Proof
This proof uses a simplified model of the Dirac sea and follows the proof in Cameron (13.3) which is attributed to Richard Borcherds. It treats the case where the power series are formal. For the analytic case, see Apostol. The Jacobi triple product identity can be expressed as
A level is a half-integer. The vacuum state is the set of all negative levels. A state is a set of levels whose symmetric difference with the vacuum state is finite. The energy of the state is
and the particle number of is
An unordered choice of the presence of finitely many positive levels and the absence of finitely many negative levels (relative to the vacuum) corresponds to a state, so the generating function for the number of states of energy with particles can be expressed as
On the other hand, any state with particles can be obtained from the lowest energy particle state, by rearranging particles: take a partition of and move the top particle up by levels, the next highest particle up by levels, etc.... The resulting state has energy, so the generating function can also be written as
where is the partition function. The uses of random partitions by Andrei Okounkov contains a picture of a partition exciting the vacuum.
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