Local gauge transformation

Now the Lagrangean associated with the nuclear motion is not invariant under a local gauge transformation. Eor this to be the case, the Lagrangean needs to include also an interaction field. This field can be represented either as a vector field (actually a four-vector, familiar from electromagnetism), or as a tensorial, YM type field. Whatever the form of the field, there are always two parts to it. First, the field induced by the nuclear motion itself and second, an externally induced field, actually produced by some other particles E, R, which are not part of the original formalism. (At our convenience, we could include these and then these would be part of the extended coordinates r, R. The procedure would then result in the appearance of a potential interaction, but not having the field. ) At a first glance, the field (whether induced internally... 

We do not include fields s and p into L-factor since they transform homogenously under local gauge transformations. 

The effect of a local gauge transformation (Sction II) on the classical B(3 field is described as... 

However, in special relativity, the number A is a function of the spacetime coordinate xP. This property defines the local gauge transformation... 

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

The derivation of Eq. (218) from Eq. (206) follows from local gauge invariance, and it is always possible to apply a local gauge transform to the vector A, the Maxwell vector potential. The ordinary derivative of the d Alembert wave equation is replaced by an 0(3) covariant derivative. The U(l) equivalent of Eq. (218) in quantum-mechanical (operator) form is Eq. (13), and Eq. (212) is the rigorously correct form of the phenomenological Eq. (25). It can be seen that Eq. (212) is richly structured in the vacuum and must be solved numerically. The vacuum currents present in Eq. (218) can be computed from the right-hand side of the wave equation (212), and these vacuum currents follow from local gauge invariance. 

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... 

So it becomes clear that the description of the vacuum in gauge theory can be developed systematically by recognizing that, in general, A is an -dimensional vector. On the U(l) level, it is one-dimensional on the 0(3) level, it is three-dimensional and so on. The internal gauge space in this development is a physical space that can be subjected to a local gauge transform to produce physical vacuum charge current densities. 

So in the general case where A is an -dimensional vector [46], a local gauge transform on this vector is represented in the vacuum by... 

So the product gMaA is the result of special relativity in the vacuum, and g is adjusted for correct units. Ryder [46] simply describes A as an additional held or potential Feynman describes it as the universal influence. Therefore, as argued in the foregoing section, both the potential and the electromagnetic held in the vacuum originate in local gauge transformation, which, in turn, originates in special relativity itself. 

It has been demonstrated already that local gauge transformation on this Lagrangian leads to Eq. (153), which contains new charge current density terms due to the Higgs mechanism. For our present purposes, however, it is clearer to use the locally invariant Lagrangian obtained from Eq. (325), specifically... 

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... 

The action is therefore not invariant under local gauge transformation. To restore invariance the four potential, A must be introduced into the pure gauge vacuum to give the Lagrangian... 

The action is still not invariant under a local gauge transformation, however, because of the presence of the term - g y A on the right-hand side of Eq. (935), a term in which... 

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... 

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