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Classical versions of q-Gaussian processes: conditional moments and Bell's inequality

Wlodzimierz Bryc

Abstract

We show that classical processes corresponding to operators what satisfy a q-commutative relation have linear regressions and quadratic conditional variances. From this we deduce that Bell's inequality for their covariances can be extended from q=-1 to the entire range -1<q<1.

Classical versions of q-Gaussian processes: conditional moments and Bell's inequality

Abstract

We show that classical processes corresponding to operators what satisfy a q-commutative relation have linear regressions and quadratic conditional variances. From this we deduce that Bell's inequality for their covariances can be extended from q=-1 to the entire range -1<q<1.

Paper Structure

This paper contains 11 sections, 8 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. $q$-Fock space
  5. $q$-Gaussian processes
  6. Conditional expectations
  7. Conditional Moments of Classical Versions
  8. Triplets
  9. Relation to processes with independent increments
  10. Bell's Inequality
  11. Examples

Key Result

theorem \oldthetheorem

If ${\mathbf Y}=\psi(g_1\otimes\dots \otimes g_m)$, ${\mathbf X}_1={\mathbf X}_{f_1},\dots,{\mathbf X}_k={\mathbf X}_{f_k}$ for some $f_i,g_j\in{\mathcal{H}}$ and $P:{\mathcal{H}}\to {\mathcal{H}}$ denotes orthogonal projection onto the span of $f_1,\dots,f_k$ then

Theorems & Definitions (12)

  • definition 1
  • theorem \oldthetheorem
  • corollary 1
  • proposition 1
  • proof
  • corollary 2
  • definition 2
  • theorem \oldthetheorem
  • lemma 1
  • proposition 2
  • ...and 2 more