Knowlpedia

Wiener Transform W

A unitary transform on Gaussian L₂ intertwining T with T^{*-1}

The Wiener transform WW is a unitary on L2(M,n)L_2(M,n) defined (on polynomials) by

(Wf)(x)=∫f(2 y+ix) dn(y).(Wf)(x)=\int f(\sqrt2\,y+i x)\,dn(y).
Remarks

Key property (paper use):

  • Intertwining (Cor. 3.1.1): W U(T) W−1=U(T∗−1)W\,\mathfrak U(T)\,W^{-1}=\mathfrak U(T^{*-1}).
  • In §4, this yields P(x)=WQ(x)W−1P(x)=WQ(x)W^{-1} for the Fock-Cook field operators.
Examples
  • In 1D, under an explicit unitary identification, WW becomes the Fourier transform.