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Symmetry Groups in Physics Part 2 Symmetry groups in physics Part 2 George Jorjadze Free University of Tbilisi Zielona Gora - 25.01.2017 G.J.| Symmetry groups in physics Lecture 3 1/24 Contents • Lorentz group • Poincare group • Symplectic group • Lie group • Lie algebra • Lie algebra representation • sl(2; R) algebra • UIRs of su(2) algebra G.J.| Symmetry groups in physics Contents 2/24 Lorentz group Now we consider spacetime symmetry groups in relativistic mechanics. The spacetime geometry here is given by Minkowski space, which is a 4d space with coordinates xµ, µ = (0; 1; 2; 3), and the metric tensor gµν = diag(−1; 1; 1; 1). We set c = 1 (see Lecture 1) and x0 is associated with time. Lorentz group describes transformations of spacetime coordinates between inertial systems. Mathematically it is given by linear transformations µ µ µ ν x 7! x~ = Λ n x ; which preserves the metric structure of Minkowski space α β Λ µ gαβ Λ ν = gµν : The matrix form of these equations reads ΛT g L = g : This relation provides 10 independent equations, since g is symmetric. Hence, Lorentz group is 6 parametric. G.J.| Symmetry groups in physics Lorentz group 3/24 From the definition of Lorentz group follows that 0 2 i i det[Λ] = ±1 and (Λ 0) − Λ 0 Λ 0 = 1. Therefore, Lorentz group splits in four non-connected parts 0 0 1: (det[Λ] = 1; Λ 0 ≥ 1); 2: (det[Λ] = 1; Λ 0 ≤ 1), 0 0 3: (det[Λ] = −1; Λ 0 ≥ 1); 4: (det[Λ] = −1; Λ 0 ≤ 1). The matrices of the first part are called proper Lorentz transformations. They form a group which is denoted by SO"(1; 3). Geometrically it is a connected domain like SO(2). However, it is not compact and needs 6 parameters. Other three parts of Lorenz group are not subgroups and they are obtained by the products Λ2 · SO"(1; 3), Λ3 · SO"(1; 3), Λ4 · SO"(1; 3), where the matrices Λ2, Λ3 and Λ4 can be choose as follows Λ2 = diag(−1; −1; 1; 1) ; Λ3 = diag(1; −1; 1; 1) ; Λ4 = diag(−1; 1; 1; 1) : They correspond to reflections of the space-time axes. Thus, it suffices to investigate the group SO"(1; 3). G.J.| Symmetry groups in physics Lorentz group 4/24 Below we show that Λ 2 SO"(1; 3) can be represented as Λ = ΛR · ΛB, where ΛR is a rotation and ΛB is a boost corresponding to the transformation to a moving inertial system. µ For a given Λ ν, the boost matrix ΛB has the following block structure 0 0 ! Λ 0 Λ l 0 0 ΛB = 0 Λ k Λ l : Λ k δkl + 0 1+Λ 0 µ So, it is defined by the first row of Λ ν only. A rotation matrix ΛR has the form 1 0l ΛR = ; 0k Rkl where 0k and 0l denote null 3-vectors written as a column and as a row, respectively, and Rkl is given by k 0 k Λ 0 Λ l Rkl = Λ l − 0 . 1+Λ 0 G.J.| Symmetry groups in physics Lorentz group 5/24 Exercise 28. Check that the given ΛR and ΛB indeed provide ΛR · ΛB = Λ. T Exercise 29. Check that ΛB is indeed a Lorentz matrix, i.e. ΛB g ΛB = g. Exercise 30. Check that the Rkl is an orthogonal matrix. αµ µ µα Hint: use the relations Λ Λαν = δν = Λνα Λ , which follow from the definition of Lorentz group. Exercise 31. Check that the boost ΛB can be represented as the exponent ΛB = exp[ηl Kl] ; where Kl are the boost generator matrices 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 K = B C;K = B C;K = B C; 1 @ 0 0 0 0 A 2 @ 1 0 0 0 A 3 @ 0 0 0 0 A 0 0 0 0 0 0 0 0 1 0 0 0 0 and ηk is related to Λ k. Find this relation and express the velocity of motion between the inertial systems through ηk. G.J.| Symmetry groups in physics Lorentz group 6/24 Poincare group Poincare group is given by the set of pairs (Λ; a), where Λ is a matrix of Lorentz group and a is a 4-vector, a = (a0; a1; a2; a3). Thus, Poincare group is 10 parametric. A group element (Λ; a) creates the transformation of Minkowski space x 7! x0 = Λx + a : Two consecutive transformations then define the multiplication rule (Λ0; a0) · (Λ; a) = (Λ0 · Λ; Λ0a + a0) : Minkowski space has a natural generalization for d-dimensional (d > 1) spacetime with metric tensor gµν = diag(−1; 1; :::; 1). 1;d−1 This space is denoted by R . 1;d−1 Generalization of Poincare group for R is straightforward. 1;d−1 d(d+1) Exercise 32. Check that Poincare group in R is 2 parametric. G.J.| Symmetry groups in physics Poincare group 7/24 Symplectic group Sp(n) This group is given by 2n × 2n real matrices S, which satisfy the condition 0 I ST ΩS = Ω ; with Ω = n n : −In 0n Here, 0n and In are n × n zero and unit matrices, respectively. Exercise 33. Check that Sp(n; R) is n(2n + 1) parametric group. Exercise 34. Check that Sp(1; R) = SL(2; R): Definition. A non-degenerated and antisymmetric matrix is called symplectic. Exercise 35. Prove that the number of rows of a symplectic matrix is even. Thus, Ω is a symplectic matrix. Note that the symplectic group Sp(n) describes linear canonical transformations in 2n-dimensional phase space. G.J.| Symmetry groups in physics Symplectic group 8/24 Lie Groups Lie group is a group that is also a manifold, where the group operations (product and inversion) are given by smooth functions. In other words, a Lie group has two structures (of a group and of a manifold) and these structures are compatible. All considered matrix groups and Poincare group are Lie groups. A Lie group is parameterized by some coordinates. The number of coordinates is called group dimension. A Lie group element g 2 G is given by g = g(u), with u = (u1; : : : ; un). Without loss of generality one can set g(0) = e. The group product defines functions f i(u; v), i = (1; : : : ; n) obtained from g(u) · g(v) = g(f(u; v)) : These functions obviously satisfy the conditions f i(u; 0) = ui = f i(0; u) ; and, therefore, their expansion in powers of u and v reads i i i i j k f (u; v) = u + v + Ajku v + ::: (1) G.J.| Symmetry groups in physics Lie Group 9/24 Let us consider a representation of G with operators Rg = R(u). These operators have a standard Taylor expansion i 1 i j R(u) = I + u Ri + 2 u u Rij + ::: (2) and they obey the identity R(u) · R(v) = R(f(u; v)) : (3) Exercise 36. Using the expansions (1)-(2) and comparing the coefficients of uivj in (3), check that k Ri Rj = AijRk + Rij : From Rij = Rji then follows k Ri Rj − Rj Ri = Cij Rk , k k k with Cij = Aij − Aji. The operators Ri are called Lie group generators. Their commutators create an algebra, which is called Lie algebra. Thus, a Lie group is related to its Lie algebra. The Lie algebra of a Lie group is denoted similarly, only without capital letters. For example, su(n) is the Lie algebra for SU(n) group. G.J.| Symmetry groups in physics Lie group 10/24 Lie algebras A Lie algebra G is a linear space with a multiplication rule, which is bilinear, antisymmetric and satisfies the Jacobi identity. The product of two vectors A 2 G and B 2 G is called the Lie bracket and it is denoted by [ A;B ]. Thus, [ A;B ] 2 G and it satisfies the conditions [λ A + B;C ] = λ [ A;C ] + [ B;C ] ; [ A;B ] = − [ B;A ] ; [[ A;B ] ;C ] + [ [ B;C ] ;A ] + [ [ C;A ] ;B ] = 0 : The numbers λ are real for real algebras and to distinguish the real and complex algebras one uses the letters R and C, respectively. Examples: 3 1. R is a Lie algebra under the vector product ~a ×~b. 2. The set of smooth functions on a phase space is a Lie algebra with respect to Poisson brackets. G.J.| Symmetry groups in physics Lie algebras 11/24 Lie algebra representation A representation of G is a linear map of G to a space of linear operators A 7! O^A , such that [ A;B ] 7! O^A O^B − O^B O^A : Let us consider the operator adA acting on G by adA(B) = [ A;B ] : The Jacobi identity provides that ad[ A;B ] = adA adB − adB adA : Thus, the map A 7! adA defines a representation of G, which is called the adjoint representation. Another example of Lie algebra representation is given by Lie group generators discussed above. G.J.| Symmetry groups in physics Lie algebra representation 12/24 Exercise 37. Let en be a basis of a Lie algebra G. Since [ em ; en ] 2 G, one has l [ em ; en ] = Cmn el ; l with some constants Cmn , which are called the structure constants of G. Check that the structure constants satisfy the relations l l Cmn = −Cnm ; k j k j k j Cmn Ckl + Cnl Ckm + Clm Ckn = 0 : m Exercise 38.
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