1 research outputs found
Quantum measures and integrals
We show that quantum measures and integrals appear naturally in any
-Hilbert space . We begin by defining a decoherence operator
and it's associated -measure operator on . We show that
these operators have certain positivity, additivity and continuity properties.
If is a state on , then D_\rho (A,B)=\rmtr\sqbrac{\rho D(A,B)} and
have the usual properties of a decoherence
functional and -measure, respectively. The quantization of a random variable
is defined to be a certain self-adjoint operator \fhat on . Continuity
and additivity properties of the map f\mapsto\fhat are discussed. It is shown
that if is nonnegative, then \fhat is a positive operator. A quantum
integral is defined by \int fd\mu_\rho =\rmtr (\rho\fhat\,). A tail-sum
formula is proved for the quantum integral. The paper closes with an example
that illustrates some of the theory.Comment: 16 page