Lecture 32.

To develop our observations from last lecture, about the quantum
field's Hamiltonian being a sum of harmonic oscillator Hamiltonians,
we put our field in a box with periodic boundary conditions. This
makes the set of allowed plane wave momenta k discrete, so that all
our momentum space integrals become sums. Particularly, this makes our
commutation relations for a_k and a_k^\adjoint go from being a
continuum generalization of harmonic oscillator commutation relations,
to being exactly the commutation relations [a_k, a_k'^\adjoint ] =
\hbar  \delta_{kk'} between creation and annihilation operators of
independent oscillators.

Without the Dirac delta functions obscuring our view, we more easily
see the interpretation discussed last time: that a_k annihilates a
particle of momentum k; a_k^\adjoint creates a particle of momentum k;
and N_k gives the number of particles with momentum k. We also saw
that the commutator term in the Hamiltonian corresponds to a sum over
the zero point energy of each oscillator mode k. This contribution to
H is present in every energy eigenstate --- which means that we can
never observe it, as we can only observe the energies of states in
comparison with some reference state whose energy we can measure. We
thus discard it --- which is equivalent to setting the energy of the
lowest energy state, the ``vacuum'', to zero.

We gave the following intuitive picture for what is going on here: a
classical field \phi has many different Fourier (or ``normal'') modes,
each of which describes a classical vibration of the field. Each mode
evolves classically in time *exactly* like a simple harmonic
oscillator of frequency \omega_k. Under canonical quantization, each
oscillator normal mode becomes quantized, giving its own harmonic
oscillator tower of excited states, labeled by the number n_k and
separated by the energy quantum \omega_k. This energy quantum is
precisely a particle with momentum k. Thus particles arise inevitably
in quantum field theory, as the quanta (or excitations) of the
classical field. Examples you are familiar with (as an axiom perhaps,
without seeing the full theory relating them) are photons as the
quanta of the electromagnetic field, and phonons as the quanta of a
lattice displacement field. For lattice displacement, we showed
earlier that the continuum limit gives a Klein-Gordon Lagrangian, with
two caveats: 1) the speed of the phonon is not generally c, the speed
of light, but depends on the parameters of nearest-neighbor
interactions; and 2) phonons --- or quantized vibrational modes ---
generally have much more going on than the quadratic interactions we
included. These complications lead (effectively) to a k-dependent
mass.  Note that in applying our machinery of canonical quantization
to lattice vibrations we are quantizing a nonrelativistic field: while
our motivation in this course has been to find a theory that is both
relativistic and quantum, canonical field quantization is a tool that
is widely applied in the nonrelativistic limit as well, particularly
for condensed matter systems where nonzero temperature makes
multiparticle states important.

We then returned to the continuum limit. We identified the commutator
term in H as the total zero point energy of all oscillator modes
k --- which becomes infinite for continuous k. Again,
we discard it in setting the zero of the energy. We discussed how the
operation of ``normal ordering'' --- or always reordering terms so
that all annihilation operators fall to the right of all creation
operators, *neglecting commutators* --- always sets the
expectation value of an observable to zero in the vacuum, since for
any a_k^\adjoint there must be an a_k further to
the right, which annihilates the vacuum.

We then began a discussion of the quantum field's Hilbert space ---
called Fock space --- which we will develop next time.

--- 
KB