I finally took the GTM test:

Saunders Mac Lane’s ‘Categories for the Working Mathematician’

“I provide an array of general ideas useful in a wide variety of fields. Starting from foundations, I illuminate the concepts of category, functor, natural transformation, and duality. I then turn to adjoint functors, which provide a description of universal constructions, an analysis of the representation of functors by sets of morphisms, and a means of manipulating direct and inverse limits.”

I tend to agree with the first sentence, but certainly did not realize that I may be performing the remaining actions.

I do like the thought of trying to illuminate concepts and providing universal construction descriptions.

1 – I have tried reading a few GTM texts and find them diificult to undertand.

I find non-GTM books easier to comprehend. Examples:

a – Terry Gannon, Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics (Cambridge Monographs on Mathematical Physics)

b – Avner Ash and Robert Gross, Fearless Symmetry: Exposing the Hidden Patterns of Numbers

c – almost anything by John Conway

2 – I recall that you have a background in electrical engineering.

A – I highly recommend reading Paul J Nahin [PhD EE, former chair, now emeritus UNH-US]. Consider:

‘Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills’.

I expected lots of phasor equations, but none.

He demonstrated Dirac [impulse], Heaviside [step] and other function ideas with calculus relying on that well known identity.

B – I found this website topic

‘Analogous Electrical and Mechanical Systems’

There is a table showing analogies beteen EE an ME type I (Force-Current) and type II (Force Voltage).

http://www.swarthmore.edu/NatSci/echeeve1/Ref/

Analogs/ElectricalMechanicalAnalogs.html

A particle with 0 dimensions would be a singularity and this would have UV divergence problems. The only physical model which makes sense is that a fundamental charge, e.g. a fermion’s innermost core (ignoring the surrounding vacuum particle creation-annihilation phenomena which affects the field) must have spatial extent, so it can’t be a 0-dimensional singularity. If it were a singularity, the energy density of the field at the centre would be infinite – allowing unphysically large virtual particles to pop into existence there (the UV divergence problem) with infinite momenta.

So looked at from a experimentally, observationally based viewpoint, clearly the core of a fermion has more than 0 spatial dimensions. The simplest case for it is to have 1 spatial dimensions, a “string-like” line.

I have no problem with investigating this at all. It’s rational, defensible physics. The mainstream goes wrong where it adds one time dimension to form a 2-d worldsheet and assume that resonate vibrations of the string (like energy levels) produce all the different possible particles, adding 8 more dimensions to include conformal field theory for supersymmetry. Instead of these speculations, people should stick to a 1-dimensional string and ask how to get it to model what we already know simply:

*how is particle spin derived? (i.e., can the 1-d be looped to form a spinning particle? yes it can!)

*can you get all known without adding unobservable extra spatial dimensions? (yes you can; vacuum polarization phenomena surrounding the particle core causes shielding and converts some of the energy of long ranged EM force into short ranged nuclear forces – when bringing 2 or 3 electron-like fermionic preons very close together into a hadron, they share the same polarized vacuum, which accounts for the difference in charge between say a downquark and an electron – you have EM field energy converted into that of weak isospin and QCD fields).