## The maths and physics of walking in the sand

I love this – which I picked up from Ian Stewart’s now slightly out-of-date (e.g., pre-proof of Fermat’s Last Theorem) and out-of-print The Problems of Mathematics (but a good read and on sale very cheaply at Amazon) – because it demonstrates the harmony of physics with maths, is based on a common experience and is also quite counter-

intuitive.

Most of us are familiar with the experience – if you walk on damp sand two things happen: firstly the area around our foot becomes suddenly dry and secondly, as we lift our foot off, the footprint fills with water. What is happening here?

Well, it turns out that the sand, before we stand on it, is in a locally optimised packing state – in other words, although the grains of sand are essentially randomly distributed they are packed together in a way that minimises (locally) the space between the grains. If they weren’t then even the smallest disturbance would force them into a better packed state and release the potential energy they store in their less efficiently packed state.

This doesn’t mean, of course, that they are packed in the most efficient way possible – just as they are randomly thrown together they fall into the locally available lowest energy state (this is the physics) which is the locally available best packing (this is the maths).

But this also means that when we stand on the sand we cannot actually be compressing it – because that would actually imply a form of perpetual motion as we created an ever lower energy state/even more efficient packing out of nothing. In fact we make the sand less efficiently compressed – the energy of our foot strike allowing the grains to reach a less compressed  packing – and, as a result, create more space for the water in the surrounding sand to rush into: hence the sand surrounding our foot becomes drier as the water drains out of it and into where we are standing.

Then, as we lift our foot, we take away the energy that was sustaining the less efficient packing and the grains of sand rearrange themselves into a more efficient packing (or – to look at it in the physical sense – release the energy stored when we stand on the sand). This more efficient packing mean less room for the water in the sand and so the space left by our foot fills with water expelled from the sand.

## A probably entirely naïve question about the principle of relativity

Surely I can quite easily design an experiment that shows the relativity principle is false.

If turn around on the spot the principle, as I understand it, asserts that I cannot build an experiment that proves it was me that moved as opposed to everything else that moved while I stayed still.

But the rest of the universe is very massive – possibly of infinite mass – and so to move it through $2\pi$ radians takes a hell of a lot more energy than moving me.

## Perhaps “you” will live forever after all

This is inspired by Max Tegmark‘s Our Mathematical Universe: My Quest for the Ultimate Nature of Reality: I have been thinking about this since I finished the book and I cannot find a convincing argument against the thesis (certainly the ones Tegmark uses in the book didn’t impress me – but perhaps I misunderstood them.)

So, let us conduct a thought experiment that might suggest “you” can live forever.

In this world we assume that you don’t do anything dangerous – such as commute to work. The only factors that could kill you are the normal processes of human ageing (and related factors such as cancer): your fate is completely determined by chemical processes in your body.

And we accept the “many worlds” view of quantum mechanics – in other words all the possible quantum states exist and so “the universe” is constantly multiplying as more and more of these worlds are created.

Now, if we accept that the chemical processes are, in the end, driven by what appears to us as stochastic (random) quantum effects – in other words chemicals react because atoms/electrons/molecules are in a particular range of energies governed by the quantum wave equation – then it must surely be the case that in one of the many worlds the nasty (to our health) reactions never happen because “randomly” it transpires that the would-be reactants are never in the right energy state at the right time.

To us in the everyday world our experience is that chemical reactions “just happen”, but in the end that is a statistically driven thing: there are billions of carbon atoms in the piece of wood we set fire to and their state is changing all the time so eventually they have the energy needed to “catch fire”. But what if, in just one quantum world of many trillions, the wood refuses to light?

So, too for us humans: in one world, the bad genetic mutations that cause ageing or cancer just don’t happen and so “you” (one of many trillions of “you”s) stays young for ever.

The obvious counter argument is: where are these forever-young people? The 300 year olds, the 3000 year olds? Leaving aside Biblical literalism, there is no evidence that such people have ever lived.

But that is surely just because this is so very, very rare that you could not possible expect to meet such a person. After all, around 70 – 100 billion humans have ever been born and each of them has around 37 trillion cells, which live for an average of a few days (probably) – so in a year perhaps 37 billion trillion cell division events – each of which could spawn a new quantum universe – take place. That means the chances of you being in the same universe as one of the immortals is pretty slim.

Yet, on the other hand, we all know someone who seems to never age as quickly as we do…

…I’d be really interested in hearing arguments against the hypothesis from within the many worlds view of quantum physics.

## Deconstructing Max Tegmark’s argument against a simulated universe

In the end Max Tegmark‘s Our Mathematical Universe: My Quest for the Ultimate Nature of Reality has proved to be something of a disappointment – somewhere along the way the science got lost and was replaced by a lot of metaphysical speculation.

I haven’t quite finished it yet – so I’ll do a fuller review when I do (and there were good bits too), but what I want to take issue with here is his case (or, perhaps more accurately, the cases he quotes with approval) against the idea that we live in some sort of great computer simulation.

I am not arguing in favour of such a view of our universe – but it certainly has its strengths – if you think computer power will keep on growing then it is pretty difficult, if we apply the basic “Copernican” principal that we are nothing special, to avoid the conclusion that we are in such a universe.

Tegmark uses two major arguments against this idea that I want to take issue with.

The first, I think, is not an argument against it at all – namely that we are more likely to be a simulation within a simulation if we accept this basic thought. Well, maybe – but this is completely untestable/falsifiable and so beyond science. (In contrast the basic idea that we are in a simulated universe is testable – for instance if we find that our universe has a “precision limit” that would be a strong pointer.)

The second is the degree of complexity of simulating a many worlds quantum multiverse. But, of course, the simulator does not need to actually “run” all those other quantum worlds at all – because it’s not a physical reality, merely a simulation. All it has to do is leave the signs (eg the traces of superposition we can detect) in our environment that such alternate universes exist, but once “decoherence” takes place those alternate universes go straight to your garbage collection routines. So too for more anything much beyond the solar system – all the simulation has to do is provide us with the signals – it doesn’t have to actually, for instance, “run” a supernova explosion in a distant galaxy.

## Why we’ll never meet aliens

Well, the answer is pretty plain: Einstein‘s theory of general relativity – which even in the last month has added to it’s already impressive list of predictive successes – tells us that to travel at the speed of light a massive body  would require an infinite amount of propulsive energy. In other words, things are too far away and travel too slow for us to ever hope to meet aliens.

But what if – and it’s a very big if – we could communicate with them, instantaneously? GR tells us massive bodies cannot travel fast, or rather along a null time line – which is what really matters if you want to be alive when you arrive at your destination – but information has no mass as such.

Intriguingly, an article in the current edition of the New Scientist looks at ways in which quantum entanglement could be used to pass information – instantaneously – across any distance at all. Quantum entanglement is one of the stranger things we can see and measure today – Einstein dismissed it as “spooky interaction at a distance” – and essentially means that we can take two similar paired particles and by measuring the state of one can instantaneously see the other part of the pair fall into a particular state (e.g., if the paired particles are electrons and we measure one’s quantum spin, the other instantly is seen to have the other spin – no matter how far away it is at the time).

Entanglement does not allow us to transmit information though, because of what the cosmologist Antony Valentini calls, in an analogy with thermodynamic “heat death”, the “quantum death” of the universe – in essence, he says that in the instants following the Big Bang physical particles dropped into a state in which – say – all electron spins were completely evenly distributed, meaning that we cannot find electrons with which to send information – just random noise.

But – he also suggests – inflation – the super-rapid expansion of the very early universe may also have left us with a very small proportion of particles that escaped “quantum death” – just as inflation meant that the universe is not completely smooth because it pushed things apart at such a rate that random quantum fluctuations were left as a permanent imprint.

If we could find such particles we could use them to send messages across the universe at infinite speed.

Perhaps we are already surrounded by such “messages”: those who theorise about intelligent life elsewhere in the universe are puzzled that we have not yet detected any signs of it, despite now knowing that planets are extremely common. That might suggest either intelligent life is very rare, or very short-lived or that – by looking at the electromagnetic spectrum – we are simply barking up the wrong tree.

Before we get too excited I have to add a few caveats:

• While Valentini is a serious and credible scientist and has published papers which show, he says, the predictive power of his theory (NB he’s not the one speculating about alien communication – that’s just me) – such as the observed characteristics of the cosmic microwave background (an “echo” of the big bang) – his views are far from the scientific consensus.
• To test the theories we would have to either be incredibly lucky or detect the decay products of a particle – the gravitino – we have little evidence for beyond a pleasing theoretical symmetry between what we know about “standard” particle physics and theories of quantum gravity.
• Even if we did detect and capture such particles they alone would not allow us to escape the confines of general relativity – as they are massive and so while they could allow two parties to theoretically communicate instantly, the parties themselves would still be confined by GR’s spacetime – communicating with aliens would require us and them in someway to use such particles that were already out there, and perhaps have been whizzing about since the big bang itself.

But we can dream!

Update; You may want to read Andy Lutomirski’s comment which, I think it’s fair to say, is a one paragraph statement of the consensus physics. I am not qualified to say he’s wrong and I’m not trying to – merely looking at an interesting theory. And I have tracked down Anthony Valentini’s 2001 paper on this too.

## In what sense do photons exist?

This is a genuine question on my part – and I would be grateful for any answers!

The inspiration for asking the question comes from Genius: Richard Feynman and Modern Physics – my current “listen while running” book – along with Feynman’s own description of radiation in QED – The Strange Theory of Light and Matter.

Feynman argues that there is no radiation without absorption: in other words a tree that falls in an empty forest does indeed make no sound (if we imagine the sound is transmitted by photons, that is).

This sounds like a gross violation of all common sense – how could a photon know when it leaves a radiating body that it is to be absorbed?

But then, general relativity comes to our rescue – because in the photon’s inertial frame the journey from radiator to absorber is instantaneous.

But how can a body that exists for no time at all, exist at all?

Then again my assumption in asking this question is that time is in some sense privileged as a dimension of spacetime. This is a pretty deep controversy in theoretical physics these days and I am not qualified to shed much light on it – but let us assume that a body can exist with a zero dimension in time but real dimensions in space, can we then have bodies which have zero dimensions in space but a real dimension in time? If so, what are they?

## The Summa Metaphysica and all that

Stumbled across a fascinating article in this weekend’s Guardian magazine about David Birnbaum and his “Summa Metaphysica” (this – Q4P2 – Principia Metaphysica (The Birnbaum Summa Metaphysics) – is listed in Amazon and appears to be the work, or a version of it).

Normally I more or less ignore the magazine supplements that come with the weekend’s papers but I was attracted this one by a graphic that mentioned the “Bohr radius” – as I had just been listening to Genius: Richard Feynman and Modern Physics while running in the gym.

To be honest, the article doesn’t do a lot to illuminate what Birnbaum  a New York jeweller who has spent a lot of his own money to promote his ideas – is about. But then, maybe that is because he’s not about very much at all: metaphysics is, after all, literally beyond science and testing.

It does tell us a lot about how Birnbaum has upset quite a few genuine scientists about how he has promoted his claims to have found an ultimate theory to explain existence. He used an imprint – Harvard Matrix – on his self-published books that seems to have left a few people at the university feeling their good name has been misappropriated, while others feel that they were used to give a veneer of credibility to a conference Birnbaum funded at Bard college this past May (though I can only admire – seriously – the Oxford chemist, Peter Atkins, who says he attended because it gave him a chance to get an expenses-paid trip to New York and because he likes a good argument).

Such as they are, Birnbaum’s ideas seem to centre on the concept of “potential” (energy? it’s not clear). It’s not mentioned in the article but, of course, the concept of a multitude of inflationary universes is also, in a sense, related to potential energy (or at least the energy that is freed when the ‘inflation’ changes state). But those ideas are, at least to some degree, testable and potentially falsifiable. By all accounts Birnbaum’s are not.

In any case I doubt there is much relation between them and inflationary cosmology either, but as cosmology/particle physics becomes more complex, then the scope for the naive (a category into which, being kind, I will place Birnbaum) as well as the exploitative – wait for the next batch of psuedo-science in the Daily Mail – grows larger.

Many scientists in these advanced fields are unhappy about the core of the “standard model” – in that it posits a very large number of supposedly “fundamental” particles. There has been disappointment as well as joy over how well the model has stood up to the LHCs explorations – a triumph of the scientific method as great as Le Verrier’s prediction of Neptune, but also a confirmation of a model that looks less than fundamental after all. If, and until, we solve some of those seeming contradictions then we are just going to have to live with the interstices of physics being filled with ideas from strange people – especially rich ones who want to be taken seriously.

(Incidentally, the Bohr radius graphic was actually a reference to an idea promoted by Jim Carter who denies the truth of quantum mechanics.)

## My one problem with Feynman’s QED

Well, as predicted, I finished off Richard Feynman‘s QED – The Strange Theory of Light and Matter in short order this morning – and it is a truly marvellous book. I just wish I had read it as an undergraduate.

My one problem with it was its explanation of “stimulated emission“. Now, as an undergraduate, I remember I understood this quite well – it came up in a discussion of MASERs (intense microwave sources in deep space) as opposed to the more familiar LASERs ifI remember correctly. But that’s a long time ago.

Perhaps I should look it all up again.

## What a brilliant book

Just over three hours ago I started reading Richard Feynman‘s QED – The Strange Theory of Light and Matter (Penguin Press Science): and now, 110 pages later, I am stunned at its brilliance.

If you are any sort of physics undergraduate you must read it. Similarly, if I was teaching ‘A’ level physics I would be handing it out to my students.

There is little maths in it and not much physics either – but as a way of explaining a high concept of physics – without cutting corners with bad analogies – it is just fantastic.

I’ve taken a break now because reading that much of a science book at more or less one sitting is not conducive to grasping all its points – but I am sure it will be finished either this evening or tomorrow morning.

## Schrödinger’s cat: for real

Quantum Mechanics is, along with General Relativity, the foundation stone of modern physics and few explanations of its importance are more famous than the “Schrödinger’s cat” thought experiment.

This seeks to explain the way “uncertainty” operates at the heart of the theory. Imagine a cat in a box with a poison gas capsule. The capsule is set off if a radioactive decay takes place. But radioactivity is governed by quantum mechanics – we can posit statistical theories about how likely the radioactive decay is to take place but we cannot be certain – unless we observe. Therefore the best way we can state of the physical state of the cat – so long as it remains unobserved – is to say it is both alive and dead.

Now physicists still argue about what happens next – the act of observing the cat. In the classical, or Copenhagen, view of quantum mechanics the “wave equation collapses” and observing forces the cat into a dead or alive state. In increasingly influential “many worlds” interpretations anything that can happen does and an infinite number of yous sees an infinite number of dead or alive cats. But that particular mind bender is not what we are about here. (NB we should also note that in the real world the cat is “observed” almost instantaneously by the molecules in the box – this is a mind experiment not a real one, except… well read on for that).

The idea of the cat being alive and dead at once is what is known as “quantum superposition” – in other words both states exist at once, with the prevalence of one state over another being determined by statistics and nothing else.

Quantum superposition is very real and detectible. You may have heard of the famous interferometer experiments where a single particle is sent through some sort of diffraction grating and yet the pattern detected is one of interference – as though the particle interfered with itself – in fact this indicates that superposed states exist.

In fact the quantum theories suggest that superposition should apply not just to single particles but to everything and every collection of things in the universe. In other words cats could and should be alive and dead at the same time. If we can find a sufficiently large object where superimposition does not work then we would actually have to rethink the quantum theories and equations which have stood us in such good stead (for instance making the computer you are reading this on possible).

And Stefan Nimmrichter of the Vienna Centre for Quantum Science Technology and Klaus Hornberger of the University of Duisberg-Essen have proposed we use this measurement – how far up the scale of superposition we have got as a way of determining just how successful quantum mechanics’s laws are (you can read their paper here).

They propose a logarithmic scale (see graph) based on the size of the object showing superposition – so the advance from the early 60s score of about 5 to today’s of about 12 might mean we can be one million times more confident in quantum theory’s universal application. (A SQUID is a very sensitive magnetometer which relies on superconductivity.)

And they say that having a 4kg ‘house cat’ be superposed in two states 10cm apart (which might be taken for a good example of lying dead versus prowling around) would require a score of about 57 – in other words about $10^{45}$ more experimental power than currently available.

That probably means no one reading this is ever likely to see a successful demonstration that Schrödinger’s cat is rather more than a thought experiment, but it does give us a target to aim at!