Classical Electromagnetism

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Equations Of Motion ◽  
Classical Mechanics ◽  
Classical Field Theory ◽  
Scalar Fields ◽  
Local Symmetry ◽  
Momentum Tensor ◽  
Global Symmetry ◽  
Complex Scalar ◽  
Symmetry Properties ◽  
Time Variations

In this chapter, Noether’s theorem as a classical field theory is presented and the properties of variations are again discussed for fields (i.e. field variations, space variations, time variations, spacetime variations), resulting in the Noether condition. Quasisymmetries and spontaneous symmetry breaking are discussed, as well as local symmetry and global symmetry. Following these definitions, Noether’s first theorem and Noether’s second theorem are developed. The classical Schrödinger field is investigated and the key equations of classical mechanics are summarised into a single Lagrangian. Symmetry properties of the field action and equations of motion are then compared. The chapter discusses the energy–momentum tensor, the Klein–Utiyama theorem, the Liouville equation and the Hamilton–Jacobi equation. It also discusses material science, special orthogonal groups and complex scalar fields.

Elements of Classical Field Theory

2021 ◽  
pp. 24-34
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Classical Field Theory ◽  
Classical Field ◽  
Lagrange Equations ◽  
Lie Group Of Transformations ◽  
Infinitesimal Transformations ◽  
Conserved Currents

The purpose of this chapter is to recall the principles of Lagrangian and Hamiltonian classical mechanics. Many results are presented without detailed proofs. We obtain the Euler–Lagrange equations of motion, and show the equivalence with Hamilton’s equations. We derive Noether’s theorem and show the connection between symmetries and conservation laws. These principles are extended to a system with an infinite number of degrees of freedom, i.e. a classical field theory. The invariance under a Lie group of transformations implies the existence of conserved currents. The corresponding charges generate, through the Poisson brackets, the infinitesimal transformations of the fields as well as the Lie algebra of the group.

Canonical quantization of free fields

2021 ◽  
pp. 70-98
Author(s):  
Iosif L. Buchbinder ◽  
Ilya L. Shapiro
Keyword(s):  
Field Theory ◽  
Quantum Theory ◽  
Electromagnetic Fields ◽  
Classical Theory ◽  
Canonical Quantization ◽  
Classical Mechanics ◽  
Scalar Fields ◽  
Complex Scalar ◽  
Spinor Fields ◽  
Free Fields

This chapter discusses canonical quantization in field theory and shows how the notion of a particle arises within the framework of the concept of a field. Canonical quantization is the process of constructing a quantum theory on the basis of a classical theory. The chapter briefly considers the main elements of this procedure, starting from its simplest version in classical mechanics. It first describes the general principles of canonical quantization and then provides concrete examples. The examples include the canonical quantization of free real scalar fields, free complex scalar fields, free spinor fields and free electromagnetic fields.

scholarly journals A General Method for Transforming Nonphysical Configurations in BPS States

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
J. R. L. Santos ◽  
A. de Souza Dutra ◽  
O. C. Winter ◽  
R. A. C. Correa
Keyword(s):  
Scalar Field ◽  
Condensed Matter ◽  
Equations Of Motion ◽  
Cosmological Parameters ◽  
Topological Defects ◽  
Scalar Fields ◽  
Complex Scalar ◽  
Complex Scalar Field ◽  
Bps States ◽  
General Method

In this work, we apply the so-called BPS method in order to obtain topological defects for a complex scalar field Lagrangian introduced by Trullinger and Subbaswamy. The BPS approach led us to compute new analytical solutions for this model. In our investigation, we found analytical configurations which satisfy the BPS first-order differential equations but do not obey the equations of motion of the model. Such defects were named nonphysical ones. In order to recover the physical meaning of these defects, we proposed a procedure which can transform them into BPS states of new scalar field models. The new models here founded were applied in the context of hybrid cosmological scenarios, where we derived cosmological parameters compatible with the observed Universe. Such a methodology opens a new window to connect different two scalar fields systems and can be implemented in several distinct applications such as Bloch Branes, Lorentz and Symmetry Breaking Scenarios, Q-Balls, Oscillons, Cosmological Contexts, and Condensed Matter Systems.

scholarly journals Gauge-independent Theory of Symmetry. I.

1964 ◽  
Vol 17 (4) ◽  
pp. 431 ◽  
Author(s):  
LJ Tassie ◽  
HA Buchdahl
Keyword(s):  
Electromagnetic Field ◽  
Single Particle ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Conserved Quantities ◽  
Continuous Symmetry ◽  
Symmetry Properties ◽  
Symmetry Transformations ◽  
Hamiltonian Forms

The invariance of a system under a given transformation of coordinates is usually taken to mean that its Lagrangian is invariant under that transformation. Consequently, whether or not the system is invariant will depend on the gauge used in describing the system. By defining invariance of a system to mean the invariance of its equations of motion, a gauge-independent theory of symmetry properties is obtained for classical mechanics in both the Lagrangian and Hamiltonian forms. The conserved quantities associated with continuous symmetry transformations are obtained. The system of a single particle moving in a given electromagnetic field is considered in detail for various symmetries of the electromagnetic field, and the appropriate conserved quantities are found.

REMARKS ON THE MODEL WITH LOCAL U(1)V×U(1)A SYMMETRY IN TWO DIMENSIONS

1991 ◽  
Vol 06 (14) ◽  
pp. 1291-1298 ◽  
Author(s):  
YOSHIYUKI WATABIKI
Keyword(s):  
Energy Momentum Tensor ◽  
Central Charge ◽  
Axial Vector ◽  
Local Symmetry ◽  
Momentum Tensor ◽  
Two Dimensions ◽  
Global Symmetry ◽  
Supersymmetric Extension ◽  
Commutator Algebra ◽  
Local Vector

We investigate a 2-dimensional model which possesses a local vector U (1)V and axial vector U (1)A symmetry. We obtain a general form of Lagrangian which possesses this local symmetry. We also investigate the global symmetry aspects of the model. The commutator algebra of the energy-momentum tensor and the currents is derived, and the central charge of the model is calculated. Supersymmetric extension of the model is also studied.

scholarly journals PALATINI FORMULATION OF MODIFIED GRAVITY WITH A NON-MINIMAL CURVATURE-MATTER COUPLING

2011 ◽  
Vol 26 (20) ◽  
pp. 1467-1480 ◽  
Author(s):  
TIBERIU HARKO ◽  
TOMI S. KOIVISTO ◽  
FRANCISCO S. N. LOBO
Keyword(s):  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Scalar Fields ◽  
Momentum Tensor ◽  
Nonlinear Dependence ◽  
Modified Theories Of Gravity ◽  
Field Equations ◽  
Test Particles ◽  
Theories Of Gravity ◽  
Minimal Curvature

We derive the field equations and the equations of motion for scalar fields and massive test particles in modified theories of gravity with an arbitrary coupling between geometry and matter by using the Palatini formalism. We show that the independent connection can be expressed as the Levi–Cività connection of an auxiliary, matter Lagrangian dependent metric, which is related with the physical metric by means of a conformal transformation. Similarly to the metric case, the field equations impose the nonconservation of the energy–momentum tensor. We derive the explicit form of the equations of motion for massive test particles in the case of a perfect fluid, and the expression of the extra-force is obtained in terms of the matter-geometry coupling functions and of their derivatives. Generally, the motion is non-geodesic, and the extra force is orthogonal to the four-velocity. It is pointed out here that the force is of a different nature than in the metric formalism. We also consider the implications of a nonlinear dependence of the action upon the matter Lagrangian.

scholarly journals SCALAR NONLUMINOUS MATTER IN GALAXIES

2002 ◽  
Vol 17 (04) ◽  
pp. 555-560 ◽  
Author(s):  
BYUNG JOO LEE ◽  
TAE HOON LEE
Keyword(s):  
Rotational Velocity ◽  
Scalar Potential ◽  
Scalar Fields ◽  
Global Symmetry ◽  
Late Time ◽  
Complex Scalar ◽  
Acceleration Of The Universe ◽  
The Universe ◽  
Static Space ◽  
Quintessence Model

As a candidate for dark matter in galaxies, we study an SU(3) triplet of complex scalar fields which are nonminimally coupled to gravity. In the spherically symmetric static space–time where the flat rotational velocity curves of stars in galaxies can be explained, we find simple solutions of scalar fields with SU(3) global symmetry broken to U(1) × U(1), in an exponential scalar potential, which will be useful in a quintessence model of the late-time acceleration of the universe.

The classical scalar field

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Scalar Field ◽  
Symmetry Breaking ◽  
Free Field ◽  
Scalar Fields ◽  
Global Symmetry ◽  
Lagrange Equations ◽  
Complex Scalar ◽  
Klein Gordon ◽  
Complex Scalar Field ◽  
The Fourier Transform

This chapter explains some of the properties of scalar fields, which are paradigmatic in relativistic field theory. It also shows how a complex scalar field can confer an effective mass to a ‘gauge’ field. The chapter first provides the Klein–Gordon equation derived from the Euler–Lagrange equations outlined in the previous chapter. It then illustrates the Fourier transform of a free field, before embarking on further discussions on complex fields, charge, and symmetry breaking. Finally, this chapter considers that the fact that global symmetry breaking leads to the appearance of a massless, and therefore long-range, scalar field is problematic because such a field is not observed experimentally. It thus takes a look at the BEH mechanism (named after its inventors, Robert Brout, François Englert, and Peter Higgs), which can make it ‘disappear’.

The ponderomotive four-momentum

1998 ◽  
Vol 76 (1) ◽  
pp. 87-94 ◽  
Author(s):  
J Gao ◽  
D Bagayoko ◽  
D -S Guo
Keyword(s):  
Light Propagation ◽  
Mass Shell ◽  
Equations Of Motion ◽  
Critical Role ◽  
Rest Mass ◽  
Classical Mechanics ◽  
Classical Field Theory ◽  
Free Form ◽  
Electron Mass ◽  
Time Component

Using classical mechanics and classical field theory, we confirm that the ponderomotive momentum exists as a counterpart of the ponderomotive energy to form the ponderomotive four-momentum when an electron interacts with a light plane wave. As a solution to the equations of motion, a time-dependent four-momentum on the electron (rest) mass shell is obtained in a coordinate-free form. By taking out the quiver motion, we find that the nonoscillating part of the four-momentum is not on the electron mass shell. We show mathematically that the decomposition of the nonoscillating four-momentum into an on-mass-shell four-momentum and a lightlike four-momentum along the light-propagation direction is unique. The time-component of the lightlike four-momentum is exactly the ponderomotive energy, while the space-component is the ponderomotive momentum. We discuss the critical role of the ponderomotive four-momentum in light–electron scattering processes due to the lightlike property and the uniqueness of the decomposition of a four-momentum. Two pieces of experimental evidence of the ponderomotive momentum are identified in the discussion. PACS No. 32.80Rm

scholarly journals On the localized quantum oscillators in a common heat bath

2017 ◽  
Vol 31 (15) ◽  
pp. 1750122
Author(s):  
Z. Nasr ◽  
F. Kheirandish
Keyword(s):  
Quantum Dynamics ◽  
Equations Of Motion ◽  
Canonical Quantization ◽  
Heat Bath ◽  
Scalar Fields ◽  
Strong Coupling Regime ◽  
Total System ◽  
Complex Scalar ◽  
Fluctuation Dissipation ◽  
Starting Point

To remedy the failure of minimal coupling method in describing the quantum dynamics of two localized Brownian oscillators interacting with a common medium, a scheme is introduced to model the medium by a continuum of complex scalar fields or equivalently two independent real scalar fields. The starting point is a Lagrangian of the total system and quantization is achieved in the framework of canonical quantization. The equations of motion, memory or response functions and fluctuation–dissipation relations are obtained. An induced force between oscillators is obtained originating from the fluctuations of the medium. Ohmic and non-Ohmic regimes are discussed and the positions of oscillators are obtained approximately in large time limit and weak coupling regime. Quantum entanglement between localized oscillators is obtained in zero temperature and strong coupling regime.

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