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TEDx Talks
Published: 2015-08-04
Source: https://www.youtube.com/watch?v=smX2lSyi2js
theoretical physics what is that make you think of maybe I physics in school or maybe you think of one of the greats like Albert Einstein maybe think of fundamental particles the elementary building blocks of our universe I'm a theoretical physicist and and I think of these things but I spent an awful lot of time think about knots but I usually want to know about knots is
whether one not is the same or different from another not what I mean by this is can the not on the right be twisted and turned around and turned into the not on the left without cutting without using scissors if you can do this we say their equivalent knots and otherwise we see the equivalent surprisingly enough this question of equivalence of knots is very important for
certain types a fundamental particles Furthermore it's important for the future of technology this is what I'm going to tell you in the next fifteen minutes to get started we need some of the results from relativity now relativity is a pretty complicated subject I'm not gonna explain much of it but one of the themes that we learn from it is that space and time are mostly the
same thing so a little story to tell to explain this it's a story of Einstein's world and his day so we have at home his work he said among on the screen and does a clock in the upper right hand corner is to keep your eye on the clock during the day so I signed starts his day he goes to work then after awhile he comes
home for lunch clock keeps ticking he goes back to work clock keeps ticking in the afternoon he decides to go to the cinema goes the cinema clock keeps ticking and then eventually he goes home well physicist would look at this and will want to treat time more similarly as space and the way we do this is we plot space on an axis and we've thought time
on another access Einstein so called world line is this dark red line which tells you where in space is he at any given time it's called his world line because it tells you where in the world is he at any given time now we can go through the day and keep your eye on the dark red ball the but the ball goes up one step every
hour as we go through the day it goes back and forth in space tracing Einstein's position so the world Linus is a convenient way of keeping track of where Einstein is at any point in the day we can do the same thing with a more complicated world so here we imagine looking down on Einstein's neighborhood from a helicopter above so I signed sources day at home
he goes to work he goes to cinema he goes back home a student on the same day starts at home goes to library goes the pub and goes home now if we follow them both on the same day isixhosa were extinguished library eyes glisten in Austin because the pop eyes and goes home student goes home search look pretty complicated but we can simplify it by looking
at the space time diagram of what happened we do that by turning the neighborhood sideways plotting time vertically and Inocybe Gianna blue vertical line at the position of every object in the neighborhood it doesn't move such as the library or the pub they stay fixed in space and they move through time I signed the students world lines move around in the neighborhood as they go through
time now you can kind of see where I'm going with this Einstein in the students run lines have wrapped around each other if you pull those tight you'll discover that you have them nodded then we need one more thing from the theory of relativity we need equals MC squared again this is a thing that I'm not going to explain to much death but roughly what it
means is that energy and mass on the same thing so we have a particle in our world like an electron that's a particle of matter now each particle of matter has an opposite particle of anti matter in the case of electronic the anti matter particles called the positron both the electron and positron have mass if you bring them together however they can annihilate each other giving
off their masks as energy usually as light energy the process works in reverse just as well you can put in the energy and get out of the massive particles now what do you do the same thing we did with iSeries neighborhood were looking down at a neighborhood we put in energy to create a particle and the anti particle we put in energy to create a particle
and the anti particle then maybe we move one particle around another we bring them back together within our lifetime and we annihilate them with the scene the energy again now if we look at that in a space time diagram it looks a little bit like this time running vertically we put in the energy we put in the energy we wrap one particle around the other and
we annihilate them and we annihilate them again you can see quite clearly here that the world lies have nodded around each other do the same thing with more particles by putting in more energy move them around is a very complicated way and bring them back together the space time diagram would look a little bit like this making a very complicated not now here's the amazing fact
upon which the rest my talk realize certain particles called and eons exist in two plus one dimensions now I should probably say what I mean by two plus one dimensions two dimensions mean was are talking about a flat surface so these particles live on flat surfaces we say plus one dimensions we mean also time so we're just saying that the particles on the flat surface move
around in time so these particles called Anne on CIGS ist with the properties at the end depend on the space time not that their world lines have formed so you can kind of see now why I'm so concerned with whether to knots of the same over there the different we can conduct an experiment by which we create some particles move them around to form a knot
and then they have some property at the end of the not then you do the experiment again create the particles make another not and I want to know with the properties of the particles at the end of the not the same as they're participate because at the end of a different not this is why I'm concerned with whether not to the same or they different now
just looking at these two simple lots and may not be obvious that sometimes it's hard to tell if two not to the same with a different is there some way to unravel one and turn it into the other without cutting fortunately mathematicians have been thinking about this problem for over a hundred years and they've cooked up some important tools to help us distinguish knots from each
other the most important tool is known as a knot invariants I'm not invariant is an algorithm that takes isn't in put a picture of a not and gives isn't output some mathematical quantity a number of polynomials some mathematical expressions of mathematical symbols the important thing about a not invariant is that equivalent knots to Nasik can be deformed into each other without cutting have to give the
same output so I have two knots I don't know if they're the same or not I put them into the algorithm if they give two different outlets I know immediately they can't be deformed into each other without cutting that they're fundamentally different not now in order to show you how these things work I'm actually gonna show you how to calculate I'm not invariant the problem here
is I have to give you warning that there's going to be mass now I've given this talk and high schools before and and nobody died so so I suspect most people can handle the amount of math and going to do but I know some people have a math phobic like the person in that slide and if that's you just close your eyes when you get scared
open them up later and everything will be okay you won't miss too much so that not invariant we're going to consider is known as the Khalfan invariant or the Jones invariant we start with the number which we call a a stands for a number in this case so the first rule of the of the company variant is that if you ever have a loop of string
a simple loop with nothing going through it we can replace that loop but the algebraic combination minus a squared minus one over a squared and that combination occurs frequently so we call it de so anyway this first rule is that if you ever have a loop of string with nothing going through it you can replace that loop with just the number de the second rule is
a harder all this rule says if you have two strings across over each other you can replace the picture with the two strings crossing over each other with the sum of two pictures in the first picture the strings go vertically in the second picture the strings or horizontally the first picture gets a coalition of a out front in the second place to get copies of one
over a out from now this may look very puzzling because you've replaced the picture with the sum of two pictures we put numbers in front of those pictures so now we're talking about adding pictures together as well as putting numbers in front of our pictures but all we're doing is we're doing math with pictures and I'll show you that it's not that hard by actually doing
a calculation we're going to do gonna take our rules and we're going to apply them into a very simple not this very simple not this figure eight looking thing while secretly we know that it's actually just a loop of string that we folded over to make it look like a figure eight but suppose we didn't know that suppose we weren't so clever to figure out that
we could just unfolded and make into a simple little we would go ahead and try to calculate the Calvin not invariant by following the algorithm so what you do is you look at the knot and you discover the two strings are crossing over each other so I've circle that in that red box now within that red box we have two strings crossing over each other so
we can apply the rule and replace this two strings crossing over each other with a sum of two pictures in the first the teachings go vertical in the second shoe strings go horizontal in the first you have a coefficient a out front in the second of coefficient won over a out front now we just fill in the rest were not exactly like it is over on
the left so now we replaced one picture by some of two pictures with appropriate coefficients now in these pictures they're still crossings are in the not down below we have on now indicate that he blew and we have to apply the Calvin rule to these crossings as well which we do exactly the same way and now we have a sum of four diagrams with appropriate coefficients
at this point we've gotten rid of all the crossings and we're left with only simple loops and simple loops by the first rule get a value of D. so each time we have a loop replace it by a factor of D. so in the first picture for example these two loops circuits D. squared the second picture is just one big loop so it's a factor of
D. and so forth at this point we are now down to only symbols are no pictures laughed so we have aids Indies so it's just some algebra at this point so we combined together some terms that we use the definition of D. being minus a square miles one owner is great to replace this by minus de we get a DQ canceling a minus DQ at the
end of the day we get D. yeah so this is why is this get yea this is exciting for two reasons first of all it's exciting because this is the end of the match the second reason is exciting is it because of the result giving us D. the reason that it's interesting that we get D. is because of the beginning what we actually started with was
just a simple loop we folded over to make it look like a figure eight but it was a simple loop and the cow can invariant of a simple loop is just de now even though we folded over to make it look more complicated when we went to this algorithm at the end of the day we get D. that's I'm not in various work we could fold
it over a hundred times made it look incredibly complicated but still it would have given us D. so if we have these two knots here and one know if they're the same or different we put them into the algorithm and we get out to different algebraic results these results don't equal each other and so we know immediately these two not a fundamentally different they cannot be
turned into each other without cutting the strands so someone gives you this not you might say well go ahead follow the algorithm and see what comes out unfortunately Calvin in various exponentially hard to calculate what I mean by that well in this picture here we had two crossings we ended up with four diagrams each time we had to evaluate a crossing we doubled the number of
diagrams if we had three crossings would eat diagrams for crossings we had sixteen diagrams and so forth and so on so in an in this not we have about a hundred crossings which would be two hundred diagrams that number is so enormous did the world's largest computer would take over a hundred years to be able to evaluate the captain in vain of this not so you
might think well if I have a complicated not a value in the Khalfan invariant maybe isn't that interesting after all but let's go back to this amazing fact these particles called and eons exist where the properties at the end depend on the space time not that their world lines have formed what does that mean well precisely the probability that the particles will annihilate at the end
of the not is proportional to the Khalfan invariant of the not square so if I have these and eons by measuring whether they annihilate I can estimate I can measure the company variant of complicated not the way you do it is you produce your Anne on you move them around to make this complicated not and then you try to annihilate and you see if they annihilate
he did many many many times so you get a very good estimate of exactly what the probability of annihilation is and so you've measured the Calvin invariant of this not why is that interesting well the reason it's interesting it's because his company variants are exponentially hard to calculate the world's biggest computer would not be able to calculate the captain in vain for that not even a
hundred years but these anti ons can do it these enhanced can solve this exponentially hard problem no that's kind of an interesting thing to know just a fundamentally that these particles have a way of calculating something that our biggest computers still can't do but maybe we're not so interested in talking in the cabinet variant of a not it turns out that this anti on computer can
do the same calculations as any so called quantum computer can do now I'm not gonna explain what a quantum computer is but roughly a quantum computer is a type of computing device that uses the odd properties of quantum mechanics to do calculations that modern computers essentially cannot do it all this particular type of quantum computer that uses and eons in knots is known as a topological