Free independence

2021 ◽  
pp. 31-48
Author(s):  
Arup Bose
Keyword(s):  
Free Independence

scholarly journals Schwinger’s picture of quantum mechanics IV: Composition and independence

2020 ◽  
Vol 17 (04) ◽  
pp. 2050058 ◽  
Author(s):  
F. M. Ciaglia ◽  
F. Di Cosmo ◽  
A. Ibort ◽  
G. Marmo
Keyword(s):  
Quantum Mechanics ◽  
Entangled State ◽  
Total System ◽  
Free Products ◽  
Algebraic Description ◽  
Free Independence

The groupoid description of Schwinger’s picture of quantum mechanics is continued by discussing the closely related notions of composition of systems, subsystems, and their independence. Physical subsystems have a neat algebraic description as subgroupoids of the Schwinger’s groupoid of the system. The groupoid picture offers two natural notions of composition of systems: Direct and free products of groupoids, that will be analyzed in depth as well as their universal character. Finally, the notion of independence of subsystems will be reviewed, finding that the usual notion of independence, as well as the notion of free independence, find a natural realm in the groupoid formalism. The ideas described in this paper will be illustrated by using the EPRB experiment. It will be observed that, in addition to the notion of the non-separability provided by the entangled state of the system, there is an intrinsic “non-separability” associated to the impossibility of identifying the entangled particles as subsystems of the total system.

Wigner’s semicircle law and free independence

Resonance ◽  
2009 ◽  
Vol 14 (10) ◽  
pp. 970-977
Author(s):  
B. V. Rajarama Bhat
Keyword(s):  
Semicircle Law ◽  
Free Independence

scholarly journals Mixtures of classical and free independence

2016 ◽  
Vol 107 (4) ◽  
pp. 445-453 ◽  
Author(s):  
Roland Speicher ◽  
Janusz Wysoczański
Keyword(s):  
Free Independence

scholarly journals Approximation of a free Poisson process by systems of freely independent particles

2018 ◽  
Vol 21 (03) ◽  
pp. 1850020 ◽  
Author(s):  
Marek Bożejko ◽  
José Luís da Silva ◽  
Tobias Kuna ◽  
Eugene Lytvynov
Keyword(s):  
Poisson Process ◽  
Radon Measure ◽  
Poisson Point Process ◽  
Particle System ◽  
Intensity Measure ◽  
Bounded Set ◽  
Lévy White Noise ◽  
Free Independence ◽  
Bounded Sets ◽  
Number Of Particles

Let [Formula: see text] be a non-atomic, infinite Radon measure on [Formula: see text], for example, [Formula: see text] where [Formula: see text]. We consider a system of freely independent particles [Formula: see text] in a bounded set [Formula: see text], where each particle [Formula: see text] has distribution [Formula: see text] on [Formula: see text] and the number of particles, [Formula: see text], is random and has Poisson distribution with parameter [Formula: see text]. If the particles were classically independent rather than freely independent, this particle system would be the restriction to [Formula: see text] of the Poisson point process on [Formula: see text] with intensity measure [Formula: see text]. In the case of free independence, this particle system is not the restriction of the free Poisson process on [Formula: see text] with intensity measure [Formula: see text]. Nevertheless, we prove that this is true in an approximative sense: if bounded sets [Formula: see text] ([Formula: see text]) are such that [Formula: see text] and [Formula: see text], then the corresponding particle system in [Formula: see text] converges (as [Formula: see text]) to the free Poisson process on [Formula: see text] with intensity measure [Formula: see text]. We also prove the following [Formula: see text]-limit: Let [Formula: see text] be a deterministic sequence of natural numbers such that [Formula: see text]. Then the system of [Formula: see text] freely independent particles in [Formula: see text] converges (as [Formula: see text]) to the free Poisson process. We finally extend these results to the case of a free Lévy white noise (in particular, a free Lévy process) without free Gaussian part.

scholarly journals Non-commutative stochastic independence and cumulants

2017 ◽  
Vol 20 (02) ◽  
pp. 1750010 ◽  
Author(s):  
Sarah Manzel ◽  
Michael Schürmann
Keyword(s):  
Lie Algebras ◽  
Generating Functions ◽  
Stochastic Independence ◽  
Fundamental Lemma ◽  
Probability Spaces ◽  
Free Independence

In a fundamental lemma we characterize “generating functions” of certain functors on the category of algebraic non-commutative probability spaces. Special families of such generating functions correspond to “unital, associative universal products” on this category, which again define a notion of non-commutative stochastic independence. Using the fundamental lemma, we prove the existence of cumulants and of “cumulant Lie algebras” for all independences coming from a unital, associative universal product. These include the five independences (tensor, free, Boolean, monotone, anti-monotone) appearing in Muraki’s classification, c-free independence of Bożejko and Speicher, the indented product of Hasebe and the bi-free independence of Voiculescu. We show how the non-commutative independence can be reconstructed from its cumulants and cumulant Lie algebras.

The Sky in an Ant’s Egg: Ġhālib’s Structural Poetics

2020 ◽  
Vol 1 (1) ◽  
pp. 53-66
Author(s):  
Frances W. Pritchett
Keyword(s):  
Structural Patterns ◽  
Free Independence ◽  
Context Free

Abstract Some of the most important structural patterns and devices used in individual ghazal verses by the famous poet Mirza Asadullah Khan ‘Ghalib’ are identified and analyzed; their literary effectiveness is illustrated with examples and discussion. In particular, the paper considers two such patterns. One set of verses have a ‘twist’ to them, such that the reader (or, ideally, hearer) is first misled or confused, then at the last possible moment is suddenly and almost explosively enlightened. Another set of verses create an inherently unresolvable ‘tangle’ of several possible meanings which cannot be either affirmed or rejected on any non-arbitrary grounds. The context-free independence of such small ghazal verses, together with their division into two formally distinct and performatively separated lines, makes for unusual poetic constraints and opportunities. The author has prepared an extensive commentarial website on the poetry of Ghalib and Mir.

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