Knowlpedia

Duality Transform D (Segal)

Unitary map identifying bosonic Fock space with Gaussian L₂(M,n)

Segal's duality transform D:S(H)→L2(M,n)D: S(H)\to L_2(M,n) is a unitary identifying the Fock space picture with the Gaussian L2L_2 picture.

Remarks

Key properties (paper use):

  • Transports the Fock–Cook operators to L2(M,n)L_2(M,n): R(z)=D(C(z)+C∗(z))∼D−1R(z)=D(C(z)+C^*(z))^{\sim}D^{-1}.
  • Makes Q(x)Q(x) a multiplication operator and P(x)P(x) its Wiener transform conjugate.
Examples
  • In 1D, this corresponds to the Hermite-function realization of the oscillator basis.