram technique and results with the existing ones in the literature. PACS: 71.10.-w,71.27.+a I. INTRODUCTION this constrained theory leads to the commutation rules of the Hubbard-operators. Next, by using path-integral The role of electronic correlations is an important and techniques, the correlation functional and effective La- open problem in solid state physics. Its close connection grangian were constructed. 1 with the phenomena of high-Tc superconductivity makes In Ref.[ 11], we found a particular family of constrained this problem relevant in present days. Lagrangians and showed that the corresponding path- One of the most popular models in the context of high- integral can be mapped to that of the slave-boson rep- 13,5 Tc superconductivity is the t J model. -
Quantum Field Theory*
Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement. -
Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions
PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 05 Tuesday, January 14, 2020 Topics: Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions. Feynman Diagrams We can use the Wick’s theorem to express the n-point function h0jT fφ(x1)φ(x2) ··· φ(xn)gj0i as a sum of products of Feynman propagators. We will be interested in developing a diagrammatic interpretation of such expressions. Let us look at the expression containing four fields. We have h0jT fφ1φ2φ3φ4g j0i = DF (x1 − x2)DF (x3 − x4) +DF (x1 − x3)DF (x2 − x4) +DF (x1 − x4)DF (x2 − x3): (1) We can interpret Eq. (1) as the sum of three diagrams, if we represent each of the points, x1 through x4 by a dot, and each factor DF (x − y) by a line joining x to y. These diagrams are called Feynman diagrams. In Fig. 1 we provide the diagrammatic interpretation of Eq. (1). The interpretation of the diagrams is the following: particles are created at two spacetime points, each propagates to one of the other points, and then they are annihilated. This process can happen in three different ways, and they correspond to the three ways to connect the points in pairs, as shown in the three diagrams in Fig. 1. Thus the total amplitude for the process is the sum of these three diagrams. PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 ⟨0|T {ϕ1ϕ2ϕ3ϕ4}|0⟩ = DF(x1 − x2)DF(x3 − x4) + DF(x1 − x3)DF(x2 − x4) +DF(x1 − x4)DF(x2 − x3) 1 2 1 2 1 2 + + 3 4 3 4 3 4 Figure 1: Diagrammatic interpretation of Eq. -
An Introduction to Supersymmetry
An Introduction to Supersymmetry Ulrich Theis Institute for Theoretical Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D–07743 Jena, Germany [email protected] This is a write-up of a series of five introductory lectures on global supersymmetry in four dimensions given at the 13th “Saalburg” Summer School 2007 in Wolfersdorf, Germany. Contents 1 Why supersymmetry? 1 2 Weyl spinors in D=4 4 3 The supersymmetry algebra 6 4 Supersymmetry multiplets 6 5 Superspace and superfields 9 6 Superspace integration 11 7 Chiral superfields 13 8 Supersymmetric gauge theories 17 9 Supersymmetry breaking 22 10 Perturbative non-renormalization theorems 26 A Sigma matrices 29 1 Why supersymmetry? When the Large Hadron Collider at CERN takes up operations soon, its main objective, besides confirming the existence of the Higgs boson, will be to discover new physics beyond the standard model of the strong and electroweak interactions. It is widely believed that what will be found is a (at energies accessible to the LHC softly broken) supersymmetric extension of the standard model. What makes supersymmetry such an attractive feature that the majority of the theoretical physics community is convinced of its existence? 1 First of all, under plausible assumptions on the properties of relativistic quantum field theories, supersymmetry is the unique extension of the algebra of Poincar´eand internal symmtries of the S-matrix. If new physics is based on such an extension, it must be supersymmetric. Furthermore, the quantum properties of supersymmetric theories are much better under control than in non-supersymmetric ones, thanks to powerful non- renormalization theorems. -
Perturbation Theory and Exact Solutions
PERTURBATION THEORY AND EXACT SOLUTIONS by J J. LODDER R|nhtdnn Report 76~96 DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS J J. LODOER ASSOCIATIE EURATOM-FOM Jun»»76 FOM-INST1TUUT VOOR PLASMAFYSICA RUNHUIZEN - JUTPHAAS - NEDERLAND DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS by JJ LODDER R^nhuizen Report 76-95 Thisworkwat performed at part of th«r«Mvchprogmmncof thcHMCiattofiafrccmentof EnratoniOTd th« Stichting voor FundtmenteelOiutereoek der Matctk" (FOM) wtihnnmcWMppoft from the Nederhmdie Organiutic voor Zuiver Wetemchap- pcigk Onderzoek (ZWO) and Evntom It it abo pabHtfMd w a the* of Ac Univenrty of Utrecht CONTENTS page SUMMARY iii I. INTRODUCTION 1 II. GENERALIZED FUNCTIONS DEFINED ON DISCONTINUOUS TEST FUNC TIONS AND THEIR FOURIER, LAPLACE, AND HILBERT TRANSFORMS 1. Introduction 4 2. Discontinuous test functions 5 3. Differentiation 7 4. Powers of x. The partie finie 10 5. Fourier transforms 16 6. Laplace transforms 20 7. Hubert transforms 20 8. Dispersion relations 21 III. PERTURBATION THEORY 1. Introduction 24 2. Arbitrary potential, momentum coupling 24 3. Dissipative equation of motion 31 4. Expectation values 32 5. Matrix elements, transition probabilities 33 6. Harmonic oscillator 36 7. Classical mechanics and quantum corrections 36 8. Discussion of the Pu strength function 38 IV. EXACTLY SOLVABLE MODELS FOR DISSIPATIVE MOTION 1. Introduction 40 2. General quadratic Kami1tonians 41 3. Differential equations 46 4. Classical mechanics and quantum corrections 49 5. Equation of motion for observables 51 V. SPECIAL QUADRATIC HAMILTONIANS 1. Introduction 53 2. Hamiltcnians with coordinate coupling 53 3. Double coordinate coupled Hamiltonians 62 4. Symmetric Hamiltonians 63 i page VI. DISCUSSION 1. Introduction 66 ?. -
Second Quantization
Chapter 1 Second Quantization 1.1 Creation and Annihilation Operators in Quan- tum Mechanics We will begin with a quick review of creation and annihilation operators in the non-relativistic linear harmonic oscillator. Let a and a† be two operators acting on an abstract Hilbert space of states, and satisfying the commutation relation a,a† = 1 (1.1) where by “1” we mean the identity operator of this Hilbert space. The operators a and a† are not self-adjoint but are the adjoint of each other. Let α be a state which we will take to be an eigenvector of the Hermitian operators| ia†a with eigenvalue α which is a real number, a†a α = α α (1.2) | i | i Hence, α = α a†a α = a α 2 0 (1.3) h | | i k | ik ≥ where we used the fundamental axiom of Quantum Mechanics that the norm of all states in the physical Hilbert space is positive. As a result, the eigenvalues α of the eigenstates of a†a must be non-negative real numbers. Furthermore, since for all operators A, B and C [AB, C]= A [B, C] + [A, C] B (1.4) we get a†a,a = a (1.5) − † † † a a,a = a (1.6) 1 2 CHAPTER 1. SECOND QUANTIZATION i.e., a and a† are “eigen-operators” of a†a. Hence, a†a a = a a†a 1 (1.7) − † † † † a a a = a a a +1 (1.8) Consequently we find a†a a α = a a†a 1 α = (α 1) a α (1.9) | i − | i − | i Hence the state aα is an eigenstate of a†a with eigenvalue α 1, provided a α = 0. -
Canonical Quantization of the Self-Dual Model Coupled to Fermions∗
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Canonical Quantization of the Self-Dual Model coupled to Fermions∗ H. O. Girotti Instituto de F´ısica, Universidade Federal do Rio Grande do Sul Caixa Postal 15051, 91501-970 - Porto Alegre, RS, Brazil. (March 1998) Abstract This paper is dedicated to formulate the interaction picture dynamics of the self-dual field minimally coupled to fermions. To make this possible, we start by quantizing the free self-dual model by means of the Dirac bracket quantization procedure. We obtain, as result, that the free self-dual model is a relativistically invariant quantum field theory whose excitations are identical to the physical (gauge invariant) excitations of the free Maxwell-Chern-Simons theory. The model describing the interaction of the self-dual field minimally cou- pled to fermions is also quantized through the Dirac-bracket quantization procedure. One of the self-dual field components is found not to commute, at equal times, with the fermionic fields. Hence, the formulation of the in- teraction picture dynamics is only possible after the elimination of the just mentioned component. This procedure brings, in turns, two new interac- tions terms, which are local in space and time while non-renormalizable by power counting. Relativistic invariance is tested in connection with the elas- tic fermion-fermion scattering amplitude. We prove that all the non-covariant pieces in the interaction Hamiltonian are equivalent to the covariant minimal interaction of the self-dual field with the fermions. The high energy behavior of the self-dual field propagator corroborates that the coupled theory is non- renormalizable. -
Solutions 4: Free Quantum Field Theory
QFT PS4 Solutions: Free Quantum Field Theory (22/10/18) 1 Solutions 4: Free Quantum Field Theory 1. Heisenberg picture free real scalar field We have Z 3 d p 1 −i!pt+ip·x y i!pt−ip·x φ(t; x) = 3 p ape + ape (1) (2π) 2!p (i) By taking an explicit hermitian conjugation, we find our result that φy = φ. You need to note that all the parameters are real: t; x; p are obviously real by definition and if m2 is real and positive y y semi-definite then !p is real for all values of p. Also (^a ) =a ^ is needed. (ii) Since π = φ_ = @tφ classically, we try this on the operator (1) to find Z d3p r! π(t; x) = −i p a e−i!pt+ip·x − ay ei!pt−ip·x (2) (2π)3 2 p p (iii) In this question as in many others it is best to leave the expression in terms of commutators. That is exploit [A + B; C + D] = [A; C] + [A; D] + [B; C] + [B; D] (3) as much as possible. Note that here we do not have a sum over two terms A and B but a sum over an infinite number, an integral, but the principle is the same. −i!pt+ip·x −ipx The second way to simplify notation is to write e = e so that we choose p0 = +!p. −i!pt+ip·x ^y y −i!pt+ip·x The simplest way however is to write Abp =a ^pe and Ap =a ^pe . -
A Supersymmetry Primer
hep-ph/9709356 version 7, January 2016 A Supersymmetry Primer Stephen P. Martin Department of Physics, Northern Illinois University, DeKalb IL 60115 I provide a pedagogical introduction to supersymmetry. The level of discussion is aimed at readers who are familiar with the Standard Model and quantum field theory, but who have had little or no prior exposure to supersymmetry. Topics covered include: motiva- tions for supersymmetry, the construction of supersymmetric Lagrangians, superspace and superfields, soft supersymmetry-breaking interactions, the Minimal Supersymmetric Standard Model (MSSM), R-parity and its consequences, the origins of supersymmetry breaking, the mass spectrum of the MSSM, decays of supersymmetric particles, experi- mental signals for supersymmetry, and some extensions of the minimal framework. Contents 1 Introduction 3 2 Interlude: Notations and Conventions 13 3 Supersymmetric Lagrangians 17 3.1 The simplest supersymmetric model: a free chiral supermultiplet ............. 18 3.2 Interactions of chiral supermultiplets . ................ 22 3.3 Lagrangians for gauge supermultiplets . .............. 25 3.4 Supersymmetric gauge interactions . ............. 26 3.5 Summary: How to build a supersymmetricmodel . ............ 28 4 Superspace and superfields 30 4.1 Supercoordinates, general superfields, and superspace differentiation and integration . 31 4.2 Supersymmetry transformations the superspace way . ................ 33 4.3 Chiral covariant derivatives . ............ 35 4.4 Chiralsuperfields................................ ........ 37 arXiv:hep-ph/9709356v7 27 Jan 2016 4.5 Vectorsuperfields................................ ........ 38 4.6 How to make a Lagrangian in superspace . .......... 40 4.7 Superspace Lagrangians for chiral supermultiplets . ................... 41 4.8 Superspace Lagrangians for Abelian gauge theory . ................ 43 4.9 Superspace Lagrangians for general gauge theories . ................. 46 4.10 Non-renormalizable supersymmetric Lagrangians . .................. 49 4.11 R symmetries........................................ -
Spin-1/2 Fermions in Quantum Field Theory
Spin-1/2 fermions in quantum field theory µ First, recall that 4-vectors transform under Lorentz transformations, Λ ν, as p′ µ = Λµ pν, where Λ SO(3,1) satisfies Λµ g Λρ = g .∗ A Lorentz ν ∈ ν µρ λ νλ transformation corresponds to a rotation by θ about an axis nˆ [θ~ θnˆ] and ≡ a boost, ζ~ = vˆ tanh−1 ~v , where ~v is the corresponding velocity. Under the | | same Lorentz transformation, a generic field transforms as: ′ ′ Φ (x )= MR(Λ)Φ(x) , where M exp iθ~·J~ iζ~·K~ are N N representation matrices of R ≡ − − × 1 1 the Lorentz group. Defining J~+ (J~ + iK~ ) and J~− (J~ iK~ ), ≡ 2 ≡ 2 − i j ijk k i j [J± , J±]= iǫ J± , [J± , J∓] = 0 . Thus, the representations of the Lorentz algebra are characterized by (j1,j2), 1 1 where the ji are half-integers. (0, 0) is a scalar and (2, 2) is a four-vector. Of 1 1 interest to us here are the spinor representations (2, 0) and (0, 2). ∗ In our conventions, gµν = diag(1 , −1 , −1 , −1). (1, 0): M = exp i θ~·~σ 1ζ~·~σ , butalso (M −1)T = iσ2M(iσ2)−1 2 −2 − 2 (0, 1): [M −1]† = exp i θ~·~σ + 1ζ~·~σ , butalso M ∗ = iσ2[M −1]†(iσ2)−1 2 −2 2 since (iσ2)~σ(iσ2)−1 = ~σ∗ = ~σT − − Transformation laws of 2-component fields ′ β ξα = Mα ξβ , ′ α −1 T α β ξ = [(M ) ] β ξ , ′† α˙ −1 † α˙ † β˙ ξ = [(M ) ] β˙ ξ , ˙ ξ′† =[M ∗] βξ† .