Sunday 19 January 2014

Truncertainty

As a member of that most exclusive group of objects - "The things which physically exist" - you are imbued with a number of characteristics. First among them is obviously the matter of existing in itself, and that is the gateway to enjoying the other benefits of your membership such as going to the pictures or witnessing crumbs.
                Proud owners of a physical forms (or even leaseholders thereof) generally have predilection for keeping tabs on the characteristic of its position, or, where that physical form or "body" is. They will say "I am in Jamaica",  "I'm in the bath", or "Most of me is in the bath, but some of me has become entwined in the towel rack due to factors beyond my control". Other body owners may even share your interest in the location and engagement of your body, evidenced by exclamations such as "Where are you?", or in more advanced cases: "Why are you in my bathroom and what are you doing with my towel rack?".
                "Where are you?" should be a relatively easy question to answer for most active members and even some distinguished alumni of the things which physically exist. After all, if you can't even point to a space which you are certain that you occupy (generally within arm's reach of your arm) then your existence must be a peculiar thing indeed, and the board of governors may have to review your membership eligibility.
                Another characteristic of this "body" you so proudly wear every day is the speed at which it is travelling. A knowledge of this can be used to determine both where you were some time ago, and where you will be at some point in the future. As such, the value of this quantity is displayed prominently in all  motorcars so that the driver may ignore it in pursuit of important tailgating commitments.
                If you are an electron, this is the stage at which things become a little ropey. Not only can you not reach the pedals or the steering wheel, but if you manage know where you are, then you have no idea how fast you are going and could end up somewhere else entirely in any amount of time. If you are watching your speedometer then you have no idea where you are at the time and will most likely become close friends with a lamp-post, the tail you were so diligently gating, or a discount furniture store. You are a victim of a brutal conspiracy, courtesy of Heisenberg and his unwavering principle of uncertainty.
                As a conspirator, Heisenberg is a fairly eminent chap. He hasn't simply driven a sheet of steel into your car such that your head can be either a) above it looking out of the windscreen, or b) below it looking at the speedometer. He's gone one step further and decided that as soon as you so much as glance at one of them then the other is sabotaged; it is ruined in a fit of quantum mechanical spite - the worst kind of spite. Your knowledge or measurement of one observable  actively precludes the precise measurement of the other. He has either replaced your windscreen with an elaborate kaleidoscope, or glued a random number generator to your speedo. For these reasons, Mr Heisenberg can be inferred to be both a skilled mechanic, and a terrible choice of MOT provider.
                That isn't the only way in which he's been a tinker about people knowing too much about something either. The safeguarding of simultaneous knowledge applies equally to lots of other things, such as the momentum of one object in each of the three spatial dimensions. Should you be so bold, so daring, so ARROGANT as to try and know all three spatial components of your momentum at the same time (god forbid) then you will be mercilessly slapped down. Two components? That's fine. We LIKE knowing two. But all three? You make me sick.
                It's not a question of terrible measurement, it's a fundamental characteristic of the universe we live in, which happens to be a Hilbert Space (this is quite distinct from the crawlspace running behind Mr Hilbert's bath). Defining each observable quantity of a system with an operator and combining them within the mathematical constraints of a Hilbert Space, i.e. reality, causes these relations to fall out of the equations like racial slurs falling out of a UKIP party conference. Some of the operators just don't commute with one another, which means that for two operators A and B, AB does not equal BA, or in other words AB - BA is not 0 (Incidentally, this is not how numbers work, so a reassuring consequence of this observation is that we are not numbers. If this doesn't seem to describe you, then please pause to re-evaluate whether you are a human or the final balance of a weekly shop at Tesco getting rather above its station in terms of literacy.). This very roughly translates to the statement that measuring A first and then B gives you a different answer to measuring B first and then A. This is because knowing one of these things defines the system in such a way that the other thing could be one of a number of values. The value of the second thing is uncertain.
                It's one of those charming eventualities we all cherish, whereby trying to understand reality by plugging something we know into our equations leads to us learning that we don't in fact know what we thought we did. And when we think we do know it?  We not only don't, but actually can't. Or something.

Thursday 16 January 2014

Trousectomy operations

It is a truth widely acknowledged that humans have a skeleton inside them. They're quite useful in a structural capacity and have received a largely positive reception, with the notable exceptions of devout contortionists and militant mollusc groups (though it is speculated that the motivation of the latter stems from jealousy).
                The skeleton is made of a number of different bones, since early non-articulated prototypes proved impractical when trying to operate heavy machinery or search under the oven for a lost pea. Sadly, the pea retention rate of human skeletons is still far below ideal, but we do keep trying.
Now imagine you have cause to not be sure which of your bones is which. You look at your forearm and think to yourself 'Is that an arm bone in there, or did my skull get the wrong post code?'. Aside from ceasing to be devastatingly stupid, you would need a foolproof method in place if you wished to find out which of a finite number of solutions is the correct one for the (eternal) question "which bone is that?".
                It is a fairly simple thing to determine which of the possible outcomes is the correct one in this case, since the eyes in your face will be able to tell you if your arm is long and thin or if it has a jaw, and from this you could infer whether or not your last surgeon really still knew what he was doing after all that sherry.
                Your spirits buoyed, you step into the cold morning air and BAM. Some bastard has shrunk you to the size of an atom while you weren't looking. Except for your eyes.
Your comparatively huge eyeballs roll around for a bit and settle down conveniently facing you but gosh! You're too small to see, so now how will you work out whether that's a finger bone or a generous portion of your spine attached to your hand. The hand in question is in your ribcage, but we deal with one issue at a time around here.
                Thankfully, all you need to do is consult a group of physicists over a 100 year period and they will find some equations to tell you that at each point around your body there is a good chance of finding a bone, and that bone is a superposition of all the bones in your body. It is all your bones at once and none of them at the same time. By now you should be questioning a) why you asked physicists a question which was clearly medical in nature and b) why you never checked on their progress in the full century you waited. You won't actually get around to asking those questions because you're too small to be heard and you're probably dead. After all, 100 years is a very long time to be alive in the best of situations, and nanoNando's is yet to open in your area so good luck eating.
                What these seemingly immortal physicists seek to do is find out which of the many possible bones is in the region they are looking at by gathering the characteristics of the region in question i.e. the state it is in (its momentum, position or number of times broken by a swan (primarily arms)) into what's known as a wavefunction describing the bone-state. They can then apply an operator which measured bone type to that wavefunction. Operators carry out operations. If that is surprising then kindly leave, but don't forget to take a gift bag.
                The bone operator is a neat little machine they've made which can take an input of the information about the area of your body as a whole and spit out an answer of which kind of bone lives in that little fleshy house. So, you apply the operation (in this case a bonectomy) to yourself and out plops an answer which will be the sum of all the possible answers, weighted by how likely each one is. So, if you're looking squarely at the end of your legs, the answer will be mostly feet and some toe. If you look at your chest there will be a lot of rib, sternum and spine going on. Similarly, if you look at an atom, you can ask your operators "if I was an electron spinning upwards with this much momentum and THIS much angular momentum" here you would be throwing your arms wide to demonstrate not only how much angular momentum you have, but also how much you don't understand that angular momentum is not measured in units of distance "then where would I be? And how much energy would I have?" and get a meaningful answer out. Unless you did the maths wrong due to advanced cretin-hood.
                In the same way that you have a finite number of bones but don't know which one is which by looking, the quantum in quantum physics means that there is a discrete set of values the thing you're trying to describe can take, be it momentum or bone name (note that if your electron has bones then you should consider reviewing whether that's an electron or a particularly small haddock).
                The quantum mechanical operator exists as a mathematical manifestation of an observable quantity, something which we can determine and is a real characteristic of the system. By applying the operator to the equation describing the state of the system, it effectively simulates the act of measurement by spitting out the possible finite number of situations in which that state can exist given the conditions you spat into it. Spitting is especially important when it comes to operators. In itself that's only about as impressive as a man eating a kilo of chicken kievs without vomiting. However, by simply applying some operators and thinking about how reality dictates the maths must behave, a group of people managed to correctly write down some maths which described completely non-intuitive situations that they couldn't possibly imagine or understand, let alone measure or observe. Whilst that is simply the job for which the men who came up with the equations were paid, the point is that applied manipulation of operators and quantum mechanics allowed a man with a piece of paper to come to a conclusion that both required and caused the construction of the Large Hadron Collider in Geneva to prove him right or wrong. The maths is so accurate and has been thought about so much that just a few pages of it has led to a 27km ring of some of the most energetic particles in universe recreating the conditions of the big bang being constructed underground in central Europe. That's a multiple-tonne-consumption-of-chicken-kievs tier achievement at least.
                But what about your disembodied eyes and the mystique surrounding your skeleton? Well it appears that the only group of people around when you were mugged by what was apparently a cartoon super-villain not only had a century to kill and confused "immediate medical attention" with "extended period of high level theoretical physics", but they managed to spend that entire hundred years replying to the single utterance you made, namely  the question "which bone is this". Their answer was: "one of the possibilities". Another job well done!