what is symmetry? immunity (of aspects of a system) to a possible change
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What is symmetry? Immunity (of aspects of a system) to a possible change. The natural language of Symmetry - Group Theory. - PowerPoint PPT PresentationTRANSCRIPT
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What is symmetry?
Immunity (of aspects of a system) to a possible change
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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]
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D isc re te g rou p sE lem en ts can b e en u m era tedE x. D ih ed ra l g rou p s (esp . D 4 )
C on tin u ou s g rou p sE lem en ts a re g en era ted b y c on tin u ou s ly
va ryin g on e o r m ore p aram ete rs .E x. L ie G rou p s
G rou p s
Ex. Of continuous group (also Lie gp.)
Group of all Rotations in 2D space - SO(2) group
1
1
cossin
sincos
2
2
y
x
y
x
cossin
sincos)(U Det(U) = 1
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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 :
(where we have generalized for N parameters).
Convention: Call A1, A2 ,etc. As the generators (local behavior determined by these).
2211)( AAii eU
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Lie Algebras
•Commutation is def as : [A,B] = AB - BA
•If generators (A i) are closed under commutation, i.e.
then they form a Lie Algebra.
kk
ijkji AfAA ,
Generators and physical reality
•Hermitian conjugate: A
take transpose of matrix and complex conjugate of elements
•U = eiA ------ if U is unitary , A must be hermitian
U U = 1 A = A
Hermitian operators ~ observables with real eigenvalues in QM
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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)
Action on operator: U U -1 (symmetry transformation)
Again, What is Symmetry?
Symmetry is the invariance of a system under the action of a group
U U -1 =
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Why use Symmetry in physics?
1. Conservation Laws (Noether’s Theorem):
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 significantCollective modes more important
•Need effective Hamiltonian
Use symmetry principles to constrain general form of effective Hamiltonian + strength parameters ~ usually fitted from experiment
For every continuous symmetry of the laws of physics, there must exist a conservation law.
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High TC Superconductivity
•The Cuprates (ex. Lanthanum + Strontium doping)
•BCS or New mechanism? - d-wave pairing with long-range order.
CuO4 lattice
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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 )
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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
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)
Ci’s are invariant under the action of the group !!
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Dynamical symmetries and Subgroup Chains
Hamiltonian
Physical implications
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•Good experimental agreement with phase diagram.
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Casimirs and the SU(4) Hamiltonian
Casimir operators
Model Hamiltonian:
Effect of parameter (p) :
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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 ]
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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 :
•So, if , then U is a constant of the motion
•Continuous compact groups can be represented by Unitary matrices.
•U can be expressed as (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
HUi
t
U
dt
dU,
0
t
U
AieU
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)()()()(!
)(''!2
)(')()()(
0
2
xfexfexfexfdx
d
n
xfxfxfxfxUf
Aidx
dii
dx
d
nn
nn
)()( 1111 xfexUf AAi
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Nature of U and A
•For any finite or (compact) infinite group, we can find Unitary matrices that represent the group elements
•U = eiA = 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
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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 + Ly
2 + Lz2
6. C commutes with all group elements ~ CU = UC ~ UCU-1 = CC is invariant under the action of the group