Prolog Text Books: –W.K.Tung, "Group Theory in Physics", World Scientific (85) –J.F.Cornwell, "Group Theory in Physics", Vol.I, AP (85) Website: Homework Grades: Exercises: 50% MidTerm:20% Final:30%

Why Group Theory? Physical problems ~ differential equations. Solutions to DE ~ linear vector / Hilbert space V. Symmetries in physical system → structures in V. →Group theory.

Applications Finite / discrete / countable groups: Molecules, crystals, defects, ions, surface absorbances,… Spectral analysis: Electronic, vibrational, rotational, … Obtain degeneracies & classifications of eigenstates effortlessly. Ditto selection rules & branching ratios. Obtain block diagonalized hamiltonians & symmetrized bases. Lie (continuous) groups: Rotation, Lorentz, Poincare groups: Angular momentum, spin,... Classification of elementary particles: Isotopic multiplets for hadrons / quarks. Construction of gauge fields & unified theories. Electroweak: SU(2)  U(1) Strong: SU(3) Unified: SU(5) Properties of special functions.

Group Theory in Physics 1. Introduction 2. Basic Group Theory 3. Group Representations 4. General Properties of Irreducible Vectors & Operators 5. Representations of the Symmetric Groups D Continuous Groups 7. Rotations in 3-D Space: The Group SO(3) 8. The Group SU(2) & SO(3) 9. Euclidean Groups in 2- & 3-D Spaces 10. The Lorentz & Poincare Groups, & Space: Time Symmetries 11. Space Inversion Invariance 12. Time Reversal Invariance 13. Finite–Dimensional Representations of the Classical Groups W.K.Tung, World Scientific (85)

Appendices I.Notations & Symbols II.Summary of Linear Vector Spaces III.Group Algebra & the Reduction of Regular Representations IV.Supplement to the Theory of the Symmetric Groups V.Clebsch-Gordan Coefficients & Spherical Harmonics VI.Rotational & Lorentz Spinors VII.Unitary Representations of Proper Lorentz Group VIII.Anti-Linear Operators

Supplementary Readings Similar to Tung but more physically oriented: –M.Hamermesh, "Group Theory & its Application to Physical Problems", Addison-Wesley (62) Molecular theories: –F.A.Cotton, "Chemical Applications of Group Theory", 2nd ed., John Wiley (71) Teasers on atomic, molecular & solid-state applications: –M.Tinkham, "Group Theory & Quantum Mechanics", McGraw Hill (64) Solid state theories: –T.Inui, et al, "Group Theory & its Applications in Physics", Springer-Verlag (90)

1. Introduction 1. Particle on a 1-D Lattice 2. Representations of the Discrete Translation Operators 3. Physical Consequences of Translational Symmetry 4. The Representation Functions and Fourier Analysis 5. Symmetry Groups of Physics Aim: To demonstrate relations between physical symmetries, group theory, & special functions ( group representation )

1.1. Particle on a 1-D Lattice b = Lattice constant Translational symmetry (discrete): H invariant under Effect on states: { |Ψ  } & { |Ψ  } physically equivalent    Symm ops are unitary Physical operators are linear  Symmetry operators are linear

 By def:   Transl symm  Similarly:   Eigenstates of T(n) is also eigenstates of H 

1.2. Representations of the Discrete Translation Operators { T(n) } or any set of symm ops satisfies (  dropped ) : Closure Existence of Identity Existence of Inverse Associativity  T d = { T(n) } forms a group Furthermore:  T d is Abelian ( commutative)  T d is unitary More accurately: { T(n) } is the realization of T d = { T(n) } on the single particle Hilbert space.

→  simultanous eigenvectors | ξ   T(n) t n (ξ ) = eigenvalue of T(n) associated with | ξ        Ansatz   For each given ξ, { t n (ξ ) } forms a representation of T d

1.3. Physical Consequences of Translational Symmetry Let the simultaneous eigenstates of H & T(n) be | E k  ( k = ξ / b ) x-representation: Any x is related to a y in the unit cell by x = n b + y–b/2 ≤ y < b/2n  Z  | x  = T(n) | y   x | =  y | T † (n)  –b/2 ≤ y < b/2  The Bloch function is periodic in x with period b Reduced Schrodinger eq:

Brillouin zone

…T(-1)T(0)T(1)… ξ…exp(i ξ)1exp(-i ξ) 0…111  m\T…T(-1)T(0)T(1)… 0…111… 11 … exp(  i 2 π/N) 1exp(i 2 π/N)… 1… 1 exp(  i 2 π/N) … 

1.4. The Representation Functions and Fourier Analysis Harmonic analysis: Functions as series of basic waves (harmonics) E.g., Fourier series/transform, orthogonal polynomial expansion, … Orthonormality Completeness Inverse For compact groups, the basis of each group representation can be chosen to form a complete orthonormal set.

1.5. Symmetry Groups of Physics Let G be the symmetry group of H [ U(g),H ] = 0  g  G  Eigenstates of H are basis vectors of rep of G Reps of G are independent of H Some applications in which group theory is indispensable: –Spectroscopy –Electronic band theory –Strong interaction / TOE Symmetry considerations can always simplify and / or clarify a problem. E.g., expansion in terms of symmetrized functions

Some Symmetries in Physics Continuous Space-Time Symmetries –Translations in space –Translations in time –Rotations in space –Lorentz transformations Discrete Space-Time Symmetries –Space Inversion (Parity) –Time Reversal –Translations & rotations in a lattice (space groups) –Permutations of identical particles –Gauge invariance & charge conservation –Internal symetries of elementary particles