mburns's Journal: A Primer on the Use of Vectors

It was Feynmann who misunderstood electromagnetic theory in his Lectures. It was an inability to properly visualize and diagram the E and B fields which caused the difficulty. His incorrect use of vectors caused him to think (visually) of the E and B fields as ephemeral under observation from observers with different motion. Now, the vector representation is indeed ephemeral because it is not correct.

But, by comparison, the correct 2-form representation of the Faraday tensor is as steady as a rock under transform, not ephemeral. The 2-sided egg-crate structure of a 2-form is as easy to imagine as a vector, and it is also much more accurate. See GRAVITATION, Chapter 4 for pretty pictures of electromagnetism.

So, when Feynmann is humble about the possible understanding of the universe, I can only think of the problems in understanding that can be easily fixed and improved.

It turns out that correct visualization and diagramming of physical objects, fields, and waves and such, is possible. This is even a foundational principle in itself. The conventional name for this is the principle of general covariance. But, the new name should be geometric reality; things are "geometric objects".

The beginning-level example for general covariance is the proper diagramming of a cross product. It is a bivector that is diagrammed with a pair of vectors, not one vector. See GRAVITATION again. I think it is vital to acquire the bivector concept as a necessary prerequisite for acquiring the concepts of the other geometric forms that are exhibited in field theory.

Neglect of the principle of general covariance has led to a malaise in the study of field theory, I think. But, diligence leads to the adoption of the Einstein-Davis and Kaluza-Klein ideas, and to the rejection of the cosmological constant and of many features of string theory.

And again, vectors are just so inadequate for the task that they are assigned to. In most cases of a given number of dimensions, the total number of components of a vector representation fail to match the total number of components of a cross product. In three dimensions, the number of independent components match, but this is only a coincidence. This is true in three dimensions only, and the nonlinearity of the scaling is an unacceptable difficulty in every number of dimensions.

So, to correctly diagram a tensor, each factor of the diagram must be linear in the scaling of that particular dimension and direction; this is the principle of general covariance. And, a vector can only represent correctly first rank contravariant tensors with a dimensionality of L, namely spacetime intervals, but also 4-momentum in the special case of classical mechanics. Nothing else is ever correctly thought of as a vector.

--
Michael J. Burns

Vectors

[mercedo]

It turns out that correct visualization and diagramming of physical objects, fields, and waves and such, is possible. This is even a foundational principle in itself.

True. But it's extremely difficult and almost impossible. It is almost impossble since nature unknown to us is always ahead of represented by these semantics.

The conventional name for this is the principle of general covariance. But, the new name should be geometric reality; things are "geometric objects".

I don't think it's very meaningful