Hmmm. John Baez promised that the plain old harmonic oscillator would have something to do withOf course it's no coincidence; the electromagnetic field is just a big bunch of harmonic oscillators, one for each ``mode'', and the formula for the energy is just like that for the harmonic oscillator.EandB. Hey wait, the energy for the harmonic for the harmonic oscillator is(p; for the electromagnetic field,^{2}+ q^{2})/2(. Coincidence? You be the judge.E^{2}+B^{2})/2

What we're doing now is like finishing school. Everyone bumps into the harmonic oscillator in quantum mechanics, but rather few get to see its full beauty. There are three main representations to learn:

- 1.
- the Schrödinger representation
- 2.
- the Heisenberg (or Fock) representation
- 3.
- the Bargmann-Segal representation

Similarly, the second should be called the ``particle'' representation.
In this we diagonalize the energy, thinking of the state as a linear combination
of states |* n*> having n ``quanta'' of energy in them.

The equivalence of these first two representations is the basis of ``wave-particle duality'' in quantum field theory.

The third representation could be called the ``complex wave'' representation.
At least that's what it's called in the book *Introduction to Algebraic
and Constructive Quantum Field Theory*, where the first representation
is called the ``real wave representation''. In some rough sense, this representation
diagonalizes the creation operators. Of course, not being self-adjoint,
the creation operators can't be diagonalized in the usual sense. There
is a nice substitute, however. In the complex wave representation, we think
of phase space as a complex vector space using the trick

and then think of states as *analytic* functions on phase space. Then
the creation operator becomes multiplication by *Z*.

Perhaps this would be the most enlightening set of names for these three representations:

- 1.
- the configuration space representation
- 2.
- the particle representation
- 3.
- the phase space representation

Anyway, if you master these three basic viewpoints on the harmonic oscillator, it's a snap to generalize to quantum field theory, at least for free quantum fields, which are just big bunches of harmonic oscillators.