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9th October 2013, 10:11 PM  #1 
Muse
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Anyone interested in Lie algebras?
I've found it difficult to find online some software to calculate what I was looking for, so I wrote it myself: SemisimpleLieAlgebras.zip I've also done that with some other mathematicalphysics stuff, like differential geometry for general relativity. I've succeeded, and I've written it in Mathematica, Python, and C++. It does:
Mathematica is overall the nicest for developing, but Python and C++ are nonproprietary. C++ was the most awkward for developing, though its Standard Template Library was a big help. The Mma and Python versions have similar speeds, with the C++ version being much faster, a factor of 45 or so at full optimization. The smallest of the "simple" Lie algebras is SU(2) ~ SO(3) ~ Sp(2). It's the algebra of quantummechanical angular momentum, and if you've ever done operator algebra with QM AM, you've worked with a Lie algebra. If you've ever worked with QM AM states, you've worked with that algebra's representations. 
9th October 2013, 11:34 PM  #2 
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lpetrich, this looks very impressive and potentially very useful. Thank you for posting it.
Have you considered submitting it to the arxiv? 
10th October 2013, 05:30 AM  #3 
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As someone who has trouble with such mathematical concepts as subtraction (what are numbers, really?), I'm happy to accept that if Sol thinks this is good work, it is good work but could anyone explain in idiotreadable terms, what it is?

10th October 2013, 05:46 AM  #4 
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10th October 2013, 07:18 PM  #5 
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I'll try to explain what a Lie algebra is. It's named after Norwegian mathematician Sophus Lie ("Lee"), who first studied that kind of mathematical entity.
It's based on group theory. An abstractalgebra group is a set of entities and a binary operation * that satisfy these properties: Associative: a*(b*c) = (a*b)*c Identity e: a*e = e*a = a Inverse: a*inv(a) = inv(a)*a = e It's easy to show that integers with addition are a group. Here is a property that some groups have: Commutative; a*b = b*a Commutative groups are often called abelian after 19th cy. Norwegian mathematician Niels Henrik Abel. Their opposite is nonabelian. Addition of integers is, of course, abelian. The simplest group is the identity group, like {0} under addition or {1} under multiplication. All the finite abelian groups are known, and all the finite "simple" groups are also known, though the classification of the latter was an enormous effort. But what leads to Lie algebras are continuous groups. These have an infinite number of elements, of course. Groups like real numbers under addition. More interesting is groups of rotations and reflections. The group of all R and R's in an ndimensional space is called O(n), after how they can be implemented as orthogonal matrices. All the pure rotations without reflections are called SO(n), after the special orthogonal matrices, those with determinant 1. For a line, SO(1) is the identity group, and O(1) is the reflection group. For a surface, the groups are SO(2) and O(2), and for 3space, they are SO(3) and O(3). For spacetime, it's trickier, but one gets a remarkable result. Lorentz boosts can be treated much like rotations, and one gets groups much like SO(4) and O(4). All these groups have finite subgroups, and for low numbers of dimensions, all of them are known. 
10th October 2013, 09:51 PM  #6 
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I am impressed even though this is way out of my league. I am passing it to a Physics teacher or so in case they have really good kids this might help. Or just who could have fun with it!!!

10th October 2013, 11:37 PM  #7 
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The groups SO(n) are connected: one can get to any element by increasing the amount of rotation from zero, the identityelement amount.
But the groups O(n) are not. They consist of two connected parts, SO(n) and the nD rotationreflections. It's a more complicated for the "Lorentz group" of spacetime rotations and boosts. It has 4 connected parts. One for pure rotations/boosts, one with space reflections, one with time reflections, and one with space and time reflections combined. The groups SO(n) and O(n) are "Lie groups". Connected Lie groups have a very nice property. Consider a rotation element. It can be produced from a large number of small rotations: D(a) = D(a/n)^{n} ~ (1 + L*a/n)^{n} where L is that rotation's generator. One can get every connectedLiegroup element from elements arbitrarily close to the identity element. The departures from identity form a "Lie algebra", and my OP is for calculating some important sorts of Lie algebras. Quantummechanical angular momentum is related to rotation generators by (angular momentum) ~ (hbar) * (rotation generators) Liealgebra elements are related by their "commutators": [L_{i},L_{j}] = L_{i}.L_{j}  L_{j}.L_{i} = sum over k of f_{ij}^{k}*L_{k} The f's are the algebra's "structure constants". One can generalize the matrices in O(n) and SO(n) from real to complex ones. One gets groups of "unitary" matrices, U(n) and SU(n). How are these groups related? U(n) = SU(n) * U(1) O(n) = SO(n) * {I, I} only for odd n. For even n, I is in SO(n) The math for the groups' matrices D: Orthogonality: D.D^{T} = I T = transpose Hermitian conjugate: D^{+} = D^{*T} Complexconjugate and transpose Unitarity: D.D^{+} = 1 There's an additional sort of matrix groups, the "symplectic" ones of real matrices: D.J.D^{T} = J where J^{T} = J (antisymmetry) J.J^{T} ~ I A convenient choice of J: {{0,I},{I,0}} The Lorentz group of spacetime symmetries is a group of real matrices that satisfy D.g.D^{T} = g where g is the spacetime metric. Since g is symmetric, one can get the Lorentz group by analytically continuing O(4). So it's just a version of O(4). 
10th October 2013, 11:54 PM  #8 
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Let's look at the generators. What constraints do they satisfy?
SO(n): L + L^{T} = 0  L is antisymmetric (i*L is Hermitian) SU(n): L + L^{+} = 0  L is antiHermitian (i*L is Hermitian) Sp(2n)  symplectic group: L.J + J.L^{T} = 0 We get some isomorphisms: SO(2) ~ U(1) SO(3) ~ SU(2) ~ Sp(2) SO(4) ~ SU(2) * SU(2) SO(5) ~ Sp(4) SO(6) ~ SU(4) What does it mean that SO(3) and SU(2) are related? The rotation group for spinor (spin1/2) wavefunctions is SU(2). They are related by SO(3) matrices being sortofsquares of SU(2) ones, so a 360degree rotation multiplies a spinor wavefunction by 1. More generally, spinor groups for nD space are groups Spin(n), though they are locally isomorphic to SO(n). Not globally, however, as we've seen for SU(2) and SO(3). 
11th October 2013, 12:20 AM  #9 
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What's with two different groups having the same algebra? That gets me into the next subject, group representations.
A representation or rep of a group is a set of matrices that follow its operation table: D(a).D(b) = D(a*b) There's one that always exists, the regular rep. Use the group's elements as its basis space, then set D(a)_{cb} = 1 if c = a*b, 0 otherwise. It may be possible to turn a rep's matrices into the a shared blockdiagonal form. The blocks can be separated out as reps on their own. If it's possible to do that reduction, then a rep is reducible, otherwise, it's irreducible, an irrep. More formally, if a matrix X satisfies X.D(a) = D(a).X for all a in the group, then for an irrep, X can only be proportional to I. It's easy to show that all the irreps of an abelian group are 1dimensional. A finite group has a finite number of irreps, an infinite group an infinite number. Here are all the irreps of U(1): D^{(k)}(a) = exp(i*k*a) for integer k and real a This ought to remind you of 2D quantummechanical angularmomentum wavefunctions. Representations can easily be extended to Lie algebras. Here, one can learn much of what one is likely to want by constructing the basis space of a rep without bothering to construct its matrices. That's what one does for 3D quantummechanical angularmomentum wavefunctions. One constructs a set of states that share some total angular momentum while varying in projected angular momentum (the "magnetic" quantum number). These states serve as a basis space. One can construct a rotation matrix for them, and while it's fairly easy for rotating around the projection axis, its very complicated in general. Spin 0: trivial Spin 1/2: SU(2) ~ Spin(3) matrices Spin 1: SO(3) matrices (derived from SU(2) ones) Higher spins: matrices derived from SU(2) ones For bigger algebras, one does a more complicated version of this procedure. 
11th October 2013, 04:52 AM  #10 
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There are four infinite families of "simple" Lie algebras and five "exceptional" ones.
A(n) = SU(n+1) B(n) = SO(2n+1) C(n) = Sp(2n) D(n) = SO(2n) G2, F4, E6, E7, E8 Strictly speaking, D2 is "semisimple" rather than simple, since it is SO(4) ~ SU(2)*SU(2) To see the connection between Lie algebras and field structure, let's start with a (relatively) simple example: spin and 3space tensors. We start with a direction vector: n = {sin(θ)*cos(φ),sin(θ)*sin(φ),cos(θ)} We compare it with the orbitalangularmomentum wavefunctions: Y(j,m,θ,φ) for angular momentum j and projection m. For a vector, it's easy, the 3 components of Y(1,...) map onto the 3 components of n. Now consider a 2tensor. N_{ij} = n_{i}*n_{j}. Y(2,...) has 5 components and N has 6 independent components. So while Y maps onto N, N has one component left over. What is it? Consider the identity matrix, I_{ij} = δ_{ij}. It's a 2index object that transforms as a scalar. So one subtracts it out of N: N' = N  (1/3)I, with the (1/3) chosen to give N' 5 independent components. One now finds an exact match. So, in general, Y(j,...) and j n's with appropriate δ's map onto each other. Thus, a jindex symmetric and traceless tensor can reasonably be said to have spin j. We can now consider a more general 2tensor. It splits up into a symmetric traceless 2tensor, an antisymmetric part that contains a vector, and a part proportional to the identity matrix that's a scalar. That exactly matches how two angular momenta combine when they both have value 1: (1) * (1) = (2) + (1) + (0) One can continue in this vein with spinors. Two spinors combine to make a vector and a scalar: (1/2) * (1/2) = (1) + (0) A vector and a spinor make a vectorspinor and a spinor: (1) * (1/2) = (3/2) + (1/2) One gets the same connection between Liealgebra reps and field structure in many other cases, even though the reps are often much more complicated. This has been a powerful tool for analyzing field structure and particle interactions, and that's the main reason that I wrote my Liealgebra software. 
11th October 2013, 11:48 AM  #11 
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Let's see what happens with spacetime. It's 4D, and the appropriate algebra is the SO(4) one. We can express its irreps as pairs of angular momenta.
A scalar is, of course, (0,0) A vector is (1/2,1/2)  2*2 = 4 components A Dirac spinor is (1/2,0) + (0,1/2)  2+2 = 4 components A Dirac spinor is reducible to two parts, one for each chirality (handedness). Let's now consider a 2tensor. One takes a product of two vectors: (1/2,1/2) * (1/2,1/2) = (1,1) + (1,0) + (0,1) + (0,0) A symmetric traceless 2tensor, the two parts of an antisymmetric 2tensor, and a scalar: 4*4 = 9 + 3 + 3 + 1 = 16 The antisymmetric 2tensor splits into two, from multiplying by the 4D antisymmetric symbol. For the electromagnetic field, the electric and magnetic fields E and B combine into E + i*B and E  i*B It's easy to show that the two combinations get Lorentz boosted separately. One can also get the two EMfield invariants from the two compoments. E + i*B^{2} = (E^{2}  B^{2}) + 2*i*(E.B) E  i*B^{2} = (E^{2}  B^{2})  2*i*(E.B) Spinors (spin1/2 wavefunctions) are rather complicated entities, but like vectors and tensors, they can be generalized to more space and spacetime dimensions, and one finds: Even (2n) space(time) dimensions: two irreps with 2^{n1} components each. Odd (2n+1) space(time) dimensions: one irrep with 2^{n} components. 
11th October 2013, 06:15 PM  #12 
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Going from spacetime and spin to quark flavors, we find some additional Liealgebra applications.
The up, down, and strange quarks all have masses smaller than the QCD colorconfinement scale, with the up and down ones being especially small. That produces some approximate flavor symmetries. The first one discovered is between the up and down quarks, "isotopic/isobaric spin" or "isospin". It's SU(2) because of the quarks' complex wavefunctions and because of their 2 flavors, and the "spin" in its name is from what it behaves like. Mesons have a quarkantiquark structure, and their isospins combine as: (1/2) * (1/2) = (1) + (0) pion and rho / eta and omega Baryons have 3 quarks, and their isospins combine as: (1/2) * (1/2) * (1/2) = ( (1) + (0) ) * (1/2) = (3/2) + 2 (1/2) delta baryon / nucleon With the strange quark, one gets the group SU(3), with the ordinary quarks being in a rep called 3, and antiquarks in a rep called 3* (properly 3 bar), because it's not equivalent to ordinary reps. In this notation, a state with spin j would be called "2j+1". There's a nice graphical technique called "Young diagrams" for constructing SU(n) reps, and a technique called the "LittlewoodRichardson rule" for finding product reps. The light quarks are * (size 3) The light antiquarks are * * (size 3*) Quarks and antiquarks combine as ** * (size 8) and * (size 1) Two quarks make combinations ** (size 6) and * * (size 3*) Adding a third one makes combinations *** (size 10) two ** * (size 8) and * * * (size 1) The spin3/2 light baryons form a 10 in the quark flavors, and the quarks' wavefunctions are symmetric in both spin and flavor. The spin1/2 light baryons form a 8 in the quark flavors, and to get their magnetic moments correct, it's necessary for the spins and flavors to have a symmetric combination. This was a big difficulty for the quark model, because it implied that quarks follow BoseEinstein statistics rather than the FermiDirac statistics that one expects from their spins. But in QCD, the quarks also have color states, with SU(3) color symmetry. Their colors combine to make a colorless combination, a state with size 1. This is antisymmetric, and that rescues quarks' FermiDirac statistics. I could crunch through all of this with the software I'd written. 
11th October 2013, 06:53 PM  #13 
Muse
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Now for symmetry breaking. Many symmetries are only approximate, and some symmetries are hidden enough to take some effort to discover. We indeed see that here.
In this thread, we first encountered it in orbitalangularmomentum wavefunctions, where we find a breakdown from rotation around all axes to rotation around some selected axis. That produces breakdown SO(3) > SO(2) or SU(2) > U(1) For the light quarks, we get a breakdown SU(3) > SU(2) * U(1) uds to isospin (ud) and strangeness (s) + something Yes, strangeness. Back in the 1950's or so, it became evident that some particles lasted much longer than what one might expect, and this strange property led to the naming of a new quantum number: strangeness. The light mesons and spin1/2 baryons: 8 > (1,0) + (0,0) + (1/2,1) + (1/2,1)  pseudo angular momentum 8 > (3,0) + (1,0) + (2,1) + (2,1)  multiplicity pion, eta, kaon, antikaon sigma, lambda, nucleon, xi The light spin3/2 baryons: 10 > (3/2,1) + (1,0) + (1/2,1) + (0,2)  pseudo angular momentum 10 > (4,1) + (3,0) + (2,1) + (1,2)  multiplicity delta, sigmax, xix, omega I'm able to crunch through all of that also. 
11th October 2013, 07:26 PM  #14 
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Have you looked at R or Octave? Seems like one or the other or both might be a better fit that python or c++. http://www.rproject.org/ http://www.gnu.org/software/octave/
(If you already considered and rejected them, I'm curious why.) 
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11th October 2013, 10:57 PM  #15 
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I'm not very familiar with either R or Octave, and I hadn't considered either. But I'm willing to take requests.
Octave is an opensource imitation of Matlab, so some of my experience there could carry over. Rosetta Code seems like a good place to look if one wants to check on how easy or hard it is to do various things in various programming languages. For instance, I didn't do a plainC version because I would have had to write an associative array: Associative arrays/Creation/C  Rosetta Code vs. Associative array/Creation  Rosetta Code and Associative array/Iteration  Rosetta Code I also had to integerize some of the calculations for Python, because its implementation of fractions is dog slow, at least in version 2.7.3. I carried that implementation over to C++, IIRC. I had to do fractions explicitly for C++: Arithmetic/Rational  Rosetta Code. I have a way of estimating how much work my code has to do  or how feasible it would be by my code's bruteforce methods. For E8, especially, some simplelooking reps are surprisingly large. The difficulty is that my code calculates complete rep bases in the CartanWeyl basis, giving results as a list of {multiplicity, root, weight}. I use Weyl's formula for the total degeneracy or multiplicity of a rep; one that does not need an explicit rep basis. To succeed, I needed to implement bignums (Arbitraryprecision integers (included)  Rosetta Code). They are built into Mma and Python, but I had to use the GNU Scientific Library for C++. I had to do a lot of Matrix multiplication  Rosetta Code, and I inverted matrices with Gaussian elimination  Rosetta Code. They are built into Mma, and I implemented both in Python and C++. At least I didn't have to do eigensystems. 
12th October 2013, 12:12 AM  #16 
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For C++ side have you looked at Boost?
http://www.boost.org/doc/libs/1_54_0.../rational.html http://www.boost.org/doc/libs/1_54_0...tml/index.html 
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12th October 2013, 12:43 AM  #17 
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Happy birthday, lpetrich, and thanks for sharing this too! Do you plan to host the source on sourceforge / github / google code or similar?

12th October 2013, 03:56 AM  #18 
Muse
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Titled links:
Rational Number Library  1.54.0 Math Toolkit  1.54.0 I like Firefox's Copy Link Text extension. Looks good. I guess I ought to benchmark my rationalnumber code vs. Boost's. I didn't find any linear algebra in the Boost library, however. I had wanted to avoid outside libraries as much as possible, but I threw in the towel about C++ bignums. I hadn't thought about that. Any good comparisons of those sites? Or should I choose one that's already hosting similar sorts of software? I checked on simplie  a simple program for Lie algebras  Google Project Hosting and it didn't have much of what I'd implemented. 
12th October 2013, 04:20 AM  #19 
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Understandable. Although you don't need full Boost, just headers and maybe implementation files. (For Pools it was sufficient...)
Was uBLAS insufficient? 
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12th October 2013, 04:40 AM  #20 
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A bit more that I want to explain. Gauge theory^{WP}, which states:
Quote:
Imagine a system that transforms with continuous parameters p that have the same value everywhere in it. The parameters are for a global or rigid symmetry. Now make the symmetry local or varying: p = p(x), x = spacetime position. One quickly gets terms in gradients of p: D(p). So one adds a gauge particle whose field varies under the symmetry transformations so as to cancel out the D(p) terms. My code is only for the symmetry groups of gauge theories, and these are Lie groups, of course. Gauge particles are multiplets in the gauge symmetry, with one member for each gaugegroup generator. That puts gauge particles in the gauge algebra's "adjoint representation". Its basis is the algebra generators themselves, and its matrices are composed from the algebra's structure constants. For SO(n), the adjoint rep is antisymmetric 2tensors. Its size is n(n1)/2. For SU(n), the adjoint rep is from a vector * a vector's conjugate  scalar. Its size is n^2  1. For Sp(2n), the adjoint rep is symmetric 2tensors. Its size is n(2n+1). The five exceptional algebras are more complicated, but their adjointrep sizes are: G2: 14, F4: 52, E6: 78, E7: 133, E8: 248 The first gauge theory discovered was electromagnetism or the photon. It has symmetry U(1). The unbroken Standard Model has gauge symmetry SU(3) * SU(2) * U(1) SU(3) = QCD, SU(2) = "weak isospin", U(1) = "weak hypercharge" The electroweak part has symmetry SU(2) * U(1) when unbroken. But its breaking does SU(2) > U(1) and mixes the two U(1)'s, with the only survivor being the electromagnetic U(1). The QCD part is not broken, but instead, confined. Its gauge particle, the gluon, has 8 color states: color*anticolor with no colorless state. Grand Unified Theories involve bigger groups: SU(5), SO(10), E6, E8, etc. I was able to work out how they break down to the Standard Model, and what reps become the Standard Model particle reps. I found it satisfying to be able to calculate that. I could also calculate which interactions are possible, something also satisfying. 
12th October 2013, 05:12 AM  #21 
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12th October 2013, 06:19 AM  #22 
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12th October 2013, 08:04 AM  #23 
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12th October 2013, 01:04 PM  #24 
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12th October 2013, 03:15 PM  #25 
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In addition to Boost (a great project I use all the time), there's also Blitz++, a highly optimized array/vector library for scientific computing. Looking up Blitz++ also led me to C++QED, which seems like something that not only might be of interest to you, but might even be something you might be able to contribute to. It includes Python bindings.

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12th October 2013, 03:17 PM  #26 
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What I mean by the antisymmetric symbol: ε_{ij...k} with n indices for n space dimensions. It is 1 for indices 1,2,3, ..., n, and also for even permutations of that sequence of indices, and 1 for odd permutations.
Let's see how a 2tensor decomposes. T_{ij} = (T_{ij} + T_{ji})/2 (symmetric: ji = ij) + (T_{ij}  T_{ji})/2 (antisymmetric: ji =  ij) n^{2} = n(n+1)/2 + n(n1)/2 This is as far as it goes for SU(n), but for SO(n), one can go further. The symmetric case decomposes T_{ij} = (T_{ij}  (1/n)*δ_{ij}*T_{kk}) (traceless) + (1/n)*δ_{ij}*T_{kk} (scalar) n(n+1)/2 = (n1)(n+2)/2 + 1 The antisymmetric case depends rather strongly on the number of dimensions. 2D: T_{ij} = (1/2)*ε_{ij}*(ε_{kl}T_{kl}) (scalar) 3D: T_{ij} = (1/2)*ε_{ijk}*(ε_{lmk}T_{lm}) (vector) 4D: T_{ij} splits in two: T_{ij} = (1/2)*ε_{ijkl}*T_{kl} (selfdual, 3vector) T_{ij} =  (1/2)*ε_{ijkl}*T_{kl} (antiselfdual, 3vector) 6 = 3 + 3 More than 4D: no decomposition One has to be more careful in the spacetime case. Instead of the multipliers being 1 and 1, they are i and i. That's what makes the E+i*B and Ei*B for the electromagnetic field. 
13th October 2013, 07:54 AM  #27 
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I see, you wrote "the 4D antisymmetric symbol" instead of the equation so these other guys reading this would more easily understand, thanks.
Maybe you could proof read the math part of my paper on parallel universes that I'm almost finished with if you get a chance. I'm certain it will raise an eyebrow or two. 
13th October 2013, 09:16 PM  #28 
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I'd be willing to do so.
Turning to Grand Unified Theories, here are some of the morediscussed gaugesymmetry groups that I've found:
With my code, I was able to verify which reps of these groups break into which StandardModel reps. SU(5):
G= gauge, H = Higgs, EF= elementary fermion conjg = conjugate, AS = antisymmetric SO(10):

18th October 2013, 04:54 AM  #29 
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The Standard Model is rather byzantine, it must be said. Its gaugesymmetry group is SU(3) * SU(2) * U(1)  QCD + electroweak (weak isospin + weak hypercharge).
Its multiplet structures I'll list as (QCD multiplicity, WIS multiplicity, WHC "charge") Gauge: (8,1,0), (1,3,0), (1,1,0) Gluon, W (weak isospin), B (weak hypercharge). The latter two make the lowenergy W+, Z, and photon. Higgs: (1,2,1/2), (1,2,1/2) Standard Model: 2nd = complex conjugate of 1st (same particle), supersymmetric extensions: separate particles Elementary fermions: Lefthanded: (3,2,1/6), (1,2,1/2), (antiparticles) (3*,1,2/3), (3*,1,1/3), (1,1,0), (1,1,1/2) quark, lepton, up quark, down quark, neutrino, electron Righthanded: (3,1,2/3), (3,1,1/3), (1,1,0), (1,1,1/2), (antiparticles) (3*,2,1/6), (1,2,1/2) up quark, down quark, neutrino, electron, quark, lepton The main difficulty with GUT's is that they exchange this complicated structure for complicated symmetry breaking. I've surveyed the literature for proposed GUT Higgs particles, and I've found quite a lot. They are also in multiplets that can get very big. SU(5): 24: adjoint: (vector * conjg vector)  scalar SO(10): 10: vector 45: adjoint: antisymmetric 2tensor 16, 16*: spinors 54: symmetric traceless 2tensor 120: AS 3tensor 210: AS 4tensor 126, 126*: AS 5tensor split by antisymmetric symbol 144, 144*: (vector * spinor)  spinor (not necessarily all of them together!) E6: Gauge multiplet: adjoint of course, with size 78. Each generation of elementary fermions, and also the Higgs particles, can fit into an irrep with size 27. Their antiparticles go into 27*, its conjugate. The GUT Higgs particles go into reps with sizes like 351, 351', 351*, 351'*, 650. But E6 has a nice property. A symmetric product of three 27's yields, among other irreps, a scalar one. So an E6 GUT can reduce the EFHiggs interactions to very simple form. One gets a similar result for SO(10), though it is not quite as simple. The downside of this EFHiggs interaction unification is that it is too successful. It does not allow crossgeneration decays. I concede my code is not able to verify the detailed workings of these GUT Higgs mechanisms, since it isn't designed for that. However, I'm able to verify how all these multiplets are constructed, and what products and powers of them contain. The sizes of some of these Higgs multiplets ought to make one suspicious, and superstring compactification (6 out of 10 spacetime dimensions becoming a tiny ball) usually involves some different GUT symmetrybreaking mechanism. 
18th October 2013, 05:53 AM  #30 
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26th October 2013, 12:06 PM  #31 
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Still trying to decide on a good opensource license.
I was reminded of The AIPS (Astronomical Image Processing System) FAQ
Quote:
G (Q1)> G1 (Q2)> G2 ... until it stops I'll have to get it into releasable form so I can upload it. 
26th October 2013, 09:07 PM  #32 
Muse
Join Date: Feb 2007
Posts: 762

There are some important sorts of Lie algebras that my code won't work on, since it's only designed for semisimple ones and their decompositions.
Consider the Euclidean group, Euc(n), the group of symmetries of a flat nspace. As elsewhere, it's often much easier to study the algebra of its generators than the full group directly, and that algebra is nspace rotations, SO(n), with nspace translations or shifts, T^n, added. The Poincaré group, the group of spacetime symmetries, is related to Euc(4), much as the Lorentz group is related to SO(4). Its reps are related to elementaryparticle states with: Translation operators ~ linear momentum Rotation operators ~ angular momentum 
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