Clarification concerning gauge theory.

[hmg]

I am not sure about the proper forum for addressing this question, so I will start here as it concerns certain fundamental concepts about the nature of a norm (unit standard), gauge and metric as applied to various field theories, which I want to make sure I understand properly. The following is my understanding, which I hope you will correct as needed.

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?

 

[eljose79]

Thank you for your question and for seeking clarification on these fundamental concepts. I will do my best to provide a clear and accurate understanding of norms, gauges, and metrics as applied to field theories.

Firstly, a norm or unit standard is a fundamental concept in mathematics and physics. It is used to define a consistent measurement scale for a particular property or quantity. In the context of field theories, norms are used to determine the intensity or density of a field. This can be done through experimental measurements or through theoretical considerations.

In the case of a lattice, the regular spacing of the lattice can serve as a norm for determining the intensity or density of the field. Similarly, in the case of discrete entities, a measure of size or radius can serve as a norm. However, as you correctly pointed out, if these entities are point-like, then the application of the inverse square law can lead to infinite values. This is where the concept of renormalization comes into play, which is a mathematical technique used to remove these infinite values and make the calculations more physically meaningful.

For continuous fields, the situation is slightly different. In the absence of an obvious source or sink, there may not be a local property that can serve as a basis for an intrinsic norm. In this case, the norm may be determined by some constraint or initial condition, such as a fundamental frequency of resonance. This norm is then used to gauge the field and make calculations.

In general relativity, the metric is dependent on the choice of a length norm, which is then normalized with the time dimension by the speed of light. This is known as the spacetime interval. Similarly, in electromagnetism, the fine structure constant serves as a norm for fundamental charge and electron rest mass. These norms are used to gauge or scale the interactions for gravity and electromagnetism, respectively.

In the quest for a unified field theory, it is important to have a common norm that can gauge the separate interactions. This is a challenging problem and is an active area of research in theoretical physics.

In summary, norms, gauges, and metrics are essential concepts in field theories. They provide a consistent measurement scale for properties and quantities, and are used to make calculations and predictions. I hope this clarifies your understanding of these concepts. Please let me know if you have any further questions.