Chapter 9+

Question 1

Derive the Klein-Gordon equation from the action.

Answer 1

Question 2

Given the Dirac action, derive the dirac eqn.

Answer 2

Question 3

What are the 3 important properties of the Lagrangian density?

Answer 3

1. Lagrangian is real.
2. Lagrangian is a lorentz scalar.
3. Lagrangian is scalar with respect to internal symmetries[phase factors, like colour or charge, not spatial symmetries].

Question 4

Sketch if V_I is cubic, quartic field, quadratic in one field but also quadratic in another.

Answer 4

Question 5

Sketch the diagram for this klein gordon action with the additional scalar field/ interaction term.
State the propagator and coupling constant.

Answer 5

The first sketch is the propagator. It describes how a single particle moves through space time. It represents a particle in isolation, its a ‘before’ and ‘after’ of an experiment, without it we have no way of showing how it moves through space and time (or just space since we say time is a dimension of space).
^^ MUST DRAW FOR INTERACTIONS

The vertex/X represents the event, it is a specific point in space-time where physics happens- where particles are created or destroyed.

Question 6

Describe the steps in finding the propagator.

Answer 6

Question 7

What are the Feynman rules of this?
State propagator and coupling const.

Answer 7

Question 8

What are the Feynmann rules of this?
State propagator and coupling const.

Answer 8

Question 9

Construct the lagrangian eqn from this.

Answer 9

Question 10

What the differences between 1 and 2?

Answer 10

Question 11

Derive the eqn of motion.

Answer 11

Question 12

What are the Feynmann rules for this?
State propagator and coupling const.

Answer 12

Remember- You can test for U(1) symmetry by transforming
phi –> e^itheta phi
phi* –> e^-itheta phi*

Propagator:
The Numerator : This part comes from the Dirac equation. It accounts for the particle’s spin and mass as it moves.The Denominator (p^2 + m^2): This is the standard “force carrier” part of the propagator, where p^2 is the square of the four-momentum and m is the mass.

^^ Only have the numerator for FERMIONS (spin 1/2 particles).
Coupling constant = q.

1 A_mu term (photon) -> 1 squiggly line

ALL Electromagnetic interactions have U(1) symmetry.

Question 13

Feynmann rules:
When do you used dashed, solid or wavy lines?

Answer 13

phi usually represents a scalar (spin 0) particle. [dashed]

psi usually represents a spinor/fermion (spin 1/2) particle. [bold with arrow]

Question 14

Answer 14

Question 15

Sketch all electroweak basis interactions:

(9.360) Electron-Photon Interaction

(9.361) Triple Gauge Boson Coupling

(9.362) Neutrino-Z Interaction

(9.363) Electron-Neutrino W Interaction

(9.364) W-Z Interaction

(9.365) Electron-Z Interaction

Answer 15

(9.360) Electron-Photon Interaction: This is the basic interaction of Quantum Electrodynamics (QED). It shows an electron ($e$) emitting or absorbing a photon ($\gamma$).

(9.361) Triple Gauge Boson Coupling: This shows the interaction between three force-carrying bosons: a $W^+$, a $W^-$, and a photon ($\gamma$). This interaction exists because the $W$ bosons themselves carry electric charge.

(9.362) Neutrino-Z Interaction: A neutrino ($\nu_e$) interacting with a neutral $Z^0$ boson. Because neutrinos have no electric charge, they do not interact with photons, only with $Z$ and $W$ bosons.

(9.363) Electron-Neutrino W Interaction: This shows a flavor-changing interaction where an electron ($e$) transforms into a neutrino ($\nu_e$) (or vice versa) by emitting or absorbing a $W^-$ boson. This is the fundamental process behind radioactive beta decay.

(9.364) W-Z Interaction: A coupling between a $W^+$, a $W^-$, and a $Z^0$ boson. Like (9.361), this is a self-interaction between the force carriers of the electroweak force.

(9.365) Electron-Z Interaction: An electron ($e$) interacting with a $Z^0$ boson. This is the “weak” version of the QED vertex shown in 9.360.

Question 16

What is Fermi’s Golden rule?

Answer 16

Question 17

Sketch ee scattering.
Parametrically, what is the amplitude of the diagram? What is the rate of the process?
Note: The rate comes from squaring the probability amplitude of Feynmann diagram.

Answer 17

• Parametrically means we look how result depends on fundamental variables of theory, rather than exact number.

Rate ~ Ie^2I^2 ~ e^4 ~ a^2

Question 18

Sketch the fourth order process of electron-electron scattering ~ e^4
State the parametric amplitude and rate in terms of alpha.

Answer 18

Amplitude~ e^4
Rate~ e^8 ~ alpha^4.

Question 19

What operator that is part of the SU(2) group is responsible for rotating a spinor by an angle theta about an axis n?.

Express in terms of sin and cos.

Why does it belong to the SU(2) group?

Answer 19

Note: We have n1, n2, n3 that corresponds to the pauli matricies sigma1, sigma2, sigma3 (x,y,z).
Theta is a real number. n_vec is a 3 dimensional vector. Once you know n and theta it completely determines operator.

It belongs to the SU(2) group:
(a) because it is unitary (Udagger U = I)
(b) as it is special, it has a determinant of one.
(c) It is a 2x2 matrix, hence (2).

Question 20

Check that U satisifies to be in the SU(2) group.

Answer 20

Question 21

Approximate rotation matrix U for theta «1.
What is commutation between pauli 1 and 2?

Answer 21

Commutation cyclic!

group = e^algebra- understanding algebra gives all info to understand the group- lie algebra.

Question 22

We think of pauli matrix algebra as one of many representations of underlying algebra.

Answer 22

Question 23

State defining properties of SU(2) lie algebra.

Answer 23

Question 24

State defining properties of SU(2) lie algebra.
Prove it.

Answer 24

Question 25

State some examples that satisfy SU(2) lie algebra. What does exp (lie algra) return.

Answer 25

Remember exp(lie algebra operator) return rotation of SU(2) group. j can be any matrix.

Question 26

Show this result for pauli matrices.

Answer 26

Note: sigma_i squared = Identity, 1 for ALL i. The nice thing about this is that squaring results in the identity matrix, which is commutable with anything.

Question 27

Test this for pauli matrices.

Answer 27

Question 28

How do we define ladder operators? What is Hermitian conjugate of j1? Hermitian conjugate of (j+)? Hermitian conjugate of j-?

Answer 28

j1, j2, j3 are Hermitian, i.e. j1 dagger = j1, j2 dagger = j2 etc. (j+)dagger = (j1 + ij2) dagger = j1 - ij2 = j-. j- dagger = j+.

Question 29

What is the lowering and raising Pauli operator?

Answer 29

Note: For lowering operator, can compute from SIGMA + dagger, don't need to do 1/2 sigma1 - i/2 sigma2.

Question 30

Why does j1 = 1/2 sigma1?

Answer 30

Question 31

Answer 31

Question 32

What is j3 Ij,m>? Continuing from this result: [j3,j±]=±j±, don't do this exercise here. Do the next part.

Answer 32

1. Highlighted green j3 Ij,m> = mIj,m> Note: we know how to move the j3 through the j+- due to the commutation relation between them.

Question 33

By definition, what do these equal?

Answer 33

Question 34

Show this holds for spinnors (use sigma lowering or raising, pauli, and eigenvector). Do it for Raising on spin down state.

Answer 34

Raising (0,1), get state +1/2. If you do lowering on +1/2, get state -1/2. Raising (0,1), get state +1/2.

Question 35

What is sigma + on spin up state?

Answer 35

Question 36

What doe eigenvalues of j3 range from?

Answer 36

By definition, m ranges from -j to j.

Question 37

Show this.

Answer 37

Question 38

Use one of these to show j^2 I j, j> = j(j+1)I j, j>

Answer 38

Question 39

Prove for Pauli (show LHS = RHS)

Answer 39

Hint: 1/2 is largest j value for j3 Note: The eigenvalue result depends on the representation being worked with. If we use T, the rotation generation matrix, we find T^2 = 2 1hat *2*identity). Pauli is j=1/2 representation, generators are j=1 representation 1(1+1)=2. The eigenvalue j(j+1) depends on which representation (which j) you’re in. For Pauli matrices (j=1/2), eigenvalue is 3/4. For the j=1 generators (like in the adjoint rep of SU(2)), eigenvalue is 1(1+1)=2. The hint just reminds you that m_"max" =j, so for j=1/2, the largest m is +1/2, not 1. So the note contrasts spin-1/2 (Pauli) with spin-1 (T matrices) to show the formula j(j+1) in action.

Question 40

Answer 40

Question 41

Question 42

Answer 42

Question 43

Construct the triplet from Ip>Ip> Hint: Apply lowering operator on top state to find lower states, keep into account not necessarily normalised.

Answer 43

(iii) check In>In> is final state. [I-_tot]^3 Ip>Ip> dproportional to I-_totIn>In> = 0+0 = 0. Hence found all 3 states.

Question 44

State full multiplet(triplet). Don't forget normalisation, done by finding inner product. Next state in form II_tot, I3>. Check I3_tot on In>Ip> form state/ I1,0>

Answer 44

Question 45

We know that there must be 4 states as degeneracy = 2j+1 (for higher isospin, I=1, 2I+1 = 3) State the 4th state (The one with I=0)

Answer 45

Note: Swapping round the 2 kets gets -ve of original, hence, singlet is anti-symmetric.