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Fractal and Fractional

Topological Subordination in Quantum Mechanics

Alexander Iomin, Ralf Metzler, Trifce Sandev

Published May 25, 2023• 4 citations
Publisher

Abstract

An example of non-Markovian quantum dynamics is considered in the framework of a geometrical (topological) subordination approach. The specific property of the model is that it coincides exactly with the fractional diffusion equation, which describes the geometric Brownian motion on combs. Both classical diffusion and quantum dynamics are described using the dilatation operator xddx. Two examples of geometrical subordinators are considered. The first one is the Gaussian function, which is due to the comb geometry. For the quantum consideration with a specific choice of the initial conditions, it corresponds to the integral representation of the Mittag–Leffler function by means of the subordination integral. The second subordinator is the Dirac delta function, which results from the memory kernels that define the fractional time derivatives in the fractional diffusion equation.

DOI
10.3390/fractalfract7060431

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An example of non-Markovian quantum dynamics is considered in the framework of a geometrical (topological) subordination approach. The specific property of the model is that it coincides exactly with the fractional diffusion equation, which describes the geometric Brownian motion on combs. Both classical diffusion and quantum dynamics are described using the dilatation operator xddx. Two examples of geometrical subordinators are considered. The first one is the Gaussian function, which is due to the comb geometry. For the quantum consideration with a specific choice of the initial conditions, it corresponds to the integral representation of the Mittag–Leffler function by means of the subordination integral. The second subordinator is the Dirac delta function, which results from the memory kernels that define the fractional time derivatives in the fractional diffusion equation.
Quantum dynamics of a particle on a two-dimensional comb structure is considered. This dynamics of a Hamiltonian system with a topologically constrained geometry leads to the non-Markovian behavior. In the framework of a rigorous analytical consideration, it is shown how a fractional time derivative appears for the relevant description of this non-Markovian quantum mechanics in the framework of fractional time Schrödinger equations. Analytical solutions for the Green functions are obtained for both conservative and periodically driven in time Hamiltonian systems.
We investigate a generalized quantum Schrödinger equation in a comb-like structure that imposes geometric constraints on spatial variables. The model is extended by the introduction of nonlocal and fractional potentials to capture memory effects in both space and time. We consider four distinct scenarios: (i) a time-dependent nonlocal potential, (ii) a spatially nonlocal potential, (iii) a combined space–time nonlocal interaction with memory kernels, and (iv) a fractional spatial derivative, which is related to distributions asymptotically governed by power laws and to a position-dependent effective mass. For each scenario, we propose solutions based on the Green’s function for arbitrary initial conditions and analyze the resulting quantum dynamics. Our results reveal distinct spreading regimes, depending on the type of non-locality and the fractional operator applied to the spatial variable. These findings contribute to the broader generalization of comb models and open new questions for exploring quantum dynamics in backbone-like structures.
A probability density of fractional (or fractal) order is defined by the probability increment pr{x < X ≤ x+dx} = pα(x)(dx)α, 0 < α < 1, and appears to be quite suitable to deal with random variables defined in a fractal space. Combining this definition with the fractional Taylor's series [Formula: see text] denotes the Mittag–Leffler function) provided by the modified Riemann–Liouville derivative, one can expand a probability calculus parallel to the standard one. This approach could be considered as a framework for the derivation of some space fractional partial differential diffusion equations in coarse-grained spaces. It is shown firstly that there is some relation between fractional probability and signed measure of probability, and secondly that when α = 1/2, there is some identity between this fractal probability and quantum probability. Shortly, a wavefunction could be thought of as a fractal probability density of order 1/2. One exhibits further relations with possibility theory and relative information. Lastly, one arrives at a new informational entropy based on the inverse of the Mittag–Leffler function.
Open quantum systems exhibit dynamics ranging from unitary evolution to irreversible dissipation. While the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation uniquely characterizes Markovian CPTP evolution, many physical platforms display non-Markovian features such as algebraic relaxation and coherence backflow. Fractional calculus provides a natural way to model such long-memory behavior through power-law temporal kernels introduced by fractional time derivatives. Here we develop a unified framework that embeds fractional master equations within the broader hierarchy of open-system formalisms. The fractional equation forms a structured subclass of memory-kernel models, reduces to the Lindblad form at unit order, and, through Bochner--Phillips subordination, admits a CPTP representation as an average over Lindblad semigroups. Its resolvent structure further connects fractional dynamics to established non-Markovian approaches, including Nakajima--Zwanzig kernels and hierarchical equations of motion, providing a compact surrogate for long-memory effects. This formulation positions fractional calculus as a rigorous and practical language for quantum dynamics with intrinsic memory, supporting both analytical insight and efficient quantum simulation.
We consider a quantum motion governed by the time-dependent Schrödinger equation on a three dimensional comb structure. We derive the corresponding fractional Schrödinger equations for the reduced probability density functions by projection of the three dimensional comb dynamics in the two- and one-dimensional configuration space. This represents another physical example of a system where fractional calculus emerges. We give closed-form solutions of the corresponding equations for the reduced probability density functions in terms of the Fox H-function, by using the Green’s function approach.
Generating functions for integrals of stochastic processes have been known in analytically closed form for just a handful of processes: namely, the Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (CIR) process and the exponential of Brownian motion. In virtue of their analytical tractability, these processes are extensively used in modelling applications. In the first part of the thesis, broad extensions of these process classes are constructed. The known models are shown to fit into a classification scheme for diffusion processes for which generating functions for integrals of the process and transition probability densities can be evaluated as integrals of hypergeometric functions against the spectral measure for certain self-adjoint operators. This scheme also extends to a class of finite-state Markov processes related to hypergeometric polynomials in the discrete series of the Askey classification tree. We also extend to quantum mechanics the technique of stochastic subordination, by means of which one can express any semi-martingale as a time-changed Brownian motion. Stochastic subordination preserves the Markov property of the process, whereas quantum subordination preserves quantum probability in the dynamics. As applications, we consider two versions of the q-deformed harmonic oscillator in both ordinary and imaginary time and show how the related q-deformed algebras can be understood as different patterns of time quantization rules. The second part of the thesis concerns the analogy between q -deformation and stochastic subordination. It is shown that the Poisson kernel of stochastic processes on chains of arbitrary length can be expressed in terms of the q-Racah polynomials, the most general q-deformed orthogonal polynomials in the discrete series of the Askey scheme. We give a new interpretation of this kernel as the transition probability density for a subordinated Markov process with only nearest neighbor hops. As an application, we give an elementary proof and extend a positivity result for a class of Poisson kernels which Gasper and Rahman established with direct methods.