A schedule of content and exercises will gradually appear below.
Assignments/Exercises
The first hand-in assignment consists of Exercises 3.5, 5.1, 8.7, 9.3 and
10.2 from the book by Srednicki. These basically repeat almost everything
we did for a real scalar field, but replacing that field with a complex
scalar field. Deadline is November 27 for now (if this turns out to be
much harder than expected we may adjust)
Notes
Notes may appear here if necessary.
Weekly comments
Week 1:
Notes:
After some motivational remarks, we explored the idea of finding a
(special) relativistic generalization of
the Schroedinger equation for a single particle. To do this we recalled
some material from special relativity, after which we "stumbled" upon the
Klein-Gordon and Dirac equations. These actually did not quite achieve our
goals, but turn out to be the classical field equations for spin 0 and
spin 1/2 particles (in the same way that the electromagnetic field is the
classical field equation for photons). We will need to quantuize these
classical field theories to deal with particles properly.
Reading:Srednicki section 1 (and anything else needed to understand that)
Week 2:
Notes:
We introduced Fock space and creation and annihilation
operators. We then reformulated the
non-relativistic quantum mechanics of
any number of particles as a field theory in terms of the creation and
annihilation operator fields. (There is a creation operator associated to
every point in space - hence we can think of these together as the
creation operator
quantum field). We then started the quantization of the Klein-Gordon
field. We found an Lorentz invariant action and Lagrangian density for the
field which reproduces the K-G-equation as the condition on the field that makes the
action's first order variation vanish. We found canonical momenta for the
field operators, found the Hamiltonian and quantized the fields by introducing
canonical equal time commutation relations.
Reading:Srednicki, rest of section 1, section 2, section 3 (and anything
else needed
to understand that)
Week 3:
Notes:
We further examined the quantization of the scalar K-G field and showed
that the Fourier coefficients of the field can, after quantization, be
interpreted as creation and annihilation operators for particles, which
are bosons due to the canonical equal time commutation relations.
The
Hamiltonian then simply gives these particles their relativistic kinetic
energies. We remark on the fact that our quantization procedure would not
have worked if the field operators were anticommuting rather than
commuting - this is a special case of the spin-statistics connection (we
briefly discussed that but defer a proper treatment to later, when we
have actually seen some particles with spin). We then discussed the LSZ
reduction formula, which reduces the amplitude for a scattering process to
an expression involving a correlation function of the field at different
positions. This should motivate us to calculate these correlation
functions. We then started to show how transition amplitudes in quantum
mechanical systems can be expressed in terms of path integrals (TBC)
Reading:Srednicki, section 3 (read it again!) section 4 (have a glance)
section 5 (have a longer glance, maybe find another treatment too)
section 6 (read and re-read carefully)
Week 4:
Notes:
We looked at how path integrals can be used to calculate transition
amplitudes and correlation functions in quantum mechanics. We considered
the general case of a path integral over paths through phase space and
showed how this reduces in simple cases to a path integral over paths
through configuration space. We showed how path integrals naturally
calculate time ordered correlation functions and how these correlation
functions can be written as functional derivatives of a partition
function with sources. We also considered how to implement different
initial and final states, particularly how we can get the initial and
final state to be the ground state (later: the vacuum).
Reading:Srednicki,
section 6 (re-read and re-re-read carefully), section 7. Start section 8
if you dare.
Week 5:
Notes:
Reading:Srednicki, section 3 (read it again!) section 4 (have a glance)
section 5 (have a longer glance, maybe find another treatment too)
section 6 (read and re-read carefully)
Week 6:
Notes:
Reading:Srednicki, section 3 (read it again!) section 4 (have a glance)
section 5 (have a longer glance, maybe find another treatment too)
section 6 (read and re-read carefully)
Week 7:
Notes:
Reading:Srednicki, section 3 (read it again!) section 4 (have a glance)
section 5 (have a longer glance, maybe find another treatment too)
section 6 (read and re-read carefully)