Gauge Transformations of the Lagrangian

It is immediately verified that L and L yield the same equations of motion by respecting the chain rule. 

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

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

The second fundamental symmetry property of the Lagrangian is established by local gauge transformations of the form... 

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,... 

Under the local gauge transformation (226) of the Lagrangian (219), the action is no longer invariant [46], and invariance must be restored by adding terms to the Lagrangian. One such term is... 

Here,, 4 ( is the vector 4-potential introduced in the vacuum as part of the covariant derivative, and therefore introduced by spacetime curvature. The electromagnetic field and the topological charge g are the results of the invariance of the Lagrangian (868) under local U(l) gauge transformation, in other words, the results of spacetime curvature. 

The effect of the local gauge transform is to introduce an extra term 8M A in the transformation of the derivatives of fields. Therefore, 8 A does not transform covariantly, that is, does not transform in the same way as A itself. These extra terms destroy the invariance of the action under the local gauge transformation, because the change in the Lagrangian is... 

This method has the advantage of formulating the theory in terms of the (observable) field intensities. The more usual procedure which starts from a lagrangian formulation22 expresses the theory in terms of the potentials Alt(x). A gauge transformation... 

U(l), whose group space is a circle. This result is another internal inconsistency, because the group space of a gauge theory is a circle, there can be no physical quantity in free space perpendicular to that plane. It is necessary but not sufficient, in this view, that the Lagrangian in U(l) field theory be invariant [6] under U(l) gauge transformation. 

Therefore the fact that 9 is arbitrary in U(l) theory compels that theory to assert that photon mass is zero. This is an unphysical result based on the Lorentz group. When we come to consider the Poincare group, as in section XIII, we find that the Wigner little group for a particle with identically zero mass is E(2), and this is unphysical. Since 9 in the U(l) gauge transform is entirely arbitrary, it is also unphysical. On the U(l) level, the Euler-Lagrange equation (825) seems to contain four unknowns, the four components of , and the field tensor H v seems to contain six unknowns. This situation is simply the result of the term 7/MV in the initial Lagrangian (824) from which Eq. (826) is obtained. However, the fundamental field tensor is defined by the 4-curl ... 

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... 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

Chemical behaviour depends on chemical potential and electromagnetic interaction. Both of these factors depend on the local curvature of space-time, commonly identified with the vacuum. Any chemical or phase transformation is caused by an interaction that changes the symmetry of the gauge field. It is convenient to describe such events in terms of a Lagrangian density which is invariant under gauge transformation and reveals the details of the interaction as a function of the symmetry. The chemically important examples of crystal nucleation and the generation of entropy by time flow will be discussed next. The important conclusion is that in all cases, the gauge field arises from a symmetry of space-time and the nature of chemical matter and interaction reduces to a function of space-time structure. 

The aim is to introduce as many vector fields W x), gauge fields that are the analogue of the photon field A, as is necessary in order to construct a Lagrangian which is invariant under the local gauge transformations specified by Bj x). 

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