## Archive for May, 2010

Posted in Uncategorized on May 29, 2010 by quantummoxie

Well, summer has come and I’ve gotten a bit lazy.  Actually, I’ve been trying – and ‘trying’ is the key word here – to relax since the semester ended.  Hence I haven’t posted in awhile.  At any rate, I have been reading, however.  In particular I’ve been working through Chris Isham’s recent paper on topos methods in formulating physical theories.  It’s pretty interesting reading and we’ve had some interesting discussions about related things over at the nForum.

But it got me thinking about something.  Now, Isham’s approach may indeed lead somewhere eventually.  In fact I think his approach is almost precisely what I had envisioned for this idea of relational empiricism that I was trying to cook up.  But the question is, what will it really serve to accomplish in the end?

Personally I think it may accomplish quite a bit, but here’s the thing.  Both GR and QM are excellent working theories.  That is, the formalism for each is used on a daily basis by scientists (and even some engineers, e.g. GPS) to accomplish a wide variety of tasks.  Thus a full theory of quantum gravity would naturally be expected to recover the essence of these formalisms (since they are different formalisms, a unified theory would not likely be able to capture all of each – the important thing, of course, is that a unified theory match experiment).

But some approaches to quantum gravity seem to be motivated solely by an effort to find a unified description of these two phenomena.  That is certainly a laudable – and possibly even an unachievable – one.  But ideally a full theory of quantum gravity ought to also offer new physical insights, perhaps explaining the unexplained.

And this is where I worry a bit about Isham’s approach.  As much as I am a major proponent of category theory, I am more in favor of a ‘radically de-abstracted’ category theory.  In other words, I love the essence of category theory but it can get so abstract at times to be almost unapproachable.  I worry that any new insights this approach will lead to could leave too much room for interpretation.  These insights will naturally be couched in the language of category theory and topos.  Being as abstract as it is, it is possible that there could be multiple routes from this language to the physical phenomena themselves making any theoretical results gleaned from this method murky at best.

Isham’s method, of course, is just an example.  And as I said I am generally in favor of it – the essence of category theory is fantastic.  But clear lines must be drawn between the objects, symbols, definitions, etc. in category and topos theory to physical phenomena so that new theoretical discoveries are not ambiguous.  Otherwise, all we’re doing is translating from one language to another.  Quantum gravity needs to have the spirit of a universal language in the manner conceived by John Wilkins – not just to facilitate communication between disparate ideas, but also to offer new insight.

Posted in Uncategorized on May 22, 2010 by quantummoxie

I’ve been insanely busy and haven’t had the chance to post anything profound lately, but I did just find that if you go to the front page of Google you can play Pac-Man.  Just click “Insert Coin.”  It’s only up for 48 hours in honor of the Pac-Man 30th anniversary.

## Catenary versus parabolic curves: math on the fly

Posted in Uncategorized on May 15, 2010 by quantummoxie

I’ve been tied up this week finishing out the semester as well as assisting with some calculations regarding some equipment to be used in the cleanup efforts in the Gulf of Mexico.  In relation to those efforts, I spent some time studying catenary and parabolic curves.  Mind you, I was doing this in “crisis mode” at a manufacturing site without many of my reference books and, ultimately, without reliable Internet access.

Specifically I was looking at these from the point-of-view of load calculations and was interested in trying to predict the behavior theoretically (of course, the eventual real-world test didn’t go as planned and my friend’s boat nearly sank because he forgot to put the drain plugs back in after the winter, but, hey…).

So considering some flexible object (like a cable) hanging from two secured points, not necessarily at equal height (e.g. a tow rope connecting a boat to a parasail).  Mechanically, the general deflection curve for something like this is given by

$y=\frac{1}{H}\int \left[\int w(x) dx\right] dx + C_{1}x + C_{2}$

where H is the horizontal component of the tensile force in the deflected object and w(x) is a loading function given per unit length.  The constants of integration must be determined from boundary conditions.

Suppose we now consider a cable (like a telephone wire) connected at the same height and subjected to a uniform load (e.g. a line of birds sitting on it).   The loading function in this case is a constant, the constants of integration can be made to be zero, and the deflection curve turns out to be

$y=\frac{w}{2H}x^2$

which is a parabolic curve.  If we assume the greatest sag h occurs in the middle, we set $y=h$ for $x=l/2$ where $l$ is the straight-line distance between the two connection points (i.e. it’s a chord), and thus

$h=\frac{w}{2H}\frac{l^2}{4}$.

We may use this to solve for H and substituting back in we can find the deflection curve entirely in terms of h and l which are both easily measurable:

$y=4\frac{hx^2}{l^2}$.

On the other hand, if we consider a cable loaded only by its own weight, we get a deflection curve

$y=\frac{H}{w}\left(\textrm{cosh}\frac{w}{H}x-1\right)$.

This is actually a catenary curve and not a parabola!  To solve for H in this instance, we again set $y=h$ for $x=l/2$ to get

$h=\frac{H}{w}\left(\textrm{cosh}\frac{wl}{2H}-1\right)$.

Unfortunately this is not easy to solve (many mechanics texts simply advise using trial and error on this).  Obviously, then, in practice the parabolic curve is much easier to work with.  So, given the parameters of the problem under which I was working, I set out to figure out where the two diverged.  Using values for w and H from a known model, I overlaid plots of the two curves and found that they began to diverge roughly at a ratio of $y/x \approx 0.46$.  This meant that as long as the ratio $h/l \le 0.23$, I could use a parabola instead of the catenary.

In my real-world model, I knew the value of l.  Solving for h from this again required trial and error (at least given that I was doing this in crisis mode on the fly without any of my reference books).  So I used a series approximation for a sine function that popped up in the equation and after a few more calculations found that for my real-world application, $h/l \approx 0.154$.  Thus I was safe modeling it as a parabola which meant finding other parameters would be much easier!

And that’s “math on the fly,” as it were…

## The nature of spin, magnetism, and monopoles

Posted in Uncategorized on May 7, 2010 by quantummoxie

Spin is a rather interesting parameter (in case you didn’t already know that).  It is generally believed to be an inherent property much like mass and charge.  Nevertheless, it does appear to affect certain systems in much the same way as a more traditional angular momentum would.  The reason, of course, that we don’t think of it in these terms – i.e. that it actually is the angular momentum of a “spinning” particle – is that the electron would need to be spinning faster than the speed of light if spin really were associated with a rotation (this was first noticed by Pauli, criticizing Ralph Kronig’s interpretation of Pauli’s “two degrees of freedom” for the electron, though neither Kronig’s suggestion nor Pauli’s response was published.  Independently, Sam Goudsmit and George Uhlenbeck made the same observation).  Ultimately this led to the proposition that the electron had no size, i.e. it was a point particle (which was the only way to have it obey relativity and still have an angular momentum).

But here’s a curious thing.  Physicists were presented with a conundrum: if spin is the angular momentum of a spinning, non-point particle then the electron would have to be spinning so fast it violated relativity.  So, should they abandon the notion of spin as an angular momentum or should they abandon the non-point particle interpretation?  In fact, over the course of time, they did both.

Part of the reason for this is surely due to the fact that we now know that whole atoms and even molecules can have a quantized spin and this simply makes no sense if spin is just plain, old angular momentum.  So we’ll agree, then, that spin isn’t just angular momentum.  On the other hand, it is clearly related since we know, for instance, that the total angular momentum of an electron in an atom can be given by the sum of the orbital angular momentum and the spin.

So where do we draw the line on the comparison between spin and angular momentum?  I say that we shouldn’t be too hasty to disassociate the two since treating it more like an angular momentum neatly explains two interesting ideas: the origin of magnetism and the non-existence of magnetic monopoles (I acknowledge the inspiration of Tom Moore regarding these ideas).

So, first, let’s assume that it does represent some kind of angular momentum behavior despite its obvious deviations from true classical angular momentum behavior.  Let’s also assume that, in all instances in which we observe spin, it is associated with something (an atom, a point-particle, etc.) that, whether or not it has any true size, possesses a spherical symmetry.  In this way we draw an analogy to a black hole which is technically a single point but that possesses a spherical symmetry in the extent of its field.

Relativity tells us that massive, rotating bodies actually “drag” spacetime around themselves in a process known as “frame-dragging” (whether spacetime truly rotates or not is debated but, given the recent results of the Gravity Probe B experiment, we take it as such since it seems the only logical way to explain why photons appear to behave differently in such a situation depending on which direction they are going).  If we take spacetime to be the gravitational field (i.e. the old “dimples on a bedsheet” interpretation which is admittedly debated) then frame dragging can be viewed as the motion of the actual field itself in a kind of twisting sense (or, perhaps more accurately, like a vortex).

Given that, but irrespective of the relationship between the electromagnetic field and gravitational field, the “rotation” of an electron ought to include rotation of its electric field.  This rotation means that there is always relative motion between an electron’s electric field and any observer and it is well-known that changing electric fields produce magnetic fields (and vice-versa).  As such, at the most fundamental level, magnetism is a purely relativistic effect.  A colleague of mine originally disputed this saying that one could simply imagine an isolated Dirac monopole doing the same thing, thus making electricity relativistic.  But Dirac monopoles have never been observed.  Furthermore, we know that stationary charge always has an electric field but not necessarily a magnetic field.  It is not clear that a similar intrinsic property exists for magnetism (though some have conjectured spin fits the bill, but it depends on how we interpret spin!).

Now, recalling that we are assuming a spherical symmetry (or ellipsoidal if we use the frame-dragging analogy), the moment any sphere is rotated on an axis, one immediately ends up with two possible directions it can spin on that single axis.  This is a simple fact of rotational symmetry and, when combined with spherical symmetry, one sees that any rotating sphere rather naturally has two poles!  Thus, if magnetism is truly nothing but a relativistic process as proposed above, then magnetic monopoles ought not exist.  Since magnetic (Dirac) monopoles have ever been found, the evidence seems to suggest that magnetism is purely relativistic and that spin is more like angular momentum than we like to think it is.

## Wavefunctions, Part Deux: a curious problem

Posted in Uncategorized on May 4, 2010 by quantummoxie

To really get a sense of the nature of the wavefunction, let’s go back to the early days of contemporary physics and look at the nature of light.  By ‘contemporary’ here I am referring roughly to post-Galilean physics.  As far back as Newton and Huygens there was a debate over the nature of light – Newton held the corpuscular (particle) view, which likely stemmed from his work on optics, while Huygens held the wave view.  The wave interpretation began to dominate the rhetoric after the implications of Young’s double-slit experiment (1801) were fully realized.  It was then that the debate began to center on just what was it that was ‘waving,’ as it were.  Maxwell, of course, answered that question by showing that light was merely the fluctuation of an electromagnetic field, though debate about this continued in the form of debate over the aether.  Suffice it to say that, once the Michelson-Morley experiment was performed and certainly by the time relativity came along, the consensus was that what was ‘waving’ was an electromagnetic field.  (Of course, ironically, in the same year he introduced special relativity, Einstein also explained the corpuscular nature of light that seemed to be suggested by certain experimental results.)

Since it had also become clear that light demonstrated corpuscular behavior as well, the wave-particle duality of light was firmly established.  Louis de Broglie then proposed, in his PhD thesis, that all matter possessed this same duality.  It took quite awhile for physicists to determine exactly what was ‘waving’ in the case of the particles of matter, but, as QFT seems to indicate, it’s just some other type of field.

But is Young’s double-slit experiment really evidence for the wave nature of light?  Passing single photons through the slits one at a time gives a statistical distribution on a detection screen.  I’m not sure of the historical roots of this, but I suspect that this is what led to the interpretation of the wavefunction as a probability distribution.  Somehow along the way, when this notion was melded with the modern formalism of quantum mechanics, the wavefunction appears to have lost its ontological status.

On the other hand, we know that we get an interference pattern regardless of the wavelength of the light which means we can get interference with radio waves as well.  In fact this is easily demonstrated using a pair of broadcasting radio antennas acting in the place of the slits.  But most people never discuss “photons” in the context of radio waves because the notion somehow seems difficult to imagine (how big would radio photons be?).  (Either they’re there and simply hard to imagine or they’re not there.  If it is the latter, at what wavelength does light suddenly ‘rid itself’ of photons?)

Given the fact that, starting with de Broglie and Einstein essentially, photons were just another particle (in this case a massless boson), it made sense to treat the wavefunction of all these particles in the same manner.  Given that it took awhile for QFT to really explain many of the other particles and the fact that the probability distribution interpretation seemed to work just fine, it seemed as if wavefunctions no longer needed any ontological status and was divorced from its association with the fluctuating field.  But by stripping them of that status we are forgetting the historical origins of the wavefunction as I’ve just outlined and we are forgetting that, in the case of light at least, and really in the case of all particles, something physical – a field – really is fluctuating.

Now, given all of that, it is not particularly clear where the quantum-classical transition comes into play here.  People often talk about ‘quantum’ versus ‘classical’ light, i.e. noting that light exhibits both quantum and classical behavior.  But, according to Rovelli, Griffiths, and others, the world is entirely quantum and classicity is merely a perception on our part.

Given all of this and assuming for the moment that wavefunction collapse is real and that the act of measurement should cause this (again, just assume this for the moment whether or not you buy it), then shouldn’t the act of listening to the radio collapse the signal wave to a particular, localized state thereby preventing anyone else from listening to it on a different radio?