"The natural language of Symmetry Group Theory"
What is symmetry? Immunity (of aspects of a system) to a possible change The natural language of Symmetry - Group Theory We need a super mathematics in which the operations are as unknown as the quantities they operate on, and a super-mathematician who does not know what he is doing when he performs these operations. Such a super-mathematics is the Theory of Groups. - Sir Arthur Stanley Eddington •GROUP = set of objects (denoted ‘G’) that can be combined by a binary operation (called group multiplication - denoted by ) •ELEMENTS = the objects that form the group (generally denoted by ‘g’) •GENERATORS = Minimal set of elements that can be used to obtain (via group multiplication) all elements of the group RULES FOR GROUPS: •Must be closed under multiplication () - if a,b are in G then ab is also in G •Must contain identity (the ‘do nothing’ element) - call it ‘E’ •Inverse of each element must also be part of group (gg -1 = E) •Multiplication must be associative - a (b c) = (a b) c [not necessarily commutative] Groups Discrete groups Continuous groups Elements can be enumerated Elements are generated by continuously Ex. Dihedral groups (esp. D4) varying one or more parameters. Ex. Lie Groups Ex. Of continuous group (also Lie gp.) Group of all Rotations in 2D space - SO(2) group x 2 cos sin x1 y 2 sin cos y1 cos sin U ( ) cos Det(U) = 1 sin Lie Groups •Lie Group: A group whose elements can be parameterized by a finite number of parameters i.e. continuous group where: 1. If g(ai) g(bi) = g(ci) then - ci is an analytical fn. of ai and bi . 2. The group manifold is differentiable. ( 1 and 2 are actually equivalent) •Group Generators: Because of above conditions, any element can be generated by a Taylor expansion and expressed as : U ( i ) e i 1 A1 2 A2 (where we have generalized for N parameters). Convention: Call A1, A2 ,etc. As the generators (local behavior determined by these). Lie Algebras •Commutation is def as : [A,B] = AB - BA •If generators (A i) are closed under commutation, i.e. Ai , A j f ijk Ak k then they form a Lie Algebra. Generators and physical reality •Hermitian conjugate: A take transpose of matrix and complex conjugate of elements U U = 1 A = A iA •U = e ------ if U is unitary , A must be hermitian Hermitian operators ~ observables with real eigenvalues in QM Symmetry : restated in terms of Group Theory State of a system: | [Dirac notation] Transformation: U| = | [Action on state] Linear Transformation: U ( | + | ) = U| + U| [distributive] Composition: U1U2( | ) = U1(U2 | ) = U1 | Transformation group: If U1 , U2 , ... , Un obey the group rules, they form a group (under composition) -1 Action on operator: U U (symmetry transformation) Again, What is Symmetry? Symmetry is the invariance of a system under the action of a group -1 U U = Why use Symmetry in physics? 1. Conservation Laws (Noether’s Theorem): For every continuous symmetry of the laws of physics, there must exist a conservation law. 2. Dynamics of system: •Hamiltonian ~ total energy operator •Many-body problems: know Hamiltonian, but full system too complex to solve •Low energy modes: All microscopic interactions not significant Collective modes more important •Need effective Hamiltonian Use symmetry principles to constrain general form of effective Hamiltonian + strength parameters ~ usually fitted from experiment High TC Superconductivity •The Cuprates (ex. Lanthanum + Strontium doping) CuO4 lattice •BCS or New mechanism? - d-wave pairing with long-range order. The procedure - 1 1. Find relevant degrees of freedom for system 2. Associate second-quantized operators with them (i.e. Combinations of creation and annihilation operators) 3. If these are closed under commutation, they form a Lie Algebra which is associated with a group ~ symmetry group of system. Subgroup: A subset of the group that satisfies the group requirements among themselves ~ G A . Direct product & subgroup chain: G = A1 A2 A3 ... if (1) elements of different subgroups commute and (2) g = a1 a2 a3 ... (uniquely ) The Procedure - 2 4. Identify the subgroups and subgroup chains ~ these define the dynamical symmetries of the system. (next slide.) 5. Within each subgroup, find products of generators that commute with all generators ~ these are Casimir operators - Ci. [Ci ,A] = 0 CiA = ACi ACiA-1 = Ci Ci’s are invariant under the action of the group !! 6. Since we know that effective Hamiltonian must (to some degree of approximation) also be invariant ~ use casimirs to construct Hamiltonian 7. The most general Hamiltonian is a linear combination of the Casimir invariants of the subgroup chains - = aiCi where the coefficients are strength parameters (experimental fit) Dynamical symmetries and Subgroup Chains Hamiltonian Physical implications •Good experimental agreement with phase diagram. Casimirs and the SU(4) Hamiltonian Casimir operators Model Hamiltonian: Effect of parameter (p) : High TC Superconductivity - SU(4) lie algebra •Physical intuition and experimental clues: Mechanism: D-wave pairing Ground states:Antiferromagnetic insulators •So, relevant operators must create singlet and triplet d-wave pairs •So, we form a (truncated) space ~ ‘collective subspace’ whose basis states are various combinations of such pairs - •We then identify 16 operators that are physically relevant: 16 operators ~ U(4) group [# generators of SU(N) = N2 ] Noether’s Theorem •If is the Hamiltonian for a system and is invariant under the action of a group U U -1 = •Operating on the right with U, U U -1 U = U •i.e. Commutator is zero U - U = 0 = [ U , ] •Quantum Mechanical equation of motion : dU U i U , H dt t U •So, if 0 , then U is a constant of the motion t •Continuous compact groups can be represented by Unitary matrices. iA •U can be expressed as U e (i.e. a Taylor expansion) •Since U is unitary, we can prove that A is Hermitian •So, A corresponds to an observable and U constant A constant •So, eigenvalues of A are constant ‘Quantum numbers’ conserved 2 Uf ( x) f ( x ) f ( x) f ' ( x) f ' ' ( x) 2! d d n n d i i f ( x) e n dx f ( x) e dx f ( x ) e i A f ( x ) n 0 n! dx i 1 A1 1 A1 Uf ( x ) e f ( x) Nature of U and A •For any finite or (compact) infinite group, we can find Unitary matrices that represent the group elements iA •U = e = exp(iA) (A - generator, - parameter) •U = unitary U U = 1 (U - Hermitian conjugate) • exp(-iA) exp(iA) = 1 • exp ( i(A - A) ) = 1 • (A - A) = 0 A = A •So, A is Hermitian and it therefore corresponds to an observable •ex. A can be Px - the generator of 1D translations •ex. A can be Lz - the generator of rotations around one axis Angular momentum theory 1. System is in state with angular momentum ~ | ~ state is invariant under 3D rotations of the system. 2. So, system obeys lie algebra defined by generators of rotation group ~ su(2) algebra ~ SU(2) group [simpler to use] 3. Commutation rule: [Lx,Ly] = i Lz , etc. 4. Maximally commuting subset of generators ~ only one generator 5. Cartan subalgebra ~ Lz Stepping operators ~ L+ = Lx + i Ly L- = Lx - i Ly Casimir operator ~ C = L2 = Lx2 + Ly2 + Lz2 6. C commutes with all group elements ~ CU = UC ~ UCU-1 = C C is invariant under the action of the group