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Keywords = quantum stochastic calculus

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15 pages, 429 KB  
Article
A Note on the Relativistic Transformation Properties of Quantum Stochastic Calculus
by John E. Gough
Entropy 2025, 27(5), 529; https://doi.org/10.3390/e27050529 - 15 May 2025
Viewed by 1057
Abstract
We present a simple argument to derive the transformation of the quantum stochastic calculus formalism between inertial observers and derive the quantum open system dynamics for a system moving in a vacuum (or, more generally, a coherent) quantum field under the usual Markov [...] Read more.
We present a simple argument to derive the transformation of the quantum stochastic calculus formalism between inertial observers and derive the quantum open system dynamics for a system moving in a vacuum (or, more generally, a coherent) quantum field under the usual Markov approximation. We argue, however, that, for uniformly accelerated open systems, the formalism must break down as we move from a Fock representation over the algebra of field observables over all of Minkowski space to the restriction regarding the algebra of observables over a Rindler wedge. This leads to quantum noise having a unitarily inequivalent non-Fock representation: in particular, the latter is a thermal representation at the Unruh temperature. The unitary inequivalence is ultimately a consequence of the underlying flat noise spectrum approximation for the fundamental quantum stochastic processes. We derive the quantum stochastic limit for a uniformly accelerated (two-level) detector and establish an open system description of the relaxation to thermal equilibrium at the Unruh temperature. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness V)
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20 pages, 1061 KB  
Article
The Algebra and Calculus of Stochastically Perturbed Spacetime with Classical and Quantum Applications
by Dragana Pilipović
Symmetry 2024, 16(1), 36; https://doi.org/10.3390/sym16010036 - 28 Dec 2023
Cited by 2 | Viewed by 2491
Abstract
We consider an alternative to dark matter as a potential solution to various remaining problems in physics: the addition of stochastic perturbations to spacetime to effectively enforce a minimum length and establish a fundamental uncertainty at minimum length (ML) scale. To explore the [...] Read more.
We consider an alternative to dark matter as a potential solution to various remaining problems in physics: the addition of stochastic perturbations to spacetime to effectively enforce a minimum length and establish a fundamental uncertainty at minimum length (ML) scale. To explore the symmetry of spacetime to such perturbations both in classical and quantum theories, we develop some new tools of stochastic calculus. We derive the generators of rotations and boosts, along with the connection, for stochastically perturbed, minimum length spacetime (“ML spacetime”). We find the metric, the directional derivative, and the canonical commutator preserved. ML spacetime follows the Lie algebra of the Poincare group, now expressed in terms of the two-point functions of the stochastic fields (per Ito’s lemma). With the fundamental uncertainty at ML scale a symmetry of spacetime, we require the translational invariance of any classical theory in classical spacetime to also include the stochastic spacetime perturbations. As an application of these ideas, we consider galaxy rotation curves for massive bodies to find that—under the Robertson–Walker minimum length theory—rotational velocity becomes constant as the distance to the center of the galaxy becomes very large. The new tools of stochastic calculus also set the stage to explore new frontiers at the quantum level. We consider a massless scalar field to derive the Ward-like identity for ML currents. Full article
(This article belongs to the Special Issue Symmetry/Asymmetry and the Dark Universe)
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13 pages, 565 KB  
Article
Characteristics of Solitary Stochastic Structures for Heisenberg Ferromagnetic Spin Chain Equation
by Munerah Almulhem, Samia Z. Hassan, Alanwood Al-buainain, Mohammed A. Sohaly and Mahmoud A. E. Abdelrahman
Symmetry 2023, 15(4), 927; https://doi.org/10.3390/sym15040927 - 17 Apr 2023
Cited by 1 | Viewed by 1672
Abstract
The impact of Stratonovich integrals on the solutions of the Heisenberg ferromagnetic spin chain equation using the unified solver approach is examined in this study. In particular, using arbitrary parameters, the traveling wave arrangements of rational, trigonometric, and hyperbolic functions are developed. The [...] Read more.
The impact of Stratonovich integrals on the solutions of the Heisenberg ferromagnetic spin chain equation using the unified solver approach is examined in this study. In particular, using arbitrary parameters, the traveling wave arrangements of rational, trigonometric, and hyperbolic functions are developed. The detailed arrangements are exceptionally critical for clarifying diverse complex wonders in plasma material science, optical fiber, quantum mechanics, super liquids and so on. Here, the Itô stochastic calculus and the Stratonovich stochastic calculus are considered. To describe the dynamic behaviour of random solutions, some graphical representations for these solutions are described with appropriate parameters. Full article
(This article belongs to the Section Mathematics)
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10 pages, 294 KB  
Article
A Stochastic Fractional Calculus with Applications to Variational Principles
by Houssine Zine and Delfim F. M. Torres
Fractal Fract. 2020, 4(3), 38; https://doi.org/10.3390/fractalfract4030038 - 1 Aug 2020
Cited by 12 | Viewed by 3877
Abstract
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, [...] Read more.
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation. Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
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21 pages, 544 KB  
Article
PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black–Scholes Equation
by Will Hicks
Entropy 2019, 21(2), 105; https://doi.org/10.3390/e21020105 - 23 Jan 2019
Cited by 7 | Viewed by 5077
Abstract
The Accardi–Boukas quantum Black–Scholes framework, provides a means by which one can apply the Hudson–Parthasarathy quantum stochastic calculus to problems in finance. Solutions to these equations can be modelled using nonlocal diffusion processes, via a Kramers–Moyal expansion, and this provides useful tools to [...] Read more.
The Accardi–Boukas quantum Black–Scholes framework, provides a means by which one can apply the Hudson–Parthasarathy quantum stochastic calculus to problems in finance. Solutions to these equations can be modelled using nonlocal diffusion processes, via a Kramers–Moyal expansion, and this provides useful tools to understand their behaviour. In this paper we develop further links between quantum stochastic processes, and nonlocal diffusions, by inverting the question, and showing how certain nonlocal diffusions can be written as quantum stochastic processes. We then go on to show how one can use path integral formalism, and PT symmetric quantum mechanics, to build a non-Gaussian kernel function for the Accardi–Boukas quantum Black–Scholes. Behaviours observed in the real market are a natural model output, rather than something that must be deliberately included. Full article
(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)
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