Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
1899411 | Reports on Mathematical Physics | 2012 | 15 Pages |
Abstract
We show that quantum measures and integrals appear naturally in any L2-Hilbert space H. We begin by defining a decoherence operator D(A, B) and its associated q-measure operator μ(A) = D(A, A) on H. We show that these operators have certain positivity, additivity and continuity properties. If Ï is a state on H, then D Ï(A, B) = tr[Ï D(A,B)] and μÏ(A)= D Ï(A,A) have the usual properties of a decoherence functional and q-measure, respectively. The quantization of a random variable f is defined to be a certain self-adjoint operator
fË on H. Continuity and additivity properties of the map
fâ¦fË are discussed. It is shown that if f is nonnegative, then
fË is a positive operator. A quantum integral is defined by
â«fdμÏ=tr(ÏfË). A tail-sum formula is proved for the quantum integral. The paper closes with an example that illustrates some of the theory.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematical Physics
Authors
S. Gudder,