Re: [Phys-L] Special Relativity for QFT

[John Denker <jsd@av8n.com>]

On 07/20/2018 12:02 AM, David Craig wrote:

> students need to be able to recognize the equations they will find in
> field theory books, the majority of which contain many expressions of
> the form x^\mu=\Lambda^\mu_\nu x^\nu and so forth.

It might be worth revising that to emphasize that the
x-components on the LHS are not the same as the
x-components on the RHS ... even though the abstract
x-vector has the same physical meaning.

There is a super-fundamental rule of algebra that says
x means the same thing each time it appears.  This is
so fundamental that it is all-too-often left unstated
and taken for granted.

Possibly constructive suggestion:

	x@B^\mu=\Lambda^\mu_\nu x@R^\nu

where @B refers the the components using the blue
basis, while @R refers to the components using the
red basis.

It must be emphasized that the vector x is a first-class
physical and mathematical object unto itself, independent
of whatever reference frame (if any) you choose.  The
expansion of x in terms of components is frame-dependent,
but x itself is not.

You might hope students would have learned that at their
mother's knee, but you can't count on it ... and oftentimes
the textbook notation tramples on the distinction.

Also, some computer languages use the word "vector" to
refer to a bucket of components, even when the "vector"
is not a well-behaved physical object, and the components
do not have any well-behaved relationship to one another.

It's sad that the grade-school definition of vector
(something with direction and magnitude) is a more
sophisticated and more physical concept than the "bucket
of components" notion.  The goal should be to introduce
components without trampling on the underlying concept.