ICACGA 2022: The Supergeometric Algebra as the Language of Physics (2)

Transcript

foreign [Music] so let me introduce again Andrew Hamilton with the continuation of his talk about super geometric algebra as the language of physics thank you again you know get another helping of me I'm afraid to say I am an astrophysicist I've been for a long time and it kind of affects how I think about things I'm not really interested in fundamental physics I'm very interested in what happens inside black holes and that forces me to engage in what happens fundamentally in physics and especially at high energies Beyond those accessible to human beings you're seeing a the same visualization that I showed you this morning so I'm interested in high energy physics to try and solve problems in black holes where the energies are more extreme than anywhere else in the universe my guiding philosophy as an astrophysicist has always been what is nature really doing and if I look around at nature I can see that all matter leptons and quarks is made of spinners you and I are made of spinners spin half particles and I also see in the standard model that all interactions all the forces of nature arise from symmetries of spinners the standard model is the product of three groups U1 cross su2 cross su3 hypercharge weak ISO Spin and color groups and general relativity is also based on two groups the Lorenz group product with the translation group so that says to me that Spinners are kind of at the heart of nature and the third thing is Spinners have the smallest spin spin half non-trivial spin half a non-trivial spin what is the what is the super geometric algebra the super geometric algebra consists of complex linear combinations of Spinners column Spinners row Spinners and their inner and outer products this conference this Workshop is about the geometric algebra and it was proved by Brower and wild in 1935 that the algebra of outer products with spinners is isomorphic to the geometric algebra of multi-vectors and that is true in arbitrary dimensions this is an outer product of a column spinner and a row spinner it makes a matrix you can also take the product of a row spinner with a column spinner and get an inner product which is a scalar but at least in the laws of matrices you can't multiply a row Spinner by a row spinner and you can't multiply a column Spinner by a column spinner and so it turns out that the super geometric algebra not only contains the geometric algebra by taking out of products but it also contains the Exclusion Principle you can't put you can't the those products were row cross row is forbidden and column cross column is forbidden look very much like the rules for creation Annihilation operators and one can in fact show that the algebra of those things is identical one of the things I want to emphasize is that the index of a spinner in N space-time Dimensions is a bit code with n over 2 bits that seems to me quite fundamental the reason why the multi-vector algebras are are products of two when you add up all the multi-vectors is they arise as a an outer product of the the spinner algebras so let's remind ourselves from of some familiar spinner examples I guess I've started off with saying a vector in N space-time Dimensions is indexed by a Cartesian index I learned about Cartesian vectors in high school and it was really an eye-opener for me to realize that I no longer needed to any geometry at all I could throw away my geometry textbook and replace it by algebra and do proofs way more quickly and I wondered why anybody had ever bothered to teach him uh algebra I'm sorry geometry I think it ought to be taught in high school that Spinners also have a beautiful structure which is to me even more beautiful which is that they're indexed by a bit code with n over two bits and they have two to the N over two complex dimensions for example a power loose spinner which is the GM it's that's the spinner part of the geometric algebra in two or three dimensions it has one bit and the bits of a powerly spinner can be either up or down something that's quite familiar the algebra of outer products of Cowley Spinners yields the geometric algebra into all three dimensions in four dimensions we live in three plus one space-time dimensions one has a direct spinner introduced by Dirac which is a essentially a relativistic version of a Paoli space a power spinner it has four over two equals two bits and four complex components and its basic Spinners consist of up up down down up down and down up and these two bits consist of a boost bit which aligns with the direction of motion of the spinner and a spin bit and if you form okay if you have a the Boost and the spin bits are aligned is called right-handed if they're anti-aligned the direct spinner is called left-handed and an electron is a direct spinner and I find it fascinating that something that's fundamental as an electron contains the structure of space time within itself so how do you about go about constructing these things foreign ERS would live naturally in an even number of Dimensions so what you do is you take an even number of Dimensions you parcel the 2N basis vectors into n Pairs and those go those vectors I've labeled gamma a plus or minus which comes from direct gamma matrices and then you when you pair them you can for com form these complex combinations these are chiral vectors right-handed or left-handed each orthonormal pair defines a complex chiral Vector so a complex structure which is essential to Quantum Mechanics is actually built into the get-go in the super geometric algebra or in Spinners in general and the way it works is that if you've got two Dimensions this is one of the pair and you rotate in that plane it'll rotate the chiral vector by e to the I Phi plus or minus and if you take a spinner they can be either spin up or spin down spin half and they rotate by by half that phase e to the plus or minus I Phi over two if you are in odd Dimensions instead of even Dimensions the standard thing that one does is one projects the odd spin algebra into one dimension lower by identifying the odd dimensional pseudoscaler with a phase Factor times the unit Matrix for example the poly pseudo scalar we routinely call the power vector sigmas their product is set equal to the imaginary times the unit Matrix spin 10 was first noticed in 1975 by Gorge and independently by friction and minkowski and it remains today the viable Grand unified group that unites the three forces of the standard model it contains the there were three grand unified groups su5 then the panty Salam group and then spin 10 which contains them all so spin 10 is the granddaddy of grand unified groups physicists tend to call it so10 it's actually the covering group of so10 that's the group of rotations of orthogonal rotations in 10 Dimensions the group is actually spin 10 because it acts is it's a symmetries of spinners spin 10 has 5 bits 10 over 2 equals five bits and 32 basis Spinners this was something that was pointed out by will check in 1998 so for example an electron it there's a Dirac electron has four components it's got a right-handed a left-handed and particle and anti-particle parts and and here it's bits it's got two weak bits and three color bits and all the leptons and quarks of a generation have five bits going up or down making two to two to the five equals 32 Spinners I've drawn a little diagram of what they are with the red green and blue Dimensions being the RGB the color bits of the strong force and then the other two I've got a I've labeled them y and z so we'll check didn't notice this maybe because he didn't write it down in the right way but if you write down all the Spinners in a table organized by the number of up bits you get this table this is the First Column is zero up bits then one up bit two up bits three four five up bits you can see various familiar things or neutrino the electron and up and down quartz complete with various colors in the case of quarks there's very striking things it should hit you between the eye on this one of those facts is that the spin 10 chirality coincides with direct chirality direct chirality is something to do with with the Lorenz group whether they spin and boost bits are aligned or anti-aligned and it happens to coincide with spin 10 chirality spin 10 chirality counts whether the number of spin 10 up bits is even or odd you can see zero or left-handed one or right-handed two or left hand and so on is this a coincidence does it have to be it didn't have to be in fact it's a relationship between two algebras the Dirac algebra it's a it's a chirality is is really the sign of the chiral operator or the pseudoscaler the Dirac pseudoscaler is the product of the four Dirac vectors the spin TENS unit of scale is the product of the spin 10 vectors two completely different algebras if you believe what the physicists say about the Dirac algebras and the spin 10 algebra is being distinct but here you have an equality of two things so that suggests that the algebras are related and not distinct as commonly assumed the other thing is that standard model transformations are all a subgroup ever su5 and su5 is a subgroup that subgroup of spin 10 that preserves the number of up bits each of these columns is an su5 multiplet so standard model Transformations can transfer form you vertically but not horizontally and if you look at the electron for example which you used to say well that's those are related by Lawrence transformations those are arrayed horizontally so standard model vertical Lorenz Transformations horizontal again it's the same saying I'm part of the same thing I'm fine one final comment that if you look carefully you see that here's the right-handed neutrino electron left-handed electron they differ by a flip of the Y bit and it turns out that electroweak symmetry braking which is a Breaking of the hyper charge and weak forces into the electromagnetic force and said adding a Time dimension is potentially capable of unifying the Dirac algebra and the spin 10 algebra which contains the standard model but you have to do a lot of work to make that happen in order to make it consistent with physics as we know it it's necessary that the Dirac and standard model algebras be commuting sub-algebras of the spin 11-1 geometric algebra that's the Coleman mandula theorem the coma Doula theorem essentially says you can only combine an internal algebra with the pronkare algebra if you do it in a trivial fashion well that's not quite what it says that's the physicist version of the theorem the theorem actually says yes can you go back in um hide the uh no I can't that doesn't give you the option here no we tried that it's still recording we just can see the uh I know that when it's because I'm running Linux and it doesn't Okay Zoom hasn't implemented that yet okay the coma mandula theorem simply requires that if you have an internal algebra and a and a geometry of space time those must be commuting sub algebras of the unified algebra that you come up with if in fact the grand unified algebra is spin 11 1 then all internal dimensions are spatial are our space-time Dimensions there's 12 of them and so the Coleman mandula theorem is satisfied trivially the relation that you end up with these are the four vectors of the Dirac algebra expressed in terms of spin 11 1 vectors I've got the six bits I've drawn them T gold y silver Z bronze and then RGB the color bits that is the this is the algebra that that you get and you find out that the inner products and the algebra of these direct vectors defined in terms of the spinner level one vectors are precisely those of the Dirac algebra moreover the Iraq algebra and spinal M1 geometric algebra satisfy automatically satisfy all the discrete symmetries that they must because spin 11 1 differs from spin three one by eight which is the bot periodicity theorem so the conclusion is the four forces of nature unify in the spin 11-1 geometric algebra and though you have not heard this idea before and yes this is an original to respond to the referee complaining that was this an original idea or not yes it is so it's interesting once you've added a t-bit this extra time bit it adds extra fields which happen to behave like some of the fields that we already know in particular the electro week Higgs field is built into the spin 11 algebra it you take one of the two time Dimensions this is actually the spatial Dimension associated with the time Direction make a bi Vector add that of that with the weak vectors and you get four things whose algebra is precisely that of the Weinberg Electro week multiplier the electric weak exit field itself is this particular one of that set of four and it carries one unit of Y charge and it breaks y Symmetry and it gives fermions their mass by flipping their white y bits completely consistent with a standard model essentially without doing anything the Higgs field that breaks the grand unifies spin 10 symmetry turns out to be the product of the the bi-vector product of the time vector and its spatial companion and that turns out to it carries one unit of T charge and is able to give the right-handed neutrino a mayorana mass by flipping their t-bit only the right-handed neutrino is allowed to do that because only it has no standard model charge and that's nice because the right-handed neutrino really needs if you give it a mass it solves the through the Seesaw mechanism it solves the problem of where neutrino masses come from this is a very specific model there are basically almost no free parameters and amongst other things you can see what additional groups before spin 11 1 the the uh the electro the standard model group unifies to and it turns out that spinner Latin one between Spin 11 1 and the standard model there is a single possible group which is the Patty Salam group spin four cross spins six and it unifies where this particular combination of coupling parameters is equal to one and that turns out from the running of the coupling parameters to be about four times ten to the 11 gev and that's simply not negotiable so there has to be another level of symmetry braking to spin for across spin six four times ten to the 11 G it's not negotiable ground unification occurs where after that first unification then occurs whether we can color groups to left are unify and that happens at 3 times 10 to the 14 gev you don't need Super symmetry you need to fine tune the three coupling parameters they automatically unify first one and they're the other one and here's a list of predictions and are there any ones that I need to say I'll mention the two at the end Grand unification is mediated by a Meyer and a mass Higgs field the mining around a mass Higgs field is available to drive cosmological inflation at the Grand unified scale and that's a non-trivial statements is a single field which is consistent with the data from the plank and there is the only evidence for for a dark matter particle is a light scale with Vanishing standard model charge so the scalar companion of a photon or a z Boson and then there's some non-predictions I don't know what causes the three generations I don't know what causes the masses of fundamental fermions and so on okay this morning I talked about the DNA of the universe the complicated DNA of Earthly life is a language written with four letters t-a-c and G Lenny saskind has proposed of the complicated DNA of the universe is included in the labyrintho convolutions of the folded up dimensions and the stringy fields that coil I must have been writing that late at night and if the present work is correct then the letters of the DNA of our universe are the six bits t y z r g and B thank you and are there any questions from the audience yes and Anthony lesably here so this looks very interesting um in terms of um predictions are you saying that you can predict that electro-week forces effectively only operate on the the left sector not the right sector things of that kind does that come in I I don't know whether that's a predictions it's part of spin 10 which is one of the ingredients I start off with yes but I I haven't seen uh okay it's fine that it it appears to work but how does it explain that so asymmetry how does that come about when you haven't put that in to start with I'm not God I I didn't design the Universe I as I say I'm an astrophysicist I I look up the ingredients that we have and what we seem to have as a standard model which unifies in spin 10 that seems to be the grandified group and then if you want to combine that with general relativity you're stuck with combining that with the Dirac group oh sorry the durac algebra and that to my complete amazement hasn't been explored before so I thought I would explore it and the answer is it works out and it works out very elegantly as to what I can say about electric week symmetry breaking is that the electroweak Hicks field is sitting there in the spin 11-1 algebra without it having to be invented if you look at papers in the recent years even this year on Grand unification in in so10 as they like to call it they routinely talk about here's my assumption about the Higgs fields and in this particular case there aren't any assumptions about Higgs Fields the Higgs Fields turn out to be they have to be part of the algebra spin 11-1 they're the scalar part of the algebra there so it's by vectors with which are lorentz scalers that's what they have to be and that fits okay perhaps I'm asking something that can't be answered here but it seems to be a priori you would have a symmetry in such a theory in an enveloping clipped algebra between left and right and in the same way that you get a Higgs which you know trips uh left and to write you would have one that goes right left and so on now why is that symmetry not present that's what I'm trying to get at well what it comes from is that spin 10 breaks down into spin four across Spin Six and spin four is a direct product of su2s two su-2s so there's an su2 left-handed su2 right-handed and if those have two different couplings they behave as two different forces that seems to be true in in our universe today so that's the way it is and it fits okay I don't have to do anything to make it work yeah you you you might have thought there should be a symmetry between left-handed but there isn't the next question thanks Andrew okay I'm gonna go back to the prediction slide okay can I ask a question about the way that the Lorenz group kind of fitted in the waste space time kind of dropped out because think you were saying that the spatial vectors were actually five vectors in your uh in your sort of 111 algebra yeah the time like is still a vector in the 111 algebra that's correct but that seems to break Lorenzen variants that you you seem to have pulled out as a preferred direction of time and well you you might have thought so but this satisfies exactly the Dirac algebra and the Lorenz invariance that you require and the Higgs fields in addition that you bring in also satisfy Lawrence and variants so it may not look to you like that but the algebra the mathematics is precisely that because your Lorenz Transformations are they're going to be mixing grade one and grade five up in your 12-dimensional space correct yes and the the Lawrence group is products of these and it turns out that they are all six-dimensional vectors so those at least all have the same grade all the by vectors have grade six but the the time vectors does seem to be special here and again I refuse to take any responsibility for this this is mathematics this is the conclusion take it or leave it just a a quick question so among the non-predictions what causes the three generations of fermions now that sounds like a pretty big problem to the setup add space for that no I don't have any space for that and the thing you should know about the generations the three generations of fermions but only one generation of Birmingham of bosons so if there is a symmetry that connects Generations it's not a gauge Symmetry and should not be part of the gauge group if there were there would there would be additional bosonic symmetries associated with it my own suspicion about where those three generations come from since they they're identical in their gauge properties as you know but uh but they differ in their their masses and if the string theorists are right that the spectrum of masses arises from excitations of the internal dimensions something like that would be consistent with what I'm showing today so that's what I think you're appealing to string theory I I love String Theory it it tells me there's a Multiverse and there's a DNA that allows the universe to reproduce I'm a big fan of string theory okay good night you want me to criticize it I can do that