Search Results (152)

The q-Heisenberg algebra ${\mathfrak{h}}_n\left(q\right)$ is a significant class of solvable polynomial algebras, and it unifies the canonical commutation relations of Heisenberg algebras and the deformation theory of quantum groups. In this paper, we employ Gröbner-Shirshov basis theory and PBW (Poincar$\acute{e}$-Birkhoff-Witt) basis techniques to systematically investigate ${\mathfrak{h}}_n\left(q\right)$. Our main results establish that: ${\mathfrak{h}}_n\left(q\right)$ possesses an iterated skew-polynomial algebra structure, and it satisfies the important homological regularity properties of being Auslander regular, Artin-Schelter regular, and Cohen-Macaulay. These findings provide deep insights into the algebraic structure of ${\mathfrak{h}}_n\left(q\right)$, while simultaneously bridging the gap between noncommutative algebra and quantum representation theory. Furthermore, our constructive approach yields computable methods for studying modules over ${\mathfrak{h}}_n\left(q\right)$, opening new avenues for further research in deformation quantization and quantum algebra. Full article

Quantum anomalies are traditionally understood as classical symmetries that fail to survive quantization, while experimental “anomalies” denote deviations between theoretical predictions and measured values. In this work, we develop a unified framework in which both phenomena can be interpreted through the lens of algebraic quantum field theory (AQFT). Building on the renormalization group viewed as an extension problem, we show that renormalization ambiguities correspond to nontrivial elements of Hochschild cohomology, giving rise to a deformation of the observable algebra $A\ast B=AB+\epsilon \omega (A,B)$, where $\omega$ is a Hochschild 2-cocycle. We interpret $\omega$ as an intrinsic algebraic curvature of the net of local algebras, namely the (local) Hochschild class that measures the obstruction to trivializing infinitesimal scheme changes by inner redefinitions under locality and covariance constraints. The transported product is associative; its first-order expansion is associative up to $\mathcal{O}\left({\epsilon }^2\right)$ while preserving the ∗-structure and Ward identities to the first order. We prove the existence of nontrivial cocycles in the perturbative AQFT setting, derive the conditions under which the deformed product respects positivity and locality, and establish the compatibility with current conservation. The construction provides a direct algebraic bridge to standard cohomological anomalies (chiral, trace, and gravitational) and yields correlated deformations of physical amplitudes. Fixing the small deformation parameter $\epsilon$ from the muon $(g-2)$ discrepancy, we propagate the framework to predictions for the electron $(g-2)$, charged lepton EDMs, and other low-energy observables. This approach reduces reliance on ad hoc form-factor parametrizations by organizing first-order scheme-induced deformations into correlation laws among low-energy observables. We argue that interpreting quantum anomalies as manifestations of algebraic curvature opens a pathway to a unified, testable account of renormalization ambiguities and their phenomenological consequences. We emphasize that the framework does not eliminate renormalization or quantum anomalies; rather, it repackages the finite renormalization freedom of pAQFT into cohomological data and relates it functorially to standard anomaly classes. Full article

This paper analyzes the modified canonical Heisenberg commutation relations or GUP, from a standard Hamiltonian point of view. For a one-dimensional system, a such modified canonical Heisenberg commutation relation is defined by the commutator between a position $\widehat{x}$ and a momentum operator $\widehat{p}$ (called the deformed momentum), which becomes a function F of the same operators: $\left[\widehat{x},\widehat{p}\right]=F(\widehat{x},\widehat{p})$, that is, the Heisenberg algebra closes itself in general in a nonlinear way. The function F also depends on a parameter that controls the deformation of the Heisenberg algebra in such a way that for a null parameter value, one recovers the usual Heisenberg algebra $\left[\widehat{x},{\widehat{p}}_0\right]=i\hslash \mathbb{I}$. Thus, it naturally raises the following questions: What does a relation of this type mean in Hamiltonian theory from a standard point of view? Is the deformed momentum the canonical variable conjugate to the position in such a relation? Moreover, what are the canonical variables in this model? The answer to these questions comes from the existence of two different phase spaces: The first one, called the non-deformed phase (which is obtained for control parameter value equal to zero), is defined by the Cartesian $\widehat{x}$ coordinate and its non-deformed conjugate momentum ${\widehat{p}}_0$, which satisfies the standard quantum mechanical Heisenberg commutation relation. The second phase space, the deformed one, is given by the deformed momentum $\widehat{p}$ and a new position coordinate $\widehat{y}$, which is its canonical conjugate variable, so $\widehat{y}$ and $\widehat{p}$ also satisfy standard commutation relations. We construct a classical canonical transformation that maps the non-deformed phase space into the deformed one for a specific class of deformation functions F. Additionally, a quantum mechanical operator transformation is found between the two non-commutative phase spaces, which allows the Schrödinger equation to be written in both spaces. Thus, there are two equivalent quantum mechanical descriptions of the same physical process associated with a deformed commutation relation. Full article

Many of the most fundamental observables—position, momentum, phase point, and spin direction—cannot be measured by an instrument that obeys the orthogonal projection postulate. Continuous-in-time measurements provide the missing theoretical framework to make physical sense of such observables. The elements of the time-dependent instrument define a group called the instrumental group (IG). Relative to the IG, all of the time dependence is contained in a certain function called the Kraus-operator density (KOD), which evolves according to a classical Kolmogorov equation. Unlike the Lindblad master equation, the KOD Kolmogorov equation is a direct expression of how the elements of the instrument (not just the total quantum channel) evolve. Shifting from continuous measurements to sequential measurements more generally, the structure of combining instruments in sequence is shown to correspond to the convolution of their KODs. This convolution promotes the IG to an involutive Banach algebra (a structure that goes all the way back to the origins of POVM and C*-algebra theory), which will be called the instrumental group algebra (IGA). The IGA is the true home of the KOD, similar to how the dual of a von Neumann algebra is the true home of the density operator. Operators on the IGA, which play the analogous role for KODs as superoperators play for density operators, are called ultraoperators and various important examples are discussed. Certain ultraoperator–superoperator intertwining relationships are also considered throughout, including the relationship between the KOD Kolmogorov equation and the Lindblad master equation. The IGA is also shown to have actually two distinct involutions: one respected by the convolution ultraoperators and the other by the quantum channel superoperators. Finally, the KOD Kolmogorov generators are derived for jump processes and more general diffusive processes. Full article

The geometrical view of the electron as a spinning bivector leads to the partitioning of the electron’s energy into internal and external. The reduced Compton wavelength, ${\overline{\lambda }}_C$, is taken as the radius of the inertial ring (a disc), while $r_e$ characterizes the EM coupling scale. Within this picture, the fine-structure constant emerges as the structural ratio $\alpha =r_e/{\overline{\lambda }}_C$. We make the partitioning explicit, derive simple ratios among moments of inertia and stored energies, and compare the Bivector Standard Model with the Standard model. Full article

The mathematical universe of the quantum topos, which is formulated on the basis of classical Boolean snapshots, delivers a neo-realist description of quantum mechanics that preserves realism. The main contribution of this article is developing formal objectivity in physical theories beyond quantum mechanics in the topos-theory approach. It will be shown that neo-realist responses to non-perturbative structures of quantum field theory do not preserve realism. In this regard, the method of Feynman graphons is applied to reframe the task of describing objectivity in quantum field theory in terms of replacing the standard Hilbert-space/operator-algebra ontology with a new context category built from a certain family of topological Hopf subalgebras of the topological Hopf algebra of renormalization as algebraic/combinatorial data tied to non-perturbative structures. This topological-Hopf-algebra ontology, which is independent of instrumentalist probabilities, enables us to reconstruct gauge field theories on the basis of the mathematical universe of the non-perturbative topos. The non-Boolean logic of the non-perturbative topos cannot be recovered by classical Boolean snapshots, which is in contrast to the quantum-topos reformulation of quantum mechanics. The article formulates a universal version of the non-perturbative topos to show that quantum field theory is a globally and locally neo-realist theory which can be reconstructed independent of the standard Hilbert-space/operator-algebra ontology. Formal objectivity of the universal non-perturbative topos offers a new route to build objective semantics for non-perturbative structures. Full article

This work investigates the stability analysis of linear control systems defined on Lie groups, with a particular focus on low-dimensional cases. Unlike their Euclidean counterparts, such systems evolve on manifolds with non-Euclidean geometry, where trajectories respect the group’s intrinsic symmetries. Stability notions, such as inner asymptotic, inner, and input–output (BIBO) stability, are studied. The qualitative behavior of solutions is shown to depend critically on the spectral decomposition of derivations associated with the drift, and on the algebraic structure of the underlying Lie algebra. We study two classes of examples in detail: Abelian and solvable two-dimensional Lie groups, and the three-dimensional nilpotent Heisenberg group. These settings, while mathematically tractable, retain essential features of non-commutativity, geometric non-linearity, and sub-Riemannian geometry, making them canonical models in control theory. The results highlight the interplay between algebraic properties, invariant submanifolds, and trajectory behavior, offering insights applicable to robotic motion planning, quantum control, and signal processing. Full article

In this paper we analyze complex systems dynamics using a multifractal framework derived from Scale Relativity Theory (SRT). By extending classical differential geometry to accommodate non-differentiable, scale-dependent behaviors, we formulate Schrödinger-type equations that describe multifractal geodesics. These equations reveal deep analogies between quantum mechanics and macroscopic complex dynamics. A key feature of this approach is the identification of hidden symmetries governed by multifractal analogs of classical groups, particularly the SL(2ℝ) group. These symmetries help explain universal dynamic behaviors such as double period dynamics, damped dynamics, modulated dynamics, or chaotic dynamics. The resulting framework offers a unified geometric and algebraic perspective on the emergence of order within complex systems, highlighting the fundamental role of fractality and scale covariance in nature. Full article

This paper explores some frame bundles and physical implications of Killing vector fields on the two-sphere ${\mathbb{S}}^2$, culminating in a novel application to Maxwell’s equations in free space. Initially, we investigate the Killing vector fields on ${\mathbb{S}}^2$ (represented by the unit sphere of ${\mathbb{R}}^3$), which generate the isometries of the sphere under the rotation group $SO\left(3\right)$. These fields, realized as functions $K_v:{\mathbb{S}}^2\to {\mathbb{R}}^3$, defined by $K_v\left(q\right)=v\times q$ for a fixed $v\in {\mathbb{R}}^3$ and any $q\in {\mathbb{S}}^2$, generate a three-dimensional Lie algebra isomorphic to $\mathfrak{so}\left(3\right)$. We establish an isomorphism $K:{\mathbb{R}}^3\to \mathcal{K}\left({\mathbb{S}}^2\right)$, mapping vectors $v=au$ (with $u\in {\mathbb{S}}^2$) to scaled Killing vector fields $a{K}_u$, and analyze its relationship with $SO\left(3\right)$ through the exponential map. Subsequently, at a fixed point $e\in {\mathbb{S}}^2$, we construct a smooth orthonormal right-handed tangent frame $f_e:{\mathbb{S}}^2\backslash \{e,-e\}\to T{\left({\mathbb{S}}^2\right)}^2$, defined as $f_e\left(u\right)=({\widehat{K}}_e\left(u\right),u\times {\widehat{K}}_e\left(u\right))$, where ${\widehat{K}}_e$ is the unit vector field of the Killing field $K_e$. We verify its smoothness, orthonormality, and right-handedness. We further prove that any smooth orthonormal right-handed frame on ${\mathbb{S}}^2\backslash \{e,-e\}$ is either $f_e$ or a rotation thereof by a smooth map $\rho :{\mathbb{S}}^2\backslash \{e,-e\}\to SO\left(3\right)$, reflecting the triviality of the frame bundle over the parallelizable domain. The paper then pivots to an innovative application, constructing solutions to Maxwell’s equations in free space by combining spherical symmetries with quantum mechanical de Broglie waves in tempered distribution wave space. The deeper scientific significance lies in bringing together differential geometry (via $SO\left(3\right)$ symmetries), quantum mechanics (de Broglie waves in Schwartz distribution theory), and electromagnetism (Maxwell’s solutions in Schwartz tempered complex fields on Minkowski space-time), in order to offer a unifying perspective on Maxwell’s electromagnetism and Schrödinger’s picture in relativistic quantum mechanics. Full article

There are two ways of linearizing the Klein–Gordon equation: Dirac’s choice, which introduces a matter–antimatter pair, and a second approach using a bivector, which Dirac did not consider. In this paper, we show that a bivector provides the classical origin of quantum spin. At high precessional frequencies, a symmetry transformation occurs in which classical reflection becomes quantum parity. We identify a classical spin-1 boson and demonstrate how bosons deliver energy, matter, and torque to a surface. The correspondence between classical and quantum domains allows spin to be identified as a quantum bivector, $i\sigma$. Using geometric algebra, we show that a classical boson has two blades, corresponding to magnetic quantum number states $m=\pm 1$. We conclude that fermions are the blades of bosons, thereby unifying both into a single particle theory. We compare and contrast the Standard Model, which uses chiral vectors as fundamental, with the Bivector Standard Model, which uses bivectors, with two hands, as fundamental. Full article

Proxy re-signature enables transitive authentication of digital identities across different domains and has significant application value in areas such as digital rights management, cross-domain certificate validation, and distributed system access control. However, most existing proxy re-signature schemes, which are predominantly based on traditional public-key cryptosystems, face security vulnerabilities and certificate management bottlenecks. While identity-based schemes alleviate some issues, they introduce key escrow concerns. Certificateless schemes effectively resolve both certificate management and key escrow problems but remain vulnerable to quantum computing threats. To address these limitations, this paper constructs an efficient post-quantum certificateless proxy re-signature scheme based on algebraic lattices. Building upon algebraic lattice theory and leveraging the Dilithium algorithm, our scheme innovatively employs a lattice basis reduction-assisted parameter selection strategy to mitigate the potential algebraic attack vectors inherent in the NTRU lattice structure. This ensures the security and integrity of multi-party communication in quantum-threat environments. Furthermore, the scheme significantly reduces computational overhead and optimizes signature storage complexity through structured compression techniques, facilitating deployment on resource-constrained devices like Internet of Things (IoT) terminals. We formally prove the unforgeability of the scheme under the adaptive chosen-message attack model, with its security reducible to the hardness of the corresponding underlying lattice problems. Full article

Conjecture $\mathcal{O}$ (and the Gamma Conjectures), introduced by Galkin, Golyshev, and Iritani stand as pivotal open problems in the quantum cohomology of Fano manifolds, bridging algebraic geometry, mathematical physics, and representation theory. These conjectures aim to decode the structural essence of quantum multiplication by uncovering profound connections between spectral properties of quantum cohomology operators and the underlying geometry of Fano manifolds. Conjecture $\mathcal{O}$ specifically investigates the spectral simplicity and eigenvalue distribution of the operator associated with the first Chern class $c_1$ in quantum cohomology rings, positing that its eigenvalues govern the convergence and asymptotic behavior of quantum products. Full article

We present novel proofs of the Riemann hypothesis by extending the standard complex Riemann zeta function into a quaternionic algebraic framework. Utilizing λ-regularization, we construct a symmetrized form that ensures analytic continuation and restores critical-line reflection symmetry, a key structural property of the Riemann ξ(s) function. This formulation reveals that all nontrivial zeros of the zeta function must lie along the critical line Re(s) = 1/2, offering a constructive and algebraic resolution to this fundamental conjecture. Our method is built on convexity and symmetrical principles that generalize naturally to higher-dimensional hypercomplex spaces. We also explore the broader implications of this framework in quantum statistical physics. In particular, the λ-regularized quaternionic zeta function governs thermodynamic properties and phase transitions in Bose–Einstein condensates. This quaternionic extension of the zeta function encodes oscillatory behavior and introduces critical hypersurfaces that serve as higher-dimensional analogues of the classical critical line. By linking the spectral features of the zeta function to measurable physical phenomena, our work uncovers a profound connection between analytic number theory, hypercomplex geometry, and quantum field theory, suggesting a unified structure underlying prime distributions and quantum coherence. Full article

Quantum channels define key objects in quantum information theory. They are represented by completely positive trace-preserving linear maps in matrix algebras. We analyze a family of quantum channels defined through the use of the Weyl operators. Such channels provide generalization of the celebrated qubit Pauli channels. Moreover, they are covariant with respective to the finite group generated by Weyl operators. In what follows, we study self-adjoint Weyl channels by providing a special Hermitian representation. For a prime dimension of the corresponding Hilbert space, the self-adjoint Weyl channels contain well-known generalized Pauli channels as a special case. We propose multipartite generalization of Weyl channels. In particular, we analyze the power of prime dimensions using finite fields and study the covariance properties of these objects. Full article

Since Hamilton proposed quaternions as a system of numbers that does not satisfy the ordinary commutative rule of multiplication, quaternion algebras have played an important role in many mathematical and physical studies. This paper introduces the generalized notion of Pauli Fibonacci polynomial quaternions, a definition that incorporates the advantages of the Fibonacci number system augmented by the Pauli matrix structure. With the concept presented in the study, it aims to provide material that can be used in a more in-depth understanding of the principles of coding theory and quantum physics, which contribute to the confidentiality needed by the digital world, with the help of quaternions. In this study, an approach has been developed by integrating the advantageous and consistent structure of quaternions used to solve the problem of system lock-up and unresponsiveness during rotational movements in robot programming, the mathematically compact and functional form of Pauli matrices, and a generalized version of the Fibonacci sequence, which is an application of aesthetic patterns in nature. Full article