DOCSLIB.ORG

III. Classical Field Theory and Canonical Quantization Hamilton's

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
III. Classical Field Theory and Canonical Quantization Hamilton's III. Classical Field Theory and Canonical Quantization Hamilton’s Principle and the Euler-Lagrange Equations 1 Noether’s Theorem and Symmetries 3 Poincar´eSymmetries 6 Space-Time Translations: Canonical Energy-Momentum Tensor 7 Lorentz Transformations: Improved Energy-Momentum Tensor 8 Canonical Quantization 14 1 III. Classical Field Theory and Canonical Quantization In the previous chapter, we saw how to construct Lorentz covariant free field equations using group theoretic techniques applied to the Lorentz group. However, we still lack a dynamical principle which allows for the construc- tion of the Poincar´egenerators including the Hamiltonian in terms of the operator fields nor a means to give a physical interpretation to the solution to the these field equations. In this chapter, we begin to address these issues. The canonical quantization approach to constructing a quantum field theory starts with a classical field theory whose dynamics is governed by classical field equations which are obtained by applying (a generalized) Hamilton’s principle to an action functional of a Lagrange density (Lagrangian). The classical fields are then promoted to become quantum operator fields satisfy- ing certain equal time canonical commutation or anti-commutation relations. Using this canonical formalism, the various symmetry generators, including the Hamiltonian, can be identified and constructed in terms of the field op- erators. As we shall see, a major advantage of basing a dynamical principle on an action is that it is Lorentz invariant while the Hamiltonian is not. Hamilton’s Principle and the Euler-Lagrange Equations The classical Lagrangian (density) characterizing the underlying dynamics is assumed to be a function of the classical fields, ϕs(~r, t), and their first space-time derivatives, ∂µϕs(~r, t): L = L(ϕs(~r, t), ∂µϕs(~r, t)) . (1) The action functional for the classical field theory is then defined as Z 4 S[ϕ] = d xL(ϕs(t, ~r), ∂µϕs(t, ~r)) , (2) Ω where Ω is the space-time volume. Here the subscript s is used to distinguish different fields. The (generalized) Hamilton principle states that the action is stationary with respect to variations of classical field, δϕs(~r, t), which vanish on the boundary of Ω but are otherwise arbitrary.1 1Rather than employing this generalized Hamilton’s principle, one could divide 3-space 2 Using functional derivatives, Hamilton’s principle takes the form δS[ϕ] 0 = δϕr(x) δϕr(x) Z y ! 4 ∂L δϕs(y) ∂L δ∂µϕs(y) = δϕr(x) d y + y Ω ∂ϕs(y) δϕr(x) ∂∂µϕs(y) δϕr(x) Z ! 4 ∂L δϕs(y) ∂L y δϕs(y) = δϕr(x) d y + y ∂µ Ω ∂ϕs(y) δϕr(x) ∂∂µϕs(y) δϕr(x) Z ! 4 ∂L y ∂L δϕs(y) = δϕr(x) d y − ∂µ y Ω ∂ϕs(y) ∂∂µϕs(y) δϕr(x) Z ! 4 y ∂L δϕs(y) + δϕr(x) d y∂µ y Ω ∂∂µϕ(y) δϕr(x) Z ! 4 ∂L y ∂L 4 = δϕr(x) d y − ∂µ y δrsδ (x − y) Ω ∂ϕs(y) ∂∂µϕs(y) Z ! 4 y ∂L 4 + δϕr(x) d y∂µ y δrsδ (x − y) Ω ∂∂µϕ(y) Z ! 4 ∂L y ∂L 4 = δϕr(x) d y − ∂µ y δrsδ (x − y) Ω ∂ϕs(y) ∂∂µϕs(y) ∂L 4 + δϕr(x)nµ y δrsδ (x − y)|y∈Σ ∂∂µϕ(y) ∂L ∂L ! = δϕr(x) − ∂µ (3) ∂ϕr(x) ∂∂µϕr(x) In obtaining the penultimate equality, we used the divergence theorem with µ µ Σ the boundary of Ω having unit (space-like) normal n (n nµ = 1), where the last equality follows since the field variations vanish on Σ. Since, other than being constrained to vanish on the boundary, each variation into discrete cells (a lattice) and decompose the field which depends on the continuum of spatial variables into a discretely numerated number of degrees of freedom, one associated with each lattice cell, all depending only on time. The Lagrange function and its associ- ated action can then be constructed. The ordinary Hamilton’s principle which states that the action is stationary with respect to variations which vanish at the temporal endpoints, but are otherwise arbitrary, can then be applied. The resultant Euler-Lagrange equa- tions so obtained reproduces the Euler-Lagrange equations obtained with the generalized Hamilton’s principle once the continuum limit is taken. 3 δϕr is arbitrary within Ω, it follows that ∂L ∂L 0 = − ∂µ (4) ∂ϕr(x) ∂∂µϕr(x) which are the Euler-Lagrange equations. The canonically conjugate momentum classical field, πs(x), is defined as ∂L πs(x) = . (5) ∂∂0ϕs(x) Noether’s Theorem and Symmetries Poincar´einvariance restricts the form of the field equations and consequently the form of the actions from which they are derived. In addition, there may be additional symmetries of the system which will also restrict the form of the action. We shall formalize these restrictions and also establish the con- nection between the presence of a continuous symmetry, the existence of a conserved current and its associated time independent charge. This is em- bodied in Noether’s theorem. Let ϕr(x) denote a multicomponent field labeled by r. A transformation which changes the space-time point (eg. translations, Lorentz transforma- tions) is referred to as a space-time transformation while a transformation which leaves the space-time point unchanged is referred to as an internal transformation. Consider the transformation parametrized by Υ which is a symmetry of the action and is continuously connected to the identity. The classical field theory transformation law is Υ Υ ϕr(x) → ϕr (x ) = Drs(Υ)ϕs(x) (6) so that Υ Υ−1 ϕ (x) = Drs(Υ)ϕs(x ) (7) where D(Υ) is a matrix representation of the transformation. Here xΥ is the transformed space-time point (if the transformation is internal, then 4 Υ xµ = xµ). For infinitesimal transformations, parametrized by real υA (cryptically Υ = 1 + υ) with |υA| << 1, the space-time point changes as µ υ µ µ X µ ∆(υ)x = (x ) − x = υA∆Ax (8) A If the transformation is internal transformation, then ∆(υ)xµ = 0. For infinitesimal transformations, the matrix representation can be written X A Drs(I + υ) = δrs + i υA(d )rs (9) A (A) with drs being υ independent numbers. The intrinsic transformation of the field is ϕr(x) → ϕr(x) + δ(υ)ϕr(x) (10) where υ δ(υ)ϕr(x) = ϕr (x) − ϕr(x) = Drs(I + υ)ϕs(x − ∆(υ)x) − ϕ(x) ! X A = δrs + i υA(d )rs ϕs(x − ∆(υ)x) − ϕr(x) A µ X A = −∆x (υ)∂µϕr(x) + i υA(d )rsϕs(x) A X µ A = υA −δrs∆Ax ∂µ + i(d )rs ϕs(x) A X = υAδAϕr(x) (11) A which identifies µ A δAϕr(x) = −δrs∆Ax ∂µ + i(d )rs ϕs(x) (12) while δ(υ)∂µϕr(x) = ∂µ δ(υ)ϕr(x) ⇒ δA∂µϕr(x) = ∂µδAϕr(x) (13) 5 Since L = L(ϕr, ∂µϕr), it follows that ∂L ∂L δ(υ)L = δ(υ)ϕr + δ(υ)(∂µϕr) ∂ϕr ∂∂µϕr ∂L ∂L = δ(υ)ϕr + ∂µδ(υ)ϕr ∂ϕr ∂∂µϕr ∂L ∂L ∂L = δ(υ)ϕr + ∂µ δ(υ)ϕr − ∂µ δ(υ)ϕr ∂ϕr ∂∂µϕr ∂∂µϕr ∂L ∂L ! ∂L ! = − ∂µ δ(υ)ϕr + ∂µ δ(υ)ϕr ∂ϕr ∂∂µϕr ∂∂µϕr δS ∂L ! = δ(υ)ϕr + ∂µ δ(υ)ϕr (14) δϕr ∂∂µϕr or δS ∂L ! δ(υ)ϕr = δ(υ)L − ∂µ δ(υ)ϕr(x) . (15) δϕr ∂∂µϕr(x) which is an algebraic identity and constitutes Noether’s theorem. This iden- tity holds for an field transformation where it is a symmetry of action or not. If the transformation is a symmetry of the classical action so that the action is invariant under the transformation,,2 then the Lagrangian is either invari- ant or transforms into a total divergence. As we shall see, for space-time symmetries, the Lagrangian transforms into a total divergence so that µ δ(υ)L = ∂µΛ (υ) (18) 2The action is invariant provided it algebraically (independent of field equations) sat- isfies δ(υ)S[ϕ] = 0 (16) where Z 4 δ δ(υ) = d x (δ(υ)ϕr(x)) δϕr(x) Z ! 4 X µ A δ = d x υA −δrs∆Ax ∂µ + i(d )rs ϕs(x) (17) δϕr(x) A 6 while for internal symmetries, it is invariant (Λµ(υ) = 0). In either case, the RHS of Noether’s theorem is the divergence of a current µ µ ∂L X µ J (υ) = Λ (υ) − δ(υ)ϕr(x) υAJA(x) (19) ∂∂µϕr(x) A and Noether’s theorem takes the form δS[ϕ] µ δ(υ)ϕr(x) = ∂µJ (υ) (20) δϕr(x) or equivalently δS[ϕ] µ δAϕr(x) = ∂µJA(x) (21) δϕr(x) After application of the Euler-Lagrange field equations: δS[ϕ] = 0, we see δϕr(x) µ µ µ µ that that the current J (υ) or JA(x) is conserved: ∂µJ (υ) = 0 = ∂µJA(x). For such a conserved current, we can define charges Z 3 0 X Q(υ) = d xJ (υ) = υAQA (22) A such that dQ(υ) Z Z dQ = d3x∂ J 0(υ) = − d3x∂ J i(υ) = 0 ⇒ A = 0 (23) dt 0 i dt where we have used the divergence theorem and set the surface terms at infinity to zero. Thus, after application of the Euler-Lagrange field equations, the Noether constructed charges Q(υ) or QA are time independent. These charges will be identified with the generators of the associated symmetries. Poincar`eSymmetries We now apply Noether’s theorem to the space-time symmetries associated with the Poincar´etransformations. 7 Space-Time Translations: Canonical Energy-Momentum Tensor The infinitesimal space-time translation is given by ∆()xµ = µ (24) µ µ P µ µ ν µ with | | << 1.
Load more

Recommended publications
  • Aspects of Symplectic Dynamics and Topology: Gauge Anomalies, Chiral Kinetic Theory and Transfer Matrices
    ASPECTS OF SYMPLECTIC DYNAMICS AND TOPOLOGY: GAUGE ANOMALIES, CHIRAL KINETIC THEORY AND TRANSFER MATRICES BY VATSAL DWIVEDI DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 2017 Urbana, Illinois Doctoral Committee: Associate Professor Shinsei Ryu, Chair Professor Michael Stone, Director of Research Professor James N. Eckstein Professor Jared Bronski Abstract This thesis presents some work on two quite disparate kinds of dynamical systems described by Hamiltonian dynamics. The first part describes a computation of gauge anomalies and their macroscopic effects in a semiclassical picture. The geometric (symplectic) formulation of classical mechanics is used to describe the dynamics of Weyl fermions in even spacetime dimensions, the only quantum input to the symplectic form being the Berry curvature that encodes the spin-momentum locking. The (semi-)classical equations of motion are used in a kinetic theory setup to compute the gauge and singlet currents, whose conservation laws reproduce the nonabelian gauge and singlet anomalies. Anomalous contributions to the hydrodynamic currents for a gas of Weyl fermions at a finite temperature and chemical potential are also calculated, and are in agreement with similar results in literature which were obtained using thermodynamic and/or quantum field theoretical arguments. The second part describes a generalized transfer matrix formalism for noninteracting tight-binding models. The formalism is used to study the bulk and edge spectra, both of which are encoded in the spectrum of the transfer matrices, for some of the common tight-binding models for noninteracting electronic topological phases of matter.
    [Show full text]
  • Constraining Conformal Field Theories with a Higher Spin Symmetry
    PUPT-2399 Constraining conformal field theories with a higher spin symmetry Juan Maldacenaa and Alexander Zhiboedovb aSchool of Natural Sciences, Institute for Advanced Study Princeton, NJ, USA bDepartment of Physics, Princeton University Princeton, NJ, USA We study the constraints imposed by the existence of a single higher spin conserved current on a three dimensional conformal field theory. A single higher spin conserved current implies the existence of an infinite number of higher spin conserved currents. The correlation functions of the stress tensor and the conserved currents are then shown to be equal to those of a free field theory. Namely a theory of N free bosons or free fermions. arXiv:1112.1016v1 [hep-th] 5 Dec 2011 This is an extension of the Coleman-Mandula theorem to CFT’s, which do not have a conventional S matrix. We also briefly discuss the case where the higher spin symmetries are “slightly” broken. Contents 1.Introduction ................................. 2 1.1. Organization of the paper . 4 2. Generalities about higher spin currents . 5 1 3. Removing operators in the twist gap, 2 <τ< 1 .................. 7 3.1. Action of the charges on twist one fields . 9 4. Basic facts about three point functions . 10 4.1. General expression for three point functions . 11 5. Argument using bilocal operators . 12 5.1. Light cone limits of correlators of conserved currents . 12 5.2.Gettinganinfinitenumberofcurrents . 15 5.3. Definition of bilocal operators . 17 5.4. Constraining the action of the higher spin charges . 19 5.5. Quantization of N˜: the case of bosons . 24 5.6.
    [Show full text]
  • Energy Is Conserved in General Relativity
    Energy is Conserved in General Relativity By Philip Gibbs Abstract: Since the early days of relativity the question of conservation of energy in general relativity has been a controversial subject. There have been many assertions that energy is not exactly conserved except in special cases, or that the full conservation law as given by Noether’s theorem reduces to a trivial identity. Here I refute each objection to show that the energy conservation law is exact, fully general and useful. Introduction Conservation of Energy is regarded as one of the most fundamental and best known laws of physics. It is a kind of meta-law that all systems of dynamics in physics are supposed to fulfil to be considered valid. Yet some physicists and cosmologists claim that the law of energy conservation and related laws for momentum actually break down in Einstein’s theory of gravity described by general relativity. Some say that these conservation laws are approximate, or only valid in special cases, or that they can only be written in non-covariant terms making them unphysical. Others simply claim that they are satisfied only by equations that reduce to a uselessly trivial identity. In fact none of those things are true. Energy conservation is an exact law in general relativity. It is fully general for all circumstances and it is certainly non-trivial. Furthermore, it is important for students of relativity to understand the conservation laws properly because they are important to further research at the forefront of physics. In earlier papers I have described a covariant formulation of energy conservation law that is perfectly general.
    [Show full text]
  • Quantum Field Theory Example Sheet 1 Michelmas Term 2011 Exercise 1
    Quantum Field Theory Example Sheet 1 Michelmas Term 2011 Solutions by: Johannes Hofmann [email protected] Laurence Perreault Levasseur [email protected] David Morris [email protected] Marcel Schmittfull [email protected] Note: In the conventions of this course, the Minkowski metric is g = diag (1, −1, −1, −1), or ‘mostly minus.’ Exercise 1 Lagrangian: " # Z a σ ∂y 2 T ∂y 2 L = dx − (1) 0 2 ∂t 2 ∂x Express y(x, t) as a Fourier series: r ∞ 2 X nπx y(x, t) = q (t) sin . (2) a n a n=1 The derivative of y(x, t) with respect to x and t is r ∞ ∂y(x, t) 2 X nπx = q˙ (t) sin (3) ∂t a n a n=1 r ∞ ∂y(x, t) 2 X nπ nπx = q (t) cos , (4) ∂x a a n a n=1 where we abbreviateq ˙ ≡ ∂q/∂t. Substituting Eqs. (3) and (4) in (1) gives 1 X Z a nπx mπx nmπ2 nπx mπx L = dx σq˙ (t)q ˙ (t) sin sin − T q (t)q (t) cos cos . (5) a n m a a a2 n m a a n,m 0 Using the orthonormality relations 2 Z a nπx mπx dx sin sin = δnm and (6) a 0 a a 2 Z a nπx mπx dx cos cos = δnm (7) a 0 a a we see that the terms with n 6= m vanish. We obtain: X Z a σ T nπ 2 L = dx q˙2 (t) − q2 (t) .
    [Show full text]
  • Energy-Momentum Current for Coframe Gravity
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Energy-momentum current for coframe gravity Yakov Itin Institute of Mathematics, Hebrew University of Jerusalem Givat Ram, Jerusalem 91904, Israel, [email protected] Abstract. The obstruction for the existence of an energy momentum tensor for the gravitational field is connected with differential-geometric features of the Riemannian manifold. It has not to be valid for alternative geometrical structures. A teleparallel manifold is defined as a parallelizable differentiable 4D-manifold endowed with a class of smooth coframe fields related by global Lorentz, i.e., SO(1; 3) transformations. In this article a general free parametric class of teleparallel models is considered. It includes a 1-parameter subclass of viable models with the Schwarzschild coframe solution. A new form of the coframe field equation is derived from the general teleparallel Lagrangian by introducing the notion of a 3-parameter conjugate field strength a. F The field equation turns out to have a form completely similar to the Maxwell field equation d a = a. By applying the Noether procedure, the source 3-form a is ∗F T T shown to be connected with the diffeomorphism invariance of the Lagrangian. Thus the source a of the coframe field is interpreted as the total conserved energy-momentum T current. The energy-momentum tensor for coframe is defined. The total energy- momentum current of a system of a coframe and a material fields is conserved. Thus a redistribution of the energy-momentum current between a material and a coframe (gravity) fields is possible in principle, unlike as in the standard GR.
    [Show full text]
  • All Possible Symmetries of Cfts
    All possible symmetries of CFTs Alexander Zhiboedov Princeton University in collaboration with Juan Maldacena Introduction Noether’s theorem: symmetries constrain the dynamics. How do the dynamics constrain possible symmetries? In the context of relativistic QFT the question can be restated as follows: What are the possible symmetries of a theory with the non-trivial S matrix? The answer was given in the paper by S.Coleman and • J.Mandula “All Possible Symmetries Of The S Matrix” in 1967 which is known as the no-go Coleman-Mandula theorem. Introduction Noether’s theorem: symmetries constrain the dynamics. How do the dynamics constrain possible symmetries? In the context of relativistic QFT the question can be restated as follows: What are the possible symmetries of a theory with the non-trivial S matrix? The answer was given in the paper by S.Coleman and • J.Mandula “All Possible Symmetries Of The S Matrix” in 1967 which is known as the no-go Coleman-Mandula theorem. Introduction Noether’s theorem: symmetries constrain the dynamics. How do the dynamics constrain possible symmetries? In the context of relativistic QFT the question can be restated as follows: What are the possible symmetries of a theory with the non-trivial S matrix? The answer was given in the paper by S.Coleman and • J.Mandula “All Possible Symmetries Of The S Matrix” in 1967 which is known as the no-go Coleman-Mandula theorem. Introduction Noether’s theorem: symmetries constrain the dynamics. How do the dynamics constrain possible symmetries? In the context of relativistic QFT the question can be restated as follows: What are the possible symmetries of a theory with the non-trivial S matrix? The answer was given in the paper by S.Coleman and • J.Mandula “All Possible Symmetries Of The S Matrix” in 1967 which is known as the no-go Coleman-Mandula theorem.
    [Show full text]
  • 4. Introducing Conformal Field Theory
    4. Introducing Conformal Field Theory The purpose of this section is to get comfortable with the basic language of two dimen- sional conformal field theory4.Thisisatopicwhichhasmanyapplicationsoutsideof string theory, most notably in statistical physics where it o↵ers a description of critical phenomena. Moreover, it turns out that conformal field theories in two dimensions provide rare examples of interacting, yet exactly solvable, quantum field theories. In recent years, attention has focussed on conformal field theories in higher dimensions due to their role in the AdS/CFT correspondence. A conformal transformation is a change of coordinates σ↵ σ˜↵(σ)suchthatthe ! metric changes by g (σ) ⌦2(σ)g (σ)(4.1) ↵ ! ↵ A conformal field theory (CFT) is a field theory which is invariant under these transfor- mations. This means that the physics of the theory looks the same at all length scales. Conformal field theories care about angles, but not about distances. Atransformationoftheform(4.1)hasadi↵erentinterpretationdependingonwhether we are considering a fixed background metric g↵, or a dynamical background metric. When the metric is dynamical, the transformation is a di↵eomorphism; this is a gauge symmetry. When the background is fixed, the transformation should be thought of as an honest, physical symmetry, taking the point σ↵ to pointσ ˜↵. This is now a global symmetry with the corresponding conserved currents. In the context of string theory in the Polyakov formalism, the metric is dynamical and the transformations (4.1) are residual gauge transformations: di↵eomorphisms which can be undone by a Weyl transformation. In contrast, in this section we will be primarily interested in theories defined on fixed backgrounds.
    [Show full text]
  • Noether's Theorem in Non-Local Field Theories
    Noether’s theorem in non-local field theories M. I. Krivoruchenko1,2,3 and A. A. Tursunov1,4 1Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research, 141980 Dubna, Russia 2Institute for Theoretical and Experimental Physics B. Cheremushkinskaya 25, 117218 Moscow, Russia 3Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia 4Institute of Physics, Silesian University in Opava, Bezruˇcovo n´am. 13, CZ-74601 Opava, Czech Republic Abstract Explicit expressions are constructed for a locally conserved vector current associated with a continuous internal symmetry and for energy-momentum and angular-momentum density tensors associated with the Poincar´egroup in field theories with higher-order derivatives and in non-local field theories. An example of non-local charged scalar field equations with broken C and CPT symmetries is considered. For this case, we find simple analytical expressions for the conserved currents. PACS numbers: 11.10.Lm, 11.30.-j, 03.65.Pm, 02.30.Lt 1 Introduction According to Noether’s theorem [1], the invariance of the Lagrangian of a physical sys- arXiv:1602.03074v1 [math-ph] 9 Feb 2016 temwith respect to continuous transformations leads to the existence of conserved charges. In its standard form, Noether’s theorem refers to local field theories with derivatives of no higher than second order in the field equations. Quantum field theories with higher derivatives are used for intermediate regularization pro- cedures (see, e.g., [2]). The low-energy regime of QCD is known to be successfully described by chiral perturbation theory based on a power-series expansion in derivatives [3, 4]. Field theories with higher derivatives are also discussed in the context of general relativity [5].
    [Show full text]
  • A Short Review on Noether's Theorems, Gauge Symmetries and Boundary
    A short review on Noether's theorems, gauge symmetries and boundary terms Max Ba~nadosand Ignacio Reyes Facultad de F´ısica,Pontificia Universidad Cat´olicade Chile, Casilla 306, Santiago, Chile August 31, 2017 arXiv:1601.03616v3 [hep-th] 30 Aug 2017 Abstract This review is dedicated to some modern applications of the remarkable paper written in 1918 by E. Noether. On a single paper, Noether discovered the crucial relation between symmetries and conserved charges as well as the impact of gauge symmetries on the equations of motion. Almost a century has gone since the publication of this work and its applications have permeated modern physics. Our focus will be on some examples that have appeared recently in the literature. This review is aim at students, not researchers. The main three topics discussed are (i) global symmetries and conserved charges (ii) local symmetries and gauge structure of a theory (iii) boundary conditions and algebra of asymptotic symmetries. All three topics are discussed through examples. Contents 1 Preface 3 2 Noether's theorem for global symmetries in classical mechanics and field theory 5 2.1 Noether's theorem in particle mechanics . .5 2.1.1 The theorem . .5 2.1.2 The `conformal' particle . 10 2.1.3 Background fields and non-conservation equations . 11 2.2 Noether's theorem in Hamiltonian mechanics: Symmetry generators and Lie algebras 13 2.2.1 The conformal particle in Hamiltonian form . 15 2.3 Noether's theorem in Field theory. Derivation and examples. 17 2.3.1 The proof . 17 2.3.2 Symmetries act on fields: Lie derivatives .
    [Show full text]
  • 3 Classical Symmetries and Conservation Laws
    3 Classical Symmetries and Conservation Laws We have used the existence of symmetries in a physical system as a guiding principle for the construction of their Lagrangians and energy functionals. We will show now that these symmetries imply the existence of conservation laws. There are different types of symmetries which, roughly, can be classi- fied into two classes: (a) spacetime symmetries and (b) internal symmetries. Some symmetries involve discrete operations, hence called discrete symme- tries, while others are continuous symmetries. Furthermore, in some theories these are global symmetries,whileinotherstheyarelocal symmetries.The latter class of symmetries go under the name of gauge symmetries.Wewill see that, in the fully quantized theory, global and local symmetries play different roles. Spacetime symmetries are the most common examples of symmetries that are encountered in Physics. They include translation invariance and rotation invariance. If the system is isolated, then time-translation is also a symme- try. A non-relativistic system is in general invariant under Galilean trans- formations, while relativistic systems, are instead Lorentz invariant. Other spacetime symmetries include time-reversal (T ), parity (P )andcharge con- jugation (C). These symmetries are discrete. In classical mechanics, the existence of symmetries has important conse- quences. Thus, translation invariance,whichisaconsequenceofuniformity of space, implies the conservation of the total momentum P of the system. Similarly, isotropy implies the conservation of the total angular momentum L and time translation invariance implies the conservation of the total energy E. All of these concepts have analogs in field theory. However, infieldtheory new symmetries will also appear which do not have an analog in the classi- 52 Classical Symmetries and Conservation Laws cal mechanics of particles.
    [Show full text]
  • Charting the Space of 3D Cfts with a Continuous Global Symmetry
    Charting the space of 3D CFTs with a continuous global symmetry Anatoly Dymarskya, Joao Penedonesb, Emilio Trevisanic, Alessandro Vichib a Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Moscow 143026 Russia c Institute of Physics, Ecole´ Polytechnique F´ed´erale de Lausanne (EPFL), Rte de la Sorge, BSP 728, CH-1015 Lausanne, Switzerland d Centro de Fisica do Porto, Universidade do Porto, Rua do Campo Alegre 687, 4169 007 Porto, Portugal Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada Abstract We study correlation functions of a conserved spin-1 current Jµ in three dimen- sional Conformal Field Theories (CFTs). We investigate the constraints imposed by permutation symmetry and current conservation on the form of three point functions hJµJνO∆;`i and the four point function hJµJνJρJσi and identify the minimal set of independent crossing symmetry conditions. We obtain a recurrence relation for conformal blocks for generic spin-1 operators in three dimensions. In the process, we improve several technical points, facilitating the use of recurrence relations. By arXiv:1705.04278v1 [hep-th] 11 May 2017 applying the machinery of the numerical conformal bootstrap we obtain universal bounds on the dimensions of certain light operators as well as the central charge. Highlights of our results include numerical evidence for the conformal collider bound and new constraints on the parameters of the critical O(2) model. The results obtained in this work apply to any unitary, parity-preserving three dimensional CFT. Contents 1 Introduction4 2 Setup 7 ± 2.1 Three point functions hJJO∆;`i .........................7 2.1.1 Special case: hJJTµνi ...........................9 2.2 Four point function hJJJJi ..........................
    [Show full text]
  • 3. Symmetries and Conservation Laws
    3. Symmetries and Conservation Laws The dynamics of interacting field theories, such as φ4 theory in Eq. (2.105), are extremely complex. The resulting equations of motion are not analytically soluble. In fact, free field theory (with the optional addition of a source term, as we will discuss) is the only field theory in four dimensions which has an analytic solution. Nevertheless, in more complicated interacting theories it is often possible to discover many important features about the solution simply by examining the symmetries of the theory. In this chapter we will look at this question in detail and develop some techniques which will allow us to extract dynamical information from the symmetries of a theory. 3.1 Classical Mechanics Let’s return to classical mechanics for a moment, where the Lagrangian is L = T V . As − a simple example, consider two particles in one dimension in a potential L = 1 m q˙2 + 1 m q˙2 V (q ,q ). (3.1) 2 1 1 2 2 2 − 1 2 The momenta conjugate to the qi’s are pi = miq˙i, and from the Euler-Lagrange equations @V @V @V p˙ = , P˙ p˙ +˙p = + . (3.2) i − @q ⌘ 1 2 − @q @q i ✓ 1 2 ◆ If V depends only on q q (that is, the particles aren’t attached to springs or anything 1 − 2 else which defines a fixed reference frame) then the system is invariant under the shift q q + ↵, and @V/@q = @V/@q ,soP˙ = 0. The total momentum of the system is i ! i 1 − 2 conserved.
    [Show full text]