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In order to make a quantitative assessment with respect to a field of a given property, be it continuous, discrete or lattice, a determination must be made as to the norm or unit standard of its intensity or density, if a scalar, such as mass or energy, and/or length measure, if a vector/tensor field. Such determination can be made on the basis of experiment by applying an arbitrary external norm to the input and resulting data and possibly on a theoretical basis by applying a norm that is intrinsic to the system of the field. I think of the application of a theoretical norm as part of the process of fixing an effective gauge.

In the case of a lattice, the regular, minimum, maximum, or average, etc. spacing of the lattice, could serve as such norm.

In the case of freely movable, regular parts of a field of discrete entities, a similar measure of size, radius, etc, could serve as a norm. However, if the parts were point-like, i.e. without extension, then any equation using the inverse square law in evaluating the interaction of such parts would tend toward infinity as the parts approached coincidence. It is my assumption that this fact enters into the problem of renormalization of QED. The selection of an appropriate norm would not necessarily prevent this, but if that norm represented a limit of approach prior to coincidence of the parts at some finite maximum intensity or density, then it would.

In the case of a field of continuous, initially uniform density, absent any obvious source or sink, there is no local property that would provide a basis for an intrinsic norm, in particular if the field is infinite in extent. If the field is finite in extent, then there may be some constraint, geometric or topological, or some initial condition, that would provide a basis for a norm and therefore for gauging the field. As with a wave bearing medium, it might possesses a fundamental frequency of resonance, for example. In such case the norm and the process of gauge fixing could be a function of these constraints or initial condition and not (necessarily) subject to some perturbative mathematical process, as would appear to be the case if based on fluctuations in field density or the parameters of local sources or sinks in providing a norm for an field of infinite extent.

As I understand it, in general relativity, the metric is dependent on the choice of a length norm, which is itself normalized with the time dimension by the speed of light, while the mass is normalized or geometrized by the factor, G/c^2 or Newton’s gravitational constant divided by the square of the speed of light. In similar fashion, in electromagnetism (and perhaps the electroweak model?) the fine structure constant normalizes fundamental charge and electron rest mass as well as charge and current.

These factors, therefore, gauge or scale the interactions for gravity and electromagnetism. An attempt to model gravity and quantum theory in a unified field, then requires a common norm for gauging the separate interactions, ideally as an inherent aspect of the field.

Am I understanding these concepts properly?