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17. Group Theory

17. Group Theory. Introduction to Group Theory Representation of Groups Symmetry & Physics Discrete Groups Direct Products Symmetric Groups Continuous Groups Lorentz Group Lorentz Covariance of Maxwell’s Equations Space Groups. 1. Introduction to Group Theory. Symmetry :

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17. Group Theory

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  1. 17. Group Theory Introduction to Group Theory Representation of Groups Symmetry & Physics Discrete Groups Direct Products Symmetric Groups Continuous Groups Lorentz Group Lorentz Covariance of Maxwell’s Equations Space Groups

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

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

  4. More Definitions Finite group : Group with a finite number n of elements. n =order of the group. Discrete group :  1-1 map between set G & a subset of the natural number. ( label of elements of G is discrete ) Continuous group with n-parameter:  1-1 map between set G & subset ofRn. Abelian group :  is commutative, i.e., Cyclic groupCnof order n: Cnis 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 .

  5. Example 17.1.1. D3 Symmetry of an Equilateral Triangle Table of gi gj for D3 gj gi Subgroups : Dihedral group Mathematica

  6. Example 17.1.2. Rotation of a Circular Disk Rotation in x-y plane by angle  :   1-D continuous abelian group.

  7. Example 17.1.3. An Abstarct Group An abstract group is defined by its multiplication table alone. Vierergruppe (4-group) :

  8. Example 17.1.4. Isomorphism & Homomorphism: C4 C4 = Group of symmetry operations of a square that can’t be flipped. abelian C4 & G are isomorphic. Subgroup:

  9. 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.

  10. Example 17.2.1. A Unitary Representation Unitary representations for :

  11. More Definitions & Properties A representation U(G) is faithful if U(G) is isomorphic to G. Every group has a trivial representation with Let U(G) be a representation of G, then is also a representation. W(G) & U(G) are equivalent representations : A representation U(G) is reducible if everyU(g) is equivalent to the sameblock diagonal form, i.e., for some We then write : W1 A representation U(G) is irreducible if it is not reducible. W2 Commuting matrices can be simultaneously digonalized  All irreducible representations (IRs) of an abelian group are 1-D. 

  12. Example 17.2.2. A Reducible Representation A reducible representation for : Using & , we get the equivalent block diagonal form

  13. Example 17.2.3. Representations of a Continuous Group Symmetry of a circular disk : G is abelian  R is reducible. Let &  Independent IRs : Only U1 & U1are faithful.

  14. 3. Symmetry & Physics Let R be a tranformation operator such as rotation or translation.   i.e., is the tranformed hamiltonian & is the transformed wave function   If H is invariant under R : i.e., is also an eigenfunction with eigenvalue E.  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 .

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

  16. Starting with any function, we can generate a basis for a d-D representation for G. Uis in general reducible, i.e., where m= number of blocks equivalent to the same IR U (). w.r.t. a basis for an IR of G. ( Shur’s lemma )   If  is an eigenfunction of H, then U is an IR.  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.

  17. multiplication table Example 17.3.1. An Even H His even in x Let  be the operator then IR Cs is abelian  All IRs are 1-D. For an arbitrary  (x) :   Mathematica  Even Odd   = basis for W

  18. Generation of IR Basis Using Schur’s lemma, one can show that (Tung, §4.2) where P( ) = projector onto the space of unitary IR U( ).  ( ) (g) = Character (trace) of U( ) (g) . n = dimension of IR. nG = 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( ).

  19. Example 17.3.2.QM: Triangular Symmetry 3 atoms at vertices Riof an equilaterial triangle : Starting with atomic s-wave function (r1) at R1: 

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

  21. Example 17.4.1. Classes of D3 Table of ga g1for D3 a g  Classes of D3are : Usually denoted as Mathematica All members of a class have the same character(trace).  Orthogonality relations : Dimensionality theorem :

  22. Normalized full representation table of D3 : Take each row (column) as vector : They’re all orthonormalized. Sum over column (row) then gives the completeness condition.

  23. Example 17.4.2. Orthogonality Relations: D3 D3 row orthogonality A1 , E: Character table of D3 Mathematica E , E: Completeness C3 , C2: C3 , C3:

  24. Example 17.4.3. Counting IRs multiplication table C4 Table of gb g1for C4 b g Character table C4

  25. Example 17.4.4. Decomposing a Reducible Representation

  26. Other Discrete Groups

  27. 5. Direct Products

  28. 6. Symmetric Groups

  29. 7. Continuous Groups

  30. 8. Lorentz Group

  31. 9. Lorentz Covariance of Maxwell’s Equations

  32. 10. Space Groups

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