Quantum measures and integrals

Details Quantum measures and integrals Quantum measures and integrals

2011-05-19

arxiv.org/abs/1105.3781v1

Physics

Mathematical Physics

16 pages

Scientific paper

We show that quantum measures and integrals appear naturally in any $L_2$-Hilbert space $H$. We begin by defining a decoherence operator $D(A,B)$ and it's associated $q$-measure operator $\mu (A)=D(A,A)$ on $H$. We show that these operators have certain positivity, additivity and continuity properties. If $\rho$ is a state on $H$, then $D_\rho (A,B)=\rmtr\sqbrac{\rho D(A,B)}$ and $\mu_\rho (A)=D_\rho (A,A)$ 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 $\fhat$ on $H$. Continuity and additivity properties of the map $f\mapsto\fhat$ are discussed. It is shown that if $f$ 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.

Affiliated with

Also associated with

No associations

Rating

Quantum measures and integrals does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Quantum measures and integrals, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Quantum measures and integrals will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-68696