Search Results (39)

Since the early twentieth century, quantum mechanics has sought an interpretation that offers a consistent worldview. In the course of that, many proposals were advanced, but all of them introduce, at some point, interpretation elements (semantics) that find no correlate in the formalism (syntactics). This distance from semantics and syntactics is one of the major reasons for finding so abstruse and diverse interpretations of the formalism. To overcome this issue, we propose an alternative stochastic interpretation, based exclusively on the formal structure of the Schrödinger equation, without resorting to external assumptions such as the collapse of the wave function or the role of the observer. We present four (mathematically equivalent) mathematical derivations of the Schrödinger equation based on four constructs: characteristic function, Boltzmann entropy, Central Limit Theorem (CLT), and Langevin equation. All of them resort to axioms already interpreted and offer complementary perspectives to the quantum formalism. The results show the possibility of deriving the Schrödinger equation from well-defined probabilistic principles and that the wave function represents a probability amplitude in the configuration space, with dispersions linked to the CLT. It is concluded that quantum mechanics has a stochastic support, originating from the separation between particle and field subsystems, allowing an objective description of quantum behavior as a mean-field theory, analogous, but not equal, to Brownian motion, without the need for arbitrary ontological entities. Full article

The conditional Feynman integral provides solutions to integral equations equivalent to heat and Schrödinger equations. The Cameron–Martin translation theorem illustrates how the Wiener measure changes under translation via Cameron–Martin space elements in abstract Wiener space. Translation theorems for analytic Feynman integrals have been established in many research articles. This study aims to present a translation theorem for the conditional function space integral of functionals on the generalized Wiener space $C_{a,b}[0,T]$ induced via a generalized Brownian motion process determined using continuous functions $a\left(t\right)$ and $b\left(t\right)$. As an application, we establish a translation theorem for the conditional generalized analytic Feynman integral of functionals on $C_{a,b}[0,T]$. We then provide explicit examples of functionals on $C_{a,b}[0,T]$ to which the conditional translation theorem on $C_{a,b}[0,T]$ can be applied. Our formulas and results are more complicated than the corresponding formulas and results in the previous research on the Wiener space $C_0[0,T]$ because the generalized Brownian motion process used in this study is neither stationary in time nor centered. In this study, the stochastic process used is subject to a drift function. Full article

Traditional brain emulation approaches often rely on classical computational models that inadequately capture the stochastic, nonlinear, and potentially coherent features of biological neural systems. In this position paper, we introduce NeuroQ a quantum-inspired framework grounded in stochastic mechanics, particularly Nelson’s formulation. By reformulating the FitzHugh–Nagumo neuron model with structured noise, we derive a Schrödinger-like equation that encodes membrane dynamics in a quantum-like formalism. This formulation enables the use of quantum simulation strategies—including Hamiltonian encoding, variational eigensolvers, and continuous-variable models—for neural emulation. We outline a conceptual roadmap for implementing NeuroQ on near-term quantum platforms and discuss its broader implications for neuromorphic quantum hardware, artificial consciousness, and time-symmetric cognitive architectures. Rather than demonstrating a working prototype, this work aims to establish a coherent theoretical foundation for future research in quantum brain emulation. Full article

This work investigates fractional stochastic Schrödinger evolution equations in a Hilbert space, incorporating complex potential symmetry and Poisson jumps. We establish the existence of mild solutions via stochastic analysis, semigroup theory, and the Mönch fixed-point theorem. Sufficient conditions for exponential stability are derived, ensuring asymptotic decay. We further explore trajectory controllability, identifying conditions for guiding the system along prescribed paths. A numerical example is provided to validate the theoretical results. Full article

The dynamics of complex systems often exhibit multifractal properties, where interactions across different scales influence their evolution. In this study, we apply the Multifractal Theory of Motion within the framework of scale relativity theory to explore synchronization phenomena in complex systems. We demonstrate that the motion of such systems can be described by multifractal Schrödinger-type equations, offering a new perspective on the interplay between deterministic and stochastic behaviors. Our analysis reveals that synchronization in complex systems emerges from the balance of multifractal acceleration, convection, and dissipation, leading to structured yet highly adaptive behavior across scales. The results highlight the potential of multifractal analysis in predicting and controlling synchronized dynamics in real-world applications. Several applications are also discussed. Full article

Understanding the dynamic nature of biological systems is fundamental to deciphering cellular behavior, developmental processes, and disease progression. Single-cell RNA sequencing (scRNA-seq) has provided static snapshots of gene expression, offering valuable insights into cellular states at a single time point. Recent advancements in temporally resolved scRNA-seq, spatial transcriptomics (ST), and time-series spatial transcriptomics (temporal-ST) have further revolutionized our ability to study the spatiotemporal dynamics of individual cells. These technologies, when combined with computational frameworks such as Markov chains, stochastic differential equations (SDEs), and generative models like optimal transport and Schrödinger bridges, enable the reconstruction of dynamic cellular trajectories and cell fate decisions. This review discusses how these dynamical system approaches offer new opportunities to model and infer cellular dynamics from a systematic perspective. Full article

This study considers a class of backward stochastic semi-linear Schrödinger equations with Poisson jumps in ${\mathbb{R}}^d$ or in its bounded domain of a $C^2$ boundary, which is associated with a stochastic control problem of nonlinear Schrödinger equations driven by Lévy noise. The approach to establish the existence and uniqueness of solutions is mainly based on the complex Itô formula, the Galerkin’s approximation method, and the martingale representation theorem. Full article

In this paper, we review and compare the stochastic quantum mechanics of Nelson and the scale relativity theory of Nottale. We consider both nonrelativistic and relativistic frameworks and include the electromagnetic field. These theories propose a derivation of the Schrödinger and Klein–Gordon equations from microscopic processes. We show their formal equivalence. Specifically, we show that the real and imaginary parts of the complex Lorentz equation in Nottale’s theory are equivalent to the Nelson equations, which are themselves equivalent to the Madelung and de Broglie hydrodynamical representations of the Schrödinger and Klein–Gordon equations, respectively. We discuss the different physical interpretations of the Nelson and Nottale theories and stress their strengths and weaknesses. We mention potential applications of these theories to dark matter. Full article

We consider the Cauchy problem for the stochastic nonlinear Schrödinger equation augmented by nonlinear energy-critical damping term arising in nonlinear optics and quantum field theory. Through examining the behavior of the momentum and energy functionals, we almost surely prove the existence and uniqueness of global solutions with continuous $H^1\left({\mathbb{R}}^d\right)$ valued paths. The results cover either defocusing nonlinearity in the full energy critical and subcritical range of exponents or focusing nonlinearity in the full subcritical range, as in the deterministic case. Full article

The simulation analogy presented in this work enhances the accessibility of abstract quantum theories, specifically the stochastic hydrodynamic model (SQHM), by relating them to our daily experiences. The SQHM incorporates the influence of fluctuating gravitational background, a form of dark energy, into quantum equations. This model successfully addresses key aspects of objective-collapse theories, including resolving the ‘tails’ problem through the definition of quantum potential length of interaction in addition to the De Broglie length, beyond which coherent Schrödinger quantum behavior and wavefunction tails cannot be maintained. The SQHM emphasizes that an external environment is unnecessary, asserting that the quantum stochastic behavior leading to wavefunction collapse can be an inherent property of physics in a spacetime with fluctuating metrics. Embedded in relativistic quantum mechanics, the theory establishes a coherent link between the uncertainty principle and the constancy of light speed, aligning seamlessly with finite information transmission speed. Within quantum mechanics submitted to fluctuations, the SQHM derives the indeterminacy relation between energy and time, offering insights into measurement processes impossible within a finite time interval in a truly quantum global system. Experimental validation is found in confirming the Lindemann constant for solid lattice melting points and the ⁴He transition from fluid to superfluid states. The SQHM’s self-consistency lies in its ability to describe the dynamics of wavefunction decay (collapse) and the measure process. Additionally, the theory resolves the pre-existing reality problem by showing that large-scale systems naturally decay into decoherent states stable in time. Continuing, the paper demonstrates that the physical dynamics of SQHM can be analogized to a computer simulation employing optimization procedures for realization. This perspective elucidates the concept of time in contemporary reality and enriches our comprehension of free will. The overall framework introduces an irreversible process impacting the manifestation of macroscopic reality at the present time, asserting that the multiverse exists solely in future states, with the past comprising the formed universe after the current moment. Locally uncorrelated projective decays of wavefunction, at the present time, function as a reduction of the multiverse to a single universe. Macroscopic reality, characterized by a foam-like consistency where microscopic domains with quantum properties coexist, offers insights into how our consciousness perceives dynamic reality. It also sheds light on the spontaneous emergence of gravity in discrete quantum spacetime evolution, and the achievement of the classical general relativity limit in quantum loop gravity and causal dynamical triangulation. The simulation analogy highlights a strategy focused on minimizing information processing, facilitating the universal simulation in solving its predetermined problem. From within, reality becomes the manifestation of specific physical laws emerging from the inherent structure of the simulation devised to address its particular issue. In this context, the reality simulation appears to employ an optimization strategy, minimizing information loss and data management in line with the simulation’s intended purpose. Full article

We consider a toy model for the study of monitored dynamics in many-body quantum systems. We study the stochastic Schrödinger equation resulting from continuous monitoring with a rate $\Gamma$ of a random Hermitian operator, drawn from the Gaussian unitary ensemble (GUE) at every time t. Due to invariance by unitary transformations, the dynamics of the eigenvalues ${\left\{{\lambda }_{\alpha }\right\}}_{\alpha =1}^n$ of the density matrix decouples from that of the eigenvectors, and is exactly described by stochastic equations that we derive. We consider two regimes: in the presence of an extra dephasing term, which can be generated by imperfect quantum measurements, the density matrix has a stationary distribution, and we show that in the limit of large size $n\to \infty$ it matches with the inverse-Marchenko–Pastur distribution. In the case of perfect measurements, instead, purification eventually occurs and we focus on finite-time dynamics. In this case, remarkably, we find an exact solution for the joint probability distribution of $\lambda$’s at each time t and for each size n. Two relevant regimes emerge: at short times $t\Gamma =O\left(1\right)$, the spectrum is in a Coulomb gas regime, with a well-defined continuous spectral distribution in the $n\to \infty$ limit. In that case, all moments of the density matrix become self-averaging and it is possible to exactly characterize the entanglement spectrum. In the limit of large times $t\Gamma =O\left(n\right)$, one enters instead a regime in which the eigenvalues are exponentially separated $\log ({\lambda }_{\alpha }/{\lambda }_{\beta })=O(\Gamma t/n)$, but fluctuations $\sim O\left(\sqrt{\Gamma t/n}\right)$ play an essential role. We are still able to characterize the asymptotic behaviors of the entanglement entropy in this regime. Full article

We study the Schrödinger equation in quantum field theory (QFT) in its functional formulation. In this approach, quantum correlation functions can be expressed as classical expectation values over (complex) stochastic processes. We obtain a stochastic representation of the Schrödinger time evolution on Wentzel–Kramers–Brillouin (WKB) states by means of the Wiener integral. We discuss QFT in a flat expanding metric and in de Sitter space-time. We calculate the evolution kernel in an expanding flat metric in the real-time formulation. We discuss a field interaction in pseudoRiemannian and Riemannian metrics showing that an inversion of the signature leads to some substantial simplifications of the singularity problems in QFT. Full article

The Feynman–Kac formula establishes a link between parabolic partial differential equations and stochastic processes in the context of the Schrödinger equation in quantum mechanics. Specifically, the formula provides a solution to the partial differential equation, expressed as an expectation value for Brownian motion. This paper demonstrates that the Feynman–Kac formula does not produce a unique solution but instead carries infinitely many solutions to the corresponding partial differential equation. Full article

The aim of this work is to investigate the influence of nonlinear multiplicative noise on the Cauchy problem of the nonlinear fractional Schrödinger equation in the non-radial case. Local well-posedness follows from estimates related to the stochastic convolution and deterministic non-radial Strichartz estimates. Furthermore, the blow-up criterion is presented. Then, with the help of Itô’s lemma and stopping time arguments, the global solution is constructed almost surely. The main innovation is that the non-radial global solution is given under fractional-order derivatives and a nonlinear noise term. Full article

The exact solutions of the nonlinear Schrödinger equation (NLSE) predict consistent novel applicable existences such as solitonic localized structures, rouge forms, and shocks that rely on physical phenomena to propagate. Theoretical explanations of randomly nonlinear new extension NLSE structure solutions have undergone stochastic mode examination. This equation enables accurate and efficient solutions capable of simulating developed solitonic structures with dynamic features. The generated random waves are a dynamically regulated system that are influenced by random water currents behaviour. It has been noticed that the stochastic parameter modulates the wave force and supplies the wave collapsing energy with related medium turbulence. It has been observed that noise effects can alter wave characteristics, which may lead to innovative astrophysics, physical density, and ocean waves. Full article