Course Information

Syllabus

 

QFT I Notes

 

QFT II Notes

 

Seminar

 

 

Monday, January 11: Squaring the spacetime Dirac equation

 

Lecture: Squaring the Dirac equation in curved spacetime

 

Scaling properties of the Einstein-Hilbert action and the scalar field action. The gravity action is not dimensionless.

QFT II Jan 11a

 

Combine the scalar curvature and a scalar field to make a gravity action which is scale invariant. Dirac used this action in his Large Numbers papers, suggesting that the scalar field be interpreted as a dynamical gravitational constant, G(t).

The resulting wave equation contains the scalar curvature.

The Dirac action generalizes to curved spacetime by changing the derivative to a covariant derivative, and modifying the volume element. We must also generalize the Dirac matrices. (See Brill & Wheeler on neutrinos)

QFT II Jan 11b

 

Squaring the covariant Dirac operator now leads to both symmetric and antisymmetric pieces. The symmetric piece gives the covariant dÕAlembertian, while the antisymmetric piece give Spin(3,1) Lorentz generators.

The antisymmetric generators induce a commutator on the covariant derivatives, which we resolve using the Ricci identity.

QFT II Jan 11c

 

We actually need the Ricci identity in the spinor representation of the Lorentz generators. This takes some work:

QFT II Jan 11d

QFT II Jan 11e

QFT II Jan 11f

QFT II Jan 11g

 

Inserting the Ricci identity for the commutator of covariant derivatives gives a pair of Spin(3,1) generators contracted with the curvature tensor.

QFT II Jan 11h

 

Since the symmetric part of any product of two Dirac matrices may be replaced by a metric, the product of 4 Dirac matrices must be a linear combination of the identity, the generators, and g5 . (That is, 0, 2 or 4 Dirac matrices, antisymmetrized.) We can find the coefficients by taking traces, but we only need one. The other terms, contracted with the curvature tensor, will vanish by symmetry.

We are left with exactly the scale-invariant field equation used by Dirac, acting on each component of the spinor.

QFT II Jan 11i

It is not unreasonable to assume, since this is the wave operator that the Dirac equation squares to, that this is the right way to think about scalar waves in spacetime.

 

 

A similar calculation using the U(1) connection does not yield a scalar wave equation, since one remaining generator still mixes components of the spinor:

QFT II Jan 11j

 

 

Wednesday, January 13: Transition amplitudes for scattering

 

Lecture: Transition amplitudes for scattering in quantum field theory

 

Scattering experiments have initial and final states that are essentially two different free field solutions. We write the Hamiltonian as a free Hamiltonian plus an interaction Hamiltonian. The two asymptotic, free states differ due to the effects of the interaction part of the Hamiltonian, which typically has a perturbative effect.

QFT II Jan 13a

 

The scattering matrix, S, connects initial and final states. Unitarity of the S matrix preserves overall probability.

QFT II Jan 13b

 

We expand S = 1 + iT, and find it convenient to compute T.

QFT II Jan 13c

QFT II Jan 13d

 

Recall that the space of physical states is found by acting with creation operators on the vacuum. The vacuum is the state annihilated by all destruction operators.

It is convenient to use time-independent, Heisenberg states instead of time-dependent Schršdinger states.

QFT II Jan 13e

 

In the Heisenberg representation, the operators become time dependent. This is convenient for field theory, where the spacetime-dependent fields become the operators. 

QFT II Jan 13f

 

Recall our solution for the creation and annihilation operators for scalar fields.

Define ŅinÓ and ŅoutÓ fields as free fields, with an additional normalization constant. This will let us take into account the scattered part of the state more easily.

QFT II Jan 13g

 

Write the normalization the other way.

QFT II Jan 13h

 

 

Consider a transition amplitude between an initial state with m particles and a final state with n particles. We want to rewrite this as a product of field operators between vacuum states.

WeÕll need a trick or two. Write the spatial integral as a spacetime integral of a time derivative.

QFT II Jan 13i

 

Now we can replace the second derivatives with respect to time using the field equation. We can then combine everything as a complete wave operator in a spacetime integral

QFT II Jan 13j

 

Substitute into the transition amplitude. We have eliminate one of the incoming momenta from the state, replacing it by a spacetime integral and a wave operator

QFT II Jan 13k

 

Friday, January 15: Transition amplitudes in terms of vacuum expectation values

 

Lecture: Transition amplitudes, continued. Amplitudes in terms of vacuum expectation values

 

(Answer to a question on solving for the connection in GR:  QFT II Jan 15b  )

 

Last time, we described transition amplitudes and the S and T matrices. We expressed the transition amplitude in terms of m initial and n final momenta, and showed how to eliminate one momentum from the initial state, replacing it with a combination of creation and annihilation operators.

 

One of these operators gives zero:

QFT II Jan 15a

 

We will need the time ordering operator, T. Acting on a product of fields, it re-orders them with the latest times to the left. We can see the need for this if we consider a perturbative expansion of the time dependence of a quantum state. At second order, we rewrite the sequenced integrals, t1 ˆ t2 ˆ t3 as half the time-ordered product of two integrals over the full time range, t1 ˆ t3 . Repeating the argument at all orders leads to a time-ordered exponential.

QFT II Jan 15c

QFT II Jan 15d

QFT II Jan 15e

QFT II Jan 15f

 

The time-ordering operator:

QFT II Jan 15g

 

As shown last time, we can replace one of the incoming momenta with raising and lowering operators;

QFT II Jan 15i

Near duplicate: QFT II Jan 15h

 

Taking the adjoint gives the same effect on an outgoing momentum. Iterating for all incoming and outgoing momenta gives a vacuum expectation:

QFT II Jan 15j

 

Note that, since we assume no final momentum equals any incoming momentum, we have actually computed the T matrix

QFT II Jan 15k

 

The vacuum expectation value is an n+m point Green function. We write its Fourier transform.

QFT II Jan 15m

 

The final expression:

QFT II Jan 15l

 

We have now written a general transition amplitude in terms of a vacuum expectation value.

 

 

Wednesday, January 20: Vacuum expectation values in terms of free fields

 

Lecture: Using the interaction representation to express transition amplitudes in terms of free fields

 

We can write the raising and lowering operators in terms of field integrals. The form here is a little more compact than what we wrote last semester, but equivalent. Check the equivalence.

QFT II Jan 20a

QFT II Jan 20b

QFT II Jan 20c

(Duplicate: QFT II Jan 20d )

 

The full time evolution of the system is given by an exponential of the Hamiltonian, but for the interaction picture, we define operators evolved by the free Hamiltonian only. Inserting an identity (written as the free field evolution operator times its inverse) into the full time evolution, we identify the operator U(t,tÕ) as the thing to be computed.

QFT II Jan 20e

QFT II Jan 20f

QFT II Jan 20g

 

U(t,tÕ) satisfies a simple differential equation

QFT II Jan 20h

 

The solution to the equation is as expected, except that it requires time ordering

QFT II Jan 20i

 

The full vacuum expectation value, written with interaction picture fields and U(t,tÕ) operators:

QFT II Jan 20j

 

We show that the product U(t, tÕ) U(tÕ, tÕÕ) = U(t, tÕÕ), that is, these operators obey a simple product rule

QFT II Jan 20k

 

Using this product rule, we eliminate almost all of the U(t, tÕ) operators, leaving a product of interaction picture fields. Remember that these interaction picture fields are computed from the free field, and the free field Hamiltonian. The only effect of the interactions is in the U(t, tÕ) operators. These, in turn, are computed from the interaction terms of the Hamiltonian.

QFT II Jan 20l

QFT II Jan 20m

Choosing the starting and finishing times judiciously gives a simple expression.

Next time we will reduce this further.

 

Friday, January 22: WickÕs theorem and Feynman propagators

 

Lecture: Wick's theorem and Feynman propagators

 

Recall the definition of the time ordering operator, T

QFT II Jan 22a

 

Last time, we expressed the transition amplitude in terms of vacuum expectation values of time ordered  products of fields, together with time evolution operators, U(t,tÕ). By a convenient choice of the initial and final times we eliminate all but one such factor, U(-°, +°). An additional phase shows up as a normalization.

QFT II Jan 22b

 

Now study the time ordered product of two fields. We can rearrange the fields using the commutation relations of the creation and annihilation operators, with the one non-vanishing commutator, D, expressed as a Fourier integral

QFT II Jan 22c

 

The simplest example of WickÕs theorem: we rewrite the time ordered product in terms of a normal ordered product and a commutator. The commutator is simply a function.

QFT II Jan 22d

 

Substituting these expressions into the vacuum expectation, the normal ordered terms vanish. The final result, D(x-y), is the Green function for the original wave equation. It is called the Feynman propagator; it acts like a heat kernel, evolving a distribution from one time to another.

QFT II Jan 22e

 

Explicit calculation of the Feynman propagator. Contour integration is used, closing in the lower half plane for x0 > y0  and the upper half plane for y0 > x0.

QFT II Jan 22f  (In lower left, a quick review of how a Green function is used)

QFT II Jan 22g

QFT II Jan 22h  (We will frequently use the Fourier transform of the propagator)

 

 

Monday, January 25: An example: 0th and 1st order perturbation of the scalar field interaction

 

Lecture: Perturbation theory and Feynman diagrams

 

Exercise: Prove Eq. 5.86:

<0| T{f1 f2 f3 f4 } |0>   =  D12 D34  +   D13 D24  +  D14 D23

 

An exercise, a statement of a little more general version of WickÕs theorem, and the reason that vacuum expectation values of normal order products of fields always vanishes:

QFT II Jan 25a

 

Feynman diagrams (far right) for two non-interacting particles

QFT II Jan 25b

 

Example: 1st and 2nd order calculations of the scalar field scattering matrix

QFT II Jan 25c (see close-ups below)

 

Proof that if the Hamiltonian is polynomial in the fields, then the interaction Hamiltonian is the same polynomial of the interacting fields

QFT II Jan 25d

 

Writing out the transition amplitude in terms of propagators at zeroth order in lambda

QFT II Jan 25e

 

A simple change of coordinates lets us compute the integrals explicitly

QFT II Jan 25f

 

Putting it all together, we see that the final result for the iT matrix vanishes when we set p2 – m2 = 0.  This is expected, because at zeroth order there is no scattering.

QFT II Jan 25g

 

Moving to order l, we get non-vanishing contributions only for those propagators that relate the original fields to fields in the interaction Hamiltonian. Otherwise, the terms vanish as above. This time, the integrals separate nicely and with a trivial change of variable each integral gives a phase times a momentum space propagator. Performing all integrals gives the same product of propagators that we need to cancel, together with a conservation-of-momentum Dirac delta function and some factors.

QFT II Jan 25h

 

Cancelling the common propagator factors, we are left with a coupling strength, factors of 2¹, and the conservation of momentum delta function. We draw the Feynman diagram corresponding to this term of the calculation.

QFT II Jan 25i

 

 

 

Wednesday, January 27: 2nd order perturbation of the cubic scalar field interaction; loops

 

Lecture: Second order perturbation of the scalar field

 

The scattering matrix for 2nd order perturbation of the scalar field with cubic interaction and the corresponding Feynman diagrams.

QFT II Jan 27a

 

Further 2nd order diagrams and the calculation for diagram 1.

QFT II Jan 27b

 

The integrals simplify quickly and we are again left with a factor, the propagator poles that cancel, and the conservation of momentum delta function.

QFT II Jan 27c

 

The delta function integral: the definition of the matrix Mfi. We will develop general rules for writing down Mfi directly.

QFT II Jan 27d

 

Where the factor of  -i  comes from (lower left).

QFT II Jan 27e

 

There should have been a factor of  (-i)2  in our original equation (note this factor in the upper right).

QFT II Jan 27f

 

At second order with the quartic interaction it is possible to form loops:

QFT II Jan 27g

 

The calculation of a loop diagram leaves us with an extra integral

QFT II Jan 27h

QFT II Jan 27i

 

We will look at this integral next time.

 

Friday, January 29: Loops and tadpoles

 

Lecture: Loops and tadpoles

 

Recall from last time how a loop diagram leads to an extra integral over a pair of propagators.

QFT II Jan 29a

 

Three similar diagrams all contribute at this order. It is simple to write down the contributions from the remaining two.

QFT II Jan 29b

 

This remaining integral diverges as the magnitude of the integrated momentum tends to infinity. To calculate the form of the divergent part, we can drop p2 relative to q2. It is most convenient to turn the integral into a Euclidean integral, and avoid the integrate directly. We can do this because the residue theorem applies to any loop containing a given pole – we can rotate our contour 90 degrees counterclockwise in the complex plane. This turns the denominator into a positive definite expression as we integrate up the imaginary energy axis.

QFT II Jan 29c

 

To do the integral, we write it in 4-dimensional spherical coordinates. The integral over angles gives 2¹2. It is the radial part which diverges. We need to do a little bit of geometry to find the volume element.

QFT II Jan 29d

QFT II Jan 29e  (repeat)

 

All of the integrals are now easy. To see the behavior of the divergence, we put a cut-off at L instead of letting the integral run all the way to infinity. The divergence is logarithmic.  Since we set p2 to zero, we canÕt immediately tell what the finite part is, though we could do it if necessary.

QFT II Jan 29f

 

We also find loops on the propagator. The divergence is easy to integrate, and now has a quadratic piece in addition to a logarithm.

QFT II Jan 29g

 

We can sum all such ŅtadpoleÓ graphs. The effect is to shift the mass in the propagator by an amount dependent upon the cutoff.

QFT II Jan 29h

 

 

Monday, February 1: Feynman rules and the Dirac propagator

 

Before writing the Feynman rules, we look at some research related questions.

Lecture: Feynman rules and the Dirac propagator

 

Discussion of various questions:

In biconformal space, there is an induced Killing metric and a natural symplectic form. These allow us to find canonically conjugate, orthogonal, metric submanifolds of twistor space. Here is a quick review of how to tell when fields, or basis forms, are canonically conjugate

QFT II Feb 01a

QFT II Feb 01b

 

A loop diagram

QFT II Feb 01c

 

Simplifying a double arrow expression

QFT II Feb 01d

 

Now, back to field theory.

 

Why the time ordered product of fields equals the Green function of the field equation

QFT II Feb 01e

 

Feynman rules for lf4  interactions of scalar fields. We can use these rules to write Mfi directly.

QFT II Feb 01f

 

Diagrams at various orders in perturbation theory.

QFT II Feb 01g

 

Exercise:  Draw all 3rd order diagrams.

 

The Dirac propagator. Find the Green function for the Dirac equation by Fourier transform.

QFT II Feb 01h

 

The final momentum space expression for the Dirac propagator:

QFT II Feb 01i

 

 

Wednesday, February 3: Feynman rules for QED; renormalization

 

Lecture: Feynman rules for QED; renormalization

 

 

Expanding the Schršdinger equation as a time-ordered exponential.

QFT II Feb 03a

 

Rewriting the double integral so both have full time range brings in the factor of ½  for the exponential

QFT II Feb 03b

 

The QED (U(1) gauge theory) action with a variable gauge parameter. The divergence of the vector potential is taken to vanish on physical states.

QFT II Feb 03c

 

We choose the simplest form of the photon propagator. The field expansions tell us what the interactions can do.

QFT II Feb 03d

 

The interaction term is a product of all three fields. This gives eight possible interactions. However, conservation of momentum rules out two of them: there must be at least one particle in the initial state, and at least one in the final state.  (Upper right: photons can interact!)

QFT II Feb 03e

 

The form of the coupling gives a gamma matrix, a unit charge, and the usual  -i. Each propagator will carry either a basis spinor or polarization vector as well.

QFT II Feb 03f

 

Renormalization.

Start from the action with fields which are now allowed to depend on the cutoff. This gives a class of theories, one for each value of the cutoff. We are free to choose the cutoff dependence of the fields and parameters, and we make the choice so that the limit of infinite cutoff has the usual measured values of the parameters (such as mass and charge; we are also free to choose a normalization of the field). At the 2-loop level, we can define the dependence of mass on cutoff so that the combination of all of the tadpoles and the cutoff dependent mass equal the usual measured mass.

QFT II Feb 03g

 

Higher orders require sterner measures. Renormalizing the field can fix the divergences from loop amplitudes.

QFT II Feb 03h

 

If we fix the parameters at one value of momentum, the fix holds for other momenta – the difference is finite.

QFT II Feb 03i

 

 

Friday, February 5: Asymptotic series (Jeff); superficial divergence

 

Lecture: Convergent versus asymptotic series; superficial divergence

 

Asymptotic series

 

A convergent series converges; asymptotic series may or may not converge:

QFT II Feb 05a

 

The exponential integral, integrated by parts to produce a series for large x, is asymptotic:

QFT II Feb 05c

 

The exact integral is bounded:

QFT II Feb 05b

 

At fixed x, the series starts to diverge for N > x. By that value of N, the series is often extremely accurate.

QFT II Feb 05d

 

Superficial divergence

 

Some Feynman diagrams. Can we tell when higher order diagrams will introduce new divergences?

QFT II Feb 05e

 

Each loop adds four factors of the momentum; each propagator subtracts two. We therefore need to count the number of internal lines and loops for any given number of vertices.

QFT II Feb 05f

 

We can express the divergence in terms of the number of vertices and the number of external lines.

QFT II Feb 05g

 

The final result can be expressed in terms of the dimension, as a power of length, of the coupling constant(s). Positive powers of length are generally non-renormalizable.

QFT II Feb 05h

 

The cosmological constant problem. Though renormalizable, the cosmological constant requires extremely fine tuning. This seems odd.

QFT II Feb 05i

 

 

Monday, February 8: Decay rates and differential cross sections

 

Lecture: Decay rates and differential cross sections

 

 

Leptons, mesons (quark-antiquark pairs) and baryons (quark triples). The production of jets in proton-antiproton colliders.

QFT II Feb 08a

QFT II Feb 08b

 

Recall the general form of the scattering matrix. We consider the decay of a single particle first, then the scattering of a pair of particles. Because the probability of interaction goes as the square of the matrix element, giving us a delta function squared, we normalize in a finite box. This allows us to replace one of the delta functions with the volume of the box. The volume will cancel before we are done.

QFT II Feb 08c

 

Compare state normalizations in non-relativistic and relativistic scattering.

QFT II Feb 08d

 

The box normalization; the decay rate; the phase space volume element

QFT II Feb 08e (overview slide)

QFT II Feb 08f (normalization)

QFT II Feb 08g (decay rate and phase volume)

 

The probability of scattering depends on the number densities of the two particles and their relative velocity. We find relativistic generalizations of these quantities.

QFT II Feb 08h

QFT II Feb 08i

QFT II Feb 08j

 

Pulling it all together:

QFT II Feb 08k

QFT II Feb 08l

QFT II Feb 08m

 

Wednesday, February 10: Mandelstam variables

 

Lecture: Mandelstam variables

 

QFT II Feb 10a

QFT II Feb 10b

QFT II Feb 10c

QFT II Feb 10d

QFT II Feb 10e

QFT II Feb 10f

QFT II Feb 10g

QFT II Feb 10h

QFT II Feb 10i

QFT II Feb 10j

QFT II Feb 10k

QFT II Feb 10l

QFT II Feb 10m

QFT II Feb 10n

 

 

Friday, February 12: Virtual and real particles

 

Lecture: Virtual and real particles

 

QFT II Feb 12a

QFT II Feb 12b

QFT II Feb 12c

QFT II Feb 12d

QFT II Feb 12e

QFT II Feb 12f

QFT II Feb 12g

QFT II Feb 12h

QFT II Feb 12i

QFT II Feb 12j

QFT II Feb 12k

 

 

Wednesday, February 17: Scattering and CoulombÕs law

 

Lecture: Scattering, decay rates, and Coulomb's law

 

QFT II Feb 17a

QFT II Feb 17b

QFT II Feb 17c

QFT II Feb 17d

QFT II Feb 17e

QFT II Feb 17f

QFT II Feb 17g

QFT II Feb 17h

QFT II Feb 17i

QFT II Feb 17j

QFT II Feb 17k

QFT II Feb 17l

QFT II Feb 17m

QFT II Feb 17n

QFT II Feb 17o

 

 

Monday, February 22: QED example: Electron positron annihilation into muon antimuon

 

Lecture: A worked example in QED: differential cross section for electron-positron annihilation giving photo-production of a muon-antimuon pair

 

The complete calculation (with some minor corrections since the lecture):  Muon photoproduction

 

QFT II Feb 22a

QFT II Feb 22b

QFT II Feb 22c

QFT II Feb 22d

QFT II Feb 22e

QFT II Feb 22f

QFT II Feb 22g

QFT II Feb 22h

QFT II Feb 22i

QFT II Feb 22j

QFT II Feb 22k

QFT II Feb 22l

QFT II Feb 22m

QFT II Feb 22n

 

Midterm Exam: Study Compton scattering. The initial state contains one photon and one electron, and the final state contains one electron and one photon. Average over the initial spin and polarization; sum over the final spin and polarization.

a)      Why is there is no first-order diagram?

b)      Draw the two second-order diagrams.

c)       Compute the differential cross-section (at second order) as a function of the angle between the incident photon and the scattered photon. Work in the rest frame of the initial electron.

d)      The result is the Klein-Nishina formula. Consider the low energy limit to derive the Thompson (total) cross-section.

 

 

 

Wednesday, March 3: QED example: Electron-positron (Bhabha) scattering

 

Friday, March 5: QED example: Electron-positron (Bhabha) scattering

 

Monday, March 8: QED example: Electron-positron (Bhabha) scattering

 

The complete calculation:  Electron-positron scattering. The Bhabha cross section.

 

 

Monday, March 22: Questions on the midterm

 

Midterm Exam: Study Compton scattering. The initial state contains one photon and one electron, and the final state contains one electron and one photon. Average over the initial spin and polarization; sum over the final spin and polarization.

 

TodayÕs class was a student-led discussion of setting up the Compton scattering problem.

 

First: Draw and label the Feynman diagrams. Also, various forms of the fermion propagator.

QFT II Mar 22a

 

Write the matrix element for each diagram. Add the matrix elements for each diagram and square. Where the vertex contribution comes from.

QFT II Mar 22b

QFT II Mar 22c

 

There are various ways to write the sum over a pair of polarization vectors. The result is a projection operator, into the two polarization directions. It is probably easiest, in this problem, to leave the sum until the kinematic stage of the problem.

QFT II Mar 22d

 

Rearranging and reducing the squared matrix element to a trace (first term only)

QFT II Mar 22e

QFT II Mar 22f

 

 

 

Wednesday, March 24

Question on the derivation of the Feynman path integral.

QFT II Mar 24a

 

 

Begin: The Standard Model

This pdf (last update April 29, 2010) contains detailed notes on the whole series of lectures below.

These notes are now essentially complete.

 

 

Wednesday, March 24: Yang-Mills gauge theory

 

We want to build an SU(n) gauge theory over flat spacetime. Start with the product of SU(n) with the PoincarŽ group and take the quotient by the product of SU(n) with the Lorentz group. This gives SU(n) and Lorentz fibers over a 4-dim manifold.

Write the Lie algebra of these groups. The Lie algebra for SU(n) can be written using a basis of nxn Hermitian matrices. For the symmetric, off-diagonal ones, we should have subtracted a trace.

QFT II Mar 24b

 

More SU(n) generators; the PoincarŽ Lie algebra

QFT II Mar 24c

 

Generators from the different sub-algebras commute. Introduce connection 1-forms dual to the generators and write the Maurer-Cartan equations.

QFT II Mar 24e

 

The resulting Maurer-Cartan equations. The fiber bundle. After modifying the connection 1-forms, we have the Cartan equations, with horizontal curvature or field strength 2-forms.

QFT II Mar 24d

 

Restrict to flat, torsion-free space and solve the spacetime structure equations. The spin connection may (with Cartesian coordinates) be chosen to be zero, and the components of the solder form may be written as differentials of the Cartesian coordinates. Our base manifold is now Minkowski space. There remains only one field – the SU(n) field strength, defined by the structure equation.

QFT II Mar 24f

 

Expand the field strength and find the components.  Listing the available tensors for writing the action. Some index conventions.

QFT II Mar 24g

 

Lorentz tensors which we can construct from the Dirac matrices and spinors. The Yang-Mills action. The covariant derivative.

QFT II Mar 24h

QFT II Mar 24i

 

Friday, March 26: Yang-Mills action; background on the electroweak interaction

 

Lecture: Action functional for SU(n) Yang-Mills

 

First, some history of the weak interaction: the 4-Fermi interaction. The need for left-handed spinors in weak interactions.

QFT II Mar 26a

QFT II Mar 26b

QFT II Mar 26c

QFT II Mar 26d

QFT II Mar 26e

 

The diagonal generators of SU(n) commute:

QFT II Mar 26f

QFT II Mar 26g

QFT II Mar 26h

 

Now, the Yang-Mills action

QFT II Mar 26i

QFT II Mar 26j

 

Monday, March 29: Tensors, covariant derivative, conserved currents

                             Begin the Standard Model

 

Lecture: Covariant derivative, conserved currents. The Standard Model

 

SU(n) and Lorentz tensors

QFT II Mar 29a

 

The covariant derivative. A derivation must be linear and Leibnitz. We check these, and check covariance. Non-Abelian covariant derivatives require one connection term for each rank of the differentiated tensor; Abelian derivations require an additive weight.

QFT II Mar 29b

QFT II Mar 29c

QFT II Mar 29d

 

The Yang-Mills action. Apply NoetherÕs theorem to find the conserved current.

QFT II Mar 29e

QFT II Mar 29f

QFT II Mar 29g

QFT II Mar 29h

 

The adjoint representation of a semi-simple Lie group.

QFT II Mar 29i

 

The Standard Model: Gauging the symmetry group

QFT II Mar 29j

QFT II Mar 29k

 

The Standard Model: A trial Lagrangian and its various problems

QFT II Mar 29l

QFT II Mar 29m

 

 

Wednesday, March 31: QCD

 

Lecture: Quantum Chromodynamics

 

The antisymmetry of the proton wave function suggests the need for a new symmetry. It must be at least 3 dimensional in order to make three-quark systems into singlet states. We invent SU(3) colors: red, blue, green.

QFT II Mar 31a

QFT II Mar 31b

 

The QCD action and the Gell-Mann form of the su(3) generators.

QFT II Mar 31c

 

Interactions between the gauge particles and the quarks arise from the covariant derivative. Each term annihilates one color and creates another.

QFT II Mar 31d

QFT II Mar 31e

QFT II Mar 31f

 

Cubic and quartic interactions of the gluons.

QFT II Mar 31g

 

Friday, April 2: Spontaneous symmetry breaking and the Higgs mechanism

 

Lecture: The Higgs Mechanism

 

Spontaneous symmetry breaking.

 

The most general renormalizable potential for a scalar field is quartic.

QFT II Apr 02a

 

We can remove some terms by field redefinitions. The resulting potential, if it has the right signs, gives a ŅMexican hatÓ potential, with a peak in the center and symmetric minima

QFT II Apr 02b

 

At low energy, the field settles near a minimum, breaking the symmetry

QFT II Apr 02c

 

We may expand the field around this minimum.

QFT II Apr 02d

 

The Higgs Mechanism

 

Consider the trial action for the electroweak symmetry, SU(2) x U(1). Introduce a complex scalar doublet. As a doublet, it will be coupled to the gauge fields

QFT II Apr 02e

 

Pick a gauge to simplify the form of the Higgs doublet

QFT II Apr 02f

QFT II Apr 02g

QFT II Apr 02h (duplicate)

QFT II Apr 02i (duplicate)

 

Expand the action around the minimum – the vacuum expectation value of the Higgs. There are terms in the action which are quadratic in the gauge fields, that are now multiplied by the vacuum expectation value of the Higgs. These give an effective mass to the gauge particles. Three of the four gauge fields acquire mass, one (the photon) remains massless.

QFT II Apr 02j

QFT II Apr 02k

QFT II Apr 02l (same, except the eraser moved)

QFT II Apr 02m

 

 

Monday, April 5: Parity violation and the Yukawa potential

 

Lecture: Parity violation and the Yukawa potential

 

Answer to question: Proof that the curvature/field strength of a gauged theory is a tensor

QFT II Apr 05a

QFT II Apr 05b

QFT II Apr 05c

 

The gauge fields couple to vector currents. The simplest way to violate parity is to take a linear combination with a pseudovector.

QFT II Apr 05d

 

The V – A interaction is just right. Left and right projection operators

QFT II Apr 05e

 

With a left lepton doublet, a right lepton singlet, and the Higgs doublet we can build the Yukawa potential. When the Higgs takes on its vacuum expectation value (vev), we get mass terms for the fermions

QFT II Apr 05f

 

An improved action for the electroweak interaction.

QFT II Apr 05g

 

Yukawa terms for the quarks. In order for the quarks to change flavor in electroweak decays, we need mixing between the different families. This was first done with the Cabibbo angle for the s and d quarks. Later, Kobayashi and Maskawa predicted a third generation of quarks when they proved that two families is insufficient to give the observed CP violation

QFT II Apr 05h

 

 

Wednesday, April 21: Mass Matrices

 

Lecture: Mass Matrices: The Cabibbo angle, the Kobayashi-Maskawa (CKM) Matrix, and the PMNS Matrix

 

First, two puzzles!

 

I. A computer memory chip of dimensions x by 2x is filled with a checkerboard of squares of side w. The alternate, empty squares provide barriers between bits. We wish to calculate the bit size, w, in terms of x and the total number of bits on the chip, N. We do this calculation in two ways:

QFT II Apr 21a

1. Write the area A of the chip in two ways:   A  =  2x2  =  2N w2    This gives w as x over the square root of N.

2. Let b be the number of bits along x.  Then there are 2b bits along the 2x side. Then the number of bits must be 2N2, and we have 2bw = x. This results in the previous formula, divided by the square root of 2.

WhatÕs wrong?

 

II. A simple calculation showing that x to any power y equals 1. WhatÕs wrong?

QFT II Apr 21b

 

The CKM matrix allows us to give different masses to the different quarks. A similar matrix is required for the leptons and neutrinos.

We now know that there exist mass differences between the neutrinos because electron neutrinos produced by fusion in the sun oscillate into muon and tau neutrinos by the time they reach Earth. It is a simple quantum problem to show that this occurs only if the neutrinos have different masses.

QFT II Apr 21c

 

For the quarks, it was realized early on that the strange quark decays into the up quark. But so far, our model only allows such interactions between two quarks in the same weak doublet.

QFT II Apr 21d

 

Cabibbo solved the problem by building the (u,d) doublet using a linear combination of d and s instead of just the d.

QFT II Apr 21e

 

When CP violation was seen in quark decays, Kobayashi and Maskawa showed that CP violation cannot occur even with a full mixing matrix between the u,d,s and c quarks that were then known. They correctly suggested the existence of a third pair of quarks. It is easier to accommodate the mixing in the Yukawa mass terms than in the interaction term of the Lagrangian.

QFT II Apr 21f

 

We can now write a fairly complete Lagrangian for the standard model. We include a second mixing matrix, shown necessary by Portocorvo and introduced by Maki, Nakagawa and Sakata, for the leptons.

QFT II Apr 21g

 

The number of free parameters of the standard model is largely determined by the CKM and PMNS matrices. There are about 23 parameters:

QFT II Apr 21h

 

Additional terms can be added to the model. For example, parity violating terms such as Joe Slansky described are possible for the quarks. Also, the theory can be supersymmetrized. Frequently, experiments test predictions of the MSSM – the Minimal Supersymmetric Standard Model.

 

 

Monday, April 26: The Feynman Path Integral

 

Lecture: Feynman Path Integral

 

Recall Juan TrujilloÕs lecture from last week, deriving the phase space path integral as the transition amplitude for a particle at (x,t) to arrive at (xÕ,tÕ).

 

Question: Sorting out details of the phase factors:

QFT II Apr 26a

 

The form of the phase space path integral. The volume form becomes an infinite product of integrals in the limit as the time spacing goes to zero.

We consider an action of the general form, L = T – V.

QFT II Apr 26b

 

To evaluate the momentum integrals, we back up a step. With the standard form of the kinetic energy, the momentum integrals are all Gaussian integrals. We complete the square, and the term we add and subtract is the kinetic energy in terms of the velocity.

QFT II Apr 26c

QFT II Apr 26d

 

The result is the path integral in configuration space, together with a divergent normalization. This peculiar factor cancels with a similar factor from the configuration space terms. The phase is the usual action functional.

Now restrict attention to a free particle. We can check that the divergent terms cancel.

QFT II Apr 26e

 

Again we have Gaussians. Each integral involves a pair of terms. The only contribution is a normalization factor, while the neighboring terms coalesce into a similar term with doubled time interval. The factors cancel and the transition amplitude is a pure phase given by the action evaluated along the classical (straight line) path, that is, the phase is HamiltonÕs principal function.

QFT II Apr 26f

 

The path integral for fields becomes a functional integral over all field configurations, often called a Ņsum over histories.Ó

QFT II Apr 26g

 

 

Wednesday, April 28: Path integrals in field theory

 

Lecture: Feynman Path Integral

 

Lecture presentation by Jeff Hazboun.

 

For field Lagrangians, the momentum typically enters quadratically, just as in the particle case. Integration over the momentum gives the usual spacetime action.

QFT II Apr 28a

 

We consider a field action with minimal coupling to a current density as source. We can decouple the source from the fields by first expanding the fields in Fourier series, then making a field redefinition.

QFT II Apr 28b

QFT II Apr 28c

 

The source terms come out of the path integral as a real phase factor, leaving the free particle action in the path integral. Inverting the Fourier transform of the currents, we find the propagator of the field.

QFT II Apr 28d

 

For gauge field theories, we must avoid counting the non-physical modes of the field corresponding to gauge transformations. By writing the functional integral as an unconstrained field integral times a delta function that imposes a general, unspecified gauge condition, we can extract a factor equal to the volume of the gauge group.

QFT II Apr 28e

 

A change of variable gives a factor of the Jacobian – the determinant of the derivative of the gauge with respect to the gauge constraint.

QFT II Apr 28f

QFT II Apr 28g

 

Friday, April 30: Mass estimates from lattice computations

 

A very polished powerpoint presentation by Nathan Piepgrass.

 

 

Thanks (alphabetically) to Jeff Hazboun, Joe Slansky, Juan Fernandez, Juan Trujillo, and Nathan Piepgrass!

 

 

 

 

Course Information

Syllabus

 

QFT I Notes

 

QFT II Notes

 

Seminar