Introduction to Group Theory

1 Introduction to Group TheoryWith Applications to Quantum Mechanicsand Solid State PhysicsRoland 2011(Lecture notes version: November 3, 2015)Please, let me know if you find misprints, errors or inaccuracies in these Winkler, NIU, Argonne, and NCTU 2011 2015 General LiteratureIJ. F. Cornwell, Group Theory in Physics(Academic, 1987)general Introduction ; discrete and continuous groupsIW. Ludwig and C. Falter,Symmetries in Physics(Springer, Berlin, 1988).general Introduction ; discrete and continuous Tung, Group Theory in Physics(World Scientific, 1985).general Introduction ; main focus on continuous groupsIL. M. Falicov, Group Theory and Its Physical Applications(University of Chicago Press, Chicago, 1966).small paperback; compact introductionIE. P. Wigner, Group Theory (Academic, 1959).classical textbook by the masterILandau and Lifshitz,Quantum Mechanics, Ch. XII (Pergamon, 1977)brief Introduction into the main aspects of Group Theory in physicsIR.

2 McWeeny,Symmetry(Dover, 2002)elementary, self-contained introductionIand many othersRoland Winkler, NIU, Argonne, and NCTU 2011 2015 Specialized LiteratureIG. L. Bir und G. E. Pikus,Symmetry and Strain-Induced Effects inSemiconductors(Wiley, New York, 1974)thorough discussion of Group Theory and its applications in solid statephysics by two pioneersIC. J. Bradley and A. P. Cracknell,The Mathematical Theory ofSymmetry in Solids(Clarendon, 1972)comprehensive discussion of Group Theory in solid state physicsIG. F. Koster et al.,Properties of the Thirty-Two Point Groups(MIT Press, 1963)small, but very helpful reference book tabulating the properties ofthe 32 crystallographic point groups (character tables, Clebsch-Gordancoefficients, compatibility relations, etc.)IA. R. Edmonds,Angular Momentum in Quantum Mechanics(Princeton University Press, 1960)comprehensive discussion of the ( Group ) Theory of angular momentumin quantum mechanicsIand many othersRoland Winkler, NIU, Argonne, and NCTU 2011 2015 These notes are dedicated toProf.

3 Dr. Ulrich R osslerfrom whom I learned Group Winkler, NIU, Argonne, and NCTU 2011 2015 Introduction and OverviewDefinition: GroupA setG={a,b,c,..}is called a Group , if there exists a Group multiplicationconnecting the elements inGin 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, ,b a 1(inverse element)Corollaries(a)e 1=e(b)a 1a=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 1a 1 Commutative (Abelian) Group (5) a,b G:a b=b a(commutatitivity)Order of a Group = number of Group elementsRoland Winkler, NIU, Argonne, and NCTU 2011 2015 ExamplesIinteger numbersZwith addition(Abelian Group , infinite order)Irational numbersQ\{0}with multiplication(Abelian Group , infinite order)Icomplex numbers{exp(2 i m/n) :m= 1,..,n}with multiplication(Abelian Group , finite order, example ofcyclic Group )Iinvertible (= nonsingular)n nmatrices with matrix multiplication(nonabelian Group , infinite order, later important for representation Theory !)

4 Ipermutations ofnobjects:Pn(nonabelian Group ,n! Group elements)Isymmetry operations (rotations, reflections, etc.) of equilateral triangle P3 permutations of numbered corners of triangle more later!I(continuous) translations inRn: (continuous) translation Group vector addition inRnIsymmetry operations of a sphereonly rotations:SO(3) = special orthogonal Group inR3= real orthogonal 3 3 matricesRoland Winkler, NIU, Argonne, and NCTU 2011 2015 Group Theory in PhysicsGroup Theory is the natural language to describesymmetriesof a physicalsystemIsymmetries correspond to conserved quantitiesIsymmetries allow us to classify quantum mechanical states representation Theory degeneracies / level splittingsIevaluation of matrix elements Wigner-Eckart , selection rules: dipole matrix elements for optical transitionsIHamiltonian Hmust beinvariantunder the symmetriesof a quantum system construct Hvia symmetry Winkler, NIU, Argonne, and NCTU 2011 2015 Group Theory in PhysicsClassical MechanicsILagrange functionL(q, q),ILagrange equationsddt( L qi)= L qii= 1.

5 ,NIIf for onej: L qj= 0 pj L qjis a conserved quantityExamplesIqjlinear coordinate translational invariance linear momentumpj= const. translation groupIqjangular coordinate rotational invariance angular momentumpj= const. rotation groupRoland Winkler, NIU, Argonne, and NCTU 2011 2015 Group Theory in PhysicsQuantum Mechanics(1) Evaluation of matrix elementsIConsider particle in potentialV(x) =V( x) evenItwo possiblities for eigenfunctions (x) e(x) even: e(x) = e( x) o(x) odd: o(x) = o( x)Ioverlapp i(x) j(x) dx= iji,j {e,o}Iexpectation value i|x|i = i(x)x i(x) dx= 0well-known explanationIproduct of two even / two odd functions is evenIproduct of one even and one odd function is oddIintegral over an odd function vanishesRoland Winkler, NIU, Argonne, and NCTU 2011 2015 Group Theory in PhysicsQuantum Mechanics(1) Evaluation of matrix elements (cont d) Group Theory provides systematic generalization of these statementsIrepresentation Theory classification of how functions and operators transformunder symmetry operationsIWigner-Eckart theorem statements on matrix elements if we know how the functionsand operators transform under the symmetries of a systemRoland Winkler, NIU, Argonne, and NCTU 2011 2015 Group Theory in PhysicsQuantum Mechanics(2)

6 Degeneracies of Energy EigenvaluesISchr odinger equation H =E ori~ t = H ILet Owithi~ t O= [ O, H] = 0 Ois conserved quantity eigenvalue equations H =E and O = O can be solved simultaneously eigenvalue Oof Ois good quantum number for Example: H atomI H=~22m( 2 r2+2r r)+ L22mr2 e2r groupSO(3) [ L2, H] = [ Lz, H] = [ L2, Lz] = 0 eigenstates nlm(r):indexl L2,m LzIreally another example for representation theoryIdegeneracy for 0 l n 1: dynamical symmetry (unique for H atom)Roland Winkler, NIU, Argonne, and NCTU 2011 2015 Group Theory in PhysicsQuantum Mechanics(3) Solid State Physicsin particular: crystalline solids, periodic assembly of atoms discrete translation invariance(i) Electrons in periodic potentialV(r)IV(r+R) =V(r) R {lattice vectors} translation operator TR: TRf(r) =f(r+R)[ TR, H] = 0 Bloch theorem k(r) = eik ruk(r) withuk(r+R) =uk(r) wave vectorkis quantum number for the discrete translation invariance,k first Brillouin zoneRoland Winkler, NIU, Argonne, and NCTU 2011 2015 Group Theory in PhysicsQuantum Mechanics(3) Solid State Physics(ii) PhononsIConsider square latticeby 90orotationIfrequencies of modes are equalIdegeneracies for particular propagation directions(iii) Theory of InvariantsIHow can we construct models for the dynamics of electronsor phonons that are compatible with given crystal symmetries?

7 Roland Winkler, NIU, Argonne, and NCTU 2011 2015 Group Theory in PhysicsQuantum Mechanics(4) Nuclear and Particle PhysicsPhysics at small length scales: strong interactionProtonmp= MeVNeutronmn= MeV}rest mass of nucleons almost equal degeneracyISymmetry: isospin Iwith[ I, Hstrong] = 0 ISU(2):proton|1212 ,neutron|12 12 Roland Winkler, NIU, Argonne, and NCTU 2011 2015 Mathematical Excursion: GroupsBasic ConceptsGroup Axioms:see aboveDefinition: SubgroupLetGbe a Group . A subsetU Gthat is itself agroup with the same multiplication asGis called a subgroup Multiplication Table:compilation of all products of Group elements complete information on mathematical structure of a (finite) groupExample:permutation groupP3e=(1 2 31 2 3)a=(1 2 32 3 1)b=(1 2 33 1 2)c=(1 2 31 3 2)d=(1 2 33 2 1)f=(1 2 32 1 3)P3ea bc d feea bc d faa b ef c dbb e ad f ccc d f e a bdd f c b e aff c d a b eI{e},{e,a,b},{e,c},{e,d},{e,f},Gare subgroups ofGRoland Winkler, NIU, Argonne, and NCTU 2011 2015 Conclusions fromGroup Multiplication TableP3ea bc d feea bc d faa b ef c dbb e ad f ccc d f e a bdd f c b e aff c d a b eISymmetry main diagonal Group is AbelianIordernofg G: smallestn>0 withgn=eI{g,g2.

8 ,gn=e}withg Gis Abelian subgroup (a cyclic Group )Iin every row / column every element appears exactly once because:Rearrangement Lemma:for any fixedg G, we haveG={g g:g G}={gg :g G} , the latter sets consist of the elements inGrearranged in :g16=g2 g g16=g g2 g1,g2,g GRoland Winkler, NIU, Argonne, and NCTU 2011 2015 Goal: Classify elements in a Group (1) Conjugate Elements and ClassesILeta G. Thenb Gis called conjugate toaif x Gwithb=x a x ais equivalence relation: a areflexive b a a bsymmetric a cb c} a btransitivea=xcx 1 c=x 1axb=ycy 1= (xy 1) 1a(xy 1)IFor fixeda, the set of all conjugate elementsC={x a x 1:x G}is called a :P3xea bc d feea bc d faea bd f cbea bf c dceb ac f ddeb af d cfeb ad c f classes{e},{a,b},{c,d,f} identityeis its own classx ex 1=e x G Abelian groups: each element is its own classx ax 1=ax x 1=a a,x G Eachb Gbelongs to one and only one class decomposeGinto classes in broad terms: similar elements form a classRoland Winkler, NIU, Argonne, and NCTU 2011 2015 Goal: Classify elements in a Group (2) Subgroups and CosetsILetU Gbe a subgroup ofGandx G.

9 The setxU {x u:u U}(the setUx) is called the left coset (right coset) general, cosets are not U, the cosetxUlacks the identity element:suppose u Uwithxu=e xU x 1=u U x=u 1 UIIfx xU, thenx U=xUany x xUcan be used to define cosetxUIIfUcontainsselements, then each coset also containsselements(due to rearrangement lemma).ITwo left (right) cosets for a subgroupUare either equal or disjoint(due to rearrangement lemma).IThus: decomposeGinto cosetsG=U xU yU ..x,y,.. / UIThus Theorem 1:Lethorder ofGLetsorder ofU G} hs NICorollary: The order of a finite Group is an integer multiple of theorders of its : Ifhprime number {e},Gare the only subgroups Gis isomorphic to cyclic groupRoland Winkler, NIU, Argonne, and NCTU 2011 2015 Goal: Classify elements in a Group (3) Invariant Subgroups and Factor Groupsconnection: classes and cosetsIA subgroupU Gcontaining only complete classes ofGis calledinvariant subgroup (aka normal subgroup).

10 ILetUbe an invariant subgroup ofGandx G xUx 1=U xU=Ux(left coset = right coset)IMultiplication of cosets of an invariant subgroupU G:x,y G:(xU) (yU) =xyU=zUwherez=xywell-defined: (xU) (yU) =x(Uy)U=xyUU=zUU=zUIAn invariant subgroupU Gand the distinct cosetsxUform a Group , called factor groupF=G/U Group multiplication: see above Uis identity element of factor Group x 1 Uis inverse forxUIEvery factor groupF=G/Uis homomorphic toG(see below).Roland Winkler, NIU, Argonne, and NCTU 2011 2015 Example: Permutation GroupP3e a b c d feeabcdfaabefcdbbeadfcccdfeabddfcbeaffcd abeinvariant subgroupU={e,a,b} one cosetcU=dU=fU={c,d,f}factor groupP3/U={U,cU}UcUUUcUcUcUUIWe can think of factor groupsG/Uas coarse-grained versions , factor groupsG/Uare a helpful intermediate step whenworking out the structure of more complicated : invariant subgroups are more useful subgroups than Winkler, NIU, Argonne, and NCTU 2011 2015 Mappings of GroupsILetGandG be two groups.