Monegasque Plasma Confinement

Joint European Torus (JET), magnetic confinement fusion Tokamak. Culham, UK.











Tunnel at Monte Carlo.

Is the universe discrete or continuous?

The Foundational Questions Institute is currently inviting submissions for an essay contest entitled Is reality digital or analog?. Now, this is essentially the same thing as asking whether the physical universe is discrete or continuous. Whilst general relativistic cosmology represents the universe to be continuous, some aspects of quantum theory are discrete, and many people working in quantum gravity clearly expect the universe to be discrete at a fundamental level.

Let's take a step back, however, and consider the question in more general terms. In particular, given that the physical universe has many levels of structure, and given that the theories which describe different levels of structure can possess radically different properties, is it even possible to know whether the universe is discrete or continuous at a fundamental level?

Before going a little further, it's necessary to introduce a philosophical concept called supervenience. This is a useful concept relating different levels of structure because, unlike allied concepts such as reduction and emergence, its definition is generally agreed upon and fairly uncontroversial. Supervenience, then, holds that any change in the states or processes at a higher level of structure, must correspond to a change in the lower level states or processes. Supervenience asserts that a lower-level state or process uniquely determines the higher-level state or process, and that there is a many-one mapping between the lower-level states/processes and the higher-level states/processes.

Now, suppose on the one hand that there is a fundamental level of structure, and the fundamental level of structure is continuous. Discrete structures can clearly supervene upon such a continuous substratum. As an intuitive example, just think of the manner in which a chess board is defined upon a continuous plane. Given a continuous substratum, one can divide it up into discrete, contiguous 'chunks', assign a discrete set of possible states to those chunks, and then define a finite set of rules by which those states change from one discrete time-step to the next. A more formal notion of such a discrete system is a cellular automaton. Even if we found that space-time is a cellular automaton at some level, it is possible that such a discrete level of structure is merely supervening upon a more fundamental continuous substratum.

This seems to be reasonably intuitive, but does the converse also hold? If we suppose that the fundamental level of structure is discrete, is it possible that continuous structures can supervene upon it? Well, in one sense, this is already well-known to be true: for example, solids and liquids are known to consist of discrete collections of atoms and molecules, yet because such systems consist of large numbers of discrete entities, they can be conveniently and approximately represented as continuous systems, described by continuous fields such as those representing pressure, stress, density, internal energy, velocity etc.

Yet this is merely a form of approximate supervenience; we know that solids, for example, are really crystalline atomic lattices, or polymer chains, and that continuum solid mechanics is merely a handy tool with a limited domain of applicability. Is it possible, however, that a continuous level of structure could exactly supervene upon a discrete substructure?

Let's think about this in more abstract, mathematical terms. The set of real numbers is said to possess the cardinality of the continuum. There is an infinite number of them, and they cannot be placed in one-to-one correspondence with the set of 'whole' numbers (1,2,3, etc), hence the continuum is said to be uncountably infinite. Within the real numbers, however, there are discrete subsets, such as the set of integers (...-2,-1,0,1,2,...), and the set of rational numbers. The set of rational numbers essentially contains those real numbers which can be given a finite decimal expansion, such as 23.45786, or a recurring infinite expansion. Numbers such as pi, which cannot be given a finite or recurring decimal expansion, are real numbers, but not rational numbers.

Now, given the set of rational numbers, the set of real numbers can be obtained from them by simply taking the limit points of all sequences of rational numbers. In other words, those real numbers such as pi, which require an infinite non-recurring decimal expansion, can be seen as the limit of an infinite sequence of rational numbers, each member of which has a finite or recurring decimal expansion. One says that the set of rational numbers is (topologically) dense in the set of reals.

Defined in this sense, the set of real numbers, a set with the cardinality of the continuum, clearly supervenes upon the set of rational numbers, a discrete set. Any change from one real number to another entails a change in the sequence of rational numbers with which it is associated. If we suppose that the fundamental level of structure in the physical world is discrete like the set of rational numbers, then it is clearly possible for continuous structures to supervene upon discrete substructures, and for the supervenience to be exact.

This example opens up a more general question for the ontology of mathematical physics: If the physical world objectively possesses a mathematical structure, then it presumably follows that it also possesses any substructure of that structure; however, does it also follow that the physical world possesses any superstructure within which that structure can be embedded? The answer to the latter question is surely 'no', for by taking a disjoint union of structures, one can embed the structure of the physical universe within a superstructure to which it is totally unrelated. The crucial additional condition which needs to be added is that of supervenience, and I propose the following:

Any structure which can be constructed from the apparent structure of the physical world, and which supervenes upon that structure, must also be said to physically exist.

The example considered above, in which one structure is densely embedded inside another, can be seen as one of the tightest supervenience relationships it is possible to define!

So, in conclusion, it seems that discrete structures can supervene on continuous structures, and conversely, continuous structures can supervene on discrete structures. Given this fact, it seems impossible to establish what the fundamental cardinality of the universe is, unless one can also ascertain that the fundamental level of structure (if there is one) has been reached. And how could we know that?

Being the helmet

McLaren, it seems, have a Human High Performance Programme. This double alliteration is slightly troubling, not least because the apparent need to qualify the programme as 'human', might be thought to imply that the 1998 World Champion Constructor is also in the process of preparing an army of androids or extra-terrestrials to fulfil some indispensable, high-performance function within the McLaren Technology Centre. It certainly places McLaren a step ahead of the opposition, who will have to rely upon mere fitness coaches, physiotherapists, and the odd sporting psychologist.

Of course, it may be that McLaren have taken on-board the lessons expounded in Clyde Brolin's 2010 book Overdrive, and recognise the performance benefits to be had when a driver enters a trance-like rhythm called 'the zone'.

In the spirit of Overdrive then, here's a particularly vivid account of what it feels like for a racing driver to be in such a state. What's most interesting in this recollection is that, whilst the driver doesn't lose his sense of having a spatial location fixed behind the eyes, he does describe an apparent loss of spatial extension. In effect, the driver seems to be describing a partial decoupling of the conscious mind from the body; there is temporal structure but no spatial structure to the driver's point of perception.

Those who fancy a challenge might wish to identify the driver.

"When I'm in that groove, I can go on forever. I wish I knew how I got into that state. I don't. I simply find myself in it...

"Then I drive out of that window in my helmet. I look through that window and what I see out of it is the sole and only thing that exists in the whole wide world; everything is happening out there in front of me. My legs and arms and every other part of me are just parts of a whole and doing what they're supposed to be doing automatically, so that I don't have to think consciously about gearing or braking or accelerating; that's all going on without any orders from me. I concentrate, intensely, on everything that's in front of me: be it a car or a corner, there's an invisible line extending from that window in my head to whatever's next. My body is in unison. It doesn't really exist; it's compacted, the whole of me is bunched up tight inside that little area of plexiglass. I'm entirely in my helmet and I think of myself as being the helmet, looking out. Everything, body or car, obeys that module.

"The sensation is not physical...I'm seeing more than I ever have before. My vision is enlarged and the sensation is purely mental."

Sorry, no clues.

The case for working with your hands

"The ex-boyfriend of an older housemate, Chas was a machinist by training. Currently he worked the parts counter at Donsco, the oldest VW speed shop in the Bay Area, in Belmont. He also built race motors for them and pitted for their off-road racing campaigns. Once a classical guitar-playing Buddhist vegetarian, he was now a gun freak and brilliant misanthropist. He still had long hair, but it was rarely released from the bun under his tweed cap."

"I once attended a conference entitled 'After the Beautiful'. The premise was a variation on 'the death of God', the supposed disenchantment of the world, and so forth. Speaking up for my own sense of enchantment, I pointed out, from the audience, the existence of beautiful human bodies. Youthful ones, in particular. This must have touched a nerve, as it was greeted with incredulous howls of outrage from some of the more senior harpies."

Matthew Crawford lived in a commune between the ages of nine and fifteen, worked as a trainee electrician and mechanic, obtained a degree in physics from UC Santa Barbara, worked in several fruitless and depressing office jobs, obtained a PhD in the history of political thought from the University of Chicago, and was briefly occupied as director of a Washington think tank, before finally returning to his true vocation, as a motorcycle mechanic. He's been around, in other words. And applying his personal experience and philosophical skills to an analysis of modern work, he's come to some very salutary conclusions.

The principal thesis of the book is that working in a manual trade is vastly preferable to working in the office. A trade, argues Crawford, has objective standards of competence, makes you responsible for your own work, is motivated not purely by the extrinsic desire for money but by the intrinsic good of fixing and building things, requires people to get outside their own heads, entails a constant acquaintance with failure that engenders honesty and humility, encourages self-reliance, and satisfies the fundamental human cognitive requirement to wield tools.

Crawford argues against the wisdom of sending so many people to university, preparing them for careers in a purported post-industrial, information economy, when most of those office jobs actually consist of mind-numbing, morally debasing, routine tasks. This is largely correct, although one might point to software engineering as an office job which also possesses many of the characteristics that Crawford finds so admirable in the trades.

The philosophy, however, is also entwined with delightful personal recollections of Crawford's life as a mechanic, hence this book could be seen as a harmonic overtone to Pirsig's Zen and the Art of Motorcycle Maintenance. The highlight of these reminiscences describe Crawford's disenchantment with academia, and his initial attempt to earn a living as a mechanic in a rented Chicago warehouse:

"There were...various litters of kittens and a rotating series of questionable individuals, usually 'in between situations', living upstairs in the unheatable, uncoolable warehouse, including one very sexy young S and M model and a pizza delivery guy who shot a man in self-defense and then skipped town, leaving behind only a Koran and a pile of porn. I'd gone from the Committee on Social Thought to this."

Quality in Analytical Laboratories

If there's two facts about the quality of an analytical laboratory that you'll never read in a Quality Manual, it's the following:

(1) Quality is primarily a function of the calibre of the people you employ, their skills and their levels of dedication.

(2) If you devise job definitions which make people's work excessively routine and repetitive, if there is insufficient opportunity to learn new skills, or to exercise discretionary judgement, and if you generally require your employees to follow the steps of a process which is prescribed for them, then those employees will lack pride in their work, will have low levels of dedication, and, unless you rigorously enforce performance metrics and have the power to sack underperforming employees, the work of those employees will generally be of a low quality.

Despite the evidential truth of these statements, the corporate management of an analytical laboratory will eagerly buy into the notion that quality is a function of management systems and processes. Such a notion not only appeals to the self-aggrandising instincts of the management, it also absolves them of responsibility for the thing they really hate doing: namely, thinking critically and imaginatively about the job definitions and career development of their own employees.

To be clear, good management processes and systems are still a necessary condition for quality. High calibre employees working within a bureaucratic, inefficient management system, will find their efforts stymied on a regular basis. The point, however, is that quality systems are not alone sufficient to ensure analytical quality.

Many managers of analytical laboratories are smart enough to realise this. They know that systems and processes won't solve the underlying problem of poor calibre staff, wafer-thin levels of talent, and levels of dedication which can only be measured with a micrometer. They despair, however, of solving this fundamental problem, and decide instead that their only goal should be to cover themselves. And in this context, the perfect way of covering your managerial backside is to introduce quality systems and processes revolving around documentation, audits and accreditation.

After doing all this, the analytical laboratory will still be as piss-poor as it was to begin with, perhaps even more so, but this is of no concern to the corporate manager, who will happily move onto his or her next role, proudly claiming to have transformed the quality of the laboratory. And they'll have the accreditation certificates to prove it.

The Quantum Theory of Formula 1

The Chapman Uncertainty Principle: The greater the performance level of a car, the lower the reliability level, and vice versa. Performance and reliability cannot be simultaneously optimised.

Quantization of income: The possible income levels of a team, prior to sponsorship revenue, occupy a discrete set of values, specified by the Concorde Agreement. If a team drops an income level, it will emit talented drivers and technical staff in response; conversely, if a team absorbs talented new drivers and technical staff, it will jump to a higher income level.

Bose-Ecclestone statistics: No pair of teams can simultaneously possess the same level of pre-sponsorship income, as specified by the Concorde Agreement.

Intrinsic spin: Each car-driver combination possesses a purely quantum property called intrinsic spin, which specifies the propensity of that car, under the control of that driver, to complete an unintended rotation about its centre of mass.

Quantum holism: The creativity of a design team cannot be uniquely determined by the reduced creativity of its constituent members.

Non-locality: Two drivers in the same team, with similar levels of performance, can become entangled even when separated by a considerable on-track distance. Thus, when one driver sets a faster lap, the other will also set a faster lap, despite the absence of direct communication between the two, and despite the frantic entreaties from the pit-wall for both drivers to 'preserve the tyres and fuel'.

The Measurement Problem: A team can enjoy a significantly higher level of performance than the other teams for only a brief period of time, before measurement-like interactions with the governing body induce a discontinuous and non-deterministic collapse of the performance envelope.

Bernie

"I've always felt that Bernie functions a bit like most people think the Mafia functions, and you can ask Bernie for anything, and if it's in Bernie's power, he'll do it. But then you're made. That's it. Then you owe him forever." (Ron Dennis, p345).

Susan Watkins's long-awaited biography of Bernie Ecclestone, published shortly before Christmas, is essentially a comprehensive list of all the deals that Bernie's done, and all the occasions on which he's out-witted people. It's extremely well researched, but short on revelation. Moreover, the latter half of the book, covering the labyrinthine politics and finance of the post-Brabham years, is about as interesting to read as a telephone directory. There's sufficient raw material here to make a good book, but the final product lacks the influence of a really good editor.

One shocking episode which the book does cover rather well, is the manner in which Bernie stitched-up Gordon Murray, the chief designer and technical director and race engineer and operational director of Brabham during the years in which they won two World Championships.

Up until the mid-1980s, Murray worked 18-hour days for Bernie, on a salary of £30,000, sometimes taking prescription drugs to cope with the exhaustion levels. Gordon apparently owned half of Brabham at this time, but eventually decided that he wanted out. A long meeting was convened with Bernie, and Murray recalls that "He convinced me that we were so far in debt that if I got my half of the company, I would be paying somebody...and he got me to sign a bit of paper resigning as a director...After fifteen years and two World Championships...'We'll pay you £30,000 and that's it'. Later he sold Brabham for five and a half million quid. My fault was believing it wouldn't happen to me...because I watched him do it to everyone else, you know, that's what he's good at...I watched and I thought 'He's never gonna do that to me after fifteen years and two World Championships', and in the end I got exactly the same treatment."

Identity criteria for Formula 1 teams

The ongoing controversy over the right to use the Lotus brand-name in Formula 1, highlights some subtle questions concerning the identity criteria for a Formula 1 team.

Companies, like people, are able to remain the same entity, despite changing substantially over time. Whilst philosophical discussions of personal identity revolve around bodily and psychological continuity, the identity of a company seems to hinge upon a combination of location, key personnel, name and ownership.

It seems fairly uncontroversial to propose that continuity of location, personnel, name and ownership, are jointly sufficient to preserve the identity of a Formula 1 team. Conversely, if all four of these conditions are violated, then it seems impossible to justify any claim for the preservation of identity; a team located in a different place, employing different people, bearing a different name, and possessing different ownership, must simply be a different team. Equally, however, none of these four conditions is individually necessary to preserve identity. Let us look at some examples to demonstrate this.

For a start, continuity of name and ownership is unnecessary to preserve identity, as clearly illustrated by the case of the 'Brackley-based team', which has, within the past decade, evolved from BAR into Honda, from Honda into Brawn, and from Brawn into Mercedes. The successive changes of name and ownership have not prevented this being considered to be a continuation of the same team. Hence, continuity of location and personnel alone are sufficient to preserve the identity of a team.

Continuity of location is also unnecessary to preserve identity, as exemplified by the Williams team, which in its current location at Grove is still clearly a continuation of the same team which was once located at Didcot. This case proves that continuity of personnel, name and ownership, is jointly sufficient to preserve the identity of a team.

The history of McLaren, meanwhile, provides another interesting case study. The merger between McLaren and Project 4 in the early 1980s effectively resulted in a change of ownership (from Teddy Mayer and Tyler Alexander to Ron Dennis and partners), and a change of location from Colnbrook to Woking. In this case, there was only continuity of name and personnel, yet it seems to be accepted that McLaren in the 1980s was a continuation of McLaren in the 1970s. It appears, then, that continuity of name and personnel alone are also sufficient to preserve the identity of a team.

Another interesting case is provided by the creation of the Arrows team from the key personnel within the Shadow team. (In fact, this is almost an example of fission, exactly the type of phenomenon which poses such difficulties in philosophical discussions of personal identity). On the basis of continuity of location, name and ownership, the Shadow team remained identifiable as the same team which existed prior to the departure of Alan Rees, Jackie Oliver, Dave Wass and Tony Southgate, even if the team had been fatally weakened in the process.

Abstracting from such cases, it might be proposed that the identity of a Formula 1 team is preserved if and only if the following criterion is satisfied:

There is continuity of: Either (location and personnel) Or (personnel and name) Or (location, name and ownership).

Irrespective of these complications, the preservation of identity seems to be impossible without some form of continuity. Setting up a new team with the same name as a famous, but defunct team, isn't sufficient to ensure preservation of identity. As the case of the Lotus name in F1 demonstrates, breaking all forms of continuity opens up the possibility of multiple entities, each claiming to be the modern representative of a famous historical team. And if there's one thing which the criteria for identitiy must ensure, it is the preservation of initial uniqueness.

Vorticity and helicity

There seem to be at least two types of vortex relevant to racing car aerodynamics: transverse vortices and longitudinal vortices. The former spin along axes which lie orthogonal to the direction of travel (as seen in the diagram on the left here), whilst the latter spin along axes parallel to the direction of travel.

Whilst transverse vortices just create drag, a longitudinal vortex can be rather useful, because the centre of such a vortex is a high velocity, low pressure region, which can be used to accelerate or direct airflow in certain directions. The Renault F1 team, for example, used a V-shaped cut in the front wing at Monza this year to create a vortex which accelerated the airflow under the car.

At first sight, there seems to be a particularly simple way to mathematically characterise the distinction between transverse and longitudinal vortices.
Given the fluid flow velocity vector field u, the vorticity vector field ω is the curl of the velocity field:



Basically, the vorticity vector points along the axis of spin, and the magnitude of the vorticity vector encodes the rate of spin. Given the vorticity vector field, mathematicians introduce several useful additional concepts: vortex lines, vortex sheets and vortex tubes. Technically speaking, vortex lines are the integral curves of the vorticity vector field; this simply means that vortex lines are curves which are tangent to the vorticity field at each point. Vortex sheets, meanwhile, are surfaces which are tangent to the vorticity field at all points. Vortex tubes are three-dimensional regions obtained by taking a 2-dimensional area orthogonal to the vorticity field, and then taking all the vortex lines through that area.

Now, the helicity h is simply defined as the inner product of the velocity and the vorticity:

h = u • ω

Thus, if the streamlines of the fluid are orthogonal to the vorticity, then the helicity is zero. This is the case with a tranverse vortex. In the case of a longitudinal vortex, the helicity is non-zero, and measures how tightly the streamlines corkscrew along a vortex tube. In fact, the helicity of a vortex tube can be defined by integrating the helicity field:





It is a theorem of inviscid fluid mechanics that the helicity of a vortex tube is preserved over time. However, if a vortex tube is stretched, (as I presume it must be when it is sucked underneath the floor of a racing car), then its cross-sectional area decreases, and the magnitude of the vorticity ω increases, lowering the pressure at the centre of the vortex.

One might therefore conclude that the stretching of longitudinal vortex tubes should be a principal objective of all racing car aerodynamicists.

King of the galaxy

"Bernie, who is the king of the world – more than the world, he is king of the galaxy." Luca di Montezemolo.

I knew it!

The Minotaur and the Monopole

In a moonlit glade of an enchanted wood, the Minotaur of Sirius-A sat and mourned the loss of his beloved Medusa, caustic tears burning a crater in the mossy ground betwixt his feet. The last survivor of a dying species, it was centuries since Daedalus-A's labyrinth had crumbled to ruin, and millennia since the Minotaur had feasted on the final consignment of virgins from Athens-A.

Such was the solitary intensity of his grief, the Minotaur resolved to be re-united with his partner by any means possible, and began furiously researching cosmology in the deserted library of Athens-A. There, he discovered that a universe just like our own - and therefore containing his dearest, alive again - could be created, without the need for an initial singularity, if one could only find a magnetic monopole.

The Minotaur, however, had no idea where to begin searching for such an item, and immediately lapsed into a pit of mythological depression, hurling himself out of the colonnaded library, and back to his friendless forest dell. There he lay forlorn in supine agony, and many days passed until, by providence, he descried above a lofty flock of migrating birds, and pondered anew the basis for such navigational feats. Thus it was that the Minotaur commenced a five-year effort, harvesting the magnetite from a billion birds, slowly implanting the precious ferrimagnetic mineral within his bovine skull.

Eventually, the harvest complete, the Minotaur tuned his magnetoreception to the faint magnetic fields which permeate the galaxy, and embarked on a thousand-year odyssey, pursuing hidden Teslatic paths between sparkling spiral arms and gigantic dust lanes, seeking hints and traces of the fabled monopole.

Finally, the grieving beast alighted upon a dark, brooding planet where magnetic field lines converged in densely-packed radial spokes. There, in a dark sulphuric cavern, between rivers of roiling magma, the Minotaur cleaved his monopole from an obsidian peninsula.

Summoning forth his cumulative, concentrated sorrow, he crushed the pulsing topological defect within his hand, compressing it until a black hole was formed. At once, a Reissner-Nordstrom space-time effloresced within, and a bubble universe blossomed with false vacuum energy inside the double black hole horizons.

The Minotaur was instantly sucked inside his own creation, his mass-energy converted to surface tension in the bubble wall. The child universe expanded and cooled, stars and galaxies formed, planets coalesced, life evolved, civilisations waxed and waned, and after billions of years, in a moonlit glade of an enchanted wood, the Minotaur of Sirius-A sat and mourned the loss of his beloved Medusa, caustic tears burning a crater in the mossy ground betwixt his feet.