2. Groups

Question 1

What is a group?

Answer 1

A group, G, is a set of objects for which a law of combination is defined

Question 2

When is there an Abelian group?

Answer 2

If all pairs of members of a group commute

Question 3

When is there a continuous group?

Answer 3

When elements can be labelled by one or more continuously varying parameters
E.g. Translations, rotations

Question 4

When is there a compact group?

Answer 4

When the range of parameters for the group is bounded
E.g. between 0 and 2pi for rotations

Question 5

When are two groups isomorphic?

Answer 5

When there is a 1-2-1 correspondence between their elements such that their products of corresponding elements correspond to each other

Question 6

What is a subgroup?

Answer 6

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

Question 7

What is a representation of a group?

Answer 7

• 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

Question 8

What is the Lie group theorem?

Answer 8

The commutators of the generators of the Lie group are a linear combination of the generators

Question 9

Give 3 examples of representations that satisfy the SO(3) group

Answer 9

1. 3x3 orthogonal rotation matrix acting on 3 vector
2. Unitary exponentiated differential operator on wavefunctions
3. Pauli spin matrices

Question 10

What is the U(1) group

Answer 10

The group of simple phase transformations
- Has a single, real constant and a single operator and is continuous and Abelian

Question 11

What are the 3 U(1) symmetries which are respected by nature

Answer 11

Baryon number
Lepton number
Electric charge

Question 12

Describe the properties of the baryon and lepton number U(1) symmetries

Answer 12

They are global symmetries
- The parameter alpha is a single parameter and is independent on space time

Question 13

Describe the properties of the electric charge U(1) symmetry

Answer 13

It is a local/gauge symmetry
- The parameter alpha is dependent on space time

Question 14

What are the lepton numbers and describe what was assumed traditionally about them

Answer 14

3 of them which correspond to their doublets
- Electron number, muon number and tau number
- All must be conserved independently

Question 15

How did the traditional way of viewing the conservation of lepton numbers change

Answer 15

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

Question 16

For the notation of an SU(n) group, what does the S and the U stand for?

Answer 16

S - special meaning det U = 1
U - Unitary

Question 17

How many parameters does a nxn complex matrix have generally, and after the requirement of unitarity?

Answer 17

2n^2 generally
n^2 parameters for unitarity and therefore n^2 generatorsS

Question 18

What are the 4 group axioms for the U(n) and SU(n) groups which involve their multiplication

Answer 18

1. The product of any two nxn unitary matrices is an nxn unitary matrix
2. The nxn identity exists and is a member of the set
3. The inverse exists as det(U) is non-0. Also a nxn matrix which is a member of the set
4. Matrix multiplication is associative

Question 19

How are the U(n) and SU(n) groups related

Answer 19

SU(n) are a subgroup of the U(n) group where for the SU(n) group:
- detU =1

Question 20

What is meant by a “fundamental” group and give examples

Answer 20

A group which is not a DIRECT product of anything else
- E.g. U(1) and SU(n)

Question 21

How are the SO(3) and the SU(2) groups related?

Answer 21

Their generators are the same so they are locally isomorphic
G_i = 1/2 Pauli spin matrix

Question 22

What type of representation do quarks, leptons and gauge bosons exist in

Answer 22

Quarks + leptons in the fundamental representation
Bosons in the adjoint representation

Question 23

Describe how the strong interaction “sees” protons and neutrons

Answer 23

The strong interaction doesn’t see their charge and does not differentiate between them
- They have almost identical masses

Question 24

Describe Heisenberg’s proposal for the protons and neutrons behavior under the strong interaction

Answer 24

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

Question 25

Describe how the isospin states of the u and d quarks break the isospin symmetry and describe how it is fixed

Answer 25

• They have different masses
• Their charges are different
• Fixed by placing a mass term in the Hamiltonian

Question 26

How are the Gell-Mann matrices formed

Answer 26

They are an extension of the Pauli matrices

Question 27

State the types of symmetries for the U(1), SU(2) and SU(3)

Answer 27

• U(1) is baryon number symmetry
• SU(2) is isospin symmetry
• SU(3) is flavour symmetry

Question 28

What was proposed that mediated the strong force?

Answer 28

The exchange of massive bosons called pions
- Virtual particles which produce the attractive force holding it together against the electromagnetic repulsion