“It really is very strange that mathematics should describe our physical world so well.  There is after all no good reason why certain mathematical functions should so precisely describe what goes on in our physical world, unless there is of course some hidden link between these two sciences.  In fact some people find this link is so peculiar that they have written papers on the subject, as Eugene Wigner first did with  “The unreasonable effectiveness of mathematics in the physical sciences”.

Einstein is said to have remarked, “The most incomprehensible thing about the Universe is that it is comprehensible.” And I think this guy knew what he was talking about.

To quote Eugene Wigner:

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.”

Is this one of those cases where one introduces complexity when it isn’t really there, or is there something deep and meaningful here?  Why should mathematics be able to describe physical events so well?  As any Mathematician will tell you, the maths is already “out there” it has an existence of its own independent of us, all we do is occasionally turn over a new stone and find a new piece of maths that had always “been in existence” independent of us.  Likewise with our physical measurements and experiments, the results of these experiments has always “been out there” we just came along at this particular point in time to uncover some of them.

If you were to apply Occam’s Razor to this problem, where Occam’s Razor states that the simplest most logical answer is usually the right one – you might be led to conclude – as some people firmly believe, that the reason mathematics so “unreasonably” describes the “real” world we live in is because we really are “living” inside a computer simulation – the Matrix had it right all along!

Thank you for listening, have a good evening, and let’s hope the program doesn’t decide to crash tonight!”

In trying to show those poor wayward Quantum Scientists where they were going wrong, Einstein came up with a number of “thought experiments” which tested and probed Quantum Mechanics to its limits.  One such thought experiment involved the simultaneous emission of two oppositely polarised photons from a source (something which can be achieved, and which has actually been practically carried out in ground-breaking experimental work by Aspect et al) and then measuring the polarisation state of each photon when separated by a distance greater than that which would allow “communication” between the two photons during the polarisation measurement.  In other words the polarisation state of each photon was measured in a time shorter than that which would allow a photon to travel between the two polarised photons being measured.

For many years I didn’t think this was much of an experiment.  For conservation reasons the two photons will be emitted in opposite polarisation states from the source, so that if at some large separation distance I measure the polarisation state of photon A and then I measure the polarisation state of photon B it is hardly surprising that I find the polarisation states are separated by 90 degrees.  This is in fact true and correct and shows that both common-sense and Quantum Mechanics agree for this special case.  Now what I cannot simply explain is that if the polarisation measurements are made for angles other than 90 degrees (and in fact 45 degrees where again common-sense and Quantum Mechanics agree) we find a discrepancy between the common-sense expected result of the polarisation measurement and the Quantum Mechanical result.  How very odd!!  As mentioned above, this experiment has been carried out practically by Alain Aspect and his team, and the experimental results agreed with …………………………Quantum Mechanics.  How extremely odd!!!!!

Now this is not the first time that a Quantum Mechanical result has gone against “common-sense”  but the repercussions of this are a little more far-reaching than in some of the other cases.  Einstein, together with Podolsky and Rosen (hence EPR) came up with this thought experiment to show an inconsistency in the Quantum Mechanical theory that required the theory to be “non-local” that is it allowed photon A to know what polarisation state photon B was in at any separation distance, even if that distance was greater than a photon could travel during the measurement time.  Einstein having created the Special Theory of Relativity would have been extremely unhappy with this possibility existing within another theory (Quantum Mechanics) – and this was the whole idea behind the EPR thought experiment – to show that the current ideas of Quantum Theory were “incomplete” as they required QM to be a non-local theory.

If you go into great detail regarding this experiment and where it “goes wrong” as far as common-sense is concerned, there are just three basic principles, one (or more) of which must be violated.  These principles are:

1)  The Reality principle.

2)  The Induction principle.

3)  The Locality principle.

Reality – regularity of phenomena is due to an underlying physical reality.

Induction – it is possible to reach conclusions valid for all systems of a given type from a consistent set of observations on a large sample of systems of that type.

Locality – if two systems have for a time been in dynamical isolation from each other, then a measurement on the first system can produce no real change in the second.

Now it is the Locality principle that was being “probed” by Einstein’s thought experiment – so it seems extremely perverse to me that out of the three possible principles that could be at fault – this is the one chosen by the Quantum Mechanical theorists to be the “joker”.

Surely in this post-Matrix age where the possibility exists that we are all part of a computer simulation, it is the Reality principle that needs to be called into question – and that the EPR Paradox is actually an extremely testing experiment into the very reality of our Universe – not simply a statement about the light-like separation of particles that had once interacted.

Please click on the image to see the animation in a new window:

I was watching a programme on the same subject, basically how alien technology was being utilised by the United States government – and part of this programme was a live link to the States and a discussion with a well known Professor who was supporting the “alien technology” thesis.  He was there telling us how the strange craft coming out of Area 51 were the direct result of reverse-engineering crashed extra-terrestrial vehicles when……… suddenly…….. we lost the TV link.  Now call me a sceptic, but I have a real problem understanding how a country that has already reverse-engineered flying saucers for its own military use can possibly have trouble maintaining a video link for a few minutes.  Or maybe the break in communications was from our side

An ordinary (planar) angle is defined by considering a circle of radius r (make r=1 for simplicity).  Now consider a length of arc on the circumference of the circle of length L, this will subtend a planar angle at the centre of the circle defined as L/r = L radians.  Now the total circumference of a circle is 2 x Pi x r so that if again we have r=1, then the total angle about the central point of a circle is 2Pi radians.  2Pi radians is therefore equivalent to 360 degrees, Pi radians is equivalent to 180 degrees, and Pi/2 radians is a right angle.  So far so good I hope.

Now let’s move onto the slightly more involved concept of a SOLID angle.  This is no more difficult in reality to the planar angle, it’s just that we don’t use it much (if at all) in every day life.

The unit of solid angle is the STERADIAN and it is defined as follows.  Consider a sphere of radius r, and consider some area on the surface of the sphere of area A.  Then the solid angle subtended by the area A at the centre of the sphere is A/r x r steradians.  The total surface area of a sphere of radius r is 4 x Pi x r x r so by using our definition of solid angle we see that the total solid angle about a point is 4 x Pi x r x r / r x r or simply 4Pi steradians (this is precisely why 4Pi turns up in the permeability of free space – but that’s another story).

Solid angle in steradians (or in square degrees) is of importance to astronomers too as it gives an indication of the size of an object in the sky – but as solid angle isn’t generally understood this also means that the apparent size of objects in the sky is also not well-understood.  When astronomers say that the Sun and Moon subtend about half a degree – they are talking PLANAR degrees and that the Sun and Moon are about half a degree in (planar) diameter.  That’s fair enough, but to put things into perspective we should know what looking out into one hemisphere means in terms of steradians (or square degrees) as it is only by looking at the “sphere of space” above us in this way that we can get some measure of how BIG our total field of view is.  A hemisphere is 2Pi steradians and if we convert this to square degrees we can get some idea of how big the celestial sphere is for an observer with a telescope with a typical field of view of 1 square degree.

We can go back to our PLANAR definition of angle to work this one out.

Pi radians = 180 degrees, so

Pi x Pi steradians = 180 x 180 square degrees, so

4Pi steradians = whole celestial sphere = 4 x 32,400/Pi square degrees  = 41,252.96 square degrees, so

2Pi steradians = celestial hemisphere = 20,626.48 square degrees.

So our observer with a 1 square degree field of view would have a roughly 1 in 20,000 chance of randomly hitting a selected object – it gives an indication of how BIG it is up there!

As a corollary:  1 steradian = 3,282.81 square degrees or equivalently 1 square degree = 3E-4 steradians.

Returning back to the Golden Solid angle!

We now consider a sphere whose surface area has been divided into two, one of area unity and one of area phi (the golden ratio or 1.618…) and the unity surface area will subtend some solid angle, let’s call it gamma, at the centre of the sphere.  In exactly the same way we define the Golden Ratio on the line, or the Golden angle for the circle, we can come up with an equation for the Golden Solid angle for the sphere:

(4Pi – gamma)/gamma = 4Pi/(4Pi – gamma) which is a quadratic in gamma which can be solved in the usual way to give:

gamma = 1.52786Pi steradians or 15757.2 square degrees.  If you (for whatever reason) wanted to take a slice through the sphere to see what PLANAR angle this solid angle corresponded to  you would get an angle of  152.7 degrees – though I’m not sure what use this information is except that it is NOT the same as the Golden planar angle of 137.5 degrees.

For completeness:  The solid angle corresponding to the Golden angle of 137.5 degrees is 1.275Pi steradians.

So I return to my original question – anyone seen the Golden Solid angle anywhere in Nature???