1.4 Symmetries and Conservation Laws Flashcards
How does a system behave under a symmetry transformation?
It is unchanged
What are the three main classes of symmetry?
Continuous, discrete and internal
State Noether’s theorem
A system which is symmetric under a symmetrical transformation naturally generates a conservation law
Give two examples of a continuous transformation
Linear translation and temporal translation
Which conservation laws arise due to the translations in continuous symmetry?
Linear translation -> momentum conservation
Temporal translation -> Energy conservation
Why do continuous symmetries arise?
Due to continuous transformations
Which fundamental forces are conserved under continuous symmetries?
All of them
What are the three discrete symmetries?
Parity, charge conjugation and time reversal
Which fundamental forces are conserved under a parity operator?
EM and strong force conserve parity, weak doesn’t
Describe how the parity operator effects the wavefunction
It inverts all the spatial axis - mirror symmetry
PΨ(x,y,z) = Ψ(-x, -y, -z)
Which fundamental forces are conserved under the charge conservation operator?
EM and strong force conserve charge conjugation
Weak doesn’t
What is charge conjugation/ C parity?
Changing the sign of all charges
- Particles change to anti particles
Which fundamental forces are conserved under time reversal?
EM and strong conserve time reversal
weak doesn’t
Describe time reversal symmetry
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)
Describe the CP operator in terms of the universe
Take the universe, flip all the spatial axis and change all particles to anti particles
Which fundamental forces are conserved under the CP operator?
Strong and EM conserve CP
Weak doesn’t
Describe the CPT operator in terms of the universe
Take the universe, flip all the spatial axis and change all particles to anti particles, run time backwards
Which fundamental forces are conserved under the CPT operator?
All of them
What are the eigenvalues of the parity operator?
+1 and -1
What does a parity eigenvalue of +1 imply?
Even parity/even wavefunction
What does a parity eigenvalue of -1 imply?
Odd parity/odd wavefunction
What is the parity of a lepton and a quark?
Parity(lepton) = 1 Parity(quark) = 1
What is the parity of a fermion, boson and thier antiparticles?
Parity(fermion) = - Parity(anti-fermion) Parity(boson) = Parity(anti-boson)
What is the parity of a photon and a gluon?
Parity(g) = Parity(photon) = -1
How can you work out the parity of a compound system of n particles?
Multiply the product of the intrinsic parities of each n particles, multiplied by (-1) ^ L (orbital anglular momentum of the compound state)
What are the eigenvalues for the C operator
+1 and -1
What is a necessary condition for a state to be an eigenstate of the C operator?
It must have net neutral charge
What is the only fundamental eigenstate of the C operator?
The photon
Does the C operator work on compound states?
Only if they are net neutral e.g. pi-0
Why arent the gluons or the neutrinos eigenstates of the C operator?
They carry colour and weak charge
What is positronium, and is it an eigenstate of the C operator?
A bound state of an electron and a positron and it is an eigenstate of the C operator
C |e+ e-> –> |e- e+>
What is the eigenvalue of the C operator when acting on a photon?
-1 or (-1)^n for n photons
What is the eigenvalue of the C operator when acting on a fermion anti-fermion pair?
(-1)^L+S |f fbar>
What is the eigenvalue of the C operator when acting on a boson anti-boson pair?
(-1)^L |b bbar>
What do we mean by J^(PC) states for notation in charge parity?
J - Total angular momentum
P - parity
C - Charge conjugation if it exists
Define internal symmetry
Symmetries that operate on the mathematical description of a system in some sort of abstract space
What is colour symmetry in SM?
SM is invariant to changes in the quark field colour in some abstract colour space
How are internal symmetries represented in the SM?
By changes in the phase fields in SM
What two important transformation is the Dirac equation invariant of?
Invariant to the internal symmetry
Invariant to the global phase transformation
Is the Dirac equation invariant to changes in local phase and why?
No as there is an extra term with the 4-derivative acting on alpha(x)
What does the Dirac equation represent?
The equation of motion for a free fermion
What do we have to add to the Dirac equation to enforce invariance to the phase rotations of matter fields?
A new field, A_mu
- Effectively adding interactions of A_mu with the wavefunction to the theory
