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Andrei Khrennikov - One of the best experts on this subject based on the ideXlab platform.
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the schrodinger robinson inequality from stochastic analysis on a Complex Hilbert Space
Physica Scripta, 2013Co-Authors: Andrei KhrennikovAbstract:We explored the stochastic analysis on a Complex Hilbert Space to show that one of the cornerstones of quantum mechanics (QM), namely Heisenberg's uncertainty relation, can be derived in the classical probabilistic framework. We created a new mathematical representation of quantum averages: as averages with respect to classical random fields. The existence of a classical stochastic model matching with Heisenberg's uncertainty relation makes the connection between classical and quantum probabilistic models essentially closer. In real physical situations, random fields are valued in the L2-Space. Hence, although we model QM and not QFT, the classical systems under consideration have an infinite number of degrees of freedom. And in our modeling, infinite-dimensional stochastic analysis is the basic mathematical tool.
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interference in the classical probabilistic model and its representation in Complex Hilbert Space
Physica E-low-dimensional Systems & Nanostructures, 2005Co-Authors: Andrei KhrennikovAbstract:Abstract The notion of a context (Complex of physical conditions, that is to say: specification of the measurement setup) is basic in this paper.We show that the main structures of quantum theory (interference of probabilities, Born's rule, Complex probabilistic amplitudes, Hilbert state Space, representation of observables by operators) are present already in a latent form in the classical Kolmogorov probability model. However, this model should be considered as a calculus of contextual probabilities. In our approach it is forbidden to consider abstract context independent probabilities: “first context and only then probability”. We construct the representation of the general contextual probabilistic dynamics in the Complex Hilbert Space. Thus dynamics of the wave function (in particular, Schrodinger's dynamics) can be considered as Hilbert Space projections of a realistic dynamics in a “preSpace”. The basic condition for representing of the preSpace-dynamics is the law of statistical conservation of energy—conservation of probabilities. In general the Hilbert Space projection of the “preSpace” dynamics can be nonlinear and even irreversible (but it is always unitary). Methods developed in this paper can be applied not only to quantum mechanics, but also to classical statistical mechanics. The main quantum-like structures (e.g., interference of probabilities) might be found in some models of classical statistical mechanics. Quantum-like probabilistic behavior can be demonstrated by biological systems. In particular, it was recently found in some psychological experiments.
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on the representation of contextual probabilistic dynamics in the Complex Hilbert Space linear and nonlinear evolutions schrodinger dynamics
Il Nuovo Cimento B, 2005Co-Authors: Andrei KhrennikovAbstract:We constructed the representation of contextual probabilistic dynamics in the Complex Hilbert Space. Thus dynamics of the wave function can be considered as Hilbert Space projections of realistic dynamics in a "preSpace".The basic condition for representing the preSpace-dynamics is the law of statistical conservation of energy-conservation of probabilities. The construction of the dynamical representation is an important step in the development of contextual statistical viewpoint to quantum processes. But the contextual statistical model is essentially more general than the quantum one. Therefore in general the Hilbert Space projection of the "preSpace" dynamics can be nonlinear and even irreversible (but it is always unitary). There were found conditions of linearity and reversibility of the Hilbert Space dynamical projection. We also found conditions for the conventional Schrodinger dynamics (including time-dependent Hamiltonians). We remark that in general even the Schrodinger dynamics is based just on the statistical conservation of energy; for individual systems the law of conservation of energy can be violated (at least in our theoretical model).
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on the representation of contextual probabilistic dynamics in the Complex Hilbert Space linear and nonlinear evolutions schrodinger dynamics
Il Nuovo Cimento, 2005Co-Authors: Andrei KhrennikovAbstract:On the representation of contextual probabilistic dynamics in the Complex Hilbert Space: linear and nonlinear evolutions, Schrodinger dynamics
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on the representation of contextual probabilistic dynamics in the Complex Hilbert Space linear and nonlinear evolutions schroedinger dynamics
arXiv: Quantum Physics, 2004Co-Authors: Andrei KhrennikovAbstract:We constructed the representation of contextual probabilistic dynamics in the Complex Hilbert Space. Thus dynamics of the wave function can be considered as Hilbert Space projections of realistic dynamics in a ``preSpace''. The basic condition for representing of the preSpace-dynamics is the law of statistical conservation of energy -- conservation of probabilities. Construction of the dynamical representation is an important step in the development of contextual statistical viewpoint to quantum processes. But the contextual statistical model is essentially more general than the quantum one. Therefore in general the Hilbert Space projection of the ``preSpace'' dynamics can be nonlinear and even irreversible (but it is always unitary). There were found conditions of linearity and reversibility of the Hilbert Space dynamical projection. We also found conditions for the conventional Schroedinger dynamics (including time-dependent Hamiltonians). We remark that in general even the Schroedinger dynamics is based just on the statistical conservation of energy; for individual systems the law of conservation of energy can be violated (at least in our theoretical model).
Paul Kallol - One of the best experts on this subject based on the ideXlab platform.
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Refined inequalities for the numerical radius of Hilbert Space operators
2021Co-Authors: Bhunia Pintu, Jana Suvendu, Paul KallolAbstract:We present some new upper and lower bounds for the numerical radius of bounded linear operators on a Complex Hilbert Space and show that these are stronger than the existing ones. In particular, we prove that if $A$ is a bounded linear operator on a Complex Hilbert Space $\mathcal{H}$ and if $\Re(A)$, $\Im(A)$ are the real part, the imaginary part of $A$, respectively, then $$ w(A)\geq\frac{\|A\|}{2} +\frac{1}{2\sqrt{2}} \Big | \|\Re(A)+\Im(A)\|-\|\Re(A)-\Im(A)\| \Big | $$ and $$ w^2(A)\geq\frac{1}{4}\|A^*A+AA^*\|+\frac{1}{4}\Big| \|\Re(A)+\Im(A)\|^2-\|\Re(A)-\Im(A)\|^2\Big|. $$ Here $w(.)$ and $\|.\|$ denote the numerical radius and the operator norm, respectively. Further, we obtain refinement of inequalities for the numerical radius of the product of two operators. Finally, as an application of the second inequality mentioned above, we obtain an improvement of upper bound for the numerical radius of the commutators of operators
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On inequalities for A-numerical radius of operators
2020Co-Authors: Bhunia Pintu, Paul Kallol, Nayak, Raj KumarAbstract:Let $A$ be a positive operator on a Complex Hilbert Space $\mathcal{H}.$ We present inequalities concerning upper and lower bounds for $A$-numerical radius of operators, which improve on and generalize the existing ones, studied recently in [A. Zamani, A-Numerical radius inequalities for semi-Hilbertian Space operators, Linear Algebra Appl. 578 (2019) 159-183]. We also obtain some inequalities for $B$-numerical radius of $2\times 2$ operator matrices where $B$ is the $2\times 2$ diagonal operator matrix whose diagonal entries are $A$. Further we obtain upper bounds for $A$-numerical radius for product of operators which improve on the existing bounds.Comment: 16 page
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$A$-Numerical radius orthogonality and parallelism of semi-Hilbertian Space operators and their applications
2020Co-Authors: Bhunia Pintu, Feki Kais, Paul KallolAbstract:In this paper, we aim to introduce and characterize the concept of numerical radius orthogonality of operators on a Complex Hilbert Space $\mathcal{H}$ which are bounded with respect to the semi-norm induced by a positive operator $A$ on $\mathcal{H}$. Moreover, a characterization of the $A$-numerical radius parallelism for $A$-rank one operators is proved. As applications of the obtained results, we obtain some $\mathbb{A}$-numerical radius inequalities of operator matrices where $\mathbb{A}$ is the operator diagonal matrix with diagonal entries are positive operator $A$. Some other related results are also investigated
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Bounds of numerical radius of bounded linear operator using $t$-Aluthge transform
2020Co-Authors: Bag Santanu, Bhunia Pintu, Paul KallolAbstract:We develop a number of inequalities to obtain bounds for the numerical radius of a bounded linear operator defined on a Complex Hilbert Space using the properties of $t$-Aluthge transform. We show that the bounds obtained are sharper than the existing bounds.Comment: 14 page
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Proper improvement of well-known numerical radius inequalities and their applications
2020Co-Authors: Bhunia Pintu, Paul KallolAbstract:New inequalities for the numerical radius of bounded linear operators defined on a Complex Hilbert Space $\mathcal{H}$ are given. In particular, it is established that if $T$ is a bounded linear operator on a Hilbert Space $\mathcal{H}$ then \[ w^2(T)\leq \min_{0\leq \alpha \leq 1} \left \| \alpha T^*T +(1-\alpha)TT^* \right \|,\] where $w(T)$ is the numerical radius of $T.$ The inequalities obtained here are non-trivial improvement of the well-known numerical radius inequalities. As an application we estimate bounds for the zeros of a Complex monic polynomial
Scott Mccullough - One of the best experts on this subject based on the ideXlab platform.
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the failure of rational dilation on a triply connected domain
Journal of the American Mathematical Society, 2005Co-Authors: Michael A Dritschel, Scott McculloughAbstract:Let R denote a domain in C with boundary B. Let X denote the closure of R. An operator T on a Complex Hilbert Space H has X as a spectral set if cr(T) C X and ||/(T)||<||/||n = sup{|/(*)|:*eJR} for every rational function / with poles off X. The expression f(T) may be inter preted either in terms of the Riesz functional calculus, or simply by writing the ra tional function / as pq~~l for polynomials p and q, in which case f(T) = p(T)q(T)~l. The operator T has a normal i?-dilation if there exists a Hilbert Space /C con taining H and a normal operator TV on K so that
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the failure of rational dilation on a triply connected domain
arXiv: Functional Analysis, 2003Co-Authors: Michael A Dritschel, Scott McculloughAbstract:For R a bounded triply connected domain with boundary consisting of disjoint Jordan loops there exists an operator T on a Complex Hilbert Space H so that the closure of R is a spectral set for T, but T does not dilate to a normal operator with spectrum in B, the boundary of R. There is considerable overlap with the construction of an example on such a domain recently obtained by Agler, Harland and Rafael using numerical computations and work of Agler and Harland.
Pekka Lahti - One of the best experts on this subject based on the ideXlab platform.
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quantum mechanics why Complex Hilbert Space
Philosophical Transactions of the Royal Society A, 2017Co-Authors: Gianni Cassinelli, Pekka LahtiAbstract:We outline a programme for an axiomatic reconstruction of quantum mechanics based on the statistical duality of states and effects that combines the use of a theorem of Soler with the idea of symmetry. We also discuss arguments favouring the choice of the Complex field. This article is part of the themed issue ‘Second quantum revolution: foundational questions’.
Vasil Penchev - One of the best experts on this subject based on the ideXlab platform.
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the isomorphism of minkowski Space and the separable Complex Hilbert Space and its physical interpretation
Social Science Research Network, 2020Co-Authors: Vasil PenchevAbstract:An isomorphism is built between the separable Complex Hilbert Space (quantum mechanics) and Minkowski Space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That isomorphism can be interpreted physically as the invariance between a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincare’s conjecture (proved by G. Perelman) hinting another way for proving it, more concise and meaningful physically. Mathematically, the isomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”.
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poincare s conjecture proved by g perelman by the isomorphism of minkowski Space and the separable Complex Hilbert Space
2019Co-Authors: Vasil PenchevAbstract:The physical interpretation of the conjecture is meant. Minkowski Space is isomorphic to a 4-ball meant in the conjecture, on the one hand, and to the separable Complex Hilbert Space utilized by quantum mechanics, on the other rand, once the axiom of choice is involved. Reversible causality relevant to quantum mechanics is sugegsted physical inreprettaion of teh conjecture in teh present context.