1.4 Symmetries and Conservation Laws

Question 1

How does a system behave under a symmetry transformation?

Answer 1

It is unchanged

Question 2

What are the three main classes of symmetry?

Answer 2

Continuous, discrete and internal

Question 3

State Noether’s theorem

Answer 3

A system which is symmetric under a symmetrical transformation naturally generates a conservation law

Question 4

Give two examples of a continuous transformation

Answer 4

Linear translation and temporal translation

Question 5

Which conservation laws arise due to the translations in continuous symmetry?

Answer 5

Linear translation -> momentum conservation

Temporal translation -> Energy conservation

Question 6

Why do continuous symmetries arise?

Answer 6

Due to continuous transformations

Question 7

Which fundamental forces are conserved under continuous symmetries?

Answer 7

All of them

Question 8

What are the three discrete symmetries?

Answer 8

Parity, charge conjugation and time reversal

Question 9

Which fundamental forces are conserved under a parity operator?

Answer 9

EM and strong force conserve parity, weak doesn’t

Question 10

Describe how the parity operator effects the wavefunction

Answer 10

It inverts all the spatial axis - mirror symmetry

PΨ(x,y,z) = Ψ(-x, -y, -z)

Question 11

Which fundamental forces are conserved under the charge conservation operator?

Answer 11

EM and strong force conserve charge conjugation

Weak doesn’t

Question 12

What is charge conjugation/ C parity?

Answer 12

Changing the sign of all charges

- Particles change to anti particles

Question 13

Which fundamental forces are conserved under time reversal?

Answer 13

EM and strong conserve time reversal

weak doesn’t

Question 14

Describe time reversal symmetry

Answer 14

Run time backwards
Symmetry under time reversal implies that the rate of a reaction forwards in time (A + B -> C + D + E) is equivalent to the rate of the reverse reaction (C + D + E -> A + B)

Question 15

Describe the CP operator in terms of the universe

Answer 15

Take the universe, flip all the spatial axis and change all particles to anti particles

Question 16

Which fundamental forces are conserved under the CP operator?

Answer 16

Strong and EM conserve CP

Weak doesn’t

Question 17

Describe the CPT operator in terms of the universe

Answer 17

Take the universe, flip all the spatial axis and change all particles to anti particles, run time backwards

Question 18

Which fundamental forces are conserved under the CPT operator?

Answer 18

All of them

Question 19

What are the eigenvalues of the parity operator?

Answer 19

+1 and -1

Question 20

What does a parity eigenvalue of +1 imply?

Answer 20

Even parity/even wavefunction

Question 21

What does a parity eigenvalue of -1 imply?

Answer 21

Odd parity/odd wavefunction

Question 22

What is the parity of a lepton and a quark?

Answer 22

Parity(lepton) = 1
Parity(quark) = 1

Question 23

What is the parity of a fermion, boson and thier antiparticles?

Answer 23

Parity(fermion) = - Parity(anti-fermion)
Parity(boson) = Parity(anti-boson)

Question 24

What is the parity of a photon and a gluon?

Answer 24

Parity(g) = Parity(photon) = -1

Question 25

How can you work out the parity of a compound system of n particles?

Answer 25

Multiply the product of the intrinsic parities of each n particles, multiplied by (-1) ^ L (orbital anglular momentum of the compound state)

Question 26

What are the eigenvalues for the C operator

Answer 26

+1 and -1

Question 27

What is a necessary condition for a state to be an eigenstate of the C operator?

Answer 27

It must have net neutral charge

Question 28

What is the only fundamental eigenstate of the C operator?

Answer 28

The photon

Question 29

Does the C operator work on compound states?

Answer 29

Only if they are net neutral e.g. pi-0

Question 30

Why arent the gluons or the neutrinos eigenstates of the C operator?

Answer 30

They carry colour and weak charge

Question 31

What is positronium, and is it an eigenstate of the C operator?

Answer 31

A bound state of an electron and a positron and it is an eigenstate of the C operator C |e+ e-> --> |e- e+>

Question 32

What is the eigenvalue of the C operator when acting on a photon?

Answer 32

-1 or (-1)^n for n photons

Question 33

What is the eigenvalue of the C operator when acting on a fermion anti-fermion pair?

Answer 33

(-1)^L+S |f fbar>

Question 34

What is the eigenvalue of the C operator when acting on a boson anti-boson pair?

Answer 34

(-1)^L |b bbar>

Question 35

What do we mean by J^(PC) states for notation in charge parity?

Answer 35

J - Total angular momentum P - parity C - Charge conjugation if it exists

Question 36

Define internal symmetry

Answer 36

Symmetries that operate on the mathematical description of a system in some sort of abstract space

Question 37

What is colour symmetry in SM?

Answer 37

SM is invariant to changes in the quark field colour in some abstract colour space

Question 38

How are internal symmetries represented in the SM?

Answer 38

By changes in the phase fields in SM

Question 39

What two important transformation is the Dirac equation invariant of?

Answer 39

Invariant to the internal symmetry | Invariant to the global phase transformation

Question 40

Is the Dirac equation invariant to changes in local phase and why?

Answer 40

No as there is an extra term with the 4-derivative acting on alpha(x)

Question 41

What does the Dirac equation represent?

Answer 41

The equation of motion for a free fermion

Question 42

What do we have to add to the Dirac equation to enforce invariance to the phase rotations of matter fields?

Answer 42

A new field, A_mu | - Effectively adding interactions of A_mu with the wavefunction to the theory