Knowlpedia

Jacobian X(T) in Shale's Gaussian Setup

The Radon–Nikodym derivative of the transformed Gaussian measure n(T) with respect to n

For T∈rGL(M)T\in rGL(M), Shale defines a new measure n(T)n(T) via pullback on L∞L_\infty, and sets

X(T)=dn(T)dn,X(T)=\frac{dn(T)}{dn},

the .

Remarks

Key properties (paper use):

  • X(T)1/2∈L2(M,n)X(T)^{1/2}\in L_2(M,n) and appears in the unitary implementer U(T)\mathfrak U(T).
  • When ∣T∣∈GL(M)1|T|\in GL(M)_1, X(T)X(T) has an explicit "Gaussian quadratic form" formula.
Examples
  • In finite dimensions, X(T)X(T) reduces to ∣det⁡T∣|\det T| times an exponential factor.