DOCSLIB.ORG

An Exact Integration of A\Phi^ 4 Quantum Field Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Exact Integration of A\Phi^ 4 Quantum Field Theory An exact integration of a φ4 quantum field theory Timothy D. Andersen (Dated: Received: date / Accepted: date) Abstract Most quantum field theories are not exactly solvable. In this paper show the statistical equiva- lence of the standard exponential path integral to products of Heaviside functions, i.e. a product of specially tuned uniform distributions. This allows exact integrations of certain quantum field theories. I apply the equivalence to calculate the exact, non-perturbative path integral for a 3+1-D scalar (real) phi-4 field theory. PACS numbers: 03.70.+k, 05.20.Gg arXiv:1306.6836v1 [physics.gen-ph] 27 Jun 2013 1 The standard or “exponential” path-integral formulation of quantum field theory is the functional integral over a Hilbert space, i = Dφe ~ S[φ], Z Z where S[φ] = d4x , the action functional in the quantum field φ, is only exactly com- L putable for a smallR number of Lagrangians such as the free field, [φ]= ∂µφ∂ φ m2φ2 + Jφ, Lfree µ − for a scalar field φ and source J both in L2 Hilbert space over Minkowski spacetime [13]. The massive, real scalar φ4 theory is one of the simplest theories in quantum field theory without an exact integration. Its Lagrangian is given by, [φ]= ∂µφ∂ φ m2φ2 λφ4 + Jφ, L µ − − where λ > 0. There are no known mathematical techniques for calculating for the φ4 Z theory. The main analytical approaches are Feynman diagrams, which give perturbations of the free particle ensemble for small coupling constant, λ, or perturbation series for strong coupling and small kinetic term[7]. The former perturbation series, however, do not converge for the quartic interaction theory and are asymptotic at best, while the latter have limited applicability in high energy physics. What we need for strong couplings and high energy is an exact solution. Under a Wick rotation, the 3 + 1-D φ4 quantum field theory becomes a 4-D statistical mechanical theory [11], − 1 (J)= Dφe ~ S[φ], Z Z where S[φ]= d4x and [φ]= ∂µφ∂ φ+m2φ2 +λφ4 Jφ. The functional is now a 4-D H H µ − H energy functional,R the amplitude is a partition function, and Planck’s reduced constant ~ Z is equivalent to temperature. The path integral is equivalent to the “canonical” ensemble. For experiments, we are only interested in how the quantization changes relative to the vacuum. Therefore, the quantity of interest is the ratio: = Z(J) . Most expectations of A Z(0) observables can be found from this ratio or a scaling of it. Perturbation theory, specifically Feynman diagrams, is a time-tested and useful technique for evaluating this ratio at high energy, but it is also tedious and tends to blow up if the coupling is too strong. Therefore, there is a strong motivation in quantum field theory 2 to calculate the ratio exactly for as many theories as possible, even ones as simple as the φ4, because of the possibility that it will lead to the development of new forms of non- perturbative calculus for making quantum predictions in QED and QCD, useful at high energies and densities where couplings become too strong. With this motivation in mind, I apply the principle of ensemble equivalence to derive an exact solution. The principle of ensemble equivalence is a common tool for simplifying computations in both statistical mechanics and quantum field theory. Let Ω (J)= dφ dφ δ(A σ [φ, J]), N 1 ··· N N − N Z be the N-dimensional microcanonical ensemble and − ~ Z (J)= dφ dφ e σN [φ]/ , N 1 ··· N Z the N-dimensional canonical ensemble, with σN the N dimensional action. By definition, two statistical ensembles are equivalent if, in the “thermodynamic” limit of infinite degrees of freedom, N , they generate identical expectations of observables. Equivalence is → ∞ typically established by an asymptotic relation (e.g., with the method of steepest-descent or saddle point method [5]) that becomes exact in the limit. The equivalence between microcanonical and canonical ensembles has been known since at least the 1930’s [10] and criteria for non-equivalence established in the 1970’s [9][12][8]. For efficiency, lattice gauge simulations frequently implement the microcanonical quantum field theory either in “demon” Monte Carlo as in the work of Creutz et al. [3][4] or the Hamiltonian flow method of Callaway et al. [1][2]. Perturbation theories for microcanonical quantum field theory have also been established [11][6]. What is relatively unknown is that the microcanonical ensemble is not the only one for which an equivalence to the canonical can be shown. As I show below, a specially tuned product of uniform distributions, i.e. an ensemble density given by a product of step or Heaviside functions, can also be shown to be equivalent to the standard quantization by invoking the multidimensional method of steepest descent. In this paper, I show the following exact equivalence: Z(J) d4k C = exp F [J] (1) Z(0) (2π)4 Z where 3 √ 2/3 2 3( h + g ) −1 1/3 1/3 F [J] = J − + λp λp (4α + h/g + g ) wg1/3 λ−1( 2α + h/g1/3 + g1/3) − × p − h q q g − 108 + w 1√3( h + g2/3)(h +4αg1/3 + g2/3) − h 2αg1/3 + g2/3 − × − i −1 1/3 2/3 −2 −1 1/3 1/3 −1 1/3 1/3 (h +4αg + g ) 8 λp ( 2α + h/g + g )+ λp (4α + h/g + g ) (2), ! × − − ! hq q i given that g α3 + 36Aαλ +6√3w, h α2 12Aλ , ≡ p ≡ − p 1 w Aλ (α2 +4Aλ ), α (k2 + m2). ≡ p p ≡ 2 The value for A is given by thep implicit equation, 1 = √3α4 + 72√3Aα2λ + 144√3A2λ2 + 24α3 Aλ + 288α(Aλ )3/2 (√3h + 12α√Aλ)g2/3 ~ p p p p − × h i λ−1( 2α + h/g1/3 + g1/3) λ−1(4αp+ h/g1/3 + g1/3) p − − − p × hq q i √ 4/3 −1 1/3 1/3 −1 1/3 1/3 2 Aλg λp ( 2α + h/g + g ) λp (4α + h/g + g ) " − − × q q −1 −1 1/3 1/3 −1 1/3 1/3 λp ( 2α + h/g + g )+ λp (4α + h/g + g ) , (3) − − # q q and λp is the renormalized or “physical” coupling constant. The proportionality constant C is a trivial scaling. N Let S[φ, J] = limN→∞ σN where σN = ǫ j=1 Lj[φj,Jj] and, working in momentum ∆k 4 space, ǫ = 1/N is the hypercubic volumeP element of the 4-D momentum space 2π ∼ which, along with N, is a regularization. Lemma 1. Given the limit exists, the canonical ensemble and an ensemble defined by the product of Heaviside functions are equivalent for particular Aj, i.e., Z[J] N dφ θ(A ǫL [φ ,J ]) C = lim j j − j j j , Z[0] N→∞ dφ θ(A ǫL [φ , 0]) j=1 R j j j j Y − R where C is a trivial proportionality constant. The constant is trivial if and only if, given a 4 Hilbert space functional O = lim →∞ O such that O O(1), N N N ∼ dφ dφ O ρ (φ ,...,φ ) O = lim 1 ··· N N θ 1 N h i N→∞ dφ dφ ρ (φ ,...,φ ) R 1 ··· N θ 1 N dφ dφ O ρ (φ ,...,φ ) = lim R 1 ··· N N exp 1 N , (4) N→∞ dφ dφ ρ (φ ,...,φ ) R 1 ··· N exp 1 N − ~ for ρ = θ(A ǫL ) and ρ = R e ǫLj / for some choice of A . θ j j − j exp j j Proof. TheQ Heaviside has numerousQ representations, but the best one to use here is, x x γ+i∞ dη −η(x0−t) θ(x x0)= dt δ(t x0)= dt e , − − − ∞ 2πi Z0 Z0 Zγ i where we have used the inverse Laplacian representation for the Dirac delta function [5]. Now, given that dφ θ(A ǫL [φ ,J ]) = j j − j j j Z Aj γ+i∞ dηj ηj (aj −ǫLj [φj ,Jj ]) dφj daj e , − ∞ 2πi Z Z0 Zγ i we can move the integrals over φj inside: Aj γ+i∞ dηj ηj aj −ηj ǫLj [φj ,Jj ]) daj e dφje = − ∞ 2πi j 0 γ i Y Z Z Z Aj γ+i∞ dηj ηj aj −ηj Fj daj e e , (5) − ∞ 2πi j 0 γ i Y Z Z −ηj Fj −ηj ǫLj [φj ,Jj] where e = dφje . Note that Fj is a function of ηj. The methodR of steepest descent applies here because the “entropy” expression increases with N, S = η (a F ) O(N). Thus, the exponent increases with N, and the integral N j j j − j ∼ can be expandedP asymptotically in N. The N-vector saddle-point η0, where ∂/∂ηSN = 0 is the maximal point of the entropy. As N , we obtain the first order asymptotic relation, →∞ γ+i∞ γ+i∞ dη1 dηN SN [η] SN [η0] e CN e , (6) − ∞ 2πi ··· − ∞ 2πi ≈ Zγ i Zγ i where CN is related to the Hessian matrix of SN /N, − 2π N/2 ∂2S 1/2 C = det N −1 N . N N − ∂η η i j Since S [η ]= ηT a ηT F , normalizing, N 0 0 − 0 A ηT a−ηT F [J] −ηT F da da C [J]e 0 0 C [J]e 0 [J] 0 1 ··· N N = N . A ηT a−ηT F [0] −ηT F [0] da da C [0]e 0 0 CN [0]e 0 R 0 1 ··· N N R 5 With all ensemble equivalence proofs, trivial scaling factors vanish in normalization (see pg.
Load more

Recommended publications
  • Radiative Corrections in Curved Spacetime and Physical Implications to the Power Spectrum and Trispectrum for Different Inflationary Models
    Radiative Corrections in Curved Spacetime and Physical Implications to the Power Spectrum and Trispectrum for different Inflationary Models Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades Doctor rerum naturalium der Georg-August-Universit¨atG¨ottingen Im Promotionsprogramm PROPHYS der Georg-August University School of Science (GAUSS) vorgelegt von Simone Dresti aus Locarno G¨ottingen,2018 Betreuungsausschuss: Prof. Dr. Laura Covi, Institut f¨urTheoretische Physik, Universit¨atG¨ottingen Prof. Dr. Karl-Henning Rehren, Institut f¨urTheoretische Physik, Universit¨atG¨ottingen Prof. Dr. Dorothea Bahns, Mathematisches Institut, Universit¨atG¨ottingen Miglieder der Prufungskommission:¨ Referentin: Prof. Dr. Laura Covi, Institut f¨urTheoretische Physik, Universit¨atG¨ottingen Korreferentin: Prof. Dr. Dorothea Bahns, Mathematisches Institut, Universit¨atG¨ottingen Weitere Mitglieder der Prufungskommission:¨ Prof. Dr. Karl-Henning Rehren, Institut f¨urTheoretische Physik, Universit¨atG¨ottingen Prof. Dr. Stefan Kehrein, Institut f¨urTheoretische Physik, Universit¨atG¨ottingen Prof. Dr. Jens Niemeyer, Institut f¨urAstrophysik, Universit¨atG¨ottingen Prof. Dr. Ariane Frey, II. Physikalisches Institut, Universit¨atG¨ottingen Tag der mundlichen¨ Prufung:¨ Mittwoch, 23. Mai 2018 Ai miei nonni Marina, Palmira e Pierino iv ABSTRACT In a quantum field theory with a time-dependent background, as in an expanding uni- verse, the time-translational symmetry is broken. We therefore expect loop corrections to cosmological observables to be time-dependent after renormalization for interacting fields. In this thesis we compute and discuss such radiative corrections to the primordial spectrum and higher order spectra in simple inflationary models. We investigate both massless and massive virtual fields, and we disentangle the time dependence caused by the background and by the initial state that is set to the Bunch-Davies vacuum at the beginning of inflation.
    [Show full text]
  • Scalar Quantum Electrodynamics
    A complex system that works is invariably found to have evolved from a simple system that works. John Gall 17 Scalar Quantum Electrodynamics In nature, there exist scalar particles which are charged and are therefore coupled to the electromagnetic field. In three spatial dimensions, an important nonrelativistic example is provided by superconductors. The phenomenon of zero resistance at low temperature can be explained by the formation of so-called Cooper pairs of electrons of opposite momentum and spin. These behave like bosons of spin zero and charge q =2e, which are held together in some metals by the electron-phonon interaction. Many important predictions of experimental data can be derived from the Ginzburg- Landau theory of superconductivity [1]. The relativistic generalization of this theory to four spacetime dimensions is of great importance in elementary particle physics. In that form it is known as scalar quantum electrodynamics (scalar QED). 17.1 Action and Generating Functional The Ginzburg-Landau theory is a three-dimensional euclidean quantum field theory containing a complex scalar field ϕ(x)= ϕ1(x)+ iϕ2(x) (17.1) coupled to a magnetic vector potential A. The scalar field describes bound states of pairs of electrons, which arise in a superconductor at low temperatures due to an attraction coming from elastic forces. The detailed mechanism will not be of interest here; we only note that the pairs are bound in an s-wave and a spin singlet state of charge q =2e. Ignoring for a moment the magnetic interactions, the ensemble of these bound states may be described, in the neighborhood of the superconductive 4 transition temperature Tc, by a complex scalar field theory of the ϕ -type, by a euclidean action 2 3 1 m g 2 E = d x ∇ϕ∗∇ϕ + ϕ∗ϕ + (ϕ∗ϕ) .
    [Show full text]
  • Asymptotically Flat, Spherical, Self-Interacting Scalar, Dirac and Proca Stars
    S S symmetry Article Asymptotically Flat, Spherical, Self-Interacting Scalar, Dirac and Proca Stars Carlos A. R. Herdeiro and Eugen Radu * Departamento de Matemática da Universidade de Aveiro and Centre for Research and Development in Mathematics and Applications (CIDMA), Campus de Santiago, 3810-183 Aveiro, Portugal; [email protected] * Correspondence: [email protected] Received: 14 November 2020; Accepted: 1 December 2020; Published: 8 December 2020 Abstract: We present a comparative analysis of the self-gravitating solitons that arise in the Einstein–Klein–Gordon, Einstein–Dirac, and Einstein–Proca models, for the particular case of static, spherically symmetric spacetimes. Differently from the previous study by Herdeiro, Pombo and Radu in 2017, the matter fields possess suitable self-interacting terms in the Lagrangians, which allow for the existence of Q-ball-type solutions for these models in the flat spacetime limit. In spite of this important difference, our analysis shows that the high degree of universality that was observed by Herdeiro, Pombo and Radu remains, and various spin-independent common patterns are observed. Keywords: solitons; boson stars; Dirac stars 1. Introduction and Motivation 1.1. General Remarks The (modern) idea of solitons, as extended particle-like configurations, can be traced back (at least) to Lord Kelvin, around one and half centuries ago, who proposed that atoms are made of vortex knots [1]. However, the first explicit example of solitons in a relativistic field theory was found by Skyrme [2,3], (almost) one hundred years later. The latter, dubbed Skyrmions, exist in a model with four (real) scalars that are subject to a constraint.
    [Show full text]
  • A Solvable Tensor Field Theory Arxiv:1903.02907V2 [Math-Ph]
    A Solvable Tensor Field Theory R. Pascalie∗ Universit´ede Bordeaux, LaBRI, CNRS UMR 5800, Talence, France, EU Mathematisches Institut der Westf¨alischen Wilhelms-Universit¨at,M¨unster,Germany, EU August 3, 2020 Abstract We solve the closed Schwinger-Dyson equation for the 2-point function of a tensor field theory with a quartic melonic interaction, in terms of Lambert's W-function, using a perturbative expansion and Lagrange-B¨urmannresummation. Higher-point functions are then obtained recursively. 1 Introduction Tensor models have regained a considerable interest since the discovery of their large N limit (see [1], [2], [3] or the book [4]). Recently, tensor models have been related in [5] and [6], to the Sachdev-Ye-Kitaev model [7], [8], [9], [10], which is a promising toy-model for understanding black holes through holography (see also [11], [12], the lectures [13] and the review [14]). In this paper we study a specific type of tensor field theory (TFT) 1. More precisely, we consider a U(N)-invariant tensor models whose kinetic part is modified to include a Laplacian- like operator (this operator is a discrete Laplacian in the Fourier transformed space of the tensor index space). This type of tensor model has originally been used to implement renormalization techniques for tensor models (see [16], the review [17] or the thesis [18] and references within) and has also been studied as an SYK-like TFT [19]. Recently, the functional Renormalization Group (FRG) as been used in [20] to investigate the existence of a universal continuum limit in tensor models, see also the review [21].
    [Show full text]
  • Arxiv:2003.01034V1 [Hep-Th] 2 Mar 2020 Im Oe.Frta Esntelna Oe a Etogta Ge a Large As the Thought in Be Solved Be Can Can Model Models Linear These the One
    Inhomogeneous states in two dimensional linear sigma model at large N A. Pikalov1,2∗ 1Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia 2Institute for Theoretical and Experimental Physics, Moscow, Russia (Dated: March 3, 2020) In this note we consider inhomogeneous solutions of two-dimensional linear sigma model in the large N limit. These solutions are similar to the ones found recently in two-dimensional CP N sigma model. The solution exists only for some range of coupling constant. We calculate energy of the solutions as function of parameters of the model and show that at some value of the coupling constant it changes sign signaling a possible phase transition. The case of the nonlinear model at finite temperature is also discussed. The free energy of the inhomogeneous solution is shown to change sign at some critical temperature. I. INTRODUCTION Two-dimensional linear sigma model is a theory of N real scalar fields and quartic O(N) symmetric interaction. The model has two dimensionful parameters: mass of the particles and coupling constant. In the limit of infinite coupling one can obtain the nonlinear O(N) arXiv:2003.01034v1 [hep-th] 2 Mar 2020 sigma model. For that reason the linear model can be thought as a generalization of the nonlinear one. These models can be solved in the large N limit, see [1] for a review. In turn O(N) sigma model is quite similar to the CP N sigma model. Recently the large N CP N sigma model was considered on a finite interval with various boundary conditions [2–8] and on circle [9–12].
    [Show full text]
  • Physics 215C: Particles and Fields Spring 2019
    Physics 215C: Particles and Fields Spring 2019 Lecturer: McGreevy These lecture notes live here. Please email corrections to mcgreevy at physics dot ucsd dot edu. Last updated: 2021/04/20, 14:28:40 1 Contents 0.1 Introductory remarks for the third quarter................4 0.2 Sources and acknowledgement.......................7 0.3 Conventions.................................8 1 Anomalies9 2 Effective field theory 20 2.1 A parable on integrating out degrees of freedom............. 20 2.2 Introduction to effective field theory.................... 25 2.3 The color of the sky............................. 30 2.4 Fermi theory of Weak Interactions..................... 32 2.5 Loops in EFT................................ 33 2.6 The Standard Model as an EFT...................... 39 2.7 Superconductors.............................. 42 2.8 Effective field theory of Fermi surfaces.................. 47 3 Geometric and topological terms in field theory actions 58 3.1 Coherent state path integrals for bosons................. 58 3.2 Coherent state path integral for fermions................. 66 3.3 Path integrals for spin systems....................... 73 3.4 Topological terms from integrating out fermions............. 86 3.5 Pions..................................... 89 4 Field theory of spin systems 99 4.1 Transverse-Field Ising Model........................ 99 4.2 Ferromagnets and antiferromagnets..................... 131 4.3 The beta function for 2d non-linear sigma models............ 136 4.4 CP1 representation and large-N ...................... 138 5 Duality 148 5.1 XY transition from superfluid to Mott insulator, and T-duality..... 148 6 Conformal field theory 158 6.1 The stress tensor and conformal invariance (abstract CFT)....... 160 6.2 Radial quantization............................. 166 6.3 Back to general dimensions......................... 172 7 Duality, part 2 179 7.1 (2+1)-d XY is dual to (2+1)d electrodynamics.............
    [Show full text]
  • Adiabatic Limits and Renormalization in Quantum Spacetime
    DOTTORATO DI RICERCA IN MATEMATICA XXXII CICLO DEL CORSO DI DOTTORATO Adiabatic limits and renormalization in quantum spacetime Aleksei Bykov A.A. 2019/2020 Docente Guida/Tutor: Prof. D. Guido Relatore: Prof. G. Morsella Coordinatore: Prof. A. Braides Contents 1 Introduction 4 2 Doplicher-Fredenhagen-Roberts Quantum Spacetime 11 2.1 Construction and basic facts . 11 2.2 Quantum fields on the DFR QST and the role of Qµν . 12 2.2.1 More general quantum fields in DFR QST . 13 2.3 Optimally localised states and the quantum diagonal map . 17 3 Perturbation theory for QFT and its non-local generalization 21 3.1 Hamiltonian perturbation theory . 22 3.1.1 Ordinary QFT . 22 3.1.2 Non-local case: fixing HI ............................. 26 3.1.3 Hamiltonian approach: fixing Hint ........................ 35 3.2 Lagrangian perturbation theories . 38 3.3 Yang-Feldman quantizaion . 41 3.4 LSZ reduction . 41 4 Feynman rules for non-local Hamiltonian Perturbation theory 45 5 Lagrangian reformulation of the Hamiltonian Feynman rules 52 6 Corrected propagator 57 7 Adiabatic limit 61 7.1 Weak adiabatic limit and the LSZ reduction . 61 7.1.1 Existence of the weak adiabatic limit . 61 7.1.2 Feynman rules from LSZ reduction . 64 7.2 Strong adiabatic limit . 65 8 Renormalization 68 8.1 Formal renormalization . 68 8.2 The physical renormalization . 71 8.2.1 Dispersion relation renormalisation . 72 8.2.2 Field strength renormalization . 72 8.3 Conluding remarks on renormalisation . 73 9 Conclusions 75 9.1 Summary of the main results . 75 9.2 Outline of further directions .
    [Show full text]
  • The Higgs Trilinear Coupling and the Scale of New Physics
    The Higgs Trilinear Coupling and the Scale of New Physics Spencer Chang Department of Physics and Institute of Theoretical Science University of Oregon, Eugene, Oregon 97403 and Department of Physics National Taiwan University, Taipei, Taiwan 10617, ROC Markus A. Luty Center for Quantum Mathematics and Physics (QMAP) University of California, Davis, California 95616 Abstract We consider modifications of the Higgs potential due to new physics at high energy scales. These upset delicate cancellations predicted by the Standard Model for processes involving Higgs bosons and longitudinal gauge bosons, and lead to a breakdown of the theory at high energies. We focus on modifications of the Higgs trilinear coupling and use the violation of tree- level unitarity as an estimate of the scale where the theory breaks down. We obtain a completely model-independent bound of <∼ 13 TeV for an order-1 modification of the trilinear. We argue that this bound can be saturated only in fine-tuned models, and the scale of new physics is likely to be much lower. The most stringent bounds are obtained from amplitudes involving multiparticle states that are not conventional scattering states. Our results arXiv:1902.05556v2 [hep-ph] 31 Mar 2020 show that a future determination of the Higgs cubic coupling can point to a well-defined scale of new physics that can be targeted and explored at future colliders. 1 Introduction Many of the couplings of the 125 GeV Higgs boson to gauge bosons and fermions have been measured at the 10% level and agree with the predictions of the Standard Model [1, 2].
    [Show full text]
  • Homework 8 – Due Friday, Nov 17Th – 8A,8B,8C
    Homework 8 – due Friday, Nov 17th – 8A,8B,8C (8A) Field redefinitions and scattering Consider a seemingly complicated theory with the basic complex scalar field B (note that |x|2 = x∗x): g L = |(∂ B)(1 + gB)|2 − m2|B + B2|2 µ 2 1. Divide the Lagrangian for B into a free part and interaction terms. Compute the elastic N + N¯ → N + N¯ scattering amplitude in the tree level approximation, imagining that you follow the LSZ formula. Tree level means that there are no loops and therefore you don’t need any counterterms. However, you will need a g2 vertex diagram as well as a diagram with two g cubic vertices. Also, you will need derivative interactions. With derivative interactions, it is not true that L = −H, and also, there are problems with commuting the time-derivatives through the normal-ordering symbol. Believe me that these two problems cancel and can be collectively ignored, as will be clear from Feynman’s path integral formulation later. Assume L = −H and be sloppy about permuting the T , ∂µ symbols. 2. Now pretend that someone brings you a secret message that says: g ψ = B + B2 2 Using this field redefinition, guess what is the Lagrangian for ψ – it is very simple! – and verify it by an actual calculation. What is the scattering amplitude for ψ quanta? This question should be supersimple. Use this simple answer as a hint to fix any combinatorial errors in the calculation with B if you have errors. 3. If you do the same calculation in the B variables or the ψ variables, will the off-shell Green’s functions of B and those of ψ be equal (yes/no)? Should they be (yes/no)? Should the on-shell amplitudes be equal (yes/no)? Comment: An advantage of the LSZ formalism is that it doesn’t matter whether you assign a field like B or a field like ψ to the particle.
    [Show full text]
  • Lectures on Quantum Field Theory
    Lectures on Quantum Field Theory Aleksandar R. Bogojevi´c1 Institute of Physics P. O. Box 57 11000 Belgrade, Yugoslavia March, 1998 1Email: [email protected] ii Contents 1 Linearity and Combinatorics 1 1.1 Schwinger–DysonEquations. 1 1.2 GeneratingFunctionals. 3 1.3 FreeFieldTheory ........................... 6 2 Further Combinatoric Structure 7 2.1 ClassicalFieldTheory ......................... 7 2.2 TheEffectiveAction .......................... 8 2.3 PathIntegrals.............................. 10 3 Using the Path Integral 15 3.1 Semi-ClassicalExpansion . 15 3.2 WardIdentities ............................. 18 4 Fermions 21 4.1 GrassmannNumbers.......................... 21 5 Euclidean Field Theory 27 5.1 Thermodynamics ............................ 30 5.2 WickRotation ............................. 31 6 Ferromagnets and Phase Transitions 33 6.1 ModelsofFerromagnets . 33 6.2 TheMeanFieldApproximation. 34 6.3 TransferMatrices............................ 35 6.4 Landau–Ginsburg Theory . 36 6.5 TowardsLoopExpansion . 38 7 The Propagator 41 7.1 ScalarPropagator ........................... 41 7.2 RandomWalk.............................. 43 iii iv CONTENTS 8 The Propagator Continued 47 8.1 The Yukawa Potential . 47 8.2 VirtualParticles ............................ 49 9 From Operators to Path Integrals 53 9.1 HamiltonianPathIntegral. 53 9.2 LagrangianPathIntegral . 55 9.3 QuantumFieldTheory. 57 10 Path Integral Surprises 59 10.1 Paths that don’t Contribute . 59 10.2 Lagrangian Measure from SD Equations . 62 11 Classical Symmetry 67 11.1 NoetherTechnique . 67 11.2 Energy-Momentum Tensors Galore . 70 12 Symmetry Breaking 73 12.1 GoldstoneBosons............................ 73 12.2 TheHiggsMechanism . 77 13 Effective Action to One Loop 81 13.1 TheEffectivePotential. 81 13.2 The O(N)Model............................ 86 14 Solitons 89 14.1 Perturbative vs. Semi-Classical . 89 14.2 ClassicalSolitons ............................ 90 14.3 The φ4 Kink .............................
    [Show full text]
  • Quantum Field Theory and Particle Physics
    Quantum Field Theory and Particle Physics Badis Ydri Département de Physique, Faculté des Sciences Université d’Annaba Annaba, Algerie 2 YDRI QFT Contents 1 Introduction and References 1 I Path Integrals, Gauge Fields and Renormalization Group 3 2 Path Integral Quantization of Scalar Fields 5 2.1 FeynmanPathIntegral. .. .. .. .. .. .. .. .. .. .. .. 5 2.2 ScalarFieldTheory............................... 9 2.2.1 PathIntegral .............................. 9 2.2.2 The Free 2 Point Function . 12 − 2.2.3 Lattice Regularization . 13 2.3 TheEffectiveAction .............................. 16 2.3.1 Formalism................................ 16 2.3.2 Perturbation Theory . 20 2.3.3 Analogy with Statistical Mechanics . 22 2.4 The O(N) Model................................ 23 2.4.1 The 2 Point and 4 Point Proper Vertices . 24 − − 2.4.2 Momentum Space Feynman Graphs . 25 2.4.3 Cut-off Regularization . 27 2.4.4 Renormalization at 1 Loop...................... 29 − 2.5 Two-Loop Calculations . 31 2.5.1 The Effective Action at 2 Loop ................... 31 − 2.5.2 The Linear Sigma Model at 2 Loop ................. 32 − 2.5.3 The 2 Loop Renormalization of the 2 Point Proper Vertex . 35 − − 2.5.4 The 2 Loop Renormalization of the 4 Point Proper Vertex . 40 − − 2.6 Renormalized Perturbation Theory . 42 2.7 Effective Potential and Dimensional Regularization . .......... 45 2.8 Spontaneous Symmetry Breaking . 49 2.8.1 Example: The O(N) Model ...................... 49 2.8.2 Glodstone’s Theorem . 55 3 Path Integral Quantization of Dirac and Vector Fields 57 3.1 FreeDiracField ................................ 57 3.1.1 Canonical Quantization . 57 3.1.2 Fermionic Path Integral and Grassmann Numbers . 58 4 YDRI QFT 3.1.3 The Electron Propagator .
    [Show full text]
  • The Standard Model of Particle Physics and Beyond
    Context Relativity - gauge theories The Standard Model of particle physics Beyond the Standard Model Summary The Standard Model of particle physics and beyond. Benjamin Fuks & Michel Rausch de Traubenberg IPHC Strasbourg / University of Strasbourg. [email protected] & [email protected] Slides available on http://www.cern.ch/fuks/esc.php 6-13 July 2011 The Standard Model of particle physics and beyond. Benjamin Fuks & Michel Rausch de Traubenberg - 1 Context Relativity - gauge theories The Standard Model of particle physics Beyond the Standard Model Summary Outline. 1 Context. 2 Special relativity and gauge theories. Action and symmetries. Poincar´eand Lorentz algebras and their representations. Relativistic wave equations. Gauge symmetries - Yang-Mills theories - symmetry breaking. 3 Construction of the Standard Model. Quantum Electrodynamics (QED). Scattering theory - Calculation of a squared matrix element. Weak interactions. The electroweak theory. Quantum Chromodynamics. 4 Beyond the Standard Model of particle physics. The Standard Model: advantages and open questions. Grand unified theories. Supersymmetry. Extra-dimensional theories. String theory. 5 Summary. The Standard Model of particle physics and beyond. Benjamin Fuks & Michel Rausch de Traubenberg - 2 Context Relativity - gauge theories The Standard Model of particle physics Beyond the Standard Model Summary Building blocks describing matter. Atom compositness. * Neutrons. * Protons. * Electrons. Proton and neutron compositness. * Naively: up and down quarks. * In reality: dynamical objects made of • Valence and sea quarks. • Gluons [see below...]. Beta decays. − * n ! p + e +ν ¯e . * Needs for a neutrino. The Standard Model of particle physics and beyond. Benjamin Fuks & Michel Rausch de Traubenberg - 3 Context Relativity - gauge theories The Standard Model of particle physics Beyond the Standard Model Summary Three families of fermionic particles [Why three?].
    [Show full text]