Feynman left us many interesting facts/tools and one of the most powerful I have ever come across was his observation that “The same equations have the same solutions”.  By this he means that equations of the same form have similar solutions.  Now this may sound like it is stating the obvious, which I guess it is, but it is an incredibly powerful tool in the right hands.

Maybe around 30 years ago I found two areas where this statement had some very interesting outcomes, but I have not been able to push the results to a final conclusion.  As I am now getting too old for this sort of young man’s thinking it is time for me to put this out for general consumption so that someone with a more agile mind/imagination can finish off this work.

The first subject area is mechanics and the derivation of a set of “Mechanical Maxwell Equations”.  I shall give you the outline of the ideas and it’s up to you to finish it off  The starting point for equations of the same form are the Newtonian gravitational force equation and the equation for the force between two charges.  Using the similarity in form you can “equate” 1/4 Pi Epsilon nought with G – I call this the equivalence of the constants.  Charge then equates to mass and distance is the same in both equations.  We have two more quantities in electromagnetism that we need an equivalent mechanical quantity for, namely B and E.  By looking for similar equations in electromagnetism and mechanics it isn’t too difficult to find that E’s equivalent is a, the linear acceleration, and that B’s equivalent is omega or angular velocity (very interesting that B should equate to omega!!).  You now have all the necessary information to create your own set of four “Mechanical Maxwell” equations.  I have done this, and luckily they turn out to be dimensionally correct – but I have no idea what they mean or what they are saying.  I will leave it up to you to tell me

The second area where “The same equations have the same solution” shows us an interesting route for new thoughts and ideas is in the area of Quantum Mechanics, specifically Hidden-Variable theory.  The form of the de-Broglie/ Bohm equation for a hidden-variable form of wave equation has the same form as the condensate wave function in superfluid physics.  So the Quantum Potential which is the source of all the trouble in de-Broglie/Bohm Hidden-Variable theory is related to a term in superfluid physics which is only important when the superfluid density varies rapidly with position, for instance near a wall or inside a vortex.  When we are dealing with the bulk fluid we can omit the term – which is extremely interesting.

So there you have it.  Two interesting areas of research.  The first might lead to a nice paper, and the second might lead to a Nobel Prize.  Over to you.

Does the EPR “paradox” tell us anything about Physics?

Well the trite answer is yes of course it does – but can we be a bit more specific?

Physics is an “after the fact” science relying on experiments and the observations on the results of those experiments.  What do I mean by “after the fact”?  I mean you set up an experiment to measure some property or event and the experiment, if successful, will give you a numerical value for that property or event, so far you have learnt absolutely nothing new – but please keep with it  By setting up various different experiments and conditions we can measure the mass, “apparent” size and the charge on an electron.  But we cannot say WHAT an electron actually is.  Why not?  Well one reason is we don’t know how to set up the experiment to do that.  The experiments we set up measure the PROPERTIES of an electron – that is all.  We can smash things together in high energy particle accelerators and see how various particles are created (or annihilated) and what the energies involved are – but these experiments say nothing about what the particles themselves are.  How can they?  They are only measuring the EFFECTS that the particles themselves create.

Mathematical Physics is great at providing predictions.  It still requires the experiment to be performed to give the mathematical predictions any substance – any “reality”.

This situation is rather like the 4-year old kid who keeps asking why? to an initial answer to a question.  Eventually we come to a stopping point.

The EPR “paradox” was thought up as a hard test for the theory of Quantum Mechanics.  Quantum Mechanics gave its answer, a nasty answer that goes (like most things in Quantum Mechanics) against common sense.  Aspect and some other brilliant experimentalists created beautiful experiments to measure the results of an EPR-type problem, and yes, Quantum Mechanics gives  us the same answer as the experiments and common sense doesn’t.  But can  you take things further than this?  Can you, with the information provided, say anything about the MECHANISMS involved?  I don’t see how this is possible.  You created the experiment given the parameters you wished to measure to confirm (or otherwise) the EPR “paradox”.  You did not set up the experiment to look at the MECHANISMS involved (and basically you wouldn’t know how to do this anyway) so why would you expect an EPR-type MEASUREMENT to give you an insight into the underlying mechanisms?  This is very much the situation of the 4-year old kid asking “Why?” one time too many.

But if this is the case, doesn’t every experiment and every part of Physics run into this very same problem?  Even the simplest Physics that we think we know EVERYTHING about?  I think it has to  - doesn’t it?

Let’s take a very quick look at Newtonian Mechanics – not much there we don’t know everything about is there?  Masses flying about, velocities, energy (potential and kinetic), acceleration, inertia – oops what’s that one?  Inertia?  What’s that then?  Well it’s the resistance to an applied force exhibited by any body possessing mass.  Yes but what IS it?  Well it’s possibly caused by the interaction of the body with all the other mass in the Universe – but actually we really don’t know.  And what is mass anyway?  I don’t think the Higgs Boson goes very far in telling us what “mass” actually is.  So even in a science that we think we know quite well and that was dealt with a couple of hundred years ago sufficiently well for us to send space probes on Grand Tours – we don’t have to go very far back with the Why? question to hit our stopping point.

A lot of these Physics problems come down to “fields” as their ultimate answer, where “fields” is a great euphemism for “I don’t know”.  When Maxwell came out with his 4 equations for electromagnetism he hit similar issues with the scientists of his day.  What were these “fields” where are the “springs” and the ether that carry these waves?  What are the mechanisms?  I guess I am also asking what the “springs” are in all our theories – and my contention is that we simply don’t know.

If we’re stuck in asking the basics of such “very simple” questions – then where do we stand in asking the biggies?  What happened before the Big Bang for instance?

The Scientific Method is a very powerful tool for explaining what we can observe, it should be, after all the experimental method has proven to be a great way to provide the hard numbers to our observations.  But for the underlying mechanisms?  What do we do for those?

I have a very strong feeling (shouldn’t really allow “feelings” to come into it – that’s getting into Metaphysics ) that we are seeing EXACTLY the same problem rearing its head that we have already seen in mathematics.  Godel’s Theorem didn’t help us much there – but at least mathematics has a name for the issue – Physics doesn’t.  How about Greg’s Enigma??

Anne Baring’s new book “The Dream of the Cosmos:  A Quest for the Soul” has a Pleiades image from the New Forest Observatory on the cover.  The Publisher has done an excellent job in putting the book together.

In 1973 when I was 19 I left home to start work at Harwell and also to take an HNC at Oxford Polytechnic (as it was then).  This was a time of great awakening for me, I had found the whole school experience to be a complete and utter waste of time and couldn’t wait for the day when I could leave.  Starting at Harwell was that day and was also the time when I discovered (for myself, no prompting from teachers) that learning could actually be a rewarding experience.  So it was in 1973 that I bought one of the many books that would transform my world-view, in a very positive way, forever.  1973 was the first publication of Carl Sagan’s “Cosmic Connection”.  Some 40 years later I have totally forgotten its contents and the original book is long gone – but I have on my desk an Anniversary issue which I am slowly re-reading, and I understand why it gave me such a buzz such a long time ago.

Today I finished reading “The Varieties of Scientific Experience” also by Carl Sagan, and Carl has done it yet again – I got the same buzz from reading this book that I got from reading the “Cosmic Connection” 40 years ago – thank you Carl

What can I say about “The Varieties of Scientific Experience”?  Firstly, I think this should be compulsory reading for GCSE schoolkids, those that have sufficient intelligence to follow the plot could go on to do good things.  Secondly, it should make an educational read for politicians and theologians alike – so maybe they wouldn’t like it very much.  Logical argument, beautifully written, and very humanistic with none of that highly annoying zealotry that we have come to associate with Richard Dawkins.  Carl – you are very sorely missed!

If you read Einstein’s little book “The Meaning of Relativity” Appendix II, you will come across a most interesting remark (by Einstein).  With regard to his General Theory of Relativity he poses the following two questions:

1)  Should one admit the appearance of singularities?

2)  Should one postulate boundary conditions?

Einstein gives the following answer regarding singularities.  “As to the first question, it is my opinion that singularities must be excluded.  It does not seem reasonable to me to introduce into a continuum theory (my bold/italics) points (or lines, etc.) for which the field equations do not hold.  Moreover, the introduction of singularities is equivalent to postulating boundary conditions (which are arbitrary from the point of view of the field equations) on ‘surfaces’ which closely surround the singularities.  Without such a postulate the theory is much too vague.”

Whoops – that’s a biggy, did you see it?  Einstein says that his continuum theory (the General Theory of Relativity) shouldn’t allow the existence of Black-Holes.  Bit odd that he was to work on Black-Hole theory not too many years later, but that is by the by   What is more to the point is that, yes, his General Theory of Relativity IS a continuum theory, it is not a Quantum theory.  As such, although it gives extremely good agreement with observation, it is undoubtedly “incorrect” in the fine detail, maybe even where Black-Holes are concerned.  Interesting!

See plenty of new Fractal renders on my Flickr Mathematical Structures page.

I would like to bring to your attention the last couple of paragraphs of my Inaugural Lecture.  This explains why I believe we live in a Matrix Universe.

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 so peculiar 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 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 you introduce 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 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!

The Big Bang was when the simulation was first started.

You can see a new bunch of fractal renders on the Scientific Artist web site:

http://www.scientificartist.com/

A recent image of Sirius shows something new in the vicinity.  A type III civilisation has constructed a Dyson sphere around our brightest star.