http://sites.science.oregonstate.edu/coursewikis/GO/ 2020-01-26T14:08:14-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/ http://sites.science.oregonstate.edu/coursewikis/GO/lib/images/favicon.ico text/html 2014-09-08T19:39:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/22non?rev=1410230340 The general $2\times2$ octonionic Hermitian matrix can be written \begin{equation} \AA = \begin{pmatrix}p& a\\ \noalign{\smallskip} \bar{a}& m\\\end{pmatrix} \end{equation} with $p,m\in\RR$ and $a\in\OO$, and satisfies its characteristic equation \begin{equation} \AA^2 - (\tr \AA) \, \AA + (\det \AA) \, \II = 0 \end{equation} where $\tr \AA$ denotes the trace of $\AA$, and where there is no difficulty with commutativity and associativity in defining the determinant of $\AA$ as usual via \begin{e… text/html 2014-09-08T20:37:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/22real?rev=1410233820 We consider the eigenvector problem $\AA v = v\lambda$ over $\OO$, and look (only) for solutions with real eigenvalues, that is, where $\lambda\in\RR$. There are no surprises in this case, or rather the only surprise is that we can not show that $\lambda$ is real, but must assume this property separately. First of all, there is only one independent octonion in $\AA$; the components of $\AA$ live in a complex subalgebra $\CC\subset\OO$. We can write \begin{equation} \AA = \begin{pmatrix}p & \b… text/html 2014-09-23T14:32:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/2square?rev=1411507920 In (ss)~9.2, we showed that the groups $\SU(2,\KK)$ are \begin{align} \SU(2,\RR)&\cong\SO(2) \\ \SU(2,\CC)&\cong\SO(3) \\ \SU(2,\HH)&\cong\SO(5) \label{su2h}\\ \SU(2,\OO)&\cong\SO(9) \end{align} for $\KK=\RR,\CC,\HH,\OO$, respectively (and where all congruences are local, that is, up to double cover). Similarly, in (ss)~9.3, we showed that the groups $\SL(2,\KK)$ are \begin{align} \SL(2,\RR)&\cong\SO(2,1) \\ \SL(2,\CC)&\cong\SO(3,1) \\ \SL(2,\HH)&\cong\SO(5,1) \\ \SL(2,\OO)&\cong\SO(9,1) \e… text/html 2014-09-07T14:51:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/33non?rev=1410126660 Without being able to solve (some version of) the characteristic equation in the $3\times3$ case, it is not possible in general to determine all the (non-real) eigenvalues of a given Hermitian octonionic matrix. It is therefore instructive to consider several explicit examples. text/html 2014-09-22T20:06:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/33real?rev=1411441560 We now turn to the $3\times3$ case. It is not immediately obvious that $3\times3$ octonionic Hermitian matrices have a well-defined characteristic equation. We therefore first review some of the properties of these matrices before turning to the eigenvalue problem. As in the $2\times2$ case, over the octonions there will be solutions of the eigenvalue problem with eigenvalues which are not real; we consider here only the real eigenvalue problem. text/html 2014-09-08T19:39:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/3psi?rev=1410230340 An essential ingredient in the construction of the Green-Schwarz superstring~[Green, Schwarz, Witten] is the spinor identity \begin{equation} \epsilon^{klm} \gamma^\mu \Psi_k \bar\Psi_l \gamma_\mu \Psi_m = 0 \label{Super} \end{equation} for anticommuting spinors $\Psi_k$, $\Psi_l$, $\Psi_m$, where $\epsilon^{klm}$ indicates total antisymmetrization. This identity can be viewed as a special case of a Fierz rearrangement. An analogous identity holds for commuting spinors $\Psi$, namely \begin… text/html 2016-05-25T14:00:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/3square?rev=1464210000 The Freudenthal--Tits magic square was originally given in terms of Lie algebras; the version shown in Table~3 lists instead particular real forms of the corresponding Lie groups. Vinberg later gave a symmetric parameterization of the Lie algebras $\alg_3(\KK_1,\KK_2)$ in this magic square in the form \begin{equation} \alg_3(\KK_1,\KK_2) = sa(3,\KK_1\otimes\KK_2) \oplus \der(\KK_1) \oplus \der(\KK_2) \label{Vin3} \end{equation} where $\KK_1$, $\KK_2$ are division algebras (or possibly thei… text/html 2014-09-08T19:40:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/aa?rev=1410230400 In order to discuss the remaining four exceptional Lie groups, we introduce the algebra of $3\times3$ octonionic Hermitian matrices, known as the Albert algebra, written as $\bH_3(\OO)$. Since the product of Hermitian matrices is not necessarily Hermitian, we introduce the Jordan product \begin{equation} \AAA \circ \BBB = {1\over2} (\AAA\BBB + \BBB\AAA) \end{equation} for elements $\AAA,\BBB\in\bH_3(\OO)$, which we henceforth refer to as Jordan matrices. It is easily checked that the Jordan p… text/html 2014-10-25T11:00:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/acknowledge?rev=1414260000 This book has its origins in the research conducted by one of us (Corinne) nearly 30 years ago. She in turn introduced her husband (Tevian) to the octonions more than 20 years ago; they have collaborated on further research in this area ever since. This book reflects Tevian's efforts to understand what Corinne has taught him; although most of the actual writing was done by Tevian, the final product has been shaped every bit as much by Corinne, however indirectly. Rob Wilson has been a part of… text/html 2014-03-24T13:04:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/calg?rev=1395691440 The complex numbers are more than just a vector space; they are also an algebra, that is you can multiply them together. How do you compute the product of complex numbers? Simply multiply it out, that is \begin{align} (a+bi) (c+di) &= (a+bi) c + (a+bi) di \nonumber\\ &= (ac-bd) + (bc+ad)i \end{align} text/html 2014-09-23T13:52:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/cayley?rev=1411505520 We have argued in (ss)~14.5--14.7 that the ordinary momentum-space (massless and massive) Dirac equation in $3+1$ dimensions can be obtained via dimensional reduction from the Weyl (massless Dirac) equation in $9+1$ dimensions. This latter equation can be written as the eigenvalue problem \begin{equation} \label{Dirac} \tilde{\PPP}\psi = 0 \end{equation} where $\PPP$ is a $2\times2$ octonionic Hermitian matrix corresponding to the 10-dimensional momentum and tilde again denotes trace reversal… text/html 2014-09-11T22:09:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/cd?rev=1410498540 We have constructed the complex numbers, the quaternions, and the octonions by doubling a smaller algebra. We have \begin{align} \CC &= \RR \oplus \RR i \\ \HH &= \CC \oplus \CC j \\ \OO &= \HH \oplus \HH \ell \end{align} We can emphasize this doubling, using a slightly different notation. A complex number $z$ is equivalent to pair of real numbers, its real and imaginary parts. So we can write \begin{equation} z = (x,y) \end{equation} corresponding in more traditional language to $z=x+iy$. C… text/html 2014-03-28T08:50:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/cgeom?rev=1396021800 Thanks to Euler's formula, \begin{equation} e^{i\theta} = \cos\theta + i\sin\theta \label{cis} \end{equation} polar coordinates can be used to write complex numbers in terms of their norm and a phase angle $\theta$. (A factor of the form $e^{i\theta}$ is called a phase.) That is, any complex number can be written in the form \begin{equation} z = r e^{i\theta} \end{equation} where \begin{equation} r = |z| \end{equation} since $|e^{i\theta}|=1$. Each complex number thus has a direction associa… text/html 2014-03-23T11:43:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/chist?rev=1395600180 Complex numbers were first used in the 16th century in order to solve cubic equations, as there are some cases with real solutions that nonetheless require the use of complex numbers in order to obtain those solutions. The recognition of the complex numbers as an object worthy of study in their own right is usually attributed to Rafael Bombelli, who in 1572 was the first to formalize the rules of complex arithmetic (and also, at the same time, the first to write down the rules for manipulating … text/html 2014-03-28T09:18:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/cintro?rev=1396023480 Begin with the real numbers, $\RR$. Add ``the'' square root of $-1$; call it $i$. You have just constructed the complex numbers, $\CC$, in the form \begin{equation} \CC = \RR \oplus \RR \, i \end{equation} That is, a complex number $z$ is a pair of real numbers ($a$,$b$), which is usually written as \begin{equation} z = a + bi \end{equation} and which can be thought of as either a point in the (complex) plane with coordinates $(a,b)$ or as a vector with components $a$ and $b$. text/html 2017-04-25T20:40:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/class?rev=1493178000 The complete classification of Lie algebras is one of the major results in mathematics, providing a description of the building blocks of all continuous symmetry groups. This classification was first outlined by the German mathematician Wilhelm Killing in the late 1800s, and the first rigorous construction was provided by French mathematician \'Elie Cartan shortly thereafter; both Killing and Cartan worked with complex Lie algebras. text/html 2012-04-11T22:05:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/complexn?rev=1334207100 Begin with the real numbers, $\RR$. Add ``the'' square root of $-1$; call it $i$. You have just constructed the complex numbers, $\CC$, in the form \begin{equation} \CC = \RR \oplus \RR \, i \end{equation} That is, a complex number $z$ is a pair of real numbers ($a$,$b$), which is usually written as \begin{equation} z = a + bi \end{equation} which can be thought of as either a point or a vector in the (complex) plane. How do you multiply complex numbers? Simply multiply it out, that is \begin… text/html 2014-09-07T15:52:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/csplit?rev=1410130320 Start with the real numbers, and apply the Cayley-Dickson process with $\epsilon=-1$. The resulting algebra is known as the split complex numbers, denoted $\CC'$, and satisfies \begin{equation} \CC' = \RR \oplus \RR L \end{equation} where \begin{equation} L^2 = 1 \end{equation} rather than $-1$. What are the properties of such numbers? text/html 2014-09-07T15:09:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/diag?rev=1410127740 The group $F_4$ is the automorphism group of both the Jordan and Freudenthal products, that is, if $\XXX,\YYY\in\HHH$, and $\phi\in F_4$, then \begin{align} \phi(\XXX\circ\YYY) &= \phi(\XXX)\circ\YYY \\ \phi(\XXX*\YYY) &= \phi(\XXX)*\YYY \\ \end{align} Loosely speaking, $F_4$ allows us to change basis in $\HHH$, without affecting normalization or inner products. We have seen in (ss)~13.6 that there is indeed an eigenvalue problem for any Jordan matrix $\XXX\in\HHH$, with exactly three real ei… text/html 2014-09-08T19:38:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/dirac?rev=1410230280 The Dirac equation describes the quantum mechanical state of a relativistic, massive, spin-$1\over2$ particle, such as the electron. The Dirac equation in 4 dimensions is usually given as \begin{equation} (i\hbar\gamma^\mu\partial_\mu - mc) \Psi = 0 \label{diraceq} \end{equation} What do these symbols mean? First of all, $m$ is the mass of the particle described by $\Psi$, $c$ is the speed of light (which we normally set to $1$) and $\hbar$ is Planck's constant (divided by $2\pi$). The notati… text/html 2012-07-04T12:17:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/diracn?rev=1341429420 The Dirac equation describes the quantum mechanical state of a relativistic, massive, spin-$1\over2$ particle, such as the electron. The Dirac equation in 4 dimensions is usually given as \begin{equation} (i\hbar\gamma^\mu\partial_\mu - mc) \Psi = 0 \label{diraceq} \end{equation} What do these symbols mean? First of all, $m$ is the mass of the particle described by $\Psi$, $c$ is the speed of light (which we normally set to $1$) and $\hbar$ is Planck's constant (divided by $2\pi$). The notati… text/html 2014-09-04T16:53:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/e6?rev=1409874780 In (ss)~11.3, we discussed the transformations that preserve the determinant and trace of Jordan matrices $\XXX\in\bH_3(\OO)$; such transformations are elements the group $F_4=\SU(3,\OO)$. It is now straightforward to remove the restriction on the trace, and consider all transformations on $\bH_3(\OO)$ that preserve the determinant; this group would deserve the name $\SL(3,\OO)$. text/html 2014-09-07T15:29:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/e7?rev=1410128940 Freudenthal provided a description of the Lie algebra $\ee_{7(-25)}$, denoted throughout this section simply as $\ee_7$. algebra,~[arXiv:1311.0341] . providing a natural symplectic interpretation of its minimal representation. We work throughout with the Lie algebra, but, given our explicit matrix description of $E_{6(-26)}$ in terms of nested operations, it is not difficult to reinterpret our discussion at the group level. In particular, since a Lie algebra is just an infinitesimal represen… text/html 2014-09-01T11:31:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/e8?rev=1409596260 It would be nice to extend our discussion to the last remaining exceptional Lie group, namely $E_8$. However, unlike the other exceptional Lie groups, $E_8$ does not have any ``small'' representations; the minimal representation of $E_8$ is the adjoint representation, with 248 elements. The minimal representation of $G_2$ is $\OO$ (7-dimensional), $F_4$ acts on the trace-free part of the Albert algebra (26-dimensional), $E_6$ acts the Albert algebra (27-dimensional), and $E_7$ acts on $\{\Pc=… text/html 2014-09-07T15:30:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/e8latt?rev=1410129000 The octonionic integers provide a natural description of the $\ee_8$ lattice, as we now show. Recall that a simple Lie algebra of dimension $d$ and rank $r$ contains a Cartan subalgebra of dimension $r$, consisting of mutually consisting elements. Over $\CC$, the remaining $d-r$ elements can be chosen to be simultaneous eigenvectors of Cartan elements, with real eigenvalues; the resulting $r$-tuples form the root diagram of the Lie algebra, which generates a lattice. text/html 2014-09-07T15:33:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/eigen?rev=1410129180 The eigenvalue problem as usually stated is to find vectors $v\ne0$ and numbers $\lambda$ that satisfy \begin{equation} \AA v = \lambda v \label{wrong} \end{equation} for a given square matrix $\AA$, which we will assume to be complex and Hermitian $(\AA^\dagger=\AA)$. The basic properties of the eigenvalue problem for such matrices are well-understood: \begin{enumerate}\item The eigenvalues of complex Hermitian matrices are real. \item Eigenvectors of a complex Hermitian matrix corresponding … text/html 2012-07-04T12:22:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/eigenn?rev=1341429720 The eigenvalue problem as usually stated is to find solutions $\lambda$, $v$ (with $v\ne0$) to the equation \begin{equation} \AA v = \lambda v \label{wrong} \end{equation} for a given square matrix $\AA$, which we will assume to be complex and Hermitian ($\AA^\dagger=\AA$). The basic properties of the eigenvalue problem for such matrices are well-understood: \begin{enumerate}\item The eigenvalues of complex Hermitian matrices are real. \item Eigenvectors of a complex Hermitian matrix correspon… text/html 2014-09-20T11:49:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/f4?rev=1411238940 As suggested in (ss)~11.2, the Albert algebra $\bH_3(\OO)$ can be regarded as a generalization of the vector space $\RR^3$ to the octonions. The elements $\XXX\in\bH_3(\OO)$ of the Albert algebra are Hermitian matrices, and have well-defined determinants. We therefore seek transformations that preserve these properties. text/html 2014-07-03T22:23:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/forms?rev=1404451380 A simple Lie algebra admits a non-degenerate inner product, called the Killing form. For complex matrix Lie algebras, the Killing form can be taken to be \begin{equation} B(X,Y) = \tr(XY) \end{equation} A vector space with a non-degenerate inner product admits an orthonormal basis $\{X_m\}$ satisfying \begin{equation} B(X_m,X_n) = \pm \delta_{mn} \end{equation} where $\delta_{mn}$ denotes the Kronecker delta, which is $1$ if $m=n$ and $0$ otherwise. text/html 2014-09-20T11:14:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/g2?rev=1411236840 What are the symmetries of the division algebras $\RR$, $\CC$, $\HH$, and $\OO$? Let's start with $\HH$. The labeling $i$, $j$, $k$ is arbitrary; any right-handed, orthonormal basis of $\Im\OO$ could be used instead of $i$, $j$, $k$. This freedom represents a geometric symmetry, corresponding to the rigid rotations on the unit sphere in the 3-dimensional vector space of imaginary quaternions. But this symmetry is also algebraic, since the choice of handedness ensures that the multiplication … text/html 2014-09-07T15:37:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/gamma?rev=1410129420 In the last chapter, we introduced the momentum-space Dirac equation, which we rewrite in the form \begin{equation} ( \gamma_\mu \, p^\mu - m ) \Psi = 0 \label{diracnew} \end{equation} where we have set $c=1$. The gamma matrices $\gamma_\mu$ satisfy \begin{equation} \{ \gamma_\mu,\gamma_\nu \} = \gamma_\mu\gamma_\nu + \gamma_\nu\gamma_\mu = 2 g_{\mu\nu} \label{gammaid} \end{equation} How do we find such matrices? text/html 2012-07-05T16:23:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/gamman?rev=1341530580 In the last chapter, we introduced the momentum-space Dirac equation, which we rewrite in the form \begin{equation} ( \gamma_\mu \, p^\mu - m ) \Psi = 0 \label{diracnew} \end{equation} where we have set $c=1$. The gamma matrices $\gamma_\mu$ satisfy \begin{equation} \{ \gamma_\mu,\gamma_\nu \} = \gamma_\mu\gamma_\nu + \gamma_\nu\gamma_\mu = 2 g_{\mu\nu} \label{gammaid} \end{equation} How do we find such matrices? text/html 2014-09-21T16:08:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/hopf?rev=1411340880 Consider a complex column vector \begin{equation} v = \begin{pmatrix}b\cr c\cr\end{pmatrix} \label{column} \end{equation} where $b,c\in\CC$. What can we do with $v$? Well, we can take its Hermitian conjugate, defined as before, namely \begin{equation} v^\dagger = \begin{pmatrix}\bar{b}& \bar{c}\cr\end{pmatrix} \end{equation} We can now multiply $v$ and $v^\dagger$ in two quite different ways. We can produce a complex number \begin{equation} v^\dagger v = |b|^2 + |c|^2 \label{scalar} \end{equa… text/html 2012-07-05T16:23:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/hopfn?rev=1341530580 Consider a complex column vector \begin{equation} v = \begin{pmatrix}b\cr c\cr\end{pmatrix} \label{column} \end{equation} where $b,c\in\CC$. What can we do with $v$? Well, we can take its Hermitian conjugate, defined as before, namely \begin{equation} v^\dagger = \begin{pmatrix}\bar{b}& \bar{c}\cr\end{pmatrix} \end{equation} We can now multiply $v$ and $v^\dagger$ in two quite different ways. We can produce a complex number \begin{equation} v^\dagger v = |b|^2 + |c|^2 \label{scalar} \end{equa… text/html 2014-03-28T11:03:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/hurwitz?rev=1396029780 A composition algebra $\KK$ possesses a norm, that is a nondegenerate quadratic form satisfying the identity \begin{equation} |pq|^2 = |p|^2|q|^2 \end{equation} for all $p,q\in\KK$. The Hurwitz Theorem, published posthumously by Adolf Hurwitz in 1923, states that the reals, complexes, quaternions, and octonions are the only real composition algebras with positive-definite norm, and hence the only composition algebras without zero divisors. That is, the only such algebras that contain $\R… text/html 2010-08-19T16:06:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/intro?rev=1282259160 text/html 2014-09-08T20:01:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/jnw?rev=1410231660 It is well-known that the Albert algebra, introduced in (ss)~11.2, is the only exceptional realization of the Jordan formulation of quantum mechanics~[Gürsey and Tze] ; this is in fact how it was first discovered. We summarize this formalism here. text/html 2014-09-08T19:59:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/jordan?rev=1410231540 In (ss)~13.2--13.5, we considered both the left and right eigenvalue problems for $2\times2$ and $3\times3$ octonionic Hermitian matrices. In the $3\times3$ octonionic case, even if we assume that the eigenvalues are real, the eigenvalue problem does not quite behave as expected. For this case, there are 6, rather than 3, real eigenvalues, which come in 2 independent families, each consisting of 3 real eigenvalues which satisfy a modified characteristic equation rather than the usual one. … text/html 2014-09-03T18:10:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/leptons?rev=1409793000 The description in the preceding sections of 10-dimensional Minkowski space in terms of Hermitian octonionic matrices is a direct generalization of the usual description of ordinary (4-dimensional) Minkowski space in terms of complex Hermitian matrices. If we fix a complex subalgebra $\CC\subset\OO$, then we single out a 4-dimensional Minkowski subspace of 10-dimensional Minkowski space. The projection of a 10-dimensional null vector onto this subspace is a causal 4-dimensional vector, which i… text/html 2014-07-03T19:23:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/liea?rev=1404440580 A Lie algebra, again named after Sophus Lie, is a vector space $\gg$ together with a binary operation \begin{align} \gg \times \gg &\longrightarrow \gg \\ (x,y) &\longmapsto [x,y] \end{align} called the Lie bracket of $x$ and $y$. The Lie bracket is bilinear and satisfies \begin{align} [x,y] + [y,x] &= 0 \label{Lie}\\ \quad[x,[y,z]] + [y,[z,x]] + [z,[x,y]] &= 0 \label{Jacobi} \end{align} where the second condition is known as the Jacobi identity. Lie algebras can be thought of as infinitesimal… text/html 2014-07-03T16:30:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/lieg?rev=1404430200 A Lie group $G$, named after the Norwegian mathematician Sophus Lie, is a group whose elements depend smoothly on some number of parameters, and on which the group operations \begin{align} G \times G &\longrightarrow G \\ (X,Y) &\longmapsto X^{-1}Y \end{align} are smooth. Most of the Lie groups considered here are matrix groups, whose elements are $n\times n$ matrices over some division algebra, and whose group operation is matrix multiplication. The simplest example is $\SO(2)$, the rotation … text/html 2014-09-12T16:17:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/lorentz?rev=1410563820 There is another kind of ``rotation'', namely the Lorentz transformations of special relativity that relate inertial reference frames. Geometrically, such transformations preserve a generalized distance, the (squared) ``interval'' between spacetime events. text/html 2012-07-05T17:50:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/lorentzn?rev=1341535800 We normally think of a vector in Minkowski spacetime in the form \begin{equation} \xx = \begin{pmatrix}t\cr x\cr y\cr z\cr\end{pmatrix} \end{equation} But there is another natural way to package these four degrees of freedom. Consider the complex matrix \begin{equation} \XX = \begin{pmatrix}t+z& x-iy\cr x+iy& t-z\cr\end{pmatrix} \label{spacetime} \end{equation} This matrix is Hermitian, that is \begin{equation} \XX^\dagger = \XX \end{equation} where the Hermitian conjugate of a matrix, denoted b… text/html 2014-09-07T16:02:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/mobius?rev=1410130920 Much of the material in this section is adapted from~[arXiv:math-ph/9905024] . The unit sphere $\SS^2\subset\RR^3$ is related to the Riemann sphere (the complex plane with a point at infinity added) via stereographic projection from the north pole, which takes the point $(x,y,z)$, with $x^2+y^2+z^2=1$, to the point \begin{equation} w = {x+iy \over 1-z} = {1+z \over x-iy} \label{SterDef} \end{equation} Under this transformation, the north pole is mapped to the point at infinity. text/html 2014-08-15T09:57:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/oalg?rev=1408121820 We define the octonionic conjugate $\bar{x}$ of an octonion $x$ as the (real) linear map which reverses the sign of each imaginary unit. Thus, \begin{equation} \bar{x} = x_1 - x_2 i - x_3 j - x_4 k - x_5 k\ell - x_6 j\ell - x_7 i\ell - x_8 \ell \end{equation} if $x=x_1+x_2i+x_3j+x_4k+x_5k\ell+x_6j\ell+x_7i\ell+x_8\ell$ as above. Direct computation shows that \begin{equation} \bar{xy} = \bar{y}\>\bar{x} \end{equation} The norm of an octonion $|x|$ is defined by \begin{equation} |x|^2 = x\… text/html 2012-07-01T16:19:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/octonionsn?rev=1341184740 What happens if we add another, independent, square root of $-1$? Call it $\ell$. Here we go again. Cayley and Graves independently discovered the octonions, denoted $\OO$, not long after Hamilton discovered the quaternions. In analogy to the previous construction of $\CC$ and $\HH$, an octonion $x$ can be thought of as a pair of quaternions, $(x_\Hone,x_\Htwo)$, so that \begin{equation} \OO = \HH \oplus \HH \, \ell \end{equation} Since we are running out of letters, we will denote $i$ times … text/html 2014-09-11T22:08:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/ogeom?rev=1410498480 As with the quaternions, the units $i$, $j$, $k$, $k\ell$, $j\ell$, $i\ell$, and $\ell$ are by no means the only square roots of $-1$. Rather, any imaginary octonion squares to a negative number, so it is only necessary to choose its norm to be 1 in order to get a square root of $-1$. The imaginary octonions of norm 1 form a 6-sphere in the 7-dimensional space of imaginary octonions. text/html 2014-05-04T20:22:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/ohist?rev=1399260120 Figure 1:The author at the Brougham Bridge in Dublin. Remember the Brougham Bridge, where Hamilton carved the quaternionic multiplication table in 1843? Well, on 21 July 2004, as the author walked along that very same canal in Dublin... (See Figure~1.) text/html 2014-09-21T16:22:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/oint?rev=1411341720 What are the octonionic integers? Based on our discussion of the quaternionic case in (ss)~12.5, we require the components of any such octonion to be either an integer or a half-integer, and the norm of any such octonion to be an integer. How many elements of this form are there? text/html 2014-05-04T20:28:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/ointro?rev=1399260480 What happens if we add another, independent, square root of $-1$? Call it $\ell$. Here we go again. In analogy to the previous construction of $\CC$ and $\HH$, an octonion $x$ can be thought of as a pair of quaternions, $(x_\Hone,x_\Htwo)$, so that \begin{equation} \OO = \HH \oplus \HH \, \ell \end{equation} Since we are running out of letters, we will denote $i$ times $\ell$ simply as $i\ell$, and similarly with $j$ and $k$. But what about the remaining products? text/html 2014-09-21T16:18:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/op1?rev=1411341480 There is another way to view $vv^\dagger$, namely as an element in projective space. Consider a pair of real numbers $(b,c)$, and identify points on the same line through the origin. This can be though of as introducing an equivalence relation of the form \begin{equation} (b,c) \sim (b\chi,c\chi) \label{equiv2} \end{equation} where $0\ne\chi\in\RR$. The resulting space can be identified with the (unit) circle of all possible directions in $\RR^2$, with antipodal points identified. This is th… text/html 2014-09-21T16:20:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/op2?rev=1411341600 In (ss)~12.2, we considered projective lines, which are equivalence classes of points in $\KK^2$. Those equivalence classes, in turn, correspond to the (normalized) ``squares'' ($vv^\dagger$) of the ``points'' ($v$). Here, we extend the discussion to projective planes, which are described in $\KK^3$. text/html 2014-03-23T10:15:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/osplit?rev=1395594900 If we repeat this process one more time, we obtain the split octonions, denoted $\OO'$, which satisfies \begin{equation} \OO' = \HH \oplus \HH L \end{equation} where we now use $I$, $J$, $K$ for the imaginary units in $\HH$. Thus, $\OO'$ consists of linear combinations of $\{1,I,J,K,KL,JL,IL,L\}$, and it again remains to work out the full multiplication table; the result is shown in Table~1. The split octonions are not associative, but they are alternative. text/html 2014-05-29T15:11:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/osub?rev=1401401460 Unlike the (ordinary) octonions, the split octonions have subalgebras that are not themselves composition algebras. The composition property itself doesn't fail; if it holds in the full algebra, it holds in all subalgebras. Rather, it is the requirement that the norm be nondegenerate that fails; these subalgebras are all null (to various degrees). text/html 2012-07-04T12:20:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/particlesn?rev=1341429600 The description in the preceding chapters of 10-dimensional Minkowski space in terms of Hermitian octonionic matrices is a direct generalization of the usual description of ordinary (4-dimensional) Minkowski space in terms of complex Hermitian matrices. If we fix a complex subalgebra $\CC\subset\OO$, then we single out a 4-dimensional Minkowski subspace of 10-dimensional Minkowski space. The projection of a 10-dimensional null vector onto this subspace is a causal 4-dimensional vector, which i… text/html 2014-09-04T22:38:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/preface?rev=1409895480 This is a book about the octonions, a bigger and better version of the complex numbers, albeit with some subtle properties. Bigger, because there are more square roots of $-1$. Better, because an octonionic formalism provides natural explanations for several intriguing results in both mathematics and physics. Subtle, because the rules are more complicated; order matters. text/html 2014-09-11T21:02:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/qalg?rev=1410494520 The quaternionic multiplication table is almost, but not quite, the vector cross product. The only difference is that imaginary quaternions square to a negative number, whereas the cross product of a vector with itself is zero. This is not a coincidence. Making the obvious identification of vectors $\vv$, $\ww$ with imaginary quaternions $v$, $w$, namely \begin{align} \vv = v_x \ii + v_y \jj + v_z \kk \longleftrightarrow v = v_x i + v_y j + v_x k \end{align} (and similarly for $\ww$), then t… text/html 2014-09-03T20:30:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/qdirac?rev=1409801400 It's time to put everything together. Insert~(9) and~(10) of (ss)12 into~(11) of (ss)12. Take advantage of the block structure by writing the Dirac (4-component) spinor $\Psi$ in terms of two Penrose/Weyl (2-component) spinors $\theta$ and $\eta$ as \begin{equation} \Psi = \begin{pmatrix}\theta\cr \noalign{\smallskip} \eta\end{pmatrix} \end{equation} This yields a particularly nice form of the Dirac equation, namely \begin{equation} \begin{pmatrix} p^0\II-p^a\SIGMA_a& -m\II\cr \noalign{\sma… text/html 2012-07-05T16:22:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/qdiracn?rev=1341530520 It's time to put everything together. Insert~(9) and~(10) of (ss)12 into~(11) of (ss)12. Take advantage of the block structure by writing the Dirac (4-component) spinor $\Psi$ in terms of two Penrose/Weyl (2-component) spinors $\theta$ and $\eta$ as \begin{equation} \Psi = \begin{pmatrix}\theta\cr \noalign{\smallskip} \eta\end{pmatrix} \end{equation} This yields a particularly nice form of the Dirac equation, namely \begin{equation} \begin{pmatrix} p^0\II-p^a\SIGMA_a& -m\II\cr \noalign{\sma… text/html 2014-04-06T17:36:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/qgeom?rev=1396830960 It is important to realize that $\pm i$, $\pm j$, and $\pm k$ are not the only square roots of $-1$. Rather, any imaginary quaternion squares to a negative number, so it is only necessary to choose its norm to be 1 in order to get a square root of $-1$. The imaginary quaternions of norm 1 form a sphere; in the above notation, this is the set of points \begin{equation} q_2^2 + q_3^2 + q_4^2 = 1 \end{equation} (and $q_1=0$). Any such unit imaginary quaternion $u$ can be used to construct a comp… text/html 2014-09-11T17:20:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/qhist?rev=1410481200 The quaternions were discovered by Sir William Rowan Hamilton in 1843, after struggling unsuccessfully to construct an algebra in three dimensions. On 16 October 1843, as Hamilton was walking along a canal in Dublin, he realized how to construct an algebra in four dimensions instead. In perhaps the most famous act of mathematical graffiti of all time, Hamilton carved the multiplication table of the quaternions, \begin{equation} i^2 = j^2 = k^2 = ijk = -1 \end{equation} onto the base of the Bro… text/html 2014-09-21T16:21:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/qint?rev=1411341660 What are integers? Over $\RR$, the answer is easy, namely the infinite set \begin{equation} \ZZ = \{ ...,-2,-1,0,1,2,... \} \end{equation} It is straightforward to extend this definition to the complex numbers, resulting in the Gaussian integers \begin{equation} \ZZ[\ell] = \ZZ\oplus\ZZ\ell = \{m+n\ell:m,n\in\ZZ\} \end{equation} where we continue to use $\ell$ rather than $i$ for the complex unit. The Gaussian integers form a lattice in two dimensions. The units of $\ZZ[\ell]$ are the element… text/html 2014-09-08T19:38:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/qintro?rev=1410230280 What happens if we add another, independent, square root of $-1$? Call it $j$. Then the big question is, what is $ij$? Hamilton eventually proposed that $k=ij$ should be yet another square root of $-1$, and that the multiplication table should be cyclic, that is \begin{align} ij = k = -ji \\ jk = i = -kj \\ ki = j = -ik \end{align} We refer to $i$, $j$, and $k$ as imaginary quaternionic units. Notice that these units anticommute! text/html 2014-09-08T19:33:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/qspin?rev=1410229980 In quantum mechanics, the eigenstates of a self-adjoint operator correspond to the physical states with particular values of the corresponding observable physical quantity. A fundamental principle of quantum mechanics further states that after making a measurement the system is ``projected'' into the corresponding eigenstate. It is therefore only eigenstates which are unaffected by the measurement process; they are projected into themselves. In particular, in order to simultaneously make two … text/html 2012-07-05T16:20:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/qspinn?rev=1341530400 In quantum mechanics, the eigenstates of a self-adjoint operator correspond to the physical states with particular values of the corresponding observable physical quantity. A fundamental principle of quantum mechanics further states that after making a measurement the system is ``projected'' into the corresponding eigenstate. It is therefore only eigenstates which are unaffected by the measurement process; they are projected into themselves. In particular, in order to simultaneously make two … text/html 2014-03-23T10:14:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/qsplit?rev=1395594840 We can repeat this process to obtain the split quaternions, denoted $\HH'$, which satisfies \begin{equation} \HH' = \CC \oplus \CC L \end{equation} In order to keep track of these different algebras, we will use $K$ rather than $i$ for the imaginary unit here. Thus, $\HH'$ consists of linear combinations of $1$, $K$, $L$, and $KL$, and it remains to work out the full multiplication table. We have \begin{align} K^2 &= -1 \qquad L^2 = +1 \\ (K)(L) &= KL = -(L)(K) \\ (KL)^2 &= KLKL = -KKLL = +1 \… text/html 2012-07-01T16:17:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/quaternionsn?rev=1341184620 What happens if we add another, independent, square root of $-1$? Call it $j$. Then the big question is, what is $ij$? Hamilton eventually proposed that $k:=ij$ should be yet another square root of $-1$, and that the multiplication table should be cyclic, that is \begin{align} ij = k = -ji \\ jk = i = -kj \\ ki = j = -ik \end{align} We refer to $i$, $j$, and $k$ as imaginary quaternionic units. Notice that these units anticommute! text/html 2014-09-06T11:06:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/references?rev=1410026760 ~~~~~~~~~~ Books on the Octonions: \begin{enumerate}\item John H. Conway and Derek A. Smith, On Quaternions and Octonions, A K Peters, Ltd., Boston, 2003. \item Geoffrey M. Dixon, Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics, Kluwer Academic Publishers, Boston, 1994. text/html 2012-07-01T23:20:34-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/referencesn?rev=1341210034 ~~~~~~~~~~ Books on the Octonions: \begin{enumerate}\item Geoffrey M. Dixon, Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics, Kluwer Academic Publishers, Boston, 1994. \item Feza G\"ursey and Chia-Hsiung Tze, On the Role of Division, Jordan, and Related Algebras in Particle Physics, World Scientific, Singapore, 1996. text/html 2014-04-19T16:54:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/rotations?rev=1397951640 Rotations in two dimensions are easily described; just specify the angle of rotation and the orientation (clockwise or counterclockwise). In three dimensions, it is also necessary to specify the axis of rotation. By convention, a positive angle of rotation about that axis implies rotation in a counterclockwise direction looking back along the axis. Thus, a rotation by $\frac\pi2$ about the North Pole corresponds to spinning the globe $\frac14$ of the way around to the east; spinning to the we… text/html 2014-07-06T08:57:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/rotationsn?rev=1404662220 Rotations in three dimensions are determined by giving the axis of rotation and the angle of rotation about that axis. But how do you specify a rotation in higher dimensions? Specifying the axis of rotation is just a way of specifying the plane in which the rotation takes place. In higher dimensions, one must specify the plane of the rotation; there is more than one ``axis'' perpendicular to any plane. text/html 2014-09-11T22:05:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/sedenions?rev=1410498300 What happens if we continue this process? We define the sedenions by considering pairs of octonions, \begin{equation} s = (p,q) \end{equation} with $p,q\in\OO$. Sedenion multiplication is defined by the Cayley-Dickson process, with $\epsilon=1$, so we have \begin{align} \bar{(a,b)} &= (\bar{a},-b) \\ (a,b)(c,d) &= (ac-\bar{d}b,da+b\bar{c}) \\ (a,b)\bar{(a,b)} &= (|a|^2+|b|^2,0) \end{align} If we define the special element \begin{equation} e = (0,1) \end{equation} then we could also write \begi… text/html 2014-09-07T16:11:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/sl2k?rev=1410131460 The construction in (ss)~9.2 can be extended to the corresponding Lorentz groups. We normally think of a vector in Minkowski spacetime in the form \begin{equation} \xx = \begin{pmatrix}t\cr x\cr y\cr z\cr\end{pmatrix} \end{equation} But there is another natural way to package these four degrees of freedom. Consider the complex matrix \begin{equation} \XX = \begin{pmatrix}t+z& x-iy\cr x+iy& t-z\cr\end{pmatrix} = t\,\II + x\,\SIGMA_x + y\,\SIGMA_y + z\,\SIGMA_z \label{spacetime} \end{equatio… text/html 2014-04-08T16:07:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/so2?rev=1396998420 From the definition in the previous section, we have \begin{equation} \SO(2) = \{ M\in\RR^{2\times 2} : M^T M = I, \det M = 1 \} \end{equation} It is easy to show that the most general element of $\SO(2)$ takes the form \begin{equation} M = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \\ \end{pmatrix} \label{2rot} \end{equation} representing a counterclockwise rotation by $\alpha$ in the $xy$ plane. It is in fact enough to check this on a basis, such as \begin{align} \ii … text/html 2014-06-24T12:07:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/so2k?rev=1403636820 Each of the division algebras corresponds to a $2k$-dimensional vector space, with positive-definite inner product. Multiplication by unit-normed elements preserves the norm, and thus induces either a rotation or a reflection on the vector space. We therefore expect to be able to represent several orthogonal groups in terms of division algebra multiplication. This expectation is correct, but there are several cases, including some surprises due to the lack of commutativity and associativity. text/html 2014-09-04T20:47:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/so3?rev=1409888820 In three dimensions, every rotation is in fact a rotation about some single axis. To specify a rotation in three dimensions, it is therefore necessary to specify this axis, and the angle about this axis through which to rotate. It takes two parameters, such as latitude and longitude, to determine the location of the axis, and a third to give the angle of rotation. These three parameters are collectively known as Euler angles. text/html 2014-09-12T16:28:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/so31?rev=1410564480 The world around us appears to have three independent directions, namely East/West, North/South, and up/down. Special relativity tells us to include time; now we have a fourth ``direction'', namely toward the future or past. It is customary to label these directions with time first, so introduce coordinates $\{t,x,y,z\}$. A spacetime vector is therefore a vector with four components, such as \begin{equation} v = \begin{pmatrix} t \\ x \\ y \\ z \end{pmatrix} \end{equation} which is also calle… text/html 2014-06-24T12:03:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/so3h?rev=1403636580 We saw in (ss)7.2 how to generate $\SO(2)$ through multiplication by a unit-normed complex number. We can try the same thing with the quaternions: Multiplication by a unit-normed quaternion should be a rotation. Since the quaternions are four-dimensional, we expect to get rotations in four dimensions. text/html 2014-09-05T11:38:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/so4?rev=1409942280 In higher dimensions, not every rotation corresponds to a single axis together with an angle of rotation about this axis; the concept of Euler angles does not generalize. The first problem is that, even for rotations in a given plane, it is not possible to associate an ``axis'' with a given rotation, as there are multiple directions orthogonal to the plane of rotation. Furthermore, it is no longer the case that every rotation corresponds to rotation in a single plane. text/html 2014-09-12T17:24:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/so42?rev=1410567840 The Lorentz group preserves ``lengths'', that is, it preserves the squared interval. We can ask instead what transformations preserve ``angles''. In Euclidean space, we can define angles in terms of the dot product, that is, the angle $\theta$ between two vectors $\vv$ and $\ww$ is given by \begin{equation} \cos\theta = \frac{\vv\cdot\ww}{|\vv|\,|\ww|} \end{equation} Any transformation that preserves length will preserve angles. But if we rescale $\vv$ and $\ww$, the angle between them still … text/html 2014-06-24T12:11:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/so4h?rev=1403637060 We return to the question of finding rotations in four dimensions. Conjugation doesn't do the job --- as we saw in (ss)~9.1.1, conjugation induces rotations in three dimensions, and single-sided multiplication isn't sufficient to induce rotations in four dimensions. Therein lies a clue; what about double-sided mlutiplication? text/html 2010-08-19T16:06:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/so64?rev=1282259160 text/html 2010-08-19T16:06:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/so7?rev=1282259160 text/html 2014-06-24T12:16:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/so7o?rev=1403637360 We showed in (ss)~9.1.1 that conjugation by a unit-normed imaginary quaternion $u$ yields a flip of imaginary quaternions about the $u$-axis. Flips can not only be used with quaternions, but also with octonions, since the expression $px\bar{p}$ involves only two directions, and hence lies in a quaternionic subalgebra of~$\OO$; there are no associativity issues here. text/html 2010-08-19T16:06:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/so8?rev=1282259160 text/html 2014-07-06T09:39:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/so8o?rev=1404664740 The transition from $\SO(7)$ to $\SO(8)$ is much the same as from $\SO(3)$ to $\SO(4)$: Flips generate $\SO(7)$, and two-sided multiplication generates the rotations with the real direction. To see the latter property, consider the transformation \begin{equation} x \longmapsto e^{\ell\alpha} x e^{\ell\alpha} \end{equation} We can divide $x\in\OO$ into a piece in the complex subalgebra containing $\ell$, and a piece orthogonal to this subalgebra. That is, since \begin{equation} \OO = \CC \oplus… text/html 2014-09-05T11:51:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/sp2?rev=1409943060 From (ss)~8.1, we have \begin{equation} \Sp(4,\RR) = \{ M\in\RR^{4\times 4} : M\Omega M^T = \Omega \} \end{equation} where \begin{equation} \Omega = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{pmatrix} \end{equation} text/html 2014-07-02T21:43:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/sp2k?rev=1404362580 In (ss)~8.1 we showed that $\Sp(2)\cong\Spin(5)$, the double cover of $\SO(5)$, and in (ss)~9.2 we showed that $\SU(2,\HH)\cong\Spin(5)$. Meanwhile, in (ss)~8.2 we showed that $\Sp(4,\RR)\cong\Spin(3,2)$, the double cover of $\SO(3,2)$. Is there a relationship between symplectic groups and the quaternions? text/html 2014-09-05T12:01:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/sp3?rev=1409943660 It is instructive to compare the description of $\Sp(4,\RR)\cong\Spin(3,2)$ given in (ss)~8.2 with the description of $\SU(2,2)\cong\Spin(4,2)$ given in (ss)~7.5. It is clear that $\Sp(4,\RR)$ is (can be identified with) a subgroup of $\SU(2,2)$; it turns out to be the ``real part'' of $\SU(2,2)$. In other words, $\SU(2,2)$ turns out to deserve the name ``$\Sp(4,\CC)$.'' A similar construction shows that $\Sp(6,\RR)$ can be interpreted as the real part of $\SU(3,3)$, which in turn deserves… text/html 2017-08-25T18:15:28-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/spin?rev=1503710128 We begin by considering angular momentum in quantum mechanics. Angular momentum is defined classically as the cross product of position and momentum, that is \begin{equation} \ll = \rr \times \pp \label{classical} \end{equation} In quantum mechanics, momentum is replaced by a differential operator, that is \begin{equation} \pp \longmapsto -i\hbar\grad \end{equation} Inserting this into~(\ref{classical}) we obtain the components of the angular momentum operator as \begin{align} L_x &= -i\hbar (y… text/html 2017-08-25T18:12:03-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/spinn?rev=1503709923 We begin by considering angular momentum in quantum mechanics. Angular momentum is defined classically as the cross product of position and momentum, that is \begin{equation} \ll = \rr \times \pp \label{classical} \end{equation} In quantum mechanics, momentum is replaced by a differential operator, that is \begin{equation} \pp \longmapsto -i\hbar\grad \end{equation} Inserting this into~($\ref{classical}$) we obtain the components of the angular momentum operator as \begin{align} L_x &= -i\hbar … text/html 2014-09-07T16:12:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/spinors?rev=1410131520 In (ss)9.3, we argued that spacetime vectors are better represented as matrices than as column vectors. What, then, do column vectors represent? In (ss)12.1, we viewed 2-component column vectors as vectors in $\RR^n$ for appropriate $n$; imposing a normalization condition led us to regard these as points in $\SS^{n-1}$. But this is really nothing more than counting the (real) degrees of freedom. text/html 2012-07-01T16:16:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/spinorsn?rev=1341184560 In (ss)5, we argued that spacetime vectors are better represented as matrices than as column vectors. What, then, do column vectors represent? In (ss)6, we viewed 2-component column vectors as vectors in $\RR^n$ for appropriate $n$; imposing a normalization condition led us to regard these as points in $\SS^{n-1}$. But this is really nothing more than counting the (real) degrees of freedom. text/html 2014-09-19T14:22:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/su2?rev=1411161720 The unitary group in two complex dimensions is defined by \begin{equation} \SU(2) = \{ M\in\CC^{2\times 2} : M^\dagger M = I, \det M = 1 \} ~~~~~~~~ \end{equation} A $2\times2$ matrix has 4 complex components, the constraint $M^\dagger M=I$ imposes 4 real conditions, and the determinant restriction adds just one more (since the other conditions already imply that $|\det M|=1$). Thus, we expect there to be three independent ``rotations'' in $\SU(2)$. A set of generators is given by \begin{alig… text/html 2014-05-08T16:47:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/su22?rev=1399592820 Just as with the orthogonal groups, we can have unitary groups that preserve an inner product with arbitrary signature. As before, we let $g$ be the diagonal matrix with signature $(p,q)$, and define \begin{equation} \SU(p,q) = \{ M\in\CC^{(p+q)\times(p+q)} : M^\dagger g M = g, \det M = 1 \} ~~~~~~~~ \end{equation} If $q=0$ (or $p=0$), we recover the ordinary unitary groups. As a nontrivial example, we consider \begin{equation} \SU(2,2) = \{ M\in\CC^{4\times4} : M^\dagger g M = g, \det M = 1 … text/html 2014-09-04T20:49:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/su2k?rev=1409888940 In (ss)~7.2, the complex group $\SU(2)=\SU(2,\CC)$ was represented in terms of $2\times2$ matrices $R_x$, $R_y$, $R_z$, acting on complex matrices of the form \begin{equation} \XX = \begin{pmatrix} 1+z & x-iy \\ x+iy & 1-z \end{pmatrix} \\ \end{equation} thus demonstrating the isomorphism $\SU(2)\cong\Spin(3)$, the double cover of $\SO(3)$. Recall that (setting $\beta=\frac\alpha2$ for convenience) \begin{equation} R_y = \begin{pmatrix} \cos\beta & -\sin\beta \\ \sin\beta & \cos\beta \e… text/html 2014-09-08T19:28:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/su2kk?rev=1410229680 The groups in Table~1 are all orthogonal groups. What is the pattern to their signatures? Let's go ahead and label the rows with the split division algebras, as shown in Table~2. Let $\kappa=|\KK|=1,2,4,8$, and for $\KK'$ keep track separately of the number of positive-normed basis units, $\kappa'_+=1,1,2,4$, and negative-normed basis units, $\kappa'_-=0,1,2,4$, with $\kappa'_++\kappa'_-=|\KK'|$. Then the groups in the table are just $\SO(\kap)$! So we need a representation of $\SO(\kap)… text/html 2014-09-19T14:30:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/su3?rev=1411162200 The unitary group in three complex dimensions is defined by \begin{equation} \SU(3) = \{ M\in\CC^{3\times 3} : M^\dagger M = I, \det M = 1 \} ~~~~~~~~ \end{equation} A $3\times3$ matrix has 9 complex components, the constraint $M^\dagger M=I$ imposes 9 real conditions, and the determinant restriction adds just one more (since the other conditions already imply that $|\det M|=1$). Thus, we expect there to be 8 independent ``rotations'' in $\SU(3)$. A set of generators is given by \begin{align}… text/html 2010-08-19T16:06:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/su33?rev=1282259160 text/html 2014-09-05T11:42:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/sym?rev=1409942520 Orthogonal and unitary transformations preserve symmetric inner products; symplectic transformations preserve an antisymmetric product. Let $\Omega$ be the $2m\times2m$ matrix with block structure \begin{equation} \Omega = \begin{pmatrix} 0 & I_m \\ -I_m & 0 \end{pmatrix} \end{equation} where $I_m$ denotes the $m\times m$ identity matrix. Then the real symmetric groups $\Sp(2m,\RR)$ are defined by \begin{equation} \Sp(2m,\RR) = \{ M\in\RR^{2m\times 2m} : M\Omega M^T = \Omega \} \end{equatio… text/html 2014-04-19T17:14:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/u1?rev=1397952840 We saw in (ss)2.4 that complex multiplication can be interpreted geometrically as a rescaling and a rotation. A pure rotation is therefore obtained by multiplying by a unit complex number. In other words, if $|w|=1$, then $|wz|=|z|$, that is, the length of $z$ is preserved under multiplication by $w$. What do unit-normed elements $w\in\CC$ look like? Since $|r e^{i\theta}|=r$, we have \begin{equation} w = e^{i\theta} \end{equation} for some $\theta$. But these are precisely the complex nu… text/html 2014-09-05T11:40:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/unitary?rev=1409942400 Unitary transformations are analogous to rotations, but over the complex numbers, rather than the reals. We start again in two dimensions. Let $v$ be a complex vector \begin{equation} v = \begin{pmatrix} w\\ z\\ \end{pmatrix} \end{equation} that is, a vector with complex components $w,z\in\CC$; we write this as $v\in\CC^2$. The squared length of $v$ is given by \begin{equation} |v|^2 = |w|^2 + |z|^2 = \bar{w}w + \bar{z}z = v^\dagger v \end{equation} where $v^\dagger$ denotes the Hermitian co… text/html 2014-09-03T20:29:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/weyl?rev=1409801340 Consider the Dirac equation~(2) of (ss)13 with $m=0$. Then equations~(5) and~(6) of (ss)13 decouple, so it is enough to consider just one, say~(5) of (ss)13. This is the Weyl equation \begin{equation} \tilde{\PPP}\psi = 0 \label{Weyl} \end{equation} where we have written $\psi$ instead of $\theta$. In matrix notation, it is straightforward to show that the momentum $p^\mu$ of a solution of the Weyl equation must be null:~(\ref{Weyl}) says that the $2\times2$ Hermitian matrix $\PPP$ has $0$ … text/html 2012-07-01T18:15:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GO/book/content/weyln?rev=1341191700 Consider the Dirac equation~(2) of (ss)13 with $m=0$. Then equations~(5) and~(6) of (ss)13 decouple, so it is enough to consider just one, say~(5) of (ss)13. This is the Weyl equation \begin{equation} \tilde{\PPP}\psi = 0 \label{Weyl} \end{equation} where we have written $\psi$ instead of $\theta$. In matrix notation, it is straightforward to show that the momentum $p^\mu$ of a solution of the Weyl equation must be null:~($\ref{Weyl}$) says that the $2\times2$ Hermitian matrix $\PPP$ has $0…