Sunday, 17 June 2012

M338 TMA03

Finished this today had to ask for a small extension due to work, but don't know if it's been granted as my tutor hasn't replied to my request. Anyway I'll submit it tomorrow and see what happens.

This block was a bit more straightforward than Block A as it concerns what most people consider topology to be namely the invariance of certain features of a solid under a transformation. Thus the usual example given is that that a cup with a handle is topologically equivalent to a ring doughnut but not a solid doughnut as they both have a hole and can be deformed into each other. There is also a discussion of the Mobius band.

I have to admit that my heart was sinking when I first started this block as it seemed to be all about visualisation and introducing boundaries in order to classify various combinations of holes and twists in a surface. As there seemed to be no systematic way of doing this it was all a bit vague.

However by the second half of block 2 an algebraic method of charactersing surfaces was introduced and I was much happier. There are a number of ways of systematically obtaining the characteristic properties of a solid and these are essentially algebraic if a bit fiddly. The TMA was mostly concerned with the algebraic aspects of the block. There are three main characteristics of a surface

The Euler Characteristic = V - E +F where V is the number of vertices E = the number of edges and F = is the number of faces. This is invariant for topologically equivalent surfaces.
The Boundary number which is the number of separarate pieces forming the boundary of a surface
The orientability number of a surface essentially the number of twists.

Associated with each surface is an N sided  polygon with arrows along each side denoting the orientation with a number of edges of the polygon having the same name.  An edge expression is obtained by  labelling the sides in order for a given direction of the arrow and labelling the side by its inverse if the arrow is pointing in an opposite direction. Various identifications give rise to different solids. Given such an edge expression one can then go onto calculate the characteristic numbers of a surface.  This process is quite straightforward but can be quite fiddly.  Also the edge equations can be reduced to canonical form which enables the classification of the surface to be made easily both directly from the edge expression and also there is a method of classifying the solids in terms of the  connected sums of constituent surfaces eg Torus's Closed disk's etc.

The final block concerned a discussion of the coloring theorems. One of the key quantities is the chromaticiy number which is related to the number of handles of a given surface and gives the minimum number of colours required to colour the parts of the surface so that no regions next to each other have the same colour. As some readers will probably know the minimum number of colours to colour a map is 4. But this wasn't actually proven until the 1970's and only by a computer so it's doubtful to old fashioned mathematicians whether it counts as a proof at all.

Question 1 concerned identifying the edge expression  of a hexagon, obtaining the edge expressions and the
characteristic numbers. This question seemed quite straightforward

Question 2 was a question on the subdivisions of a surface for a given Euler characteristic this is quite straightforward

Question 3 The bulk of the TMA with a whopping 40% of the marks concerned obtaining a single edge expression from a number of constituent ones, then deducing the characteristic numbers. Then peforming a canonical transformation and obtained the connected sum form for the surface and then using that to deduce the characteristic numbers. Fortunately the numbers seemed to tie up so I'm consisitent if not correct.

Question 4 The final question involved obtaining some inequalities for the number of handles h in terms of the chromaticity number. This seemed quite straigthforward or at least the first part. However the last part which asked us to find the range of h for a large chromaticity numbers. On the face of it this seemed quite straightforward but the sting in the tail was that dreaded phrase "justifying your answers fully" as there were 10 marks for this part. I'm sure we were supposed to do more than solve the inequality for the various values of h. But I couldn't see what.

So on the whole I think I've done reasonably well assuming my tutor accepts my late submission but that last part has me worried that I've missed something quite fundamental.

Sunday, 3 June 2012

Pilot Waves for and against part 2

So as promised here is a basic summary of the pros and cons of De-Broglies Bohm's Pilot Wave approach to quantum mechanics. A really good summary of which can be found in these lectures by Mike Towler.

So here are the pros

 i) Contrary to the claim by the Copenhagen interpretation it has been shown that it is possible to define a definite trajectory for particles it's greatest success must be it's explanation of the two slit experiment. Sanity seems to be restored we no longer have to claim that a massive particle such as an electron or a bucky ball splits in two as it interacts with a screen which has a number of slits.

ii) It brings the physics back to quantum mechanics, instead of losing contact with the world of particles and their interactions as the Dirac Formalism is apt to, it does provide a causal explanation for many quantum phenomenon. Mind you the Dirac formalism is really elegant and still encapsulates the essence of many quantum mechanical problems even if it is all quite abstract.

iii) It avoids the problem of measurement, with all the inherent problems of the world around us being created by an act of measurement.

iv) It now deserves a place as an equivalent mathematical formulation of quantum mechanics (at least for Non relativistic problems). The fact that more and more papers are being published using Bohmian mechanics has to be something. I think it's fair to say that other alternative interpretations such as the Many Worlds Interpretation have not yet reached the same degree of maturity. I would argue that once a theory of physics has reached the stage where people can 'Shut up and calculate' as the Pilot Wave theory has done then it has a right to be treated as mainstream physics.

It is quite astonishing that at Solvay in 1927, when the basics of the theory was laid out by De-Broglie, that it wasn't considered at least as an alternative equivalent mathematical reformulation of quantum mechanics just as Heisenberg's matrix mechanics was.  All the formulations of quantum mechanics had their problems of interpretation and it is at least arguable that the Pilot Wave theory has a well defined procedure for relating it's mathematical formalism to empirical results.

Now for the cons

i) It makes great play, that for it the wavefunction is a real field, something akin to an electric field or gravitaional one. Unfortunately this implies acceptance of the reality of 3N+1 dimensional configuration space,  where N is the number of particles considered and 1 represents time. Let me explain a bit more the wave function of a manybody particle system is a function of all the coordinates of the particles considered eg  for two particles with positions r1 and r2 the time independent of the wave function of the system is now a function of r1 and r2 that is we have $$\psi(x1,y1,z1,x2,y2,z2)$$ this is quite different from classical physics, for example the electric  field produced by two charges, at a given point, is still a function of 3 dimensional space and not 6 dimensional space. The question then, for those who would see the wave function as a real field, is just what is the relationship between the 3N+1 dimensional space of the wave function of an N body system, (it's so called configuration space), and our 4 dimensional space time. If you claim as Towler seems to at the end of his 6th lecture, that this just a mathematical description then you cannot claim as your theory does, that the wavefunction of quantum mechanics is real, that removes one of the main motivations for the Pilot wave theory. Some clarity is required here.

ii) It seems not to be relativistically covariant, this would imply that Einstein's theory of relativity sits uneasily in this theory. I doubt whether many physicists would welcome back the introduction of a real ether and the replacement of Minkowski space time with preferred Lorentzian frames, the idea that bodies really do contract as they approach the speed of light, (rather than just being an artefact of the relative positions of two observers). See lecture 5 of the Towler lectures for more detail I for one am not convinced.

iii) As yet it seems to be difficult to extend it to relativistic particle physics especially the treatment of fermions that means for example all the current developments in particle physics are shut off in this interpretaton.

So overall I think the Pilot Wave theory, has definitely achieved quite a lot, but it still has a lot of catching up to do with the standard formulation of quautum mechanics. Does moving the problems in the interpretation of quantum mechanics to the relationship between the 3N+1 configuration space and our own 4 dimensional space time raise more questions than it answers. I don't know. Maybe these problems will be resolved at least the pilot wave theory  has earnt the right to be heard as an alternative to the standard view and I for one want to learn more about it.