Lagrangian gauge invariance

The electromagnetic field may now formally be interpreted as the gauge field which must be introduced to ensure invariance under local U( 1) gauge transformation. In the most general case the field variables are introduced in terms of the Lagrangian density of the field, which itself is gauge invariant. In the case of the electromagnetic field, as before,... 

A condition is imposed on one of the four components of so that there are only three free components. However, the Lagrangian leading to the Proca equation is not gauge invariant due to the presence of a mass term [15]... 

It is well known that the Proca equation [6], Eq. (809), for a massive photon is not gauge-invariant because the Lagrangian (827) corresponding to it is not gauge-invariant. In SI units, this Lagrangian is... 

In formulating QED a least-action principle involving a Lagrangian is often used [9,18,20]. This involves the potentials in various forms. Not only is relativistic invariance (Lorenz potentials) desired, but also gauge invariance. At least in the current state of QED, gauge invariance is included as a fundamental part [21,22]. 

The Lagrangian (109) is not gauge-invariant, so Eq. (113) is not gauge-invariant. However, the foregoing illustrates the method of functional variation that will be used throughout this section. 

Considering a local gauge transformation of the Lagrangian (145) produces the gauge-invariant Lagrangian ... 

The only assumption therefore is that the Maxwell vector potential A exists in the physical internal space of 0(3) symmetry. The gauge-invariant Lagrangian (205) can be developed as... 

The total Lagrangian if I X I if2 is now invariant under the local gauge transformation because of the introduction of the 4-potential A, which couples to the current of the complex A of the pure gauge vacuum. The field A also contributes to the Lagrangian, and since if + ifj + if2 is invariant, an extra term if3 appears, which must also be gauge-invariant. This can be so only if the electromagnetic field is introduced... 

This paper presents an account of the dynamics of electric charges coupled to electromagnetic fields. The main approximation is to use non-relativistic forms for the charge and current density. A quantum theory requires either a Lagrangian or a Hamiltonian formulation of the dynamics in atomic and molecular physics the latter is almost universal so the main thrust of the paper is the development of a general Hamiltonian. It is this Hamiltonian that provides the basis for a recent demonstration that the S-matrix on the energy shell is gauge-invariant to all orders of perturbation theory. 

Because field quantization falls outside the scope of the present text, the discussion here has been limited to properties of classical fields that follow from Lorentz and general nonabelian gauge invariance of the Lagrangian densities. Treating the interacting fermion field as a classical field allows derivation of symmetry properties and of conservation laws, but is necessarily restricted to a theory of an isolated single particle. When this is extended by field quantization, so that the field amplitude rjr becomes a sum of fermion annihilation operators, the theory becomes applicable to the real world of many fermions and of physical antiparticles, while many qualitative implications of classical gauge field theory remain valid. 

The Lagrangian is invariant under the global gauge transformation... 

One of the most important properties of the Lagrangian (2.1) is its gauge invariance A gauge transformation of the photon field,... 

It is useful for the following discussion to consider the symmetries of the Lagrangian (2.1) in order to analyse the conservation laws of a system characterised by (2.1) on the most general level, i.e. without further specifying F", and their consequences for the structure of a density functional approach to (2.1). We first consider continuous symmetries which in the field theoretical context are usually discussed on the basis of Noether s theorem (see e.g. [26, 28]). The most obvious symmetry of the Lagrangian (2.1), its gauge invariance (2.9), directly reflects current conservation,... 

An immediate consequence of the local gauge invariance of the Lagrangian is current conservation,... 

This interaction Lagrangian density may depend explicitly on the space-time coordinates x and the 4-velocity u via the charge-current density T. However, as far as only the equation of motion for the electrodynamic field is concerned they do not represent dynamical variables. Lorentz invariance of this interaction term is obvious, and gauge invariance of the corresponding action is a direct consequence of the continuity equation for the charge-current density f, cf. Eq. (3.162),... 

In this chapter we examine some of the problems inherent in the current-current form of the weak interaction Lagrangian, and are led to the idea of the weak force being mediated by the exchange of vector mesons. This picture too runs into diflSiculties which, however, can be alleviated in gauge-invariant theories. The latter are introduced and discussed at some length. 

We now turn to the much more subtle question of local non-Abelian gauge invariance. The first generalization of SU 2) to locally gauge invariant Lagrangians is due to Yang and Mills (1954) [a detailed account can be found in Taylor (1976)], but the treatment applies to any group with a finite number of generators [see Abers and Lee (1973)]. 

According to Section 2.3 the locally gauge invariant Lagrangian for the coupling of the gauge bosons (GB) to the scalars (5) is... 

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