Rotation-invariant operators on white noise functionals (Q811508)

A full description of rotation-invariant operators acting on the white noise functionals and a group-theoretic characterization of the infinite dimensional Laplacians (Gross Laplacian and the number operator) are obtained. These results make a remarkable contrast with the classical result for a finite dimensional case. The discussion is based upon the so-called white noise calculus which has been developed as an infinite dimensional analogue of Schwartz' distribution theory. Let \(T\) be a Riemannian manifold with a smooth measure \(\nu\) and put \(H=L^ 2(T,\nu)\). Let \(A\) be a positive selfadjoint operator in \(H\) with \(A^{-1}\) being of Hilbert-Schmidt type. Then in a standard way we obtain a Gelfand triple: \(E=\bigcap_{p=0}^ \infty\text{Dom}(A^ p)\subset H\subset E^*\). Let \(\mu\) be the standard Gaussian measure on \(E^*\) and put \((L^ 2)=L^ 2(E^*,\mu)\). Since the second quantized operator \(\Gamma(A)\) becomes a selfadjoint operator in \((L^ 2)\), we obtain a new Gelfand triple: \((E)=\bigcap_{p=0}^ \infty\text{Dom}(\Gamma(A)^ p)\subset(L^ 2)\subset(E)^*\). Let \(O(E;H)\) be the infinite dimensional rotation group, namely, the automorphism group of the Gelfand triple \(E\subset H\subset E^*\). An operator \(\Xi:(E)\to(E)^*\) is called rotation-invariant if \(\Gamma(g)^*\Xi\Gamma(g)=\Xi\) for all \(g\in O(E;H)\). For \(t\in T\) let \(\partial_ t\) denote the Gâteaux derivative along the delta function \(\delta_ t\in E^*\). It is a continuous derivation on \((E)\) (often called an annihilation operator) and the adjoint \(\partial_ t^*\) (a creation operator) is a continuous operator on \((E)^*\). It is known that the Gross Laplacian and the number operator are expressed as follows: \[ \Delta_ G=\int_ T\partial_ t^ 2dt, \qquad N=\int_ T\partial_ t^*\partial_ t dt, \] which are obvious analogies of the finite dimensional Laplacian: \(\sum_{j=1}^ n\partial_ j^ 2=-\sum_{j=0}^ n\partial_ j^*\partial_ j\) when it acts on \({\mathcal S}(\mathbb{R}^ n)\). The main results are: (1) Every rotation-invariant operator is generated by \(N\), \(\Delta_ G\) and \(\Delta_ G^*\). (2) If an operator on \((E)\) is rotation-invariant, it is generated by \(N\) and \(\Delta_ G\). The proof is divided into two steps: (i) Establishment of an integral expression of an operator \(\Xi\) in terms of \(\partial_ t\) and \(\partial_ t^*\) with distributions as integral kernels. (ii) Characterization of rotation-invariant distributions. In the finite dimensional case the multiplication by \(\sum_{j=1}^ n x_ j^ 2\) is also \(O(n)\)-invariant. The white noise analogy is given by \[ \Phi(x)=\int_ T\colon x(t)^ 2\colon dt, \qquad x\in E^*, \] where \(\colon\cdot\colon\) is the Wick-ordring (or renormalization). Since \(\Phi\in(E)^*\), it is regarded as a multiplication operator form \((E)\) into \((E)^*\). However, contrary to the finite dimensional case, this operator is related to the infinite dimensional Laplacians: \(\Phi=2N+\Delta_G+\Delta_G^*\).