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 ===Observations===
-In textbooks on quantum mechanics, it is commonly pointed out that the main distinguishing feature of quantum theory is the presence of ''incompatible observables.'' We recall that two observables <math>A</math> <math>B</math> and  are incompatible if it is impossible to assign values to them jointly. In the probabilistic model, this leads to impossibility to determine their joint probability distribution (JPD). The basic examples of incompatible observables are position and momentum of a quantum system, or spin (or polarization) projections onto different axes. In the mathematical formalism, incompatibility is described as noncommutativity of Hermitian operators <math>\hat{A}</math> and  <math>\hat{B}</math>  representing observables, i.e.,  <math>[\hat{A},\hat{B}]\neq0</math>
+Dans les manuels de mécanique quantique, il est communément souligné que la principale caractéristique distinctive de la théorie quantique est la présence d'observables incompatibles. Rappelons que deux observables <math>A</math> <math>B</math> et  sont incompatibles s'il est impossible de leur attribuer des valeurs conjointement. Dans le modèle probabiliste, cela conduit à l'impossibilité de déterminer leur distribution de probabilité conjointe (JPD). Les exemples de base d'observables incompatibles sont la position et la quantité de mouvement d'un système quantique, ou les projections de spin (ou de polarisation) sur différents axes. Dans le formalisme mathématique, l'incompatibilité est décrite comme la non-commutativité des opérateurs hermitiens <math>\hat{A}</math> et <math>\hat{B}</math> représentant les observables, c'est-à-dire <math>[\hat{A},\hat{B}]\neq0</math>
+Nous nous référons ici au modèle original et toujours fondamental et largement utilisé des observables quantiques, Von Neumann 1955<ref>Von Neumann J. Mathematical Foundations of Quantum Mechanics Princeton Univ. Press, Princeton, NJ, USA (1955)</ref> (Section 3.2).
+L'incompatibilité-non-commutativité est largement utilisée en physique quantique et les observables physiques de base, comme par exemple les projections de position et d'impulsion, de spin et de polarisation, sont traditionnellement représentées dans ce paradigme, par des opérateurs hermitiens. Nous pointons également de nombreuses applications de cette approche à la cognition, à la psychologie, à la prise de décision (Khrennikov, 2004a,<ref>Khrennikov A. Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena, Ser.: Fundamental Theories of Physics, Kluwer, Dordreht(2004)</ref> Busemeyer et Bruza, 2012,<ref name=":10">Busemeyer J., Bruza P. Quantum Models of Cognition and Decision Cambridge Univ. Press, Cambridge(2012)</ref> Bagarello, 2019<ref>Bagarello F. Quantum Concepts in the Social, Ecological and Biological Sciences Cambridge University Press, Cambridge (2019)</ref>) (voir notamment l'article (Bagarello et al., 2018<ref>Bagarello F., Basieva I., Pothos E.M., Khrennikov A. Quantum like modeling of decision making: Quantifying uncertainty with the aid of heisenberg-robertson inequality J. Math. Psychol., 84 (2018), pp. 49-56</ref>) qui est consacré à la quantification de la Relations d'incertitude de Heisenberg dans la prise de décision). Pourtant, ce n'est peut-être pas assez général pour notre objectif - à la modélisation de type quantique en biologie, aucun type de biostatistique non classique ne peut être facilement délégué au modèle d'observations de von Neumann. Par exemple, même des effets cognitifs très basiques ne peuvent pas être décrits d'une manière cohérente avec le modèle d'observation standard (Khrennikov et al., 2014,<ref>Khrennikov A., Basieva I., DzhafarovE.N., Busemeyer J.R. Quantum models for psychological measurements: An unsolved problem. PLoS One, 9 (2014), Article e110909</ref> Basieva et Khrennikov, 2015<ref>Basieva I., Khrennikov A. On the possibility to combine the order effect with sequential reproducibility for quantum measurements Found. Phys., 45 (10) (2015), pp. 1379-1393</ref>).
-Here we refer to the original and still basic and widely used model of quantum observables, Von Neumann 1955<ref>Von Neumann J. Mathematical Foundations of Quantum Mechanics Princeton Univ. Press, Princeton, NJ, USA (1955)</ref> (Section 3.2).
-Incompatibility–noncommutativity is widely used in quantumphysics and the basic physical observables, as say position and momentum, spin and polarization projections, are traditionally represented in this paradigm, by Hermitian operators. We also point to numerous applications of this approach to cognition, psychology, decision making (Khrennikov, 2004a<ref>Khrennikov A. Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena, Ser.: Fundamental Theories of Physics, Kluwer, Dordreht(2004)</ref>, Busemeyer and Bruza, 2012<ref name=":10">Busemeyer J., Bruza P. Quantum Models of Cognition and Decision Cambridge Univ. Press, Cambridge(2012)</ref>, Bagarello, 2019<ref>Bagarello F. Quantum Concepts in the Social, Ecological and Biological Sciences Cambridge University Press, Cambridge (2019)</ref>) (see especially article (Bagarello et al., 2018<ref>Bagarello F., Basieva I., Pothos E.M., Khrennikov A. Quantum like modeling of decision making: Quantifying uncertainty with the aid of heisenberg-robertson inequality J. Math. Psychol., 84 (2018), pp. 49-56</ref>) which is devoted to quantification of the Heisenberg uncertainty relations in decision making). Still, it may be not general enough for our purpose — to quantum-like modeling in biology, not any kind of non-classical bio-statistics can be easily delegated to von Neumann model of observations. For example, even very basic cognitive effects cannot be described in a way consistent with the standard observation model (Khrennikov et al., 2014<ref>Khrennikov A., Basieva I., DzhafarovE.N., Busemeyer J.R. Quantum models for psychological measurements: An unsolved problem. PLoS One, 9 (2014), Article e110909</ref>, Basieva and Khrennikov, 2015<ref>Basieva I., Khrennikov A. On the possibility to combine the order effect with sequential reproducibility for quantum measurements Found. Phys., 45 (10) (2015), pp. 1379-1393</ref>).
 We shall explore more general theory of observations based on ''quantum instruments'' (Davies and Lewis, 1970<ref name=":3" />, Davies, 1976<ref name=":4" />, Ozawa, 1984<ref name=":5" />, Yuen, 1987<ref name=":6" />, Ozawa, 1997<ref name=":7" />, Ozawa, 2004<ref name=":8" />, Okamura and Ozawa, 2016<ref name=":9" />) and find useful tools for applications to modeling of cognitive effects (Ozawa and Khrennikov, 2020a<ref>Ozawa M., Khrennikov A. Application of theory of quantum instruments to psychology: Combination of question order effect with response replicability effect Entropy, 22 (1) (2020), pp. 37.1-9436</ref>, Ozawa and Khrennikov, 2020b<ref>Ozawa M., Khrennikov A. Modeling combination of question order effect, response replicability effect, and QQ-equality with quantum instruments (2020) </ref>). We shall discuss this question in Section 3 and illustrate it with examples from cognition and molecular biology in Sections 6, 7. In the framework of the quantum instrument theory, the crucial point is not commutativity vs. noncommutativity of operators symbolically representing observables, but the mathematical form of state’s transformation resulting from the back action of (self-)observation. In the standard approach, this transformation is given by an orthogonal projection on the subspace of eigenvectors corresponding to observation’s output. This is ''the projection postulate.'' In quantum instrument theory, state transformations are more general.