From relativistic QM to QFT

Question 1

from relativistic QM to QFT

What is the requirement for a detailed theory for calculating M_if?

Answer 1

We have to construct a Hamiltonian that is Lorentz invariant and compatible with other symmetries of nature as well.

Question 2

relativistic QM

How can we generalize QM to fit special relativity?

Answer 2

By substituting the relativistic energy-momentum relation: E^2 – p^2 = m^2 into the Schrödinger equation, the Klein-Gordon equation can be constructed.

(kocka op. + m^2)𝛙(t,x) = 0

Question 3

relativistic QM

What are the difficulties with solving the Klein-Gordon equation?

Answer 3

1. Lack of covariant probability current that gives a positive-definite probability density. From the KG eq. and its conjugate a probability current that is a conserved four-vector quantity van be constructed but its 0th probability densitiy component is not positive-definite.
2. Existence of negative-energy solutions. The solutions are plane waves with corresponding four-momenta that satisfy the mass shell condition. Only the 0th components are bounded by the condition, so unbounded negative-energy states can also be present, which contradicts the observed stability of matter.

Question 4

relativistic QM

What’s the general solution of the KG eq. and what does it represent physically?

Answer 4

Obtaining the solution:

1. Going over to the momentum space by a Fourier-transform.
2. Finding the algebraic equation that restricts the momentum space solution (non-vanishing only on the mass shell).
3. Implementing 2. by setting the momentum space solution to a specific function.
4. Substituting 3. into the real space solution.
5. Introducing a(p) and b(p) complex amplitudes (along with the invariant one-particle phase space element).

Physical meaning:

• the a(p) term: positive energy solution of spatial momentum p
• the b(p) term: negative energy solution of spatial momentum –p

Question 5

relativistic QM

How can the spin description be included and the problems with the solutions of the KG eq. be eliminated?

Answer 5

For 1/2-spin fermions the equation should be first order in the time derivative (to eliminate negative-energy states) and also in the spatial derivatives (to keep the Lorentz invariance). Furthermore the new eq. should imply the KG eq.

1. The form of the solution: insert képlet.
2. Finding a and b to reproduce the KG eq.
3. The solutions: 4x4 matrices: γ matrices and their relations.
4. Upgrading 𝛙 to a bispinor with spinors as components.
5. Result: Dirac equation

The four-vector current constructed from this is a good probability current that gives a positive-definite probability density. The problem of negative energies is still not solved though.

Question 6

relativistic QM

How can the Hamiltonian be constructed from the Dirac equation?

Answer 6

From the Dirac eq. the Dirac Hamiltonian can be obtained.

insert képlet

Question 7

relativistic QM

How can the energy eigenfunctions be derived for the Dirac equation?

Answer 7

The plane wave solutions of the Dirac eq. correspond to the eigenfunctions of the Dirac Hamiltonian with energy p^0. These must satisfy two equations to which there are four solutions: two with positive and two with negative energy.

1. Separating the positive and negative energy solutions.
2. Introducing the u and v bispinors, that satisfy certain equations.
3. Constructing u and v with spinors ξ and η.
4. Choosing an orthonormal pair of spinors χ(s) and χ~(s) with which u and v can be “finalized”.
5. Writing the completeness relations.

Question 8

relativistic QM

How can we explain the two spin states of an electron?

Answer 8

The explanation is the double degeneracy of the energy levels. Taking the low energy limit in the positive-energy solution, the two surviving components can be interpreted as the two components of the electron wave function, corresponding to the spin up and spin down states.

Question 9

relativistic QM

What’s the physical meaning of the negative energy solutions?

Answer 9

The initial idea is that we can interpret the negative-energy solutions as a positive-energy solution travelling backwards in time. This can be taken further by interpreting this as a particle traveling forwards in time but that has the exact opposite of every conserved charge of the original particle. These particles are the antiparticles: same mass, same spin but opposite charges.

The travelling backwards in time part is quite unsatisfactory anyway.

Question 10

relativistic QM

What is chirality?

Answer 10

The right- and left-handedness of states. It can be constructed with the 5th gamma matrix.

• for positive energy states: chirality = helicity
• for negative energy states: chirality = opposite helicity

E.g.: neutrinos are left-handed, antineutrinos are right-handed.

helicity: the spin component that’s in the direction of motion

Question 11

relativistic QM

What are the general solutions of the Dirac equation?

Answer 11

The plane wave solutions form a complete set of solutions of the Dirac eq. The most general solution is taking their linear combination with generally complex coefficients: insert képlet
The two terms correspond to the positive- and negative- energy states or to a particle and antiparticle each.

Question 12

a sketch of QFT

How can we recontextualize QM to get rid of its problems?

Answer 12

Problems: negative probabilities from the wave functions of scalar particles, negative energy states that are interpreted as antiparticles but the concept that that invokes such as travelling backwards in time are iffy at best. The finiteness of the speed of light also prevents us from describing interactions via a potential relativistically.

Solution: getting rid of the pointlike particles and intorducing fields whose evolution only depends on fields and their derivatives.

• particles = localised excitations of fields
• particle interactions = ripples in a mediator field

Question 13

a sketch of QFT

What’s second quantisation? How does the quantisation of fields work?

Answer 13

Promoting the probability field of the wave function to a type of field that acts as an operator on the (Hilbert) space of states that can create or annihilate particles.

So in essence, the amplitudes of the field normal modes are now operators that can excite and add particles to the vacuum state or erase them from there.

Question 14

a sketch of QFT

How do creation/annihilation operators work for bosons/fermions?

Answer 14

insert formulas for the wave function, Hamiltonian, electric charge

Bosons:

• operators obey commutation relations
• the order doesn’t matter when writing a multiparticle state

Fermions:

• operators obey anticommutation relations
• the order doesn’t matter when writing a multiparticle state up to a sign (particle operators before antiparticle ones by convention)

Question 15

a sketch of QFT

How can the EM field be described by QFT?

Answer 15

In QFT it’s the photon field: insert képlet

• it includes the two physical polarizations of the photons
• the operators mist obey the commutation relations since the photon is a spin-1 boson
• the theory reflects that the photon is its own antiparticle

Question 16

a sketch of QFT

How does QFT naturally lead to introducing antiparticles?

Answer 16

The positive-energy part annihilates a particle, removing some amount of charge from the state, and the negative-energy part creates a particle, adding some amount of charge to the state. In order to assign the same charge to the two components, one then needs the creation operator appearing in the negative-energy part to create a particle with opposite charge to that associated with the other creation operator.

Question 17

a sketch of QFT

How can particle interactions be described in QFT? Example?

Answer 17

We have to include creation and annihilation operators using fields in the interaction Hamiltonian, that change the state and number of particles during an interaction, corresponding to the emissions and absorptions to have locality and symmetries under control.

Example: elastic EM interaction of an electron and positron

• one particle is annihilated and two, including the photon, are created
• if we don’t see the photon: the net effect is teh interactio between the two particles whose state is changed by the EM force mediated by the photon

Question 18

a sketch of QFT

How are conservation laws explained in QFT?

Answer 18

Violations of energy are acceptable if they are smaller than 1/(2Δt) according to the uncertainty principle. This is the uncertainty of the energy corresponding to the time required to observe the process – we would not be able to observe them.

The explanation to this is that the exchanged particle is not on the mass-shell (it could not be observed anyway), these are the virtual particles.

Finally, enforcing conservation laws comes from the symmetries of the systems, which already imply the existence of conserved quantities. So we need to biuld interactions that respect the desired symmetries and this is what field make easier.

If we treat the interaction Hamiltonian as a perturbation of the free Hamiltonian, the interaction of real particles through the exchange of virtual particles comes naturally.

Question 19

Feynman diagrams

What are Feynman diagrams?

Answer 19

They are associated to each and all of the infinitely many possible ways that a physical process can take place.

For the visual aid:
With each factor of V_I(t), we associate a graph with as many lines as fields (and so creation and annihilation operators) in the interacting Hamiltonian, with a different type of line for each type of field (and so type of particle). All these lines meet at a point, which we also call interaction vertex.

Question 20

Feynman diagrams

What are the constituents of a Feyman diagram? What are their contributions? What’s the principle?

Answer 20

The way this works in practice is that one computes the matrix elements of the S operator between initial and final multiparticle states by expanding Dyson’s formula for the S-matrix and computing term by term the various contributions.

We are essentially pairing one creation and one annihilation operator among those appearing in the fields and in the initial and final state vectors, so describing the propagation of the corresponding particle from one point of the graph to another.

The constituents:

• external line: lines paired with the incoming and outgoing particles of the same type, contributes a wave function
• internal line: he remaining lines of each type are further paired with each other in all possible ways, contributes a propagator
• vertex: contributes a coupling constant

Conserved quantities are conserved at each vertex.

Question 21

Feynman diagrams

What are the properties of interaction vertices for the fundamental interactions?

Answer 21

EM (QED):

• only one interaction vertex, differrent variations cannot be considered in isolation
• coupling constant: electric charge of the type of particle involved
• conserved quantities: energy, momentum, angular momentum, electric charge, particle number of each type
• all electrically charged particles interact throught this type of vertex

Strong (QCD):

• colour: extra degree of freedom of quarks and gluons
• coupling constant: g(s)
• conserved quantities: energy, momentum, angular momentum, electric charge, flavour, colour
• gluons self-interact
• only quarks, antiquarks and gluons are affected by strong interactions

Weak:

• there are two types of vertices: charged current (W+/- bosons), neutral current (Z boson)
• coupling constants: g(w) (charged), g(w)0 + suitable flavour-dependent coefficient (neutral), the g values are proportional to e via the Weinberg angle
• conserved quantities: energy, momentum, angular momentum, electric charge, particle type (neutral), lepton family number (charged)
• every elementary particle, except for gluons, is affected by weak interactions

Question 22

Feynman diagrams

What’s the CKM matrix? What motivates the need for it?

Answer 22

Cabibbo-Kobayashi-Maskawa matrix: a unitary matrix, that determines how flavours mix.

This comes from the fact that the quarks strong (mass) eigenstates don’t coincide with the weak ones. When quarks take part in the weak interaction there could be an analogy for how only quarks in the same family couple (like with leptonic processes) but this is not the case; a quark can turn into a linear combination of other quarks and this is what the matrix describes.

Question 23

a sketch of QFT

How does QFT explain the range of interaction for particles?

Answer 23

It shows that the range of interaction is indeed determined by the inverse mass of the corresponding mediator.

The factor corresponding to an internal line in a Feynman diagram is in a sense the relativistic analogue of the interaction (Yukawa) potential: it represents the interaction as the exchange of a particle, hence the 1/(p^2 + M^2) factor (this is the momentum space version of the Yukawa potential).

• this can be shown from solving the static limit of the KG equation in the presence of a charged static source
• the Yukawa field becomes a classical field describing the instantaneous interaction of matter particles that are coupled to it

Question 24

a sketch of QFT

How can we compare the strength of interactions? What’s the “result”?

Answer 24

Comparing the lifetimes of unstable
particles whose decays are governed by different interactions or looking at the cross sections from scattering processes. Γ and the scattering cross section is typically proportional to the fourth power of the coupling constant, hence the comparison.

physical process —» contribution to transition amplitude —» transition probability —»cross section, decay width —» lifetime (lifetime ~ |amplitude|^2)

The hierarchy:
strong > EM > weak
(-23 ~ -20; ~-16; -13 ~ 3)

Question 25

a sketch of QFT

Why is the weak interaction weak? Why is the range of lifetimes so large for weak decay processes?

Answer 25

The reason is NOT a weak coupling constant. The main difference between the two interactions is that while the photon is massless, the intermediate vector bosons are very massive.

At energy &laquo_space;M(vector boson): the coupling constant is the Fermi contstant and due to the varying M masses of the processes, g(w) has a large range of values, hence the large range of lifetimes. The interaction is weak not because of a small coupling constant g(w), but because of
the large mass of the mediators that lead to a small effective coupling.

At energy > M(vector boson): weak and electromagnetic interactions become comparable, p^2&raquo_space; M(vector boson)^2 dominates the propagator and the weak interaction becomes much stronger than the electromagnetic one

Question 26

basic physical processes and Feyman rules

What are the steps for computing the scattering amplitude? What are the rules?

Answer 26

To describe an actual physical process one has to properly combine two or more of the three-particle vertices in which the matter particles appear. Pairing lines with initial and final particles means essentially that we choose which annihilation or creation operator does the job of removing a particle from the initial or final state.

• when the partaking particles are indistinguishable, both possibilities of assigning the operators has to be taken into account
• momentum is conserved at each vertex

Steps:

1. Assigning a momentum to every line, labeling the vertices.
2. Assigning wave functions and vertex factors corresponding to their quantum numbers to each of the fermionic lines going backwards.

• starting from the final state for fermions and from initial state for antifermions
• fermionic lines can start and and in initial/final states as well

3. Including the factor of the propagator and considering conservation of four-momentum.
4. Putting everything together (subtracting the potentially multiple possibilities).

The vertex label means the possible polarizations of the photon.

Question 27

basic physical processes and Feyman rules

How can we calculate the scattering cross section from the scatering amplitude?

Answer 27

1. Calculating the transition pobability using certain identities.
2. Applying Casimir’s trick: averaging over the initial spins and summing over the final spins.
3. Putting the result and the expression for the two-body phase space into the formula for the differential cross section.
4. Making estimationd for the CM frame.

The total cross section can be computed by integrating over the solid angle and deviding by 2.

Question 28

basic physical processes and Feyman rules

What are the 7 interactions from combining EM vertices for electrons and positrons?

Answer 28

• electron-electron scattering (Møller scattering)
• electron-positron scattering (Bhabha scattering)
• positron-positron scattering
• electron-positron annihilation
• electron-positron pair creation
• Compton scattering of photons on electrons
• Compton scattering of photons on positrons

Question 29

basic physical processes and Feyman rules

What distinguishes diagrams contributing to the same process? How should the contributions be considered?

Answer 29

The distingushing factor is the topology, so only the diagrams with different topologies should be considered.

• certain diagrams require the inclusion of suitable symmetry factors: one should count in how many ways the contractions of the lines originating from the vertices can be made for a given diagram topology, (1/n! from the expansion of Dyson’s formula)
• loops can form as well (the vertices here don’t necessarily conserve momentum)
• diagrams differing by the exchange of external bosonic/fermionic lines should be added to/subtracted from each other
• we will need a limited number of diagrams, as the introduction of more interaction vertices will further suppress the contribution of the other
  diagrams

Question 30

basic physical processes and Feyman rules

What are the Feynman rules for strong interactions?

Answer 30

We have to take colour and the self-interaction of gluons into account.

• different vertex factor based on the colour states of the quarks and the gluons (t(ij)^a)
• different gluon propagator: it carries colour from one vertex to another
• Casimir’s trick should be applied for the initial and final colours as well
• gluon self-interaction vertices lead to completely different physics

Question 31

basic physical processes and Feyman rules

What are the Feynman rules for weak interactions? Example diagrams?

Answer 31

The new rules required here are those for the interaction vertex and for the propagator of
the mediator bosons.

• the propagator for W and Z bosons is the same apart from their mass
• the interaction vertex here changes the type of fermion (lepton or quark) and includes a projector that sekects only one of the chiralities of the fermions (the reason behind the parity violations in weak interactions)
• for a low energy approximation (see other card): the two vertices connected by the propagator join together, leading to a point-like four-fermion interaction vertex, with effective coupling given by the Fermi constant