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The concept of subgroups in group theory, providing examples and theorems. It also covers the product of groups, the inner product, and matrix representations. based on a research paper from Roland Winkler and collaborators from NIU, Argonne, and NCTU, published between 2011 and 2015.
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Roland Winkler rwinkler@niu.edu
August 2011
(Lecture notes version: November 3, 2015)
Please, let me know if you find misprints, errors or inaccuracies in these notes. Thank you.
I (^) J. F. Cornwell, Group Theory in Physics (Academic, 1987) general introduction; discrete and continuous groups
I (^) W. Ludwig and C. Falter, Symmetries in Physics (Springer, Berlin, 1988). general introduction; discrete and continuous groups
I (^) W.-K. Tung, Group Theory in Physics (World Scientific, 1985). general introduction; main focus on continuous groups
I (^) L. M. Falicov, Group Theory and Its Physical Applications (University of Chicago Press, Chicago, 1966). small paperback; compact introduction
I (^) E. P. Wigner, Group Theory (Academic, 1959). classical textbook by the master
I (^) Landau and Lifshitz, Quantum Mechanics, Ch. XII (Pergamon, 1977)
brief introduction into the main aspects of group theory in physics
I (^) R. McWeeny, Symmetry (Dover, 2002) elementary, self-contained introduction
I (^) and many others
A set G = {a, b, c,.. .} is called a group, if there exists a group multiplication connecting the elements in G in the following way
(1) a, b ∈ G : c = a b ∈ G (closure)
(2) a, b, c ∈ G : (ab)c = a(bc) (associativity)
(3) ∃ e ∈ G : a e = a ∀a ∈ G (identity / neutral element)
(4) ∀a ∈ G ∃ b ∈ G : a b = e, i.e., b ≡ a−^1 (inverse element)
(a) e−^1 = e
(b) a−^1 a = a a−^1 = e ∀a ∈ G (left inverse = right inverse)
(c) e a = a e = a ∀a ∈ G (left neutral = right neutral)
(d) ∀a, b ∈ G : c = a b ⇔ c−^1 = b−^1 a−^1
(5) ∀a, b ∈ G : a b = b a (commutatitivity)
Group theory is the natural language to describe symmetries of a physical system
I (^) symmetries correspond to conserved quantities
I (^) symmetries allow us to classify quantum mechanical states
I (^) evaluation of matrix elements ⇒ Wigner-Eckart theorem e.g., selection rules: dipole matrix elements for optical transitions
I (^) Hamiltonian ˆH must be invariant under the symmetries of a quantum system ⇒ construct ˆH via symmetry arguments
I (^)...
I (^) Lagrange function L(q, q˙),
I (^) Lagrange equations d dt
∂ q˙i
∂qi
i = 1,... , N
I (^) If for one j : ∂L ∂qj
= 0 ⇒ pj ≡
∂ q˙j
is a conserved quantity
I (^) qj linear coordinate
I (^) qj angular coordinate
Group theory provides systematic generalization of these statements
I (^) representation theory ≡ classification of how functions and operators transform under symmetry operations
I (^) Wigner-Eckart theorem ≡ statements on matrix elements if we know how the functions and operators transform under the symmetries of a system
I (^) Schr¨odinger equation Hˆψ = E ψ or iℏ∂t ψ = Hˆψ
I (^) Let ˆO with iℏ∂t Oˆ = [ Oˆ, Hˆ] = 0 ⇒ Oˆ is conserved quantity
⇒ eigenvalue equations Hˆψ = E ψ and ˆOψ = λ (^) Oˆ ψ can be solved simultaneously
⇒ eigenvalue λ (^) Oˆ of ˆO is good quantum number for ψ
2 2 m
∂r 2
r
∂r
2 mr 2
e^2 r
⇒ group SO(3)
⇒ [Lˆ^2 , Hˆ] = [Lˆz , Hˆ] = [Lˆ^2 , Lˆz ] = 0 ⇒ eigenstates ψnlm(r): index l ↔ Lˆ^2 , m ↔ ˆLz I (^) really another example for representation theory I (^) degeneracy for 0 ≤ l ≤ n − 1: dynamical symmetry (unique for H atom)
(ii) Phonons
I (^) Consider square lattice
by 90o
rotation
I (^) frequencies of modes are equal I (^) degeneracies for particular propagation directions
(iii) Theory of Invariants I (^) How can we construct models for the dynamics of electrons or phonons that are compatible with given crystal symmetries?
Physics at small length scales: strong interaction
Proton mp = 938.28 MeV
Neutron mn = 939.57 MeV
rest mass of nucleons almost equal ∼ degeneracy
I (^) Symmetry: isospin ˆI with [ˆI , Hˆstrong] = 0
I (^) SU(2): proton | 1212 〉, neutron | 12 − 12 〉
P 3 e a b c d f e e a b c d f a a b e f c d b b e a d f c c c d f e a b d d f c b e a f f c d a b e
I (^) Symmetry w.r.t. main diagonal ⇒ group is Abelian
I (^) order n of g ∈ G: smallest n > 0 with g n^ = e
I (^) {g , g 2 ,... , g n^ = e} with g ∈ G is Abelian subgroup (a cyclic group)
I (^) in every row / column every element appears exactly once because:
Rearrangement Lemma: for any fixed g ′^ ∈ G, we have G = {g ′g : g ∈ G} = {gg ′^ : g ∈ G} i.e., the latter sets consist of the elements in G rearranged in order. proof: g 1 6 = g 2 ⇔ g ′g 1 6 = g ′g 2 ∀g 1 , g 2 , g ′^ ∈ G
(1) Conjugate Elements and Classes
I (^) Let a ∈ G. Then b ∈ G is called conjugate to a if ∃ x ∈ G with b = x ax−^1.
Conjugation b ∼ a is equivalence relation:
} ⇒ a ∼ b transitive a^ =^ xcx
− (^1) ⇒ c = x− (^1) ax b = ycy −^1 = (xy −^1 )−^1 a(xy −^1 )
I (^) For fixed a, the set of all conjugate elements C = {x ax−^1 : x ∈ G} is called a class.
Example: P 3 x e a b c d f e e a b c d f a e a b d f c b e a b f c d c e b a c f d d e b a f d c f e b a d c f ⇒ classes {e}, {a, b}, {c, d, f }
(3) Invariant Subgroups and Factor Groups
connection: classes and cosets I (^) A subgroup U ⊂ G containing only complete classes of G is called invariant subgroup (aka normal subgroup). I (^) Let U be an invariant subgroup of G and x ∈ G ⇔ x Ux−^1 = U ⇔ x U = Ux (left coset = right coset) I (^) Multiplication of cosets of an invariant subgroup U ⊂ G: x, y ∈ G : (x U) (y U) = xy U = z U where z = xy
well-defined: (x U) (y U) = x (U y ) U = xy U U = z U U = z U I (^) An invariant subgroup U ⊂ G and the distinct cosets x U form a group, called factor group F = G/U
e a b c d f e e a b c d f a a b e f c d b b e a d f c c c d f e a b d d f c b e a f f c d a b e
invariant subgroup U = {e, a, b}
⇒ one coset cU = dU = f U = {c, d, f }
factor group P 3 /U = {U, cU} U cU U U cU cU cU U
I (^) We can think of factor groups G/U as coarse-grained versions of G.
I (^) Often, factor groups G/U are a helpful intermediate step when working out the structure of more complicated groups G.
I (^) Thus: invariant subgroups are “more useful” subgroups than other subgroups.
