2. Groups Flashcards
What is a group?
A group, G, is a set of objects for which a law of combination is defined
When is there an Abelian group?
If all pairs of members of a group commute
When is there a continuous group?
When elements can be labelled by one or more continuously varying parameters
E.g. Translations, rotations
When is there a compact group?
When the range of parameters for the group is bounded
E.g. between 0 and 2pi for rotations
When are two groups isomorphic?
When there is a 1-2-1 correspondence between their elements such that their products of corresponding elements correspond to each other
What is a subgroup?
A subset of a group which still satisfies the same group axioms
- E.g. the SO(2) group which represent the 2D group of rotations are a subset of the SO(3) group which represent the 3D group of rotations
What is a representation of a group?
- It is a set of linear operators in which for each member of the group, a, there is an operator O(a)
- For pairs of elements, a,b, there is a pair of operators O(a)O(b) whose product corresponds to the operator for the product ab
What is the Lie group theorem?
The commutators of the generators of the Lie group are a linear combination of the generators
Give 3 examples of representations that satisfy the SO(3) group
- 3x3 orthogonal rotation matrix acting on 3 vector
- Unitary exponentiated differential operator on wavefunctions
- Pauli spin matrices
What is the U(1) group
The group of simple phase transformations
- Has a single, real constant and a single operator and is continuous and Abelian
What are the 3 U(1) symmetries which are respected by nature
Baryon number
Lepton number
Electric charge
Describe the properties of the baryon and lepton number U(1) symmetries
They are global symmetries
- The parameter alpha is a single parameter and is independent on space time
Describe the properties of the electric charge U(1) symmetry
It is a local/gauge symmetry
- The parameter alpha is dependent on space time
What are the lepton numbers and describe what was assumed traditionally about them
3 of them which correspond to their doublets
- Electron number, muon number and tau number
- All must be conserved independently
How did the traditional way of viewing the conservation of lepton numbers change
Evidence of neutrino oscillations showed the flavour eigenstates were not mass eigenstates
- Single lepton numbers can be violated due to propagating neutrinos, however the sum of the lepton numbers is still conserved
For the notation of an SU(n) group, what does the S and the U stand for?
S - special meaning det U = 1
U - Unitary
How many parameters does a nxn complex matrix have generally, and after the requirement of unitarity?
2n^2 generally
n^2 parameters for unitarity and therefore n^2 generatorsS
What are the 4 group axioms for the U(n) and SU(n) groups which involve their multiplication
- The product of any two nxn unitary matrices is an nxn unitary matrix
- The nxn identity exists and is a member of the set
- The inverse exists as det(U) is non-0. Also a nxn matrix which is a member of the set
- Matrix multiplication is associative
How are the U(n) and SU(n) groups related
SU(n) are a subgroup of the U(n) group where for the SU(n) group:
- detU =1
What is meant by a “fundamental” group and give examples
A group which is not a DIRECT product of anything else
- E.g. U(1) and SU(n)
How are the SO(3) and the SU(2) groups related?
Their generators are the same so they are locally isomorphic
G_i = 1/2 Pauli spin matrix
What type of representation do quarks, leptons and gauge bosons exist in
Quarks + leptons in the fundamental representation
Bosons in the adjoint representation
Describe how the strong interaction “sees” protons and neutrons
The strong interaction doesn’t see their charge and does not differentiate between them
- They have almost identical masses
Describe Heisenberg’s proposal for the protons and neutrons behavior under the strong interaction
He proposed that the two particles are different states of the same particle
- The strong interaction theory must therefore be invariant under exchange of these particle types
- There are two eigenstates of the nucleon
Describe how the isospin states of the u and d quarks break the isospin symmetry and describe how it is fixed
- They have different masses
- Their charges are different
- Fixed by placing a mass term in the Hamiltonian
How are the Gell-Mann matrices formed
They are an extension of the Pauli matrices
State the types of symmetries for the U(1), SU(2) and SU(3)
- U(1) is baryon number symmetry
- SU(2) is isospin symmetry
- SU(3) is flavour symmetry
What was proposed that mediated the strong force?
The exchange of massive bosons called pions
- Virtual particles which produce the attractive force holding it together against the electromagnetic repulsion
