Symmetry Groups in Physics Part 1

Home , Symmetry (geometry), Symmetry (physics)

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 ∈ 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 ∈ G, such that for any g ∈ G

e · g = g · e = g .

• Inverse element. Any g ∈ 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 ∈ 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 ∈ R is −x.

2. Positive real numbers R+ with standard multiplication form a group. Its unit element is 1 and the inverse of a ∈ 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 ψ | U φ i = h ψ | φ i.

The product of two transformations is deﬁned as their composition

(U1 · U2) | ψ i = U1 (U2 | ψ 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 |z| = 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 ∈ [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 ∈ 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 ∈ 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 ∈ GL and f2 ∈ GL, then f1 · f2 (x) = f1(f2(x)), for any x ∈ 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 ∈ 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 ∈ 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) | fg(y) i = h x | y i, for all g ∈ G. Examples: 2 1. R as a linear space of 2-vectors ~x = (x1, x2) with scalar product h ~x | ~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 ∈ C with scalar product h z | 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 ∈ U(1) and z ∈ 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 ∈ 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.

G.J.— Symmetry groups in physics GL(n) group 12/21 SL(n) group SL(n) is a subgroup of GL(n). Its elements are n × n matrices with unit determinant and this set is a subgroup because of

det[M · N] = det[M] · det[N] .

Similarly to GL(n), one can consider the groups SL(n, R) and SL(n, C). Obviously, they are n2 − 1 and 2n2 − 2 parametric groups, respectively. Due to Jacobi’s formula, exp[A] ∈ SL(n), when A is a traceless matrix. a b Exercise 12. If A = , with a2 + bc > 0, check that c −a

a b sinh β exp = cosh β I + A, c −a β √ where β = a2 + bc and I is the unit matrix. Generalize this formula for a2 + bc ≤ 0.

G.J.— Symmetry groups in physics SL(n) group 13/21 U(N) group

This group is a subgroup of GL(n, C) given by the unitary matrices. A matrix U is called unitary if U † U = UU † = I,

† † ∗ where U , deﬁned by Ukl = Ulk, is called Hermitian conjugated to U. Exercise 13. Check that U(n) is n2 parametric group. A matrix h is called Hermitian if h† = h. Exercise 14. Check that if h is Hermitian, then exp[ih] ∈ U(n). Exercise 15. Check that n × n Hermitian matrices form n2 dimension real linear space. Exercise 16. Check that the unitary matrices preserve the scalar product in n C , i.e. h U(z) | U(w) i = h z | w i, where n X ∗ h z | w i = zk wk . k=1

G.J.— Symmetry groups in physics U(N) group 14/21 SU(n) group

This group is given as the intersection of SL(n, C) and U(n), which means that M ∈ SU(n), when M is unitary and det[M] = 1. Exercise 17. Check that SU(n) is n2 − 1 parametric group. It is clear that if h is Hermitian and traceless, then exp[ih] ∈ SU(n). Exercise 18. Check that the number of n × n traceless Hermitian matrices is n2 − 1. For example, for n = 2 one can choose the matrices 0 1 0 −i 1 0 σ = , σ = , σ = , 1 1 0 2 i 0 3 0 −1 which are called Pauli matrices. Exercise 19. Check that Pauli matrices satisfy the identities

σk σl = δklI + iklm σm ,

where I is the unit matrix, δkl and klm are Kroneker and Levi-Chivita symbols, respectively.

G.J.— Symmetry groups in physics SU(N) group 15/21 SU(2) group Exercise 20. Using the result of Exercise 19, check that sin α exp [iα σ ] = cos α I + [iα σ ] , k k α k k

p 2 2 2 with α = α1 + α2 + α3 . a b Let us represent M ∈ SU(2) in the form M = . c d d −b One then gets M −1 = , since det[M] = 1. −c a Becouse of M −1 = M †, one obtains d = a∗, c = −b∗ and M takes the form a b M = , with |a|2 + |b|2 = 1 . −b∗ a∗

3 4 The last equation deﬁnes 3d sphere S embedded in R . Thus, geometrically, SU(2) group is identiﬁed with S3.

G.J.— Symmetry groups in physics SU(2) group 16/21 O(N) group O(n) is the group of n × n real orthogonal matrices.

M matrix is called orthogonal if Mki Mkj = δij. Thus, O(n) is a subgroup of GL(n, R) deﬁned by the matrix relations M T · M = M · M T = I,

T T where M is the transposed matrix to M, given by Mkl = Mlk. n(n−1) Exercise 21. Check that O(n) is 2 parametric group. Exercise 22. Check that rows (and columns) of an orthogonal matrix M are orthonormal. Exercise 23. Check that the orthogonal matrices preserve the scalar n product in R , i.e. h M(x) | M(y) i = h x | y i, where n X h x | y i = xk yk . k=1

G.J.— Symmetry groups in physics O(n) group 17/21 SO(n) group From the definition of O(n) follows that det[M] = ±1, for M ∈ O(n). Since determinant is a continuous function of the matrix elements, O(n) group contains two non-connected parts: one with unit determinant and another with determinant equal to -1. Orthogonal matrices with unit determinant form SO(n) group. The corresponding matrices are called rotations. Exercise 24. Check that SO(2) group is completely described by the matrices M(θ) of Exercise 2. The set of orthogonal matrices with determinant equal to −1 can be written as P · SO(n), where P corresponds to the reflection of the first axe. For example, in O(2) one gets P = diag(−1, 1), and

− cos θ sin θ P · SO(2) = . sin θ cos θ

Exercise 25. Check that exp[A] ∈ SO(n) if AT = −A.

G.J.— Symmetry groups in physics SO(n) group 18/21 SO(3) group Note that SO(1) group contains only one element, which is number 1. SO(2) group elements are given by the matrices M(θ) (see Exercise 2) and they can be written as exponents presented in Exercise 10. 3 Now we consider SO(3) matrices. They correspond to rotations in R . The rotation matrices around x, y and z axes, respectively, are

 1 0 0   cos β 0 sin β  R1(α) =  0 cos α − sin α  ,R2(β) =  0 1 0  , 0 sin α cos α − sin β 0 cos β

 cos γ − sin γ 0  R3(γ) =  sin γ cos γ 0  . 0 0 1 An arbitrary SO(3) group element can be written as a product of these three matrices. Therefore, the rotation angles (α, β, γ) can be treated as group parameters.

G.J.— Symmetry groups in physics SO(3) group 19/21 A more common parameterization of the SO(3) rotations is given by the Euler angels. These angels describe the orientation of a rotated system with respect to the initial system, and this orientation is obtained by the following consecutive rotations

R(ϕ, φ, θ) = R3(ϕ) · R1(φ) · R3(θ) . To get the unique parameterization of the orientation of the rotated system one has to choose the angles within the intervals: 0 ≤ ϕ < 2π, 0 ≤ φ ≤ π, 0 ≤ θ < 2π. Using the form of the matrices R1 and R3, one ﬁnds R(ϕ, φ, θ) = cos ϕ cos θ − sin ϕ cos φ sin θ sin ϕ cos φ cos θ + cos ϕ sin θ sin ϕ sin φ ! − sin ϕ cos θ − cos ϕ cos φ sin θ cos ϕ cos φ cos θ − sin ϕ sin θ cos ϕ sin φ . sin φ sin θ − sin φ cos θ cos φ Exercise 26. Check that the matrix R(ϕ, φ, θ) indeed has this form. Exercise 27. Check the orthonormality of the columns and rows in R(ϕ, φ, θ) and calculate its determinant.

G.J.— Symmetry groups in physics SO(3) group 20/21 Future plan In the next lecture we introduce Lorentz and Poincare groups as symmetry transformations of spacetime in relativistic theories.

We also consider the symplectic group as the group of phase space linear canonical transformations in Hamiltonian mechanics.

We then discuss general structures of Lie groups and Lie algebras.

We introduce sl(2, R) algebra as a simple and remarkable example and analyzed its structure in more detail.

Finally, we construct ﬁnite dimensional UIRs of spin group and present a UIR of sl(2, R) algebra.

G.J.— Symmetry groups in physics Future plan 21/21