Bookworms
 | If I were a Springer-Verlag Graduate Text in Mathematics, I would be W.B.R. Lickorish's An Introduction to Knot Theory. I am an introduction to mathematical Knot Theory; the theory of knots and links of simple closed curves in three-dimensional space. I consist of a selection of topics which graduate students have found to be a successful introduction to the field. Three distinct techniques are employed; Geometric Topology Manoeuvres, Combinatorics, and Algebraic Topology. Which Springer GTM would you be? The Springer GTM Test |
posted by Kea | 1:18 PM
7 Comments:
I'm Robin Hartshorne's Algebraic Geometry. :)
I could have guessed!
I can't believe it. How did he do that. I ended up as "Foundations of Differentiable Manifolds and Lie Groups", a book that I either own or should. DM and Lie groups were my two favorite classes when I was a math grad student at the University of Washington. And it's the basis of geometric algebra.
The first four pages fromeach of the first four chapters in "An Introduction to Knot Theory" by W. B. Raymond Lickorish is readable online in the Google Book Search preview here. how it starts off, matter has to be something like a 1 dimensional (line-like) string in 3 spatial dimensions (plus time dimension(s)) in order to get around singularities.
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).
BTW, "problems" with particle spin in quantum mechanics are exaggerated and mainly occur when you make false assumptions. E.g. 't Hooft points out that an electron's equator would have to spin at a speed of 137*c if it's a sphere of classical electron radius. (The classical electron radius is clearly related to the IR cutoff range, not the UV cutoff or core size of a fermion.) In addition the amount of spin for fermions is widely cited as a crazy problem that is impossible to think about physically (half integer for fermions means they have a rotational symmetry when rotated by 720 degrees, like a Mobius strip). However, if the underlying entity in an electron core is a 1-d Heaviside-Poynting vector or electromagnetic energy current (which has electric and magnetic field lines both perpendicular to each other and to the direction of of the 1-d line of propagation which I'm taking to be the fundamental "string"), things work out well: the E field lines seen from a large distance obey Gauss' law, while the B field lines give a magnetic dipole as seen from a great distance, the motion of the energy which is the 1-d string gives spin, and a rotation of the plane of polarization as the energy travels around the small loop which is the electron core gives the electron half-integer spin because you have to wait for two revolutions to get back where you started (this is a little like a Mobius strip, a loop of paper with half a twist so a line drawn around it has a length of twice the circumference of the loop, because both "sides" are the joined, i.e. there is only one side). The usual arguments against physical understanding of quantum mechanics are the source of stringy error, because instead of trying to understand the real known physical problems, people think (wrongly) they are dead ends (following Bohr's philosophy, not Einstein's) and go down a real dead end instead - mainstream M-theory. E.g. the uncertainty principle is just the result of many body interference on electrons etc.: virtual particles appear in intense fields and deflect real particles severely on small (atomic, subatomic) distances, causing chaotic orbits. These virtual particles play the part of air molecules in the analogy with Brownian motion: air molecules make dust particles smaller than 5 microns jiggle chaotically, but on larger scales the effects cancel out statistically because an equal number of air molecules hits each side of a big rock! There's no cleverness here. Virtual particles which appear between IR and UV cutoff energies (i.e. distances from around 1 fermimetre down to grain size of the vacuum) introduce chaos. In the double-slit experiment, it's probably the electrons in the small slits which introduce the chaos. Feynman points this out in his book QED: in a small space, the individual feynman diagram interactions become important and introduce uncertainty, but on large scales the statistical sum or path integral involves large numbers of events and reduces to the classical approximation. This isn't rocket science, it's basic obvious stuff being blocked out by stupidity that not even Feynman's charm could bypass.
Hi Kea,
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
Hi Kea,
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.
Post a Comment
<< Home