17. Group Theory 1.Introduction to Group Theory 2.Representation of Groups 3.Symmetry & Physics 4.Discrete Groups 5.Direct Products 6.Symmetric Groups 7.Continuous Groups 8.Lorentz Group 9.Lorentz Covariance of Maxwell’s Equations 10.Space Groups

1.Introduction to Group Theory Symmetry : 1.Spatial symmetry of crystals ~ X-ray diffraction patterns. 2.Spatial symmetry of molecules ~ Selection rules in vibrational spectra. 3.Symmetry of periodic systems ~ e-properties: energy bands, conductivity, … Invariance under transformations : 1.Linear displacement ~Conservation of (linear) momentum. 2.Rotation ~Conservation of angular momentum. 3.Between (inertial) frames ~General (special) relativity. Theories of elementary particles begin with symmetries & conservation laws. Group theory was invented to handle symmetries & invariance.

3.Identity 1.Closure 2.Associativity Definition of a Group A group { G,  } is a set G with a multiplication  such that  a, b, c  G, 4.Inverse Group { G,  } is usually called simply group G and a  b, ab. Two easily proved theorems : 1.Every a  1 is unique. 2. Rearrangement theorem Refs:W.K.Tung, “Group Theory in Physics” (85) M.Tinkham, “Group Theory & QM” (64)

Abelian group :  is commutative, i.e., More Definitions Discrete group :  1-1 map between set G & a subset of the natural number. ( label of elements of G is discrete ) Finite group : Group with a finite number n of elements. n = order of the group. Continuous group with n -parameter :  1-1 map between set G & subset of R n. Cyclic group C n of order n : C n is abelian Group {G,  } is homomorphic to group { H,  } :  a map f : G  H that preserves multiplications, i.e.,  If f is 1-1 onto ( f  1 exists ), then {G,  } and { H,  } are isomorphic. Subgroup of group {G,  } : Subset of G that is closed under .

Example D 3 Symmetry of an Equilateral Triangle Dihedral group Subgroups : gigi gjgj Table of g i g j for D 3 Mathematica

Example Rotation of a Circular Disk Rotation in x-y plane by angle  :  1-D continuous abelian group. 

Example An Abstarct Group Vierergruppe (4-group) : An abstract group is defined by its multiplication table alone.

Example Isomorphism & Homomorphism: C 4 C 4 = Group of symmetry operations of a square that can’t be flipped. C 4 & G are isomorphic. Subgroup: abelian

2.Representation of Groups A representation of a group is a set of linear transformations on a vector space that obey the same multiplication table as the group. Matrix representation : Representation in which the linear transformations tak the form of invertible matrices ( done by choosing a particular basis for the vector space ). Unitary representation : Representation by unitary matrices. Every matrix representation is isomorphic to a unitary reprsentation.

Example A Unitary Representation Unitary representations for :

W2W2 Let U(G) be a representation of G, then is also a representation. More Definitions & Properties A representation U(G) is faithful if U(G) is isomorphic to G. Every group has a trivial representation with W(G) & U(G) are equivalent representations : A representation U(G) is reducible if every U(g) is equivalent to the same block diagonal form, i.e., for some A representation U(G) is irreducible if it is not reducible. We then write : All irreducible representations (IRs) of an abelian group are 1-D. Commuting matrices can be simultaneously digonalized  W1W1 

Example A Reducible Representation A reducible representation for : Using&, we get the equivalent block diagonal form

Example Representations of a Continuous Group Symmetry of a circular disk : Let &  G is abelian  R is reducible. Independent IRs : Only U 1 & U  1 are faithful.

i.e., is also an eigenfunction with eigenvalue E. i.e., is the tranformed hamiltonian & is the transformed wave function 3.Symmetry & Physics Let R be a tranformation operator such as rotation or translation.   If H is invariant under R :   possibility of degeneracy. Actual degeneracy depends on the symmetry group of H & can be calculated, without solving the Schrodinger eq., by means of the representation theory. 

i.e., is a representation of G on the space spanned by .  Starting with any function , we can generate a set  Or, in matrix form : Next, we orthonormalize S using, say, the Gram-Schmidt scheme, to get  = basis that spans an d –D space.

U is in general reducible, i.e., where m  = number of blocks equivalent to the same IR U (  ). Starting with any function , we can generate a basis for a d-D representation for G.  For arbitary , we can take one state from each U (  ) block to get a basis to set up a matrix eigen-equation of H to calculate E.  w.r.t. a basis for an IR of G. ( Shur’s lemma )  If  is an eigenfunction of H, then U is an IR.

Example An Even H H is even in x  Let  be the operatorthen multiplication table IR C s is abelian  All IRs are 1-D. For an arbitrary  (x) :    Even Odd   = basis for W Mathematica

where P (  ) = projector onto the space of unitary IR U (  ).  (  ) (g) = Character (trace) of U (  ) (g). n  = dimension of IR. n G = order of G. R(g) = operator corresponding to g. For any f (x),, if not empty, is the ith basis vector for the IR U (  )., if not empty, is a basis vector for the IR U (  ). Generation of IR Basis Using Schur’s lemma, one can show that (Tung, §4.2)

Example QM: Triangular Symmetry 3 atoms at vertices R i of an equilaterial triangle : Starting with atomic s-wave function  (r 1 ) at R 1 : 

4.Discrete Groups For any a  G, the set is called a class of G. Rearrangement theorem  A class can be generated by any one of its members. ( a can be any member of C ). C is usually identified by one of its elements. Classes :

Example Classes of D 3 Mathematica g a Table of g a g  1 for D 3  Classes of D 3 are : Usually denoted as All members of a class have the same character(trace).  Orthogonality relations : Dimensionality theorem :

Normalized full representation table of D 3 : Take each row (column) as vector : They’re all orthonormalized. Sum over column (row) then gives the completeness condition.

Example Orthogonality Relations: D 3 D3D3 Character table of D 3 Mathematica A 1, E : E, E : C 3, C 2 : C 3, C 3 : row orthogonality Completeness

Example Counting IRs multiplication table C4C4 g b Table of g b g  1 for C 4 C4C4 Character table

Example Decomposing a Reducible Representation

Other Discrete Groups

5.Direct Products

6.Symmetric Groups

7.Continuous Groups

8.Lorentz Group

9.Lorentz Covariance of Maxwell’s Equations

10.Space Groups