Lecture 33.
Handouts:
24. Homework 8 Solutions
25. Exam 2 (Takehome)
We discussed the Fock space of the quantized Klein-Gordon field, then canonical quantization of other boson fields.
The Fock space, which contains all physical states of the quantum
field, is spanned by the energy eigenstates. We constructed these as
in the simple harmonic oscillator case, building them up by applying
creation operators (which we have for all momenta 
) to the
vacuum (that is, to the state annihilated by all creation operators).
Since we interpret the quantum with energy 
as a
particle of momentum , we interpret the vacuum as a
no-particle state, and the energy eigenstates created by multiple
applications of creation operators, as multiparticle states. We set
the normalization of our eigenstates, finding (because of the simple
harmonic oscillator commutation relations) that our tower of states
goes up forever; that is, we always obtain a physical state with more
particles by applying a creation operator. This is unlike the angular
momentum case, where (for each j) we always found a top state that
could be raised no further, as the raising operator annihilated it.
Since we can have multiple particles in a particular state ,
the particles we have obtained by canonical quantization are
necessarily bosons.
We then discussed canonical quantization of the complex Klein-Gordon field. Because the field can be complex, complex conjugate plane wave solutions need not have exactly conjugate coefficients. The extra set of allowed coefficients become, on quantization, creation operators for a whole new set of particles. We showed that these new particles have the same mass, but opposite Noether charge (associated with the field's phase symmetry), to the original particles. Thus a complex fields automatically gives rise to quantum states with arbitrary numbers of particles and antiparticles.
We then discussed, for completeness, the electromagnetic field. Here
gauge transformations greatly complicate the story. To uniquely
associate the gauge field 
with a physically observed 
and 
field, we chose a gauge which reduced to only
two transverse components (transverse is a fancy word for orthogonal
to the direction 
of motion). These components could be taken
to be right and left polarization of the light. With that done,
canonical quantization goes through as usual, giving a Fock space of
multiphoton states of both polarizations, with photons being bosons.
Thus the intuitive picture we have developed of fields and their
particle quanta applies to photons as well. Had we not chosen a unique
gauge first, we would have obtained more types of photons than are
physical, as an artifact of including more field components than are
physical. However, in choosing a unique gauge -- and getting a
quantum theory with only physical states -- we gave up both the gauge
invariance and Lorentz invariance of our quantized theory. Nowadays,
we actually use another way of quantizing the electromagnetic field,
which gives up neither.
-- KB