Casimir Torque for a Perfectly Conducting Wedge: a Canonical Quantum Field Theoretical Approach
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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 ........................... -
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. -
Physical Masses and the Vacuum Expectation Value
View metadata, citation and similar papers at core.ac.uk brought to you by CORE PHYSICAL MASSES AND THE VACUUM EXPECTATION VALUE provided by CERN Document Server OF THE HIGGS FIELD Hung Cheng Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139, U.S.A. and S.P.Li Institute of Physics, Academia Sinica Nankang, Taip ei, Taiwan, Republic of China Abstract By using the Ward-Takahashi identities in the Landau gauge, we derive exact relations between particle masses and the vacuum exp ectation value of the Higgs eld in the Ab elian gauge eld theory with a Higgs meson. PACS numb ers : 03.70.+k; 11.15-q 1 Despite the pioneering work of 't Ho oft and Veltman on the renormalizability of gauge eld theories with a sp ontaneously broken vacuum symmetry,a numb er of questions remain 2 op en . In particular, the numb er of renormalized parameters in these theories exceeds that of bare parameters. For such theories to b e renormalizable, there must exist relationships among the renormalized parameters involving no in nities. As an example, the bare mass M of the gauge eld in the Ab elian-Higgs theory is related 0 to the bare vacuum exp ectation value v of the Higgs eld by 0 M = g v ; (1) 0 0 0 where g is the bare gauge coupling constant in the theory.We shall show that 0 v u u D (0) T t g (0)v: (2) M = 2 D (M ) T 2 In (2), g (0) is equal to g (k )atk = 0, with the running renormalized gauge coupling constant de ned in (27), and v is the renormalized vacuum exp ectation value of the Higgs eld de ned in (29) b elow. -
B1. the Generating Functional Z[J]
B1. The generating functional Z[J] The generating functional Z[J] is a key object in quantum field theory - as we shall see it reduces in ordinary quantum mechanics to a limiting form of the 1-particle propagator G(x; x0jJ) when the particle is under thje influence of an external field J(t). However, before beginning, it is useful to look at another closely related object, viz., the generating functional Z¯(J) in ordinary probability theory. Recall that for a simple random variable φ, we can assign a probability distribution P (φ), so that the expectation value of any variable A(φ) that depends on φ is given by Z hAi = dφP (φ)A(φ); (1) where we have assumed the normalization condition Z dφP (φ) = 1: (2) Now let us consider the generating functional Z¯(J) defined by Z Z¯(J) = dφP (φ)eφ: (3) From this definition, it immediately follows that the "n-th moment" of the probability dis- tribution P (φ) is Z @nZ¯(J) g = hφni = dφP (φ)φn = j ; (4) n @J n J=0 and that we can expand Z¯(J) as 1 X J n Z Z¯(J) = dφP (φ)φn n! n=0 1 = g J n; (5) n! n ¯ so that Z(J) acts as a "generator" for the infinite sequence of moments gn of the probability distribution P (φ). For future reference it is also useful to recall how one may also expand the logarithm of Z¯(J); we write, Z¯(J) = exp[W¯ (J)] Z W¯ (J) = ln Z¯(J) = ln dφP (φ)eφ: (6) 1 Now suppose we make a power series expansion of W¯ (J), i.e., 1 X 1 W¯ (J) = C J n; (7) n! n n=0 where the Cn, known as "cumulants", are just @nW¯ (J) C = j : (8) n @J n J=0 The relationship between the cumulants Cn and moments gn is easily found. -
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. -
Two Dimensional Supersymmetric Models and Some of Their Thermodynamic Properties from the Context of Sdlcq
TWO DIMENSIONAL SUPERSYMMETRIC MODELS AND SOME OF THEIR THERMODYNAMIC PROPERTIES FROM THE CONTEXT OF SDLCQ DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Yiannis Proestos, B.Sc., M.Sc. ***** The Ohio State University 2007 Dissertation Committee: Approved by Stephen S. Pinsky, Adviser Stanley L. Durkin Adviser Gregory Kilcup Graduate Program in Robert J. Perry Physics © Copyright by Yiannis Proestos 2007 ABSTRACT Supersymmetric Discrete Light Cone Quantization is utilized to derive the full mass spectrum of two dimensional supersymmetric theories. The thermal properties of such models are studied by constructing the density of states. We consider pure super Yang–Mills theory without and with fundamentals in large-Nc approximation. For the latter we include a Chern-Simons term to give mass to the adjoint partons. In both of the theories we find that the density of states grows exponentially and the theories have a Hagedorn temperature TH . For the pure SYM we find that TH at infi- 2 g Nc nite resolution is slightly less than one in units of π . In this temperature range, q we find that the thermodynamics is dominated by the massless states. For the system with fundamental matter we consider the thermal properties of the mesonic part of the spectrum. We find that the meson-like sector dominates the thermodynamics. We show that in this case TH grows with the gauge coupling. The temperature and coupling dependence of the free energy for temperatures below TH are calculated. As expected, the free energy for weak coupling and low temperature grows quadratically with the temperature. -
Quantum Mechanics Propagator
Quantum Mechanics_propagator This article is about Quantum field theory. For plant propagation, see Plant propagation. In Quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. In Feynman diagrams, which calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the diagram. They also can be viewed as the inverse of the wave operator appropriate to the particle, and are therefore often called Green's functions. Non-relativistic propagators In non-relativistic quantum mechanics the propagator gives the probability amplitude for a particle to travel from one spatial point at one time to another spatial point at a later time. It is the Green's function (fundamental solution) for the Schrödinger equation. This means that, if a system has Hamiltonian H, then the appropriate propagator is a function satisfying where Hx denotes the Hamiltonian written in terms of the x coordinates, δ(x)denotes the Dirac delta-function, Θ(x) is the Heaviside step function and K(x,t;x',t')is the kernel of the differential operator in question, often referred to as the propagator instead of G in this context, and henceforth in this article. This propagator can also be written as where Û(t,t' ) is the unitary time-evolution operator for the system taking states at time t to states at time t'. The quantum mechanical propagator may also be found by using a path integral, where the boundary conditions of the path integral include q(t)=x, q(t')=x' . -
The Quantum Vacuum and the Cosmological Constant Problem
The Quantum Vacuum and the Cosmological Constant Problem S.E. Rugh∗and H. Zinkernagel† Abstract - The cosmological constant problem arises at the intersection be- tween general relativity and quantum field theory, and is regarded as a fun- damental problem in modern physics. In this paper we describe the historical and conceptual origin of the cosmological constant problem which is intimately connected to the vacuum concept in quantum field theory. We critically dis- cuss how the problem rests on the notion of physical real vacuum energy, and which relations between general relativity and quantum field theory are assumed in order to make the problem well-defined. Introduction Is empty space really empty? In the quantum field theories (QFT’s) which underlie modern particle physics, the notion of empty space has been replaced with that of a vacuum state, defined to be the ground (lowest energy density) state of a collection of quantum fields. A peculiar and truly quantum mechanical feature of the quantum fields is that they exhibit zero-point fluctuations everywhere in space, even in regions which are otherwise ‘empty’ (i.e. devoid of matter and radiation). These zero-point arXiv:hep-th/0012253v1 28 Dec 2000 fluctuations of the quantum fields, as well as other ‘vacuum phenomena’ of quantum field theory, give rise to an enormous vacuum energy density ρvac. As we shall see, this vacuum energy density is believed to act as a contribution to the cosmological constant Λ appearing in Einstein’s field equations from 1917, 1 8πG R g R Λg = T (1) µν − 2 µν − µν c4 µν where Rµν and R refer to the curvature of spacetime, gµν is the metric, Tµν the energy-momentum tensor, G the gravitational constant, and c the speed of light. -
The Minkowski Quantum Vacuum Does Not Gravitate
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by KITopen Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 946 (2019) 114694 www.elsevier.com/locate/nuclphysb The Minkowski quantum vacuum does not gravitate Viacheslav A. Emelyanov Institute for Theoretical Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany Received 9 April 2019; received in revised form 24 June 2019; accepted 8 July 2019 Available online 12 July 2019 Editor: Stephan Stieberger Abstract We show that a non-zero renormalised value of the zero-point energy in λφ4-theory over Minkowski spacetime is in tension with the scalar-field equation at two-loop order in perturbation theory. © 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction Quantum fields give rise to an infinite vacuum energy density [1–4], which arises from quantum-field fluctuations taking place even in the absence of matter. Yet, assuming that semi- classical quantum field theory is reliable only up to the Planck-energy scale, the cut-off estimate yields zero-point-energy density which is finite, though, but in a notorious tension with astro- physical observations. In the presence of matter, however, there are quantum effects occurring in nature, which can- not be understood without quantum-field fluctuations. These are the spontaneous emission of a photon by excited atoms, the Lamb shift, the anomalous magnetic moment of the electron, and so forth [5]. This means quantum-field fluctuations do manifest themselves in nature and, hence, the zero-point energy poses a serious problem. -
The Higgs Vacuum Is Unstable
The Higgs vacuum is unstable Archil Kobakhidze and Alexander Spencer-Smith ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Sydney, NSW 2006 E-mails: [email protected], [email protected] Abstract So far, the experiments at the Large Hadron Collider (LHC) have shown no sign of new physics beyond the Standard Model. Assuming the Standard Model is correct at presently available energies, we can accurately extrapolate the theory to higher energies in order to verify its validity. Here we report the results of new high precision calcula- tions which show that absolute stability of the Higgs vacuum state is now excluded. Combining these new results with the recent observa- tion of primordial gravitational waves by the BICEP Collaboration, we find that the Higgs vacuum state would have quickly decayed dur- ing cosmic inflation, leading to a catastrophic collapse of the universe into a black hole. Thus, we are driven to the conclusion that there must be some new physics beyond the Standard Model at energies below the instability scale Λ 109 GeV, which is responsible for the I ∼ stabilisation of the Higgs vacuum. arXiv:1404.4709v2 [hep-ph] 22 Apr 2014 1 Introduction The Higgs mechanism [1, 2, 3, 4] employed within the Standard Model pro- vides masses for all the elementary particles, except neutrinos. These masses (mi) are universally proportional to the respective constants of interaction between the elementary particles and the Higgs boson: mi = λiv, where −1/2 v √2G 246 GeV is the Higgs vacuum expectation value, which, ≡ F ≈ in turn, is defined through the measurements of the Fermi constant GF in 1 weak processes (e.g., week decays of muon). -
HIGGS BOSONS: THEORY and SEARCHES Updated May 2010 by G
–1– HIGGS BOSONS: THEORY AND SEARCHES Updated May 2010 by G. Bernardi (LPNHE, CNRS/IN2P3, U. of Paris VI & VII), M. Carena (Fermi National Accelerator Laboratory and the University of Chicago), and T. Junk (Fermi National Accelerator Laboratory). I. Introduction Understanding the mechanism that breaks electroweak sym- metry and generates the masses of all known elementary par- ticles is one of the most fundamental problems in particle physics. The Higgs mechanism [1] provides a general framework to explain the observed masses of the W ± and Z gauge bosons by means of charged and neutral Goldstone bosons that end up as the longitudinal components of the gauge bosons. These Goldstone bosons are generated by the underlying dynamics of electroweak symmetry breaking (EWSB). However, the fun- damental dynamics of the electroweak symmetry breaking are unknown. There are two main classes of theories proposed in the literature, those with weakly coupled dynamics - such as in the Standard Model (SM) [2] - and those with strongly coupled dynamics. In the SM, the electroweak interactions are described by a gauge field theory based on the SU(2)L×U(1)Y symmetry group. The Higgs mechanism posits a self-interacting complex doublet of scalar fields, and renormalizable interactions are arranged such that the neutral component of the scalar doublet acquires a vacuum expectation value v ≈ 246 GeV, which sets the scale of EWSB. Three massless Goldstone bosons are generated, which are absorbed to give masses to the W ± and Z gauge bosons. The remaining component of the complex doublet becomes the Higgs boson - a new fundamental scalar particle. -
Quantization of Scalar Field
Quantization of Scalar Field Wei Wang 2017.10.12 Wei Wang(SJTU) Lectures on QFT 2017.10.12 1 / 41 Contents 1 From classical theory to quantum theory 2 Quantization of real scalar field 3 Quantization of complex scalar field 4 Propagator of Klein-Gordon field 5 Homework Wei Wang(SJTU) Lectures on QFT 2017.10.12 2 / 41 Free classical field Klein-Gordon Spin-0, scalar Klein-Gordon equation µ 2 @µ@ φ + m φ = 0 Dirac 1 Spin- 2 , spinor Dirac equation i@= − m = 0 Maxwell Spin-1, vector Maxwell equation µν µν @µF = 0;@µF~ = 0: Gravitational field Wei Wang(SJTU) Lectures on QFT 2017.10.12 3 / 41 Klein-Gordon field scalar field, satisfies Klein-Gordon equation µ 2 (@ @µ + m )φ(x) = 0: Lagrangian 1 L = @ φ∂µφ − m2φ2 2 µ Euler-Lagrange equation @L @L @µ − = 0: @(@µφ) @φ gives Klein-Gordon equation. Wei Wang(SJTU) Lectures on QFT 2017.10.12 4 / 41 From classical mechanics to quantum mechanics Mechanics: Newtonian, Lagrangian and Hamiltonian Newtonian: differential equations in Cartesian coordinate system. Lagrangian: Principle of stationary action δS = δ R dtL = 0. Lagrangian L = T − V . Euler-Lagrangian equation d @L @L − = 0: dt @q_ @q Wei Wang(SJTU) Lectures on QFT 2017.10.12 5 / 41 Hamiltonian mechanics Generalized coordinates: q ; conjugate momentum: p = @L i j @qj Hamiltonian: X H = q_i pi − L i Hamilton equations @H p_ = − ; @q @H q_ = : @p Time evolution df @f = + ff; Hg: dt @t where f:::g is the Poisson bracket. Wei Wang(SJTU) Lectures on QFT 2017.10.12 6 / 41 Quantum Mechanics Quantum mechanics Hamiltonian: Canonical quantization Lagrangian: Path