Knots And Strings 17-Jul-90

Theories of everything have to be able to link together all the forces of nature (gravity, electro-magnetism, and the strong and weak nuclear forces), and to do so in a framework compatible with general relativity and quantum theory.

In the 1930s, Quantum Electrodynamics (QED) was developed, which combined electrons with photons as the mediators of the electro-magnetic force.

The 'Electro-Weak' theory of Glashow, Weinberg and Salam in the late 60s and early 70s linked together electro-magnetism and the weak nuclear force, Quantum Chromodynamics (QCD) used quark theory to nicely explain the strong force, and attempts were made to combine these in Grand Unified Theories (GUTs). Unfortunately, GUTs seem easier to formulate than to obtain evidence for; this is mainly due to the very high energies and long life times associated with these theories.

Combining all the four forces is something that has proved to be very intractable, since severe mathematical difficulties arise when attempts are made to quantise relativity theory. These difficulties were effectively inherited from earlier classical theories, because 'point like' particles - which was the assumed structure of the electron - give rise to infinite values of mass and charge. Initially, some clever mathematical juggling in the shape of renormalization eventually managed to avoid these (and other) problems in QED, but they remain a thorn in the mathematical flesh of other quantised field theories.

String theory seems potentially to offer a way out of the dilemma, since it is possible to formulate it in such a way as to eliminate the mathematical problems when dealing with gravity. In fact, string theory developed from an early attempt by Gabrielle Veneziano to deal with very short lived Hadrons. Workers in this field were puzzled by unwanted solutions which described a zero mass particle, it was only much later that it was realised that this unwanted particle could be a 'graviton' - the particle which mediates the force of gravity.

The 'Super' in Super strings refers to the use of 'Super Symmetry', which developed as a mathematical method of using a number of additional dimensions to match up Fermions with Bosons. Without this symmetry, there seems to be no underlying connection between the Fermions (particles with odd spin), and Bosons (particles with even spin).

The problem is that string theorists seem uncertain at present exactly how many extra dimensions are needed to really describe reality. Early work along these lines were content with a mere ten space dimensions (eleven including time), then it seemed that twenty six dimensions were needed to describe Bosons, later on the number went down to nine (plus one) when Fermions were added to the Bosons. Even then, the symmetries were too big, generating an unwarranted number of particles, so now there are the 'Heterotic' string theories (does anyone else remember 'Heterodyne' Radio Receivers ? - and 'Superhet'), which actually combine the theories with differing numbers of dimensions into one (by internalizing the extra dimensions). This way it seems as if good old four dimensional space time may be enough after all. As if this wasn't enough, string theorists are also unsure whether strings are open, or closed up in little loops. The general feeling seems to be that they are closed, but then again...

The plethora of theories is redeemed somewhat by the observation that they are actually differing solutions based on a limited set of theory types.

Other ideas to come (partially) out of string theory are the possible existence of 'shadow matter' which would have a gravitational effect on the real stuff, but would other wise be undetectable, and maybe a fifth force of nature (as if they haven't got enough trouble with four). The latest experimental results seem to rule out the fifth force, but the jury is still out on shadow matter. Results from the new super high energy colliders are keenly awaited to see if they can throw light in the shadows.

Recent developments have involved the exploration of two dimensional surfaces swept out by strings, from which a natural physics of 2 D space time emerge. Other groups are working on two dimensional membranes (or many dimensional 'p-branes'), which are very tricky mathematically. In the other direction, something of a breakthrough has been made by formulating an exact solution (in place of the normal series approximation methods) for strings of less than one dimension - the idea of fractional or fractal dimensions has been explored by Benoit Mandelbrot.

So much for the introduction to string theories, now for some ideas of my own for your entertainment and delectation.

I don't know about you, but in my experience, strings are very good at getting themselves tangled up. In fact I would like to postulate a corollary to Murphy's Law which is that 'Unattended strings instantly tie themselves in knots'.

Now, as far as I know, none of the string theories actually mention knots, which I find very surprising. Apparently, all that these strings do is wiggle about in a (possibly infinite) number of vibration modes, the base modes corresponding to the common or garden elementary particles. The wiggling goes on in the however many there are extra dimensions, but not apparently in Time. Wiggling in Time is something that may come easy to Michael Jackson, but seems to strike physicists dumb with horror. Personally, I can take it or leave it; I mean, if space time itself can be quantized, or stringy, or 'foamy' (as physicist John Wheeler describes it in his 'Jiffy World' description of Planck scale space time), then I am quite prepared to believe in things wobbling about it time.

To get back to knots, there is talk of strings possibly winding themselves round in little coils, which sounds more like the kind of string I come across, and possibly having twists in it. But what of knots I ask ? Knots do, as it happens, form groups; and as we all know, groups are heavily involved in all manner of fundamental particle physics (including string theory).

Another item of interest that rapidly gets itself into tangled knots is a Rubik Cube. So what, do I hear you ask? Well, I can recommend you to Douglas Hofstadter's 'Metamagical Themas'. This book was his follow on to 'Godel, Escher and Bach', and is a collection of articles written for Scientific American, with much feedback from readers. A chapter called 'Magic Cubology', describes how move sequences of a Rubik Cube form groups, and that there can be drawn a surprisingly close match between states of a Rubik Cube, and the known zoo of fundamental particles. One Solomon W. Golomb (a mathematician) identified quarks, and anti quarks, as 1/3 twists of a corner 'cubie' (one of the 27 components of the cube). It turns out that no sequence of moves can twist a corner cubie by 1/3, and leave all other cubies unchanged - hence quark confinement. More than this, it is possible to give two corner cubies opposite 1/3 twists, or three corners 1/3 twists in the same direction. Thus a cube with oppositely twisted corners is equated to a meson (quark - anitiquark pair), and one with three corners twisted in the same direction is a baryon (a quark triplet).

In a further chapter entitled 'On Crossing the Rubicon' (he is fond of puns), observations on a modified cube by Tony Durham shows how twists correspond to baryon number. Twist sequences performed on corners are equated to protons, neutrons and three types of meson. Durham's cube model accounts for spin, integral electric charge, up and down quarks, and the weak and strong nuclear forces. OK, it doesn't account for the conservation of charge and baryon number, nor does it have quark 'colours', but then nothing's perfect.

Now what, I ask you, in the world of strings could correspond to all this twisting and tangling thats going on in a Rubik Cube ? - Knots, of course, that's what. As a mere layman, it seems quite obvious to me that strings without knots are like spaghetti without bolognese. There are, I admit, a few snags; for example, knots are somewhat static items to describe what increasingly seems to be the dynamic nature of ultimate reality (here I refer the interested reader to Fritjof Capra's 'The Tao of Physics' - a book popular with many people, but not other physicists). But yet, I see no reason why knots shouldn't wiggle about as much as strings do - after all, how come they got knotted in the first place ? In any event, I think its time that knots took their rightful place in fundamental physics; I look forward to hearing from the theoretical boys in due course.

Most of the stuff on strings came from 'Superstrings' by Paul Davies et al., but I also had quite a long chat with Michael Green, who is one of the U.K.'s top string men.

Apparently, the p-brane idea is now flavour of the month in string land, and has been generalised to M-theory. Personally, I do not think the end is in sight just yet.