GAME2020 3. Professor Anthony Lasenby. A new language for physics. (new audio!)

Transcript

okay well thanks very much and thanks to stephen for the invitation for coming here um so as he said i'm actually an astrophysicist and cosmologist and i got into geometric algebra over 30 years ago and i use it a lot for things i actually do in gravitation and cosmology and what i want to do today is to give you a sort of overview of where we've got to in applying geometric algebra in physics and it's an incredibly useful approach to the mathematics of physics the point is it allows you to use a common language in a huge variety of contexts so there's just just starting off with a partial list of all the different uh mathematical um bits of apparatus which it can deal with and subsume and show how similar they are and you just need one apparatus the geometric algebra to work with it so it includes complex variables vectors quaternions matrix theory differential forms tensor calculus spinners twisters they're all subsumed in this common approach so this gives you very great efficiency you can quickly get into new areas i've been able to get into areas i never thought i'd be able to approach because the mathematics just look too difficult but you don't need new mathematics to get into these areas you can use geometric algebra and that's just an incredible efficiency and gives you the opportunity to actually get to grips with the concepts in these areas without the maths being totally unfamiliar and it does tend to suggest new geometrical and i'll say therefore physically clear and coordinate independent ways of looking at things and you'll see that a few times as we go through some applications it's not just in a sense it's a mathematical tool it's a conceptual tool and so you can think in a slightly different way about a problem and maybe get some geometrical type inside that you didn't have before which will introduce some new ideas you could have about the physics so um what i want to do today is to introduce a few aspects of it in more detail and this will be principally applications to electromagnetism quantum mechanics and gravity and in fact you could take this as a sort of subtitle for the lecture the four fundamental forces of nature how do we treat these in geometric algebra and does it give us some new insights and just a reminder of course these are the four forces electromagnetism that's the force which keeps atoms uh together the electrons in their orbits the weak interaction that's responsible for radioactivity the strong interaction is what binds nuclei together and therefore incredibly important to give us the building blocks of atoms and then gravity which stands rather apart from the others it seems but we'll show today how you can treat it in a way that makes it really look much more similar um now there may there's just a few people here maybe who haven't been to the preceding lectures um but i felt because of that i want to give a short introduction to geometric algebra itself so apologies if you sat through this several times i will go quite rapidly through this first bit just explaining the very basics of geometric algebra so we know that in complex numbers as a unit unit imaginary i is properties i squared equals minus one and of course this was a mathematical problem for several hundred years people worried what is this thing that could square to minus 1 because no ordinary number does that and so in the end they settled on an axiomatic definition and just asserted there is a number with this property and complex numbers are the form x plus i y x y ordinary numbers but consider this suppose we've got two directions in space a and b these are just vectors as usual and suppose you had a language and in which you could use vectors as words and string together meaningful phrases and sentences with them so a b b a b etc would all be meaningful phrases and then you have just two rules if a and b are perpendicular then a b is minus b a so they uh anti-commute if a and b are parallel in the same sense then a b is just the product of their lengths so here's your rule for multiplying vectors in this case and just this does an amazing amount of mathematics suppose you have two vectors unit vectors of right angles so they square to one and they anti-compute then this is the basic calculation now of geometric algebra what we're just about to do right everyone should make sure they uh go through this for themselves it's very simple look at e1 e2 squared that's e1 e2 e182 we can anti-commute these two in the middle to get a minus sign then we have minus e1 e1 e2 e2 which is -1 and then suddenly this sort of cuts through hundreds of years of mathematical problems you have an object a geometrical object which squares to minus one and you can now see that complex numbers are essentially objects of the form x plus e1 e2 y and we call this c1a2 a bivector and we think of it as an oriented plane segment uh swept out and going from e1 to e2 i've drawn the more suddenly here but more generally given any two vectors a and b you can form a which b where you sweep out over the angle between them and this builds up to further dimensions like this you start with a scalar here's a vector directed line segment here's what we've just been doing and of course these bi-vector planes have a associated orientation and then you can carry on take one of these parallelograms you formed sweep it out over a further vector here c and get an oriented volume so you can keep going to as many dimensions as you've got so let's look at this more formally you've got a vector space for the usual inner product and we've introduced this outer wedge product to produce this bivector and the essence of geometric algebra this is what clifford did he united these two things into the single product called a b so a b is a dot b plus a wedge b and the key thing is that this product's invertible that's why it's so good individually you cannot invert these you can't strip off a at the left but here you can you can multiply on the left and remove the a from this so in a proper development what you do is you take the geometric product itself because a b is primary and then you write in terms of it the dot product has a half a b plus b a uh the ant commutator and a which b is a half a b minus b a so that's how you get things going aximatically so moving up to 3d which we already did when we saw how we swept out that parallelogram along a further vector so now we have three vectors all squaring to plus one they're all mutually orthogonal and now we can form their wedge product all three of them that gives us uh i the pseudo scalar for the space it's an oriented volume element for the space and it's quite nice that if you take i times any vector you get a bi vector because imagine we've got high times e1 then the e1 e1 vanishes you just get left with uh e2 that should be e3 sorry that's wrong uh ie2 is e3 e1 i3 is e1e2 and now there's something quite interesting because we squared e1 e2 which was the pseudoscaler of 2d space we got minus 1. what happens if we square this thing we found which is a tri vector which is a pseudoscaler with a 3d space you work it out you get minus one so this is quite instructive we've now got another real geometric object which squares to minus one and the general lesson is there are many such objects with square to minus one in physics people have tended to reach for little i right this uninterpreted unit imaginary they've uh reached for that whenever they've had something squaring to minus one and so it's ubiquitous in physics but what this approach makes clear to you find out that in several cases these little eyes are masking something else they're incorporating a bit of geometry which you could have been using and smuggling in this unit imaginary when it shouldn't be there quite possibly and in fact it means you seldom have any need for complex numbers okay we call the highest grade object in the space the pseudoscaler is unique up to scale and the two further things we need to just be very clear on are reflections and rotations uh as you've seen several times reflections very easy to implement in geometric algebra if you've got a vector a you want to reflect it in the direction n to produce new vector a dashed is really very nice you get this sandwich product a dashed is minus n a n and just to give one more little example again this is something you should work through for yourselves of uh you know a basic thing in geometric algebra why is that true well let's take two n wedge a times n we know n by j times two is that a minus a n n squared is one so this is n a minus a and so if i now add into here n dot a inside that bracket then these things combine to give me just n a so this thing is 2 n a n but i'm adding 2 a dot n n to this like that you look at this look at this equation and that equation is minus n a n is a minus two n dot a n so that's the essence of the sort of calculations you can do in geometric algebra they're really quite simple once you've got these basic uh rules and this is a really nice result so as you've heard from other speakers um rotations you can get to these very quickly because there's a fact that two reflections for rotation and so let's reflect in n and then in m we do that like that you can see that what piles up on this side is n times m but you get the reverse order on the other side and so you define a rotor which does this to be in this case the m times n rotations are given by a goes to r a r reverse what's this r reverse that's where you reverse the order of these objects inside it so r reverses nm if r is m n and so of course that means that r times r reverse is one which is the other criterion for a rotor so this works in spaces any dimensional signature which is incredible it's a real unification and it works for all grades of multi-vectors so uh it doesn't matter what a is whether it's a vector a bi vector tri vector however whatever grade it's got exactly the same thing happens to it under rotations which is really useful [Music] and then finally on this first bit uh we can have a general rotor by exponentiating a bivector in 3d as we've said we can get the bi vectors by taking the i times a vector and so in 3d it's sort of accidental uh symmetry of the space in a way that you could always write a bi vector like this as the dual of a vector and so of course this was the reason for decades of uh misunderstanding about rotations what you should always do is thinking the rotation as being a plane i bi vector oriented segment of a plane and not a direction this n which has come in here it's only in 3d that accidentally we end up with something that um can be dual to a direction so the bi vector b is the plane of rotation and the rotor is then a scalar plus a bivector as you see uh just here um this ties in perfectly with the quartenions here's a quarternion here's its definitions and we get a complete map between the two like this all you have to worry about is that the j in quaternions is actually minus ie2 so people doing quartalions versus people doing this e1 e2 e3 algebra there's actually a change in the handedness between the two okay that was the introduction now let's go to space-time so this is the effectively a new bit so in space time we want to construct the geometric algebra corresponding to it we'll be uh looking at uh preserving this invariant interval so this is the equivalent in euclidean geometry right of x squared plus y squared plus z squared when you go to relativity the thing you want to preserve under transformations is now here s squared is c squared t squared minus x squared minus y squared minus z squared we're going to work in natural units where c equals one so i think you won't see another c anywhere else in this presentation it's far more sensible for us to have c equals one speed of light so now we need four vectors and looking at this we can see how they need to square e naught squared the time vector is going to be one e one squared is minus one e naught dot e j uh zero and e i dot e j is minus the kronecker delta what's going on there we're just saying that the spatial vectors because of this signature we've used here i'm going to square it to minus one and everything's orthogonal and you can summarize that in this way uh in relativity you normally use suffixes greek indices which run from zero to three call those mu and u then e mu dot e nu is the diagonal plus minus one is minus that's called the minkowski metric uh and written e to mu u let's now start forming by vectors from these there's six pi vectors in the algebra that you can get and they divide into two types so there's one type containing e naught and they're these e i wedge e naught for i equals one two three and those not containing e naught and they're like this e i wedge e j and um how these differ why they're uh singles out in this way is they have different squares so here's a general result for any pair of vectors a and b with a dot b equals zero then a wedge b squared because they're orthogonal we can write that as a b so we have a b a b if they're orthogonal we can flip the a and b here we get minus a b a and then that's equal to uh minus a squared times b squared because the b in the middle just becomes b squared and we've got the a outside so that's minus a squared b squared so going back to these uh bi vectors here we can easily work out what they square to and the ones that don't have the e naught you square those up you use this rule here they go to minus one and because we've effectively flipped sine we our euclidean part we've got the opposite signature this just has generally done in special relativity it's the time bit which has the positive signature these are actually euclidean all right and they're just the ordinary uh they generate disorder rotations in a plane as we've been discussing the time like bivectors the ones which got the e naught in they square to plus one and they generate a hyperbolic geometry so when you exponentiate these like this one i've got here e to the alpha e one e naught you expand that out it goes to cosh alpha plus shine alpha e one e naught whereas if i exponentiated uh something without an e naught it would go to a cos of the sign and of course this is crucial to the treatment of the lorentz transformations and we'll look at that in a bit more detail soon how about the highest grade element of the space well that's the pseudo scalar i which is e naught e1 e2 e3 uh it's grade four now any grade four element is always going to reverse to itself so i reversed you work it out that's equal to i what does this one square to because uh we should work out for each element we get to what is square is to understand it a bit better i squared is the same as i i reverse you work that out and again we get minus one so here's yet another thing that squares to minus one but again is the geometrical entity so if you multiply a bivector any of these bivectors we're talking about by i then you get a grade of four minus two this is duality uh which is two and so you get another bi vector so i uh transforms by vectors to pi vectors and in doing that it provides a map between bi vectors with positive and negative square so just an example here's a spatial vivect sorry his uh a bi vector involving um the e naught so this is time like so i e uh one e naught is the same as e one e naught i it uh i anti-commutes with each vector so when you've got two vectors there's a bi vector it commutes i commutes with all even grade elements and you work that out you get minus e2 e3 and this is one of these what we call spatial bi vectors which square to minus one d squared plus one and of course that makes sense because one's got i in you know there hasn't and the i squares to minus one so how about vectors and what and tri vectors there's four of these and they get interchanged by duality again so i moves by vectors to bi vectors when you multiply and if you start with a vector like here we've got i e naught that's the same as e one e1 e2 e3 with a minus sign and i've already said i anticipates with vectors and tri vectors and always commutes with the even grade elements now we've called everything e so far what we're going to do before carrying on is we're going to settle on some given uh fixed cartesian frame of vectors in which physics you can think of this as a laboratory frame and we're going to rename our emu as gamma mu right so you'll see very shortly why we do that why we call these things gaomu and we're basically done now we now have available the basic tools for relativistic physics which we call the space-time algebra and here's the ingredients we have one scalar four vectors the gamma muse mu zero to three six by vectors four tri vectors dual to the uh vectors and a single pseudoscaler as shown there and uh yeah so we're using this new name let's take the dot product of gamma mu times gamma u and we know that's the uh minkowski uh metric the e to me mu that we talked about before that's how we set everything up and so in particular let's expand that out we know that gamma mu dot gamma u is gamma mu u plus gamma u mu and the dot product has a half so bring the two up here and here's why we call them gammas because for people who do relativistic quantum mechanics this is the defining relation of the gamma matrices that dirac uh invented well he didn't actually invent them in this point in this gamma form but um they're the key to doing relativistic quantum mechanics and so we've got to the direct matrix algebra and that explains the notation because um these things will be matrices in relativistic quantum mechanics uh and they're all called gammas but for us they're just vectors they're not a set of matrices in iso space they are actual vectors and so you get this is the first sort of astonishing thing that happens these matrix representations for relativistic quantum particles which um you know people in the field extremely familiar with turn out to be just the basis vectors of um 4d space which is very striking that that happens now exposure in some inertial frame uh in that frame it's possible to define a set of relative vectors and these are space-time areas swept out while moving along the velocity vector of the frame so i've illustrated that here the velocity vector of our frame is gamma naught we start with a vector gamma i we sweep it out as time passes it gets swept up here to some new gamma i and in that sweeping out we develop a bivector which is an oriented plane segment and we label that sigma i all right so this is the definition sigma i is gamma i times gamma law so the spatial gamma vectors multiplied on the right by gamma naught give you this sigma high and of course they're actually space-time bi vectors that they can function as spatial vectors in the frame orthogonals gamma and again this is quite interesting idea that um as human beings we're we're very we're not used at all the idea that we're actually shooting along a time axis at the speed of light at any moment that's what we're primarily doing we're moving extremely rapidly up the time axis and so these vectors which we have around us you know cartesian basis vectors in this room are actually being swept really rapidly and so it's very interesting that in the 4d way of thinking about that we they're bi vectors because that's sweeping out but coming back down the other way we can think we can use the gamma naught to define these things that are actually vectors for all intents and purposes in this room so what about these sigma i's it's easy to show from what we've already find that they satisfy this sigma is sigma j again we just take this anti-combination product sort that out that comes to since it only involves the spatial gamma i as it comes to delta i j so their dot product is delta i j and if you work out the anti-comet of the sorry the commutator uh it's easy to show that's the alternating symbol epsilon i j k times i sigma k so actually these sigmas we've defined are precisely satisfying the same relations as the poly spin radiuses so i've done this an unusual way around people normally do paulie quantum physics and pauly spin and then you work up to the drag equation here we're starting with uh direct physics and then coming down and finding oh yes we can define these relative vectors which satisfy all the properties we need for the pali spin matrices this displays the matrices explicitly the hats indicating on these quantities that they are matrices so okay i've said this is the same as the um algebra of the pali spin matrices what you can also observe are these relations are precisely the same as relations for an e1 e2 e3 type frame with things squaring to plus one uh in 3d space so actually we've recovered the geometric algebra of 3d space in this relative space form we've got it just by basically multiplying on the right by gamma law and now a particularly nice feature of this is that there's a bit of economy that happens here the pseudo scalar of the 3d space which is sigma 1 sigma 2 sigma 3 you work that out and it comes to the same as the pseudoscaler of the 4d space it rearranges to give this thing we called i so basically the 3d sub algebra shares the same pseudoscaler as space time which is quite an interesting realization so we've got this picture here that we've ended up with uh where we can have at the 4d level the full algebra one the gammas sigma i i sigma i so there's six all together of those i gamma mu and i and then if we just take the even space so the scalar the bivectors and the pseudoscaler as we drop down these ones here the sigma eyes uh end up here as the basis vectors so odd elements of the 3d space and these ones here become the bivectors so this is very neat it means that again starting at 40 is the way to go because then we come down and find out that we have this algebra for 3d space using this even parts of what we had in 4d space and that's an observer dependent split which makes sense because each you know three frame we work with is relative to some observer we'll have a velocity now let's look at uh the rents transformations which is where relativity started effectively so how you'll probably if you've worked with florence transformations you'll probably have seen them looking like this which is not very friendly but basically it's all seen as a coordinate transformation you have some coordinates t x y z for an event in one frame and you want to calculate those coordinates in a new frame this shows it for just the t and x components and the apparatus you need is this gamma factor it's not to do with any of the other gammas we've had that's one minus b squared all inverse square rooted and beta's the scalar velocity so b over c so that's what it looks like and when geometric algebra starts to work on this the first thing you think of is what's invariant what's the geometrical object and so the geometrical object is a position vector of where some event occurs and it shouldn't matter which frame you use to express that so that's got to be the same in a frame emu or in a frame emu dashed and so in particular xmu emu equals xmu dash emu dashed now i'm just going to give a little aside here because it's relevant to what happens a little bit the relation of coordinates to these frames comes from the notion of reciprocal frame this is another extremely important part of geometric algebra in physics given a frame emu you can define the reciprocal frame e upstairs mu right this one was downstairs this is upstairs like this you start with your downstairs frame and you demand that the upstairs frame when it's dotted into it like that gives you the kronecker delta so you've got also you've got orthogonality to the downstairs frame in the upstairs frame and it's normalized orthogonality because it's one on things that um indices that match so in particular the gammas that we've been looking at since they square to something negative or that square to minus one uh in their downstairs frame the upstairs frame introduces a minus whereas the gamma law doesn't flip when you go upstairs and so with these definitions of a reciprocal frame then it turns out you get upstairs coordinates like this where you just dot the position vector with the upstairs frame the reciprocal frame so in particular the time is e upstairs zero dot position vector and uh the time dashed will be e upstairs uh zero dashed dot the same x all right so in this approach we just think of events in space time and they're labeled with the position vector and then it's up to us how we peel off coordinates from that position vector um as i'll show you shortly it's this whole thing is very useful for working with curvilinear coordinates if you happen to have those and that articulates extremely well with geometric calculus okay so concentrating on the zero one components we've got t e zero plus x c one must be t dashed e0 dash plus x dash v one dashed now using these uh relations here the lawrence relations it turns out if i then write uh use those in these expressions here you can peel off the e naught dashed is this gamma factor times e naught plus b to e one and e one dashed is gamma e one plus b turkey lord now i regard this as totally a backwards way of doing it because this is the thing that is actually obvious because where this comes from is jumping uh i'm just going to jump to this to show you it comes from taking your initial frame and acting on it with a rotor to get the new frame where the rotor in this particular case if we have a velocity in the uh one direction is e to the alpha e one e naught over two it's a hyperbolic transformation we looked at before and this just goes through making sure how one understands how alpha relates to beta beta is the ordinary velocity it turns out that that's the same as tange of this hyperbolic angle alpha and so what's this gamma thing that we had back here okay what's the gamma that is shown just here that is one minus tan squared alpha to the minus half which is cosh alpha and so that means the e naught is uh which we were able to peel off from here by going in this direction that is equal to cos alpha e naught plus sine alpha e one and that is the same as e to the l for e one a naught times e naught similarly the other one the e one dashed is equal to this times e one and if you look at the fact that the e two and e three frames don't change at all you deduce the transformation in this form of course that's because we're starting with the lorentz transformation itself that look quite complicated by far the easiest thing is always only to work with this sort of approach where you rotate your frame vectors and then you can read off things like this if you absolutely have to um let me give you an example of that in in a moment about how useful this approach where you start like this it is basically the same rotor prescription works for boosts as well as rotations boosts where you've got the velocity in a particular direction and it makes space-time a unified entity and you can generalize all this to r equals e to the b where b is any bi vector in the space time algebra and this gives you fully general lorenz transformations and again given any object in the algebra you just rotate it in the same way m dashed is r m are reversed so let's try and actually use this uh in something that you you may have come across and this is the context of lorenz transformations in electromagnetism so electromagnetism uh deals with uh when you look at it relativistically is a thing called the faraday so i'm just writing here the conventional faraday tensor which is an anti-symmetric 4x4 tensor we'll see shortly what it really is in geometric algebra so as a matrix it's got these components uh where what's happening here is you've got the e field they go on the top row and then down the first column and then on these in these positions here the um positions which are off again off the diagonal because this is anti-symmetric we have the magnetic field bx by bz in that particular arrangement now that is how people doing relativistic electrodynamics think of the field strength tensor the faraday they think of it as this matrix you often see this but it hides the natural complex structure in our version f is a space-time bi vector and you generate it from the electric and magnetic fields e and b which are relative vectors in the three space orthogonal to the time axis so i'm going to use these bold letters for a relative vector and that means since the relative vectors in this uh space orthogonal to the gamma naught therefore they're interpreted in the sda they're actually bi vectors so these e and b fields are actually bi-vectors and what's the faraday it's just this it's e plus ib where i is a pseudoscaler for 4d space and this gives suddenly all sorts of wonderful intuition about what's going on with electric and magnetic fields they were never re properly vectors they're these relative vectors which become bi vectors in the full 4d field and when you're thinking about the difference between them it's because these ones involve the gamma naught and these ones being the form ib don't involve the gamma law that gives you the difference between magnetic and electric fields if you want to go back to the matrix elements you can get them very simply like this the component set mu u uh gamma nu which gamma mu dot it with f and this is very characteristic you take a geometric algebra thing like here f and if you really want to get components out of it in some sort of standard tensor calculus type way you can and it's generally done by things like this dotting with the the gammas now how can we uh recover e and b individually we can do it if we know gamma naught if we know the velocity uh vector of the frame and that's because if we surround f with gamma naught you get because it can't pass through the e without flipping because he has got a gamma naught in it plus something uh else then it flips to minus c whereas this one it passes straight through and so you can recover the e and i b individually like this is a half f minus gamma naught f gamma naught ib is a half half plus gamma l have gamma naught just hold on to this thing about gamma n or f gamma to right near the end we'll find out this is crucial in the strong force that we talked about that binds nuclei together anyway here the split into emb depends on an observer's velocity and if you had a different observer moving with a different uh speed like this so if v is r gamma or r reverse they'd have a co-moving frame gamma mu dashed given by you take all the frame vectors you rotate them in the same way what would they see for the electric field well you can work this out because they've got a sigma i dashed dotted with f again here's the recipe there's just a single geometric entity sitting there but if you insist on knowing what its values in a particular frame you just dot with that frame you work that out that's the same as back in the original system the sigma eyes the back rotated faraday so all we have to do to get the new uh electric field strength is bank rotate the faraday and dot it with the original sigma eyes so you get exactly the same transformation laws for vectors i this rotor thing it's very efficient and for example let's carry out a case let's have stationary charges and the gamma dot frame set up a field like this just pure electric f is e e x sigma one plus e y sigma two we have a second observer they have velocity tangent alpha and gamma one direction and so the relevant road trips one we talked about before is e alpha sigma one over two and let's apply that to f so we take that we apply it backwards as i've just shown you to get the components of the new frame and so let's put that row to either side of this the sigma one will just float straight through that we won't see it and then the ease will um because we've got an r reverse here we one of these got a minus one's got a plus so you just get no change in the sigma one component which is the e field in the x direction the other one though as it goes through the sigma two you pile up an e to the minus alpha sigma one um and so if we write this in terms of caution shine we can identify then that the electric field is unchanged in the y direction we've got a cosh right expanding out this uh exponential of the sigma one uh for the y field and we've got a new bit because we've got uh when i expand out this exponential as caution shine the sigma one sigma two they're going to hit each other and give me an i sigma three what's an i sigma three that's a magnetic field in the z direction and so in this very elementary way you find out that the motion induces because you had a y field in the electric uh sorry y electric field that induces magnetic field in the z direction so you can really understand why you've got uh this happening and it comes down to with the clifford algebra very basic that sigma one sigma two is i sigma three is why this um z component of a magnetic field is induced now this is this sort of thing um is very nice i find because it's completely sort of rock solid conceptually we know that all we're doing is this and then you work it out and you get some definite dance and you feel confident in it this is really quite different from what you get if you've ever tried this in conventional approach these sort of calculations take pages to do in standard uh tensor uh component approaches and you generally don't feel very confident at the end so actually it really is the case this is definitely much simpler than working with tensors okay so um that was the first little look at electromagnetism and come back to that shortly but to do that we're going to need an additional entity which is the derivative operator and how do we do that we do that using this reciprocal frame we discussed up we said if e downstairs mu is a frame then the reciprocal frame e upstairs nu is defined like this which we went through and just to emphasize here these upstairs things are vectors just like the e mu they live in the same space people sometimes get confused about this and think oh well you're defining a one form space but not these are just vectors in the original space that they have this relationship with the downstairs frame so let's define a vector differential operator will use e mu equals these gamma mu says cartesian frame to start off with his definition the gradient operator is the gamma mu upstairs d by d x mu like that where the x mu are the coordinates and i'll often write d by dx mute in this simple way of partial with just a mu at the bottom that's d by dx mu and you can see that this operator combines the pieces of a derivative obviously in the partials and uh geometric algebra in that it's got these gammas um now so as things go on and maybe some of it starts to look more complicated just as a quick encouragement with just this object and the sta that we've done so far you can basically do all of electromagnetism and quantum mechanics right through to electro weak theory without introducing any new mathematics all right so it's quite a strong claim but uh i think you'll find it's true so just to say the definition of gamma here not gamma sorry the derivative operator did not you need to use rectangular coordinates and the orthonormal system gamma mu you can do this for any arbitrary coordinate system x mu for example suppose we were working with um spherical polars in space time that would be t r theta phi would be our x naught x one x two x three that would be fine as because what you can do you can get these reciprocal frames in a very nice way um uh from either the position vector x or from the coordinates x mu how does that work it turns out the downstairs frame is the partial derivatives of the position vector x the upstairs frame is the gradient the the derivative operator acting on the coordinates right um so you can write this out to yourself have a go at this define like this all right so i've got coordinates and i express my x um position vector in terms of them the downstairs frame emu is the d by dx mu or position vector the upstairs frame is the gradient of the coordinate they will satisfy this reciprocal frame thing and in that setup the geometric object grad is given by e mu d by dx mu which we can write like that so it becomes really easy as soon because you've got access to this reciprocal frame it's just no worse than you had with the gammas and i should just say when you've got this reciprocal frame and you want to get the components of a vector it couldn't be simpler the upstairs and downstairs components of a vector are a upstairs sorry the vector is a a upstairs mu is a dot e mu and a downstairs mu is a dot emu downstairs now says here these statements look trivial they're actually enough to do everything associated with vector calculus in curvilinear coordinate systems i don't know if anyone's actually tried to do this you get this often for example in fluid dynamics and also some electromagnetic cases where you want to work in some general curvilinear coordinate systems and the tensor calculus identities you need to do that that's generally pretty uh difficult and can take pages of work this cuts through the whole thing and i've written everything you need to do it right here okay so this is one of the um most useful bits of the geometric outer approach uh that that one can have that is just widely applicable if you don't you know once you've got this in your head um then you know you never need worry about things like curvilinear coordinate systems and any special identities that you might think you need for that it all goes away okay so what we're going to do now is we're going to split up the gamma i keep calling gamma i'm sorry this would be del or nabla if i was giving it a greek name let's just call it the derivative and we're going to split that up by multiplying it on the right by gamma norm this will cast it into a relative form let's do that you find out that that's d by dt minus sigma i d by d i this bit here is quite clearly since the sigma i are their own reciprocal frames a square to plus one that's dt minus what i call a bold grad the bulk grad is what i've got here sigma i d by d i so that's the relative 3d vector derivative again that's very useful uh you can use that and fluid dynamics and things like that if you're just in a 3d application in that context okay what i'm doing again is backwards right i started with the lorenz transformation and move forward to show you all with just rotors whereas in reality what you should do is start with the rotors and derive the lawrence transformation from that here i'm going to start up with the standard split up form of maxwell's equations written in uh using these operators and here they are right before maxwell's equations you can rewrite that in this form here right they reassemble using the properties of the geometry product to that well you notice that it's reassembled to bring in this e plus ib that we said was the faraday j is the current which we had coming in this equation here j bull j is a 3d current we now define a 4d current like this we write scalar plus relative vector j and we put a gamma naught on the right that then reassembles to being a space-time current vector j and you get this beautiful single covariance equation for maxwell's theory right this is the whole of all maxwell's theory and electromagnetism all in that equation right uh grad f equals j that's very beautiful does it actually help you at all and it really does and it's this crucial thing that the geometric product has which is the inversibility so the advantage here is not merely notational just as the geometric product is invertible unlike the separate dot on which product the geometric product for the derivative is invertible by our greens functions where the separate things that this would have this would have a grad dot which would be a divergence and a grad wedge you can't invert either of those so this led to a new method of solving uh some electromagnetic equations which crest around and i worked on this is an example i don't have the animation for this one i'll show you an animation of another thing soon but the basic idea here was that we laid down a set of mirrors in a plane and we illuminated this with a beam of radiation and this shows the mirrors reflecting uh that radiation that doesn't sound very hard the point is this is exact this has got all the diffraction effects coming in you can see some diffraction effects occurring in this and the particular feature about this was that we set this up so the user person wanted to play with this could change the angle at which the beam went in and hit these mirrors just in real time and then you could just watch this thing respond immediately in real time and that was possible because we're able to sort of take off this grad and define an underlying structure which was precomputed and then you could just supply the beam of radiation to it and that was effectively a new technique okay so here's a second example with electromagnetism uh i want to consider radiation from a moving charge and this in this particular area is david hesters who pioneered the techniques on this some years ago so these equations are basically due to him and the final form of this faraday that you get to is i think definitely the most compact and informative of any that have ever been achieved if you've ever read the fireman lectures on physics he starts one of the volumes saying uh he's achieved this really nice compact form for the radiation from a moving charge i think this one is much better and more intuitive so now don't worry if you don't follow the details i just want you to get a little idea of what goes into it so one of the maxwell's equations if we peel off the tri-vector part is grad wedge f equals zero that has the indication you can introduce a vector potential like this f is grad wedge a is because the grad word wedge grad is zero that you can do that you then go into what's called a particular gauge div of a equals zero that's just a gauge choice for the a to make things simple so that because it makes it simple because then f is just grad of a and so if you want to calculate grad of f which is going to be the input current then that's just del squared of a so you've got a nice wave equation in this setup the idea is that you have an observer a position vector x and you have a particle going along some path which we write as x naught of tau the x the naught here means that's particle the tau is the proper time of the particle as it moves along the the path and we've drawn this light cone here because what you get at position x corresponds to where the part of the particle hits your backward light cone that's where any information about the particle has to reach you along this null vector i call it big x which goes from one position to another so big x is position vector x for the observer minus the x more to the particle that has to be null now the first beautiful thing is again i don't know if you've ever been through this the lena adventure potential is a solution to this uh problem but it's quite difficult often to um get it right and notate conventionally this is the full answer in geometric algebra a is q for pi that's just units here's a relevant bit the velocity of the particle b at this point over x dot v is that's the full lienhood vector potential which solves this problem and this is true no matter how complicated the motion well we want f the faraday which is grad a and you need a few differential identities and there's one really interesting one that comes out i won't go through the derivation but basically because x squared is zero if you differentiate that you find this rather interesting relation here you find that x squared equals zero tells you that the grad of the proper time is equal to this x over x dot b and that's something a bit strange here because what i mean by the grad of this proper time how do i take how do i interpret a proper time as a field which i can take the gradient of well the point is if i go back here at every x as i move around in space here you intersect a backward light cone uh your backward light cone intersects the path of the particle at a different tile and so as you move around that towel changes and so it's like having a field and that's you can find this gradient just here um this is just an example that not everything that's a field has to be interpreted as being a sort of physically real thing which carries energy and so on and this is called an adjunct field when this sort of thing happens well proceeding what we do is we define the acceleration by vector for the particle that turns out to be v dot wedge v which is actually the same as v dot b these are all orthogonal and here's the result for f that i promised so this is the complete radiation field of that particle and it's ignore the denominator for a moment it's x wedge v plus a half x omega v x on the omega v as a reminder is this acceleration by vector this first bit the x v over the denominator is x dot v squared that's actually just the coulomb field uh in the rest frame of the charge so this is not radiation this is the coulomb bit this bit here is the radiation uh it's q over four pi half x omega v x over x dot v cubed and how do i know that's radiation and what i'm going to do here again illustrates some of the efficiency and ease of use of the geometric algebra here the way you prove its radiation is to show that there's a contribution that it makes to the energy at infinity so we have to work out the energy of this radiation and you need something called the stress energy tensor or energy momentum tensor to do that which in this approach just becomes a linear function of some input vector uh a and here's the linear function it's very beautiful it's the reflection of the input input vector to the strategy stress energy tensor in the faraday itself you take minus a half faf that returns a vector and that tells you the flow of energy again if you've ever seen the conventional forms in terms of calculus for this stress energy tensor you'll know they're really complicated so this is uh you know really very pleasant to work with instead it turns out that bit drops off as one over distance squared and so when you're integrating it over a sphere which goes up in area as the distance squared you get something that doesn't vanish and that's this radiation component here so what you can do is um along the lines of what i was telling you we've done with the mirrors where we changed the input radiation field in real time what you can do is have a go let me just find the demonstration so this is going to calculate the faraday field for particle that we're going to control where that particle is by using um a mouse in 2d to drag around where the particle is so let's have a look at that annoyingly it loses the place there it is so let's get rid of the current cursor what you're seeing here it's a bit hard to see this is the mouse position and that's where the charge is okay and we're moving that charge around at will and what's following it is the radiation field you can see that if there's always a lag because as the charge is moved the observer at that point doesn't find out about it until a light travel time uh delay has occurred now we're starting to do something interesting we're starting to wind up the radiation into a circular form we're going to make it tighter and tighter and eventually becomes pretty much like synchrotron radiation so doing that computation is something that uh you know is really quite tough to do conventionally but you can just write down really simple equations as i've shown you and it just takes you know a few lines to be able to do that in this geometric algebra formulation i'm not sure my son would like emphasizing how [Music] easy it is so he actually did that when he was still at school uh as a whole simulation just using some simple geometric algebra programs so you know it really is something that anyone can try once you've got that in hand okay so let's go back here right so that was electromagnetism now let's get started if people are up to it on quantum mechanics okay so electromagnetism one of the four forces we can now move off towards um the ultimate aim being electroweak and strong forces along the way also will include gravity so what does spacetime algebra bring to this um we find that the algebraic structure of wave mechanics arises naturally from the geometric algebra of space-time it allows you to reformulate standard quantum mechanics in a more geometrical way and suggest new lines of interpretation and we'll get started by looking at what are called powly spinners this works conventionally by regarding these powly matrices as being matrix operators on column vectors all right i don't know how many would have done this but in basic quantum mechanics quantum mechanics of spin you introduce these things here which are called paulie spinners these are column vectors they have entries psi one and psi two each of which has got uh is complex and so you've got two real degrees of freedom in each so there's four real degrees of freedom uh overall and then you operate on them with these pali spin matrices that we already looked at okay so you have the structure of matrix acting on a column vector now something quite remarkable happens in the geometric algebra approach you can replace both objects by elements of the same algebra and so space-time objects and the relations between them can replace all these single particle quantum statements all right so what we have to understand is how to model the parallel indirect spinners within the spacetime algebra now this is a bit detailed probably i just wanted to show you there's concrete stuff underneath this what we do is we peel off the two degrees of freedom in psi one call that a naught and as it turns out a3 the two degrees of freedom in site two which we call minus a2 plus ia1 and then the translation from the conventional pauli column spinner through to the sta object psi is as shown here right i've told you what the elements are in here they line up like this this gives you a nor the scalar bit plus a k i sigma k so we have scalar plus bivector is the psi and this is common exactly the same thing happens for the direct case it's the even sub-algebra of uh the space which is actually the spinners which is an amazing revelation actually because spinners are assumed to be you know living in some completely different uh space different part of mathematics it turns out they're just the even elements of the geometric algebra of whatever dimension space you're in so for example spin up state is just scalar one spin down is minus size sigma two and the action of the quantum operators on the states has got an analogous operation on the multiple vector psi it happens like this this is the translation this powerless spin matrix in the k hat on psi is the same as operating with sigma k our sigma k on the left and a sigma 3 on the right so you always use sigma 3 on the right but the what you're doing at the left varies with the operator and you need this segment here on the right to ensure that this thing stays in the even sub algebra and verifying that this is all correct is basically a matter of computation for example if i apply sigma 1 hat you get this column vector that translates through to this equivalent which you can compute is exactly the same as sigma 1 psi sigma 3 which goes in with this thing here so again this is something i mean i don't know if you've ever thought about venturing into the quantum mechanics of spin it's you know increasingly important technologically this gives you a means of doing it just using the same geometric algebra you would use in you know for example in computer graphics well this is actually a simpler version this is just the 3d ordinary algebra so we need a translation for multiplication by the unit imaginary so the unit imaginary in quantum mechanics is not thought to have any geometrical meaning we can find out what it does mean if we take these three pauly spin matrices multiply them together then we find out that you get i down the diagonal let's see what it gives here so the translation of that would be sigma 1 sigma 2 sigma 3 on the left and of course sigma 3 cubed on the right that boils down to psi i sigma 3 so we discover that multiplication by the unit imaginary conventionally is the same as putting i sigma 3 on the right so this is very interesting this is yet another version of um something that squares to -1 and it has a different inter geometrical interpretation it's multiplication on the right by i sigma 3 and exactly the same happens in the direct case so now you carry on you can define a scalar if you take psi times psi reverse in 3d for something that's just got even elements in that always gives you a scalar and then you find this rather magical thing the spinner side decomposes into this scalar square rooted times a rotor the rotor is rotated minus half times psi and this multivector satisfies our our reverse as one and in this approach powly spinners are simply unnormalized rotors so you start realizing that the quantum mechanics you've been doing which looks as though it's written in entirely sort of foreign mathematics compared to the mathematics of ordinary three-dimensional space actually it's using just the same things ordinary rotors in 3d space and other concepts become quite simple for example the hermitian a joint which you need if you're going to form the inner product of quantum states quite generally it corresponds to reversion followed by reflection in the time axis that's what you discover in geometric algebra if we move the time axis gamma naught through this side because it's even nothing happens so you just find that the hermitian adjoin is equal to the reverse in this case which is nice and simple here's how you do inner products in quantum mechanics and this is how it translates through to the spacetime algebra and i won't go into the details of this but basically it projects out the one and the i sigma 3 components of psi reverse phi these angle brackets on the right uh hugo was using them the swallowing with a little number down here to indicate that you're pulling off a particular grade uh the convention i use is that if you're pulling off the scalar part you don't bother putting the zero to the right but basically is it just the scalar part of the thing inside the brackets so this can start making sense of some quantum mechanical things that look rather um difficult so let's form the expectation value of the spin and k direction that's where you sandwich the sigma k pauli matrix between um versions of the psi right so this is the same stage browsing what's the spin in this stage quantum mechanically you work that out you find it's the same as the scalar part of sigma k times psi sigma 3 psi reverse well that thing there is a vector let's define it as a vector s and what you find is that taking this expectation is the same as finding the kth component of s so the quantum mechanical apparatus which looks really forbidding is actually telling you oh just find the case component which is very interesting so um here's i'm just reiterating that basically if i pull out the uh the row i can find that my s the spin vector is r sigma 3 r reverse and then scaled in this way so for example suppose um i rotate my laboratory apparatus through a new rotation our lord so s would go to arnold s on or reverse that composes with the r embedded here to tell you the argos that arnold are and so this tells us about the how we have this thing which is very mysterious in quantum mechanics about the transformation laws of spinners we can understand now that they transform differently from other things they only transform on one side right so cyber go to arnold side because the r builds up there because the spinner is itself a rotor and so that's why it only transforms on one side so this explains the spin half nature of spin sort of wave functions is because they're actually rotors you might be worried that picking out sigma 3 breaks the rotational symmetry of the theory but actually this is not the case and we can give an example from rigid body mechanics um to illustrate that what's going on is it's exactly like right let's just go back here we're rotating a fixed vector sigma 3 to get this angular momentum vector of uh the particle and that's exactly like in rigid body mechanics we set up some fiducial fixed copy reference copy of our body and then we use a rotor to move it off to whatever orientation it has and of course in rigid body mechanics there'll be some position path that it follows at the center of mass follows as well which we call x not t for the purposes i've got here the important part is that you take x as some coordinate position vector in the reference copy of the body how do you get over here you actually use this rotor to move the fixed reference thing through to the outside world and it's exactly the same here right we move the fixed sigma 3 through to where the s is actually pointing um and so here's our picture of what the uh poundly spinner does it's based on instruction to take this fixed frame here which has a signal on sigma 2 and sigma 3 and it transforms it to the spin vector and these other two vectors i haven't gone into it here but one of these other two vectors well they don't matter individually what matters is rotations about this we saw that there was something that rotated about it that was the imaginary which was i sigma 3 so where these vectors lie in the plane is the phase of the wave function the overall global phase and so we've got an understanding then of the things that seem really quantum mechanical the spin vector and the global phase just in terms of where these axes end up now what i'd like to do now is just say a few more words about the bridge of body mechanics because i want to show you a solution uh within rigidbody mechanics which is going to parallel exactly uh a solution that we have in the pali theory so uh hugo was talking about this sort of thing earlier and the conformal geometric algebra we can extract an inertia tensor um given information about where the material in the body is situated the inertia tensor is defined like this it operates on by vectors and it returns another bivector because this thing here the x dot b effectively gives you velocity x wedge that gives you the angular momentum and the row tells you where the material is concentrated that's shown in here so the body rotates at angular frequency mode b momentum density is this and the angular momentum density is x wedge you integrate to get the total expressed to the reference body that's all in the reference body you have to move it off to where the body actually is in attitude and that's given by this rotor r we talked about just now which rotates the uh angle momentum of the reference body off to give you the space angular momentum so for a free case with no external couple you can differentiate this with respect to time you should get zero and it turns out you can get this in just a few lines that i on the time derivative of omega b is minus omega b cross i omega b equals zero uh this cross i've defined it here is a cross b is a half a b minus b a in geometric algebra it's very useful uh to have um a half in front of the commutator we call it the commentator product so this thing here which looks really simple and is really simple that's the whole of the euler equations of rotational dynamics and anyone who's done rotational dynamics will know that the order equation is very subtle and difficult this is how quickly we can get to it and you'll see how quickly we can solve it so so it's not just that you get a quick derivation but solutions um similar to what hugo was saying this morning you align the body frame with the principal axes so that your moments of inertia uh can just be written simply like this is numbers i k one to three and uh we'll call that i sub k right so that's moment of inertia about the case principal axis and from that you can define two constant precession rates and directions you take the angular momentum the space angle of momentum which of course is constant you divide it by i1 and then you take i e3 which is we know rotations about the body sort of principal axis and you get to this rate uh omega r coming out but the important bit is you've got one bind vector there one by of x there you find the euler equation then becomes its rotor equation r dot is minus a half omega l this one times r minus half r omega r and the key about this you see one part is transforming on the left one part is transforming on the right and you can immediately integrate that independently each side to get what's shown here here's the initial attitude and then this one here makes it rotate about the i sigma 3 axis in the body this is an internal transformation exactly like we were getting in quantum mechanics and this is an external one like that and fully describes the motion of a symmetric top um i'm just going to risk one more video to show you this in operation so let's find that so this video is impressive for a particular reason so it's not actually a video it's a program written for windows in 1999 which still works right now and it was written by christian powers that some of you will know christian quite well uh who was a student of jane of mine joan blazenby's of mine in cambridge and uh has contributed a lot to geometric algebra i just want to show you this program still working perfectly all this time and the several demonstrations this is the spinning top one and if i move this down this is a full slider control i can change all the rates there we go you can see it a bit better there what i wanted to demonstrate with this i don't know how well you can see it that there's a little picture actually a christian sitting down on the top here and there's two frequencies involved in this you can count the rate at which this top is moving around by where the you know the procession axis sorry its main axis is processing around okay count when that gets back to where it was and then count where the picture gets back to where it was and the two are different and that's the two rates one internal there's an internal one busy rotating it about the body axis here and then there's the external rate the external procession moving this whole body axis around so there are two rates buried in that which come out perfectly in that geometric algebra view okay so let's get back to this and the thing i just want to show you is how you can link this immediately with quantum mechanics so right this next bit pro this is probably one of the hardest bits of what i'm talking about that's because i actually need to give you a bit of standard quantum mechanics which which really doesn't look very nice what we're going to do is suppose that a particle is placed in a magnetic field we're not worrying about the spatial dynamics we're just looking at a particle spin and magnetic field conventionally what you do is the magnetic field is b and the particle has magnemetic moment mu you write down uh hamiltonian like that this gamma is what's called the gyromagnetic ratio it shows the way the particle responds to the ambient magnetic field and you then introduce a coefficient which is complex multiplying the up state and the coefficient which is another one complex multiplying the down state and that spans your state of possible quantum states you then write down the schrodinger equation and you find that's the hamiltonian is equal to ihd by dt of the state now this is actually quite hard to analyze conventionally because it involves a pair of coupled differential equations for alpha and beta which are general complex quantities so of course people have done this but it's not very pleasant instead let's see what the schrodinger equation looks like in the ga formulation you go through this you do translate it as i've been saying you get this really simple equation psi dot is one half the gyro magnetic ratio times pseudoscaler times the ambient field times psi and we can just plug our psi being row to the half times the rotor into that and you get this and i'll multiply on the right by side reverse i get psi dot psi reverse turns out is one half rho dot plus rho r dot r reverse and that's equal to this uh by vector ib now it may not be immediately obvious but this combination r dot reverse is always a bivector okay that's because um if you reverse it you can show very quickly from the fact that r r reverse is one that um the reverse is minus itself so it's definitely a pi vector so you have bivector equals bivector plus scalar oh so that scale in the smallish and you find immediately that the row was irrelevant all that matters is the rotor because the row is fixed and so you end up with this really simple rotor equation r dot is a half gamma ibr so the quantum theory of a spin half particle and magnetic field reduces to a simple rotor equation and recovering this explains the difficulty of the traditional analysis based on a pair of coupled equations because it doesn't know anything about the underlying rotor nature of what's going on and so a solution is just a medius and a constant field you just have the single-sided thing here that size function of t is e to the um effectively the i times the magnetic field times t over two so that's really uh quite simple so basically the spin vector just processes and the ib plane at a constant rate now i want to make the transition to relativistic theory now i'm not going to go into this in as much detail i'm just going to tell you that in the relativistic theory has been half particles things are very similar in a bit when we get to the electro weak theory and the strong force i'll show you a bit more about the specific nature of the direct wave function you can just assume it consists of all elements of the even sub-algebra of the space-time algebra so we can do the same decomposition what you find is that so it's the even sub algebra that's got eight elements six of those go in this lorentz row to r which is exactly the same that we talked about before that's the form e to the bi vector there's six five vectors so that's got six degrees of freedom here's the row to the half we had in the powder case here's the new bit it turns out that when you go to resistic theory there's an extra duality parameter e to the ib per over two and in simple cases like plane waves you find that this tells you whether you've got particles or anti-particles that's what it controls but it's it is more mysterious for example in a hydrogen atom you can work out this decomposition of the direct wave function and you find that parts of the wave function are particle-like but in other parts it's got a little element of anti-particle so i have to say no one really understands what the beta term in here means and if anyone you know is interested in trying to take this uh particular thing further um there's definitely work to do to figure out what beta means in general so i'm not saying everything's solved here right there's no explanation conventionally because they don't even know it exists but we know it exists and we don't know exactly what it does now you get exactly the same picture of a rigid body rotation so now you've got the four vectors the gammas that we talked about and the dirac wave function transforms it through to a new set and we still have the spin vector we have these two connected to the phase but now because we've got a time axis we get a new thing and it's this j is psi gamma or psi reverse that should say reversing it sorry that's a mistake uh equals rho it turns out that same as rho are gammon or r reverse and so you just rotate the gamma naught thing and what you get is called the direct current we'll talk about its properties in a moment spin vector is the same as before rho r gamma 3 in this case r reverse so that's what the wave function does and what it looks like what's the equation that it obeys and so this is the dirac equation and here it is in the form first proposed by david hestoners it's really incredibly simple grants psi i sigma 3 here's this multiplication by i um the equivalent of multiplication i minus the charge times the electromagnetic potential times psi equals m psi gamma naught and that's the pooled drag equation coupled to electromagnetism all in a very simple form and only using elements of the space-time algebra we don't need to step out of that at all to do that you can show quickly that that equation implies the current is conserved um and so that tells you you can't create or destroy um uh fermions at the level of a single particle tyranic equation these are that would be the high energy multi-particle process not covered by this single particle equation now the time like component of j is positive definite and conventionally it's interpreted as probability density which is fine and a normalized wave function has this because you're interpreting it as a probability density you want that to happen and conservation of j implies that the probability density flows along non-intersecting stream lines which is very useful for visualization and i don't think i've got time to go into this in detail but i just want to give you a little flavor of the very different view of what happens in quantum mechanics that this approach leads to you've probably heard of the stern gerlach apparatus and people measuring spin the idea is that you take a beam of particles and you you prepared them in some state that isn't specifically spin up or spin down it's got a general mixture of spins and they go into the stengerlach apparatus which creates an inhomogeneous magnetic field and you find that everything that comes out out is either spin up or spin down and the general idea in quantum mechanics is that what's happening is that before you did your measurement this is regarded as a measurement machine before you did your measurement you've got the particle exists in the superposition of two states you do your measurements his wave function collapses to one of the eigen functions uh corresponding to the measurement process so there's this whole idea of collapse of the wave function and measurement as being a process which is forcing something to be either one or the other even though it was neither to start with so what actually happens here is something that you just couldn't do normally um it turns out that if you put a wave packet in so this is something that is representing a distribution of particles here it is at the bottom as time goes on it is split up spatially into two parts each traveling in a different direction here's the streamlines of the particles as they exit the stern girl and cavaratus and here's a picture of the spins of these particles and you can see so these this is it one side naturally uh of the um apparatus this is the other side and what you find is that the uh spin directions are aligned by the instrument gradually to be either up or down and this gives you a totally different view of what's happening in the measurement process you're not measuring at all this stern garlic device is a polarizer which is taking a continuously distributed initial quantity and gradually transforming it as it propagates through the instrument into either spin up or spin down so it looks entirely classical is what i'm trying to say there's no collapse of the wave function there's an entirely classical evolution of this with a perfectly good spin available at each stage and this is not liked this view is not liked by many people working uh in quantum physics but the thing is if you work out at any stage what would be the predictions for the number of particles that are spin out versus spin down you get exactly the same results as standard quantum mechanics so this is something that can sit underneath standard quantum mechanics and actually um replace it with some entirely causal deterministic picture as shown just here anyway um now let's see how we're doing okay are we brave enough to try some gravity yes okay so gravity is one of the four forces and normally it's considered very different from the other forces that we've been talking about electromagnetism electro weak strong force and we in fact in this approach we want to make it as much like um those forces as possible so these other forces not gravity they're normally described in terms of a yang mills type gauge theory these are unified in quantum chromodynamics this thing here is what the strong force is to do with right the forces which bind the nucleus are called uh are worked out in something called quantum chromodynamics the chromo means that color is involved and that's because the they're meant to exchange uh different particles which are assigned this label an arbitrary color exchange particles uh to change their color that's i'm just explaining why it says chromo so these gauge theories sit in a flat space-time background and they're of this particular type it's called young mills time now ordinary gravity knows nothing of that it doesn't have any yang mil structure i'm talking about conventional gravity and it certainly doesn't sit in a flat space-time background it sits in a as you know a curved space-time background so the aim here is to have a gauge theory of gravity which is as much like yang mills type gauge theory as possible and expressed in a flat space time so this is pretty radical okay we are stepping away conceptually a great deal from general relativity well if it's a gauge theory well what is a gauge theory it's where you have a symmetry which is global symmetry of the problem it could be something like global rotational symmetry something like that and you choose to make it local and you say right in our case we're going to choose the fact that you can carry out lorentz rotations at a point in any way you wish and the theory should survive that happening so it sort of relativizes everything and the theory should still be okay and the answer is what the only way you can do that is by introducing extra forces and the forces corresponding to localizing these global symmetries so why should we want to do this and you find that the dirac equation direct spinners is probably the easiest place to start so if i take spinners psi one of x and psi two of x right these are two spinner fields um a sample physical statement would be that psi one of x is the same as psi two of x at all space time positions x that would be a nice physical statement which makes sense and of course it's independent of where we place the fields in the space-time algebra i could equally well introduce two new fields psi one dashed of x equals psi one the old one at some new position x dashed and same for side two and then these new fields where i've done this arbitrary remapping of the space it should have been precisely the same physical content as the original so an arbitrary remapping of space shouldn't make a difference same as tree effect on fields with a space-time rotor my um statement that psi one is equal to psi two should be the same if i stick an r in front of each because i haven't changed anything by doing it if i do it to both because the psi one dash equals psi two dashed should have the same physical content as the original equation and that's all fine i can do all that where it doesn't work is for derivatives um i'm not going to go through this in detail let's just look at this one thing here which is about what happens to rotations if i've got a derivative so i've got r operating on psi i wish to differentiate it that's the same as grad r times psi plus oh i have to differentiate through this r for this grad to hit this psi and what that certainly isn't equal to is a rotated grad of psi so this is the key requirement in a gauge theory for local covariance as it's called i've got to be able to in this case move the rotor through the grad and come out the other sorry through the ground here come out the other side so that the derivative of the rotated object should be the rotated derivative and that clearly doesn't work well what you find is that you get around that sort of problem and there's an equivalent one for uh the derivative under this position remapping by introducing two gauge fields they're called h o barb a which is a vector field an omega of a which is a bi-vector field and they these are functions of a uh vector argument a and you define these to have the properties so that these bad bits we didn't like in terms of the derivatives not working anymore are undone and you get something that transforms properly and for example for the omega field the bivector field that will work provided the new biovector field uh rotates you just certainly expect that but also has this inhomony homogeneous bit here given my minus two a dot grad of r reverse i said this structure uh will always give you a bivector so it grows a new bit of bivector just here and that takes off the bad bits in this equation here so that's what we're doing and it turns out then that you can uh wrap that up with the directional derivative in a particular direction this direction a which we're using is the argument and this thing here this is the fundamental uh derivative in this version of gravity you just have we call it a covariant derivative of a dot grad plus omega a cross this omega a cross is this commutator product and so this looks odd well maybe not odd anymore you're used to this by now this is a scalar this is a bivector operator but it turns out that the properties of the cross operator mean that the omega cross just returns the same grade as it operates on that means that d a this thing we just defined here is always a scalar operator and in fact it satisfies the leibniz rule for derivatives um so that's the main part of what we need we've actually already reached the essence of gravity in this approach just in that really simple expression um actually there's one more thing that we need because i just want to convince you very briefly that you could do some of this quite easily yourself but you're if you're looking at this you'll say oh what's this thing i've defined something here the multi-vector derivative with respect to a d by d a comes into this theory i haven't got time uh for the details but the multi-vector derivative by vector a is just this it's just like the gradient one we had before you have an e upstairs mu you now differentiate by the components of a in the downstairs frame and this is a really nice quantity i i think leo talked about it yesterday and he showed you these two rules um if you're taking the dot of a with some greater object then the da returns are the grade times the object you dualize that to this outer product you get n minus r and it's the dimension space but the thing i want to show you here is this rule which is quite a surprising one if you differentiate through a gradar object you get -1 to the r dimension the space minus 2 times the grade times ar and so that means that if you're in 4d space time and you differentiate a vector when there's a bi vector in between so if r is two you get four minus two times two which is zero and so you're gonna get zero if differentiate through a uh bivector and it turns out this to be key to why electromagnetism is actually a massless theory uh i why the photon has no mass is because of that relation and it's key to being able to demonstrate how the riemann tensor for a black hole works the riemann tensor is got by taking these derivatives i've defined and taking the commutator on any object you get this beautiful result but the commutator dadb on any multi-vector is equal to the riemann tensor which is a function of a and b times m in fact the riemann tensor quite generally it's very interesting it maps by vectors to bivectors it's just like the inertia tensor in doing that so i'm not going to go through the details of this because i just want to show you what the equations of motion of the theory are here we are so sorry to jump to this but these are the full equations of the theory right this is gravity d a of this riemann tensor evaluated on a which b is should be zero and this uh covariant derivative of the h function field the wedge part of that times curl should be zero and that's it and this does a huge amount of the work of ordinary tense calculus an incredibly small space so for example all the symmetries of riemann that one encounters conventionally are encoded in the grade 3 part of this the d a wedge r of a wedge b and if you want to relate this to what people have said about gravity in the past this equation here means in this particular approach you've got zero torsion um that that would relate it for you so what do black holes look like in this approach it's always exciting to think right i've got something that looks really simple here what does it do for black holes and you can treat it exactly like electromagnetism so in electromagnetism if you're dealing with point charges for the first time you'd need to choose a gauge for this a field and you'd work from there you want here a choice of gauge for we start with the h function it covers all the space except possibly for a singularity of the origin and just like electromagnetism you'd expect the field strength tensor to be same and not depend on the choice of gauge so i'm going to show you two of these choices of h which are really good they both have advantages neither correspond to what people ordinarily do at all if you try to translate it through to to metrics um the first one i just have to define two things so er is the union unit radial vector we've got spherical symmetry so we work with a radial vector e t is the gamma naught as usual but i define this new vector for one of these it's called e t minus e r uh and written e minus and the point about this this is null all right if you multiply that out because that squares to plus one the squares to minus one he is null and anyway here's two really good choices of h which we eventually found h of a is a minus the square root of two m over r m as the mass of the black hole a dot er times e t and the other one is a plus m over r a dot this e minus times e minus these look quite different but both very simple we call the first one the newtonian gauge a lot of the physics looks very newtonian like in the gauge and the second one because of this null vector it turns out it's really good for treating the motion of photons if you want to do that now these would be like the a fields and electromagnetism so we would want that the field strength tense so you derive from them we should know nothing about gauge should be the same that's exactly what you get you find that the riemann tensor is this mapping of bivectors r of b is minus m over 2r cubed r is the distance you are from the origin b plus this rather neat double-sided sigma r three sigma rb sigma r and what's sigma r that's the radial equivalent of a space um you know space-time sigma it's e radius times time so here's the riemann which is incredibly compact and it's so nice because you could immediately check that the field equation d a of r a which b which we talked about has satisfied we're actually just going to do it so we want d a vowel a which b so the b in here is replaced by a wedge b so i've written a doubt we want d a of a h b plus 3 sigma r a which b sigma r will not be zero using that rule for the derivative when you're wedging with something that first one gives three b all right that was this rule back here right so we've got four minus one which is three um just here we expand out the a wedge b is a b minus a dot b and then here's what i was talking about this crucial thing i'm differentiating uh the a through this um by vector sigma r so that vanishes so i just get left with d a of a dot b again you apply the rule you just get a dot uh you just get b but now you've got three of those work it out sigma r squared is one you get zero so that's the complete derivation of and showing that the einstein field equations our equivalent of them is satisfied in this step so it's really impressive as regards compactness and the use of working now even more impressive is doing the same for a rotating black hole um so this is the riemann tensor for a rotating black hole called a curve black hole and of course we believe most if not all the black holes in the universe will have some angular momentum and so this is probably applicable to all the black holes out there and what you can see is i've made had to make one change here this r goes to r plus i l cos theta l is the angular momentum in the black hole theta is the ordinary spherical polar theta and this i is our space-time pseudoscaler and so this is all still fitting nicely in the bivector algebra because remember when i take this upstairs and expand it in some way the i will hit these b's and that will still be by vector so i am still returning a bi vector from this if you want to check that this satisfies the equations i said you don't need to do any extra work because um the the d a thing anti-commutes with i and so the bits that involve i when you expand down this cube uh the dda just passes through you get a flip assign but since it's zero that's fine so actually we've already done the work to show that this satisfies the equations um so it turns out this is quite certainly the most compact form of the remove occur i think anyone's ever seen if you look if very few authors will ever try to write down the riemann components in the care case and they take pages if they try and do it i mean chandrasekhar for example in the famous book did have a go of this no one has any idea outside geometric algebra but it's actually really simple uh of this form okay um there's a tiny bit more on gravity which i just wanted to get in which is convinced you gravitational waves are really simple as well so there's a very abbreviated version of gravitational waves but hopefully is useful and we just need to write down an h function which we're going to try out and see if it works and it turns out to be the analog for um this planar case that you want for a plane gravitational wave the one we wrote down and said was good for photons above here it is e minus is now uh e t minus um uh yes sorry that that's the one sorry that's the one that we had before where e minus was the e t minus e r what we do here is we have a new vector turns out to be more convenient to work with this plus version which is e t plus e z he says the direction of propagation right so i think you can see a pretty good analogy between these two h of a is a minus a half h a dot e plus c plus that's all you need you plug this in and you find that the riemann is jumping forward a tiny bit this the g of eta is just some scalar pulse function that you put in that describes the amplitude of the wave as a function of time that's just a scalar and you find that the riemann sort of machine has a bi vector sat in the middle of a another bi-vector sandwich so it's just like the sigma r b sigma r we had before but the bi vector is e plus e perpendicular uh where well what are these e plus we defined here that's the direction of propagation and whilst the z perpendicular that's perpendicular to the direction of the wave and is actually the polarization direction of the wave so this is really neat you take the wave direction you take a perpendicular which is a polarization direction you form the bivector between them and that's what sandwiches this be and it gives you a really good idea about why it is that um gravitational uh waves have the polarization that they do and they react very strangely according to how people ordinarily think about this of rotating frames because the gravitational wave polarization responds at two times the rate you'd expect but we can see that straight away from here's our input b in uh coming in that bivector sandwich is the output b and it turns out the output angle is the input angle plus twice this polarization direction and that actually corresponds to the unusual nature of the polarization of gravitational waves and we can see that coming out directly in this picture there's a little derivation here of how it is that this riemann tensor satisfies the equation it had to again the key bit is you differentiate through a bivector just here and you see there's a bi vector in the way that being zero means when you get a contribution from this bit and when you work out what that bit gives because c plus is null it manages so that's really nice and something i can point out is that uh doing it this way which by the way is completely exact normally gravitational waves are done in a linear approximation everything i've shown you here is completely exact and it turns out that because people had only been working with a linear approximation they missed off something rather important i call this velocity memory and so conventionally after a wave passes through for example a ring of particles so you have a ring of particles here this shows how they respond when the wave goes through and when the waves pass conventionally they're meant to just be sitting there and nothing's happening to them just the way it's gone but what i found is that actually the wave imparts a net velocity to a test particle that persists after the wave has passed and you get these rather beautiful things if you start with the ring of particles just here and let the wave act upon it afterwards the particles can move off on these trajectories shown here they're very tiny velocities but the question is whether they will be observable in future experiments uh particularly experiments done with uh spaceborne instruments um and so that's actually an active question i know other people have now picked this up that this is being pointed out and are investigating whether there could be effects seen in future experiments and i should say that there's another group that i should have put a reference to here that working on this independently and conventionally and their first publication on this coincided with when i first talked about the effect was about three years ago um and i'd only been working on this area this particular area for a few years but it's interesting that it took uh basically about 70 years in the conventional picture to be able to see that this actually this effect existed so there's an interesting one a great deal of research is likely to concentrate on this in the near future now probably if i i'll try and speak no more than another 10 minutes okay because i do want to get to electro weak and strong i just want to give you the flavor of this because it's really interesting so the starting point in this approach is the direct spinner one finds that's always really the best place to start and we saw gravitational forces arising by gauging the rotor transformations inside the left right this was our starting point and we did gauging on that r where arsenal are once rota where do electro weak forces come from it appears they arise from gauging the rotor transformations at the right and also it's not just the rotors there's one extra bit which is duality transformations e to the alpha i so how's this work we need to introduce something that's already sitting there in our algebra that i haven't mentioned yet which are item potents or projectors these things are made from the sigmas and you have a projector p plus which is one half one plus sigma three why do i call it a projector or item code and we'll multiply it out and it comes back to itself because sigma three squared is one you can define another one a half one minus sigma three and again that squares to itself and these enable you to create what are called left and right handed vial spinners and you can promote a powly spinner phi the type we've been talking about to fire one by multiplying by one of these so phi which is apparently goes to five into a half plus one one plus sigma three this is very useful because it means you can now apply a full lorentz rotor at the left a full thing involving boost and so on without that it would take you outside the status power spinners which only respond properly to spatial rotations so putting these on the right means you can do full remnants of rotations and now here is a full direct spinner with the type we were talking about before okay written in terms of two poly spinners and you make it up with one of these uh p minus things and one of these p plus things and the phi and the omega here are um pally spinners and we interpolate an i sigma 2 uh term just here because it turns out to be useful in the moment so there's four degrees of freedom real degrees of freedom there are four there so there's eight overall which is right so how does electro weak theory work it's got two sectors these left and right sectors that we just talked about which are these vowel spinners half one minus sigma three half one plus sigma three and it seems neutrinos which you'll have heard about only appear in the left sector and so here's a neutrino wave function the superscript p means uh that's a power spinner okay so here's the full neutrino spinner which is one of these times one minus seven three electrons have both the left and the right sector and so here's the electron uh left sector his electron right center spect and uh right sector and they again have uh their root powly spinners like that now this is really surprising how this works um that this is um you know it's odd that nature does it this way but this is how it goes you combine the two left-handed components into a single direct wave function like this you add the left-handed component of the electron with the left-handed component of the neutrino and you get this thing here and this isigma2 is the thing that converts the left-handed neutrino wave function into something you can use as a right-hand part of a folder acts better what is the gauging the gauging is such that you'll try to leave the direct current the left hand direct current uh psi l gamma naught gamma psi l reverse you try to leave that invariant and that picks out the set of bi vectors which compute with gamma law well we know that's i sigma 1 i sigma 2 and i sigma 3 and of course the scalar i which reverses to itself so actually we've we're there because the isigma one two and three parts is then just a spatial rotor r and that's defines what's called the su-2 part of electric transformations and the action of the pseudoscaler is like a phase rotation so is u1 um okay so that means uh that we've got mapping for what these transformations are just in these very simple space-time algebra terms and an odd thing is that nature seems only to take advantage of this su-2 part the spatial transformations for the left-handed wave functions the right-handed wave functions only see the duality transformations by the pseudoscaler so what you need then as a final object is a way of coupling the left and right sectors of the theory together and this is the role of the higgs field and you might be quite surprised to realize you can understand higgs field which i'm sure you've heard about really quickly it just pops out straight away here so since the spinners are transforming on the right okay that was what the electro weak did you can form invariant in the products between them uh like this you can take the scalar part of some theta times psi reverse because if they transform on the right we have theta r then r reverse psi which just goes back to what it was and so the higgs field is then a powly spinner h which is a scale spatial rotor which you use in the middle of such a product to provide a term which dynamically couples the left and the right sectors so psi left is coupled with cybrite via the higgs field like that and because the i said this right sex it doesn't transform under su-2 it's clear that this higgs field must provide a missing r and so it transforms as h goes to hr now i want that to be if you know if i wanted that to be a full rents boost i'd be in trouble because h is a powly spinner and um that's a problem but luckily it's not because of course the only transformations i need are these spatial ones so i've got a spatial uh transformation h r here which means that this just stays apparently spinner and that's finally well when i first saw this this finally explained to me what a higgs particle was it it's a powly spinner but it doesn't respond to any space-time transformations and that means that it's treated as though it's a lorentz scalar but it is actually a powerless spinner um so that is what the the higgs field is and then finally uh i'm not going to go through this but strong forces we can start fitting into the same idea we just need one more thing to bring in the strong forces which is this that we have a uh pauli sorry a drag spinner psi which as well as being a function of position x is also a function of a linear function of a bivector and this sounds odd to start with but of course it's very much like a spinner version of riemann tensor the riemann tensor was a function of position but you shoved in a bivector in a linear fashion you got a bivector output and it looks as though that's what's happening with the strong force because you just need to uh allow your direct weight function to be a function of this bi vector b and then what are the transformations of the strong force the color force that we talked about it turns out that all you're trying to do is keep the inner product of uh hermitian in the product of that put itself invariant i said the hermitian inner product would be gamma f gamma naught because f reverses to itself so when you take the submission thing you just get f back this is the thing we need to remain invariant and it turns out that you can go through and find all the things that keep that invariant and they correspond to generators of a group called su-3 and i won't go through it now so i've taken up too much of your time just want to show you one really nice thing that it ends up with and the generators uh they're called e in this approach that um you work with three of them boil down to being nothing more than spatial rotations of this input by vector and this is a full finite form for a transformation within the accessories 3d theory that's normally quite hard to get but it just boils down to those spatial rotations the other lots not quite as pretty this is the full form uh for the um other sets of transformations but it's reasonably pretty and it's all totally expressed in space-time algebra so the basic thing that i want to end with is this that we can do all of these forces we believe wholly with sta entities there's only one problem with this that is only dealing with one generation of particles you've probably heard this three generation particles there's a pair of up down quarks there's um then charmed and strange and top and bottom they've formed three parallel generations this system only seems to have room for one generation so what are these three generations and i'm fairly convinced that this will involve stepping outside the sta to things that are more like the conformal geometric algebra and in particular the spaces uh either cl41 which i call the wandi up control geometric algebra or the full conformal geometric algebra itself so i just wanted to end with this that these may be the space or the spaces we need for all the generation of particles and uh maybe this will shed light on issues why the left hand and right hand sectors are different to electro weak because there's no explanation in what i've shown you why that is but i think you'd like to know that um for you know it's good to know that an algebra that you're already using for conformal geometric ground where 3d space this one cl41 may actually be the key to all the forces and everything can be done within that which will be quite nice okay thank you for listening [Applause]