Quantum Riesz Fractional Derivative

• Quantum Riesz fractional derivative is a pseudo-differential operator that extends traditional differentiation to fractional orders, capturing nonlocal effects in quantum mechanics.
• It employs Fourier symbol representations to modify spectral properties, resulting in algebraic tunneling, anomalous diffusion, and altered dispersion relations.
• Its use in quantum mechanics and quantum field theory enables effective regularization, lattice modeling, and insights into fractional quantum cosmology and tunneling phenomena.

The quantum Riesz fractional derivative is a pseudo-differential operator central to fractional quantum mechanics and nonlocal quantum field theory. It generalizes the standard Laplacian to non-integer order differentiation, encoding Lévy-flight kinetics and spatial nonlocality via power-law kernels or their Fourier symbols. The operator appears in various settings: space-fractional Schrödinger equations, QFT regularization, fractional lattice models, and quantum cosmology. Its mathematical definition, spectral properties, and physical implications differ significantly from canonical local derivatives, allowing for phenomena such as anomalous diffusion, algebraic tunneling, and modified quantum statistics.

1. Mathematical Definition and Representations

The Riesz fractional derivative of order $\alpha$ (typically $0<\alpha\le 2$, with quantum applications favoring $1<\alpha\le 2$) is defined on $\mathbb{R}^n$ either in coordinate space or momentum space.

• Hypersingular integral (coordinate space, $\mathbb{R}^n$):

$$(-\Delta)^{\alpha/2}\phi(x) = C_{n,\alpha}\,\text{P.V.} \int_{\mathbb{R}^n} [\phi(x)-\phi(y)]\,|x-y|^{-(n+\alpha)}\,d^n y$$

where $$C_{n,\alpha} = 2^{\alpha-1}\alpha\Gamma\left(\frac{n+\alpha}{2}\right)/(\pi^{n/2}\Gamma(1-\alpha/2))$$ and P.V. denotes the Hadamard principal value (Tarasov, 2018, Oliveira et al., 2010, Herrmann, 2012).

• Fourier symbol (momentum space):

$$\mathcal{F}\{(-\Delta)^{\alpha/2}\phi\}(k) = |k|^{\alpha}\,\mathcal{F}\{\phi\}(k)$$

so plane waves diagonalize the operator: $(-\Delta)^{\alpha/2}e^{ikx} = |k|^{\alpha}e^{ikx}$ (Tarasov, 2018).

• One-dimensional Riesz derivative:

$$D^\alpha_{\rm Riesz}f(x) = -\frac{\Gamma(1+\alpha)}{\pi} \sin\left(\frac{\pi\alpha}{2}\right) \int_0^\infty \frac{f(x+\xi)-2f(x)+f(x-\xi)}{\xi^{1+\alpha}} d\xi$$

(Herrmann, 2012, Herrmann, 2013).

Alternative differential (local) representations as infinite series exist (binomial or hypergeometric forms), converging on plane waves and emphasizing formal locality (Herrmann, 2013).

2. Spectral, Algebraic, and Physical Properties

• Linearity and scaling: $(-\Delta)^{\alpha/2}$ is linear, homogeneous, and under $x\to\lambda x$ scales as $\lambda^{-\alpha}$.
• Semigroup property: $$( -\Delta )^{\alpha/2}( -\Delta )^{\beta/2} = ( -\Delta )^{(\alpha+\beta)/2}$$ for suitable $\alpha,\beta$ (Tarasov, 2018).
• Self-adjointness and positivity: On $L^2(\mathbb{R}^n)$, $( -\Delta )^{\alpha/2}$ is self-adjoint and positive semi-definite.
• Nonlocality: The integral kernel's power-law tails ensure every point $x$ "samples" $\phi(y)$ over the entire domain with algebraic decay.
• Spectral measure: Continuous spectrum with eigenfunctions $e^{ikx}$ and eigenvalues $|k|^\alpha$.
• Integer-order limits: For $\alpha=2$, the Riesz derivative reduces to the standard Laplacian; for $\alpha\to1$ the limit is not smooth and does not recover the ordinary first derivative operator directly (Bayin, 2016).

3. Incorporation in Quantum Field Theory and Regularization

Fractional derivative regularization replaces the d'Alembertian ($\Box$) by its fractional power, $(\Box)^{\alpha/2}$:

• Propagator modification: The free propagator in momentum space becomes $$G_\alpha(k) = 1/[ (k^2)^{\alpha/2} - m^2 + i\epsilon ]$$.
• Loop integrals: E.g., one-loop self-energy in $\phi^4$ theory:

$$\Sigma_\alpha(0) = g \int \frac{d^n q}{(2\pi)^n} \frac{1}{ [ (q^2)^{\alpha/2} - m^2 + i\epsilon ] }$$

Reduces UV divergence, rendering integrals convergent for $\mathrm{Re}\, \alpha > n$ (Tarasov, 2018).

• Physical meaning: $\alpha$ controls nonlocality; $\alpha=2$ is local QFT, $\alpha<2$ introduces nonlocal kinetic terms.
• Analytic continuation: The procedure is analogous to dimensional regularization but leaves $n$ fixed, altering only the order of the kinetic operator.
• Generalization: Applicable to lattice models, gauge theories, and gravitational settings through discretization via $|k|^\alpha$ (Tarasov, 2018).

4. Quantum Fractional Mechanics: Schrödinger Equation and Observables

• Fractional Schrödinger equation (FSE):

$$i\hbar \frac{\partial}{\partial t}\psi(x,t) = D_\alpha (-\hbar^2 \Delta )^{\alpha/2}\psi(x,t) + V(x)\psi(x,t)$$

($1<\alpha\le 2$) (Oliveira et al., 2010, Bayin, 2016).

• Dispersion: Plane-wave solutions yield $E_k = |k|^\alpha$, interpolating between ultra-relativistic $\alpha=1$ and standard quadratic $\alpha=2$.
• Physical interpretation: $\alpha$ is the Lévy index governing the quantum path-integral measure; for $\alpha<2$ one obtains heavy-tailed, nonlocal propagators.

Table: Key spectral consequences in fractional quantum systems

System	Spectrum (α=2, local)	Spectrum (α<2, fractional)
Free particle	$E_k = k^2$	$E_k = |k|^{\alpha}$
Infinite potential well	$E_n \propto n^2$	$E_n \propto n^{\alpha}$
Harmonic oscillator	$E_n = \hbar\omega(n+1/2)$	No discrete $E_n$; $k$-dependent metastable eigenvalues

Excited state construction in fractional oscillators entails Riesz–Feller Hermite polynomials and inverse Fourier transforms leading to non-Gaussian, heavy-tailed wavefunctions (Rosu et al., 2020, Boumali et al., 2024).

5. Boundary Conditions, Finite Domains, and Lattice Formulations

• Finite periodic domains: The fractional Laplacian acquires an $L$-periodic kernel

$$(-\Delta)^{\alpha/2}_L u(x) = \int_0^L K_L^{(\alpha)}(|x-y|) [u(x)-u(y)] dy$$

$K_L^{(\alpha)}(r)$ is expressed via Hurwitz–ζ functions; the operator is self-adjoint on $L^2([0,L])$ with periodic boundary conditions, eigenfunctions are plane waves $e^{ik_lx}/\sqrt{L}$ with $k_l=2\pi l/L$, and eigenvalues $-|k_l|^\alpha$ (Michelitsch et al., 2014).

• Lattice models: Discrete fractional Laplacian matrices converge to continuum Riesz operators in the $h\to0$ limit, with scaling dictated by particle mass $\mu\sim h$ and frequency $\Omega_\alpha^2\sim h^{-\alpha}$.
• Boundary effects: In bounded domains (infinite wells), nonlocality causes wavefunction pile-up near boundaries and modifies energy scaling (Herrmann, 2012).

6. Applications in Tunneling, Quantum Cosmology, and Information Measures

• Tunneling: Fractional equations with Riesz derivatives permit zero-energy tunneling across delta potentials; for $1<\alpha<2$, transmission as $E\to0$ is $T_0 = \cos^2(\pi/\alpha)$, contrasting with standard $T_0=0$ for $\alpha=2$ (Oliveira et al., 2010).
• Quantum cosmology: In fractional Wheeler–DeWitt equations, a decrease in $\alpha$ suppresses tunneling probability for universe creation; $\alpha$ and cosmological constant $\Lambda$ trade off in their effect on tunneling rates (Canedo et al., 19 Mar 2025).
• Quantum information: Fisher information and Shannon entropy in fractional oscillators quantify the impact of nonlocality; fractional Fisher information involves the gradient $D^{\alpha/2}\rho(x)$, directly sensitive to the power-law decay of wavefunction tails (Boumali et al., 2024).

7. Locality, Uniqueness, and Physical Interpretation

• Local vs. nonlocal representations: Integral forms are strictly nonlocal, requiring global data; differential infinite-series forms offer quasi-locality but are equivalent on Fourier bases (Herrmann, 2013).
• Limitations: Smoothly taking $\alpha\to 1$ is not possible within the Riesz definition—there is a discontinuity at $\alpha=1$, and no direct correspondence to the ordinary first derivative.
• Physical implications: The Riesz fractional derivative fundamentally alters the quantum dynamics, enabling Lévy-flight statistics, nonlocal quantum transport, anomalous diffusion, non-Gaussian eigenstates, and modified UV behavior in quantum field theoretical models.

---

Relevant works include Tarasov (Tarasov, 2018) for QFT regularization; Herrmann (Herrmann, 2012, Herrmann, 2013) for fractional Schrödinger boundary problems and differential representations; Michelitsch et al. (Michelitsch et al., 2014) for lattice and periodic structures; Patra (Patra, 2019) for similarity analysis and Fourier solutions; Boumali et al. (Boumali et al., 2024) for quantum information dynamics; and fractional cosmology applications in (Canedo et al., 19 Mar 2025).