Entropy and the Segal-Bargmann transform

Abstract

The Segal-Bargmann transform A is a unitary isomorphism of the quantum mechanical configuration space L2(Rn,ν) onto the Segal-Bargmann, or phase, space HL2(Cn,μ) of entire functions square integrable with respect to μ. Here μ and ν are Gaussian measures, and for z∈Cn and measurable f:Rn→CAf(z) :=∫Rn dν(t)f(t)exp(−z2/2+√z⋅t) if the integral converges. Af is defined for all f∈Lp(Rn,ν) if 1<p⩽∞, but A does not preserve Lp structure if p≠2. Specifically, it is proven here that A is a compact map from Lp(Rn,ν) to Lq(Cn,μ) if 1⩽q<2 and p>1+q/2 (which implies p>q) and in this case ‖A‖p→q⩽((1+q/2−p′q/2)(1−q/2))−n/2q where p′ is the dual of p. Moreover, a calculation with test functions shows that ‖A‖p→q=∞ if q>2 or p<1+q/2 (which is implied by p<q). Using an improved operator norm estimate (analogous to the Hausdorff-Young inequality) we prove Hirschman type entropy inequalities for f∈L2(Rn,ν), (p−1−2−1)S(f )⩽(q−1−2−1)S(Af )+nCp,q‖f‖22, where 1⩽q<2, p>1+q/2, Cp,q is given explicitly as a function of p and q and S(g) is the entropy of g. Examples given here show A is not entropy preserving. So these inequalities provide constraints on the change in entropy. Similarly, we get energy-entropy, or log-Sobolev, inequalities a1S(Af )+a2S(f )⩽a3‖f‖22+a4〈f,Nf〉, for f in a dense subspace of L2(Rn,ν). Here 〈f,Nf〉 is the Dirichlet form of the energy operator N in the configuration space and the ai can have various signs and are also given explicitly. The principal results of this article are generalized Heisenberg uncertainty relations, namely they are constraints on Af, the phase space representation, given information about f, the configuration space representation.