Tensor field strength

We shall explore how the electric and magnetic fields fit into the explicitly covariant framework developed so far. Since the fields E and B are physical observables and therefore gauge invariant quantities, we have to construct a gauge invariant quantity based on the gauge field A. The simplest choice is the so-called yield strength tensor F = (fl ) defined by 

As for every contravariant Lorentz tensor its indices can be lowered by application of the metric g, 

The four-dimensional totally antisymmetric (pseudo-)tensor been 

In terms of the dual field strength tensor the homogeneous Maxwell equations 

For V = 0 the homogeneous Maxwell equation (3.175) states the absence of magnetic monopoles, cf. Eq. (2.97) and for v = i this equation represents Faraday s law of induction. 

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The weak angles are defined trigonometrically by the terms g/(g2 + g 2) and g /(g2 + g2). This means that the field strength tensor satisfies... 

Here G"v and F v are elements of the field strength tensors for the two SU(2) principal bundles. So far the theory is entirely parallel to the basic standard model of electroweak intereactions. In further work the Dirac and Yukawa Lagrangians that couple the Higgs field to the leptons and quarks will be included. It will then be pointed out how this will modify the B field. The < )4 field may be written according to a small displacement in the vacuum energy ... 

If these new fields are inserted in the Langrange density (30) and the field strength tensors are expanded, then one obtains the following Lagrange density [74]... 

We close this section on the field strength tensor by a discussion of two important invariants of the electromagnetic field, which are both Lorentz and gauge invariant. They can be evaluated by direct calculation and read. 

We are now in a position to discuss the transformation behavior of electric and magnetic fields under a change of the coordinate system, i.e., their transformation properties under Lorentz transformations. Since the electric and magnetic fields do not represent Lorentz 4-vectors, we have to analyze the transformation property of the field strength tensor Fi instead. 

Upon reflection from the surface, the local field strength tensor L has to be... 

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