Group theory. March 7, 2016

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1 Group theory March 7, 2016 Nearly all of the cetral symmetries of moder physics are group symmetries, for simple a reaso. If we imagie a trasformatio of our fields or coordiates, we ca look at liear versios of those trasformatios. Such liear trasformatios may be represeted by matrices, ad therefore (as we shall see) eve fiite trasformatios may be give a matrix represetatio. But matrix multiplicatio has a importat property: associativity. We get a group if we couple this property with three further simple observatios: (1) we expect two trasformatios to combie i such a way as to give aother allowed trasformatio, (2) the idetity may always be regarded as a ull trasformatio, ad (3) ay trasformatio that we ca do we ca also udo. These four properties (associativity, closure, idetity, ad iverses) are the defiig properties of a group. 1 Fiite groups Defie: A group is a pair G {S, } where S is a set ad is a operatio mappig pairs of elemets i S to elemets i S (i.e., : S S S. This implies closure) ad satisfyig the followig coditios: 1. Existece of a idetity: e S such that e a a e a, a S. 2. Existece of iverses: a S, a 1 S such that a a 1 a 1 a e. 3. Associativity: a, b, c S, a (b c) (a b) c a b c We cosider several examples of groups. 1. The simplest group is the familiar boolea oe with two elemets S {0, 1} where the operatio is additio modulo two. The the multiplicatio table is simply The elemet 0 is the idetity, ad each elemet is its ow iverse. This is, i fact, the oly two elemet group, for suppose we pick ay set with two elemets, S {a, b}. The multiplicatio table is of the form a b a b Oe of these must be the idetity; without loss of geerality we choose a e. The a b a a b b b 1

2 Fially, sice b must have a iverse, ad its iverse caot be a, we must fill i the fial spot with the idetity, thereby makig b its ow iverse: a b a a b b b a Comparig to the boolea table, we see that a simple reamig, a 0, b 1 reproduces the boolea group. Such a oe-to-oe mappig betwee groups that preserves the group product is called a isomorphism. 2. Let G {Z, +}, the itegers uder additio. For all itegers a, b, c we have a + b R (closure); 0 + a a + 0 a (idetity); a + ( a) 0 (iverse); a + (b + c) (a + b) + c (associativity). Therefore, G is a group. The itegers also form a group uder additio mod p, where p is ay iteger (Recall that a b mod p if there exists a iteger such that a b + p). 3. Let G {R, +}, the real umbers uder additio. For all real umbers a, b, c we have a + b R (closure); 0 + a a + 0 a (idetity); a + ( a) 0 (iverse); a + (b + c) (a + b) + c (associativity). Therefore, G is a group. Notice that the ratioals, Q, do ot form a group uder additio because they do ot close uder additio: π Exercise: Fid all groups (up to isomorphism) with three elemets. Fid all groups (up to isomorphism) with four elemets. Of course, the itegers form a much icer object tha a group. The form a complete Archimedea field. But for our purposes, they form oe of the easiest examples of yet aother object: a Lie group. 2 Lie groups Defie: A Lie group is a group which is also a maifold. Essetially, this meas that a Lie group is a group i which the elemets ca be labeled by a fiite set of cotiuous labels. Qualitatively, a maifold is a space that is smooth eough that if we look at ay sufficietly small regio, it looks just like a small regio of R ; the dimesio is fixed over the etire maifold. We will ot go ito the details of maifolds here, but istead will look at eough examples to get across the geeral idea. The real umbers form a Lie group because each elemet of R provides its ow label! Sice oly oe label is required, R is a 1-dimesioal Lie group. The way to thik of R as a maifold is to picture the real lie. Some examples: 1. The vector space R uder vector additio is a -dim Lie group, sice each elemet of the group may be labeled by real umbers. 2. Let s move to somethig more iterestig. The set of o-degeerate liear trasformatios of a real, -dimesioal vector space form a Lie group. This oe is importat eough to have its ow ame: GL(; R), or more simply, GL() where the field (usually R or C) is uambiguous. The GL stads for Geeral Liear. The trasformatios may be represeted by matrices with ozero determiat. Sice for ay A GL(; R) we have det A 0, the matrix A is ivertible. The idetity is the idetity matrix, ad it is ot too hard to prove that matrix multiplicatio is always associative. Sice each A ca be writte i terms of 2 real umbers, GL() has dimesio 2. GL() is a example of a Lie group with more tha oe coected compoet. We ca imagie startig with the idetity elemet ad smoothly varyig the parameters that defie the group elemets, thereby sweepig out curves i 2

3 the space of all group elemets. If such cotiuous variatio ca take us to every group elemet, we say the group is coected If there remai elemets that caot be coected to the idetity by such a cotiuous variatio (actually a curve i the group maifold), the the group has more tha oe compoet. GL() is of this form because as we vary the parameters to move from elemet to elemet of the group, the determiat of those elemets also varies smoothly. But sice the determiat of the idetity is 1 ad o elemet ca have determiat zero, we ca ever get to a elemet that has egative determiat. The elemets of GL() with egative determiat are related to those of positive determiat by a discrete trasformatio: if we pick ay elemet of GL() with egative determiat, ad multiply it by each elemet of GL() with positive determiat, we get a ew elemet of egative determiat. This shows that the two compoets of GL() are i 1 to 1 correspodece. I odd dimesios, a suitable 1 to 1 mappig is give by 1, which is called the parity trasformatio. 3. We will be cocered with Lie groups that have liear represetatios. This meas that each group elemet may be writte as a matrix ad the group multiplicatio is correctly give by the usual form of matrix multiplicatio. Sice GL() is the set of all liear, ivertible trasformatios i -dimesios, all Lie groups with liear represetatios must be subgroups of GL(). Liear represetatios may be characterized by the vector space that the trasformatios act o. This vector space is also called a represetatio of the group. We ow look at two pricipled ways of costructig such subgroups. The simplest subgroup of GL() removes the secod compoet to give a coected Lie group. I fact, it is useful to factor out the determiat etirely, because the operatio of multiplyig by a costat commutes with every other trasformatio of the group. I this way, we arrive at a simple group, oe i which each trasformatio has otrivial effect o some other trasformatios. For a geeral matrix A GL() with positive determiat, let A (det A) 1  The det  1. Sice det ( ˆB) det  det ˆB 1 the set of all  closes uder matrix multiplicatio. We also have det  1 1, ad det 1 1, so the set of all  forms a Lie group. This group is called the Special Liear group, SL(). Frequetly, the most useful way to characterize a group is by a set of objects that group trasformatios leave ivariat. I this way, we produce the orthogoal, uitary ad symplectic groups: Theorem: Cosider the subset of GL(; R) that leaves a fixed matrix M ivariat uder a similarity trasformatio: H { A A GL(), AMA t M } The H is also a Lie group. Proof: First, H is closed, sice if the the product AB is also i H because AMA t M BMB t M (AB) M(AB) t (AB) M(B t A t ) A ( BMB t) A t AMA t M The idetity is preset because IMI t M 3

4 ad if A leaves M ivariat the so does A 1. To see this, otice that (A t ) 1 ( A 1) t because the traspose of (A) 1 A I is A t ( (A) 1) t I Sice it is easy to show (exercise!) that iverses are uique, this shows that of A t. Usig this, we start with M AMA t ad multiply o the left by A 1 ad o the right by (A t ) 1 : A 1 AMA t ( A t) 1 A 1 M ( A t) 1 M A 1 M ( A t) 1 M A 1 M ( A 1) t ( (A) 1) t must be the iverse The last lie is the statemet that A 1 leaves M ivariat, ad is therefore i H. Fially, we still have the associative matrix product, so H is a group, cocludig our proof. Now, fix a (odegeerate) matrix M ad cosider the group that leaves M ivariat. Suppose M has o particular symmetry. We may oethelsss separate it ito its symmetric ad atisymmetric parts: The, for ay A i H, AMA t M implies M 1 ( M + M t ) M s + M a ( M M t ) A (M s + M a ) A t (M s + M a ) (1) The traspose of this equatio must also hold, A ( Ms t + Ma t ) A t ( Ms t + Ma t ) (2) A (M s M a ) A t (M s M a ) (3) so addig ad subtractig eqs.(1) ad (3) gives two idepedet costraits o A : AM s A t M s AM a A t M a Sice the symmetric ad atisymmetric parts are idepedetly preserved, they give subgroups H s ad H a of G by demadig preservatio of M s or M a aloe. If M is symmetric, the we ca always choose a basis for the vector space o which the trasformatios act such that M is diagoal; ideed we ca go further, for rescalig the basis we ca make every diagoal elemet ito +1 or 1. Therefore, ay symmetric M may be put i the form 1... M (p,q) ij 1 1 (4)

5 where there are p terms +1 ad q terms 1. We ca use M as a pseudo-metric; i compoets, for ay vector v i, p v, v M ij v i v j ( v i ) 2 p+q ( v i ) 2 i1 Notice that this icludes the O(3, 1) Loretz metric of the previous sectio, as well as the O(3) case of Euclidea 3-space. I geeral, the subgroup of GL() leavig M p,q ivariat is termed O(p, q), the pseudoorthogoal group i p + q dimesios. The sigature of M is s p q, ocassioally simply stated as sigature (p, q). Now suppose M is atisymmetric. This case arises i classical Hamiltoia dyamics, where we have caoically cojugate variables satisfyig fudametal Poisso bracket relatios, ip+1 {q i, q j } xπ {p i, p j } xπ 0 {p i, q j } xπ {q i, p j } xπ δ ij If we defie a sigle set of coordiates icludig both p i ad q i, ξ a (q i, p j ) where if i, j 1, 2,..., the a 1, 2,..., 2, the the fudametal brackets may be writte i terms of a atisymmetric matrix Ω ab as { ξ a, ξ b} Ω ab where ( ) Ω ab 0 δ ij δ ij Ω ba (5) 0 Caoical trasformatios are precisely the coordiate trasformatios that preserve the fudametal brackets. At each poit, caoical trasformatios comprise a group of trasformatios which preserve Ω ab. I geeral, the subgroup of GL() preservig a atisymmetric matrix is called the symplectic group. We have a similar result here as for the (pseudo-) orthogoal groups we ca always choose a basis for the vector space that puts the ivariat matrix Ω ab i the form give i eq.(5). Notice that the form give i eq.(5) is ecessarily eve dimesioal i phase space there are equal umbers of positio ad mometum coordiates. Let M be atisymmetric ad of odd dimesio. The, writig out the determiat ad trasposig each copy of M gives det M ε i1i 2 i 2k+1 ε j1j2 j 2k+1 M i1j 1 M i2j 2... M i2k+1 j 2k+1 ( 1) 2k+1 ε i1i 2 i 2k+1 ε j1j2 j 2k+1 M j1i 1 M j2i 2... M j2k+1 i 2k+1 ( 1) 2k+1 det M ad we have det M 0. M therefore has a zero eigevalue, ad is equivalet to a atisymmetric matrix of the ext lower, eve dimesio. Therefore, the symplectic group always has a eve dimesioal represetatio. The otatio for the symplectic groups is Sp(2). For either the orthogoal or symplectic groups, we ca cosider the uit determiat subgroups. Especially importat are the resultig special orthogoal groups, SO(p, q). We give oe particular example that will be useful to illustrate Lie algebras i the ext sectio. The very simplest case of a orthogoal group is O(2), leavig ( ) M 0 1 ivariat. Equivaletly, O(2) leaves the Euclidea orm x, x M ij x i x j x 2 + y 2 5

6 ivariat. The form of O(2) trasformatios is the familiar set of rotatio matrices, ( ) cos θ si θ A(θ) si θ cos θ ad we see that every group elemet is labeled by a cotiuous parameter θ lyig i the rage θ [0, 2π). The group maifold is the set of all of the group elemets regarded as a geometric object. From the rage of θ we see that there is oe group elemet for every poit o a circle the group maifold of O(2) is the circle. Note the iverse of A(θ) is just A( θ) ad the idetity is A(0). Note that all of the trasformatios of O(2) already have uit determiat, so that SO(2) ad O(2) are isomorphic. ( ) Exercise: Fid SO (1, 1), the group of trasformatios leavig M ivariat. 3 Lie algebras If we wat to work with more complicated Lie groups, workig directly with the trasformatio matrices becomes prohibitively difficult. Istead, most of the iformatio we eed to kow about the group is already preset i the ifiitesimal trasformatios. Ulike group multiplicatio, for which the ivariace coditio AMA 1 is a quadratic system, the combiatio of the ifiitesimal trasformatios is liear. This is why, i the previous sectio, we worked with ifiitesimal Loretz trasformatios. Here we ll start with a simpler case to develop some of the ideas further. Let s begi with the example of O(2). Cosider those trasformatios that are close to the idetity. Sice the idetity is A(0), these will be the trasformatios A(ε) with ε 1. Expadig i a Taylor series, we keep oly terms to first order: ( ) cos ε si ε A(ε) si ε cos ε ( ) 1 + ε ( 1 ε ε 1 ( ) The oly iformatio here besides the idetity is the matrix, but remarkably, this is eough to recover the whole group! For geeral Lie groups, we get oe geerator for each cotiuous parameter labelig the group elemets. The set of all liear combiatios of these geerators is a vector space called the Lie algebra of the group. We will give the full defiig set of properties of a Lie algebra below. Imagie iteratig this ifiitesimal group elemet may times. Applyig A(ε) times rotates the plae by a agle ε : ( ( )) A(ε) (A(ε)) 1 + ε Expadig the power o the right usig the biomial expasio, ( ) ( ) k A(ε) ε k 1 k k ε 0 To make the equality rigorous, we must take the limit as ε 0 ad, holdig the product ε θ fiite. The: ( ) ( ) k A(θ) lim ε k ε 0,ε θ k ( ) k! lim ε k k! ( k)! 6 )

7 ( ( 1) ( k + 1) lim ε k ε 0 k! 1 ( ( ) 1 1 lim ) 1 k 1 ( (ε) k ε 0 k! ( ) k 1 k! θk (( ) ) exp θ where i the last step we defie the expoetial of a matrix to be the power series i the secod to last lie. Quite geerally, sice we kow how to take powers of matrices, we ca defie the expoetial of ay matrix, M, by its power series: 1 exp M k! M k (6) Next, we check that the expoetial ( form ) of A(θ) actually is the origial class of trasformatios. To do this we first examie powers of : ( ) 2 ( ) 1 ( ) 3 ( ) ( ) 3 1 The ( eve terms ) are plus or mius the idetity, while the odd terms are always proportioal to the geerator,. Therefore, we divide the power series ito eve ad odd parts, ad remove the matrices from the sums: ( ) k 1 A(θ) θ k k! ( ) 2m 1 ( ) 2m+1 θ 2m 1 + θ 2m+1 (2m)! (2m + 1)! m0 m0 ( ) ( 1) m ( ) ( 1) m 1 (2m)! θ2m + (2m + 1)! θ2m+1 m0 m0 ( ) 1 cos θ + si θ ( ) cos θ si θ si θ cos θ The geerator has give us the whole group back. To begi to see the power of this techique, let s look at O(3), or the subgroup of SO(3) of elemets with uit determiat. A matrix is a elemet of O(3) if ad oly if A t A 1, so we have det ( A t) det (A) det (1) (det (A)) 2 1 ) k ) k 7

8 so det A ±1. Defiig the parity trasformatio to be 1 P 1 1 (7) ad let B be ay elemet of O (3) with det B 1. The det P B 1, so that A P B is a elemet of SO (3). Coversely, for every elemet, A, of SO(3), B P A is a correspodig elemet of O (3) with determiat 1. Therefore, every elemet of O (3) is of the form A or P A, where A is i SO(3). Because P is a discrete trasformatio ad ot a cotiuous set of trasformatios, O(3) ad SO(3) have the same Lie algebra. The geerators of O(3) (ad SO(3)) may be foud from the property of leavig the Euclidea metric 1 g ij 1 1 ivariat: g ij A i ma j g m Just as i the Loretz case i the previous chapter, this is equivalet to preservig the proper legth of vectors. Thus, the trasformatio y i A i mx m is a rotatio if it preserves Euclidea legth, Substitutig, we get g ij y i y j g ij x i x j g m x m x ( g ij A i mx m) ( A j x ) ( g ij A i ma j ) x m x Sice x m is arbitrary, we ca tur this ito a relatio betwee the trasformatios ad the metric, g m, but we have to be careful with the symmetry sice x m x x x m. It is ot a problem here because both sets of coefficiets are also symmetric, Therefore, we ca strip off the xs ad write g m g m g ij A i ma j g ji A j ma i g ji A i A j m g ij A i A j m g m g ij A i ma j (8) This is the most coveiet form of the defiitio of the group to use i fidig the Lie algebra. For future referece, we ote that the iverse to g ij is writte as g ij ; it is also the idetity matrix. As i the 2-dimesioal case, we look at trasformatios close to the idetity. Let A i j δ i j + ε i j where all compoets of ε i m are small. The ( g m g ij δ i m + ε i ( m) δ j + ε j ) ( g ij δm i + g ij ε i ( m) δ j + ε j ) ( g mj + g ji ε i ( m) δ j + ε j ) (g mj + ε jm ) ( δ j + ε j ) g mj δ j + ε jm δ j + g mj ε j + ε jm ε j g m + ε m + ε m + O(ε 2 ) 8

9 Droppig the secod order term ad cacellig g m o the left ad right, we see that the geerators ε m must be atisymmetric: ε m ε m (9) We are dealig with 3 3 matrices here, but ote the power of idex otatio! There is actually othig i the preceedig calculatio that is specific to 3, ad we could draw all the same coclusios up to this poit for O (p, q). For the 3 3 case, every atisymmetric matrix is of the form A(a, b, c) a 0 a b a 0 c b c b c ad therefore a liear combiatio of the three geerators J J J (10) 0 Notice that ay three idepedet, atisymmetric matrices could serve as the geerators. We begi to see why the Lie algebra is defied as the etire vector space I fact, the Lie algebra has three defiig properties. v v 1 J 1 + v 2 J 2 + v 3 J 3 Defie: A Lie algebra is a fiite dimesioal vector space V together with a biliear, atisymmetric (commutator) product satisfyig 1. For all u, v V, the product [u, v] [v, u] w is i V. 2. All u, v, w V satisfy the Jacobi idetity [u, [v, w]] + [v, [w, u]] + [w, [u, v]] 0 These properties may be expressed i terms of a basis. Let {J a a 1,..., } be a vector basis for V. The we may compute the commutators of the basis, [J a, J b ] w ab where for each a ad each b, w ab is some vector i V. We may expad each w ab i the basis as well, c w ab cab J c c for some costats cab. The c c c ab cba are called the Lie structure costats. The basis the satisfies, c [J a, J b ] c ab J c 9

10 which is sufficiet, usig liearity, to determie the commutators of all elemets of the algebra, [u, v] [ u a J a, v b J b ] u a v b [J a, J b ] u a v b c w c J c w c ab J c Exercise: Show that the commutatio relatios of the three O(3) geerators, J i, give i eq.(10) are give by k [J i, J j ] εij J k (11) where ε tesor, k ij g km ε ijm, ad ε ijm is the 3-dimesioal versio of the totally atisymmetric Levi-Civita ε 123 ε 231 ε ε 132 ε 321 ε with all other compoets vaishig. See our discussio of ivariat tesors i the sectio o special relativity for further properties of the Levi-Civita tesors. I particular, you will eed ε ijk ε im δ j mδ k δ j δ k m. Notice that most of the calculatios above for O(3) actually apply to ay of the pseudo-orthogoal groups O(p, q), ad some to every Lie algebra. We explore this geeral case i the ext Sectio, the prove some geeral properties of Lie algebras ad Lie groups. 4 The special orthogoal groups I the geeral case, the form of the geerators is still give by eq.(9), with g m replaced by M m (p,q) Droppig the (p, q) label, we have ( M m M ij δ i m + ε i ( m) δ j + ε j ) leadig to M m + M i ε i m + M mj ε j ε m M i ε i m ε m M mj ε j of eq.(4). The doubly covariat geerators ε m are still atisymmetric. The oly differece is that the idices are lowered with the (p, q) metric M m istead of g m. Aother differece occurs whe we compute the Lie algebra because i -dimesios we o loger have the coveiet form, ε ijm, for the Levi-Civita tesor. The Levi-Civita tesor i -dimesios has idices, ad does t simplify the Lie algebra expressios. Istead, we choose the followig set of atisymmetric matrices as geerators: [ ε (rs)] m (δ r mδ s δ r δ s m) The (rs) idices tell us which geerator we are talkig about, while the m ad idices are the matrix compoets. To compute the Lie algebra, we eed the mixed form of the geerators, [ε (rs)] m M mk [ ε (rs)] k M mk δ r kδ s M mk δ r δ s k M mr δ s M ms δ r 10

11 We ca ow compute the commutators, [[ ε (uv)], [ε (rs)]] m [ε (uv)] m [ε (rs)] k [ε (rs)] m [ε (uv)] k k k (M mu δk v M mv δk u ) ( M kr δ s M ks δ) r (M mr δk s M ms δk) r ( M ku δ v M kv δ u ) M mu M vr δ s M mu M vs δ r M mv M ur δ s + M mv M us δ r M mr M su δ v + M ms M ru δ v + M mr M sv δ u M ms M rv δ u M vr M mu δ s M vs M mu δ r M ur M mv δ s + M us M mv δ r M su M mr δ v + M ru M ms δ v + M sv M mr δ u M rv M ms δ u Rearragig to collect the terms as geerators, ad otig that each must have the free m ad idices, we get [[ ε (uv)], [ε (rs)]] m M vr (M mu δ s M ms δ) u M vs (M mu δ r M mr δ) u M ur (M mv δ s M ms δ) v + M us (M mv δ r M mr δ) v [ M vr ε (us)] m [ M vs ε (ur)] m [ M ur ε (vs)] m [ + M us ε (vr)] m Fially, we ca drop the matrix idices. It is importat that we ca do this, because it demostrates that the Lie algebra is a relatioship amog the differet geerators that does ot deped o whether the operators are writte as matrices or ot. The result, valid for ay O (p, q)) is [ ε (uv), ε (rs)] M vr ε (us) M vs ε (ur) M ur ε (vs) + M us ε (vr) (12) We will eed this result whe we study the Dirac matrices. Exercies: Show that the O(p, q) Lie algebra i eq.(12) reduces to the O(3) Lie algebra i eq.(11) whe (p, q) (3, 0). (Hit: Multiply eq.(12) by ε uvw ε rst ad use J i 1 2 ε ijkε (jk). Notice that M m is just g m ). 5 The relatioship betwee Lie algebras ad Lie groups The ifiitesimal geerators of ay Lie group form a Lie algebra, ad coversely, the properties of a Lie algebra guaratee that expoetiatig the algebra gives a Lie group. To see this, let s work from the group side. We have group elemets that deped o cotiuous parameters, so we ca expad g (a, b,..., c) ear the idetity i a Taylor series, g(x 1,..., x ) 1 + g x a xa g 2 x a x b xa x b J a x a K abx a x b +... Here the coefficiet matrices J a are the geerators of the group ad give a basis for the Lie algebra. Next we look at the cosequeces of each of the group properties o the ifiitesimal geerators, J a. Closure First, there exists a group product, which must close: g(x a 1)g ( x b ) 2 g(x a 3) (1 + J a x a ) (1 + J a x a ) 1 + J a x a J a x a 1 + J a x a J a x a so that at liear order, J a x a 1 + J a x a 2 J a x a 3 This requires the geerators to combie liearly uder additio ad scalar multiplicatio, so they form the basis for a vector space. 11

12 Idetity Next, the group must have a idetity operator. This just meas that the zero vector lies i the space of geerators, sice g(0,..., 0) J a 0 a. Iverse For iverses, we have g(x a 1)g 1 ( x b ) 2 1 (1 + J a x a ) (1 + J a x a ) J a x a 1 + J a x a 2 1 so that x a 2 x a 1, guarateeig a additive iverse i the space of geerators. These properties together make the set {x a J a } a vector space. Exercise: Show to secod order that the iverse of g 1 + J a x a K abx a x b +... is g 1 1 J b x b (J aj b + J b J a K ab ) x a x b +... Usig closure ad the existece of iverses, we ca derive the commutatio relatios for the Lie algebra. For this, cosider the (closed!) product of group elemets g 1 g 2 g 1 1 g 1 2 g 3 We compute each side of this equality i a Taylor series to secod order. For the idividual group elemets we write g J a x a K abx a x b g 1 1 J b x b (J aj b + J b J a K ab ) x a x b g J b y b K bcy b y c g J b y b (J aj b + J b J a K ab ) y a y b g J a z a (x, y) K abz a (x, y)z b (x, y) For the multiple product, to secod order, g 1 g 2 g1 1 g 1 2 (1 + J a x a + 12 ) K abx a x (1 b + J b y b + 12 ) K bcy b y c ( ( 1 J c x c + J c J d 1 ) ) ( ( 2 K cd x c x d 1 J d y d + J d J e 1 ) ) 2 K de y d y e (1 + J b x b + J b y b + J a J b x a y b + 12 K bcy b y c + 12 ) K abx a x b (1 J d x d J d y d + J d J e y d y e + J c J d x c y d + J c J d x c x d 12 K dey d y e 12 ) K cdx c x d 1 J d x d J d y d + J d J e y d y e + J c J d x c y d + J c J d x c x d 1 2 K dey d y e 1 2 K cdx c x d + J b x b + J b y b J b J d x d x b J b J d x d y b J b J d y d x b J b J d y d y b + J a J b x a y b K bcy b y c K abx a x b 12

13 Collectig terms, g 1 g 2 g J d x d + J b x b + J b y b J d y d +J c J d x c x d J b J d x b x d +J c J d x c y d J b J d y b x d J b J d x b y d + J a J b x a y b +J d J e y d y e J b J d y b y d K bcy b y c 1 2 K dey d y e K abx a x b 1 2 K cdx c x d 1 + J c J d x c y d J b J d y b x d 1 + [J c, J d ] x c y d ad equatig to g J a z a (x, y) +, the idetity cacels ad we are left with [J c, J d ] x c y d J a z a (x, y) Sice x c ad y d are arbitrary, z a must be biliear i them, z a x c y d c a cd ad we have derived the presece of a commutator product for the Lie algebra, [J c, J d ] c a cd J a Associativity Fially, the Lie group is associative: if we have three group elemets, g 1, g 2 ad g 3, the g 1 (g 2 g 3 ) (g 1 g 2 ) g 3 Expadig to first order, this simply implies associativity for the geerators themselves J a (J b J c ) (J a J b ) J c together with a weaker coditio, the Jacobi idetity, for the commutator product. First expad [J a, [J b, J c ]] [J a, J b J c J c J b ] J a (J b J c ) J a (J c J b ) (J b J c ) J a + (J c J b ) J a Now, permutig abc cyclically ad collectig terms gives [J a, [J b, J c ]] + [J b, [J c, J a ]] + [J c, [J a, J b ]] J a (J b J c ) (J a J b ) J c J a (J c J b ) + (J a J c ) J b (J b J c ) J a + J b (J c J a ) 0 + (J c J b ) J a J c (J b J a ) J b (J a J c ) + (J b J a ) J c +J c (J a J b ) (J c J a ) J b From the fial arragemet of the terms, we see that the Jacobi relatio is satisfied idetically as a cosequece of the associativity of the group multiplicatio. Therefore, the defiitio of a Lie algebra is a ecessary cosequece of beig built from the ifiitesimal geerators of a Lie group. Coversely, we may build a Lie group from ay Lie algebra as a limit of ifiitely may ifiitesimal trasformatios. To prove this, start with a ifiitesimal but otherwise arbitrary elemet of the Lie algebra, g (ε, w a ) 1 + εw a J a 13

14 ad take the limit lim (1 + ε 0 εwa J a ) while holdig λ ε equal to 1. The the argumet we used for O (2) still goes through. Usig the biomial expasio, lim (1 + ε 0 εwa J a ) ( k ) 1 k (εw a J a ) k! ( k)!k! εk (w a J a ) k 1 ( 1) ( 2) ( k + 1) k! k (ε) k (w a J a ) k ( 1 k! ) ( 1 2 ) ( 1 k 1 ) (1) k (w a J a ) k Takig the limit, the product of a fiitie umber k of terms, each approachig 1 is 1, so lim (1 + ε 0 εwa J a ) 1 k! (wa J a ) k exp (w a J a ) ad each ifiitesimal elemet of the Lie algebra expoetiates to give a fiite trasformatio, g (w a ) e wa J a It may be show that the properties of the Lie algebra are sufficiet to guaratee that g (w a ) is a elemet of a Lie group. The correspodece betwee Lie groups ad Lie algebras is ot oe to oe, because i geeral several Lie groups may share the same Lie algebra. However, groups with the same Lie algebra are related i a simple way. Our example above of the relatioship betwee O(3) ad SO(3) is typical these two groups are related by a discrete symmetry. Sice discrete symmetries do ot participate i the computatio of ifiitesimal geerators, they do ot chage the Lie algebra. The cetral result is this: For every Lie algebra there is a uique maximal Lie group called the coverig group such that every Lie group sharig the same Lie algebra is the quotiet of the coverig group by a discrete symmetry group. This result suggests that whe examiig a group symmetry of ature, we should always look at the coverig group i order to extract the greatest possible symmetry. Followig this suggestio for Euclidea 3-space ad for Mikowski space leads us directly to the use of spiors. I the ext Sectios, we discuss spiors i three ways. The first two make use of coveiet tricks that work i low dimesios (2, 3 ad 4), ad provide easy ways to hadle rotatios ad Loretz trasformatios. The third treatmet is begis with Dirac s developmet of the Dirac equatio, which leads us ultimately to the itroductio of Clifford algebras. 14