Spreadability for Quantum Stochastic Processes, with an Application to Boolean Commutation Relations.

Vitonofrio Crismale,Francesco Fidaleo,Maria Elena Griseta

doi:10.3390/e22050532

Vitonofrio Crismale, Francesco Fidaleo + Show 1 more

Open Access

https://doi.org/10.3390/e22050532

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Abstract

In order to manage spreadability for quantum stochastic processes, we study in detail the structure of the involved monoids acting on the index-set of all integers , that is that generated by left and right hand-side partial shifts, the monoid of all strictly increasing maps whose range has finite complement, and finally the collection of all strictly increasing maps of . We show that such three monoids are strictly ordered, and the second-named one is the semidirect product between the first and the action of generated by the one-step shift. Even if the definition of a spreadable stochastic process is provided in terms of the invariance of the finite joint distributions under the natural action of the last monoid on the indices, we see that spreadability can be directly stated in terms of invariance with respect to the action of the first monoid. Concerning the stochastic processes involving the concrete boolean -algebra generated by the annihilators acting on the boolean Fock space (i.e., the concrete -algebra satisfying the boolean commutation relations), we study their spreadability directly in terms of the invariance under the monoid generated by all strictly increasing maps whose range has finite complement because, for this case, such an investigation appears more direct and manageable. Finally, we present the version of the Ryll–Nardzewski theorem for the boolean case, establishing that spreadable, exchangeable and stationary stochastic processes coincide, and describing their common structure.

Highlights

Stochastic processes invariant under distributional symmetries have been intensively studied in classical probability theory, and their natural applications to statistical mechanics and other applied fields deeply encouraged this investigation
The definition of a spreadable stochastic process is provided in terms of the invariance of the finite joint distributions under the natural action of the monoid of strictly increasing maps on Z, here we show that spreadability can be directly stated in terms of invariance with respect to the action of the monoid, denoted by IZ, generated by left and right hand-side partial shifts on the integers
The investigation of the so-called quantum probability started with the seminal paper [22]

Summary

Introduction

Stochastic processes invariant under distributional symmetries have been intensively studied in classical probability theory, and their natural applications to statistical mechanics and other applied fields deeply encouraged this investigation. The reader is referred to [4,10] for more details about this conceptual point Using these results, in [5] spreadability was investigated for stochastic processes arising from the so-called monotone commutation relations. It provides another structure to study spreadability, as we show that spreading invariant states are exactly those invariant under the action of JZ This statement appears interesting in our successive investigation about boolean stochastic processes, since JZ offers a more flexible analysis in that case. It is extended to its closure in the uniform topology (i.e., the C ∗ -algebra of the compact linear operators), which coincides with the (concrete) boolean C ∗ -algebra, and the action is completely described in this case This result is aimed to yield the structure of boolean spreading invariant states, given in the last part of the notes. We briefly summarise some open problems for further investigations

Preliminaries

Stochastic Processes and Their Symmetries

Monoids of Increasing Maps

Spreading Invariant States on the Boolean Algebra

Conclusions