Heisenberg Algebra and Fock Models

The Heisenberg Algebra

Heisenberg (Oscillator) Algebra

The Heisenberg algebra or Oscillator Algebra is a complex Lie algebra with a basis and Lie bracket defined by Here is the reduced Planck constant, and is often set to in mathematical treatments.

The Heisenberg algebra exists by the following construction:

Proposition

A central extension of the abelian algebra of trigonometric polynomials on the circle, i.e., functions with finite Fourier series, forms a Heisenberg algebra through the Lie bracket where is a central element.

Proof Let us denote the standard basis elements as . Then Setting and gives the desired isomorphism.

Introduce the bosonic Fock space , the space of polynomials in infinitely many variables. Given , define the representation of through for all and some . Usually we set for all .

Proposition

The representation is irreducible for all and such that for all .

Proof Any polynomial in can be reduced to a constant by applying for sufficiently large . Hence, every nonzero subrepresentation must contain the constants. By repeatedly applying ​, one generates all other polynomials, showing that the subrepresentation is the entire space .

Oscillator Representation of

Subalgebra in

In , we can define a subalgebra which is spanned by where if and otherwise (this is called the normal ordering). Then this subalgebra is isomorphic to the Virasoro algebra through the map , and .

Oscillator Representation of

We now define a family of representations of for by for all . This is called the oscillator representation of .

Proposition

The operators satisfy

1. ;
2. ;
3. If and , then .

Therefore is a highest weight representation of with central charge and highest weight . Moreover, it is unitary if .

Proof

Corollary

If and , then the irreducible highest weight representation of is unitarizable. In general, if and for some , then is unitarizable.

Proof Since is unitarizable for all , given and , let and . Then and is a unitary highest weight representation of with central charge and highest weight .

is also unitarizable for and , but we do not have an explicit oscillator construction yet. This is still an open problem in mathematics.

In fact, the irreducible highest representation