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AUSTRALIAN NATIONAL UNIVERSITY FACULTY OF SCIENCE 2007 SYMMETRIES, GROUPS AND ALGEBRAS IN PHYSICS. SERGEY SERGEEV Abstract. Unfinished. A compromise between understanding in physics and formal defini- tions in mathematics. A lot of practical exercises with operators. SYMMETRIES, GROUPS AND ALGEBRAS IN PHYSICS. 1.1 Lecture 1. Group SO(3) and its algebra. Mathematical notion of a symmetry group comes from the invariance of physics laws with respect to a choice of a frame of reference. In particular, the geometry of Euclidean space and therefore a form of Newtonian equations of motion do not depend on a particular choice of a coordinate basis. Let e1, e2, . . . , en be an orthogonal basis of n-dimensional Euclidean space En: the matrix of scalar products is (Pythagoras theorem) (1.1) (ej , ek) = δj,k . Any vector in En has the unique decomposition (1.2) x = x1e1 + x2e2 + · · ·+ xnen = ∑ i xiei . Coefficients xi are the components of vector x with respect to the given basis ei. Let e′1, e ′ 2, . . . , e ′ n be another orthogonal basis of En obtained by a smooth turn of the initial one. According to (1.2), one may express e′j in the terms of ej : (1.3) e′j = ∑ k uj,kek or, in matrix form, e ′ = u · e , u = ||uj,k|| . The orthogonality conditions (1.1) for the new basis give (1.4) (e′j , e ′ k) = ∑ l,m (ujlel, ukmem) = ∑ lm ujlukmδl,m = ∑ m ujmukm = δjk ⇔ uT · u = 1 where 1 = ||δjk|| is the unity matrix. Physicists usually consider the transformations of coordinates of a vector, there exists one-to-one correspondence between the transformation of the basis and transformation of coordinates: (1.5) x = ∑ k xkek ≡ ∑ j x′je ′ j = ∑ k (∑ j x′jujk ) ek ⇒ xk = ∑ j x′jujk or x ′ j = ∑ k ujkxk We see, the transformation of the basis ej and transformation of the coordinates xj coincide – this is a particular feature of Euclidian metric. We consider the basis e′j as a result of a smooth rotation of the basis e. In its turn, we may rotate the basis e′j and obtain the third basis e ′′ j . If basis e ′′ is decomposed with respect to e′, e′′ = u2 ·e′, and e′ is decomposed with respect to e, e′ = u1 ·e, then e′′ = u ·e, u = u2 ·u1. The orthogonality is conserved at each step: (1.6) uT1 · u1 = 1 , uT2 · u2 = 1 ⇒ (u2u1)T · (u2u1) = uT1 uT2 u2u1 = 1 . 1.2 SERGEY SERGEEV This is the structure of (matrix) group: A set G is called the (matrix) group if 1). It is defined the product on G: for any u1, u2 ∈ G their product u1 · u2 ∈ G; 2). There exists the unity element 1 ∈ G: u · 1 = 1 · u = u, for any u ∈ G; 3). Any element has its inverse: for any u ∈ G there is u−1 ∈ G, u · u−1 = u−1 · u = 1. G is the matrix Lie group if it is a smooth variety. The group SO(n) (special orthogonal) is the set of n by n real matrices u satisfying the conditions (1.7) uT · u = 1 and det(u) = 1 . The last condition, det(u) = 1, just excludes the reflections. Turn now to detailed examples. The simplest case is E2. For the turn of the frame (e1, e2) by the angle θ the Pythagoras theorem reads e1 e2 e′1 e′2 cos θ sin θ e′1 = cos θe1 + sin θe2 e′2 = − sin θe1 + cos θe2 In the matrix form (1.8) e′ = u(θ) · e , u(θ) = ( cos θ sin θ − sin θ cos θ ) . You can verity, u(θ)Tu(θ) = 1 and det[u(θ)] = 1. Moreover, the group multiplication gives u(θ)u(θ′) = u(θ + θ′), it means that u(θ) is some exponent. The exponent of a matrix is defined by the series expansion (1.9) eA def = 1+A+ 1 2 A2 + · · ·+ 1 n! An + · · · Using the expansion of exponent and definitions of cosine and sine as power series, one may verify (1.10) u(θ) = eθJ , where J = ( 0 1 −1 0 ) , J2 = −1 . Easy way to get the matrix J is to consider u(θ) with small θ: cos θ = 1 − θ2/2 + . . . and sin θ = θ − θ3/6 + . . . , then u(θ) = 1 + θJ + . . . . SYMMETRIES, GROUPS AND ALGEBRAS IN PHYSICS. 1.3 Turn now to E3. There exists a particular parameterization of the element u (1.3) of orthogonal three-dimensional group in the terms of Euler angles ψ, θ, ϕ corresponding to a decomposition of u into simple rotations (1.8): (1.11) u = 1 0 0 0 cosψ sinψ 0 − sinψ cosψ ︸ ︷︷ ︸ u1(ψ) · cos θ 0 − sin θ 0 1 0 sin θ 0 cos θ ︸ ︷︷ ︸ u2(θ) · cosϕ sinϕ 0 − sinϕ cosϕ 0 0 0 1 ︸ ︷︷ ︸ u3(ϕ) . Almost like (1.10), this matrix may be rewritten in exponential form, (1.12) u1(ψ) = e iψT1 ≃ ψ→0 1+iψT1 , u2(θ) = e iθT2 ≃ θ→0 1+iθT2 , u3(ϕ) = e iϕT3 ≃ ϕ→0 1+iϕT3 , where (1.13) T1 = 0 0 0 0 0 −i 0 i 0 , T2 = 0 0 i 0 0 0 −i 0 0 , T3 = 0 −i 0 i 0 0 0 0 0 . We used the Euler decomposition in order to extract the matrices Tj “generating” the rota- tions around jth axes. Next question is: how to multiply the exponents for u = eiψT1eiθT2eiϕT3 and how to understand the group multiplications from the point of view of exponents ? The answer is an analytical formula – the Baker-Campbell-Hausdorff identity1. Let two matrices have an exponential form, u1 = eA and u2 = eB. Then their product is (1.15) eA · eB = eC(A,B) , C(A,B) ≡ A+B + 12 [A,B] + 1 12 ( [[A,B], B] + [A, [A,B]] ) − 1720 · · · where [A,B] is the commutator: (1.16) [A,B] def = AB −BA . Expression for C(A,B) is a well defined infinite series. The key feature of (1.15) is that all summands after A+B are commutator, commutators of commutators and so on. 1More famous Baker-Campbell-Hausdorff identity is (1.14) eA ·B · e−A = B + ∞∑ n=1 1 n! [A, [A, . . . [A︸ ︷︷ ︸ n times , B] . . . ]] 1.4 SERGEY SERGEEV Let us check then, what matrices produce the commutators of the generators Tj . Easy computations give (1.17) [T1, T2] = iT3 , [T2, T3] = iT1 , [T3, T1] = iT2 , what means that the set of Tj is closed under commutation, therefore (1.11) is (1.18) u = eiψT1+iθT2+iϕT3+ i 2 θϕT1+ i 2 ψϕT2+ i 2 ψθT3+··· = ei(ω1T1+ω2T2+ω3T3) , ωi = ωi(ψ, θ, ϕ) . As well, if group matrices u1 and u2 have such exponential structure, u1 · u2 would have analogous exponential structure since the set of Tj is closed under commutation and due to the BCH identity. The “logarithm” of (1.15) or of (1.18) gives the notion of Lie algebra: A set of operators L is the Lie algebra if for any A and B from L 1). their linear combination aA + bB belongs to L (a, b are numbers) and 2). their commutator [A,B] also belongs to L. The first property here means that the Lie algebra is a linear space, i.e. one may choose a basis Jα, α = 1, . . . , n where n is the dimension of algebra. When the basis is chosen, the algebra is defined by the commutation rules (1.19) [Jα,Jβ] = ∑ γ cαβ;γJγ . The set of numbers cαβ;γ is called the structure constants. Note two basic properties of the commutator, its anti-symmetry and Jacobi identity: (1.20) [A,B] = −[B,A] and [A, [B,C]] + [B, [C,A]] + [C, [A,B]] = 0 . In our example T1, T2, T3 is the basis, and (1.17) may be rewritten in (1.19) form: (1.21) [Ti, Tj ] = ∑ k iϵijkTk , where ϵαβγ is the completely antisymmetric “tensor” whose non-zero elements are (1.22) ϵ123 = ϵ231 = ϵ312 = 1 , ϵ213 = ϵ132 = ϵ321 = −1 . Note, the matrices (1.13) may be written with the help of ϵijk as (1.23) (Ti)jk = −iϵijk . The group element u (1.18) for infinitesimal ω may be written in the matrix form as (1.24) u = 1+ i ∑ i ωiTi + o(ω) ⇔ ujk = δjk + ∑ i ωiϵijk + o(ω) , ωi → 0 . SYMMETRIES, GROUPS AND ALGEBRAS IN PHYSICS. 1.5 Exercises (1) Prove exponential formulas (1.12) for (1.11) and (1.13). Use Taylor expansion defi- nition of the exponents. (2) Show in matrices (1.11,1.13) that eiϕT3T1e −iϕT3 = cosϕ T1 − sinϕ T2 , eiϕT3T2e−iϕT3 = cosϕ T2 + sinϕ T1 . (3) Prove the second Baker-Campbell-Hausdorf identity (1.14) using the Taylor expansion of eϕABe−ϕA in ϕ. SYMMETRIES, GROUPS AND ALGEBRAS IN PHYSICS. 2.1 Lecture 2. Symmetries in classical physics The first lecture introduced a notion of a continuous symmetry group on an example of rotation group SO(3). Such prescription is the universal one. For a set of naively introduced coordinates in classical mechanics or fields in a field theory2 we may consider a group of linear transformations (2.1) Ai → A′i = D∑ j=1 uijAj preserving the form of equations of motion. Conditions for u may be quite different, such transformations are not only rotations in Euclidean spaces. Presumably, matrix u depends on a set of parameters ω, and the tangent space near u = 1, (2.2) u = 1+ i ∑ α ωαJα + o(ω) , Jα ∈ L , gives an algebra L of symmetry group and defines its structure constants (1.19).