P501A Quantum Theory & Quantum Fields

Instructor: Adam Ritz
Office: Elliott 103
Office hours: Tuesday afternoons
Email: aritz@uvic.ca
Lectures: Directed Studies format

This is a one semester graduate course in introductory quantum field theory, focussing on quntum electrodynamics:
  • Quantization of free scalar fields
  • Quantization of free spinor fields
  • Interacting fields: Feynman diagrams and perturbation theory 
  • Quantum electrodynamics: basic scattering processes

See the course syllabus for further details.

Preparation: A working knowledge of upper-level quantum mechanics, and various aspects of classical field theory (specifically electrodynamics) will be assumed, as covered in the fall semester graduate courses PHYS 500A and PHYS 502A.

This is a graduate course covering introductory quantum field theory and the course syllabus is listed below:

  • Introduction and Overview
    • quantum mechanics, special relativity and causality
  • Free Scalar Fields
    • Canonical Quantization
    • Vacuum, Particles, Fock space
    • Propagators
  • Interacting Fields
    • Interaction Picture
    • Wick's Theorem
    • Feynman Diagrams
    • Tree-level Scattering, Cross sections, Decay rates
    • *Correlators
  • Fermionic fields
    • Spinor Representations and Fields
    • Dirac Action, Dirac Equation
    • Solutions: Dirac, Weyl and Majorana fields
    • Fermionic quantization
    • Propagators and Feynman Rules
  • Quantum Electrodynamics
    • Quantization of the EM field; Gauge fixing
    • Coupling to matter and QED
    • Feynman rules and Scattering
    • *A first look at radiative corrections
*Covered if time permits

There is no required textbook for this course. However, the topics to be studied are covered very well in a number of places. For example, here are some good modern textbooks covering quantum field theory:

  • An Introduction to Quantum Field Theory, M. Peskin and D. Shroeder
  • Quantum Field Theory, M. Srednicki

The course will broadly follow part I of Peskin & Shroeder, but similar material is covered by Srednicki, and a number of other QFT texts are available in the library. Keep in mind that QFT is a large subject, and all textbooks will cover more material than this course.

An older but still useful textbook, which covers a number of calculational details:

  • Quantum Field Theory, F. Mandl and G. Shaw

The following important reference text can be useful for understanding many of the subtleties 

  • The Quantum Theory of Fields, I and II, S. Weinberg
Further online material for the course, including:
  • course notes
  • assignment sheets
  • sample solutions

will be available at the PHYS 501A course page in CourseSpaces
The course will assessed according to the following three components:
  • Assignments: 40%
  • Take-home Final: 30%
  • Project: 30%

There will be ~5 assignments during the semester. The first 4 will take the conventional form, and you will generally have about 2 weeks to complete each of them. They form an integral part of the course, used to expand on the material in the lectures in various ways. The final assignment will be in the form of a take-home exam.

The final assessment component will be an individual project, requiring some background research, and leading to a 30 minute presentation during the April exam period. Possible projects will be outlined on this website as the semester progresses. The basic idea will be to review a classic paper or topic in quantum field theory, describe the results and explore the more recent extensions and applications. 

The final grade follows the University's universal conversion between letter and percentage grades:

  • A+  (90-100)
  • A    (85-89)
  • A-   (80-84)
  • B+  (77-79)
  • B    (73-76)
  • B-   (70-72)
  • C+  (65-69)
  • C    (60-64)
  • D    (50-59)
  • F    (0-49)

If the application of this scheme would result in grades deemed by the instructor to be inconsistent with the University's grading descriptions (to be found on p.38 of the current Undergraduate Calendar), percentages will be assigned which are consistent with them.

NB: Use of calculators in exams

On all examinations the only acceptable calculator is the Sharp EL-510R. This calculator can be bought in the Bookstore for about $10. DO NOT bring any other calculator to the examinations.

The final assignment for the course will be an individual project focussed on a particular process or calculation involving QFT. Assessment will be via a 30 (25+5) minute presentation to the class during a mini-conference to be held during the exam period. You should try to prepare your seminar carefully so that it covers the basic physical picture, some of the relevant calculational details, and that you focus on what you found to be the most important or interesting aspects. (Here are some helpful guidelines on preparing and presenting talks.)

The precise choice of topic is open, provided it relates directly to one or more aspects of quantum field theory, but should be reasonably well-defined. Some possible topics are outlined below, drawn from a number of areas of physics in which quantum field-theoretic tools play an important role: particle physics, condensed matter physics, cosmology, or more formal aspects of QFT (*'d topics are somewhat more technical). Of course, you are free to choose something else - just come and talk to me to get the topic approved.

  • Unitarity and W-boson scattering

The 2->2 scattering of massive vector bosons exihibits an interesting scaling with center-of-mass energy, and can be used to motivate the need for new physics (e.g. a Higgs boson) at high scales in order to preserve unitarity (i.e conservation of probability). Issues to discuss:

  • (A few) details of the 2->2 scattering cross-section
  • Implications for the Standard Model via WW->WW scattering

Some references:

  • The Higgs Hunter's Guide , John F. Gunion, Howard Haber, Sally Dawson, Gordon Kane, Google Books

  • Renormalization in QED(*)

Various loop amplitudes in QED are naively divergent if the momentum in the loop is allowed to be arbitrarily large. However, all these divergences can be accounted for by carefully defining the physical (renormalized) mass and charge of the electron (and positron). Issues to discuss:

  • Regularization of amplitudes
  • Renormalization constants in QED, renormalized mass and charge, and the RG equations

Some references:

  • An Introduction to Quantum Field Theory , Peskin and Schroeder

  • Fermi Surfaces and Field Theory

In condensed matter systems, the Fermi surface plays an important role in determining the dominant low-energy dynamics and thus the physical properties of metallic and doped semiconductor materials. Many of these properties can be understood field-theoretically through the way the Fermi surface controls the relevance or irrelevance of particular electronic interactions. Issues to discuss:

  • Properties of the Fermi surface, relevant/marginal/irrelevant interactions
  • Implications for specific materials.

Some references:

  • Field theory of graphene

Graphene, a one-atom thik layer of carbon, has attracted a lot of attention in recent years (including a Nobel Prize) due to it varied properties and applications. The physics of electrons in graphene is interesting as it exhibits features such as a relativistic (Dirac-type) dispersion relation. Issues to discuss:

  • Properties of the electron dynamics in graphene, Dirac points
  • Applications.

Some references:

  • Quantum field theory in graphene,I.V. Fialkovsky and D.V. Vassilevich, arXiv:1111.3017

  • Parity and Forward-Backward Asymmetries

Forward-backward asymmetries in e.g. b-bbar production provide a nice illustration of parity violation in the Standard Model, due to the interplay of vector and axial-vector (e.g. b-bbar-Z) couplings. These asymmetries were well measured at LEP for example. Issues to discuss:

  • Details of the contributing tree-level amplitudes, and definition of the asymmetry
  • Implications for the Standard Model and its chiral structure.

Some references:

  • Neutrino Oscillations and the MSW Effect

Neutrino flavour eigenstates undergo order-one oscillations over long baselines. This effect is particularly interesting in matter, where the MSW effect may become important. This project should discuss:

  • The origin of neutrino oscillations in vacuum
  • The MSW effect

Some references:

  • Neutrinoless Double Beta Decay

All charged leptons in the Standard Model are Dirac fermions. However, neutrinos may also have a Majorana mass term and one of the few experimental observables that can distinguish between these possibilities is neutrinoless double beta decay. The distinction is important for a number of reasons, and thus a number of experiments are currently in development aiming to probe this signal with increasing levels of precision. This project should discuss:

  • The distinction between Dirac and Majorana neutrinos, and lepton number violation
  • Probes for neutrinoless double beta decay, possible sources of a signal, and its implications

Some references:

  • Double Beta Decay, Majorana Neutrinos, and Neutrino Mass , F. Avignone, S. Elliott, J. Engel , 0708.1033

  • Lepton Flavor Violation

In the Standard Model (with massless neutrinos), there is a global symmetry leading to individually conserved lepton numbers for each flavour (electron, muon, and tau). Neutrino masses break this symmetry and allow for lepton flavor violation (LFV) but at a tiny level. However, new physics processes can lead to much larger sources of LFV making it a prominant channel in which to probe the Standard Model. This project should discuss:

  • Lepton flavor violation, and why it is suppressed in the Standard Model
  • New physics sources, e.g. extra Higgs bosons

Some references:

  • Lepton flavor violating decays of supersymmetric Higgs bosons , A. Brignole and A. Rossi, hep-ph/0304081

  • Grand unification and proton decay

The scale-dependence of the three gauge couplings in the Standard Model, when extrapolated to high energies, suggests that they nearly meet at a `unification' scale of about 10^16 GeV. This has led to speculaton about the existence of a grand unified theory of particle interactions at such high scales. This requires new degrees of freedom, which when integrated out generate irrelevant interactions which have physical consequences. This project should address:

  • Brief discussion of the scale-dependence of couplings, and the new degrees of freedom required in a GUT
  • Predictions for irrelevant interactions at low energies with experimental consequences e.g. proton decay

Some references:

  • The hierarchy problem

The scalar or Higgs sector of the Standard Model is highly sensitive to quantum corrections. The UV sensitivity of the Higgs mass in particular is dubbed the "hierarchy problem", and is often used to motivate new physics at the electroweak scale. This project should discuss the origin of the hierarchy problem, with regard to the nature of the Higgs mass as a relevant operator:

  • Origin of the hierarchy problem, i.e. Higgs mass term as a relevant operator:
  • Possible implications of the hierarchy problem, e.g. expectations for new physics

Some references:

  • The Higgs Hunter's Guide , John F. Gunion, Howard Haber, Sally Dawson, Gordon Kane, Google Books

  • QFT in de Sitter space and CMB fluctuations

Our treatment of canonical quantization of a scalar field can be repeated in curved space. In the context of inflation, this spacetime is de Sitter space and one finds that quantum fluctuations have a characteristic size, which can be probed by observations of the large-scale amplitude of temperature fluctuations in the CMB (measured by the COBE satellite in the 90's). This project should discuss:

  • The calculation of the scale of the fluctuations in de Sitter
  • (Briefly) how this leads to the temperature fluctuations in the CMB

Some references:

  • TASI Lectures on Inflation, W. Kinney, 0902.1529

  • Lamb Shift

The measurement of the energy splitting between the 2s and 2p states of atomic hydrogen by Lamb and Retherford is one of the key pieces of data that led to the development of quantum electrodynamics in the late 1940's. This effect arises at 1-loop level, and was first explained by Bethe. Issues to discuss:

  • Some details of the calculation (in modern QFT language)
  • Origin of the `Bethe log'

Some references:

  • Techniques in analytic lamb shift calculations , U. Jentschura, physics/0509143

  • Critical points and conformal field theories(*)

Second-order phase transitions occur at scale-invariant critical points, which determine the scaling behaviour of `nearby' QFTs. Under rather weak assumptions, scale-invariance implies conformal invariance and fixed points are described in general by quantum field theories constrained by conformal invariance - `conformal field theories' (CFTs). Issues to discuss:

  • The symmetries, and constraints imposed on correlation functions, of CFTs
  • CFTs in 2D, e.g the CFT(s) associated with the Ising fixed point (and its generalizations), and applications

Some references:

  • Conformal Field Theory, P. Di Francesco, P. Mathieu, and D. Senechal (first part only)
  • Statistical Field Theory, Vol II , C. Itzykson and J.-M. Drouffe

  • MHV amplitudes(*)

While Feynman diagrams provide us with a powerful tool for perturbative quantum field theory, there are situations where the final result of the calculation is vastly simpler than the intermediate stages. This is the case with a special class of multi-particle gluon (or photon) scattering amplitudes called maximally helicty violating (or MHV) amplitudes. This suggests that there is further structure which is hidden by the Feynman diagram expansion. This project should discuss:

  • The use of the helicity basis
  • Definition of helicity and MHV amplitudes and some simple examples

Some references:

  • Multi-Parton Amplitudes in Gauge Theories, M. Mangano and S. Parke, hep-th/0509223

  • Jets and collinear singularities(*)

The amplitude for radiating a soft (i.e. low energy) photon (or gluon) from a charged particle appears to diverge as the energy tends to zero. This infrared divergence is the basis of the formation of jets. This project should discuss:

  • The origin of collinear singularities in QED
  • (Qualitatively) how they lead to jets (in the context of QCD)

Some references:

  • Peskin & Schroeder (Sec 6.5 and 17.1,17.2)

  • Fermion/Boson duality in two-dimensional QFTs(*)

In 1+1 dimensions, the scalar and spinor representations of the Lorentz group are not so different (as one might imagine because there are no `rotations' in one spatial dimension). This leads to the possibility of an equivalence between theories of bosonic and fermionic fields, as exemplified by the case of the sine-Gordon (bosonic) and Thirring (fermionic) models. This project should address:

  • Discussion of fermions in 1+1D, and of the sine-Gordon and Thirring QFTs.
  • Some details of the mapping between the two theories.

Some references:

Additional Quantum Mechanics topics

Some additional topics are listed below, focussing more on quantum mechanics, with links to the original papers. As above, you're encouraged to dig further into the literature and explore the subsequent implications and applications.