The Weil Representation

The Weyl Representation

The Weyl Representation

For some , define the Weyl representation of on the bosonic Fock space :

where is some normalization constant, is the second quantization of , and and are defined as follows: for some orthonormal basis of , and , are the matrix elements of and w.r.t this basis.

Note that this definition is valid because both and are Hilbert-Schmidt, so

Proposition

For any , we have the following intertwining property holds:

Proof It is equivalent to show that Since we assume that , there are no shifting terms (i.e., ) To compute the left hand side, we need to compute the following six conjugates for any :

1. ;
2. ;
3. ;
4. ;
5. ;
6. .

First consider . By the BCH formula, we know that Consider the commutator , for any there holds Let for some , then we have where the second last equality holds because is symmetric. Therefore, by linearity, we have which commutes with any creation operator, hence the higher order commutator terms in the BCH formula vanish. So On the other hand, because creation operators commute with each other.
5 and 6 are completely similar, For the second quantization , on each monomial , we have Therefore, , and taking adjoints on the both sides gives .
Finally, we can derive the conjugation of :