Coulomb Gauge

9 Key excerpts on "Coulomb Gauge"

• 

  eBook - PDF

  Quantum Field Theory

  An Integrated Approach

  • Eduardo Fradkin(Author)
  • 2021(Publication Date)
  • Princeton University Press

    (Publisher)

  Unlike the case of a particle forced to move on the surface of a sphere, the constraints that we have to impose when quantizing Maxwellian electrodynamics do not change the energy spectrum. This is because we can reduce the number of degrees of freedom to be quantized by taking advantage of the gauge invariance of the classical theory. This procedure is called gauge fixing . For example, the classical equation of motion ∂ 2 A μ − ∂ μ (∂ ν A ν ) = 0 (9.19) in the Coulomb Gauge, A 0 = 0 and · A = 0, becomes ∂ 2 A j = 0 (9.20) However, the Coulomb Gauge is not compatible with the Poisson bracket, A j ( x ) , j , ( x ) PB = δ jj δ( x − x ) (9.21) since the spatial divergence of the delta function does not vanish. It follows that the quantization of the theory in the Coulomb Gauge is achieved at the price of a modification of the commutation relations. Since the classical theory is gauge invariant, we can always fix the gauge without any loss of physical content. The procedure of gauge fixing is attractive, because the number of independent variables is greatly reduced. A standard approach to the quantization of a gauge theory is to fix the gauge first, at the classical level, and to quantize later. However, some problems arise immediately. For instance, in most gauges (e.g., the Coulomb Gauge), Lorentz invariance is lost, or at least it is manifestly so. Thus, although the Coulomb Gauge (also known as the radiation or transverse gauge) spoils Lorentz invariance, it has the attractive feature that the nature of the physical states (the photons) is quite transparent. In section 9.2, we will see that the quantization of the theory in this gauge has some peculiarities. Another standard choice is the Lorentz gauge ∂ μ A μ = 0 (9.22) whose main appeal is its manifest covariance. The quantization of the system in this gauge follows the method developed by Suraj Gupta and Konrad Bleuer.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Advanced Classical Electromagnetism

  • Robert Wald(Author)
  • 2022(Publication Date)
  • Princeton University Press

    (Publisher)

  CHAPTER 9 Electromagnetism as a Gauge Theory As I have emphasized throughout this book, the fundamental fields in electromag-netism should be taken to be the potentials A μ = ( − φ/ c , A 1 , A 2 , A 3 ) , not the field strengths. This chapter elucidates this view by explaining how the electromagnetic field can naturally be viewed as a “gauge field.” Section 9.1 gives a Lagrangian formulation of electromagnetism and discuss its fundamental couplings to charged matter. We will see that the fundamental couplings involve A μ in an essential way and cannot be written in terms of the field strengths. Section 9.2 shows how the electromagnetic field can be viewed as a “connection” that provides a notion of differentiation of charged fields. In this way, electromagnetism can be viewed as a simple example of a more general class of theories known as Yang-Mills theories. Finally, section 9.3 shows that this new view-point naturally would allow for the presence of magnetic monopoles, provided that the monopole charge is suitably quantized. 9.1 Lagrangian for the Electromagnetic Field and Its Interactions It is possible—and extremely useful—to give a Lagrangian formulation of electro-magnetism. By a Lagrangian formulation of a field theory, we mean providing a local function L of the fields and their derivatives—called the Lagrangian density —such that the field equations are obtained by extremizing the action S = L dtd 3 x = 1 c L d 4 x (9.1) (where d 4 x = dx 0 dx 1 dx 2 dx 3 = cdtd 3 x ) with respect to all field variations that vanish outside a bounded spacetime region. A Lagrangian formulation a theory provides the theory with additional structure, 1 which plays an essential role in quantization. It also implies a relationship between symmetries of the theory and conservation laws.

  Sign up to read

  Learn more about book
• 

  No longer available |Learn more

  Particle Physics Properties

  • (Author)
  • 2014(Publication Date)
  • White Word Publications

    (Publisher)

  The complete Lagrangian for the gauge theory is now ________________________ WORLD TECHNOLOGIES ________________________ An example: Electrodynamics As a simple application of the formalism developed in the previous sections, consider the case of electrodynamics, with only the electron field. The bare-bones action which generates the electron field's Dirac equation is The global symmetry for this system is The gauge group here is U(1), just the phase angle of the field, with a constant θ . Localising this symmetry implies the replacement of θ by θ( x ). An appropriate covariant derivative is then Identifying the charge e with the usual electric charge (this is the origin of the usage of the term in gauge theories), and the gauge field A ( x ) with the four-vector potential of electromagnetic field results in an interaction Lagrangian where J μ ( x ) is the usual four vector electric current density. The gauge principle is therefore seen to naturally introduce the so-called minimal coupling of the electromagnetic field to the electron field. Adding a Lagrangian for the gauge field A μ ( x ) in terms of the field strength tensor exactly as in electrodynamics, one obtains the Lagrangian which is used as the starting point in quantum electrodynamics. Mathematical formalism Gauge theories are usually discussed in the language of differential geometry. Mathematically, a gauge is just a choice of a (local) section of some principal bundle. A gauge transformation is just a transformation between two such sections. ________________________ WORLD TECHNOLOGIES ________________________ Although gauge theory is dominated by the study of connections (primarily because it's mainly studied by high-energy physicists), the idea of a connection is not central to gauge theory in general.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Gauge Theories in Particle Physics: A Practical Introduction, Volume 1

  From Relativistic Quantum Mechanics to QED, Fourth Edition

  • Ian J R Aitchison, Anthony J.G. Hey(Authors)
  • 2012(Publication Date)
  • CRC Press

    (Publisher)

  56 2. Electromagnetism as a Gauge Theory via the replacement (2.44) 3 . As before, the method clearly generalizes to the four-dimensional case. 2.6 Comments on the gauge principle in electromagnetism Comment (i) A properly sceptical reader may have detected an important sleight of hand in the previous discussion. Where exactly did the electromagnetic charge appear from? The trouble with our argument as so far presented is that we could have defined fields A and V so that they coupled equally to all particles – instead we smuggled in a factor q . Actually we can do a bit better than this. We can use the fact that the electromagnetic charge is absolutely conserved to claim that there can be no quantum mechanical interference between states of different charge q . Hence different phase changes are allowed within each ‘sector’ of definite q : ψ ' = exp(i qχ ) ψ (2.61) let us say. When this becomes a local transformation, χ → χ ( x , t ), we shall need to cancel a term q ∇ χ , which will imply the presence of a ‘ − q A ’ term, as required. Note that such an argument is only possible for an absolutely conserved quantum number q – otherwise we cannot split up the states of the system into non-communicating sectors specified by different values of q . Reversing this line of reasoning, a conservation law such as baryon number conservation, with no related gauge field, would therefore now be suspected of not being absolutely conserved. We still have not tied down why q is the electromagnetic charge and not some other absolutely conserved quantum number. A proper discussion of the reasons for identifying A μ with the electromagnetic potential and q with the particle’s charge will be given in chapter 7 with the help of quantum field theory. Comment (ii) Accepting these identifications, we note that the form of the interaction con-tains but one parameter, the electromagnetic charge q of the particle in ques-tion.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Gauge Fields

  An Introduction To Quantum Theory, Second Edition

  • L. D. Faddeev, A. A. Slavnov(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  1 Introduction:Fundamentals of Classical Gauge Field Theory

  1.1  Basic Concepts and Notation

  The theory of gauge fields at present represents the widely accepted theoretical basis of elementary particle physics. Indeed, the most elaborate model of field theory, quantum electrodynamics, is a particular case of the gauge theory. Further, models of weak interactions have acquired an elegant and self-consistent formulation in the framework of gauge theories. The phenomenological four-fermion interaction has been replaced by the interaction with an intermediate vector particle, the quantum of the Yang-Mills field. Existing experimental data along with the requirement of gauge invariance led to the prediction of weak neutral currents and of new quantum numbers for hadrons.

  Phenomenological quark models of strong interactions also have their most natural foundation in the framwork of a gauge theory known as quantum chromodynamics. This theory provides a unique possibility of describing, in the framework of quantum field theory, the phenomenon of asymptotic freedom. This theory also affords hopes of explaining quark confinement, although this question is not quite clear.

  Finally, the extension of the gauge principle may lead to the gravitational interaction also being placed in the general scheme of Yang-Mills fields.

  So the possibility arises of explaining, on the basis of one principle, all the hierarchy of interactions existing in nature. The term unified field theory, discredited sometime ago, now acquires a new reality in the framework of gauge field theories. In the formation of this picture a number of scientists took part. Let us mention some of the key dates.

  In 1953 C. N. Yang and R. L. Mills, for the first time, generalized the principle of gauge invariance of the interaction of electric charges to the case of interacting isospins. In their paper, they introduced a vector field, which later became known as the Yang-Mills field, and within the framework of the classical field theory its dynamics was developed.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Advanced Electromagnetism: Foundations: Theory And Applications

  • Terence William Barrett, Dale M Grimes(Authors)
  • 1995(Publication Date)
  • World Scientific

    (Publisher)

  Here are three simple ones offered as appetizers. In the classical electron radius problem, expressing the self-energy e 2 /2r in terms of the electric potential as —eV/2, requires the use of the Coulomb Gauge V = — e/r; no other choice is left. This gauge is selected as an integration condition, via Einstein's mass-energy equivalence law. Stating that, in energy units, the electron's rest mass is some 511,000 electron-volts (strictly speaking, positron-volts) makes sense in the Coulomb Gauge only. This shows that the choice of gauge and of integration condition are mutually implicative, both being dictated by the underlying physical model. A related remark holds for the mutual energy: the weight of a vessel containing gaseous or liquid hydrogen includes the atomic mass defect; the additive constant in Coulomb's binding energy is thus weighed as zero. As a second example, consider a permanent current loop of intensity / at equilibrium with its magnetostatic field. If the expression chosen for its self-energy is the (gauge-invariant) one, w = i j * = i i ^ A . d l , (l) there remains the fact that barycenter's position depends on the linear distribution of the vector potentials; therefore, at each point of the loop, A must have a definite value. Einstein's mass-energy equivalence dictates that a definite answer be found. From the far-action expressed of the mutual energy of a pair of current elements,

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Gauge Theories Of Strong, Weak, And Electromagnetic Interactions

  • Chris Quigg(Author)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  THE IDEA OF GAUGE INVARIANCE DOI: 10.1201/9780429034978-3

  3.1 Historical Preliminaries

  We turn now to a discussion of the theory of electrodynamics, which is both the simplest gauge theory and the most familiar. The foundations for our present understanding of the subject were laid down by Maxwell in 1864 in his equations unifying the electric and magnetic interactions. The electromagnetic potential that one is led to introduce in order to generate fields that comply with Maxwell’s equations by construction is not uniquely defined. The resulting freedom to choose many potentials that describe the same electromagnetic fields has come to be called gauge invariance. We shall see that the gauge invariance of electromagnetism can be phrased in terms of a continuous symmetry of the Lagrangian, which leads, through Noether’s theorem, to the conservation of electric charge and to other important consequences. Although it is clearly possible to regard gauge invariance as simply an outcome of Maxwell’s unification, one may wonder whether a greater importance might not attach to the symmetry itself and thus be led to investigate the degree to which Maxwell’s equations might be seen to follow from the symmetry. Indeed, the idea of gauge invariance as a dynamical principle arose from efforts by Hermann Weyl1 to find a geometric basis for both gravitation and electromagnetism. Weyl’s attempts to unify the fundamental interactions of his day through the requirement of invariance under a space–time-dependent change of scale were unsuccessful. His terminology, Eichinvarianz (Eich = gauge or standard of calibration), has nevertheless survived, and his original program is worth recalling.

  Consider the change in a function f(x) between the point xμ and the point xμ + dxμ

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Gravitation

  • A R Prasanna(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 9 Gravity as a Gauge Theory 9.1 Introduction Developments in physics in the twentieth century were heavily based on the principle of symmetry and the associated group structure. Newtonian physics, which was the basis of the classical physics, is governed by the Galilean invari-ance of its associated laws. Maxwell’s electromagnetism, which did not respect Galilean invariance, had to wait for special relativity to amalgamate with laws of motion, to form electrodynamics, which, in fact, was the first unification of two forces. However, the Lorentz invariance, which has been the cornerstone of post–relativity physics, both for macro and micro–physics along with quan-tum principles is considered as a very basic requirement for laws of physics. However, with the advent of general relativity and covariance of physical laws the paths of micro and macro physics separated, with gravitation ruling the roost in macro world explaining the cosmic structure of the universe. On the other side, quantum physics dominated the attention of the physi-cists, who built a beautiful edifice of the elementary particle world explaining almost all experimental results of the twentieth century. Here, along with Lorentz invariance, another invariance was assumed which is the gauge invari-ance. Though the gauge invariance has played a very prominent role in con-structing theories for the microworld, arguments have been sounded, doubt-ing, whether it is a symmetry as expressed by other mathematical structures. Apart from the basic entities like momentum, energy, and angular momentum, which are all associated with the inertial properties of matter, the other physi-cally well understood property associated with the electromagnetism, another fundamental force, is the electric charge, that was known experimentally to be a conserved quantity since the late eighteenth century.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Gravitation

  • A R Prasanna(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 9 Gravity as a Gauge Theory 9.1  Introduction

  Developments in physics in the twentieth century were heavily based on the principle of symmetry and the associated group structure. Newtonian physics, which was the basis of the classical physics, is governed by the Galilean invariance of its associated laws. Maxwell’s electromagnetism, which did not respect Galilean invariance, had to wait for special relativity to amalgamate with laws of motion, to form electrodynamics, which, in fact, was the first unification of two forces. However, the Lorentz invariance, which has been the cornerstone of post–relativity physics, both for macro and micro–physics along with quantum principles is considered as a very basic requirement for laws of physics. However, with the advent of general relativity and covariance of physical laws the paths of micro and macro physics separated, with gravitation ruling the roost in macro world explaining the cosmic structure of the universe.

  On the other side, quantum physics dominated the attention of the physicists, who built a beautiful edifice of the elementary particle world explaining almost all experimental results of the twentieth century. Here, along with Lorentz invariance, another invariance was assumed which is the gauge invariance. Though the gauge invariance has played a very prominent role in constructing theories for the microworld, arguments have been sounded, doubting, whether it is a symmetry as expressed by other mathematical structures. Apart from the basic entities like momentum, energy, and angular momentum, which are all associated with the inertial properties of matter, the other physically well understood property associated with the electromagnetism, another fundamental force, is the electric charge, that was known experimentally to be a conserved quantity since the late eighteenth century. Is there a symmetry associated with this? Remarkably coincident events seem to be the discovery of Noethers theorem in 1918, and Hermmann Weyls attempts to unify gravity and electromagnetism, and the consequent discovery of the gauge symmetry, which implies the conservation of electric charge. Interesting and absorbing discussion of these developments have been narrated in the articles by Jackson and Okun [23] , Afriat [7] , and Brading [8] , which give useful perspectives of the historical developments in the discovery of gauge invariance. As has been discussed already in section 3.4 , Maxwell‘s laws of electromagnetism which are Lorentz invariant, can be expressed in terms of a four–vector potential, Ai , representing the three– vector potential,

  A →

  and φ, the scalar potential. These two generate the magnetic field,

  H =

  ∇ →

  ×

  A →

  , and the electric field,

  E = −

  1 c

  ∂

  A →

  ∂ t

  − ∇ ϕ

  Gauge invariance demands that one can change the potential, by adding the space–time derivative of any arbitrary function f (xi ) without changing these fields, as given by

  A ′

  i

  =

  A i

  +

  ∂ f

  ∂

  x i

  . It is very simple to see that the forms of

  E →

  and

  H →

  will not change as the extra term in

  H →

  is a Curl of a gradient which is zero, and in

  E →

  , the extra term cancels, between the two expressions after taking into consideration the Lorentz signature of the four–metric ηij . In section 3.4.1

  Sign up to read

  Learn more about book