Michael Weiss: Roll over  Schmotons  John Baez: Perhaps the

John Baez: The Most Enlightening Set of Names

Before I dig into the business of working out what coherent states look like in the basis of eigenstates of the harmonic oscillator Hamiltonian, let me comment on one thing:

Hmmm. John Baez promised that the plain old harmonic oscillator would have something to do with E and B. Hey wait, the energy for the harmonic for the harmonic oscillator is (p² + q²)/2; for the electromagnetic field, (E² + B²)/2. Coincidence? You be the judge.

Of 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. B is sort of like the ``position'' and E is sort of like the ``momentum''. But let's leave the details for later on.

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

The first should really be called the ``wave'' representation. In this representation we diagonalize the position operator, thinking of the state as a function on position space.

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

Z = q + ip

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

Everyone who studies quantum mechanics learns about the first two representations. The third, while in many ways the most beautiful, is somewhat less widely known. We won't get into it much here. The only reason I mention it is that it's lurking in the background whenever you relate quantum mechanics to the classical phase space and drag in the Z = q + ip trick.

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.

3/10/1998