Symmetry groups in physics Part 1 George Jorjadze Free University of Tbilisi Zielona Gora - 24.01.2017 G.J.| Symmetry groups in physics Lecture 2 1/21 Contents • Introduction • Group axioms • Examples of groups • Subgroup • Group homomorphism • Group representation • Matrix exponent • GL(n) and SL(n) groups • U(n) and SU(n) groups • SU(2) group • O(n) and SO(n) groups • SO(3) group G.J.| Symmetry groups in physics Contents 2/21 Introduction Group theory is a mathematical tool that investigates symmetries of physical systems and provides effective description of physical phenomena. The methods of group theory allow to handle the consequences of symmetries even without the dynamical content of a physical problem. Physical systems have discrete or/and continuous symmetries. Examples of discrete symmetries: • Space inversion ~x 7! −~x, • Time inversion t 7! −t, • Charge conjugation e 7! −e, • Permutations of identical particles. Examples of continuous symmetries are: • Space-time translations, • Space rotations, • Lorentz transformations, • Rescaling of space-time coordinates. G.J.| Symmetry groups in physics Introduction 3/21 Group axioms A group G is a set of elements g, h, ... with a binary operation g · h 2 G, which is called product and it satisfies the following axioms: • Asscociativity. For any three elements (g; h; k) from G one has g · (h · k) = (g · h) · k : • Unit element. There is an element e 2 G, such that for any g 2 G e · g = g · e = g : • Inverse element. Any g 2 G has an inverse element, denoted by g−1, with the property g · g−1 = g−1 · g = e : Exercise 1. Show that G contains only one unit element; and g 2 G has only one inverse element. In general, the group product is not commutative (see examples). G.J.| Symmetry groups in physics Group axioms 4/21 Examples of groups 1. Real numbers R form a group if the group product is defined as the sum of two numbers. Then, the unit element is 0 and the inverse element of x 2 R is −x. 2. Positive real numbers R+ with standard multiplication form a group. Its unit element is 1 and the inverse of a 2 R+ is 1=a. 3. Non-degenerated n × n matrices form a group under the matrix multiplications. The group unit element here is the unit matrix. 4. Unitary transformations of a Hilbert space H form a group. These transformations preserve the transition amplitudes, i.e. the scalar product in H h U j U φ i = h j φ i: The product of two transformations is defined as their composition (U1 · U2) j i = U1 (U2 j i) : G.J.| Symmetry groups in physics Examples of groups 5/21 5. Galilei transformations relate space-time coordinates of inertial systems in Newtonian mechanics t 7! t~= t + a ; xi 7! x~i = Rij xj + ai + vi t ; (i; j = 1; 2; 3): Here, a is a time-translation parameter, Rij is a rotation matrix, ai is a vector for space translations and vi is a velocity between the inertial systems. Galilei transformations form a group, where the group product is defined as a composition of two transformations, similarly to the previous example. This group is called Galilei group. µ µ µ ν µ 6. Poincare transformations, x 7! x~ = Λ ν x + a ; relate space-time coordinates of inertial systems in relativistic mechanics and they form Poincare group. We will discuss this group later in more detail. Note that the groups in the first two examples are commutative (g · h = h · g for any pair (g; h)), while others are non-commutative. In the last three examples groups are given as symmetry transformations of physical systems. Thus, groups naturally arise in physics. G.J.| Symmetry groups in physics Examples of groups 6/21 Subgroup Definition. A subset H of a group G is called subgroup, if H is a group under the product given in G. Examples: 1. Non-zero complex numbers z 6= 0 form a group under the standard multiplications. Complex numbers with unit norm jzj = 1 form its subgroup, which is denoted by U(1). 2. Space-time translations and space rotations form subgroups of Galilei group. The boosts also form its subgroup. Exercise 2. Check that the matrices cos θ − sin θ M(θ) = ; θ 2 [0; 2π) ; sin θ cos θ form a subgroup of the group of 2 × 2 non-degenerated matrices. Find the matrix M −1(θ). These group is denoted by SO(2). G.J.| Symmetry groups in physics Subgroup 7/21 Group homomorphism Definition. A map F : G 7! G~ from a group G to a group G~ is called homomorphism, if F (g · h) = F (g) · F (h) : An invertible homomorphism is called isomorphism and the isomorphism of a group to itself is called automorphism. Examples: Let us consider the group of real numbers x 2 R (see Example 1, p.5) and introduce two maps a) x 7! ex, b) x 7! eix: Exercise 3. Check that: a) provides isomorphisms of R to R+ , b) is a homomorphism of R into U(1), which is not an isomorphism. −1 Exercise 4. Check that the map Fh(g) = h · g · h , defined by a group element h 2 G, is an automorphism of G. It is called inner automorphism. Exercise 5. Check that the map θ 7! −θ is an automorphism of SO(2) (see Exercise 2), but it is not inner an automorphism. G.J.| Symmetry groups in physics Group homomorphism 8/21 Group representation Let us consider a linear space L and its one-to-one linear maps f : L 7! L. The set of all one-to-one linear maps form a group, denoted by GL. The group product in GL is defined as a composition of two maps: if f1 2 GL and f2 2 GL, then f1 · f2 (x) = f1(f2(x)), for any x 2 L. Definition. A homomorphism of a group G in some GL is a representation of G. Thus, a representation of G is a set of non-degenerated linear operators fg : L 7! L, labeled by the group elements g 2 G, which satisfy the operator relation fg · fh = fg·h . Definition. Representation of G is called irreducible if L has no subspace, apart from trivial, which is invariant under the action of all operators fg. Definition. Let fg and f~g be two representations of a group G in linear spaces L and L~, respectively. These representations are called equivalent, −1 if there exists an invertible map R : L 7! L~ such that f~g = R fg R , for all g 2 G. G.J.| Symmetry groups in physics Group representation 9/21 Definition. Representation of G is called unitary if it preserves the scalar product in L, i.e. if h fg(x) j fg(y) i = h x j y i; for all g 2 G. Examples: 2 1. R as a linear space of 2-vectors ~x = (x1; x2) with scalar product h ~x j ~y i = x1y1 + x2y2. The SO(2) matrices M(θ), given in Exercise 2, 2 define linear maps of R , which can be treated as a representation. Exercise 6. Check that it is an unitary and irreducible representation (UIR). 1 2. C is a 1-dimensional linear complex space of complex numbers z 2 C with scalar product h z j w i = z∗ w. Let us consider a linear representation 1 iθ iθ 1 of U(1) in C given by e z, where e 2 U(1) and z 2 C . Exercise 7. Check that this defines UIR of U(1). 3. The groups U(1) and SO(2) are obviously isomorphic. Hence, a representation of U(1) is a representation of SO(2) and vice versa. Exercise 6 then provides an example of UIR for U(1). 2 1 Exercise 8. Find an invertible map from R to C that makes the UIR's of U(1) given in Exercises 6 and 7 equivalent to each other. G.J.| Symmetry groups in physics Group representation 10/21 Matrix exponent Let us consider a n × n matrix A. Its exponent is defined by the series A2 An exp[A] = I + A + + ··· + + ··· 2! n! Exercise 9. Prove that this series converges for any matrix A. Exercise 10. Calculate the matrix exponents and check the identities 0 a 1 a 0 b cosh b sinh b exp = ; exp = ; 0 0 0 1 b 0 sinh b cosh b 0 −θ cos θ − sin θ exp = : θ 0 sin θ cos θ Exercise 11. Prove the Jacobi's formula for the determinant of matrix exponent det (exp[A]) = eTr[A] : G.J.| Symmetry groups in physics Matrix exponent 11/21 GL(n) group GL(n) group is given by n × n non-degenerated matrices. This group was already introduced on page 5 as an example of groups. If the matrix elements of M 2 GL(n) are only real numbers the group is denoted by GL(n; R), and it is denoted by GL(n; C), if the matrix elements are complex. The condition detM 6= 0 restricts the domain of possible values of the matrix elements of M but not their number. Therefore, GL(n; R) and 2 2 GL(n; C) are n and 2n parametric groups, respectively. For example, GL(1; R) = R+ [ R− and GL(1; C) is a complex plane without the origin. Note that the notation GL(n) is related to the fact that this group is associated with one-to-one general linear transformations of n-dimensional vector space.